research article generalized kalman-yakubovich-popov...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 854631 9 pageshttpdxdoiorg1011552013854631

Research ArticleGeneralized Kalman-Yakubovich-Popov Lemma Based I-PDController Design for Ball and Plate System

Shuichi Mochizuki1 and Hiroyuki Ichihara2

1 Graduate School of Science and Technology Meiji University 1-1-1 Higashimita Tama-ku Kawasaki-shi Kanagawa 214-8571 Japan2Department of Science and Engineering Meiji University 1-1-1 Higashimita Tama-ku Kawasaki-shi Kanagawa 214-8571 Japan

Correspondence should be addressed to Hiroyuki Ichihara ichiharamessemeijiacjp

Received 23 June 2013 Accepted 9 September 2013

Academic Editor Baocang Ding

Copyright copy 2013 S Mochizuki and H IchiharaThis is an open access article distributed under theCreative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

The ball-on-plate balancing system has a camera that captures the ball position and a plate whose inclination angles are limitedThispaper proposes a PID controller design method for the ball and plate system based on the generalized Kalman-Yakubovich-Popovlemma The design method has two features first the structure of the controller called I-PD prevents large input signals againstmajor changes in the reference signal second a low-pass filter is introduced into the feedback loop to reduce the influence of themeasurement noise produced by the camera Both simulations and experiments are used to evaluate the effectiveness of the designmethod

1 Introduction

The ball and plate [1] is an unstable underactuated nonlinearsystem that has double integrators at the origin and that hastwo control inputs against four degrees of freedom (DOF) Acamera located above the plate captures the position of theball and two motors manipulate the inclination angles of theplate to keep the ball on the plateThe ball and plate system isan extension of the ball and beam system [2] from one to twodimensions The system has demonstrated various controllerdesignmethods for positioning and trajectory tracking of theball proportional integral derivative (PID) control [3] fuzzycontrol [4] neural network control [3 5] and model predic-tive control [6] In particular PID control has the benefitsof simple implementation and fewer hardware requirementsand it has been applied in many successful designs [7]Because PID control enables us a limited performanceoptimizing the parameters in a PID controller satisfyingdesign specifications is an important subject for study In thecontroller design of the ball and plate it requires to considerlimitation of the inclination angles with good transient andsteady-state responses Although proportional and derivativecontrollers are required to improve transient responses ajump in the reference signal generates a large input signal

that reaches the limitation angle that degrades the transientresponses [8] In addition ball position data from the camerainclude measurement noise that also degrades the steady-state responses

To overcome the above issues this paper proposes aPID controller design method for the ball and plate systemby open-loop shaping based on the generalized Kalman-Yakubovich-Popov (GKYP) lemma [9]TheGKYP lemma is ageneralization of the standard KYP lemma [10] which estab-lishes the equivalence between a frequency domain inequality(FDI) for a transfer function and a linear matrix inequality(LMI) associatedwith its state-space realizationThe standardKYP lemma is available for the infinite frequency rangewhile the generalized one can limit the frequency range tobe (semi) finite By introducing the GKYP lemma to PIDcontroller design design specifications by FDIs in the finitefrequency ranges for the open-loop transfer function resultin LMIs [9] The GKYP lemma gives a systematic open-loop shaping design method through optimization to realizedesirable transient and steady-state responses In this visualfeedback system we introduce a low-pass filter Since thefilter gives freedom in optimization it allows better steady-state responses and reduces the influence of themeasurementnoise To prevent large input signals fromP andD controllers

2 Journal of Applied Mathematics

Figure 1 2D Ball Balancer

we adopt the I-PD (integral-proportional derivative) struc-ture whose design is still in the framework of the GKYPlemma because the open-loop transfer functions of PID andI-PD structures are fundamentally the same

The paper is organized as follows The description of theball and plate system including its modeling and the mea-surement noise is presented in Section 2 The GKYP lemmabased I-PD controller design method with a low-pass filteris provided in Section 3 The design of the I-PD controller isdescribed in Section 4 Simulation and experimental resultsare presented in Sections 5 and 6 respectively Finally inSection 7 we present our conclusions

The notation used is standard For a matrix 119872 thetranspose and complex conjugate transpose are denoted by119872⊤ and 119872lowast respectively For a Hermitian matrix 119872 119872 ≻

(⪰) 0 and 119872 ≺ (⪯) 0 denote positive (semi) definitenessand negative (semi) definiteness respectivelyThe symbolH

119899

stands for the set of 119899 times 119899 Hermitian matrices The subscript119899 is omitted if 119899 = 2 The real and imaginary parts of 119872are denoted by R(119872) and I(119872) For matrices Φ and 119875Φotimes119875 denotes the Kronecker productL119909(119905) represents theLaplace transform of a signal 119909(119905)

2 Ball and Plate System

The ball and plate a QUANSER 2D Ball Balancer is shownin Figure 1 The system consists of a plate a ball an overheadcamera and two servo units The plate is allowed to swivelin both the 119883- and 119884-directions The overhead CMOSdigital camera a Point Grey Research Inc FFMV-03M2C-CSmeasures the position of the ball The two servo units locatedunder the plate are QUANSER SRV02 devices each of whichhas a peak time of approximately 200 ([ms]) and an overshootof approximately 5 Each of the devices is connected to aside of the plate through a two DOF gimbal The samplingtime of the control system and the frame rate provided bythe camera are 1 ([ms]) and 60 ([fps]) respectively Thus the

Motorgear

Loadgear

Potentiometergear

Ball

Plate

Supportbeam Bottom

support plate

r arm

120572

120579x

(xb yb)

x

Ltbl

0

Figure 2 The ball and plate system

image information is renewed approximately every 17 ([ms])There is a constant time delay of less than 60 ([ms]) betweenthemeasurement of the ball position and themanipulation ofthe servo units in the visual feedback system

21 Modeling The119883-direction of the ball and plate system isillustrated in Figure 2 We assume that the ball is completelysymmetric and homogeneous and does not slip on the plateand that all frictions are neglected The plate rotates in the119883119884-Cartesian coordinates with the origin at the center of theplate The equations of motion are

(119898119887+119868119887

1199032

119887

) 119887minus 119898119887(1199091198872+ 119910119887 120573) + 119898

119887119892 sin120572 = 0

(119898119887+119868119887

1199032

119887

) 119910119887minus 119898119887(119910119887

1205732+ 119909119887 120573) + 119898

119887119892 sin120573 = 0

(1)

where (119909119887 119910119887) is the position of the ball on the plate 120572 and

120573 are the inclination angles of the plate to the 119883- and 119884-axisrespectively 119898

119887is the mass of the ball 119903

119887is the radius of the

ball 119892 is the gravitational acceleration and 119868119887is the inertia of

the ball In Figure 2 120579119909represents the angle of the load gear

The relationship between 120572 and 120579119909is as follows

sin120572 =2 sin 120579

119909119903arm

119871 tbl (2)

where 119871 tbl is the length of the side of the plate and 119903arm is thelength between the joint and the center of the load gear Therelationship of 120573 and 120579

119910is the same as (2) since both gear

systems have the same hardware and the plate is symmetricalThe numerical values of the constant parameters in theequations of motion and (2) are shown in Table 1 Since 120579

119909

and 120579119910are limited as

minus30 [∘] le 120579

119909 120579119910 le 30 [

∘] (3)

from (2) the working ranges of 120572 and 120573 are

minus53 [∘] le 120572 120573 le 53 [

∘] (4)

