Research Article119867

infinController Design for Asynchronous Multirate

Sampled-Data Systems

Xiaoqiang Sun and Weijie Mao

Department of Control Science and Engineering Zhejiang University Hangzhou 310027 China

Correspondence should be addressed to Weijie Mao wjmaozjueducn

Received 29 October 2013 Revised 22 December 2013 Accepted 6 January 2014 Published 23 February 2014

Academic Editor Ilse C Cervantes

Copyright copy 2014 X Sun and W MaoThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper considers the analysis and synthesis of a linear discrete asynchronous multirate sampled-data system An119867infincontroller

based on an observer is proposed which guarantees the stability of the closed system andmakes the119867infinnorm of the closed system

less than a given attenuation level To improve the performance further a tradeoff strategy is appliedThat is the exogenous signalssampled at different rates are lifted to an appropriate signal rate while the endogenous signals are not lifted for avoiding the causalconstraint and the dimension multiplied again The controller is obtained by solving the corresponding matrix inequality whichcan be calculated by Matlab Finally an example is presented to demonstrate the validity of these methods

1 Introduction

Sampled-data control systems which provide digital controltechniques and emergedwith the development of computershave been long studied by many researchers [1 2] Multiratesampled-data control systems are required when the signalsof interest are sampled at different rates compared withothers There are two primary reasons for this requirementFirst it has been shown that multirate sampling can improvethe closed-loop control performance significantly Secondtechnological or economic constraints in some real appli-cations necessitate the use of control schemes where sensormeasurements and control inputs have to be performedat different sampling rates Figure 1 shows an ideal modelof a standard multirate sampled-data control system Forexample in hybrid electric vehicles a hierarchical controlleris designed to decide the torque demands ofmotor generatorinternal combustion engine andmechanical brake accordingto the driverrsquos torque demand speed of vehicle and batteryrsquosstate of charge (SOC) where the SOC is estimated by batterymanagement system (BMS) and the speed of vehicle is fed bysensor [3] As a large-scale and complex system the sensorsand control inputs of subsystems are difficult to be performed

at the same sampling rates thus a multirate sampled-datasystem will be a better choice for modeling

Multirate systems have been studied since the late 1950sresulting in the development and application of varioussampling methods Classical approaches include the liftingmethod 119867

2119867infin

control [4 5] LMI method [6] predictioncontrol LQG control iterative control [7] and parameterestimation [8] Among these the lifting method [9] is a vitaltool for studying multirate systems whereby such systemscan be changed into single-rate systems This method canimprove the performance of a plant to some extent [10]However during the lifting process the dimensions of thesystem will be multiplied and it is increasingly difficult toanalyze the new system effectively especially systems withvery different sampling rates which can lead to the ldquomodelexplosionrdquo problem 119867

2119867infin

control is a useful method foranalyzing systems and for synthesizing control problemsbased on the lifting technique and LMI tools can be usedto solve the inequalities [11] By the way in real applicationsfast-input and slow-output is a good way of improving theperformance of the system and this mechanism and itsresults have been discussed in previous studies [12 13]

Most of lifting methods require that all signals should besampled and held during the same time slot [1 2 9 10 14 15]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 757282 12 pageshttpdxdoiorg1011552014757282

2 Mathematical Problems in Engineering

That is at a specific 119896

119878 (119896119879) = diag [11986810 sdot sdot sdot 0]

1198791

[11986820 sdot sdot sdot 0]

1198792

[119868119901

0 sdot sdot sdot 0]119879119901

119867 (119896119879) = diag [119868

1sdot sdot sdot 1198681]

1015840

1198791

[1198682sdot sdot sdot 1198682]

1015840

1198792

[119868119902sdot sdot sdot 119868119902]

1015840

119879119902

(1)

where 119868119894 119894 = 1 2 119901(119902) are unit matrices and 119879 119879

119894

119879 and

119879119894 will be defined in Section 2 However in most

actual situations this condition cannot be satisfied becausethe overall system lacks a slot when the entire signal can besampled or held and there is a class of multirate systems thateach output of them has its own frequency of measurementand each input has its own frequency of updating [16ndash19] Asystem with this type of sampling mechanism is referred toas asynchronous system in this paper The traditional liftingtechnique is not applicable for the sampling frequencies of thesampling and holding elements are incommensurate [4 20]

LQG control based on an observer is an effective methodfor reducing the dimensions of a multirate system with-out using the lifting technique [14 21] An LQG optimalcontroller derived is periodic and can guarantee the overallclosed-loop stability To compensate for the skewed inputsignal caused by slow-output feedback only the input signalis supplemented by the observer However in the previousresults the controller and observer are designed separatelywhich may result in conservativeness

Based on the methods mentioned above the design ofa controller for a multirate sampled-data system that doesnot multiply the dimensions necessitates the synthesis ofvarious appropriate measures In this paper we propose anobserver-based 119867

infinmethod for analysis and synthesis of

linear discrete asynchronousmultirate sampled-data systemsIn contrast to the traditional lifting method the controlinputs and measured outputs are not lifted to avoid thedimension multiplication and causal constraints Howeverthe exogenous system signals are lifted rationally to changethe system into a fast-input-slow-output system The con-troller can be obtained by solving an inequality with LMItools of Matlab Although the inequality is nonlinear it canbe calculated linearly under a constraint [22] The approachproposed in this paper is different from the observer-basedLQG method in [21] Here the controller and observer areobtained simultaneously by solving amatrix inequality underthe same optimal performance constraint thus reducing theconservativeness and improving the system performanceIn addition the concept of asynchronism in this paper isdifferent from those in [23 24] and so forth where thesampling at sensors is assumed to be asynchronized withtime-varying delay and it falls into the framework of time-delay systems not the multirate sampled-data systems

The remainder of this paper is organized as followsSection 2 provides a description of the multirate systemand its background including lemmas and definitions In

w z

u y

G

K

H1Hq

Tm1Tn1Tnq Tmp

SpS1middot middot middot middot middot middot

Figure 1 Multirate sampled-data system

Signal 1Signal 2Signal 3

0 1 2 3 4 5 6 7 8 9

Time slot

Figure 2 Asynchronous signals from the sampler or the holder

Section 3 the main results obtained using the synthesismethods are presented In Section 4 a simulated numericalexample is provided to demonstrate the effectiveness of themethod Finally Section 5 concludes the paper

2 System Descriptions

Definition 1 A multirate sampled-data system is called anasynchronous system if the signals from its sampler or holdercannot be obtained simultaneously during any time slot

For example as shown in Figure 2 Signal 1 and Signal 2are the signals of the sampler and the holder with period 3respectively while Signal 3 is the signal with period 2 Thusthey cannot be obtained simultaneously during any time slotbecause the odd period signals appear alternately In Figure 2the filled circles indicate that the signal is sampled or held inthe slot whereas the hollow cylinders indicate that the signalis not sampled or held Therefore this system cannot satisfythe assumptionmentioned inmost previous studies where allsignals can be sampled or held during the same time slot

We consider a linear discrete asynchronous multiratesampled-data system

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 1198721119908 (119896)

119911 (119896) = 119862119909 (119896) +1198722119908 (119896)

119910 (119896) = 119878 (119896) 119911 (119896)

(2)

where 119896 isin N 119909(119896) isin R119899 119906(119896) isin R119902 119911(119896) isin R119901 119908(119896) isin

R119897 119909(0) = 1199090 |1198722| = 0 119878(119896) is a sampling device 119910(119896) is the

output measured vector and 1199090is the initial state

Assume that the 119894th component 119906119894(119896) 119894 = 1 2 119902 of

the input vector 119906(119896) can be modified every 119879119894time instant

Mathematical Problems in Engineering 3

119879119894isin Z+ Then 119906(119896) is the output of the following discrete-

time period system which is defined as the input-holdingmechanism

V (119896 + 1) = (119868 minus 119867 (119896)) V (119896) + 119867 (119896) 119903 (119896)

119906 (119896) = (119868 minus 119867 (119896)) V (119896) + 119867 (119896) 119903 (119896)

(3)

where

119867(119896) = diag ℎ1(119896) ℎ

2(119896) ℎ

119902(k)

ℎ119894(119896) =

1 119896 = 119895119879119894+ 120591119894

0 119896 = 119895119879119894+ 120591119894

119895 integer(4)

where V(119896) = [V1(119896) V

2(119896) sdot sdot sdot V

119902(119896)]

1015840 is a new statevariable and 119903(119896) = [119903

1(119896) 1199032(119896) sdot sdot sdot 119903

119902(119896)]

1015840 is a new inputvariable The integer 0 le 120591

119894lt119879119894(119894 = 1 2 119902) describes the

skew of the inputs-holding mechanismThen matrix119867(sdot) and system (2) have a period of 119879 and

119879 = lcm

119894=12119902

119879119894 (5)

Similarly the output sampling mechanism is given as

119878 (119896) = diag V1(119896) V2(119896) V

119901(119896) (6)

where

V119894(119896) =

1 119896 = 119895119879119894+ 120591119894

0 119896 = 119895119879119894+ 120591119894

119895 integer (7)

where 0 le 120591119894lt 119879119894(119894 = 1 2 119901) indicates the skew of the

outputs mechanismMatrix 119878(sdot) has a 119879-period and

119879 = lcm

119894=12119901

119879119894 (8)

After combining (2) and (3) we obtain a new sampling-holding system

120585 (119896 + 1) = Φ (119896) 120585 (119896) + Γ (119896) 119903 (119896) + Ψ119908 (119896)

119911 (119896) = Υ120585 (119896) + 1198722119908 (119896)

119910 (119896) = Δ (119896) 120585 (119896) + Ω (119896)119908 (119896)

(9)

where

120585 (119896) = [

119909 (119896)

V (119896)] Φ (119896) = [

119860 119861 (119868 minus 119867 (119896))

0 119868 minus 119867 (119896)]

Γ (119896) = [

119861

119868]119867 (119896) Ψ = [

1198721

0]

Υ = [119862 0] Δ (119896) = [119878 (119896) 119862 0]

Ω (119896) = 119878 (119896)1198722

(10)

Based on the 119879 of matrix 119867(sdot) and the 119879 of matrix 119878(sdot)

system (9) has the period of

119879 = lcm 119879 119879 (11)

The state observer of system (2) is

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119871 (119896) sdot [119910 (119896) minus 119878 (119896) 119862119909 (119896)]

(12)

where 119871(119896) asymptotically stabilizes system (12)The observer-based controller is chosen as

119903 (119896) = 119870 (119896)120585 (119896) (13)

where 120585(119896) = [119909(119896)

V(119896) ]The state error is defined as

119890 (119896) = 119909 (119896) minus 119909 (119896) (14)

According to (2) (12) and (14) the following equation canbe obtained

119890 (119896 + 1) = [119860 minus 119871 (119896) 119878 (119896) 119862] 119890 (119896)

+ (1198721minus 119871 (119896) 119878 (119896)119872

2) 119908 (119896)

(15)

Combining (9) (13) and (15) yields

[

120585 (119896 + 1)

119890 (119896 + 1)] = 119860[

120585 (119896)

119890 (119896)] + 119861119908 (119896)

119911 (119896) = 119862 [

120585 (119896)

119890 (119896)] + 119863119908 (119896)

(16)

where

119860 =[

[

[

Φ (119896) + Γ (119896)119870 (119896) minusΓ (119896)119870 (119896) [

1198682

0]

0 119860 minus 119871 (119896) 119878 (119896) 119862

]

]

]

119861 = [

Ψ

1198721minus 119871 (119896) 119878 (119896)119872

2

]

119862 = [Υ 0] 119863 = 1198722

(17)

Lemma 2 (see [21 25] ) If

(i) the pair (119860 119861) is stabilizable(ii) the pair (119860119872

1) is stabilizable

(iii) the pair (119860 119862) is detectable(iv) for any pair of distinct eigenvalues of 119860 120582

119894and 120582

119895

|120582119894| ge 1 |120582

119895| ge 1 it follows that 120582119879

119894= 120582

119879

119895

(v) there is no eigenvalue 120582 of 119860 120582 = 1 such that 120582119879 = 1

then the pairs (Φ(119896) Γ(119896)) and (Φ(119896) Ψ) are controllable andthe pair (Φ(119896) Υ) is detectable

4 Mathematical Problems in Engineering

Lemma 3 (see [10] (the Bounded Real Lemma) ) For a givenscalar 120574 gt 0 if there is a matrix 119883 = 119883

1015840

gt 0 that satisfies thematrix inequality

[

[

[

[

119883 119860119883 119861 0

1198831198601015840

119883 0 1198831198621015840

1198611015840

0 120574119868 1198631015840

0 119862119883 119863 120574119868

]

]

]

]

gt 0 (18)

then the system (16) is asynchronously stable and it achieves aspecified attenuation level 120574 gt 0 such that 119879

119911119908infin

lt 120574

3 Main Results

For the linear discrete asynchronous multirate sampled-data system (9) the exogenous parts of the system arelifted to improve its performance To avoid multiplying thedimensions of the control inputs and measured outputs wedo not lift them thereby avoiding causal constraints of thetraditional lifting method

If we assume that there is an underlying clock with a baseperiod of 120591 then system (9) represents a fast discrete system

with a subperiod of 120591119898 for the discrete process describedin [9] The sampler 119878

119886samples the continuous signal 119908 for a

discrete 119886-period and the holder 119867119887holds the output signal

119911 during period 119887 The rates of the sampler and the holdercan be set at different flexible rates that is 119886 = 119887 If weselect (13) as the new input 119870(119896) is the controller of system(9)

These concepts are described in Figure 3 where

119898 = lcm 119886 119887 (19)

Theorem 4 Consider the sampled-data system (9) with thelifting process shown in Figure 3 For a given attenuationlevel 120574 gt 0 if there exist matrices 119883

1(119896) = 119883

1015840

1(119896) gt

0 1198832(119896) = 119883

1015840

2(119896) gt 0 119870(119896) and 119871(119896) (119896 = 0 119873

denotes the iteration time) which are suboptimal solutions ofthe inequalities below then the closed system of (9) with thecontroller (13) is asynchronously stable and satisfies the 119867

infin

norm constraint 119879119911119908infin

lt 120574

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831(119896) 0 119860

119889+ 1198612119889119870 (119896) minus119861

2119889119870 (119896) [

1198682

0] 119861

10

lowast 1198832(119896) 0 119860 minus 119871 (119896) 119878 (119896) 119862 (119872

1minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 119883minus1

1(119896) 0 0 119862

1015840

1+ 1198701015840

(119896)1198631015840

12

lowast lowast lowast 119883minus1

2(119896) 0 minus [119868

20]1198701015840

(119896)1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0 (20)

Proof After lifting system (9) according to Figure 3 thetransform function of a completely discrete system with aperiod of 120591119898 which is partly contained within the dot-dashline is

119866 = [

119871119898

0

0 119868] [

119878119891

0

0 119878 (119896)

] [

11986611

11986612

11986621

11986622

] [

119867119891

0

0 119867 (119896)

] [119871minus1

1198980

0 119868

]

=[

[

11987111989811987811989111986611119867119891119871minus1

119898

11987111989811987811989111986612119867(119896)

119878 (119896) 11986621119867119891119871minus1

119898

119878 (119896) 11986622119867(119896)

]

]

(21)

Based on the above we can derive minimal state-spacerepresentations 119892(120582) for 119866 as follows based on the transferfunction theory given in [9 15] However in contrast to thetheory 119867 and 119878 are not blocks of unit matrices because

sampling and holding are not synchronous that is the entiresignal cannot be sampled or held at the same time

119892 (120582) = [

11987511

11987512

11987521

11987522

] (22)

Next we calculate the minimal state-space realization of(22) as follows(a) Transfer Function for 119875

11 Note that119875

11= 11987111989811987811989111986611119867119891119871minus1

119898

which comes directly from the theory in [8] and its corre-sponding state model is

11(120582) = [

119860119889

1198611

1198621

11986311

] (23)

where

119860119889=

[

[

119860119898

119898

sum

119895=1

119860119898minus119895

119861 (119868 minus 119867 (119896))

0 119868 minus 119867 (119896)

]

]

1198611= [

119860119898minus1

1198721119860119898minus2

1198721sdot sdot sdot 119872

1

0 0 sdot sdot sdot 0

]

Mathematical Problems in Engineering 5

1198621=

[

[

[

[

[

119862 0

119862119860 0

119862119860119898minus1

0

]

]

]

]

]

11986311=

[

[

[

[

[

1198722

0 sdot sdot sdot 0

1198621198721

1198722

sdot sdot sdot 0

d

119862119860119898minus2

1198721119862119860119898minus3

1198721sdot sdot sdot 119872

2

]

]

]

]

]

(24)

(b) Transfer Function for 11987512 As119867(119896) in 119875

12= 11987111989811987811989111986612119867(119896)

is not an 119898-blocks unit matrix the theory cannot be applieddirectly However note that

119878119891119867(119896) = 119871

minus1

119898

[

[

[

[

[

119868

119868

119868

]

]

]

]

]119898

119867(119896) (25)

Then

11987512= 11987111989811987811989111986612119867119891119871minus1

119898

[

[

[

[

[

119868

119868

119868

]

]

]

]

]119898

119867(119896) (26)

The corresponding state model can be obtained as

12(120582) = [

119860119889

1198612119889

1198621

11986312

] (27)

where 119860119889and 119862

1have been defined previously and

1198612119889

=[

[

119898

sum

119894=1

119860119898minus119894

119861

119868

]

]

119867 (119896)

11986312=

[

[

[

[

[

[

[

[

[

[

0

0 +

119898

sum

119895=119898

119862119860119898minus119895

119861

0 +

119898

sum

119895=2

119862119860119898minus119895

119861

]

]

]

]

]

]

]

]

]

]

119867(119896)

(28)

(c) Transfer Function for 11987521 Similarly because 119878(119896) is also not

an119898-blocks unit matrix 11987521can be changed to

11987521= 119878 (119896) [119868 0 sdot sdot sdot 0]

11989811987111989811987811989111986621119867119891119871minus1

119898 (29)

Thus the state model can be derived as

21(120582) = [

119860119889

1198611

1198622

11986321

] (30)

where 119860119889and 119861

1were defined previously and

1198622= Δ (119896)

11986321= [Ω (119896) 0 sdot sdot sdot 0]

(31)

(d) Transfer Function for 11987522 Like to parts 119887 and 119888 the state

description of 11987522is

22(120582) = [

119860119889

1198612119889

1198622

11986322

] (32)

where 119860119889 1198612119889 and 119862

2were defined previously while

11986322= 0 (33)

Finally formula (23) is represented using the state-spaceform as follows

120585 (119896 + 1) = 119860119889120585 (119896) + 119861

1119908 (119896) + 119861

2119889119903 (119896)

119911 (119896) = 1198621120585 (119896) + 119863

11119908 (119896) + 119863

12119903 (119896)

119910 (119896) = 1198622120585 (119896) + 119863

21119908 (119896)

(34)

where the underlining below 119911 and 119908 indicates that thesignals are lifted

According to Lemma 2 the controller119870(119896) exists and canstabilize the system

Let weierp(119896) = [1205851015840

(119896) 1198901015840

(119896)]

1015840 and after combining (13) (15)and (34) we obtain

weierp (119896 + 1) = 119860119888weierp (119896) + 119861

119888119908 (119896)

119911 (119896) = 119862119888weierp (119896) + 119863

119888119908 (119896)

(35)

where

119860119888=

[

[

[

119860119889+ 1198612119889119870 (119896) minus119861

2119889119870 (119896) [

1198682

0]

0 119860 minus 119871 (119896) 119878 (119896) 119862

]

]

]

119861119888= [

1198611

(1198721minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

119898

]

119862119888= [119862

1+ 11986312119870 (119896) minus119863

12119870 (119896) [

1198682

0]]

119863119888= 11986311

(36)

In the light of [5] we set

119883 (119896) = [

1198831(119896) 0

0 1198832(119896)

] (37)

6 Mathematical Problems in Engineering

By substituting119883 in Lemma 3 with119883(119896) we can obtain

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831(119896) 0 (119860

119889+ 1198612119889119870 (119896))119883

1(119896) minus119861

2119889119870 (119896) [

1198682

0

]1198832(119896) 119861

10

lowast 1198832(119896) 0 (119860 minus 119871 (119896) 119878 (119896) 119862)119883

2(119896) (119872

1minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 1198831(119896) 0 0 119883

1(119896) (119862

1015840

1+ 1198701015840

(119896)1198631015840

12)

lowast lowast lowast 1198832(119896) 0 minus119883

2(119896) [1198682

0]1198701015840

(119896)1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0

(38)

Then according to [26] after multiplying the left- andright-hand sides of inequality (38) by diag119868 119868 119883minus1

1(119896)

119883minus1

2(119896) 119868 119868 we can obtain inequality (20)According to quadratic stability theory the system should

be stable at each step 119896 Thus for a discrete linear time-invariant system the eigenvalues of the closed system shouldlie within a unit circle

From Theorem 4 if matrix inequalities (20) can besolved there exists a state feedback 119867

infincontroller that can

asymptotically stabilize closed system (35) of enlarged system(34) which satisfies the performance index Since system(34) is lifted from the original system (9) and we know thatthe lifting operator is an isometric isomorphism that isthe feedback stability and system norm can be preservedthen system (16) which is the closed system of (9) isasynchronously stable and it satisfies the119867

infinnorm constraint

119879119911119908infin

lt 120574

The synchronous case can be regarded as a specialexample of the asynchronous case In this case the elementsof sampling and holder are unit matrices with the assumption

that the whole signals of sampling and holding can beobtained at the beginning of every period then the system (9)turns into a linear discrete synchronous multirate sampled-data system with period 120591 as follows

120585 (119896 + 1) = Φ120585 (119896) + Γ119903 (119896) + Ψ119908 (119896)

119911 (119896) = Υ120585 (119896) + 1198722119908 (119896)

119910 (119896) = Δ120585 (119896) + Ω119908 (119896)

(39)

As119867 and 119878 are unit matrices that is constant now Φ ΓΔ andΩ related to119867 and 119878 are also constant Theorem 4 canbe applied to system (39) straightforwardly

Corollary 5 Consider the sampled-data system (39) with thelifting process shown in Figure 3 For a given attenuation level120574 gt 0 if there exist matrices 119883

1= 1198831015840

1gt 0 119883

2= 1198831015840

2gt

0 119870 and 119871 which are suboptimal solutions of the inequalitiesbelow then the closed system of (39) with the controller (13)is synchronously uniformly stable and it satisfies the119867

infinnorm

constraint 119879119911119908infin

lt 120574

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831

0 119860119889+ 1198612119889119870 minus119861

2119889119870[

1198682

0] 119861

10

lowast 1198832

0 119860 minus 119871119862 (1198721minus 119871119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 119883minus1

10 0 119862

1015840

1+ 1198701015840

1198631015840

12

lowast lowast lowast 119883minus1

20 minus [119868

20]1198701015840

1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0 (40)

Remark 6 Let us compare the designed observer (12) withthat in [21] where the observer (predictor) is

(119896 + 1 | 119896) = 119860 (119896 | 119896 minus 1) + 119861119906 (119896)

+ 119871 (119896) sdot [120577 (119896) minus 119862 (119896) (119896 | 119896 minus 1)]

120577 (119896) =

119879

lowast

sum

119894=1

119910 (119896 minus 119894) 119879

lowast

= max119894=1119901

119879119894

(41)The sum of outputs is used to avoid the singularity butmay cause more noise The observer applied in this paper is

Mathematical Problems in Engineering 7

w z

u yG

K

Tm1 Tmp

middot middot middot middot middot middotTn1Tnq H1Hq SpS1

Sa Lminus1mLb HbHf Sf Lminus1aLm

Figure 3 System lifting and controller designing

simpler that is just the current output is introduced in theobserver This will result in better tracking performance andsmaller state ripple than [21] where the past outputs otherthan current output are used for feedback

Remark 7 InTheorem 4 and Corollary 5 due to the existenceof inverse matrices 119883minus1

1and 119883

minus1

2 inequalities (20) and (40)

are not LMIs and cannot be directly solved by the LMI toolHowever notice that the matrices and their inverse matricesappear in pairs we can solve the inequalities by iterationon LMIs according to Cone Complementarity LinearizationAlgorithm [22]

4 Numerical Example with Simulation

Consider the original plant described by

119888

(119905) = 119860119888

119909119888

(119905) + 119861119888

119906119888

(119905) + 119872119888

1119908119888

(119905)

119911119888

(119905) = 119862119888

119909119888

(119905) + 119872119888

2119908119888

(119905)

(42)

where

119860119888

= [

0 1

minus6 minus5] 119861

119888

= [

0 1

1 12] 119872

119888

1= [

1

06]

119862119888

= [02 01] 119872119888

2= 001

(43)

If the fast discretization formulae follow119860 = 119890119860119888(120591119898) 119861 =

int

120591119898

0

119890119860119888120590

119889120590119861119888 1198721= int

120591119898

0

119890119860119888120590

119889120590119872119888

1 119862 = 119862

119888 and 1198722=

119872119888

2 then system (2) can be obtained The initial state is 119909

0=

[30 20]If 1198791=

1198792= 2 the input-holding mechanism can be

described as

119867(0) = [

1 0

0 0] 119867 (1) = [

0 0

0 1] (44)

If 1198791= 3 and 119879

2= 1 the output-sampling matrices are

assumed to be

119878 (0) = 1 119878 (1) = 0 119878 (2) = 0 (45)

Therefore this is a linear discrete asynchronous dual-ratesampled-data system [18] which applies a fast-input-slow-output mechanism for 119879 lt 119879

0 5 10 15 20 25 30

0

5

10

15

20

25

30

minus5

Real stateObserved state

Figure 4 Plant state 1199091(119896)

0

5

10

15

20

minus25

minus20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 5 Plant state 1199092(119896)

0001002003004005006007008

minus0010 5 10 15 20 25 30

Figure 6 Control signal 1199061(119896)

8 Mathematical Problems in Engineering

0

05

minus3

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 7 Control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 8 Output

For comparison we introduce a previous method [21]where the system is not lifted according to the pure LQGtheory and also let 119908