If the angular velocities and 120573 are relatively low theapproximations

120573 = 0 2= 0 120573

2= 0 (5)

Journal of Applied Mathematics 3

Table 1 Parameters of the ball and plate system

Parameters Numerical values119898119887

00252 [kg]119903119887

00170 [m]119892 981 [ms2]119868119887

289 times 10minus6 [kgsdotm]

119871 tbl 0275 [m]119903arm 00254 [m]

are often used Linearizing the equations of motion at 120579119909= 0

and 120579119910= 0 we have

(119898119887+119868119887

1199032

119887

) 119887+2119898119887119892119903arm119871 tbl

120579119909= 0

(119898119887+119868119887

1199032

119887

) 119910119887+2119898119887119892119903arm119871 tbl

120579119910= 0

(6)

Since the axes are independent of each other we can focuson one axis for example the119883-axis For the input 120579

119909and the

output 119909119887 the transfer function is given by

119875 (119904) =119883119887(119904)

Θ119909(119904)

=

119870bap

1199042 (7)

where119883119887(119904) = L119909

119887(119905) Θ

119909(119904) = L120579

119909(119905) and

119870bap = minus2119898119887119892119903arm119903

2

119887

119871 tbl (1198981198871199032

119887+ 119868119887) (8)

22 Measurement Noise In this visual feedback system thereis inevitable noise from the camera To examine the noiselevel and frequencies we observed the error signal betweena fixed ball position and a measurement signal The resultsare shown in Figure 3 where the upper part represents a timehistory of the error signal including noise and the lower partrepresents the fast Fourier transform (FFT) analysis of theerror signal The noise level in the error signals is relativelyhigh at frequencies over 20 ([rads])

3 I-PD Control by GKYP Lemma

This section describes an I-PD controller design methodbased on the GKYP lemma The feedback control system isshown in Figure 4 where a filter is introduced into the controlsystem

31 Low-Pass Filter In the previous section we showed thatthemeasurement noise degrades the control performance Toreduce the influence of the noise a low-pass filter is availablein the controller design According to the noise propertiesthat we observed it is sufficient to introduce a first-order low-pass filter into the output of the measurement such that

119865 (119904) =120583

119904 + 120583 (9)

where 120583 ([rads]) is the cut-off frequency

0 1 2 3 4 5 6 7 8Time (s)

Am

plitu

de (m

)

002040608

112141618

2

Frequency (rads)

Pow

er

minus49

minus485

minus48

minus475times10minus3

100 101 102

Figure 3 Time history and FFT analysis results of measumentsignal

r e

minus minus

I-PD controlleru

Actuator Ball and platesystem

Low-passfilter

I

P + D

Figure 4 Feedback control system of the ball and plate

32 I-PD Controller In the standard PID control majorchanges in the reference signals cause large input signals tobe generated by the proportional and derivative actions in thecontroller that saturate the actuator Indeed it is difficult totune the parameters in the PID controller (Figure 4) such thatthe actuator in this visual feedback system is not saturatedThe control system with the I-PD controller (Figure 4) hasa structure whose inner loop includes the proportional andderivative actions [7 8] In this structure the integral actionalone acts on the error signal and prevents large signals beinginput to the actuator The control input 119906 can be written as

119906 = minus119870119901119910 +

119870119894

119904(119903 minus 119910) minus

119870119889119904

120591119904 + 1119910 (10)

where 120591(gt 0) is the parameter to approximate the differen-tiator by a proper transfer function119870

119901119870119894 and119870

119889represent

the proportional integral and derivative gains respectivelyThe open-loop transfer function is

119871 (119904) = 119865 (119904) 119875 (119904)119870 (119904) (11)

where

119870 (119904) = 119870119901+119870119894

119904+

119870119889119904

120591119904 + 1= 119870119901(1 +

1

119879119894119904+

119879119889119904

120591119904 + 1) (12)

119879119894and 119879

119889are the integral time and derivative time respec-

tively It should be noted that the open-loop transfer function


of the I-PD structure is the same as that of the standardPID structure To tune the parameters in the I-PD controllerwe employ an open-loop shaping that realizes a desirablefrequency response of the closed-loop system

33 Generalized KYP Lemma It is known that design spec-ifications for an open-loop transfer function can be reducedto LMIs through the GKYP lemma [9] We briefly review thislemma in the case of continuous-time systems

The design specification consists of a frequency range anda desired property in that range The frequency range can berepresented by

Λ (ΦΨ) = 119904 isin C | 120590 (119904 Φ) = 0 120590 (119904 Ψ) ge 0 (13)

where ΦΨ isin H

120590 (119904 Φ) = [119904lowast1]Φ[

119904

1] = 0 (14)

The equality constraint in (13) distinguishes betweencontinuous-time and discrete-time specifications Since weaddress continuous-time systems in this paper we use Φsuch that

Φ = [0 1

1 0] (15)

The inequality constraint 120590(119904 Ψ) ge 0 in (13) sets a frequencyrangeΩ For example a low frequency range is written as

Ω = 120596 | 120596 le 120603119897 = 120596 | 120590 (119895120596 Ψ) ge 0 (16)

where

Ψ = [0 minus119895

119895 2120603119897

] (17)

Table 2 presents a summary of the choice of Ψ versus a typeof the frequency range Ω where 120603

ℓ 120603ℎ 1206031 and 120603

2are real

positive numbers and 120603119888= (1206031+ 1206032)2 On the other hand

the desired property in a specific frequency range can berepresented by

[119871 (119895120596) 119868]Π [119871lowast(119895120596)

119868] ≺ 0 (18)

where

Π = [Π11

Π12

Π21

Π22

] isin H119898+119901

Π11isin H119901 Π ⪰ 0 (19)

119898 and 119901 are the input and output numbers of 119871(119904) respec-tively For SISO systems consider the requirement that 119871(119895120596)in a frequency range is on the half plane under a straight lineThat is 119871(119895120596) is under the straight line such that

119886R [119871 (119895120596)] + 119887I [119871 (119895120596)] lt 119888 (20)

that is equivalent to (18) with

Π = [0 119886 minus 119895119887

119886 + 119895119887 minus2119888] isin H

2 (21)

Table 2 Relation between Ψ andΩ for continuous time

Ψ Ω

[0 minus119895

119895 2120603ℓ

] 120596 le 120603ℓ(low)

[minus1 119895120603

119888

minus11989512060311988812060311206032

] 1206031le 120596 le 120603

2(middle)

[0 119895

minus119895 minus2120603ℎ

] 120603ℎle 120596 (high)

This requirement is designed to reduce sensitivity in a lowfrequency range Another requirement is that 119871(119895120596) in afrequency range is in the interior of the circle of radius 119903withthe center at 119888 That is 119871(119895120596) is in the circle such that

1003816100381610038161003816119871 (119895120596) minus 1198881003816100381610038161003816

2

lt 1199032 (22)

which is equivalent to (18) with

Π = [1 minus119888

lowast

minus119888 |119888|2minus 1199032] isin H

2 (23)

This requirement is designed to guarantee robustness in ahigh frequency range Under these preparations the gener-alized KYP lemma [9] is expressed as follows

Lemma 1 Let 119871(119904) be 119862(119904119868 minus 119860)minus1119861 + 119863 Λ(ΦΨ) in (13) andΠ in (18) are given Assume that det(119904119868 minus 119860) = 0 for all 119904 isin ΛThen the finite frequency condition

[119871 (119904) 119868]Π [119871lowast(119904)