1= 1199082= 119908 be Gaussian white noise

amplitude is increased to 10 which is much bigger to thesystem The results obtained using this method are shown inFigures 4 5 6 7 and 8 where 120591 = 12 s

Next the synthesis method described in Theorem 4is applied to the same plant The simulations obtain thetransients of the system shown in Figures 9 10 11 12 and 13where 120574 = 02 119886 = 1 and 119887 = 1 while119898 = 1 (monorate)

A comparison of Figures 4ndash8 with Figures 9ndash13 showsclearly that the 119867

infinmethod performs better than LQG

the controlled states are consistent with the observed stateswith smaller ripples especially state 119909

2(119896) the biggest ripple

of which is almost half of the LQG method And theoutput of119867

infinmethod has better performance of disturbance

attenuation than LQG methodWhen 120591 = 06 s and 119898 = 1 (monorate) applying

Theorem 4 again we can get Figures 14 15 16 17 and 18Then the lifting number is modified to119898 = 2 (multirate) theresults shown in Figures 19 20 21 22 and 23 are obtained

Comparing Figures 14ndash18 with Figures 19ndash23 when thelifting amplitude increases from 1 to 2 the controlled statesare almost consistent with the observed states with smallerripples here the biggest ripple of state 119909

2(119896) is almost half of

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 9 Plant state 1199091(119896)

0

5

10

15

20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 10 Plant state 1199092(119896)

0

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 11 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

07

0 5 10 15 20 25 30

Figure 12 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 13 Output

Table 1 Minimum119867infinattenuation level

120591 119898 = 1 119898 = 2 119898 = 4

12 sec 00974 00959 0094206 sec 00904 00863 00856

the monorate And the output of the multirate case has betterperformance of disturbance attenuation than the monoratecase

Furthermore we change the lifting amplitude to 119898 = 4adjust the underlying period of system (2) to 120591 = 06 andcan draw the similar pictures here we omit them but only listtheir minimum119867

infinattenuation levels to compare them with

different cases see Table 1The inequality acquired using the theorem is almost

linear except in a few terms Although these terms are justthe inverses of their corresponding terms they could easily becalculated by LMI tool ofMatlab under a constraintThus it isfar easier to solve the controller and observer under the same119867infinconstraint comparing withmost of the existingmethods

which use highly complex formulae

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 14 Plant state 1199091(119896)

0

5

10

15

20

Real stateObserved state

minus5

minus30

minus25

minus20

minus15

minus10

0 5 10 15 20 25 30

Figure 15 Plant state 1199092(119896)

0

0 5 10 15 20 25 30

minus3

minus25

minus35

minus4

minus45

minus2

minus15

minus1

minus05

Figure 16 Plant control signal 1199061(119896)

10 Mathematical Problems in Engineering

0

02

04

06

08

1

12

14

0 5 10 15 20 25 30

Figure 17 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30minus1

Figure 18 Output

5 Conclusion

Motivated by a desire to provide a better solution for thedesign of a controller for a discrete-time asynchronous mul-tirate sampled-data system we developed synthesis methodsin this paper to improve performance and guarantee closed-loop transient behavior via observer-based 119867

infincontrol

Furthermore an extended lifting technique was adoptedto change the original system into a fast-input-slow-outputdiscrete system thereby improving its performance Finallywe presented an example that demonstrates the effectivenessof thesemethods in controlling a discrete-time asynchronousmultirate sampled-data system whose performance is thusimproved considerably

In this paper we focus on the standard 119867infin

control ofasynchronous multirate sampled-data system and the inputnoise discussed here is Gaussian white for simplicity As iswell known practical input signals often have finite frequency(FF) so it is reasonable to consider the 119867

infinperformance in

finite frequency range for sampled-data systems As in [27ndash31] where Kalman-Yakubovic-Popov (KYP) lemma is usedto establish the equivalence between a frequency domaininequality (FDI) and a linear matrix inequality it is a valuablework to extend the results in this paper to the FF frameworkbased on KYP lemma This will be the direction for futuredevelopment of this paper

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Real stateObserved state

minus10

minus5

Figure 19 Plant state 1199091(119896)

0

5

10

15

20

0 5 10 15 20 25 30

Real stateObserved state

minus20

minus15

minus10

minus5

Figure 20 Plant state 1199092(119896)

005

0 5 10 15 20 25 30

minus3

minus35

minus4

minus25

minus2

minus15

minus1

minus05

Figure 21 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 11

0

02

04

06

08

1

12

0 5 10 15 20 25 30minus02

Figure 22 Plant control signal 1199062(119896)

minus10 5 10 15 20 25 30

0

1

2

3

4

5

6

7

8

9

Figure 23 Output

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Zhejiang Provincial Natural Sci-ence Foundation of China under Grant no LR12F03002 andNational Natural Science Foundation of China (NSFC) underGrant no 61074045

References

[1] M F Sagfors H T Toivonen and B Lennartson ldquoState-spacesolution to the period multirate 119867

infincontrol problem a lifting

approachrdquo IEEE Transaction on Automatic Control vol 37 no5 pp 2345ndash2350 2000

[2] X Zhao Y Yao and JMa ldquoOverview ofmultirate sampled-datacontrol systemrdquo Journal of Harbin Institute of Technology vol 9no 4 pp 405ndash410 2002

[3] Y Fukada ldquoSlip-angle estimation for vehicle stability controlrdquoVehicle System Dynamics vol 32 no 4 pp 375ndash388 1999

[4] T Chen and L Qiu ldquo119867infindesign of general multirate sampled-

data control systemsrdquo Automatica vol 30 no 7 pp 1139ndash11521994

[5] Z Wang B Huang and H Unbehauen ldquoRobust 119867infin

observerdesign of linear state delayed systems with parametric uncer-tainty the discrete-time caserdquo Automatica vol 35 no 6 pp1161ndash1167 1999

[6] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Control Theory vol 15 ofSIAM Studies in Applied Mathematics SIAM Philadelphia PaUSA 1994

[7] S Patra and A Lanzon ldquoA closed-loop data based test forrobust performance improvement in iterative identification andcontrol redesignsrdquo Automatica vol 48 no 10 pp 2710ndash27162012

[8] J Ding Y Shi H Wang and F Ding ldquoA modified stochasticgradient based parameter estimation algorithm for dual-ratesampled-data systemsrdquo Digital Signal Processing vol 20 no 4pp 1238ndash1247 2010

[9] T W Chen and B Francis Optimal Sampled-Data ControlSystems Springer London UK 1995

[10] L Shen and M J Er ldquoMultiobjectives design of a multirateoutput controllerrdquo in Proceedings of the IEEE InternationalConference on Control Applications (CCA rsquo99) pp 193ndash198August 1999

[11] M Chilali and P Gahinet ldquo119867infin

design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996

[12] S Mo X Chen J Zhao J Qian and Z Shao ldquoA two-stagemethod for identification of dual-rate systems with fast inputand very slow outputrdquo Industrial and Engineering ChemistryResearch vol 48 no 4 pp 1980ndash1988 2009

[13] M De la Sen and S Alonso-Quesada ldquoModel matching viamultirate sampling with fast sampled input guaranteeing thestability of the plant zeros extensions to adaptive controlrdquo IETControl Theory amp Applications vol 1 no 1 pp 210ndash225 2007

[14] M F Sagfors and H T Toivonen ldquo119867infin

and LQG control ofasynchronous sampled-data systemsrdquo Automatica vol 33 no9 pp 1663ndash1668 1997

[15] S Lopez-lopez Optimal Hinfin

Design of Causal Multirate Con-trollers and Filters Mechanical and Aerospace EngineeringUniversity of California Irvine Calif USA 2010

[16] U Borison ldquoSelf-tuning regulators for a class of multivariablesystemsrdquo Automatica vol 15 no 2 pp 209ndash215 1979

[17] V S Ritchey and G F Franklin ldquoA stability criterion forasynchronous multirate linear systemsrdquo IEEE Transactions onAutomatic Control vol 34 no 5 pp 529ndash535 1989

[18] B Shen Z Wang and X Liu ldquoSampled-data synchronizationcontrol of dynamical networks with stochastic samplingrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2644ndash2650 2012

[19] S Longo G Herrmann and P Barber ldquoRobust scheduling ofsampled-data networked control systemsrdquo IEEE Transactionson Control Systems Technology vol 20 no 6 pp 1613ndash1621 2011

[20] P Voulgaris ldquoControl of asynchronous sampled data systemsrdquoIEEE Transactions on Automatic Control vol 39 no 7 pp 1451ndash1455 1994

[21] P Colaneri R Scattolini and N Schiavoni ldquoLQG optimalcontrol of multirate sampled-data systemsrdquo IEEE Transactionson Automatic Control vol 37 no 5 pp 675ndash682 1992

[22] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

12 Mathematical Problems in Engineering

[23] A Seuret ldquoStability analysis of networked control systems withasynchronous sampling and input delayrdquo in Proceedings of theAmerican Control Conference (ACC rsquo11) pp 533ndash538 July 2011

[24] B Tavassoli ldquoStability of nonlinear networked control systemsover multiple communication links with asynchronous sam-plingrdquo IEEE Transaction on Automatic Control 2013

[25] L Yang and J M Li ldquoSufficient and necessary conditions ofcontrollability and observability of a class of linear switchingsystemrdquo Systems Engineering and Electronics vol 25 no 5 pp588ndash590 2003

[26] W-J Mao ldquoObserver-based energy decoupling of linear time-delay systemsrdquo Journal of Zhejiang University vol 37 no 5 pp499ndash503 2003

[27] T Iwasaki and S Hara ldquoGeneralized KYP lemma unifiedfrequency domain inequalities with design applicationsrdquo IEEETransactions on Automatic Control vol 50 no 1 pp 41ndash592005

[28] H Gao and X Li ldquo119867infinfiltering for discrete-time state-delayed

systems with finite frequency specificationsrdquo IEEE Transactionson Automatic Control vol 56 no 12 pp 2935ndash2941 2011

[29] X Li H Gao and CWang ldquoGeneralized Kalman-Yakubovich-Popov lemma for 2-D FM LSS modelrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3090ndash3103 2012

[30] X Li and H Gao ldquoRobust finite frequency 119867infin

filtering foruncertain 2-D systems the FM model caserdquo Automatica vol49 no 8 pp 2446ndash2452 2013

[31] X W Li and H J Gao ldquoA heuristic approach to static output-feedback controller synthesis with restricted frequency-domainspecificationsrdquo IEEE Transaction on Automatic Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

That is at a specific 119896

119878 (119896119879) = diag [11986810 sdot sdot sdot 0]

1198791

[11986820 sdot sdot sdot 0]

1198792

[119868119901

0 sdot sdot sdot 0]119879119901

119867 (119896119879) = diag [119868

1sdot sdot sdot 1198681]

1015840

1198791

[1198682sdot sdot sdot 1198682]

1015840

1198792

[119868119902sdot sdot sdot 119868119902]

1015840

119879119902

(1)

where 119868119894 119894 = 1 2 119901(119902) are unit matrices and 119879 119879

119894

119879 and

119879119894 will be defined in Section 2 However in most

actual situations this condition cannot be satisfied becausethe overall system lacks a slot when the entire signal can besampled or held and there is a class of multirate systems thateach output of them has its own frequency of measurementand each input has its own frequency of updating [16ndash19] Asystem with this type of sampling mechanism is referred toas asynchronous system in this paper The traditional liftingtechnique is not applicable for the sampling frequencies of thesampling and holding elements are incommensurate [4 20]

LQG control based on an observer is an effective methodfor reducing the dimensions of a multirate system with-out using the lifting technique [14 21] An LQG optimalcontroller derived is periodic and can guarantee the overallclosed-loop stability To compensate for the skewed inputsignal caused by slow-output feedback only the input signalis supplemented by the observer However in the previousresults the controller and observer are designed separatelywhich may result in conservativeness

Based on the methods mentioned above the design ofa controller for a multirate sampled-data system that doesnot multiply the dimensions necessitates the synthesis ofvarious appropriate measures In this paper we propose anobserver-based 119867

infinmethod for analysis and synthesis of

linear discrete asynchronousmultirate sampled-data systemsIn contrast to the traditional lifting method the controlinputs and measured outputs are not lifted to avoid thedimension multiplication and causal constraints Howeverthe exogenous system signals are lifted rationally to changethe system into a fast-input-slow-output system The con-troller can be obtained by solving an inequality with LMItools of Matlab Although the inequality is nonlinear it canbe calculated linearly under a constraint [22] The approachproposed in this paper is different from the observer-basedLQG method in [21] Here the controller and observer areobtained simultaneously by solving amatrix inequality underthe same optimal performance constraint thus reducing theconservativeness and improving the system performanceIn addition the concept of asynchronism in this paper isdifferent from those in [23 24] and so forth where thesampling at sensors is assumed to be asynchronized withtime-varying delay and it falls into the framework of time-delay systems not the multirate sampled-data systems

The remainder of this paper is organized as followsSection 2 provides a description of the multirate systemand its background including lemmas and definitions In

w z

u y

G

K

H1Hq

Tm1Tn1Tnq Tmp

SpS1middot middot middot middot middot middot

Figure 1 Multirate sampled-data system

Signal 1Signal 2Signal 3

0 1 2 3 4 5 6 7 8 9

Time slot

Figure 2 Asynchronous signals from the sampler or the holder

Section 3 the main results obtained using the synthesismethods are presented In Section 4 a simulated numericalexample is provided to demonstrate the effectiveness of themethod Finally Section 5 concludes the paper

2 System Descriptions

Definition 1 A multirate sampled-data system is called anasynchronous system if the signals from its sampler or holdercannot be obtained simultaneously during any time slot

For example as shown in Figure 2 Signal 1 and Signal 2are the signals of the sampler and the holder with period 3respectively while Signal 3 is the signal with period 2 Thusthey cannot be obtained simultaneously during any time slotbecause the odd period signals appear alternately In Figure 2the filled circles indicate that the signal is sampled or held inthe slot whereas the hollow cylinders indicate that the signalis not sampled or held Therefore this system cannot satisfythe assumptionmentioned inmost previous studies where allsignals can be sampled or held during the same time slot

We consider a linear discrete asynchronous multiratesampled-data system

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 1198721119908 (119896)

119911 (119896) = 119862119909 (119896) +1198722119908 (119896)

119910 (119896) = 119878 (119896) 119911 (119896)

(2)

where 119896 isin N 119909(119896) isin R119899 119906(119896) isin R119902 119911(119896) isin R119901 119908(119896) isin

R119897 119909(0) = 1199090 |1198722| = 0 119878(119896) is a sampling device 119910(119896) is the

output measured vector and 1199090is the initial state

Assume that the 119894th component 119906119894(119896) 119894 = 1 2 119902 of

the input vector 119906(119896) can be modified every 119879119894time instant

Mathematical Problems in Engineering 3

119879119894isin Z+ Then 119906(119896) is the output of the following discrete-

time period system which is defined as the input-holdingmechanism

V (119896 + 1) = (119868 minus 119867 (119896)) V (119896) + 119867 (119896) 119903 (119896)

119906 (119896) = (119868 minus 119867 (119896)) V (119896) + 119867 (119896) 119903 (119896)

(3)

where

119867(119896) = diag ℎ1(119896) ℎ

2(119896) ℎ

119902(k)

ℎ119894(119896) =

1 119896 = 119895119879119894+ 120591119894

0 119896 = 119895119879119894+ 120591119894

119895 integer(4)

where V(119896) = [V1(119896) V

2(119896) sdot sdot sdot V

119902(119896)]

1015840 is a new statevariable and 119903(119896) = [119903

1(119896) 1199032(119896) sdot sdot sdot 119903

119902(119896)]

1015840 is a new inputvariable The integer 0 le 120591

119894lt119879119894(119894 = 1 2 119902) describes the

skew of the inputs-holding mechanismThen matrix119867(sdot) and system (2) have a period of 119879 and

119879 = lcm

119894=12119902

119879119894 (5)

Similarly the output sampling mechanism is given as

119878 (119896) = diag V1(119896) V2(119896) V

119901(119896) (6)

where

V119894(119896) =

1 119896 = 119895119879119894+ 120591119894

0 119896 = 119895119879119894+ 120591119894

119895 integer (7)

where 0 le 120591119894lt 119879119894(119894 = 1 2 119901) indicates the skew of the

outputs mechanismMatrix 119878(sdot) has a 119879-period and

119879 = lcm

119894=12119901

119879119894 (8)

After combining (2) and (3) we obtain a new sampling-holding system

120585 (119896 + 1) = Φ (119896) 120585 (119896) + Γ (119896) 119903 (119896) + Ψ119908 (119896)

119911 (119896) = Υ120585 (119896) + 1198722119908 (119896)

119910 (119896) = Δ (119896) 120585 (119896) + Ω (119896)119908 (119896)

(9)

where

120585 (119896) = [

119909 (119896)

V (119896)] Φ (119896) = [

119860 119861 (119868 minus 119867 (119896))

0 119868 minus 119867 (119896)]

Γ (119896) = [

119861

119868]119867 (119896) Ψ = [

1198721

0]

Υ = [119862 0] Δ (119896) = [119878 (119896) 119862 0]

Ω (119896) = 119878 (119896)1198722

(10)

Based on the 119879 of matrix 119867(sdot) and the 119879 of matrix 119878(sdot)

system (9) has the period of

119879 = lcm 119879 119879 (11)

The state observer of system (2) is

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119871 (119896) sdot [119910 (119896) minus 119878 (119896) 119862119909 (119896)]

(12)

where 119871(119896) asymptotically stabilizes system (12)The observer-based controller is chosen as

119903 (119896) = 119870 (119896)120585 (119896) (13)

where 120585(119896) = [119909(119896)

V(119896) ]The state error is defined as

119890 (119896) = 119909 (119896) minus 119909 (119896) (14)

According to (2) (12) and (14) the following equation canbe obtained

119890 (119896 + 1) = [119860 minus 119871 (119896) 119878 (119896) 119862] 119890 (119896)

+ (1198721minus 119871 (119896) 119878 (119896)119872

2) 119908 (119896)

(15)

Combining (9) (13) and (15) yields

[

120585 (119896 + 1)

119890 (119896 + 1)] = 119860[

120585 (119896)

119890 (119896)] + 119861119908 (119896)

119911 (119896) = 119862 [

120585 (119896)

119890 (119896)] + 119863119908 (119896)

(16)

where

119860 =[

[

[

Φ (119896) + Γ (119896)119870 (119896) minusΓ (119896)119870 (119896) [

1198682

0]

0 119860 minus 119871 (119896) 119878 (119896) 119862

]

]

]

119861 = [

Ψ

1198721minus 119871 (119896) 119878 (119896)119872

2

]

119862 = [Υ 0] 119863 = 1198722

(17)

Lemma 2 (see [21 25] ) If

(i) the pair (119860 119861) is stabilizable(ii) the pair (119860119872

1) is stabilizable

(iii) the pair (119860 119862) is detectable(iv) for any pair of distinct eigenvalues of 119860 120582

119894and 120582

119895

|120582119894| ge 1 |120582

119895| ge 1 it follows that 120582119879

119894= 120582

119879

119895

(v) there is no eigenvalue 120582 of 119860 120582 = 1 such that 120582119879 = 1

then the pairs (Φ(119896) Γ(119896)) and (Φ(119896) Ψ) are controllable andthe pair (Φ(119896) Υ) is detectable

4 Mathematical Problems in Engineering

Lemma 3 (see [10] (the Bounded Real Lemma) ) For a givenscalar 120574 gt 0 if there is a matrix 119883 = 119883

1015840

gt 0 that satisfies thematrix inequality

[

[

[

[

119883 119860119883 119861 0

1198831198601015840

119883 0 1198831198621015840

1198611015840

0 120574119868 1198631015840

0 119862119883 119863 120574119868

]

]

]

]

gt 0 (18)

then the system (16) is asynchronously stable and it achieves aspecified attenuation level 120574 gt 0 such that 119879

119911119908infin

lt 120574

3 Main Results

For the linear discrete asynchronous multirate sampled-data system (9) the exogenous parts of the system arelifted to improve its performance To avoid multiplying thedimensions of the control inputs and measured outputs wedo not lift them thereby avoiding causal constraints of thetraditional lifting method

If we assume that there is an underlying clock with a baseperiod of 120591 then system (9) represents a fast discrete system

with a subperiod of 120591119898 for the discrete process describedin [9] The sampler 119878

119886samples the continuous signal 119908 for a

discrete 119886-period and the holder 119867119887holds the output signal

119911 during period 119887 The rates of the sampler and the holdercan be set at different flexible rates that is 119886 = 119887 If weselect (13) as the new input 119870(119896) is the controller of system(9)

These concepts are described in Figure 3 where

119898 = lcm 119886 119887 (19)

Theorem 4 Consider the sampled-data system (9) with thelifting process shown in Figure 3 For a given attenuationlevel 120574 gt 0 if there exist matrices 119883

1(119896) = 119883

1015840

1(119896) gt

0 1198832(119896) = 119883

1015840

2(119896) gt 0 119870(119896) and 119871(119896) (119896 = 0 119873

denotes the iteration time) which are suboptimal solutions ofthe inequalities below then the closed system of (9) with thecontroller (13) is asynchronously stable and satisfies the 119867

infin

norm constraint 119879119911119908infin

lt 120574

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831(119896) 0 119860

119889+ 1198612119889119870 (119896) minus119861

2119889119870 (119896) [

1198682

0] 119861

10

lowast 1198832(119896) 0 119860 minus 119871 (119896) 119878 (119896) 119862 (119872

1minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 119883minus1

1(119896) 0 0 119862

1015840

1+ 1198701015840

(119896)1198631015840

12

lowast lowast lowast 119883minus1

2(119896) 0 minus [119868

20]1198701015840

(119896)1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0 (20)

Proof After lifting system (9) according to Figure 3 thetransform function of a completely discrete system with aperiod of 120591119898 which is partly contained within the dot-dashline is

119866 = [

119871119898

0

0 119868] [

119878119891

0

0 119878 (119896)

] [

11986611

11986612

11986621

11986622

] [

119867119891

0

0 119867 (119896)

] [119871minus1

1198980

0 119868

]

=[

[

11987111989811987811989111986611119867119891119871minus1

119898

11987111989811987811989111986612119867(119896)

119878 (119896) 11986621119867119891119871minus1

119898

119878 (119896) 11986622119867(119896)

]

]

(21)

Based on the above we can derive minimal state-spacerepresentations 119892(120582) for 119866 as follows based on the transferfunction theory given in [9 15] However in contrast to thetheory 119867 and 119878 are not blocks of unit matrices because

sampling and holding are not synchronous that is the entiresignal cannot be sampled or held at the same time

119892 (120582) = [

11987511

11987512

11987521

11987522

] (22)

Next we calculate the minimal state-space realization of(22) as follows(a) Transfer Function for 119875

11 Note that119875

11= 11987111989811987811989111986611119867119891119871minus1

119898

which comes directly from the theory in [8] and its corre-sponding state model is

11(120582) = [

119860119889

1198611

1198621

11986311

] (23)

where

119860119889=

[

[

119860119898

119898

sum

119895=1

119860119898minus119895

119861 (119868 minus 119867 (119896))

0 119868 minus 119867 (119896)

]

]

1198611= [

119860119898minus1

1198721119860119898minus2

1198721sdot sdot sdot 119872

1

0 0 sdot sdot sdot 0

]

Mathematical Problems in Engineering 5

1198621=

[

[

[

[

[

119862 0

119862119860 0

119862119860119898minus1

0

]

]

]

]

]

11986311=

[

[

[

[

[

1198722

0 sdot sdot sdot 0

1198621198721

1198722

sdot sdot sdot 0

d

119862119860119898minus2

1198721119862119860119898minus3

1198721sdot sdot sdot 119872

2

]

]

]

]

]

(24)

(b) Transfer Function for 11987512 As119867(119896) in 119875

12= 11987111989811987811989111986612119867(119896)

is not an 119898-blocks unit matrix the theory cannot be applieddirectly However note that

119878119891119867(119896) = 119871

minus1

119898

[

[

[

[

[

119868

119868

119868

]

]

]

]

]119898

119867(119896) (25)

Then

11987512= 11987111989811987811989111986612119867119891119871minus1

119898

[

[

[

[

[

119868

119868

119868

]

]

]

]

]119898

119867(119896) (26)

The corresponding state model can be obtained as

12(120582) = [

119860119889

1198612119889

1198621

11986312

] (27)

where 119860119889and 119862

1have been defined previously and

1198612119889

=[

[

119898

sum

119894=1

119860119898minus119894

119861

119868

]

]

119867 (119896)

11986312=

[

[

[

[

[

[

[

[

[

[

0

0 +

119898

sum

119895=119898

119862119860119898minus119895

119861

0 +

119898

sum

119895=2

119862119860119898minus119895

119861

]

]

]

]

]

]

]

]

]

]

119867(119896)

(28)

(c) Transfer Function for 11987521 Similarly because 119878(119896) is also not

an119898-blocks unit matrix 11987521can be changed to

11987521= 119878 (119896) [119868 0 sdot sdot sdot 0]

11989811987111989811987811989111986621119867119891119871minus1

119898 (29)

Thus the state model can be derived as

21(120582) = [

119860119889

1198611

1198622

11986321

] (30)

where 119860119889and 119861

1were defined previously and

1198622= Δ (119896)

11986321= [Ω (119896) 0 sdot sdot sdot 0]

(31)

(d) Transfer Function for 11987522 Like to parts 119887 and 119888 the state

description of 11987522is

22(120582) = [

119860119889

1198612119889

1198622

11986322

] (32)

where 119860119889 1198612119889 and 119862

2were defined previously while

11986322= 0 (33)

Finally formula (23) is represented using the state-spaceform as follows

120585 (119896 + 1) = 119860119889120585 (119896) + 119861

1119908 (119896) + 119861

2119889119903 (119896)

119911 (119896) = 1198621120585 (119896) + 119863

11119908 (119896) + 119863

12119903 (119896)

119910 (119896) = 1198622120585 (119896) + 119863

21119908 (119896)

(34)

where the underlining below 119911 and 119908 indicates that thesignals are lifted

According to Lemma 2 the controller119870(119896) exists and canstabilize the system

Let weierp(119896) = [1205851015840

(119896) 1198901015840

(119896)]

1015840 and after combining (13) (15)and (34) we obtain

weierp (119896 + 1) = 119860119888weierp (119896) + 119861

119888119908 (119896)

119911 (119896) = 119862119888weierp (119896) + 119863

119888119908 (119896)

(35)

where

119860119888=

[

[

[

119860119889+ 1198612119889119870 (119896) minus119861

2119889119870 (119896) [

1198682

0]

0 119860 minus 119871 (119896) 119878 (119896) 119862

]

]

]

119861119888= [

1198611

(1198721minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

119898

]

119862119888= [119862

1+ 11986312119870 (119896) minus119863

12119870 (119896) [

1198682

0]]

119863119888= 11986311

(36)

In the light of [5] we set

119883 (119896) = [

1198831(119896) 0

0 1198832(119896)

] (37)

6 Mathematical Problems in Engineering

By substituting119883 in Lemma 3 with119883(119896) we can obtain

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831(119896) 0 (119860

119889+ 1198612119889119870 (119896))119883

1(119896) minus119861

2119889119870 (119896) [

1198682

0

]1198832(119896) 119861

10

lowast 1198832(119896) 0 (119860 minus 119871 (119896) 119878 (119896) 119862)119883

2(119896) (119872

1minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 1198831(119896) 0 0 119883

1(119896) (119862

1015840

1+ 1198701015840

(119896)1198631015840

12)

lowast lowast lowast 1198832(119896) 0 minus119883

2(119896) [1198682

0]1198701015840

(119896)1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0

(38)

Then according to [26] after multiplying the left- andright-hand sides of inequality (38) by diag119868 119868 119883minus1

1(119896)

119883minus1

2(119896) 119868 119868 we can obtain inequality (20)According to quadratic stability theory the system should

be stable at each step 119896 Thus for a discrete linear time-invariant system the eigenvalues of the closed system shouldlie within a unit circle

From Theorem 4 if matrix inequalities (20) can besolved there exists a state feedback 119867

infincontroller that can

asymptotically stabilize closed system (35) of enlarged system(34) which satisfies the performance index Since system(34) is lifted from the original system (9) and we know thatthe lifting operator is an isometric isomorphism that isthe feedback stability and system norm can be preservedthen system (16) which is the closed system of (9) isasynchronously stable and it satisfies the119867

infinnorm constraint

119879119911119908infin

lt 120574

The synchronous case can be regarded as a specialexample of the asynchronous case In this case the elementsof sampling and holder are unit matrices with the assumption

that the whole signals of sampling and holding can beobtained at the beginning of every period then the system (9)turns into a linear discrete synchronous multirate sampled-data system with period 120591 as follows

120585 (119896 + 1) = Φ120585 (119896) + Γ119903 (119896) + Ψ119908 (119896)

119911 (119896) = Υ120585 (119896) + 1198722119908 (119896)

119910 (119896) = Δ120585 (119896) + Ω119908 (119896)

(39)

As119867 and 119878 are unit matrices that is constant now Φ ΓΔ andΩ related to119867 and 119878 are also constant Theorem 4 canbe applied to system (39) straightforwardly

Corollary 5 Consider the sampled-data system (39) with thelifting process shown in Figure 3 For a given attenuation level120574 gt 0 if there exist matrices 119883

1= 1198831015840

1gt 0 119883

2= 1198831015840

2gt

0 119870 and 119871 which are suboptimal solutions of the inequalitiesbelow then the closed system of (39) with the controller (13)is synchronously uniformly stable and it satisfies the119867

infinnorm

constraint 119879119911119908infin

lt 120574

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831

0 119860119889+ 1198612119889119870 minus119861

2119889119870[

1198682

0] 119861

10

lowast 1198832

0 119860 minus 119871119862 (1198721minus 119871119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 119883minus1

10 0 119862

1015840

1+ 1198701015840

1198631015840

12

lowast lowast lowast 119883minus1

20 minus [119868

20]1198701015840

1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0 (40)

Remark 6 Let us compare the designed observer (12) withthat in [21] where the observer (predictor) is

(119896 + 1 | 119896) = 119860 (119896 | 119896 minus 1) + 119861119906 (119896)

+ 119871 (119896) sdot [120577 (119896) minus 119862 (119896) (119896 | 119896 minus 1)]

120577 (119896) =

119879

lowast

sum

119894=1

119910 (119896 minus 119894) 119879

lowast

= max119894=1119901

119879119894

(41)The sum of outputs is used to avoid the singularity butmay cause more noise The observer applied in this paper is

Mathematical Problems in Engineering 7

w z

u yG

K

Tm1 Tmp

middot middot middot middot middot middotTn1Tnq H1Hq SpS1

Sa Lminus1mLb HbHf Sf Lminus1aLm

Figure 3 System lifting and controller designing

simpler that is just the current output is introduced in theobserver This will result in better tracking performance andsmaller state ripple than [21] where the past outputs otherthan current output are used for feedback

Remark 7 InTheorem 4 and Corollary 5 due to the existenceof inverse matrices 119883minus1

1and 119883

minus1

2 inequalities (20) and (40)

are not LMIs and cannot be directly solved by the LMI toolHowever notice that the matrices and their inverse matricesappear in pairs we can solve the inequalities by iterationon LMIs according to Cone Complementarity LinearizationAlgorithm [22]

4 Numerical Example with Simulation

Consider the original plant described by

119888

(119905) = 119860119888

119909119888

(119905) + 119861119888

119906119888

(119905) + 119872119888

1119908119888

(119905)

119911119888

(119905) = 119862119888

119909119888

(119905) + 119872119888

2119908119888

(119905)

(42)

where

119860119888

= [

0 1

minus6 minus5] 119861

119888

= [

0 1

1 12] 119872

119888

1= [

1

06]

119862119888

= [02 01] 119872119888

2= 001

(43)

If the fast discretization formulae follow119860 = 119890119860119888(120591119898) 119861 =

int

120591119898

0

119890119860119888120590

119889120590119861119888 1198721= int

120591119898

0

119890119860119888120590

119889120590119872119888

1 119862 = 119862

119888 and 1198722=

119872119888

2 then system (2) can be obtained The initial state is 119909

0=

[30 20]If 1198791=

1198792= 2 the input-holding mechanism can be

described as

119867(0) = [

1 0

0 0] 119867 (1) = [

0 0

0 1] (44)

If 1198791= 3 and 119879

2= 1 the output-sampling matrices are

assumed to be

119878 (0) = 1 119878 (1) = 0 119878 (2) = 0 (45)

Therefore this is a linear discrete asynchronous dual-ratesampled-data system [18] which applies a fast-input-slow-output mechanism for 119879 lt 119879

0 5 10 15 20 25 30

0

5

10

15

20

25

30

minus5

Real stateObserved state

Figure 4 Plant state 1199091(119896)

0

5

10

15

20

minus25

minus20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 5 Plant state 1199092(119896)

0001002003004005006007008

minus0010 5 10 15 20 25 30

Figure 6 Control signal 1199061(119896)

8 Mathematical Problems in Engineering

0

05

minus3

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 7 Control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 8 Output

For comparison we introduce a previous method [21]where the system is not lifted according to the pure LQGtheory and also let 119908

1= 1199082= 119908 be Gaussian white noise

amplitude is increased to 10 which is much bigger to thesystem The results obtained using this method are shown inFigures 4 5 6 7 and 8 where 120591 = 12 s

Next the synthesis method described in Theorem 4is applied to the same plant The simulations obtain thetransients of the system shown in Figures 9 10 11 12 and 13where 120574 = 02 119886 = 1 and 119887 = 1 while119898 = 1 (monorate)

A comparison of Figures 4ndash8 with Figures 9ndash13 showsclearly that the 119867

infinmethod performs better than LQG

the controlled states are consistent with the observed stateswith smaller ripples especially state 119909

2(119896) the biggest ripple

of which is almost half of the LQG method And theoutput of119867

infinmethod has better performance of disturbance

attenuation than LQG methodWhen 120591 = 06 s and 119898 = 1 (monorate) applying

Theorem 4 again we can get Figures 14 15 16 17 and 18Then the lifting number is modified to119898 = 2 (multirate) theresults shown in Figures 19 20 21 22 and 23 are obtained

Comparing Figures 14ndash18 with Figures 19ndash23 when thelifting amplitude increases from 1 to 2 the controlled statesare almost consistent with the observed states with smallerripples here the biggest ripple of state 119909

2(119896) is almost half of

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 9 Plant state 1199091(119896)

0

5

10

15

20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 10 Plant state 1199092(119896)

0

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 11 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

07

0 5 10 15 20 25 30

Figure 12 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 13 Output

Table 1 Minimum119867infinattenuation level

120591 119898 = 1 119898 = 2 119898 = 4

12 sec 00974 00959 0094206 sec 00904 00863 00856

the monorate And the output of the multirate case has betterperformance of disturbance attenuation than the monoratecase

Furthermore we change the lifting amplitude to 119898 = 4adjust the underlying period of system (2) to 120591 = 06 andcan draw the similar pictures here we omit them but only listtheir minimum119867

infinattenuation levels to compare them with

different cases see Table 1The inequality acquired using the theorem is almost

linear except in a few terms Although these terms are justthe inverses of their corresponding terms they could easily becalculated by LMI tool ofMatlab under a constraintThus it isfar easier to solve the controller and observer under the same119867infinconstraint comparing withmost of the existingmethods

which use highly complex formulae

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 14 Plant state 1199091(119896)

0

5

10

15

20

Real stateObserved state

minus5

minus30

minus25

minus20

minus15

minus10

0 5 10 15 20 25 30

Figure 15 Plant state 1199092(119896)

0

0 5 10 15 20 25 30

minus3

minus25

minus35

minus4

minus45

minus2

minus15

minus1

minus05

Figure 16 Plant control signal 1199061(119896)

10 Mathematical Problems in Engineering

0

02

04

06

08

1

12

14

0 5 10 15 20 25 30

Figure 17 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30minus1

Figure 18 Output

5 Conclusion

Motivated by a desire to provide a better solution for thedesign of a controller for a discrete-time asynchronous mul-tirate sampled-data system we developed synthesis methodsin this paper to improve performance and guarantee closed-loop transient behavior via observer-based 119867

infincontrol

Furthermore an extended lifting technique was adoptedto change the original system into a fast-input-slow-outputdiscrete system thereby improving its performance Finallywe presented an example that demonstrates the effectivenessof thesemethods in controlling a discrete-time asynchronousmultirate sampled-data system whose performance is thusimproved considerably

In this paper we focus on the standard 119867infin

control ofasynchronous multirate sampled-data system and the inputnoise discussed here is Gaussian white for simplicity As iswell known practical input signals often have finite frequency(FF) so it is reasonable to consider the 119867

infinperformance in

finite frequency range for sampled-data systems As in [27ndash31] where Kalman-Yakubovic-Popov (KYP) lemma is usedto establish the equivalence between a frequency domaininequality (FDI) and a linear matrix inequality it is a valuablework to extend the results in this paper to the FF frameworkbased on KYP lemma This will be the direction for futuredevelopment of this paper

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Real stateObserved state

minus10

minus5

Figure 19 Plant state 1199091(119896)

0

5

10

15

20

0 5 10 15 20 25 30

Real stateObserved state

minus20

minus15

minus10

minus5

Figure 20 Plant state 1199092(119896)

005

0 5 10 15 20 25 30

minus3

minus35

minus4

minus25

minus2

minus15

minus1

minus05

Figure 21 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 11

0

02

04

06

08

1

12

0 5 10 15 20 25 30minus02

Figure 22 Plant control signal 1199062(119896)

minus10 5 10 15 20 25 30

0

1

2

3

4

5

6

7

8

9

Figure 23 Output

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Zhejiang Provincial Natural Sci-ence Foundation of China under Grant no LR12F03002 andNational Natural Science Foundation of China (NSFC) underGrant no 61074045

References

[1] M F Sagfors H T Toivonen and B Lennartson ldquoState-spacesolution to the period multirate 119867

infincontrol problem a lifting

approachrdquo IEEE Transaction on Automatic Control vol 37 no5 pp 2345ndash2350 2000

[2] X Zhao Y Yao and JMa ldquoOverview ofmultirate sampled-datacontrol systemrdquo Journal of Harbin Institute of Technology vol 9no 4 pp 405ndash410 2002

[3] Y Fukada ldquoSlip-angle estimation for vehicle stability controlrdquoVehicle System Dynamics vol 32 no 4 pp 375ndash388 1999

[4] T Chen and L Qiu ldquo119867infindesign of general multirate sampled-

data control systemsrdquo Automatica vol 30 no 7 pp 1139ndash11521994

[5] Z Wang B Huang and H Unbehauen ldquoRobust 119867infin

observerdesign of linear state delayed systems with parametric uncer-tainty the discrete-time caserdquo Automatica vol 35 no 6 pp1161ndash1167 1999

[6] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Control Theory vol 15 ofSIAM Studies in Applied Mathematics SIAM Philadelphia PaUSA 1994

[7] S Patra and A Lanzon ldquoA closed-loop data based test forrobust performance improvement in iterative identification andcontrol redesignsrdquo Automatica vol 48 no 10 pp 2710ndash27162012

[8] J Ding Y Shi H Wang and F Ding ldquoA modified stochasticgradient based parameter estimation algorithm for dual-ratesampled-data systemsrdquo Digital Signal Processing vol 20 no 4pp 1238ndash1247 2010

[9] T W Chen and B Francis Optimal Sampled-Data ControlSystems Springer London UK 1995

[10] L Shen and M J Er ldquoMultiobjectives design of a multirateoutput controllerrdquo in Proceedings of the IEEE InternationalConference on Control Applications (CCA rsquo99) pp 193ndash198August 1999

[11] M Chilali and P Gahinet ldquo119867infin

design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996

[12] S Mo X Chen J Zhao J Qian and Z Shao ldquoA two-stagemethod for identification of dual-rate systems with fast inputand very slow outputrdquo Industrial and Engineering ChemistryResearch vol 48 no 4 pp 1980ndash1988 2009

[13] M De la Sen and S Alonso-Quesada ldquoModel matching viamultirate sampling with fast sampled input guaranteeing thestability of the plant zeros extensions to adaptive controlrdquo IETControl Theory amp Applications vol 1 no 1 pp 210ndash225 2007

[14] M F Sagfors and H T Toivonen ldquo119867infin

and LQG control ofasynchronous sampled-data systemsrdquo Automatica vol 33 no9 pp 1663ndash1668 1997

[15] S Lopez-lopez Optimal Hinfin

Design of Causal Multirate Con-trollers and Filters Mechanical and Aerospace EngineeringUniversity of California Irvine Calif USA 2010

[16] U Borison ldquoSelf-tuning regulators for a class of multivariablesystemsrdquo Automatica vol 15 no 2 pp 209ndash215 1979

[17] V S Ritchey and G F Franklin ldquoA stability criterion forasynchronous multirate linear systemsrdquo IEEE Transactions onAutomatic Control vol 34 no 5 pp 529ndash535 1989

[18] B Shen Z Wang and X Liu ldquoSampled-data synchronizationcontrol of dynamical networks with stochastic samplingrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2644ndash2650 2012

[19] S Longo G Herrmann and P Barber ldquoRobust scheduling ofsampled-data networked control systemsrdquo IEEE Transactionson Control Systems Technology vol 20 no 6 pp 1613ndash1621 2011

[20] P Voulgaris ldquoControl of asynchronous sampled data systemsrdquoIEEE Transactions on Automatic Control vol 39 no 7 pp 1451ndash1455 1994

[21] P Colaneri R Scattolini and N Schiavoni ldquoLQG optimalcontrol of multirate sampled-data systemsrdquo IEEE Transactionson Automatic Control vol 37 no 5 pp 675ndash682 1992

[22] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

12 Mathematical Problems in Engineering

[23] A Seuret ldquoStability analysis of networked control systems withasynchronous sampling and input delayrdquo in Proceedings of theAmerican Control Conference (ACC rsquo11) pp 533ndash538 July 2011

[24] B Tavassoli ldquoStability of nonlinear networked control systemsover multiple communication links with asynchronous sam-plingrdquo IEEE Transaction on Automatic Control 2013

[25] L Yang and J M Li ldquoSufficient and necessary conditions ofcontrollability and observability of a class of linear switchingsystemrdquo Systems Engineering and Electronics vol 25 no 5 pp588ndash590 2003

[26] W-J Mao ldquoObserver-based energy decoupling of linear time-delay systemsrdquo Journal of Zhejiang University vol 37 no 5 pp499ndash503 2003

[27] T Iwasaki and S Hara ldquoGeneralized KYP lemma unifiedfrequency domain inequalities with design applicationsrdquo IEEETransactions on Automatic Control vol 50 no 1 pp 41ndash592005

[28] H Gao and X Li ldquo119867infinfiltering for discrete-time state-delayed

systems with finite frequency specificationsrdquo IEEE Transactionson Automatic Control vol 56 no 12 pp 2935ndash2941 2011

[29] X Li H Gao and CWang ldquoGeneralized Kalman-Yakubovich-Popov lemma for 2-D FM LSS modelrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3090ndash3103 2012

[30] X Li and H Gao ldquoRobust finite frequency 119867infin

filtering foruncertain 2-D systems the FM model caserdquo Automatica vol49 no 8 pp 2446ndash2452 2013

[31] X W Li and H J Gao ldquoA heuristic approach to static output-feedback controller synthesis with restricted frequency-domainspecificationsrdquo IEEE Transaction on Automatic Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Function Spaces

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

119879119894isin Z+ Then 119906(119896) is the output of the following discrete-

time period system which is defined as the input-holdingmechanism

V (119896 + 1) = (119868 minus 119867 (119896)) V (119896) + 119867 (119896) 119903 (119896)

119906 (119896) = (119868 minus 119867 (119896)) V (119896) + 119867 (119896) 119903 (119896)

(3)

where

119867(119896) = diag ℎ1(119896) ℎ

2(119896) ℎ

119902(k)

ℎ119894(119896) =

1 119896 = 119895119879119894+ 120591119894

0 119896 = 119895119879119894+ 120591119894

119895 integer(4)

where V(119896) = [V1(119896) V

2(119896) sdot sdot sdot V

119902(119896)]

1015840 is a new statevariable and 119903(119896) = [119903

1(119896) 1199032(119896) sdot sdot sdot 119903

119902(119896)]

1015840 is a new inputvariable The integer 0 le 120591

119894lt119879119894(119894 = 1 2 119902) describes the

skew of the inputs-holding mechanismThen matrix119867(sdot) and system (2) have a period of 119879 and

119879 = lcm

119894=12119902

119879119894 (5)

Similarly the output sampling mechanism is given as

119878 (119896) = diag V1(119896) V2(119896) V

119901(119896) (6)

where

V119894(119896) =

1 119896 = 119895119879119894+ 120591119894

0 119896 = 119895119879119894+ 120591119894

119895 integer (7)

where 0 le 120591119894lt 119879119894(119894 = 1 2 119901) indicates the skew of the

outputs mechanismMatrix 119878(sdot) has a 119879-period and

119879 = lcm

119894=12119901

119879119894 (8)

After combining (2) and (3) we obtain a new sampling-holding system

120585 (119896 + 1) = Φ (119896) 120585 (119896) + Γ (119896) 119903 (119896) + Ψ119908 (119896)

119911 (119896) = Υ120585 (119896) + 1198722119908 (119896)

119910 (119896) = Δ (119896) 120585 (119896) + Ω (119896)119908 (119896)

(9)

where

120585 (119896) = [

119909 (119896)

V (119896)] Φ (119896) = [

119860 119861 (119868 minus 119867 (119896))

0 119868 minus 119867 (119896)]

Γ (119896) = [

119861

119868]119867 (119896) Ψ = [

1198721

0]

Υ = [119862 0] Δ (119896) = [119878 (119896) 119862 0]

Ω (119896) = 119878 (119896)1198722

(10)

Based on the 119879 of matrix 119867(sdot) and the 119879 of matrix 119878(sdot)

system (9) has the period of

119879 = lcm 119879 119879 (11)

The state observer of system (2) is

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119871 (119896) sdot [119910 (119896) minus 119878 (119896) 119862119909 (119896)]

(12)

where 119871(119896) asymptotically stabilizes system (12)The observer-based controller is chosen as

119903 (119896) = 119870 (119896)120585 (119896) (13)

where 120585(119896) = [119909(119896)

V(119896) ]The state error is defined as

119890 (119896) = 119909 (119896) minus 119909 (119896) (14)

According to (2) (12) and (14) the following equation canbe obtained

119890 (119896 + 1) = [119860 minus 119871 (119896) 119878 (119896) 119862] 119890 (119896)

+ (1198721minus 119871 (119896) 119878 (119896)119872

2) 119908 (119896)

(15)

Combining (9) (13) and (15) yields

[

120585 (119896 + 1)

119890 (119896 + 1)] = 119860[

120585 (119896)

119890 (119896)] + 119861119908 (119896)

119911 (119896) = 119862 [

120585 (119896)

119890 (119896)] + 119863119908 (119896)

(16)

where

119860 =[

[

[

Φ (119896) + Γ (119896)119870 (119896) minusΓ (119896)119870 (119896) [

1198682

0]

0 119860 minus 119871 (119896) 119878 (119896) 119862

]

]

]

119861 = [

Ψ

1198721minus 119871 (119896) 119878 (119896)119872

2

]

119862 = [Υ 0] 119863 = 1198722

(17)

Lemma 2 (see [21 25] ) If

(i) the pair (119860 119861) is stabilizable(ii) the pair (119860119872

1) is stabilizable

(iii) the pair (119860 119862) is detectable(iv) for any pair of distinct eigenvalues of 119860 120582

119894and 120582

119895

|120582119894| ge 1 |120582

119895| ge 1 it follows that 120582119879

119894= 120582

119879

119895

(v) there is no eigenvalue 120582 of 119860 120582 = 1 such that 120582119879 = 1

then the pairs (Φ(119896) Γ(119896)) and (Φ(119896) Ψ) are controllable andthe pair (Φ(119896) Υ) is detectable

4 Mathematical Problems in Engineering

Lemma 3 (see [10] (the Bounded Real Lemma) ) For a givenscalar 120574 gt 0 if there is a matrix 119883 = 119883

1015840

gt 0 that satisfies thematrix inequality

[

[

[

[

119883 119860119883 119861 0

1198831198601015840

119883 0 1198831198621015840

1198611015840

0 120574119868 1198631015840

0 119862119883 119863 120574119868

]

]

]

]

gt 0 (18)

then the system (16) is asynchronously stable and it achieves aspecified attenuation level 120574 gt 0 such that 119879

119911119908infin

lt 120574

3 Main Results

For the linear discrete asynchronous multirate sampled-data system (9) the exogenous parts of the system arelifted to improve its performance To avoid multiplying thedimensions of the control inputs and measured outputs wedo not lift them thereby avoiding causal constraints of thetraditional lifting method

If we assume that there is an underlying clock with a baseperiod of 120591 then system (9) represents a fast discrete system

with a subperiod of 120591119898 for the discrete process describedin [9] The sampler 119878

119886samples the continuous signal 119908 for a

discrete 119886-period and the holder 119867119887holds the output signal

119911 during period 119887 The rates of the sampler and the holdercan be set at different flexible rates that is 119886 = 119887 If weselect (13) as the new input 119870(119896) is the controller of system(9)

These concepts are described in Figure 3 where

119898 = lcm 119886 119887 (19)

Theorem 4 Consider the sampled-data system (9) with thelifting process shown in Figure 3 For a given attenuationlevel 120574 gt 0 if there exist matrices 119883

1(119896) = 119883

1015840

1(119896) gt

0 1198832(119896) = 119883

1015840

2(119896) gt 0 119870(119896) and 119871(119896) (119896 = 0 119873

denotes the iteration time) which are suboptimal solutions ofthe inequalities below then the closed system of (9) with thecontroller (13) is asynchronously stable and satisfies the 119867

infin

norm constraint 119879119911119908infin

lt 120574

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831(119896) 0 119860

119889+ 1198612119889119870 (119896) minus119861

2119889119870 (119896) [

1198682

0] 119861

10

lowast 1198832(119896) 0 119860 minus 119871 (119896) 119878 (119896) 119862 (119872

1minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 119883minus1

1(119896) 0 0 119862

1015840

1+ 1198701015840

(119896)1198631015840

12

lowast lowast lowast 119883minus1

2(119896) 0 minus [119868

20]1198701015840

(119896)1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0 (20)

Proof After lifting system (9) according to Figure 3 thetransform function of a completely discrete system with aperiod of 120591119898 which is partly contained within the dot-dashline is

119866 = [

119871119898

0

0 119868] [

119878119891

0

0 119878 (119896)

] [

11986611

11986612

11986621

11986622

] [

119867119891

0

0 119867 (119896)

] [119871minus1

1198980

0 119868

]

=[

[

11987111989811987811989111986611119867119891119871minus1

119898

11987111989811987811989111986612119867(119896)

119878 (119896) 11986621119867119891119871minus1

119898

119878 (119896) 11986622119867(119896)

]

]

(21)

Based on the above we can derive minimal state-spacerepresentations 119892(120582) for 119866 as follows based on the transferfunction theory given in [9 15] However in contrast to thetheory 119867 and 119878 are not blocks of unit matrices because

sampling and holding are not synchronous that is the entiresignal cannot be sampled or held at the same time

119892 (120582) = [

11987511

11987512

11987521

11987522

] (22)

Next we calculate the minimal state-space realization of(22) as follows(a) Transfer Function for 119875

11 Note that119875

11= 11987111989811987811989111986611119867119891119871minus1

119898

which comes directly from the theory in [8] and its corre-sponding state model is

11(120582) = [

119860119889

1198611

1198621

11986311

] (23)

where

119860119889=

[

[

119860119898

119898

sum

119895=1

119860119898minus119895

119861 (119868 minus 119867 (119896))

0 119868 minus 119867 (119896)

]

]