119868] ≺ 0 forall119904 isin Λ (ΦΨ) (24)

holds if and only if there exist Hermitianmatrices119875 and119876 suchthat the matrix inequality condition

[[[

[

Γ [119861

119863]Π11

Π11[119861

119863]

lowast

minusΠ11

]]]

]

≺ 0 (25)

is satisfied where

Γ = [119860 119868

119862 0] (Φ⊤otimes 119875 + Ψ

⊤otimes 119876) [

119860 119868

119862 0]

⊤

+ [

[

0 119861Π12

Πlowast

12119861⊤119863Π12+ Πlowast

12119863⊤+ Π22

]

]

(26)

Equation (25) is affine with respect to 119861119863 119875119876 andΠ22

In the case where 119861 and119863 have affine design parameters (25)is an LMI

4 Control System Design

41 Filter Design Considering the control performance andthe noise level we set the cut-off frequency in (9) as 120583 =

20 ([rads]) We examine the effectiveness of this filter in


LPF

0 1 2 3 4 5 6 7 8Time (s)

Am

plitu

de (m

)

002040608

112141618

2

Frequency (rads)

Pow

er

minus49

minus485

minus48

minus475

times10minus3

100 101 102

Raw data

Figure 5 Time history and FFT analysis result of measurementsignal with low-pass filter

the same experimental setup as given in Section 22 Figure 5shows the spectral analysis results of the measurement andfiltered signals which are represented by the dotted and solidcurves respectively From these results it can be seen that thenoise at frequencies over 20 ([rads]) has been reduced

42 State-Space Realization of Open-Loop Transfer FunctionTo obtain an LMI based on the GKYP lemma a state-spacerealization of 119871(119904) is required to be affine with respect to aset of the design parameters 120588 = (119870

119901 119870119894 119870119889) If we fix 120591 in

119870(119904) at 10times10minus2 the design parameters appear affinely in thenumerator of119870(119904) Indeed the controllable canonical form of119870(119904) is written as

[119860119896119861119896(120588)

119862119896119863119896(120588)

] =

[[[[[[

[

0 0119870119894

120591

1 minus1

120591119870119894minus119870119889

1205912

0 1 119870119901+119870119889

120591

]]]]]]

]

(27)

Realizations of 119875(119904) and 119865(119904) are also written as

[119860119901

119861119901

119862119901119863119901

] = [

[

0 1 0

0 0 119870bap1 0 0

]

]

(28)

[119860119891

119861119891

119862119891

119863119891

] = [minus120583 1

120583 0] (29)

respectively By combining these realizations (27)ndash(29) weobtain a realization of 119871(119904) as

119871 (119904) = [119860 119861 (120588)

119862 119863 (120588)] (30)

where

119860 = [

[

119860119896

0 0

119862119896119861119901

119860119901

0

119862119896119863119901119861119891119862119901119861119891119860119891

]

]

119861 = [

[

119861119896(120588)

119863119896(120588) 119861119901

119863119896(120588)119863119901119861119891

]

]

119862 = [119862119896119863119901119861119891 119862119901119861119891119862119891] 119863 = 119863

119896(120588)119863119901119863119891

(31)

Consequently the state-space realization of 119871(119904) is affine withrespect to 120588

43 Specifications To shape the Nyquist plot of 119871(119904) werequire the following FDI specifications

minus2R [119871 (119895120596)] +I [119871 (119895120596)] gt 120574ℓforall120596 le 08 (32)

I [119871 (119895120596)] lt 120574119898 25 le

forall120596 le 28 (33)

1003816100381610038161003816119871 (119895120596)1003816100381610038161003816 lt 120574ℎ

forall120596 ge 10 (34)

Specification (32) with a large 120574ℓ(gt 0) ensures sensitivity

reduction in the low frequency range by making the gain of119871(119904) high Specification (33) requires the Nyquist plot to beoutside a circle with its center at the point minus1 + 1198950 so thata certain stability margin is guaranteed Specification (34)with a small 120574

ℎensures robustness against the unmodeled

dynamics that typically exists in the high frequency rangeIn addition to the above basic specifications we also

require the following FDIs that improve the property oftrajectory tracking

4R [119871 (119895120596)] +I [119871 (119895120596)] lt 1205741forall120596 ge 03 (35)

4R [119871 (119895120596)] +I [119871 (119895120596)] gt 1205742forall120596 ge 03 (36)

Since the integral action aloneworks on the error between theoutput and the reference signals the property of trajectorytracking depends directly on the integrator Here we focuson the corner angular frequency 120596

119868by the integral action

in the I-PD controller which is given by 120596119868= 1119879

119894 where

119879119894= 119870119901119870119894 We have a strong integral action and the error is

corrected quickly when the corner angular frequency is highwhile too high a corner angular frequency causes overshootand hunting Thus we impose restrictions for the phase ofthe I-PD controller It should be noted that the phase atlower frequencies is about minus90 ([deg]) while the phase atthe corner angular frequency is about 0 [deg] If we find thefrequency at a specific phase from minus90 ([deg]) to 0 ([deg])the corner angular frequency is greater than that frequencySpecifications (35) and (36) restrict the phase of the I-PDcontroller as well as the open-loop transfer function so thatthe corner angular frequency is greater than the frequency atthe lowest point in the frequency range 03 ([rads])

44 I-PD Controller Design We design an I-PD controller bymaximizing 120574

ℓsubject to Specifications (32)ndash(36) where

(120574119898 120574ℎ 1205741 1205742) = (minus1 05 1 minus100) (37)


0 50 100 150 200

0

50

100

150

Real axis

Imag

inar

y ax

is

The upper limit

Low frequency

The lower limit

minus150

minus100

minus50

minus150minus200 minus100 minus50

Figure 6 Nyquist plot

That is the optimization problem is

max119870119894 119870119901119870119889 120574ℓ

120574ℓ

subject to (19)ndash(36) and (37)

(38)

Each of the design Specifications (32)ndash(36) is reduced to anLMI condition through Lemma 1 with the realization (30)The Specification (32) is modified to 120576 le forall120596 le 08 where 120576 =10 times 10

minus4 because 119871(119904) includes the origin poles that preventus from taking 120596 = 0 Then the LMI optimization problemis to maximize 120574

ℓsubject to all these LMI conditions where

119870119894 119870119901 and 119870

119889are the common decision variables while

1198751 119875

5and 119876

1 119876

5appear in the LMIs as independent

decision variables It should be noted that 120574ℓappears alone in

Π22in the LMI condition (25) corresponding to (36) In this

sense 120574ℓis also an independent decision variable It should

also be noted that 119875 and 119876 in (25) appear in each of the LMIconditions as the independent decision variables

To solve this LMI optimization problem we use YALMIPR20120806 [11] an LMI parser and SPDT3 version 40 [12]an LMI solver on MATLAB R2011b The resulting optimalparameters in the I-PD controller and 120574

ℓare

(119870119901 119870119894 119870119889 120574ℓ) = (92859 107806 4138 119464)

(39)

The Nyquist plots are shown in Figures 6 and 7 where 119871(119895120596)satisfies the design specifications given in Section 43 Since 120574

ℓ

is maximized sensitivity is reduced in the frequency rangeThe Bode plots of 119865(119904)119875(119904) 119871(119904) and 119870(119904) are shown inFigure 8 where the corner angular frequency 120596

119868of the I-PD

controller is larger than 03 ([rads])

5 Simulation Results

The I-PD controller whose design is described in the previoussection was evaluated by a simulation of the step response Tocompare the response with that of a standard PID controller