1198611= [

119860119898minus1

1198721119860119898minus2

1198721sdot sdot sdot 119872

1

0 0 sdot sdot sdot 0

]

Mathematical Problems in Engineering 5

1198621=

[

[

[

[

[

119862 0

119862119860 0

119862119860119898minus1

0

]

]

]

]

]

11986311=

[

[

[

[

[

1198722

0 sdot sdot sdot 0

1198621198721

1198722

sdot sdot sdot 0

d

119862119860119898minus2

1198721119862119860119898minus3

1198721sdot sdot sdot 119872

2

]

]

]

]

]

(24)

(b) Transfer Function for 11987512 As119867(119896) in 119875

12= 11987111989811987811989111986612119867(119896)

is not an 119898-blocks unit matrix the theory cannot be applieddirectly However note that

119878119891119867(119896) = 119871

minus1

119898

[

[

[

[

[

119868

119868

119868

]

]

]

]

]119898

119867(119896) (25)

Then

11987512= 11987111989811987811989111986612119867119891119871minus1

119898

[

[

[

[

[

119868

119868

119868

]

]

]

]

]119898

119867(119896) (26)

The corresponding state model can be obtained as

12(120582) = [

119860119889

1198612119889

1198621

11986312

] (27)

where 119860119889and 119862

1have been defined previously and

1198612119889

=[

[

119898

sum

119894=1

119860119898minus119894

119861

119868

]

]

119867 (119896)

11986312=

[

[

[

[

[

[

[

[

[

[

0

0 +

119898

sum

119895=119898

119862119860119898minus119895

119861

0 +

119898

sum

119895=2

119862119860119898minus119895

119861

]

]

]

]

]

]

]

]

]

]

119867(119896)

(28)

(c) Transfer Function for 11987521 Similarly because 119878(119896) is also not

an119898-blocks unit matrix 11987521can be changed to

11987521= 119878 (119896) [119868 0 sdot sdot sdot 0]

11989811987111989811987811989111986621119867119891119871minus1

119898 (29)

Thus the state model can be derived as

21(120582) = [

119860119889

1198611

1198622

11986321

] (30)

where 119860119889and 119861

1were defined previously and

1198622= Δ (119896)

11986321= [Ω (119896) 0 sdot sdot sdot 0]

(31)

(d) Transfer Function for 11987522 Like to parts 119887 and 119888 the state

description of 11987522is

22(120582) = [

119860119889

1198612119889

1198622

11986322

] (32)

where 119860119889 1198612119889 and 119862

2were defined previously while

11986322= 0 (33)

Finally formula (23) is represented using the state-spaceform as follows

120585 (119896 + 1) = 119860119889120585 (119896) + 119861

1119908 (119896) + 119861

2119889119903 (119896)

119911 (119896) = 1198621120585 (119896) + 119863

11119908 (119896) + 119863

12119903 (119896)

119910 (119896) = 1198622120585 (119896) + 119863

21119908 (119896)

(34)

where the underlining below 119911 and 119908 indicates that thesignals are lifted

According to Lemma 2 the controller119870(119896) exists and canstabilize the system

Let weierp(119896) = [1205851015840

(119896) 1198901015840

(119896)]

1015840 and after combining (13) (15)and (34) we obtain

weierp (119896 + 1) = 119860119888weierp (119896) + 119861

119888119908 (119896)

119911 (119896) = 119862119888weierp (119896) + 119863

119888119908 (119896)

(35)

where

119860119888=

[

[

[

119860119889+ 1198612119889119870 (119896) minus119861

2119889119870 (119896) [

1198682

0]

0 119860 minus 119871 (119896) 119878 (119896) 119862

]

]

]

119861119888= [

1198611

(1198721minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

119898

]

119862119888= [119862

1+ 11986312119870 (119896) minus119863

12119870 (119896) [

1198682

0]]

119863119888= 11986311

(36)

In the light of [5] we set

119883 (119896) = [

1198831(119896) 0

0 1198832(119896)

] (37)

6 Mathematical Problems in Engineering

By substituting119883 in Lemma 3 with119883(119896) we can obtain

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831(119896) 0 (119860

119889+ 1198612119889119870 (119896))119883

1(119896) minus119861

2119889119870 (119896) [

1198682

0

]1198832(119896) 119861

10

lowast 1198832(119896) 0 (119860 minus 119871 (119896) 119878 (119896) 119862)119883

2(119896) (119872

1minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 1198831(119896) 0 0 119883

1(119896) (119862

1015840

1+ 1198701015840

(119896)1198631015840

12)

lowast lowast lowast 1198832(119896) 0 minus119883

2(119896) [1198682

0]1198701015840

(119896)1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0

(38)

Then according to [26] after multiplying the left- andright-hand sides of inequality (38) by diag119868 119868 119883minus1

1(119896)

119883minus1

2(119896) 119868 119868 we can obtain inequality (20)According to quadratic stability theory the system should

be stable at each step 119896 Thus for a discrete linear time-invariant system the eigenvalues of the closed system shouldlie within a unit circle

From Theorem 4 if matrix inequalities (20) can besolved there exists a state feedback 119867

infincontroller that can

asymptotically stabilize closed system (35) of enlarged system(34) which satisfies the performance index Since system(34) is lifted from the original system (9) and we know thatthe lifting operator is an isometric isomorphism that isthe feedback stability and system norm can be preservedthen system (16) which is the closed system of (9) isasynchronously stable and it satisfies the119867

infinnorm constraint

119879119911119908infin

lt 120574

The synchronous case can be regarded as a specialexample of the asynchronous case In this case the elementsof sampling and holder are unit matrices with the assumption

that the whole signals of sampling and holding can beobtained at the beginning of every period then the system (9)turns into a linear discrete synchronous multirate sampled-data system with period 120591 as follows

120585 (119896 + 1) = Φ120585 (119896) + Γ119903 (119896) + Ψ119908 (119896)

119911 (119896) = Υ120585 (119896) + 1198722119908 (119896)

119910 (119896) = Δ120585 (119896) + Ω119908 (119896)

(39)

As119867 and 119878 are unit matrices that is constant now Φ ΓΔ andΩ related to119867 and 119878 are also constant Theorem 4 canbe applied to system (39) straightforwardly

Corollary 5 Consider the sampled-data system (39) with thelifting process shown in Figure 3 For a given attenuation level120574 gt 0 if there exist matrices 119883

1= 1198831015840

1gt 0 119883

2= 1198831015840

2gt

0 119870 and 119871 which are suboptimal solutions of the inequalitiesbelow then the closed system of (39) with the controller (13)is synchronously uniformly stable and it satisfies the119867

infinnorm

constraint 119879119911119908infin

lt 120574

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831

0 119860119889+ 1198612119889119870 minus119861

2119889119870[

1198682

0] 119861

10

lowast 1198832

0 119860 minus 119871119862 (1198721minus 119871119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 119883minus1

10 0 119862

1015840

1+ 1198701015840

1198631015840

12

lowast lowast lowast 119883minus1

20 minus [119868

20]1198701015840

1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0 (40)

Remark 6 Let us compare the designed observer (12) withthat in [21] where the observer (predictor) is

(119896 + 1 | 119896) = 119860 (119896 | 119896 minus 1) + 119861119906 (119896)

+ 119871 (119896) sdot [120577 (119896) minus 119862 (119896) (119896 | 119896 minus 1)]

120577 (119896) =

119879

lowast

sum

119894=1

119910 (119896 minus 119894) 119879

lowast

= max119894=1119901

119879119894

(41)The sum of outputs is used to avoid the singularity butmay cause more noise The observer applied in this paper is

Mathematical Problems in Engineering 7

w z

u yG

K

Tm1 Tmp

middot middot middot middot middot middotTn1Tnq H1Hq SpS1

Sa Lminus1mLb HbHf Sf Lminus1aLm

Figure 3 System lifting and controller designing

simpler that is just the current output is introduced in theobserver This will result in better tracking performance andsmaller state ripple than [21] where the past outputs otherthan current output are used for feedback

Remark 7 InTheorem 4 and Corollary 5 due to the existenceof inverse matrices 119883minus1

1and 119883

minus1

2 inequalities (20) and (40)

are not LMIs and cannot be directly solved by the LMI toolHowever notice that the matrices and their inverse matricesappear in pairs we can solve the inequalities by iterationon LMIs according to Cone Complementarity LinearizationAlgorithm [22]

4 Numerical Example with Simulation

Consider the original plant described by

119888

(119905) = 119860119888

119909119888

(119905) + 119861119888

119906119888

(119905) + 119872119888

1119908119888

(119905)

119911119888

(119905) = 119862119888

119909119888

(119905) + 119872119888

2119908119888

(119905)

(42)

where

119860119888

= [

0 1

minus6 minus5] 119861

119888

= [

0 1

1 12] 119872

119888

1= [

1

06]

119862119888

= [02 01] 119872119888

2= 001

(43)

If the fast discretization formulae follow119860 = 119890119860119888(120591119898) 119861 =

int

120591119898

0

119890119860119888120590

119889120590119861119888 1198721= int

120591119898

0

119890119860119888120590

119889120590119872119888

1 119862 = 119862

119888 and 1198722=

119872119888

2 then system (2) can be obtained The initial state is 119909

0=

[30 20]If 1198791=

1198792= 2 the input-holding mechanism can be

described as

119867(0) = [

1 0

0 0] 119867 (1) = [

0 0

0 1] (44)

If 1198791= 3 and 119879

2= 1 the output-sampling matrices are

assumed to be

119878 (0) = 1 119878 (1) = 0 119878 (2) = 0 (45)

Therefore this is a linear discrete asynchronous dual-ratesampled-data system [18] which applies a fast-input-slow-output mechanism for 119879 lt 119879

0 5 10 15 20 25 30

0

5

10

15

20

25

30

minus5

Real stateObserved state

Figure 4 Plant state 1199091(119896)

0

5

10

15

20

minus25

minus20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 5 Plant state 1199092(119896)

0001002003004005006007008

minus0010 5 10 15 20 25 30

Figure 6 Control signal 1199061(119896)

8 Mathematical Problems in Engineering

0

05

minus3

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 7 Control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 8 Output

For comparison we introduce a previous method [21]where the system is not lifted according to the pure LQGtheory and also let 119908

1= 1199082= 119908 be Gaussian white noise

amplitude is increased to 10 which is much bigger to thesystem The results obtained using this method are shown inFigures 4 5 6 7 and 8 where 120591 = 12 s

Next the synthesis method described in Theorem 4is applied to the same plant The simulations obtain thetransients of the system shown in Figures 9 10 11 12 and 13where 120574 = 02 119886 = 1 and 119887 = 1 while119898 = 1 (monorate)

A comparison of Figures 4ndash8 with Figures 9ndash13 showsclearly that the 119867

infinmethod performs better than LQG

the controlled states are consistent with the observed stateswith smaller ripples especially state 119909

2(119896) the biggest ripple

of which is almost half of the LQG method And theoutput of119867

infinmethod has better performance of disturbance

attenuation than LQG methodWhen 120591 = 06 s and 119898 = 1 (monorate) applying

Theorem 4 again we can get Figures 14 15 16 17 and 18Then the lifting number is modified to119898 = 2 (multirate) theresults shown in Figures 19 20 21 22 and 23 are obtained

Comparing Figures 14ndash18 with Figures 19ndash23 when thelifting amplitude increases from 1 to 2 the controlled statesare almost consistent with the observed states with smallerripples here the biggest ripple of state 119909

2(119896) is almost half of

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 9 Plant state 1199091(119896)

0

5

10

15

20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 10 Plant state 1199092(119896)

0

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 11 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

07

0 5 10 15 20 25 30

Figure 12 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 13 Output

Table 1 Minimum119867infinattenuation level

120591 119898 = 1 119898 = 2 119898 = 4

12 sec 00974 00959 0094206 sec 00904 00863 00856

the monorate And the output of the multirate case has betterperformance of disturbance attenuation than the monoratecase

Furthermore we change the lifting amplitude to 119898 = 4adjust the underlying period of system (2) to 120591 = 06 andcan draw the similar pictures here we omit them but only listtheir minimum119867

infinattenuation levels to compare them with

different cases see Table 1The inequality acquired using the theorem is almost

linear except in a few terms Although these terms are justthe inverses of their corresponding terms they could easily becalculated by LMI tool ofMatlab under a constraintThus it isfar easier to solve the controller and observer under the same119867infinconstraint comparing withmost of the existingmethods

which use highly complex formulae

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 14 Plant state 1199091(119896)

0

5

10

15

20

Real stateObserved state

minus5

minus30

minus25

minus20

minus15

minus10

0 5 10 15 20 25 30

Figure 15 Plant state 1199092(119896)

0

0 5 10 15 20 25 30

minus3

minus25

minus35

minus4

minus45

minus2

minus15

minus1

minus05

Figure 16 Plant control signal 1199061(119896)

10 Mathematical Problems in Engineering

0

02

04

06

08

1

12

14

0 5 10 15 20 25 30

Figure 17 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30minus1

Figure 18 Output

5 Conclusion

Motivated by a desire to provide a better solution for thedesign of a controller for a discrete-time asynchronous mul-tirate sampled-data system we developed synthesis methodsin this paper to improve performance and guarantee closed-loop transient behavior via observer-based 119867

infincontrol

Furthermore an extended lifting technique was adoptedto change the original system into a fast-input-slow-outputdiscrete system thereby improving its performance Finallywe presented an example that demonstrates the effectivenessof thesemethods in controlling a discrete-time asynchronousmultirate sampled-data system whose performance is thusimproved considerably

In this paper we focus on the standard 119867infin

control ofasynchronous multirate sampled-data system and the inputnoise discussed here is Gaussian white for simplicity As iswell known practical input signals often have finite frequency(FF) so it is reasonable to consider the 119867

infinperformance in

finite frequency range for sampled-data systems As in [27ndash31] where Kalman-Yakubovic-Popov (KYP) lemma is usedto establish the equivalence between a frequency domaininequality (FDI) and a linear matrix inequality it is a valuablework to extend the results in this paper to the FF frameworkbased on KYP lemma This will be the direction for futuredevelopment of this paper

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Real stateObserved state

minus10

minus5

Figure 19 Plant state 1199091(119896)

0

5

10

15

20

0 5 10 15 20 25 30

Real stateObserved state

minus20

minus15

minus10

minus5

Figure 20 Plant state 1199092(119896)

005

0 5 10 15 20 25 30

minus3

minus35

minus4

minus25

minus2

minus15

minus1

minus05

Figure 21 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 11

0

02

04

06

08

1

12

0 5 10 15 20 25 30minus02

Figure 22 Plant control signal 1199062(119896)

minus10 5 10 15 20 25 30

0

1

2

3

4

5

6

7

8

9

Figure 23 Output

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Zhejiang Provincial Natural Sci-ence Foundation of China under Grant no LR12F03002 andNational Natural Science Foundation of China (NSFC) underGrant no 61074045

References

[1] M F Sagfors H T Toivonen and B Lennartson ldquoState-spacesolution to the period multirate 119867

infincontrol problem a lifting

approachrdquo IEEE Transaction on Automatic Control vol 37 no5 pp 2345ndash2350 2000

[2] X Zhao Y Yao and JMa ldquoOverview ofmultirate sampled-datacontrol systemrdquo Journal of Harbin Institute of Technology vol 9no 4 pp 405ndash410 2002

[3] Y Fukada ldquoSlip-angle estimation for vehicle stability controlrdquoVehicle System Dynamics vol 32 no 4 pp 375ndash388 1999

[4] T Chen and L Qiu ldquo119867infindesign of general multirate sampled-

data control systemsrdquo Automatica vol 30 no 7 pp 1139ndash11521994

[5] Z Wang B Huang and H Unbehauen ldquoRobust 119867infin

observerdesign of linear state delayed systems with parametric uncer-tainty the discrete-time caserdquo Automatica vol 35 no 6 pp1161ndash1167 1999

[6] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Control Theory vol 15 ofSIAM Studies in Applied Mathematics SIAM Philadelphia PaUSA 1994

[7] S Patra and A Lanzon ldquoA closed-loop data based test forrobust performance improvement in iterative identification andcontrol redesignsrdquo Automatica vol 48 no 10 pp 2710ndash27162012

[8] J Ding Y Shi H Wang and F Ding ldquoA modified stochasticgradient based parameter estimation algorithm for dual-ratesampled-data systemsrdquo Digital Signal Processing vol 20 no 4pp 1238ndash1247 2010

[9] T W Chen and B Francis Optimal Sampled-Data ControlSystems Springer London UK 1995

[10] L Shen and M J Er ldquoMultiobjectives design of a multirateoutput controllerrdquo in Proceedings of the IEEE InternationalConference on Control Applications (CCA rsquo99) pp 193ndash198August 1999

[11] M Chilali and P Gahinet ldquo119867infin

design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996

[12] S Mo X Chen J Zhao J Qian and Z Shao ldquoA two-stagemethod for identification of dual-rate systems with fast inputand very slow outputrdquo Industrial and Engineering ChemistryResearch vol 48 no 4 pp 1980ndash1988 2009

[13] M De la Sen and S Alonso-Quesada ldquoModel matching viamultirate sampling with fast sampled input guaranteeing thestability of the plant zeros extensions to adaptive controlrdquo IETControl Theory amp Applications vol 1 no 1 pp 210ndash225 2007

[14] M F Sagfors and H T Toivonen ldquo119867infin

and LQG control ofasynchronous sampled-data systemsrdquo Automatica vol 33 no9 pp 1663ndash1668 1997

[15] S Lopez-lopez Optimal Hinfin

Design of Causal Multirate Con-trollers and Filters Mechanical and Aerospace EngineeringUniversity of California Irvine Calif USA 2010

[16] U Borison ldquoSelf-tuning regulators for a class of multivariablesystemsrdquo Automatica vol 15 no 2 pp 209ndash215 1979

[17] V S Ritchey and G F Franklin ldquoA stability criterion forasynchronous multirate linear systemsrdquo IEEE Transactions onAutomatic Control vol 34 no 5 pp 529ndash535 1989

[18] B Shen Z Wang and X Liu ldquoSampled-data synchronizationcontrol of dynamical networks with stochastic samplingrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2644ndash2650 2012

[19] S Longo G Herrmann and P Barber ldquoRobust scheduling ofsampled-data networked control systemsrdquo IEEE Transactionson Control Systems Technology vol 20 no 6 pp 1613ndash1621 2011

[20] P Voulgaris ldquoControl of asynchronous sampled data systemsrdquoIEEE Transactions on Automatic Control vol 39 no 7 pp 1451ndash1455 1994

[21] P Colaneri R Scattolini and N Schiavoni ldquoLQG optimalcontrol of multirate sampled-data systemsrdquo IEEE Transactionson Automatic Control vol 37 no 5 pp 675ndash682 1992

[22] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

12 Mathematical Problems in Engineering

[23] A Seuret ldquoStability analysis of networked control systems withasynchronous sampling and input delayrdquo in Proceedings of theAmerican Control Conference (ACC rsquo11) pp 533ndash538 July 2011

[24] B Tavassoli ldquoStability of nonlinear networked control systemsover multiple communication links with asynchronous sam-plingrdquo IEEE Transaction on Automatic Control 2013

[25] L Yang and J M Li ldquoSufficient and necessary conditions ofcontrollability and observability of a class of linear switchingsystemrdquo Systems Engineering and Electronics vol 25 no 5 pp588ndash590 2003

[26] W-J Mao ldquoObserver-based energy decoupling of linear time-delay systemsrdquo Journal of Zhejiang University vol 37 no 5 pp499ndash503 2003

[27] T Iwasaki and S Hara ldquoGeneralized KYP lemma unifiedfrequency domain inequalities with design applicationsrdquo IEEETransactions on Automatic Control vol 50 no 1 pp 41ndash592005

[28] H Gao and X Li ldquo119867infinfiltering for discrete-time state-delayed

systems with finite frequency specificationsrdquo IEEE Transactionson Automatic Control vol 56 no 12 pp 2935ndash2941 2011

[29] X Li H Gao and CWang ldquoGeneralized Kalman-Yakubovich-Popov lemma for 2-D FM LSS modelrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3090ndash3103 2012

[30] X Li and H Gao ldquoRobust finite frequency 119867infin

filtering foruncertain 2-D systems the FM model caserdquo Automatica vol49 no 8 pp 2446ndash2452 2013

[31] X W Li and H J Gao ldquoA heuristic approach to static output-feedback controller synthesis with restricted frequency-domainspecificationsrdquo IEEE Transaction on Automatic Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Lemma 3 (see [10] (the Bounded Real Lemma) ) For a givenscalar 120574 gt 0 if there is a matrix 119883 = 119883

1015840

gt 0 that satisfies thematrix inequality

[

[

[

[

119883 119860119883 119861 0

1198831198601015840

119883 0 1198831198621015840

1198611015840

0 120574119868 1198631015840

0 119862119883 119863 120574119868

]

]

]

]

gt 0 (18)

then the system (16) is asynchronously stable and it achieves aspecified attenuation level 120574 gt 0 such that 119879

119911119908infin

lt 120574

3 Main Results

For the linear discrete asynchronous multirate sampled-data system (9) the exogenous parts of the system arelifted to improve its performance To avoid multiplying thedimensions of the control inputs and measured outputs wedo not lift them thereby avoiding causal constraints of thetraditional lifting method

If we assume that there is an underlying clock with a baseperiod of 120591 then system (9) represents a fast discrete system

with a subperiod of 120591119898 for the discrete process describedin [9] The sampler 119878

119886samples the continuous signal 119908 for a

discrete 119886-period and the holder 119867119887holds the output signal

119911 during period 119887 The rates of the sampler and the holdercan be set at different flexible rates that is 119886 = 119887 If weselect (13) as the new input 119870(119896) is the controller of system(9)

These concepts are described in Figure 3 where

119898 = lcm 119886 119887 (19)

Theorem 4 Consider the sampled-data system (9) with thelifting process shown in Figure 3 For a given attenuationlevel 120574 gt 0 if there exist matrices 119883

1(119896) = 119883

1015840

1(119896) gt

0 1198832(119896) = 119883

1015840

2(119896) gt 0 119870(119896) and 119871(119896) (119896 = 0 119873

denotes the iteration time) which are suboptimal solutions ofthe inequalities below then the closed system of (9) with thecontroller (13) is asynchronously stable and satisfies the 119867

infin

norm constraint 119879119911119908infin

lt 120574

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831(119896) 0 119860

119889+ 1198612119889119870 (119896) minus119861

2119889119870 (119896) [

1198682

0] 119861

10

lowast 1198832(119896) 0 119860 minus 119871 (119896) 119878 (119896) 119862 (119872

1minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 119883minus1

1(119896) 0 0 119862

1015840

1+ 1198701015840

(119896)1198631015840

12

lowast lowast lowast 119883minus1

2(119896) 0 minus [119868

20]1198701015840

(119896)1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0 (20)

Proof After lifting system (9) according to Figure 3 thetransform function of a completely discrete system with aperiod of 120591119898 which is partly contained within the dot-dashline is

119866 = [

119871119898

0

0 119868] [

119878119891

0

0 119878 (119896)

] [

11986611

11986612

11986621

11986622

] [

119867119891

0

0 119867 (119896)

] [119871minus1

1198980

0 119868

]

=[

[

11987111989811987811989111986611119867119891119871minus1

119898

11987111989811987811989111986612119867(119896)

119878 (119896) 11986621119867119891119871minus1

119898

119878 (119896) 11986622119867(119896)

]

]

(21)

Based on the above we can derive minimal state-spacerepresentations 119892(120582) for 119866 as follows based on the transferfunction theory given in [9 15] However in contrast to thetheory 119867 and 119878 are not blocks of unit matrices because

sampling and holding are not synchronous that is the entiresignal cannot be sampled or held at the same time

119892 (120582) = [

11987511

11987512

11987521

11987522

] (22)

Next we calculate the minimal state-space realization of(22) as follows(a) Transfer Function for 119875

11 Note that119875

11= 11987111989811987811989111986611119867119891119871minus1

119898

which comes directly from the theory in [8] and its corre-sponding state model is

11(120582) = [

119860119889

1198611

1198621

11986311

] (23)

where

119860119889=

[

[

119860119898

119898

sum

119895=1

119860119898minus119895

119861 (119868 minus 119867 (119896))

0 119868 minus 119867 (119896)

]

]

1198611= [

119860119898minus1

1198721119860119898minus2

1198721sdot sdot sdot 119872

1

0 0 sdot sdot sdot 0

]

Mathematical Problems in Engineering 5

1198621=

[

[

[

[

[

119862 0

119862119860 0

119862119860119898minus1

0

]

]

]

]

]

11986311=

[

[

[

[

[

1198722

0 sdot sdot sdot 0

1198621198721

1198722

sdot sdot sdot 0

d

119862119860119898minus2

1198721119862119860119898minus3

1198721sdot sdot sdot 119872

2

]

]

]

]

]

(24)

(b) Transfer Function for 11987512 As119867(119896) in 119875

12= 11987111989811987811989111986612119867(119896)

is not an 119898-blocks unit matrix the theory cannot be applieddirectly However note that

119878119891119867(119896) = 119871

minus1

119898

[

[

[

[

[

119868

119868

119868

]

]

]

]

]119898

119867(119896) (25)

Then

11987512= 11987111989811987811989111986612119867119891119871minus1

119898

[

[

[

[

[

119868

119868

119868

]

]

]

]

]119898

119867(119896) (26)

The corresponding state model can be obtained as

12(120582) = [

119860119889

1198612119889

1198621

11986312

] (27)

where 119860119889and 119862

1have been defined previously and

1198612119889

=[

[

119898

sum

119894=1

119860119898minus119894

119861

119868

]

]

119867 (119896)

11986312=

[

[

[

[

[

[

[

[

[

[

0

0 +

119898

sum

119895=119898

119862119860119898minus119895

119861

0 +

119898

sum

119895=2

119862119860119898minus119895

119861

]

]

]

]

]

]

]

]

]

]

119867(119896)

(28)

(c) Transfer Function for 11987521 Similarly because 119878(119896) is also not

an119898-blocks unit matrix 11987521can be changed to

11987521= 119878 (119896) [119868 0 sdot sdot sdot 0]