0 1

0

05

1

15

Real axis

Imag

inar

y ax

is

Middle frequency

High frequency

minus15

minus1

minus05

minus5 minus4 minus3 minus2 minus1

120596 = 102 (rads)

120596 = 30 (rads)

Figure 7 Nyquist plot (magnification)

0

100

200

Gai

n (d

B)

0

90

Phas

e (de

g)

Frequency (rads)

minus300

minus200

minus100

minus270

minus180

minus90

10minus2 100 102 104

P(s) + F(s)

L(s)

K(s)

Frequency (rads)10minus2 100 102 104

Figure 8 Bode plot

we used the PID controller whose gain parameters are thesame as those of the I-PD controller Since both feedbacksystems have the same open-loop transfer function theirfeedback properties must be the same provided that eachinput signal does not saturate The simulation results of thestep response are shown in Figure 9 where the upper andlower parts are the output and input signals respectivelyThe solid curves represent the responses given by the I-PDcontroller while the dashed curves represent those by the PIDcontroller One can see that the input signal given by the PIDcontroller is saturated while that by the I-PD controller is notsaturated and satisfies the limitation (3) The output signalgiven by the I-PD controller settles down to the desired valuewithout any overshoot

It should be noted that the gain parameters in thedesigned controller are not tuned with the I-PD structure


0 1 2 3 4 50

005

01

015

Am

plitu

de (m

)

0 1 2 3 4 5

0102030

Time (s)

Ang

le (d

eg)

I-PDPID

minus30

minus20

minus10

Time (s)

Figure 9 Step response simulation results

0 05 1 15 2 25 3 35 4 45 50

005

01

015

Time (s)

Am

plitu

de (m

)

0 05 1 15 2 25 3 35 4 45 5

0102030

Am

plitu

de (r

ad)

minus30

minus20

minus10

Time (s)

I-PDPID

Figure 10 Step response experimental results

In our experience it is difficult not to saturate the inputlimitation for the PID structure using any design method

6 Experimental Results

This section evaluates the I-PD controller whose designis given in Section 4 through an experiment of the stepresponse The PID controller used in Section 4 was alsoevaluated for comparison The results of trajectory trackingcontrol by the I-PD controller were also evaluated

61 Step Response Experiment The results of the stepresponse experiment are shown in Figure 10 where thedescription of the figure is the same as that of Figure 9 In thisexperiment the influence of the time delay appeared and theinput signals were slightly larger than those obtained in thesimulations The rise and settling time results are howeveralmost the same as those obtained in the simulation

62 Trajectory Tracking Experiments We tested two kindsof trajectories for tracking control a square and a circular


0 002 004

0

002

004

006

Reference

minus006minus006

minus004

minus004

minus002

minus002 0 002 004

0

002

004

006

minus006minus006

minus004

minus004

minus002

minus002

yb

(m)

xb (m)

I-PD controlyb

(m)

xb (m)

ReferenceI-PD control

Figure 11 Experimental results of tracking control

0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

Time (s)

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)

Time (s)Time (s)


Controller outputActuator output

xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)

Figure 12 Experimental results of square response

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)Time (s)

Time (s)Time (s)

xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)



Figure 13 Experimental results of circle response


trajectory The side of the square trajectory was 01 ([m])and the radius of the circular trajectory was 005 ([m]) Theresults are shown in Figure 11 where the left part shows atrajectory of the square trajectory tracking control and theright part shows a trajectory of the circular trajectory trackingcontrol The time histories of the ball and input angles areshown in Figures 12 and 13 In the square trajectory trackingcontrol experiment the responses were similar to those inthe step response experiment except for a slight vibrationSuch vibration phenomena are noticeable in the responsesof circular trajectory tracking control in particular the casewhen the input signal is relatively small The reason for thesephenomena could be the friction of the ball against the plateor a backlash of the gear system

7 Conclusions

This paper applied the GKYP lemma to an open-looptransfer function including an I-PD controller and a noisereduction filter for the ball and plate systemThemultiple FDIspecifications for the finite frequency ranges were satisfiedby a solution of the LMI optimization problem The solutionincludes the optimal parameters in the I-PD controller Thefirst-order low-pass filter reduced the noise in the high fre-quency range and improved the steady-state response Bothsimulations and experiments evaluated the effectiveness ofthe designmethod by comparing the standard PID controller

The PI-D (proportional integral-derivative) control sys-tem which moved the derivative controller to the innerfeedback loop also has the same open-loop transfer functionas the standard PID controller Thus the approach in thispaper can also be applied to the PI-D controller

References

[1] MMoarref M Saadat and G Vossoughi ldquoMechatronic designand position control of a novel ball and plate systemrdquo inProceedings of the Mediterranean Conference on Control andAutomation (MEDrsquo 08) pp 1071ndash1076 June 2008

[2] P Kokotovic ldquoThe joy of feedback nonlinear and adaptiverdquoIEEE Control Systems Magazine vol 12 no 3 pp 7ndash17 1992

[3] F Zheng X Li X Qian and S Wang ldquoModeling and PIDneural network research for the ball and plate systemrdquo inProceedings of the International Conference on Electronics Com-munications and Control (ICECC rsquo11) pp 331ndash334 September2011

[4] X Fan N Zhang and S Teng ldquoTrajectory planning andtracking of ball andplate systemusing hierarchical fuzzy controlschemerdquo Fuzzy Sets and Systems vol 144 no 2 pp 297ndash3122004

[5] M Bai Y Tian and Y Wang ldquoDecoupled fuzzy sliding modecontrol to ball and plate systemrdquo in Proceedings of the 2ndInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo11) pp 685ndash690 July 2011

[6] F Borrelli Constrained Optimal Control of Linear and HybridSystems vol 290 of Lecture Notes in Control and InformationSciences Springer 2003

[7] K J Astrom and T Hagglund Advanced PID Control Instru-mentation Systems 2005

[8] Q Li and Z Kemin Introduction to Feedback Control PrenticeHall 2009

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

[10] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemmardquo Sys-tems amp Control Letters vol 28 no 1 pp 7ndash10 1996

[11] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD rsquo11)Taipei Taiwan 2004

[12] K C TohM J Todd and RH Tutuncu ldquoSDPT3mdashaMATLABsoftware package for semidefinite programming version 13rdquoOptimization Methods and Software vol 11 no 1 pp 545ndash5811999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Figure 1 2D Ball Balancer

we adopt the I-PD (integral-proportional derivative) struc-ture whose design is still in the framework of the GKYPlemma because the open-loop transfer functions of PID andI-PD structures are fundamentally the same

The paper is organized as follows The description of theball and plate system including its modeling and the mea-surement noise is presented in Section 2 The GKYP lemmabased I-PD controller design method with a low-pass filteris provided in Section 3 The design of the I-PD controller isdescribed in Section 4 Simulation and experimental resultsare presented in Sections 5 and 6 respectively Finally inSection 7 we present our conclusions

The notation used is standard For a matrix 119872 thetranspose and complex conjugate transpose are denoted by119872⊤ and 119872lowast respectively For a Hermitian matrix 119872 119872 ≻

(⪰) 0 and 119872 ≺ (⪯) 0 denote positive (semi) definitenessand negative (semi) definiteness respectivelyThe symbolH

119899

stands for the set of 119899 times 119899 Hermitian matrices The subscript119899 is omitted if 119899 = 2 The real and imaginary parts of 119872are denoted by R(119872) and I(119872) For matrices Φ and 119875Φotimes119875 denotes the Kronecker productL119909(119905) represents theLaplace transform of a signal 119909(119905)

2 Ball and Plate System

The ball and plate a QUANSER 2D Ball Balancer is shownin Figure 1 The system consists of a plate a ball an overheadcamera and two servo units The plate is allowed to swivelin both the 119883- and 119884-directions The overhead CMOSdigital camera a Point Grey Research Inc FFMV-03M2C-CSmeasures the position of the ball The two servo units locatedunder the plate are QUANSER SRV02 devices each of whichhas a peak time of approximately 200 ([ms]) and an overshootof approximately 5 Each of the devices is connected to aside of the plate through a two DOF gimbal The samplingtime of the control system and the frame rate provided bythe camera are 1 ([ms]) and 60 ([fps]) respectively Thus the

Motorgear

Loadgear

Potentiometergear

Ball

Plate

Supportbeam Bottom

support plate

r arm

120572

120579x

(xb yb)

x

Ltbl

0

Figure 2 The ball and plate system

image information is renewed approximately every 17 ([ms])There is a constant time delay of less than 60 ([ms]) betweenthemeasurement of the ball position and themanipulation ofthe servo units in the visual feedback system

21 Modeling The119883-direction of the ball and plate system isillustrated in Figure 2 We assume that the ball is completelysymmetric and homogeneous and does not slip on the plateand that all frictions are neglected The plate rotates in the119883119884-Cartesian coordinates with the origin at the center of theplate The equations of motion are

(119898119887+119868119887

1199032

119887

) 119887minus 119898119887(1199091198872+ 119910119887 120573) + 119898

119887119892 sin120572 = 0

(119898119887+119868119887

1199032

119887

) 119910119887minus 119898119887(119910119887

1205732+ 119909119887 120573) + 119898

119887119892 sin120573 = 0

(1)

where (119909119887 119910119887) is the position of the ball on the plate 120572 and

120573 are the inclination angles of the plate to the 119883- and 119884-axisrespectively 119898

119887is the mass of the ball 119903

119887is the radius of the

ball 119892 is the gravitational acceleration and 119868119887is the inertia of

the ball In Figure 2 120579119909represents the angle of the load gear

The relationship between 120572 and 120579119909is as follows

sin120572 =2 sin 120579

119909119903arm

119871 tbl (2)

where 119871 tbl is the length of the side of the plate and 119903arm is thelength between the joint and the center of the load gear Therelationship of 120573 and 120579

119910is the same as (2) since both gear

systems have the same hardware and the plate is symmetricalThe numerical values of the constant parameters in theequations of motion and (2) are shown in Table 1 Since 120579

119909

and 120579119910are limited as

minus30 [∘] le 120579

119909 120579119910 le 30 [

∘] (3)

from (2) the working ranges of 120572 and 120573 are

minus53 [∘] le 120572 120573 le 53 [

∘] (4)

If the angular velocities and 120573 are relatively low theapproximations

120573 = 0 2= 0 120573

2= 0 (5)




00252 [kg]119903119887

00170 [m]119892 981 [ms2]119868119887


119871 tbl 0275 [m]119903arm 00254 [m]


and 120579119910= 0 we have

(119898119887+119868119887

1199032

119887

) 119887+2119898119887119892119903arm119871 tbl

120579119909= 0

(119898119887+119868119887

1199032

119887

) 119910119887+2119898119887119892119903arm119871 tbl

120579119910= 0

(6)


119909and the


119875 (119904) =119883119887(119904)

Θ119909(119904)

=

119870bap

1199042 (7)

where119883119887(119904) = L119909

119887(119905) Θ

119909(119904) = L120579

119909(119905) and

119870bap = minus2119898119887119892119903arm119903

2

119887

119871 tbl (1198981198871199032

119887+ 119868119887) (8)





119865 (119904) =120583

119904 + 120583 (9)


0 1 2 3 4 5 6 7 8Time (s)

Am

plitu

de (m

)

002040608

112141618

2

Frequency (rads)

Pow

er

minus49

minus485

minus48


100 101 102


r e

minus minus

I-PD controlleru


Low-passfilter

I

P + D



119906 = minus119870119901119910 +

119870119894

119904(119903 minus 119910) minus

119870119889119904

120591119904 + 1119910 (10)


119901119870119894 and119870

119889represent


119871 (119904) = 119865 (119904) 119875 (119904)119870 (119904) (11)

where

119870 (119904) = 119870119901+119870119894

119904+

119870119889119904

120591119904 + 1= 119870119901(1 +

1

119879119894119904+

119879119889119904

120591119904 + 1) (12)

119879119894and 119879







Λ (ΦΨ) = 119904 isin C | 120590 (119904 Φ) = 0 120590 (119904 Ψ) ge 0 (13)

where ΦΨ isin H

120590 (119904 Φ) = [119904lowast1]Φ[

119904

1] = 0 (14)


Φ = [0 1

1 0] (15)


Ω = 120596 | 120596 le 120603119897 = 120596 | 120590 (119895120596 Ψ) ge 0 (16)

where

Ψ = [0 minus119895

119895 2120603119897

] (17)


ℓ 120603ℎ 1206031 and 120603

2are real



[119871 (119895120596) 119868]Π [119871lowast(119895120596)

119868] ≺ 0 (18)

where

Π = [Π11

Π12

Π21

Π22

] isin H119898+119901

Π11isin H119901 Π ⪰ 0 (19)


119886R [119871 (119895120596)] + 119887I [119871 (119895120596)] lt 119888 (20)


Π = [0 119886 minus 119895119887

119886 + 119895119887 minus2119888] isin H

2 (21)


Ψ Ω

[0 minus119895

119895 2120603ℓ

] 120596 le 120603ℓ(low)

[minus1 119895120603

119888

minus11989512060311988812060311206032

] 1206031le 120596 le 120603

2(middle)

[0 119895


] 120603ℎle 120596 (high)


1003816100381610038161003816119871 (119895120596) minus 1198881003816100381610038161003816

2

lt 1199032 (22)


Π = [1 minus119888

lowast


2 (23)



[119871 (119904) 119868]Π [119871lowast(119904)

119868] ≺ 0 forall119904 isin Λ (ΦΨ) (24)


[[[

[

Γ [119861

119863]Π11

Π11[119861

119863]

lowast

minusΠ11

]]]

]

≺ 0 (25)

is satisfied where

Γ = [119860 119868

119862 0] (Φ⊤otimes 119875 + Ψ

⊤otimes 119876) [

119860 119868

119862 0]

⊤

+ [

[

0 119861Π12

Πlowast

12119861⊤119863Π12+ Πlowast

12119863⊤+ Π22

]

]

(26)







LPF

0 1 2 3 4 5 6 7 8Time (s)

Am

plitu

de (m

)

002040608

112141618

2

Frequency (rads)

Pow

er

minus49

minus485

minus48

minus475

times10minus3

100 101 102

Raw data




119901 119870119894 119870119889) If we fix 120591 in


[119860119896119861119896(120588)

119862119896119863119896(120588)

] =

[[[[[[

[

0 0119870119894

120591

1 minus1

120591119870119894minus119870119889

1205912

0 1 119870119901+119870119889

120591

]]]]]]

]

(27)


[119860119901

119861119901

119862119901119863119901

] = [

[

0 1 0

0 0 119870bap1 0 0

]

]

(28)

[119860119891

119861119891

119862119891

119863119891

] = [minus120583 1

120583 0] (29)


119871 (119904) = [119860 119861 (120588)