11989811987111989811987811989111986621119867119891119871minus1

119898 (29)

Thus the state model can be derived as

21(120582) = [

119860119889

1198611

1198622

11986321

] (30)

where 119860119889and 119861

1were defined previously and

1198622= Δ (119896)

11986321= [Ω (119896) 0 sdot sdot sdot 0]

(31)

(d) Transfer Function for 11987522 Like to parts 119887 and 119888 the state

description of 11987522is

22(120582) = [

119860119889

1198612119889

1198622

11986322

] (32)

where 119860119889 1198612119889 and 119862

2were defined previously while

11986322= 0 (33)

Finally formula (23) is represented using the state-spaceform as follows

120585 (119896 + 1) = 119860119889120585 (119896) + 119861

1119908 (119896) + 119861

2119889119903 (119896)

119911 (119896) = 1198621120585 (119896) + 119863

11119908 (119896) + 119863

12119903 (119896)

119910 (119896) = 1198622120585 (119896) + 119863

21119908 (119896)

(34)

where the underlining below 119911 and 119908 indicates that thesignals are lifted

According to Lemma 2 the controller119870(119896) exists and canstabilize the system

Let weierp(119896) = [1205851015840

(119896) 1198901015840

(119896)]

1015840 and after combining (13) (15)and (34) we obtain

weierp (119896 + 1) = 119860119888weierp (119896) + 119861

119888119908 (119896)

119911 (119896) = 119862119888weierp (119896) + 119863

119888119908 (119896)

(35)

where

119860119888=

[

[

[

119860119889+ 1198612119889119870 (119896) minus119861

2119889119870 (119896) [

1198682

0]

0 119860 minus 119871 (119896) 119878 (119896) 119862

]

]

]

119861119888= [

1198611

(1198721minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

119898

]

119862119888= [119862

1+ 11986312119870 (119896) minus119863

12119870 (119896) [

1198682

0]]

119863119888= 11986311

(36)

In the light of [5] we set

119883 (119896) = [

1198831(119896) 0

0 1198832(119896)

] (37)

6 Mathematical Problems in Engineering

By substituting119883 in Lemma 3 with119883(119896) we can obtain

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831(119896) 0 (119860

119889+ 1198612119889119870 (119896))119883

1(119896) minus119861

2119889119870 (119896) [

1198682

0

]1198832(119896) 119861

10

lowast 1198832(119896) 0 (119860 minus 119871 (119896) 119878 (119896) 119862)119883

2(119896) (119872

1minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 1198831(119896) 0 0 119883

1(119896) (119862

1015840

1+ 1198701015840

(119896)1198631015840

12)

lowast lowast lowast 1198832(119896) 0 minus119883

2(119896) [1198682

0]1198701015840

(119896)1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0

(38)

Then according to [26] after multiplying the left- andright-hand sides of inequality (38) by diag119868 119868 119883minus1

1(119896)

119883minus1

2(119896) 119868 119868 we can obtain inequality (20)According to quadratic stability theory the system should

be stable at each step 119896 Thus for a discrete linear time-invariant system the eigenvalues of the closed system shouldlie within a unit circle

From Theorem 4 if matrix inequalities (20) can besolved there exists a state feedback 119867

infincontroller that can

asymptotically stabilize closed system (35) of enlarged system(34) which satisfies the performance index Since system(34) is lifted from the original system (9) and we know thatthe lifting operator is an isometric isomorphism that isthe feedback stability and system norm can be preservedthen system (16) which is the closed system of (9) isasynchronously stable and it satisfies the119867

infinnorm constraint

119879119911119908infin

lt 120574

The synchronous case can be regarded as a specialexample of the asynchronous case In this case the elementsof sampling and holder are unit matrices with the assumption

that the whole signals of sampling and holding can beobtained at the beginning of every period then the system (9)turns into a linear discrete synchronous multirate sampled-data system with period 120591 as follows

120585 (119896 + 1) = Φ120585 (119896) + Γ119903 (119896) + Ψ119908 (119896)

119911 (119896) = Υ120585 (119896) + 1198722119908 (119896)

119910 (119896) = Δ120585 (119896) + Ω119908 (119896)

(39)

As119867 and 119878 are unit matrices that is constant now Φ ΓΔ andΩ related to119867 and 119878 are also constant Theorem 4 canbe applied to system (39) straightforwardly

Corollary 5 Consider the sampled-data system (39) with thelifting process shown in Figure 3 For a given attenuation level120574 gt 0 if there exist matrices 119883

1= 1198831015840

1gt 0 119883

2= 1198831015840

2gt

0 119870 and 119871 which are suboptimal solutions of the inequalitiesbelow then the closed system of (39) with the controller (13)is synchronously uniformly stable and it satisfies the119867

infinnorm

constraint 119879119911119908infin

lt 120574

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831

0 119860119889+ 1198612119889119870 minus119861

2119889119870[

1198682

0] 119861

10

lowast 1198832

0 119860 minus 119871119862 (1198721minus 119871119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 119883minus1

10 0 119862

1015840

1+ 1198701015840

1198631015840

12

lowast lowast lowast 119883minus1

20 minus [119868

20]1198701015840

1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0 (40)

Remark 6 Let us compare the designed observer (12) withthat in [21] where the observer (predictor) is

(119896 + 1 | 119896) = 119860 (119896 | 119896 minus 1) + 119861119906 (119896)

+ 119871 (119896) sdot [120577 (119896) minus 119862 (119896) (119896 | 119896 minus 1)]

120577 (119896) =

119879

lowast

sum

119894=1

119910 (119896 minus 119894) 119879

lowast

= max119894=1119901

119879119894

(41)The sum of outputs is used to avoid the singularity butmay cause more noise The observer applied in this paper is

Mathematical Problems in Engineering 7

w z

u yG

K

Tm1 Tmp

middot middot middot middot middot middotTn1Tnq H1Hq SpS1

Sa Lminus1mLb HbHf Sf Lminus1aLm

Figure 3 System lifting and controller designing

simpler that is just the current output is introduced in theobserver This will result in better tracking performance andsmaller state ripple than [21] where the past outputs otherthan current output are used for feedback

Remark 7 InTheorem 4 and Corollary 5 due to the existenceof inverse matrices 119883minus1

1and 119883

minus1

2 inequalities (20) and (40)

are not LMIs and cannot be directly solved by the LMI toolHowever notice that the matrices and their inverse matricesappear in pairs we can solve the inequalities by iterationon LMIs according to Cone Complementarity LinearizationAlgorithm [22]

4 Numerical Example with Simulation

Consider the original plant described by

119888

(119905) = 119860119888

119909119888

(119905) + 119861119888

119906119888

(119905) + 119872119888

1119908119888

(119905)

119911119888

(119905) = 119862119888

119909119888

(119905) + 119872119888

2119908119888

(119905)

(42)

where

119860119888

= [

0 1

minus6 minus5] 119861

119888

= [

0 1

1 12] 119872

119888

1= [

1

06]

119862119888

= [02 01] 119872119888

2= 001

(43)

If the fast discretization formulae follow119860 = 119890119860119888(120591119898) 119861 =

int

120591119898

0

119890119860119888120590

119889120590119861119888 1198721= int

120591119898

0

119890119860119888120590

119889120590119872119888

1 119862 = 119862

119888 and 1198722=

119872119888

2 then system (2) can be obtained The initial state is 119909

0=

[30 20]If 1198791=

1198792= 2 the input-holding mechanism can be

described as

119867(0) = [

1 0

0 0] 119867 (1) = [

0 0

0 1] (44)

If 1198791= 3 and 119879

2= 1 the output-sampling matrices are

assumed to be

119878 (0) = 1 119878 (1) = 0 119878 (2) = 0 (45)

Therefore this is a linear discrete asynchronous dual-ratesampled-data system [18] which applies a fast-input-slow-output mechanism for 119879 lt 119879

0 5 10 15 20 25 30

0

5

10

15

20

25

30

minus5

Real stateObserved state

Figure 4 Plant state 1199091(119896)

0

5

10

15

20

minus25

minus20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 5 Plant state 1199092(119896)

0001002003004005006007008

minus0010 5 10 15 20 25 30

Figure 6 Control signal 1199061(119896)

8 Mathematical Problems in Engineering

0

05

minus3

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 7 Control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 8 Output

For comparison we introduce a previous method [21]where the system is not lifted according to the pure LQGtheory and also let 119908

1= 1199082= 119908 be Gaussian white noise

amplitude is increased to 10 which is much bigger to thesystem The results obtained using this method are shown inFigures 4 5 6 7 and 8 where 120591 = 12 s

Next the synthesis method described in Theorem 4is applied to the same plant The simulations obtain thetransients of the system shown in Figures 9 10 11 12 and 13where 120574 = 02 119886 = 1 and 119887 = 1 while119898 = 1 (monorate)

A comparison of Figures 4ndash8 with Figures 9ndash13 showsclearly that the 119867

infinmethod performs better than LQG

the controlled states are consistent with the observed stateswith smaller ripples especially state 119909

2(119896) the biggest ripple

of which is almost half of the LQG method And theoutput of119867

infinmethod has better performance of disturbance

attenuation than LQG methodWhen 120591 = 06 s and 119898 = 1 (monorate) applying

Theorem 4 again we can get Figures 14 15 16 17 and 18Then the lifting number is modified to119898 = 2 (multirate) theresults shown in Figures 19 20 21 22 and 23 are obtained

Comparing Figures 14ndash18 with Figures 19ndash23 when thelifting amplitude increases from 1 to 2 the controlled statesare almost consistent with the observed states with smallerripples here the biggest ripple of state 119909

2(119896) is almost half of

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 9 Plant state 1199091(119896)

0

5

10

15

20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 10 Plant state 1199092(119896)

0

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 11 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

07

0 5 10 15 20 25 30

Figure 12 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 13 Output

Table 1 Minimum119867infinattenuation level

120591 119898 = 1 119898 = 2 119898 = 4

12 sec 00974 00959 0094206 sec 00904 00863 00856

the monorate And the output of the multirate case has betterperformance of disturbance attenuation than the monoratecase

Furthermore we change the lifting amplitude to 119898 = 4adjust the underlying period of system (2) to 120591 = 06 andcan draw the similar pictures here we omit them but only listtheir minimum119867

infinattenuation levels to compare them with

different cases see Table 1The inequality acquired using the theorem is almost

linear except in a few terms Although these terms are justthe inverses of their corresponding terms they could easily becalculated by LMI tool ofMatlab under a constraintThus it isfar easier to solve the controller and observer under the same119867infinconstraint comparing withmost of the existingmethods

which use highly complex formulae

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 14 Plant state 1199091(119896)

0

5

10

15

20

Real stateObserved state

minus5

minus30

minus25

minus20

minus15

minus10

0 5 10 15 20 25 30

Figure 15 Plant state 1199092(119896)

0

0 5 10 15 20 25 30

minus3

minus25

minus35

minus4

minus45

minus2

minus15

minus1

minus05

Figure 16 Plant control signal 1199061(119896)

10 Mathematical Problems in Engineering

0

02

04

06

08

1

12

14

0 5 10 15 20 25 30

Figure 17 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30minus1

Figure 18 Output

5 Conclusion

Motivated by a desire to provide a better solution for thedesign of a controller for a discrete-time asynchronous mul-tirate sampled-data system we developed synthesis methodsin this paper to improve performance and guarantee closed-loop transient behavior via observer-based 119867

infincontrol

Furthermore an extended lifting technique was adoptedto change the original system into a fast-input-slow-outputdiscrete system thereby improving its performance Finallywe presented an example that demonstrates the effectivenessof thesemethods in controlling a discrete-time asynchronousmultirate sampled-data system whose performance is thusimproved considerably

In this paper we focus on the standard 119867infin

control ofasynchronous multirate sampled-data system and the inputnoise discussed here is Gaussian white for simplicity As iswell known practical input signals often have finite frequency(FF) so it is reasonable to consider the 119867

infinperformance in

finite frequency range for sampled-data systems As in [27ndash31] where Kalman-Yakubovic-Popov (KYP) lemma is usedto establish the equivalence between a frequency domaininequality (FDI) and a linear matrix inequality it is a valuablework to extend the results in this paper to the FF frameworkbased on KYP lemma This will be the direction for futuredevelopment of this paper

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Real stateObserved state

minus10

minus5

Figure 19 Plant state 1199091(119896)

0

5

10

15

20

0 5 10 15 20 25 30

Real stateObserved state

minus20

minus15

minus10

minus5

Figure 20 Plant state 1199092(119896)

005

0 5 10 15 20 25 30

minus3

minus35

minus4

minus25

minus2

minus15

minus1

minus05

Figure 21 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 11

0

02

04

06

08

1

12

0 5 10 15 20 25 30minus02

Figure 22 Plant control signal 1199062(119896)

minus10 5 10 15 20 25 30

0

1

2

3

4

5

6

7

8

9

Figure 23 Output

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Zhejiang Provincial Natural Sci-ence Foundation of China under Grant no LR12F03002 andNational Natural Science Foundation of China (NSFC) underGrant no 61074045

References

[1] M F Sagfors H T Toivonen and B Lennartson ldquoState-spacesolution to the period multirate 119867

infincontrol problem a lifting

approachrdquo IEEE Transaction on Automatic Control vol 37 no5 pp 2345ndash2350 2000

[2] X Zhao Y Yao and JMa ldquoOverview ofmultirate sampled-datacontrol systemrdquo Journal of Harbin Institute of Technology vol 9no 4 pp 405ndash410 2002

[3] Y Fukada ldquoSlip-angle estimation for vehicle stability controlrdquoVehicle System Dynamics vol 32 no 4 pp 375ndash388 1999

[4] T Chen and L Qiu ldquo119867infindesign of general multirate sampled-

data control systemsrdquo Automatica vol 30 no 7 pp 1139ndash11521994

[5] Z Wang B Huang and H Unbehauen ldquoRobust 119867infin

observerdesign of linear state delayed systems with parametric uncer-tainty the discrete-time caserdquo Automatica vol 35 no 6 pp1161ndash1167 1999

[6] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Control Theory vol 15 ofSIAM Studies in Applied Mathematics SIAM Philadelphia PaUSA 1994

[7] S Patra and A Lanzon ldquoA closed-loop data based test forrobust performance improvement in iterative identification andcontrol redesignsrdquo Automatica vol 48 no 10 pp 2710ndash27162012

[8] J Ding Y Shi H Wang and F Ding ldquoA modified stochasticgradient based parameter estimation algorithm for dual-ratesampled-data systemsrdquo Digital Signal Processing vol 20 no 4pp 1238ndash1247 2010

[9] T W Chen and B Francis Optimal Sampled-Data ControlSystems Springer London UK 1995

[10] L Shen and M J Er ldquoMultiobjectives design of a multirateoutput controllerrdquo in Proceedings of the IEEE InternationalConference on Control Applications (CCA rsquo99) pp 193ndash198August 1999

[11] M Chilali and P Gahinet ldquo119867infin

design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996

[12] S Mo X Chen J Zhao J Qian and Z Shao ldquoA two-stagemethod for identification of dual-rate systems with fast inputand very slow outputrdquo Industrial and Engineering ChemistryResearch vol 48 no 4 pp 1980ndash1988 2009

[13] M De la Sen and S Alonso-Quesada ldquoModel matching viamultirate sampling with fast sampled input guaranteeing thestability of the plant zeros extensions to adaptive controlrdquo IETControl Theory amp Applications vol 1 no 1 pp 210ndash225 2007

[14] M F Sagfors and H T Toivonen ldquo119867infin

and LQG control ofasynchronous sampled-data systemsrdquo Automatica vol 33 no9 pp 1663ndash1668 1997

[15] S Lopez-lopez Optimal Hinfin

Design of Causal Multirate Con-trollers and Filters Mechanical and Aerospace EngineeringUniversity of California Irvine Calif USA 2010

[16] U Borison ldquoSelf-tuning regulators for a class of multivariablesystemsrdquo Automatica vol 15 no 2 pp 209ndash215 1979

[17] V S Ritchey and G F Franklin ldquoA stability criterion forasynchronous multirate linear systemsrdquo IEEE Transactions onAutomatic Control vol 34 no 5 pp 529ndash535 1989

[18] B Shen Z Wang and X Liu ldquoSampled-data synchronizationcontrol of dynamical networks with stochastic samplingrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2644ndash2650 2012

[19] S Longo G Herrmann and P Barber ldquoRobust scheduling ofsampled-data networked control systemsrdquo IEEE Transactionson Control Systems Technology vol 20 no 6 pp 1613ndash1621 2011

[20] P Voulgaris ldquoControl of asynchronous sampled data systemsrdquoIEEE Transactions on Automatic Control vol 39 no 7 pp 1451ndash1455 1994

[21] P Colaneri R Scattolini and N Schiavoni ldquoLQG optimalcontrol of multirate sampled-data systemsrdquo IEEE Transactionson Automatic Control vol 37 no 5 pp 675ndash682 1992

[22] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

12 Mathematical Problems in Engineering

[23] A Seuret ldquoStability analysis of networked control systems withasynchronous sampling and input delayrdquo in Proceedings of theAmerican Control Conference (ACC rsquo11) pp 533ndash538 July 2011

[24] B Tavassoli ldquoStability of nonlinear networked control systemsover multiple communication links with asynchronous sam-plingrdquo IEEE Transaction on Automatic Control 2013

[25] L Yang and J M Li ldquoSufficient and necessary conditions ofcontrollability and observability of a class of linear switchingsystemrdquo Systems Engineering and Electronics vol 25 no 5 pp588ndash590 2003

[26] W-J Mao ldquoObserver-based energy decoupling of linear time-delay systemsrdquo Journal of Zhejiang University vol 37 no 5 pp499ndash503 2003

[27] T Iwasaki and S Hara ldquoGeneralized KYP lemma unifiedfrequency domain inequalities with design applicationsrdquo IEEETransactions on Automatic Control vol 50 no 1 pp 41ndash592005

[28] H Gao and X Li ldquo119867infinfiltering for discrete-time state-delayed

systems with finite frequency specificationsrdquo IEEE Transactionson Automatic Control vol 56 no 12 pp 2935ndash2941 2011

[29] X Li H Gao and CWang ldquoGeneralized Kalman-Yakubovich-Popov lemma for 2-D FM LSS modelrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3090ndash3103 2012

[30] X Li and H Gao ldquoRobust finite frequency 119867infin

filtering foruncertain 2-D systems the FM model caserdquo Automatica vol49 no 8 pp 2446ndash2452 2013

[31] X W Li and H J Gao ldquoA heuristic approach to static output-feedback controller synthesis with restricted frequency-domainspecificationsrdquo IEEE Transaction on Automatic Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

1198621=

[

[

[

[

[

119862 0

119862119860 0

119862119860119898minus1

0

]

]

]

]

]

11986311=

[

[

[

[

[

1198722

0 sdot sdot sdot 0

1198621198721

1198722

sdot sdot sdot 0

d

119862119860119898minus2

1198721119862119860119898minus3

1198721sdot sdot sdot 119872

2

]

]

]

]

]

(24)

(b) Transfer Function for 11987512 As119867(119896) in 119875

12= 11987111989811987811989111986612119867(119896)

is not an 119898-blocks unit matrix the theory cannot be applieddirectly However note that

119878119891119867(119896) = 119871

minus1

119898

[

[

[

[

[

119868

119868

119868

]

]

]

]

]119898

119867(119896) (25)

Then

11987512= 11987111989811987811989111986612119867119891119871minus1

119898

[

[

[

[

[

119868

119868

119868

]

]

]

]

]119898

119867(119896) (26)

The corresponding state model can be obtained as

12(120582) = [

119860119889

1198612119889

1198621

11986312

] (27)

where 119860119889and 119862

1have been defined previously and

1198612119889

=[

[

119898

sum

119894=1

119860119898minus119894

119861

119868

]

]

119867 (119896)

11986312=

[

[

[

[

[

[

[

[

[

[

0

0 +

119898

sum

119895=119898

119862119860119898minus119895

119861

0 +

119898

sum

119895=2

119862119860119898minus119895

119861

]

]

]

]

]

]

]

]

]

]

119867(119896)

(28)

(c) Transfer Function for 11987521 Similarly because 119878(119896) is also not

an119898-blocks unit matrix 11987521can be changed to

11987521= 119878 (119896) [119868 0 sdot sdot sdot 0]

11989811987111989811987811989111986621119867119891119871minus1

119898 (29)

Thus the state model can be derived as

21(120582) = [

119860119889

1198611

1198622

11986321

] (30)

where 119860119889and 119861

1were defined previously and

1198622= Δ (119896)

11986321= [Ω (119896) 0 sdot sdot sdot 0]

(31)

(d) Transfer Function for 11987522 Like to parts 119887 and 119888 the state

description of 11987522is

22(120582) = [

119860119889

1198612119889

1198622

11986322

] (32)

where 119860119889 1198612119889 and 119862

2were defined previously while

11986322= 0 (33)

Finally formula (23) is represented using the state-spaceform as follows

120585 (119896 + 1) = 119860119889120585 (119896) + 119861

1119908 (119896) + 119861

2119889119903 (119896)

119911 (119896) = 1198621120585 (119896) + 119863

11119908 (119896) + 119863

12119903 (119896)

119910 (119896) = 1198622120585 (119896) + 119863

21119908 (119896)

(34)

where the underlining below 119911 and 119908 indicates that thesignals are lifted

According to Lemma 2 the controller119870(119896) exists and canstabilize the system

Let weierp(119896) = [1205851015840

(119896) 1198901015840

(119896)]

1015840 and after combining (13) (15)and (34) we obtain

weierp (119896 + 1) = 119860119888weierp (119896) + 119861

119888119908 (119896)

119911 (119896) = 119862119888weierp (119896) + 119863

119888119908 (119896)

(35)

where

119860119888=

[

[

[

119860119889+ 1198612119889119870 (119896) minus119861

2119889119870 (119896) [

1198682

0]

0 119860 minus 119871 (119896) 119878 (119896) 119862

]

]

]

119861119888= [

1198611

(1198721minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

119898

]

119862119888= [119862

1+ 11986312119870 (119896) minus119863

12119870 (119896) [

1198682

0]]

119863119888= 11986311

(36)

In the light of [5] we set

119883 (119896) = [

1198831(119896) 0

0 1198832(119896)

] (37)

6 Mathematical Problems in Engineering

By substituting119883 in Lemma 3 with119883(119896) we can obtain

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831(119896) 0 (119860

119889+ 1198612119889119870 (119896))119883

1(119896) minus119861

2119889119870 (119896) [

1198682

0

]1198832(119896) 119861

10

lowast 1198832(119896) 0 (119860 minus 119871 (119896) 119878 (119896) 119862)119883

2(119896) (119872

1minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 1198831(119896) 0 0 119883

1(119896) (119862

1015840

1+ 1198701015840

(119896)1198631015840

12)

lowast lowast lowast 1198832(119896) 0 minus119883

2(119896) [1198682

0]1198701015840

(119896)1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0

(38)

Then according to [26] after multiplying the left- andright-hand sides of inequality (38) by diag119868 119868 119883minus1

1(119896)

119883minus1

2(119896) 119868 119868 we can obtain inequality (20)According to quadratic stability theory the system should

be stable at each step 119896 Thus for a discrete linear time-invariant system the eigenvalues of the closed system shouldlie within a unit circle

From Theorem 4 if matrix inequalities (20) can besolved there exists a state feedback 119867

infincontroller that can

asymptotically stabilize closed system (35) of enlarged system(34) which satisfies the performance index Since system(34) is lifted from the original system (9) and we know thatthe lifting operator is an isometric isomorphism that isthe feedback stability and system norm can be preservedthen system (16) which is the closed system of (9) isasynchronously stable and it satisfies the119867

infinnorm constraint

119879119911119908infin

lt 120574

The synchronous case can be regarded as a specialexample of the asynchronous case In this case the elementsof sampling and holder are unit matrices with the assumption

that the whole signals of sampling and holding can beobtained at the beginning of every period then the system (9)turns into a linear discrete synchronous multirate sampled-data system with period 120591 as follows

120585 (119896 + 1) = Φ120585 (119896) + Γ119903 (119896) + Ψ119908 (119896)

119911 (119896) = Υ120585 (119896) + 1198722119908 (119896)

119910 (119896) = Δ120585 (119896) + Ω119908 (119896)

(39)

As119867 and 119878 are unit matrices that is constant now Φ ΓΔ andΩ related to119867 and 119878 are also constant Theorem 4 canbe applied to system (39) straightforwardly

Corollary 5 Consider the sampled-data system (39) with thelifting process shown in Figure 3 For a given attenuation level120574 gt 0 if there exist matrices 119883

1= 1198831015840

1gt 0 119883

2= 1198831015840

2gt

0 119870 and 119871 which are suboptimal solutions of the inequalitiesbelow then the closed system of (39) with the controller (13)is synchronously uniformly stable and it satisfies the119867

infinnorm

constraint 119879119911119908infin

lt 120574

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831

0 119860119889+ 1198612119889119870 minus119861

2119889119870[

1198682

0] 119861

10

lowast 1198832

0 119860 minus 119871119862 (1198721minus 119871119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 119883minus1

10 0 119862

1015840

1+ 1198701015840

1198631015840

12

lowast lowast lowast 119883minus1

20 minus [119868

20]1198701015840

1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0 (40)

Remark 6 Let us compare the designed observer (12) withthat in [21] where the observer (predictor) is

(119896 + 1 | 119896) = 119860 (119896 | 119896 minus 1) + 119861119906 (119896)

+ 119871 (119896) sdot [120577 (119896) minus 119862 (119896) (119896 | 119896 minus 1)]

120577 (119896) =

119879

lowast

sum

119894=1

119910 (119896 minus 119894) 119879

lowast

= max119894=1119901

119879119894

(41)The sum of outputs is used to avoid the singularity butmay cause more noise The observer applied in this paper is