119862 119863 (120588)] (30)

where

119860 = [

[

119860119896

0 0

119862119896119861119901

119860119901

0

119862119896119863119901119861119891119862119901119861119891119860119891

]

]

119861 = [

[

119861119896(120588)

119863119896(120588) 119861119901

119863119896(120588)119863119901119861119891

]

]

119862 = [119862119896119863119901119861119891 119862119901119861119891119862119891] 119863 = 119863

119896(120588)119863119901119863119891

(31)




I [119871 (119895120596)] lt 120574119898 25 le

forall120596 le 28 (33)

1003816100381610038161003816119871 (119895120596)1003816100381610038161003816 lt 120574ℎ

forall120596 ge 10 (34)






4R [119871 (119895120596)] +I [119871 (119895120596)] lt 1205741forall120596 ge 03 (35)

4R [119871 (119895120596)] +I [119871 (119895120596)] gt 1205742forall120596 ge 03 (36)




119894 where





(120574119898 120574ℎ 1205741 1205742) = (minus1 05 1 minus100) (37)


0 50 100 150 200

0

50

100

150

Real axis

Imag

inar

y ax

is

The upper limit

Low frequency

The lower limit

minus150

minus100

minus50




max119870119894 119870119901119870119889 120574ℓ

120574ℓ


(38)




119870119894 119870119901 and 119870


1198751 119875

5and 119876

1 119876







ℓare

(119870119901 119870119894 119870119889 120574ℓ) = (92859 107806 4138 119464)

(39)


ℓ


119868of the I-PD




0 1

0

05

1

15

Real axis

Imag

inar

y ax

is

Middle frequency

High frequency

minus15

minus1

minus05


120596 = 102 (rads)

120596 = 30 (rads)


0

100

200

Gai

n (d

B)

0

90

Phas

e (de

g)

Frequency (rads)

minus300

minus200

minus100

minus270

minus180

minus90

10minus2 100 102 104

P(s) + F(s)

L(s)

K(s)


Figure 8 Bode plot




0 1 2 3 4 50

005

01

015

Am

plitu

de (m

)

0 1 2 3 4 5

0102030

Time (s)

Ang

le (d

eg)

I-PDPID

minus30

minus20

minus10

Time (s)


0 05 1 15 2 25 3 35 4 45 50

005

01

015

Time (s)

Am

plitu

de (m

)

0 05 1 15 2 25 3 35 4 45 5

0102030

Am

plitu

de (r

ad)

minus30

minus20

minus10

Time (s)

I-PDPID








0 002 004

0

002

004

006

Reference

minus006minus006

minus004

minus004

minus002

minus002 0 002 004

0

002

004

006

minus006minus006

minus004

minus004

minus002

minus002

yb

(m)

xb (m)

I-PD controlyb

(m)

xb (m)



0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

Time (s)

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)

Time (s)Time (s)



xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)


0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)Time (s)

Time (s)Time (s)

xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)






7 Conclusions



References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












00252 [kg]119903119887

00170 [m]119892 981 [ms2]119868119887


119871 tbl 0275 [m]119903arm 00254 [m]


and 120579119910= 0 we have

(119898119887+119868119887

1199032

119887

) 119887+2119898119887119892119903arm119871 tbl

120579119909= 0

(119898119887+119868119887

1199032

119887

) 119910119887+2119898119887119892119903arm119871 tbl

120579119910= 0

(6)


119909and the


119875 (119904) =119883119887(119904)

Θ119909(119904)

=

119870bap

1199042 (7)

where119883119887(119904) = L119909

119887(119905) Θ

119909(119904) = L120579

119909(119905) and

119870bap = minus2119898119887119892119903arm119903

2

119887

119871 tbl (1198981198871199032

119887+ 119868119887) (8)





119865 (119904) =120583

119904 + 120583 (9)


0 1 2 3 4 5 6 7 8Time (s)

Am

plitu

de (m

)

002040608

112141618

2

Frequency (rads)

Pow

er

minus49

minus485

minus48


100 101 102


r e

minus minus

I-PD controlleru


Low-passfilter

I

P + D



119906 = minus119870119901119910 +

119870119894

119904(119903 minus 119910) minus

119870119889119904

120591119904 + 1119910 (10)


119901119870119894 and119870

119889represent


119871 (119904) = 119865 (119904) 119875 (119904)119870 (119904) (11)

where

119870 (119904) = 119870119901+119870119894

119904+

119870119889119904

120591119904 + 1= 119870119901(1 +

1

119879119894119904+

119879119889119904

120591119904 + 1) (12)

119879119894and 119879







Λ (ΦΨ) = 119904 isin C | 120590 (119904 Φ) = 0 120590 (119904 Ψ) ge 0 (13)

where ΦΨ isin H

120590 (119904 Φ) = [119904lowast1]Φ[

119904

1] = 0 (14)


Φ = [0 1

1 0] (15)


Ω = 120596 | 120596 le 120603119897 = 120596 | 120590 (119895120596 Ψ) ge 0 (16)

where

Ψ = [0 minus119895

119895 2120603119897

] (17)


ℓ 120603ℎ 1206031 and 120603

2are real



[119871 (119895120596) 119868]Π [119871lowast(119895120596)

119868] ≺ 0 (18)

where

Π = [Π11

Π12

Π21

Π22

] isin H119898+119901

Π11isin H119901 Π ⪰ 0 (19)


119886R [119871 (119895120596)] + 119887I [119871 (119895120596)] lt 119888 (20)


Π = [0 119886 minus 119895119887

119886 + 119895119887 minus2119888] isin H

2 (21)


Ψ Ω

[0 minus119895

119895 2120603ℓ

] 120596 le 120603ℓ(low)

[minus1 119895120603

119888

minus11989512060311988812060311206032

] 1206031le 120596 le 120603

2(middle)

[0 119895


] 120603ℎle 120596 (high)


1003816100381610038161003816119871 (119895120596) minus 1198881003816100381610038161003816

2

lt 1199032 (22)


Π = [1 minus119888

lowast


2 (23)



[119871 (119904) 119868]Π [119871lowast(119904)

119868] ≺ 0 forall119904 isin Λ (ΦΨ) (24)


[[[

[

Γ [119861

119863]Π11

Π11[119861

119863]

lowast

minusΠ11

]]]

]

≺ 0 (25)

is satisfied where

Γ = [119860 119868

119862 0] (Φ⊤otimes 119875 + Ψ

⊤otimes 119876) [

119860 119868

119862 0]

⊤

+ [

[

0 119861Π12

Πlowast

12119861⊤119863Π12+ Πlowast

12119863⊤+ Π22

]

]

(26)







LPF

0 1 2 3 4 5 6 7 8Time (s)

Am

plitu

de (m

)

002040608

112141618

2

Frequency (rads)

Pow

er

minus49

minus485

minus48

minus475

times10minus3

100 101 102

Raw data




119901 119870119894 119870119889) If we fix 120591 in


[119860119896119861119896(120588)

119862119896119863119896(120588)

] =

[[[[[[

[

0 0119870119894

120591

1 minus1

120591119870119894minus119870119889

1205912

0 1 119870119901+119870119889

120591

]]]]]]

]

(27)


[119860119901

119861119901

119862119901119863119901

] = [

[

0 1 0

0 0 119870bap1 0 0

]

]

(28)

[119860119891

119861119891

119862119891

119863119891

] = [minus120583 1

120583 0] (29)


119871 (119904) = [119860 119861 (120588)