Mathematical Problems in Engineering 7

w z

u yG

K

Tm1 Tmp

middot middot middot middot middot middotTn1Tnq H1Hq SpS1

Sa Lminus1mLb HbHf Sf Lminus1aLm

Figure 3 System lifting and controller designing

simpler that is just the current output is introduced in theobserver This will result in better tracking performance andsmaller state ripple than [21] where the past outputs otherthan current output are used for feedback

Remark 7 InTheorem 4 and Corollary 5 due to the existenceof inverse matrices 119883minus1

1and 119883

minus1

2 inequalities (20) and (40)

are not LMIs and cannot be directly solved by the LMI toolHowever notice that the matrices and their inverse matricesappear in pairs we can solve the inequalities by iterationon LMIs according to Cone Complementarity LinearizationAlgorithm [22]

4 Numerical Example with Simulation

Consider the original plant described by

119888

(119905) = 119860119888

119909119888

(119905) + 119861119888

119906119888

(119905) + 119872119888

1119908119888

(119905)

119911119888

(119905) = 119862119888

119909119888

(119905) + 119872119888

2119908119888

(119905)

(42)

where

119860119888

= [

0 1

minus6 minus5] 119861

119888

= [

0 1

1 12] 119872

119888

1= [

1

06]

119862119888

= [02 01] 119872119888

2= 001

(43)

If the fast discretization formulae follow119860 = 119890119860119888(120591119898) 119861 =

int

120591119898

0

119890119860119888120590

119889120590119861119888 1198721= int

120591119898

0

119890119860119888120590

119889120590119872119888

1 119862 = 119862

119888 and 1198722=

119872119888

2 then system (2) can be obtained The initial state is 119909

0=

[30 20]If 1198791=

1198792= 2 the input-holding mechanism can be

described as

119867(0) = [

1 0

0 0] 119867 (1) = [

0 0

0 1] (44)

If 1198791= 3 and 119879

2= 1 the output-sampling matrices are

assumed to be

119878 (0) = 1 119878 (1) = 0 119878 (2) = 0 (45)

Therefore this is a linear discrete asynchronous dual-ratesampled-data system [18] which applies a fast-input-slow-output mechanism for 119879 lt 119879

0 5 10 15 20 25 30

0

5

10

15

20

25

30

minus5

Real stateObserved state

Figure 4 Plant state 1199091(119896)

0

5

10

15

20

minus25

minus20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 5 Plant state 1199092(119896)

0001002003004005006007008

minus0010 5 10 15 20 25 30

Figure 6 Control signal 1199061(119896)

8 Mathematical Problems in Engineering

0

05

minus3

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 7 Control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 8 Output

For comparison we introduce a previous method [21]where the system is not lifted according to the pure LQGtheory and also let 119908

1= 1199082= 119908 be Gaussian white noise

amplitude is increased to 10 which is much bigger to thesystem The results obtained using this method are shown inFigures 4 5 6 7 and 8 where 120591 = 12 s

Next the synthesis method described in Theorem 4is applied to the same plant The simulations obtain thetransients of the system shown in Figures 9 10 11 12 and 13where 120574 = 02 119886 = 1 and 119887 = 1 while119898 = 1 (monorate)

A comparison of Figures 4ndash8 with Figures 9ndash13 showsclearly that the 119867

infinmethod performs better than LQG

the controlled states are consistent with the observed stateswith smaller ripples especially state 119909

2(119896) the biggest ripple

of which is almost half of the LQG method And theoutput of119867

infinmethod has better performance of disturbance

attenuation than LQG methodWhen 120591 = 06 s and 119898 = 1 (monorate) applying

Theorem 4 again we can get Figures 14 15 16 17 and 18Then the lifting number is modified to119898 = 2 (multirate) theresults shown in Figures 19 20 21 22 and 23 are obtained

Comparing Figures 14ndash18 with Figures 19ndash23 when thelifting amplitude increases from 1 to 2 the controlled statesare almost consistent with the observed states with smallerripples here the biggest ripple of state 119909

2(119896) is almost half of

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 9 Plant state 1199091(119896)

0

5

10

15

20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 10 Plant state 1199092(119896)

0

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 11 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

07

0 5 10 15 20 25 30

Figure 12 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 13 Output

Table 1 Minimum119867infinattenuation level

120591 119898 = 1 119898 = 2 119898 = 4

12 sec 00974 00959 0094206 sec 00904 00863 00856

the monorate And the output of the multirate case has betterperformance of disturbance attenuation than the monoratecase

Furthermore we change the lifting amplitude to 119898 = 4adjust the underlying period of system (2) to 120591 = 06 andcan draw the similar pictures here we omit them but only listtheir minimum119867

infinattenuation levels to compare them with

different cases see Table 1The inequality acquired using the theorem is almost

linear except in a few terms Although these terms are justthe inverses of their corresponding terms they could easily becalculated by LMI tool ofMatlab under a constraintThus it isfar easier to solve the controller and observer under the same119867infinconstraint comparing withmost of the existingmethods

which use highly complex formulae

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 14 Plant state 1199091(119896)

0

5

10

15

20

Real stateObserved state

minus5

minus30

minus25

minus20

minus15

minus10

0 5 10 15 20 25 30

Figure 15 Plant state 1199092(119896)

0

0 5 10 15 20 25 30

minus3

minus25

minus35

minus4

minus45

minus2

minus15

minus1

minus05

Figure 16 Plant control signal 1199061(119896)

10 Mathematical Problems in Engineering

0

02

04

06

08

1

12

14

0 5 10 15 20 25 30

Figure 17 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30minus1

Figure 18 Output

5 Conclusion

Motivated by a desire to provide a better solution for thedesign of a controller for a discrete-time asynchronous mul-tirate sampled-data system we developed synthesis methodsin this paper to improve performance and guarantee closed-loop transient behavior via observer-based 119867

infincontrol

Furthermore an extended lifting technique was adoptedto change the original system into a fast-input-slow-outputdiscrete system thereby improving its performance Finallywe presented an example that demonstrates the effectivenessof thesemethods in controlling a discrete-time asynchronousmultirate sampled-data system whose performance is thusimproved considerably

In this paper we focus on the standard 119867infin

control ofasynchronous multirate sampled-data system and the inputnoise discussed here is Gaussian white for simplicity As iswell known practical input signals often have finite frequency(FF) so it is reasonable to consider the 119867

infinperformance in

finite frequency range for sampled-data systems As in [27ndash31] where Kalman-Yakubovic-Popov (KYP) lemma is usedto establish the equivalence between a frequency domaininequality (FDI) and a linear matrix inequality it is a valuablework to extend the results in this paper to the FF frameworkbased on KYP lemma This will be the direction for futuredevelopment of this paper

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Real stateObserved state

minus10

minus5

Figure 19 Plant state 1199091(119896)

0

5

10

15

20

0 5 10 15 20 25 30

Real stateObserved state

minus20

minus15

minus10

minus5

Figure 20 Plant state 1199092(119896)

005

0 5 10 15 20 25 30

minus3

minus35

minus4

minus25

minus2

minus15

minus1

minus05

Figure 21 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 11

0

02

04

06

08

1

12

0 5 10 15 20 25 30minus02

Figure 22 Plant control signal 1199062(119896)

minus10 5 10 15 20 25 30

0

1

2

3

4

5

6

7

8

9

Figure 23 Output

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Zhejiang Provincial Natural Sci-ence Foundation of China under Grant no LR12F03002 andNational Natural Science Foundation of China (NSFC) underGrant no 61074045

References

[1] M F Sagfors H T Toivonen and B Lennartson ldquoState-spacesolution to the period multirate 119867

infincontrol problem a lifting

approachrdquo IEEE Transaction on Automatic Control vol 37 no5 pp 2345ndash2350 2000

[2] X Zhao Y Yao and JMa ldquoOverview ofmultirate sampled-datacontrol systemrdquo Journal of Harbin Institute of Technology vol 9no 4 pp 405ndash410 2002

[3] Y Fukada ldquoSlip-angle estimation for vehicle stability controlrdquoVehicle System Dynamics vol 32 no 4 pp 375ndash388 1999

[4] T Chen and L Qiu ldquo119867infindesign of general multirate sampled-

data control systemsrdquo Automatica vol 30 no 7 pp 1139ndash11521994

[5] Z Wang B Huang and H Unbehauen ldquoRobust 119867infin

observerdesign of linear state delayed systems with parametric uncer-tainty the discrete-time caserdquo Automatica vol 35 no 6 pp1161ndash1167 1999

[6] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Control Theory vol 15 ofSIAM Studies in Applied Mathematics SIAM Philadelphia PaUSA 1994

[7] S Patra and A Lanzon ldquoA closed-loop data based test forrobust performance improvement in iterative identification andcontrol redesignsrdquo Automatica vol 48 no 10 pp 2710ndash27162012

[8] J Ding Y Shi H Wang and F Ding ldquoA modified stochasticgradient based parameter estimation algorithm for dual-ratesampled-data systemsrdquo Digital Signal Processing vol 20 no 4pp 1238ndash1247 2010

[9] T W Chen and B Francis Optimal Sampled-Data ControlSystems Springer London UK 1995

[10] L Shen and M J Er ldquoMultiobjectives design of a multirateoutput controllerrdquo in Proceedings of the IEEE InternationalConference on Control Applications (CCA rsquo99) pp 193ndash198August 1999

[11] M Chilali and P Gahinet ldquo119867infin

design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996

[12] S Mo X Chen J Zhao J Qian and Z Shao ldquoA two-stagemethod for identification of dual-rate systems with fast inputand very slow outputrdquo Industrial and Engineering ChemistryResearch vol 48 no 4 pp 1980ndash1988 2009

[13] M De la Sen and S Alonso-Quesada ldquoModel matching viamultirate sampling with fast sampled input guaranteeing thestability of the plant zeros extensions to adaptive controlrdquo IETControl Theory amp Applications vol 1 no 1 pp 210ndash225 2007

[14] M F Sagfors and H T Toivonen ldquo119867infin

and LQG control ofasynchronous sampled-data systemsrdquo Automatica vol 33 no9 pp 1663ndash1668 1997

[15] S Lopez-lopez Optimal Hinfin

Design of Causal Multirate Con-trollers and Filters Mechanical and Aerospace EngineeringUniversity of California Irvine Calif USA 2010

[16] U Borison ldquoSelf-tuning regulators for a class of multivariablesystemsrdquo Automatica vol 15 no 2 pp 209ndash215 1979

[17] V S Ritchey and G F Franklin ldquoA stability criterion forasynchronous multirate linear systemsrdquo IEEE Transactions onAutomatic Control vol 34 no 5 pp 529ndash535 1989

[18] B Shen Z Wang and X Liu ldquoSampled-data synchronizationcontrol of dynamical networks with stochastic samplingrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2644ndash2650 2012

[19] S Longo G Herrmann and P Barber ldquoRobust scheduling ofsampled-data networked control systemsrdquo IEEE Transactionson Control Systems Technology vol 20 no 6 pp 1613ndash1621 2011

[20] P Voulgaris ldquoControl of asynchronous sampled data systemsrdquoIEEE Transactions on Automatic Control vol 39 no 7 pp 1451ndash1455 1994

[21] P Colaneri R Scattolini and N Schiavoni ldquoLQG optimalcontrol of multirate sampled-data systemsrdquo IEEE Transactionson Automatic Control vol 37 no 5 pp 675ndash682 1992

[22] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

12 Mathematical Problems in Engineering

[23] A Seuret ldquoStability analysis of networked control systems withasynchronous sampling and input delayrdquo in Proceedings of theAmerican Control Conference (ACC rsquo11) pp 533ndash538 July 2011

[24] B Tavassoli ldquoStability of nonlinear networked control systemsover multiple communication links with asynchronous sam-plingrdquo IEEE Transaction on Automatic Control 2013

[25] L Yang and J M Li ldquoSufficient and necessary conditions ofcontrollability and observability of a class of linear switchingsystemrdquo Systems Engineering and Electronics vol 25 no 5 pp588ndash590 2003

[26] W-J Mao ldquoObserver-based energy decoupling of linear time-delay systemsrdquo Journal of Zhejiang University vol 37 no 5 pp499ndash503 2003

[27] T Iwasaki and S Hara ldquoGeneralized KYP lemma unifiedfrequency domain inequalities with design applicationsrdquo IEEETransactions on Automatic Control vol 50 no 1 pp 41ndash592005

[28] H Gao and X Li ldquo119867infinfiltering for discrete-time state-delayed

systems with finite frequency specificationsrdquo IEEE Transactionson Automatic Control vol 56 no 12 pp 2935ndash2941 2011

[29] X Li H Gao and CWang ldquoGeneralized Kalman-Yakubovich-Popov lemma for 2-D FM LSS modelrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3090ndash3103 2012

[30] X Li and H Gao ldquoRobust finite frequency 119867infin

filtering foruncertain 2-D systems the FM model caserdquo Automatica vol49 no 8 pp 2446ndash2452 2013

[31] X W Li and H J Gao ldquoA heuristic approach to static output-feedback controller synthesis with restricted frequency-domainspecificationsrdquo IEEE Transaction on Automatic Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

By substituting119883 in Lemma 3 with119883(119896) we can obtain

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831(119896) 0 (119860

119889+ 1198612119889119870 (119896))119883

1(119896) minus119861

2119889119870 (119896) [

1198682

0

]1198832(119896) 119861

10

lowast 1198832(119896) 0 (119860 minus 119871 (119896) 119878 (119896) 119862)119883

2(119896) (119872

1minus 119871 (119896) 119878 (119896)119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 1198831(119896) 0 0 119883

1(119896) (119862

1015840

1+ 1198701015840

(119896)1198631015840

12)

lowast lowast lowast 1198832(119896) 0 minus119883

2(119896) [1198682

0]1198701015840

(119896)1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0

(38)

Then according to [26] after multiplying the left- andright-hand sides of inequality (38) by diag119868 119868 119883minus1

1(119896)

119883minus1

2(119896) 119868 119868 we can obtain inequality (20)According to quadratic stability theory the system should

be stable at each step 119896 Thus for a discrete linear time-invariant system the eigenvalues of the closed system shouldlie within a unit circle

From Theorem 4 if matrix inequalities (20) can besolved there exists a state feedback 119867

infincontroller that can

asymptotically stabilize closed system (35) of enlarged system(34) which satisfies the performance index Since system(34) is lifted from the original system (9) and we know thatthe lifting operator is an isometric isomorphism that isthe feedback stability and system norm can be preservedthen system (16) which is the closed system of (9) isasynchronously stable and it satisfies the119867

infinnorm constraint

119879119911119908infin

lt 120574

The synchronous case can be regarded as a specialexample of the asynchronous case In this case the elementsof sampling and holder are unit matrices with the assumption

that the whole signals of sampling and holding can beobtained at the beginning of every period then the system (9)turns into a linear discrete synchronous multirate sampled-data system with period 120591 as follows

120585 (119896 + 1) = Φ120585 (119896) + Γ119903 (119896) + Ψ119908 (119896)

119911 (119896) = Υ120585 (119896) + 1198722119908 (119896)

119910 (119896) = Δ120585 (119896) + Ω119908 (119896)

(39)

As119867 and 119878 are unit matrices that is constant now Φ ΓΔ andΩ related to119867 and 119878 are also constant Theorem 4 canbe applied to system (39) straightforwardly

Corollary 5 Consider the sampled-data system (39) with thelifting process shown in Figure 3 For a given attenuation level120574 gt 0 if there exist matrices 119883

1= 1198831015840

1gt 0 119883

2= 1198831015840

2gt

0 119870 and 119871 which are suboptimal solutions of the inequalitiesbelow then the closed system of (39) with the controller (13)is synchronously uniformly stable and it satisfies the119867

infinnorm

constraint 119879119911119908infin

lt 120574

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198831

0 119860119889+ 1198612119889119870 minus119861

2119889119870[

1198682

0] 119861

10

lowast 1198832

0 119860 minus 119871119862 (1198721minus 119871119872

2) [1198681198970 sdot sdot sdot 0]

1198980

lowast lowast 119883minus1

10 0 119862

1015840

1+ 1198701015840

1198631015840

12

lowast lowast lowast 119883minus1

20 minus [119868

20]1198701015840

1198631015840

12

lowast lowast lowast lowast 120574119868 1198631015840

11

lowast lowast lowast lowast lowast 120574119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

gt 0 (40)

Remark 6 Let us compare the designed observer (12) withthat in [21] where the observer (predictor) is

(119896 + 1 | 119896) = 119860 (119896 | 119896 minus 1) + 119861119906 (119896)

+ 119871 (119896) sdot [120577 (119896) minus 119862 (119896) (119896 | 119896 minus 1)]

120577 (119896) =

119879

lowast

sum

119894=1

119910 (119896 minus 119894) 119879

lowast

= max119894=1119901

119879119894

(41)The sum of outputs is used to avoid the singularity butmay cause more noise The observer applied in this paper is

Mathematical Problems in Engineering 7

w z

u yG

K

Tm1 Tmp

middot middot middot middot middot middotTn1Tnq H1Hq SpS1

Sa Lminus1mLb HbHf Sf Lminus1aLm

Figure 3 System lifting and controller designing

simpler that is just the current output is introduced in theobserver This will result in better tracking performance andsmaller state ripple than [21] where the past outputs otherthan current output are used for feedback

Remark 7 InTheorem 4 and Corollary 5 due to the existenceof inverse matrices 119883minus1

1and 119883

minus1

2 inequalities (20) and (40)

are not LMIs and cannot be directly solved by the LMI toolHowever notice that the matrices and their inverse matricesappear in pairs we can solve the inequalities by iterationon LMIs according to Cone Complementarity LinearizationAlgorithm [22]

4 Numerical Example with Simulation

Consider the original plant described by

119888

(119905) = 119860119888

119909119888

(119905) + 119861119888

119906119888

(119905) + 119872119888

1119908119888

(119905)

119911119888

(119905) = 119862119888

119909119888

(119905) + 119872119888

2119908119888

(119905)

(42)

where

119860119888

= [

0 1

minus6 minus5] 119861

119888

= [

0 1

1 12] 119872

119888

1= [

1

06]

119862119888

= [02 01] 119872119888

2= 001

(43)

If the fast discretization formulae follow119860 = 119890119860119888(120591119898) 119861 =

int

120591119898

0

119890119860119888120590

119889120590119861119888 1198721= int

120591119898

0

119890119860119888120590

119889120590119872119888

1 119862 = 119862

119888 and 1198722=

119872119888

2 then system (2) can be obtained The initial state is 119909

0=

[30 20]If 1198791=

1198792= 2 the input-holding mechanism can be

described as

119867(0) = [

1 0

0 0] 119867 (1) = [

0 0

0 1] (44)

If 1198791= 3 and 119879

2= 1 the output-sampling matrices are

assumed to be

119878 (0) = 1 119878 (1) = 0 119878 (2) = 0 (45)

Therefore this is a linear discrete asynchronous dual-ratesampled-data system [18] which applies a fast-input-slow-output mechanism for 119879 lt 119879

0 5 10 15 20 25 30

0

5

10

15

20

25

30

minus5

Real stateObserved state

Figure 4 Plant state 1199091(119896)

0

5

10

15

20

minus25

minus20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 5 Plant state 1199092(119896)

0001002003004005006007008

minus0010 5 10 15 20 25 30

Figure 6 Control signal 1199061(119896)

8 Mathematical Problems in Engineering

0

05

minus3

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 7 Control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 8 Output

For comparison we introduce a previous method [21]where the system is not lifted according to the pure LQGtheory and also let 119908

1= 1199082= 119908 be Gaussian white noise

amplitude is increased to 10 which is much bigger to thesystem The results obtained using this method are shown inFigures 4 5 6 7 and 8 where 120591 = 12 s

Next the synthesis method described in Theorem 4is applied to the same plant The simulations obtain thetransients of the system shown in Figures 9 10 11 12 and 13where 120574 = 02 119886 = 1 and 119887 = 1 while119898 = 1 (monorate)

A comparison of Figures 4ndash8 with Figures 9ndash13 showsclearly that the 119867

infinmethod performs better than LQG

the controlled states are consistent with the observed stateswith smaller ripples especially state 119909

2(119896) the biggest ripple

of which is almost half of the LQG method And theoutput of119867

infinmethod has better performance of disturbance

attenuation than LQG methodWhen 120591 = 06 s and 119898 = 1 (monorate) applying

Theorem 4 again we can get Figures 14 15 16 17 and 18Then the lifting number is modified to119898 = 2 (multirate) theresults shown in Figures 19 20 21 22 and 23 are obtained

Comparing Figures 14ndash18 with Figures 19ndash23 when thelifting amplitude increases from 1 to 2 the controlled statesare almost consistent with the observed states with smallerripples here the biggest ripple of state 119909

2(119896) is almost half of

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 9 Plant state 1199091(119896)

0

5

10

15

20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 10 Plant state 1199092(119896)

0

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 11 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

07

0 5 10 15 20 25 30

Figure 12 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 13 Output

Table 1 Minimum119867infinattenuation level

120591 119898 = 1 119898 = 2 119898 = 4

12 sec 00974 00959 0094206 sec 00904 00863 00856

the monorate And the output of the multirate case has betterperformance of disturbance attenuation than the monoratecase

Furthermore we change the lifting amplitude to 119898 = 4adjust the underlying period of system (2) to 120591 = 06 andcan draw the similar pictures here we omit them but only listtheir minimum119867

infinattenuation levels to compare them with

different cases see Table 1The inequality acquired using the theorem is almost

linear except in a few terms Although these terms are justthe inverses of their corresponding terms they could easily becalculated by LMI tool ofMatlab under a constraintThus it isfar easier to solve the controller and observer under the same119867infinconstraint comparing withmost of the existingmethods

which use highly complex formulae

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 14 Plant state 1199091(119896)

0

5

10

15

20

Real stateObserved state

minus5

minus30

minus25

minus20

minus15

minus10

0 5 10 15 20 25 30

Figure 15 Plant state 1199092(119896)

0

0 5 10 15 20 25 30

minus3

minus25

minus35

minus4

minus45

minus2

minus15

minus1

minus05

Figure 16 Plant control signal 1199061(119896)

10 Mathematical Problems in Engineering

0

02

04

06

08

1

12

14

0 5 10 15 20 25 30

Figure 17 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30minus1

Figure 18 Output

5 Conclusion

Motivated by a desire to provide a better solution for thedesign of a controller for a discrete-time asynchronous mul-tirate sampled-data system we developed synthesis methodsin this paper to improve performance and guarantee closed-loop transient behavior via observer-based 119867

infincontrol

Furthermore an extended lifting technique was adoptedto change the original system into a fast-input-slow-outputdiscrete system thereby improving its performance Finallywe presented an example that demonstrates the effectivenessof thesemethods in controlling a discrete-time asynchronousmultirate sampled-data system whose performance is thusimproved considerably

In this paper we focus on the standard 119867infin

control ofasynchronous multirate sampled-data system and the inputnoise discussed here is Gaussian white for simplicity As iswell known practical input signals often have finite frequency(FF) so it is reasonable to consider the 119867

infinperformance in

finite frequency range for sampled-data systems As in [27ndash31] where Kalman-Yakubovic-Popov (KYP) lemma is usedto establish the equivalence between a frequency domaininequality (FDI) and a linear matrix inequality it is a valuablework to extend the results in this paper to the FF frameworkbased on KYP lemma This will be the direction for futuredevelopment of this paper

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Real stateObserved state

minus10

minus5

Figure 19 Plant state 1199091(119896)

0

5

10

15

20

0 5 10 15 20 25 30

Real stateObserved state

minus20

minus15

minus10

minus5

Figure 20 Plant state 1199092(119896)

005

0 5 10 15 20 25 30

minus3

minus35

minus4

minus25

minus2

minus15

minus1

minus05

Figure 21 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 11

0

02

04

06

08

1

12

0 5 10 15 20 25 30minus02

Figure 22 Plant control signal 1199062(119896)

minus10 5 10 15 20 25 30

0

1

2

3

4

5

6

7

8

9

Figure 23 Output

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Zhejiang Provincial Natural Sci-ence Foundation of China under Grant no LR12F03002 andNational Natural Science Foundation of China (NSFC) underGrant no 61074045

References

[1] M F Sagfors H T Toivonen and B Lennartson ldquoState-spacesolution to the period multirate 119867

infincontrol problem a lifting

approachrdquo IEEE Transaction on Automatic Control vol 37 no5 pp 2345ndash2350 2000

[2] X Zhao Y Yao and JMa ldquoOverview ofmultirate sampled-datacontrol systemrdquo Journal of Harbin Institute of Technology vol 9no 4 pp 405ndash410 2002

[3] Y Fukada ldquoSlip-angle estimation for vehicle stability controlrdquoVehicle System Dynamics vol 32 no 4 pp 375ndash388 1999

[4] T Chen and L Qiu ldquo119867infindesign of general multirate sampled-

data control systemsrdquo Automatica vol 30 no 7 pp 1139ndash11521994

[5] Z Wang B Huang and H Unbehauen ldquoRobust 119867infin

observerdesign of linear state delayed systems with parametric uncer-tainty the discrete-time caserdquo Automatica vol 35 no 6 pp1161ndash1167 1999

[6] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Control Theory vol 15 ofSIAM Studies in Applied Mathematics SIAM Philadelphia PaUSA 1994

[7] S Patra and A Lanzon ldquoA closed-loop data based test forrobust performance improvement in iterative identification andcontrol redesignsrdquo Automatica vol 48 no 10 pp 2710ndash27162012

[8] J Ding Y Shi H Wang and F Ding ldquoA modified stochasticgradient based parameter estimation algorithm for dual-ratesampled-data systemsrdquo Digital Signal Processing vol 20 no 4pp 1238ndash1247 2010

[9] T W Chen and B Francis Optimal Sampled-Data ControlSystems Springer London UK 1995

[10] L Shen and M J Er ldquoMultiobjectives design of a multirateoutput controllerrdquo in Proceedings of the IEEE InternationalConference on Control Applications (CCA rsquo99) pp 193ndash198August 1999

[11] M Chilali and P Gahinet ldquo119867infin

design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996

[12] S Mo X Chen J Zhao J Qian and Z Shao ldquoA two-stagemethod for identification of dual-rate systems with fast inputand very slow outputrdquo Industrial and Engineering ChemistryResearch vol 48 no 4 pp 1980ndash1988 2009

[13] M De la Sen and S Alonso-Quesada ldquoModel matching viamultirate sampling with fast sampled input guaranteeing thestability of the plant zeros extensions to adaptive controlrdquo IETControl Theory amp Applications vol 1 no 1 pp 210ndash225 2007

[14] M F Sagfors and H T Toivonen ldquo119867infin

and LQG control ofasynchronous sampled-data systemsrdquo Automatica vol 33 no9 pp 1663ndash1668 1997

[15] S Lopez-lopez Optimal Hinfin

Design of Causal Multirate Con-trollers and Filters Mechanical and Aerospace EngineeringUniversity of California Irvine Calif USA 2010

[16] U Borison ldquoSelf-tuning regulators for a class of multivariablesystemsrdquo Automatica vol 15 no 2 pp 209ndash215 1979

[17] V S Ritchey and G F Franklin ldquoA stability criterion forasynchronous multirate linear systemsrdquo IEEE Transactions onAutomatic Control vol 34 no 5 pp 529ndash535 1989

[18] B Shen Z Wang and X Liu ldquoSampled-data synchronizationcontrol of dynamical networks with stochastic samplingrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2644ndash2650 2012

[19] S Longo G Herrmann and P Barber ldquoRobust scheduling ofsampled-data networked control systemsrdquo IEEE Transactionson Control Systems Technology vol 20 no 6 pp 1613ndash1621 2011

[20] P Voulgaris ldquoControl of asynchronous sampled data systemsrdquoIEEE Transactions on Automatic Control vol 39 no 7 pp 1451ndash1455 1994

[21] P Colaneri R Scattolini and N Schiavoni ldquoLQG optimalcontrol of multirate sampled-data systemsrdquo IEEE Transactionson Automatic Control vol 37 no 5 pp 675ndash682 1992

[22] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

12 Mathematical Problems in Engineering

[23] A Seuret ldquoStability analysis of networked control systems withasynchronous sampling and input delayrdquo in Proceedings of theAmerican Control Conference (ACC rsquo11) pp 533ndash538 July 2011

[24] B Tavassoli ldquoStability of nonlinear networked control systemsover multiple communication links with asynchronous sam-plingrdquo IEEE Transaction on Automatic Control 2013

[25] L Yang and J M Li ldquoSufficient and necessary conditions ofcontrollability and observability of a class of linear switchingsystemrdquo Systems Engineering and Electronics vol 25 no 5 pp588ndash590 2003

[26] W-J Mao ldquoObserver-based energy decoupling of linear time-delay systemsrdquo Journal of Zhejiang University vol 37 no 5 pp499ndash503 2003

[27] T Iwasaki and S Hara ldquoGeneralized KYP lemma unifiedfrequency domain inequalities with design applicationsrdquo IEEETransactions on Automatic Control vol 50 no 1 pp 41ndash592005

[28] H Gao and X Li ldquo119867infinfiltering for discrete-time state-delayed

systems with finite frequency specificationsrdquo IEEE Transactionson Automatic Control vol 56 no 12 pp 2935ndash2941 2011

[29] X Li H Gao and CWang ldquoGeneralized Kalman-Yakubovich-Popov lemma for 2-D FM LSS modelrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3090ndash3103 2012

[30] X Li and H Gao ldquoRobust finite frequency 119867infin

filtering foruncertain 2-D systems the FM model caserdquo Automatica vol49 no 8 pp 2446ndash2452 2013

[31] X W Li and H J Gao ldquoA heuristic approach to static output-feedback controller synthesis with restricted frequency-domainspecificationsrdquo IEEE Transaction on Automatic Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

w z

u yG

K

Tm1 Tmp

middot middot middot middot middot middotTn1Tnq H1Hq SpS1

Sa Lminus1mLb HbHf Sf Lminus1aLm

Figure 3 System lifting and controller designing

simpler that is just the current output is introduced in theobserver This will result in better tracking performance andsmaller state ripple than [21] where the past outputs otherthan current output are used for feedback

Remark 7 InTheorem 4 and Corollary 5 due to the existenceof inverse matrices 119883minus1

1and 119883

minus1

2 inequalities (20) and (40)

are not LMIs and cannot be directly solved by the LMI toolHowever notice that the matrices and their inverse matricesappear in pairs we can solve the inequalities by iterationon LMIs according to Cone Complementarity LinearizationAlgorithm [22]

4 Numerical Example with Simulation

Consider the original plant described by

119888

(119905) = 119860119888

119909119888

(119905) + 119861119888

119906119888

(119905) + 119872119888

1119908119888

(119905)

119911119888

(119905) = 119862119888

119909119888

(119905) + 119872119888

2119908119888

(119905)

(42)

where

119860119888

= [

0 1

minus6 minus5] 119861

119888

= [

0 1

1 12] 119872

119888

1= [

1

06]

119862119888

= [02 01] 119872119888

2= 001

(43)

If the fast discretization formulae follow119860 = 119890119860119888(120591119898) 119861 =

int

120591119898

0

119890119860119888120590

119889120590119861119888 1198721= int

120591119898

0

119890119860119888120590

119889120590119872119888

1 119862 = 119862

119888 and 1198722=

119872119888

2 then system (2) can be obtained The initial state is 119909

0=

[30 20]If 1198791=

1198792= 2 the input-holding mechanism can be

described as

119867(0) = [

1 0

0 0] 119867 (1) = [

0 0

0 1] (44)

If 1198791= 3 and 119879

2= 1 the output-sampling matrices are

assumed to be

119878 (0) = 1 119878 (1) = 0 119878 (2) = 0 (45)

Therefore this is a linear discrete asynchronous dual-ratesampled-data system [18] which applies a fast-input-slow-output mechanism for 119879 lt 119879

0 5 10 15 20 25 30

0

5

10

15

20

25

30

minus5

Real stateObserved state

Figure 4 Plant state 1199091(119896)

0

5

10

15

20

minus25

minus20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 5 Plant state 1199092(119896)

0001002003004005006007008

minus0010 5 10 15 20 25 30

Figure 6 Control signal 1199061(119896)

8 Mathematical Problems in Engineering

0

05

minus3

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 7 Control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 8 Output

For comparison we introduce a previous method [21]where the system is not lifted according to the pure LQGtheory and also let 119908

1= 1199082= 119908 be Gaussian white noise

amplitude is increased to 10 which is much bigger to thesystem The results obtained using this method are shown inFigures 4 5 6 7 and 8 where 120591 = 12 s

Next the synthesis method described in Theorem 4is applied to the same plant The simulations obtain thetransients of the system shown in Figures 9 10 11 12 and 13where 120574 = 02 119886 = 1 and 119887 = 1 while119898 = 1 (monorate)

A comparison of Figures 4ndash8 with Figures 9ndash13 showsclearly that the 119867

infinmethod performs better than LQG

the controlled states are consistent with the observed stateswith smaller ripples especially state 119909

2(119896) the biggest ripple

of which is almost half of the LQG method And theoutput of119867

infinmethod has better performance of disturbance

attenuation than LQG methodWhen 120591 = 06 s and 119898 = 1 (monorate) applying

Theorem 4 again we can get Figures 14 15 16 17 and 18Then the lifting number is modified to119898 = 2 (multirate) theresults shown in Figures 19 20 21 22 and 23 are obtained

Comparing Figures 14ndash18 with Figures 19ndash23 when thelifting amplitude increases from 1 to 2 the controlled statesare almost consistent with the observed states with smallerripples here the biggest ripple of state 119909

2(119896) is almost half of

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 9 Plant state 1199091(119896)

0

5

10

15

20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 10 Plant state 1199092(119896)

0

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 11 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

07

0 5 10 15 20 25 30

Figure 12 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 13 Output

Table 1 Minimum119867infinattenuation level

120591 119898 = 1 119898 = 2 119898 = 4

12 sec 00974 00959 0094206 sec 00904 00863 00856

the monorate And the output of the multirate case has betterperformance of disturbance attenuation than the monoratecase

Furthermore we change the lifting amplitude to 119898 = 4adjust the underlying period of system (2) to 120591 = 06 andcan draw the similar pictures here we omit them but only listtheir minimum119867

infinattenuation levels to compare them with

different cases see Table 1The inequality acquired using the theorem is almost

linear except in a few terms Although these terms are justthe inverses of their corresponding terms they could easily becalculated by LMI tool ofMatlab under a constraintThus it isfar easier to solve the controller and observer under the same119867infinconstraint comparing withmost of the existingmethods

which use highly complex formulae

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 14 Plant state 1199091(119896)

0

5

10

15

20

Real stateObserved state

minus5

minus30

minus25

minus20

minus15

minus10

0 5 10 15 20 25 30

Figure 15 Plant state 1199092(119896)

0

0 5 10 15 20 25 30

minus3

minus25

minus35

minus4

minus45

minus2

minus15

minus1

minus05

Figure 16 Plant control signal 1199061(119896)

10 Mathematical Problems in Engineering

0

02

04

06

08

1

12

14

0 5 10 15 20 25 30

Figure 17 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30minus1

Figure 18 Output

5 Conclusion

Motivated by a desire to provide a better solution for thedesign of a controller for a discrete-time asynchronous mul-tirate sampled-data system we developed synthesis methodsin this paper to improve performance and guarantee closed-loop transient behavior via observer-based 119867

infincontrol

Furthermore an extended lifting technique was adoptedto change the original system into a fast-input-slow-outputdiscrete system thereby improving its performance Finallywe presented an example that demonstrates the effectivenessof thesemethods in controlling a discrete-time asynchronousmultirate sampled-data system whose performance is thusimproved considerably

In this paper we focus on the standard 119867infin

control ofasynchronous multirate sampled-data system and the inputnoise discussed here is Gaussian white for simplicity As iswell known practical input signals often have finite frequency(FF) so it is reasonable to consider the 119867

infinperformance in

finite frequency range for sampled-data systems As in [27ndash31] where Kalman-Yakubovic-Popov (KYP) lemma is usedto establish the equivalence between a frequency domaininequality (FDI) and a linear matrix inequality it is a valuablework to extend the results in this paper to the FF frameworkbased on KYP lemma This will be the direction for futuredevelopment of this paper

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Real stateObserved state

minus10

minus5

Figure 19 Plant state 1199091(119896)

0

5

10

15

20

0 5 10 15 20 25 30

Real stateObserved state

minus20

minus15

minus10

minus5

Figure 20 Plant state 1199092(119896)

005

0 5 10 15 20 25 30

minus3

minus35

minus4

minus25

minus2

minus15

minus1

minus05

Figure 21 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 11

0

02

04

06

08

1

12

0 5 10 15 20 25 30minus02

Figure 22 Plant control signal 1199062(119896)

minus10 5 10 15 20 25 30

0

1

2

3

4

5

6

7

8

9

Figure 23 Output

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Zhejiang Provincial Natural Sci-ence Foundation of China under Grant no LR12F03002 andNational Natural Science Foundation of China (NSFC) underGrant no 61074045

References

[1] M F Sagfors H T Toivonen and B Lennartson ldquoState-spacesolution to the period multirate 119867

infincontrol problem a lifting

approachrdquo IEEE Transaction on Automatic Control vol 37 no5 pp 2345ndash2350 2000

[2] X Zhao Y Yao and JMa ldquoOverview ofmultirate sampled-datacontrol systemrdquo Journal of Harbin Institute of Technology vol 9no 4 pp 405ndash410 2002

[3] Y Fukada ldquoSlip-angle estimation for vehicle stability controlrdquoVehicle System Dynamics vol 32 no 4 pp 375ndash388 1999

[4] T Chen and L Qiu ldquo119867infindesign of general multirate sampled-

data control systemsrdquo Automatica vol 30 no 7 pp 1139ndash11521994

[5] Z Wang B Huang and H Unbehauen ldquoRobust 119867infin

observerdesign of linear state delayed systems with parametric uncer-tainty the discrete-time caserdquo Automatica vol 35 no 6 pp1161ndash1167 1999

[6] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Control Theory vol 15 ofSIAM Studies in Applied Mathematics SIAM Philadelphia PaUSA 1994

[7] S Patra and A Lanzon ldquoA closed-loop data based test forrobust performance improvement in iterative identification andcontrol redesignsrdquo Automatica vol 48 no 10 pp 2710ndash27162012

[8] J Ding Y Shi H Wang and F Ding ldquoA modified stochasticgradient based parameter estimation algorithm for dual-ratesampled-data systemsrdquo Digital Signal Processing vol 20 no 4pp 1238ndash1247 2010

[9] T W Chen and B Francis Optimal Sampled-Data ControlSystems Springer London UK 1995

[10] L Shen and M J Er ldquoMultiobjectives design of a multirateoutput controllerrdquo in Proceedings of the IEEE InternationalConference on Control Applications (CCA rsquo99) pp 193ndash198August 1999

[11] M Chilali and P Gahinet ldquo119867infin

design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996

[12] S Mo X Chen J Zhao J Qian and Z Shao ldquoA two-stagemethod for identification of dual-rate systems with fast inputand very slow outputrdquo Industrial and Engineering ChemistryResearch vol 48 no 4 pp 1980ndash1988 2009

[13] M De la Sen and S Alonso-Quesada ldquoModel matching viamultirate sampling with fast sampled input guaranteeing thestability of the plant zeros extensions to adaptive controlrdquo IETControl Theory amp Applications vol 1 no 1 pp 210ndash225 2007

[14] M F Sagfors and H T Toivonen ldquo119867infin

and LQG control ofasynchronous sampled-data systemsrdquo Automatica vol 33 no9 pp 1663ndash1668 1997

[15] S Lopez-lopez Optimal Hinfin

Design of Causal Multirate Con-trollers and Filters Mechanical and Aerospace EngineeringUniversity of California Irvine Calif USA 2010

[16] U Borison ldquoSelf-tuning regulators for a class of multivariablesystemsrdquo Automatica vol 15 no 2 pp 209ndash215 1979

[17] V S Ritchey and G F Franklin ldquoA stability criterion forasynchronous multirate linear systemsrdquo IEEE Transactions onAutomatic Control vol 34 no 5 pp 529ndash535 1989

[18] B Shen Z Wang and X Liu ldquoSampled-data synchronizationcontrol of dynamical networks with stochastic samplingrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2644ndash2650 2012

[19] S Longo G Herrmann and P Barber ldquoRobust scheduling ofsampled-data networked control systemsrdquo IEEE Transactionson Control Systems Technology vol 20 no 6 pp 1613ndash1621 2011

[20] P Voulgaris ldquoControl of asynchronous sampled data systemsrdquoIEEE Transactions on Automatic Control vol 39 no 7 pp 1451ndash1455 1994

[21] P Colaneri R Scattolini and N Schiavoni ldquoLQG optimalcontrol of multirate sampled-data systemsrdquo IEEE Transactionson Automatic Control vol 37 no 5 pp 675ndash682 1992

[22] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

12 Mathematical Problems in Engineering

[23] A Seuret ldquoStability analysis of networked control systems withasynchronous sampling and input delayrdquo in Proceedings of theAmerican Control Conference (ACC rsquo11) pp 533ndash538 July 2011

[24] B Tavassoli ldquoStability of nonlinear networked control systemsover multiple communication links with asynchronous sam-plingrdquo IEEE Transaction on Automatic Control 2013

[25] L Yang and J M Li ldquoSufficient and necessary conditions ofcontrollability and observability of a class of linear switchingsystemrdquo Systems Engineering and Electronics vol 25 no 5 pp588ndash590 2003

[26] W-J Mao ldquoObserver-based energy decoupling of linear time-delay systemsrdquo Journal of Zhejiang University vol 37 no 5 pp499ndash503 2003

[27] T Iwasaki and S Hara ldquoGeneralized KYP lemma unifiedfrequency domain inequalities with design applicationsrdquo IEEETransactions on Automatic Control vol 50 no 1 pp 41ndash592005

[28] H Gao and X Li ldquo119867infinfiltering for discrete-time state-delayed

systems with finite frequency specificationsrdquo IEEE Transactionson Automatic Control vol 56 no 12 pp 2935ndash2941 2011

[29] X Li H Gao and CWang ldquoGeneralized Kalman-Yakubovich-Popov lemma for 2-D FM LSS modelrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3090ndash3103 2012

[30] X Li and H Gao ldquoRobust finite frequency 119867infin

filtering foruncertain 2-D systems the FM model caserdquo Automatica vol49 no 8 pp 2446ndash2452 2013

[31] X W Li and H J Gao ldquoA heuristic approach to static output-feedback controller synthesis with restricted frequency-domainspecificationsrdquo IEEE Transaction on Automatic Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0

05

minus3

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 7 Control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 8 Output

For comparison we introduce a previous method [21]where the system is not lifted according to the pure LQGtheory and also let 119908

1= 1199082= 119908 be Gaussian white noise

amplitude is increased to 10 which is much bigger to thesystem The results obtained using this method are shown inFigures 4 5 6 7 and 8 where 120591 = 12 s

Next the synthesis method described in Theorem 4is applied to the same plant The simulations obtain thetransients of the system shown in Figures 9 10 11 12 and 13where 120574 = 02 119886 = 1 and 119887 = 1 while119898 = 1 (monorate)

A comparison of Figures 4ndash8 with Figures 9ndash13 showsclearly that the 119867

infinmethod performs better than LQG

the controlled states are consistent with the observed stateswith smaller ripples especially state 119909

2(119896) the biggest ripple

of which is almost half of the LQG method And theoutput of119867

infinmethod has better performance of disturbance

attenuation than LQG methodWhen 120591 = 06 s and 119898 = 1 (monorate) applying

Theorem 4 again we can get Figures 14 15 16 17 and 18Then the lifting number is modified to119898 = 2 (multirate) theresults shown in Figures 19 20 21 22 and 23 are obtained

Comparing Figures 14ndash18 with Figures 19ndash23 when thelifting amplitude increases from 1 to 2 the controlled statesare almost consistent with the observed states with smallerripples here the biggest ripple of state 119909

2(119896) is almost half of

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 9 Plant state 1199091(119896)

0

5

10

15

20

minus15

minus10

minus5

0 5 10 15 20 25 30

Real stateObserved state

Figure 10 Plant state 1199092(119896)

0

minus25

minus2

minus15

minus1

minus05

0 5 10 15 20 25 30

Figure 11 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

07

0 5 10 15 20 25 30

Figure 12 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 13 Output

Table 1 Minimum119867infinattenuation level

120591 119898 = 1 119898 = 2 119898 = 4

12 sec 00974 00959 0094206 sec 00904 00863 00856

the monorate And the output of the multirate case has betterperformance of disturbance attenuation than the monoratecase

Furthermore we change the lifting amplitude to 119898 = 4adjust the underlying period of system (2) to 120591 = 06 andcan draw the similar pictures here we omit them but only listtheir minimum119867

infinattenuation levels to compare them with

different cases see Table 1The inequality acquired using the theorem is almost

linear except in a few terms Although these terms are justthe inverses of their corresponding terms they could easily becalculated by LMI tool ofMatlab under a constraintThus it isfar easier to solve the controller and observer under the same119867infinconstraint comparing withmost of the existingmethods

which use highly complex formulae

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 14 Plant state 1199091(119896)

0

5

10

15

20

Real stateObserved state

minus5

minus30

minus25

minus20

minus15

minus10

0 5 10 15 20 25 30

Figure 15 Plant state 1199092(119896)

0

0 5 10 15 20 25 30

minus3

minus25

minus35

minus4

minus45

minus2

minus15

minus1

minus05

Figure 16 Plant control signal 1199061(119896)

10 Mathematical Problems in Engineering

0

02

04

06

08

1

12

14

0 5 10 15 20 25 30

Figure 17 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30minus1

Figure 18 Output

5 Conclusion

Motivated by a desire to provide a better solution for thedesign of a controller for a discrete-time asynchronous mul-tirate sampled-data system we developed synthesis methodsin this paper to improve performance and guarantee closed-loop transient behavior via observer-based 119867

infincontrol

Furthermore an extended lifting technique was adoptedto change the original system into a fast-input-slow-outputdiscrete system thereby improving its performance Finallywe presented an example that demonstrates the effectivenessof thesemethods in controlling a discrete-time asynchronousmultirate sampled-data system whose performance is thusimproved considerably

In this paper we focus on the standard 119867infin

control ofasynchronous multirate sampled-data system and the inputnoise discussed here is Gaussian white for simplicity As iswell known practical input signals often have finite frequency(FF) so it is reasonable to consider the 119867

infinperformance in

finite frequency range for sampled-data systems As in [27ndash31] where Kalman-Yakubovic-Popov (KYP) lemma is usedto establish the equivalence between a frequency domaininequality (FDI) and a linear matrix inequality it is a valuablework to extend the results in this paper to the FF frameworkbased on KYP lemma This will be the direction for futuredevelopment of this paper

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Real stateObserved state

minus10

minus5

Figure 19 Plant state 1199091(119896)

0

5

10

15

20

0 5 10 15 20 25 30

Real stateObserved state

minus20

minus15

minus10

minus5

Figure 20 Plant state 1199092(119896)

005

0 5 10 15 20 25 30

minus3

minus35

minus4

minus25

minus2

minus15

minus1

minus05

Figure 21 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 11

0

02

04

06

08

1

12

0 5 10 15 20 25 30minus02

Figure 22 Plant control signal 1199062(119896)

minus10 5 10 15 20 25 30

0

1

2

3

4

5

6

7

8

9

Figure 23 Output

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Zhejiang Provincial Natural Sci-ence Foundation of China under Grant no LR12F03002 andNational Natural Science Foundation of China (NSFC) underGrant no 61074045

References

[1] M F Sagfors H T Toivonen and B Lennartson ldquoState-spacesolution to the period multirate 119867

infincontrol problem a lifting

approachrdquo IEEE Transaction on Automatic Control vol 37 no5 pp 2345ndash2350 2000

[2] X Zhao Y Yao and JMa ldquoOverview ofmultirate sampled-datacontrol systemrdquo Journal of Harbin Institute of Technology vol 9no 4 pp 405ndash410 2002

[3] Y Fukada ldquoSlip-angle estimation for vehicle stability controlrdquoVehicle System Dynamics vol 32 no 4 pp 375ndash388 1999

[4] T Chen and L Qiu ldquo119867infindesign of general multirate sampled-

data control systemsrdquo Automatica vol 30 no 7 pp 1139ndash11521994

[5] Z Wang B Huang and H Unbehauen ldquoRobust 119867infin

observerdesign of linear state delayed systems with parametric uncer-tainty the discrete-time caserdquo Automatica vol 35 no 6 pp1161ndash1167 1999

[6] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Control Theory vol 15 ofSIAM Studies in Applied Mathematics SIAM Philadelphia PaUSA 1994

[7] S Patra and A Lanzon ldquoA closed-loop data based test forrobust performance improvement in iterative identification andcontrol redesignsrdquo Automatica vol 48 no 10 pp 2710ndash27162012

[8] J Ding Y Shi H Wang and F Ding ldquoA modified stochasticgradient based parameter estimation algorithm for dual-ratesampled-data systemsrdquo Digital Signal Processing vol 20 no 4pp 1238ndash1247 2010

[9] T W Chen and B Francis Optimal Sampled-Data ControlSystems Springer London UK 1995

[10] L Shen and M J Er ldquoMultiobjectives design of a multirateoutput controllerrdquo in Proceedings of the IEEE InternationalConference on Control Applications (CCA rsquo99) pp 193ndash198August 1999

[11] M Chilali and P Gahinet ldquo119867infin

design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996

[12] S Mo X Chen J Zhao J Qian and Z Shao ldquoA two-stagemethod for identification of dual-rate systems with fast inputand very slow outputrdquo Industrial and Engineering ChemistryResearch vol 48 no 4 pp 1980ndash1988 2009

[13] M De la Sen and S Alonso-Quesada ldquoModel matching viamultirate sampling with fast sampled input guaranteeing thestability of the plant zeros extensions to adaptive controlrdquo IETControl Theory amp Applications vol 1 no 1 pp 210ndash225 2007

[14] M F Sagfors and H T Toivonen ldquo119867infin

and LQG control ofasynchronous sampled-data systemsrdquo Automatica vol 33 no9 pp 1663ndash1668 1997

[15] S Lopez-lopez Optimal Hinfin

Design of Causal Multirate Con-trollers and Filters Mechanical and Aerospace EngineeringUniversity of California Irvine Calif USA 2010

[16] U Borison ldquoSelf-tuning regulators for a class of multivariablesystemsrdquo Automatica vol 15 no 2 pp 209ndash215 1979

[17] V S Ritchey and G F Franklin ldquoA stability criterion forasynchronous multirate linear systemsrdquo IEEE Transactions onAutomatic Control vol 34 no 5 pp 529ndash535 1989

[18] B Shen Z Wang and X Liu ldquoSampled-data synchronizationcontrol of dynamical networks with stochastic samplingrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2644ndash2650 2012

[19] S Longo G Herrmann and P Barber ldquoRobust scheduling ofsampled-data networked control systemsrdquo IEEE Transactionson Control Systems Technology vol 20 no 6 pp 1613ndash1621 2011

[20] P Voulgaris ldquoControl of asynchronous sampled data systemsrdquoIEEE Transactions on Automatic Control vol 39 no 7 pp 1451ndash1455 1994

[21] P Colaneri R Scattolini and N Schiavoni ldquoLQG optimalcontrol of multirate sampled-data systemsrdquo IEEE Transactionson Automatic Control vol 37 no 5 pp 675ndash682 1992

[22] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

12 Mathematical Problems in Engineering

[23] A Seuret ldquoStability analysis of networked control systems withasynchronous sampling and input delayrdquo in Proceedings of theAmerican Control Conference (ACC rsquo11) pp 533ndash538 July 2011

[24] B Tavassoli ldquoStability of nonlinear networked control systemsover multiple communication links with asynchronous sam-plingrdquo IEEE Transaction on Automatic Control 2013

[25] L Yang and J M Li ldquoSufficient and necessary conditions ofcontrollability and observability of a class of linear switchingsystemrdquo Systems Engineering and Electronics vol 25 no 5 pp588ndash590 2003