119862 119863 (120588)] (30)

where

119860 = [

[

119860119896

0 0

119862119896119861119901

119860119901

0

119862119896119863119901119861119891119862119901119861119891119860119891

]

]

119861 = [

[

119861119896(120588)

119863119896(120588) 119861119901

119863119896(120588)119863119901119861119891

]

]

119862 = [119862119896119863119901119861119891 119862119901119861119891119862119891] 119863 = 119863

119896(120588)119863119901119863119891

(31)




I [119871 (119895120596)] lt 120574119898 25 le

forall120596 le 28 (33)

1003816100381610038161003816119871 (119895120596)1003816100381610038161003816 lt 120574ℎ

forall120596 ge 10 (34)






4R [119871 (119895120596)] +I [119871 (119895120596)] lt 1205741forall120596 ge 03 (35)

4R [119871 (119895120596)] +I [119871 (119895120596)] gt 1205742forall120596 ge 03 (36)




119894 where





(120574119898 120574ℎ 1205741 1205742) = (minus1 05 1 minus100) (37)


0 50 100 150 200

0

50

100

150

Real axis

Imag

inar

y ax

is

The upper limit

Low frequency

The lower limit

minus150

minus100

minus50




max119870119894 119870119901119870119889 120574ℓ

120574ℓ


(38)




119870119894 119870119901 and 119870


1198751 119875

5and 119876

1 119876







ℓare

(119870119901 119870119894 119870119889 120574ℓ) = (92859 107806 4138 119464)

(39)


ℓ


119868of the I-PD




0 1

0

05

1

15

Real axis

Imag

inar

y ax

is

Middle frequency

High frequency

minus15

minus1

minus05


120596 = 102 (rads)

120596 = 30 (rads)


0

100

200

Gai

n (d

B)

0

90

Phas

e (de

g)

Frequency (rads)

minus300

minus200

minus100

minus270

minus180

minus90

10minus2 100 102 104

P(s) + F(s)

L(s)

K(s)


Figure 8 Bode plot




0 1 2 3 4 50

005

01

015

Am

plitu

de (m

)

0 1 2 3 4 5

0102030

Time (s)

Ang

le (d

eg)

I-PDPID

minus30

minus20

minus10

Time (s)


0 05 1 15 2 25 3 35 4 45 50

005

01

015

Time (s)

Am

plitu

de (m

)

0 05 1 15 2 25 3 35 4 45 5

0102030

Am

plitu

de (r

ad)

minus30

minus20

minus10

Time (s)

I-PDPID








0 002 004

0

002

004

006

Reference

minus006minus006

minus004

minus004

minus002

minus002 0 002 004

0

002

004

006

minus006minus006

minus004

minus004

minus002

minus002

yb

(m)

xb (m)

I-PD controlyb

(m)

xb (m)



0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

Time (s)

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)

Time (s)Time (s)



xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)


0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)Time (s)

Time (s)Time (s)

xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)






7 Conclusions



References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Λ (ΦΨ) = 119904 isin C | 120590 (119904 Φ) = 0 120590 (119904 Ψ) ge 0 (13)

where ΦΨ isin H

120590 (119904 Φ) = [119904lowast1]Φ[

119904

1] = 0 (14)


Φ = [0 1

1 0] (15)


Ω = 120596 | 120596 le 120603119897 = 120596 | 120590 (119895120596 Ψ) ge 0 (16)

where

Ψ = [0 minus119895

119895 2120603119897

] (17)


ℓ 120603ℎ 1206031 and 120603

2are real



[119871 (119895120596) 119868]Π [119871lowast(119895120596)

119868] ≺ 0 (18)

where

Π = [Π11

Π12

Π21

Π22

] isin H119898+119901

Π11isin H119901 Π ⪰ 0 (19)


119886R [119871 (119895120596)] + 119887I [119871 (119895120596)] lt 119888 (20)


Π = [0 119886 minus 119895119887

119886 + 119895119887 minus2119888] isin H

2 (21)


Ψ Ω

[0 minus119895

119895 2120603ℓ

] 120596 le 120603ℓ(low)

[minus1 119895120603

119888

minus11989512060311988812060311206032

] 1206031le 120596 le 120603

2(middle)

[0 119895


] 120603ℎle 120596 (high)


1003816100381610038161003816119871 (119895120596) minus 1198881003816100381610038161003816

2

lt 1199032 (22)


Π = [1 minus119888

lowast


2 (23)



[119871 (119904) 119868]Π [119871lowast(119904)

119868] ≺ 0 forall119904 isin Λ (ΦΨ) (24)


[[[

[

Γ [119861

119863]Π11

Π11[119861

119863]

lowast

minusΠ11

]]]

]

≺ 0 (25)

is satisfied where

Γ = [119860 119868

119862 0] (Φ⊤otimes 119875 + Ψ

⊤otimes 119876) [

119860 119868

119862 0]

⊤

+ [

[

0 119861Π12

Πlowast

12119861⊤119863Π12+ Πlowast

12119863⊤+ Π22

]

]

(26)







LPF

0 1 2 3 4 5 6 7 8Time (s)

Am

plitu

de (m

)

002040608

112141618

2

Frequency (rads)

Pow

er

minus49

minus485

minus48

minus475

times10minus3

100 101 102

Raw data




119901 119870119894 119870119889) If we fix 120591 in


[119860119896119861119896(120588)

119862119896119863119896(120588)

] =

[[[[[[

[

0 0119870119894

120591

1 minus1

120591119870119894minus119870119889

1205912

0 1 119870119901+119870119889

120591

]]]]]]

]

(27)


[119860119901

119861119901

119862119901119863119901

] = [

[

0 1 0

0 0 119870bap1 0 0

]

]

(28)

[119860119891

119861119891

119862119891

119863119891

] = [minus120583 1

120583 0] (29)


119871 (119904) = [119860 119861 (120588)

119862 119863 (120588)] (30)

where

119860 = [

[

119860119896

0 0

119862119896119861119901

119860119901

0

119862119896119863119901119861119891119862119901119861119891119860119891

]

]

119861 = [

[

119861119896(120588)

119863119896(120588) 119861119901

119863119896(120588)119863119901119861119891

]

]

119862 = [119862119896119863119901119861119891 119862119901119861119891119862119891] 119863 = 119863

119896(120588)119863119901119863119891

(31)




I [119871 (119895120596)] lt 120574119898 25 le

forall120596 le 28 (33)

1003816100381610038161003816119871 (119895120596)1003816100381610038161003816 lt 120574ℎ

forall120596 ge 10 (34)






4R [119871 (119895120596)] +I [119871 (119895120596)] lt 1205741forall120596 ge 03 (35)

4R [119871 (119895120596)] +I [119871 (119895120596)] gt 1205742forall120596 ge 03 (36)




119894 where





(120574119898 120574ℎ 1205741 1205742) = (minus1 05 1 minus100) (37)


0 50 100 150 200

0

50

100

150

Real axis

Imag

inar

y ax

is

The upper limit

Low frequency

The lower limit

minus150

minus100

minus50




max119870119894 119870119901119870119889 120574ℓ

120574ℓ


(38)




119870119894 119870119901 and 119870


1198751 119875

5and 119876

1 119876







ℓare

(119870119901 119870119894 119870119889 120574ℓ) = (92859 107806 4138 119464)

(39)


ℓ


119868of the I-PD




0 1

0

05

1

15

Real axis

Imag

inar

y ax

is

Middle frequency

High frequency

minus15

minus1

minus05


120596 = 102 (rads)