[26] W-J Mao ldquoObserver-based energy decoupling of linear time-delay systemsrdquo Journal of Zhejiang University vol 37 no 5 pp499ndash503 2003

[27] T Iwasaki and S Hara ldquoGeneralized KYP lemma unifiedfrequency domain inequalities with design applicationsrdquo IEEETransactions on Automatic Control vol 50 no 1 pp 41ndash592005

[28] H Gao and X Li ldquo119867infinfiltering for discrete-time state-delayed

systems with finite frequency specificationsrdquo IEEE Transactionson Automatic Control vol 56 no 12 pp 2935ndash2941 2011

[29] X Li H Gao and CWang ldquoGeneralized Kalman-Yakubovich-Popov lemma for 2-D FM LSS modelrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3090ndash3103 2012

[30] X Li and H Gao ldquoRobust finite frequency 119867infin

filtering foruncertain 2-D systems the FM model caserdquo Automatica vol49 no 8 pp 2446ndash2452 2013

[31] X W Li and H J Gao ldquoA heuristic approach to static output-feedback controller synthesis with restricted frequency-domainspecificationsrdquo IEEE Transaction on Automatic Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

07

0 5 10 15 20 25 30

Figure 12 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

minus10 5 10 15 20 25 30

Figure 13 Output

Table 1 Minimum119867infinattenuation level

120591 119898 = 1 119898 = 2 119898 = 4

12 sec 00974 00959 0094206 sec 00904 00863 00856

the monorate And the output of the multirate case has betterperformance of disturbance attenuation than the monoratecase

Furthermore we change the lifting amplitude to 119898 = 4adjust the underlying period of system (2) to 120591 = 06 andcan draw the similar pictures here we omit them but only listtheir minimum119867

infinattenuation levels to compare them with

different cases see Table 1The inequality acquired using the theorem is almost

linear except in a few terms Although these terms are justthe inverses of their corresponding terms they could easily becalculated by LMI tool ofMatlab under a constraintThus it isfar easier to solve the controller and observer under the same119867infinconstraint comparing withmost of the existingmethods

which use highly complex formulae

0

5

10

15

20

25

30

0 5 10 15 20 25 30minus5

Real stateObserved state

Figure 14 Plant state 1199091(119896)

0

5

10

15

20

Real stateObserved state

minus5

minus30

minus25

minus20

minus15

minus10

0 5 10 15 20 25 30

Figure 15 Plant state 1199092(119896)

0

0 5 10 15 20 25 30

minus3

minus25

minus35

minus4

minus45

minus2

minus15

minus1

minus05

Figure 16 Plant control signal 1199061(119896)

10 Mathematical Problems in Engineering

0

02

04

06

08

1

12

14

0 5 10 15 20 25 30

Figure 17 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30minus1

Figure 18 Output

5 Conclusion

Motivated by a desire to provide a better solution for thedesign of a controller for a discrete-time asynchronous mul-tirate sampled-data system we developed synthesis methodsin this paper to improve performance and guarantee closed-loop transient behavior via observer-based 119867

infincontrol

Furthermore an extended lifting technique was adoptedto change the original system into a fast-input-slow-outputdiscrete system thereby improving its performance Finallywe presented an example that demonstrates the effectivenessof thesemethods in controlling a discrete-time asynchronousmultirate sampled-data system whose performance is thusimproved considerably

In this paper we focus on the standard 119867infin

control ofasynchronous multirate sampled-data system and the inputnoise discussed here is Gaussian white for simplicity As iswell known practical input signals often have finite frequency(FF) so it is reasonable to consider the 119867

infinperformance in

finite frequency range for sampled-data systems As in [27ndash31] where Kalman-Yakubovic-Popov (KYP) lemma is usedto establish the equivalence between a frequency domaininequality (FDI) and a linear matrix inequality it is a valuablework to extend the results in this paper to the FF frameworkbased on KYP lemma This will be the direction for futuredevelopment of this paper

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Real stateObserved state

minus10

minus5

Figure 19 Plant state 1199091(119896)

0

5

10

15

20

0 5 10 15 20 25 30

Real stateObserved state

minus20

minus15

minus10

minus5

Figure 20 Plant state 1199092(119896)

005

0 5 10 15 20 25 30

minus3

minus35

minus4

minus25

minus2

minus15

minus1

minus05

Figure 21 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 11

0

02

04

06

08

1

12

0 5 10 15 20 25 30minus02

Figure 22 Plant control signal 1199062(119896)

minus10 5 10 15 20 25 30

0

1

2

3

4

5

6

7

8

9

Figure 23 Output

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Zhejiang Provincial Natural Sci-ence Foundation of China under Grant no LR12F03002 andNational Natural Science Foundation of China (NSFC) underGrant no 61074045

References

[1] M F Sagfors H T Toivonen and B Lennartson ldquoState-spacesolution to the period multirate 119867

infincontrol problem a lifting

approachrdquo IEEE Transaction on Automatic Control vol 37 no5 pp 2345ndash2350 2000

[2] X Zhao Y Yao and JMa ldquoOverview ofmultirate sampled-datacontrol systemrdquo Journal of Harbin Institute of Technology vol 9no 4 pp 405ndash410 2002

[3] Y Fukada ldquoSlip-angle estimation for vehicle stability controlrdquoVehicle System Dynamics vol 32 no 4 pp 375ndash388 1999

[4] T Chen and L Qiu ldquo119867infindesign of general multirate sampled-

data control systemsrdquo Automatica vol 30 no 7 pp 1139ndash11521994

[5] Z Wang B Huang and H Unbehauen ldquoRobust 119867infin

observerdesign of linear state delayed systems with parametric uncer-tainty the discrete-time caserdquo Automatica vol 35 no 6 pp1161ndash1167 1999

[6] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Control Theory vol 15 ofSIAM Studies in Applied Mathematics SIAM Philadelphia PaUSA 1994

[7] S Patra and A Lanzon ldquoA closed-loop data based test forrobust performance improvement in iterative identification andcontrol redesignsrdquo Automatica vol 48 no 10 pp 2710ndash27162012

[8] J Ding Y Shi H Wang and F Ding ldquoA modified stochasticgradient based parameter estimation algorithm for dual-ratesampled-data systemsrdquo Digital Signal Processing vol 20 no 4pp 1238ndash1247 2010

[9] T W Chen and B Francis Optimal Sampled-Data ControlSystems Springer London UK 1995

[10] L Shen and M J Er ldquoMultiobjectives design of a multirateoutput controllerrdquo in Proceedings of the IEEE InternationalConference on Control Applications (CCA rsquo99) pp 193ndash198August 1999

[11] M Chilali and P Gahinet ldquo119867infin

design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996

[12] S Mo X Chen J Zhao J Qian and Z Shao ldquoA two-stagemethod for identification of dual-rate systems with fast inputand very slow outputrdquo Industrial and Engineering ChemistryResearch vol 48 no 4 pp 1980ndash1988 2009

[13] M De la Sen and S Alonso-Quesada ldquoModel matching viamultirate sampling with fast sampled input guaranteeing thestability of the plant zeros extensions to adaptive controlrdquo IETControl Theory amp Applications vol 1 no 1 pp 210ndash225 2007

[14] M F Sagfors and H T Toivonen ldquo119867infin

and LQG control ofasynchronous sampled-data systemsrdquo Automatica vol 33 no9 pp 1663ndash1668 1997

[15] S Lopez-lopez Optimal Hinfin

Design of Causal Multirate Con-trollers and Filters Mechanical and Aerospace EngineeringUniversity of California Irvine Calif USA 2010

[16] U Borison ldquoSelf-tuning regulators for a class of multivariablesystemsrdquo Automatica vol 15 no 2 pp 209ndash215 1979

[17] V S Ritchey and G F Franklin ldquoA stability criterion forasynchronous multirate linear systemsrdquo IEEE Transactions onAutomatic Control vol 34 no 5 pp 529ndash535 1989

[18] B Shen Z Wang and X Liu ldquoSampled-data synchronizationcontrol of dynamical networks with stochastic samplingrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2644ndash2650 2012

[19] S Longo G Herrmann and P Barber ldquoRobust scheduling ofsampled-data networked control systemsrdquo IEEE Transactionson Control Systems Technology vol 20 no 6 pp 1613ndash1621 2011

[20] P Voulgaris ldquoControl of asynchronous sampled data systemsrdquoIEEE Transactions on Automatic Control vol 39 no 7 pp 1451ndash1455 1994

[21] P Colaneri R Scattolini and N Schiavoni ldquoLQG optimalcontrol of multirate sampled-data systemsrdquo IEEE Transactionson Automatic Control vol 37 no 5 pp 675ndash682 1992

[22] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

12 Mathematical Problems in Engineering

[23] A Seuret ldquoStability analysis of networked control systems withasynchronous sampling and input delayrdquo in Proceedings of theAmerican Control Conference (ACC rsquo11) pp 533ndash538 July 2011

[24] B Tavassoli ldquoStability of nonlinear networked control systemsover multiple communication links with asynchronous sam-plingrdquo IEEE Transaction on Automatic Control 2013

[25] L Yang and J M Li ldquoSufficient and necessary conditions ofcontrollability and observability of a class of linear switchingsystemrdquo Systems Engineering and Electronics vol 25 no 5 pp588ndash590 2003

[26] W-J Mao ldquoObserver-based energy decoupling of linear time-delay systemsrdquo Journal of Zhejiang University vol 37 no 5 pp499ndash503 2003

[27] T Iwasaki and S Hara ldquoGeneralized KYP lemma unifiedfrequency domain inequalities with design applicationsrdquo IEEETransactions on Automatic Control vol 50 no 1 pp 41ndash592005

[28] H Gao and X Li ldquo119867infinfiltering for discrete-time state-delayed

systems with finite frequency specificationsrdquo IEEE Transactionson Automatic Control vol 56 no 12 pp 2935ndash2941 2011

[29] X Li H Gao and CWang ldquoGeneralized Kalman-Yakubovich-Popov lemma for 2-D FM LSS modelrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3090ndash3103 2012

[30] X Li and H Gao ldquoRobust finite frequency 119867infin

filtering foruncertain 2-D systems the FM model caserdquo Automatica vol49 no 8 pp 2446ndash2452 2013

[31] X W Li and H J Gao ldquoA heuristic approach to static output-feedback controller synthesis with restricted frequency-domainspecificationsrdquo IEEE Transaction on Automatic Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0

02

04

06

08

1

12

14

0 5 10 15 20 25 30

Figure 17 Plant control signal 1199062(119896)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30minus1

Figure 18 Output

5 Conclusion

Motivated by a desire to provide a better solution for thedesign of a controller for a discrete-time asynchronous mul-tirate sampled-data system we developed synthesis methodsin this paper to improve performance and guarantee closed-loop transient behavior via observer-based 119867

infincontrol

Furthermore an extended lifting technique was adoptedto change the original system into a fast-input-slow-outputdiscrete system thereby improving its performance Finallywe presented an example that demonstrates the effectivenessof thesemethods in controlling a discrete-time asynchronousmultirate sampled-data system whose performance is thusimproved considerably

In this paper we focus on the standard 119867infin

control ofasynchronous multirate sampled-data system and the inputnoise discussed here is Gaussian white for simplicity As iswell known practical input signals often have finite frequency(FF) so it is reasonable to consider the 119867

infinperformance in

finite frequency range for sampled-data systems As in [27ndash31] where Kalman-Yakubovic-Popov (KYP) lemma is usedto establish the equivalence between a frequency domaininequality (FDI) and a linear matrix inequality it is a valuablework to extend the results in this paper to the FF frameworkbased on KYP lemma This will be the direction for futuredevelopment of this paper

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Real stateObserved state

minus10

minus5

Figure 19 Plant state 1199091(119896)

0

5

10

15

20

0 5 10 15 20 25 30

Real stateObserved state

minus20

minus15

minus10

minus5

Figure 20 Plant state 1199092(119896)

005

0 5 10 15 20 25 30

minus3

minus35

minus4

minus25

minus2

minus15

minus1

minus05

Figure 21 Plant control signal 1199061(119896)

Mathematical Problems in Engineering 11

0

02

04

06

08

1

12

0 5 10 15 20 25 30minus02

Figure 22 Plant control signal 1199062(119896)

minus10 5 10 15 20 25 30

0

1

2

3

4

5

6

7

8

9

Figure 23 Output

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Zhejiang Provincial Natural Sci-ence Foundation of China under Grant no LR12F03002 andNational Natural Science Foundation of China (NSFC) underGrant no 61074045

References

[1] M F Sagfors H T Toivonen and B Lennartson ldquoState-spacesolution to the period multirate 119867

infincontrol problem a lifting

approachrdquo IEEE Transaction on Automatic Control vol 37 no5 pp 2345ndash2350 2000

[2] X Zhao Y Yao and JMa ldquoOverview ofmultirate sampled-datacontrol systemrdquo Journal of Harbin Institute of Technology vol 9no 4 pp 405ndash410 2002

[3] Y Fukada ldquoSlip-angle estimation for vehicle stability controlrdquoVehicle System Dynamics vol 32 no 4 pp 375ndash388 1999

[4] T Chen and L Qiu ldquo119867infindesign of general multirate sampled-

data control systemsrdquo Automatica vol 30 no 7 pp 1139ndash11521994

[5] Z Wang B Huang and H Unbehauen ldquoRobust 119867infin

observerdesign of linear state delayed systems with parametric uncer-tainty the discrete-time caserdquo Automatica vol 35 no 6 pp1161ndash1167 1999

[6] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Control Theory vol 15 ofSIAM Studies in Applied Mathematics SIAM Philadelphia PaUSA 1994

[7] S Patra and A Lanzon ldquoA closed-loop data based test forrobust performance improvement in iterative identification andcontrol redesignsrdquo Automatica vol 48 no 10 pp 2710ndash27162012

[8] J Ding Y Shi H Wang and F Ding ldquoA modified stochasticgradient based parameter estimation algorithm for dual-ratesampled-data systemsrdquo Digital Signal Processing vol 20 no 4pp 1238ndash1247 2010

[9] T W Chen and B Francis Optimal Sampled-Data ControlSystems Springer London UK 1995

[10] L Shen and M J Er ldquoMultiobjectives design of a multirateoutput controllerrdquo in Proceedings of the IEEE InternationalConference on Control Applications (CCA rsquo99) pp 193ndash198August 1999

[11] M Chilali and P Gahinet ldquo119867infin

design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996

[12] S Mo X Chen J Zhao J Qian and Z Shao ldquoA two-stagemethod for identification of dual-rate systems with fast inputand very slow outputrdquo Industrial and Engineering ChemistryResearch vol 48 no 4 pp 1980ndash1988 2009

[13] M De la Sen and S Alonso-Quesada ldquoModel matching viamultirate sampling with fast sampled input guaranteeing thestability of the plant zeros extensions to adaptive controlrdquo IETControl Theory amp Applications vol 1 no 1 pp 210ndash225 2007

[14] M F Sagfors and H T Toivonen ldquo119867infin

and LQG control ofasynchronous sampled-data systemsrdquo Automatica vol 33 no9 pp 1663ndash1668 1997

[15] S Lopez-lopez Optimal Hinfin

Design of Causal Multirate Con-trollers and Filters Mechanical and Aerospace EngineeringUniversity of California Irvine Calif USA 2010

[16] U Borison ldquoSelf-tuning regulators for a class of multivariablesystemsrdquo Automatica vol 15 no 2 pp 209ndash215 1979

[17] V S Ritchey and G F Franklin ldquoA stability criterion forasynchronous multirate linear systemsrdquo IEEE Transactions onAutomatic Control vol 34 no 5 pp 529ndash535 1989

[18] B Shen Z Wang and X Liu ldquoSampled-data synchronizationcontrol of dynamical networks with stochastic samplingrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2644ndash2650 2012

[19] S Longo G Herrmann and P Barber ldquoRobust scheduling ofsampled-data networked control systemsrdquo IEEE Transactionson Control Systems Technology vol 20 no 6 pp 1613ndash1621 2011

[20] P Voulgaris ldquoControl of asynchronous sampled data systemsrdquoIEEE Transactions on Automatic Control vol 39 no 7 pp 1451ndash1455 1994

[21] P Colaneri R Scattolini and N Schiavoni ldquoLQG optimalcontrol of multirate sampled-data systemsrdquo IEEE Transactionson Automatic Control vol 37 no 5 pp 675ndash682 1992

[22] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

12 Mathematical Problems in Engineering

[23] A Seuret ldquoStability analysis of networked control systems withasynchronous sampling and input delayrdquo in Proceedings of theAmerican Control Conference (ACC rsquo11) pp 533ndash538 July 2011

[24] B Tavassoli ldquoStability of nonlinear networked control systemsover multiple communication links with asynchronous sam-plingrdquo IEEE Transaction on Automatic Control 2013

[25] L Yang and J M Li ldquoSufficient and necessary conditions ofcontrollability and observability of a class of linear switchingsystemrdquo Systems Engineering and Electronics vol 25 no 5 pp588ndash590 2003

[26] W-J Mao ldquoObserver-based energy decoupling of linear time-delay systemsrdquo Journal of Zhejiang University vol 37 no 5 pp499ndash503 2003

[27] T Iwasaki and S Hara ldquoGeneralized KYP lemma unifiedfrequency domain inequalities with design applicationsrdquo IEEETransactions on Automatic Control vol 50 no 1 pp 41ndash592005

[28] H Gao and X Li ldquo119867infinfiltering for discrete-time state-delayed

systems with finite frequency specificationsrdquo IEEE Transactionson Automatic Control vol 56 no 12 pp 2935ndash2941 2011

[29] X Li H Gao and CWang ldquoGeneralized Kalman-Yakubovich-Popov lemma for 2-D FM LSS modelrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3090ndash3103 2012

[30] X Li and H Gao ldquoRobust finite frequency 119867infin

filtering foruncertain 2-D systems the FM model caserdquo Automatica vol49 no 8 pp 2446ndash2452 2013

[31] X W Li and H J Gao ldquoA heuristic approach to static output-feedback controller synthesis with restricted frequency-domainspecificationsrdquo IEEE Transaction on Automatic Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0

02

04

06

08

1

12

0 5 10 15 20 25 30minus02

Figure 22 Plant control signal 1199062(119896)

minus10 5 10 15 20 25 30

0

1

2

3

4

5

6

7

8

9

Figure 23 Output

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Zhejiang Provincial Natural Sci-ence Foundation of China under Grant no LR12F03002 andNational Natural Science Foundation of China (NSFC) underGrant no 61074045

References

[1] M F Sagfors H T Toivonen and B Lennartson ldquoState-spacesolution to the period multirate 119867

infincontrol problem a lifting

approachrdquo IEEE Transaction on Automatic Control vol 37 no5 pp 2345ndash2350 2000

[2] X Zhao Y Yao and JMa ldquoOverview ofmultirate sampled-datacontrol systemrdquo Journal of Harbin Institute of Technology vol 9no 4 pp 405ndash410 2002

[3] Y Fukada ldquoSlip-angle estimation for vehicle stability controlrdquoVehicle System Dynamics vol 32 no 4 pp 375ndash388 1999

[4] T Chen and L Qiu ldquo119867infindesign of general multirate sampled-

data control systemsrdquo Automatica vol 30 no 7 pp 1139ndash11521994

[5] Z Wang B Huang and H Unbehauen ldquoRobust 119867infin

observerdesign of linear state delayed systems with parametric uncer-tainty the discrete-time caserdquo Automatica vol 35 no 6 pp1161ndash1167 1999

[6] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Control Theory vol 15 ofSIAM Studies in Applied Mathematics SIAM Philadelphia PaUSA 1994

[7] S Patra and A Lanzon ldquoA closed-loop data based test forrobust performance improvement in iterative identification andcontrol redesignsrdquo Automatica vol 48 no 10 pp 2710ndash27162012

[8] J Ding Y Shi H Wang and F Ding ldquoA modified stochasticgradient based parameter estimation algorithm for dual-ratesampled-data systemsrdquo Digital Signal Processing vol 20 no 4pp 1238ndash1247 2010

[9] T W Chen and B Francis Optimal Sampled-Data ControlSystems Springer London UK 1995

[10] L Shen and M J Er ldquoMultiobjectives design of a multirateoutput controllerrdquo in Proceedings of the IEEE InternationalConference on Control Applications (CCA rsquo99) pp 193ndash198August 1999

[11] M Chilali and P Gahinet ldquo119867infin

design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996

[12] S Mo X Chen J Zhao J Qian and Z Shao ldquoA two-stagemethod for identification of dual-rate systems with fast inputand very slow outputrdquo Industrial and Engineering ChemistryResearch vol 48 no 4 pp 1980ndash1988 2009

[13] M De la Sen and S Alonso-Quesada ldquoModel matching viamultirate sampling with fast sampled input guaranteeing thestability of the plant zeros extensions to adaptive controlrdquo IETControl Theory amp Applications vol 1 no 1 pp 210ndash225 2007

[14] M F Sagfors and H T Toivonen ldquo119867infin

and LQG control ofasynchronous sampled-data systemsrdquo Automatica vol 33 no9 pp 1663ndash1668 1997

[15] S Lopez-lopez Optimal Hinfin

Design of Causal Multirate Con-trollers and Filters Mechanical and Aerospace EngineeringUniversity of California Irvine Calif USA 2010

[16] U Borison ldquoSelf-tuning regulators for a class of multivariablesystemsrdquo Automatica vol 15 no 2 pp 209ndash215 1979

[17] V S Ritchey and G F Franklin ldquoA stability criterion forasynchronous multirate linear systemsrdquo IEEE Transactions onAutomatic Control vol 34 no 5 pp 529ndash535 1989

[18] B Shen Z Wang and X Liu ldquoSampled-data synchronizationcontrol of dynamical networks with stochastic samplingrdquo IEEETransactions on Automatic Control vol 57 no 10 pp 2644ndash2650 2012

[19] S Longo G Herrmann and P Barber ldquoRobust scheduling ofsampled-data networked control systemsrdquo IEEE Transactionson Control Systems Technology vol 20 no 6 pp 1613ndash1621 2011

[20] P Voulgaris ldquoControl of asynchronous sampled data systemsrdquoIEEE Transactions on Automatic Control vol 39 no 7 pp 1451ndash1455 1994

[21] P Colaneri R Scattolini and N Schiavoni ldquoLQG optimalcontrol of multirate sampled-data systemsrdquo IEEE Transactionson Automatic Control vol 37 no 5 pp 675ndash682 1992

[22] L El Ghaoui F Oustry and M AitRami ldquoA cone complemen-tarity linearization algorithm for static output-feedback andrelated problemsrdquo IEEE Transactions on Automatic Control vol42 no 8 pp 1171ndash1176 1997

12 Mathematical Problems in Engineering

[23] A Seuret ldquoStability analysis of networked control systems withasynchronous sampling and input delayrdquo in Proceedings of theAmerican Control Conference (ACC rsquo11) pp 533ndash538 July 2011

[24] B Tavassoli ldquoStability of nonlinear networked control systemsover multiple communication links with asynchronous sam-plingrdquo IEEE Transaction on Automatic Control 2013

[25] L Yang and J M Li ldquoSufficient and necessary conditions ofcontrollability and observability of a class of linear switchingsystemrdquo Systems Engineering and Electronics vol 25 no 5 pp588ndash590 2003

[26] W-J Mao ldquoObserver-based energy decoupling of linear time-delay systemsrdquo Journal of Zhejiang University vol 37 no 5 pp499ndash503 2003

[27] T Iwasaki and S Hara ldquoGeneralized KYP lemma unifiedfrequency domain inequalities with design applicationsrdquo IEEETransactions on Automatic Control vol 50 no 1 pp 41ndash592005

[28] H Gao and X Li ldquo119867infinfiltering for discrete-time state-delayed

systems with finite frequency specificationsrdquo IEEE Transactionson Automatic Control vol 56 no 12 pp 2935ndash2941 2011

[29] X Li H Gao and CWang ldquoGeneralized Kalman-Yakubovich-Popov lemma for 2-D FM LSS modelrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3090ndash3103 2012

[30] X Li and H Gao ldquoRobust finite frequency 119867infin

filtering foruncertain 2-D systems the FM model caserdquo Automatica vol49 no 8 pp 2446ndash2452 2013

[31] X W Li and H J Gao ldquoA heuristic approach to static output-feedback controller synthesis with restricted frequency-domainspecificationsrdquo IEEE Transaction on Automatic Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[23] A Seuret ldquoStability analysis of networked control systems withasynchronous sampling and input delayrdquo in Proceedings of theAmerican Control Conference (ACC rsquo11) pp 533ndash538 July 2011

[24] B Tavassoli ldquoStability of nonlinear networked control systemsover multiple communication links with asynchronous sam-plingrdquo IEEE Transaction on Automatic Control 2013

[25] L Yang and J M Li ldquoSufficient and necessary conditions ofcontrollability and observability of a class of linear switchingsystemrdquo Systems Engineering and Electronics vol 25 no 5 pp588ndash590 2003

[26] W-J Mao ldquoObserver-based energy decoupling of linear time-delay systemsrdquo Journal of Zhejiang University vol 37 no 5 pp499ndash503 2003

[27] T Iwasaki and S Hara ldquoGeneralized KYP lemma unifiedfrequency domain inequalities with design applicationsrdquo IEEETransactions on Automatic Control vol 50 no 1 pp 41ndash592005

[28] H Gao and X Li ldquo119867infinfiltering for discrete-time state-delayed

systems with finite frequency specificationsrdquo IEEE Transactionson Automatic Control vol 56 no 12 pp 2935ndash2941 2011

[29] X Li H Gao and CWang ldquoGeneralized Kalman-Yakubovich-Popov lemma for 2-D FM LSS modelrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3090ndash3103 2012

[30] X Li and H Gao ldquoRobust finite frequency 119867infin

filtering foruncertain 2-D systems the FM model caserdquo Automatica vol49 no 8 pp 2446ndash2452 2013

[31] X W Li and H J Gao ldquoA heuristic approach to static output-feedback controller synthesis with restricted frequency-domainspecificationsrdquo IEEE Transaction on Automatic Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of