120596 = 30 (rads)


0

100

200

Gai

n (d

B)

0

90

Phas

e (de

g)

Frequency (rads)

minus300

minus200

minus100

minus270

minus180

minus90

10minus2 100 102 104

P(s) + F(s)

L(s)

K(s)


Figure 8 Bode plot




0 1 2 3 4 50

005

01

015

Am

plitu

de (m

)

0 1 2 3 4 5

0102030

Time (s)

Ang

le (d

eg)

I-PDPID

minus30

minus20

minus10

Time (s)


0 05 1 15 2 25 3 35 4 45 50

005

01

015

Time (s)

Am

plitu

de (m

)

0 05 1 15 2 25 3 35 4 45 5

0102030

Am

plitu

de (r

ad)

minus30

minus20

minus10

Time (s)

I-PDPID








0 002 004

0

002

004

006

Reference

minus006minus006

minus004

minus004

minus002

minus002 0 002 004

0

002

004

006

minus006minus006

minus004

minus004

minus002

minus002

yb

(m)

xb (m)

I-PD controlyb

(m)

xb (m)



0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

Time (s)

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)

Time (s)Time (s)



xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)


0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)Time (s)

Time (s)Time (s)

xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)






7 Conclusions



References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










LPF

0 1 2 3 4 5 6 7 8Time (s)

Am

plitu

de (m

)

002040608

112141618

2

Frequency (rads)

Pow

er

minus49

minus485

minus48

minus475

times10minus3

100 101 102

Raw data




119901 119870119894 119870119889) If we fix 120591 in


[119860119896119861119896(120588)

119862119896119863119896(120588)

] =

[[[[[[

[

0 0119870119894

120591

1 minus1

120591119870119894minus119870119889

1205912

0 1 119870119901+119870119889

120591

]]]]]]

]

(27)


[119860119901

119861119901

119862119901119863119901

] = [

[

0 1 0

0 0 119870bap1 0 0

]

]

(28)

[119860119891

119861119891

119862119891

119863119891

] = [minus120583 1

120583 0] (29)


119871 (119904) = [119860 119861 (120588)

119862 119863 (120588)] (30)

where

119860 = [

[

119860119896

0 0

119862119896119861119901

119860119901

0

119862119896119863119901119861119891119862119901119861119891119860119891

]

]

119861 = [

[

119861119896(120588)

119863119896(120588) 119861119901

119863119896(120588)119863119901119861119891

]

]

119862 = [119862119896119863119901119861119891 119862119901119861119891119862119891] 119863 = 119863

119896(120588)119863119901119863119891

(31)




I [119871 (119895120596)] lt 120574119898 25 le

forall120596 le 28 (33)

1003816100381610038161003816119871 (119895120596)1003816100381610038161003816 lt 120574ℎ

forall120596 ge 10 (34)






4R [119871 (119895120596)] +I [119871 (119895120596)] lt 1205741forall120596 ge 03 (35)

4R [119871 (119895120596)] +I [119871 (119895120596)] gt 1205742forall120596 ge 03 (36)




119894 where





(120574119898 120574ℎ 1205741 1205742) = (minus1 05 1 minus100) (37)


0 50 100 150 200

0

50

100

150

Real axis

Imag

inar

y ax

is

The upper limit

Low frequency

The lower limit

minus150

minus100

minus50




max119870119894 119870119901119870119889 120574ℓ

120574ℓ


(38)




119870119894 119870119901 and 119870


1198751 119875

5and 119876

1 119876







ℓare

(119870119901 119870119894 119870119889 120574ℓ) = (92859 107806 4138 119464)

(39)


ℓ


119868of the I-PD




0 1

0

05

1

15

Real axis

Imag

inar

y ax

is

Middle frequency

High frequency

minus15

minus1

minus05


120596 = 102 (rads)

120596 = 30 (rads)


0

100

200

Gai

n (d

B)

0

90

Phas

e (de

g)

Frequency (rads)

minus300

minus200

minus100

minus270

minus180

minus90

10minus2 100 102 104

P(s) + F(s)

L(s)

K(s)


Figure 8 Bode plot




0 1 2 3 4 50

005

01

015

Am

plitu

de (m

)

0 1 2 3 4 5

0102030

Time (s)

Ang

le (d

eg)

I-PDPID

minus30

minus20

minus10

Time (s)


0 05 1 15 2 25 3 35 4 45 50

005

01

015

Time (s)

Am

plitu

de (m

)

0 05 1 15 2 25 3 35 4 45 5

0102030

Am

plitu

de (r

ad)

minus30

minus20

minus10

Time (s)

I-PDPID








0 002 004

0

002

004

006

Reference

minus006minus006

minus004

minus004

minus002

minus002 0 002 004

0

002

004

006

minus006minus006

minus004

minus004

minus002

minus002

yb

(m)

xb (m)

I-PD controlyb

(m)

xb (m)



0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

Time (s)

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)

Time (s)Time (s)



xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)


0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)Time (s)

Time (s)Time (s)

xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)






7 Conclusions



References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200

0

50

100

150

Real axis

Imag

inar

y ax

is

The upper limit

Low frequency

The lower limit

minus150

minus100

minus50




max119870119894 119870119901119870119889 120574ℓ

120574ℓ


(38)




119870119894 119870119901 and 119870


1198751 119875

5and 119876

1 119876







ℓare

(119870119901 119870119894 119870119889 120574ℓ) = (92859 107806 4138 119464)

(39)


ℓ


119868of the I-PD




0 1

0

05

1

15

Real axis

Imag

inar

y ax

is

Middle frequency

High frequency

minus15

minus1

minus05


120596 = 102 (rads)

120596 = 30 (rads)


0

100

200

Gai

n (d

B)

0

90

Phas

e (de

g)

Frequency (rads)

minus300

minus200

minus100

minus270

minus180

minus90

10minus2 100 102 104

P(s) + F(s)

L(s)

K(s)


Figure 8 Bode plot




0 1 2 3 4 50

005

01

015

Am

plitu

de (m

)

0 1 2 3 4 5

0102030

Time (s)

Ang

le (d

eg)

I-PDPID

minus30

minus20

minus10

Time (s)


0 05 1 15 2 25 3 35 4 45 50

005

01

015

Time (s)

Am

plitu

de (m

)

0 05 1 15 2 25 3 35 4 45 5

0102030

Am

plitu

de (r

ad)

minus30

minus20

minus10

Time (s)

I-PDPID








0 002 004

0

002

004

006

Reference

minus006minus006

minus004

minus004

minus002

minus002 0 002 004

0

002

004

006

minus006minus006

minus004

minus004

minus002

minus002

yb

(m)

xb (m)

I-PD controlyb

(m)

xb (m)



0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

Time (s)

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)

Time (s)Time (s)



xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)


0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)Time (s)

Time (s)Time (s)

xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)






7 Conclusions



References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 50

005

01

015

Am

plitu

de (m

)

0 1 2 3 4 5

0102030

Time (s)

Ang

le (d

eg)

I-PDPID

minus30

minus20

minus10

Time (s)


0 05 1 15 2 25 3 35 4 45 50

005

01

015

Time (s)

Am

plitu

de (m

)

0 05 1 15 2 25 3 35 4 45 5

0102030

Am

plitu

de (r

ad)

minus30

minus20

minus10

Time (s)

I-PDPID








0 002 004

0

002

004

006

Reference

minus006minus006

minus004

minus004

minus002

minus002 0 002 004

0

002

004

006

minus006minus006

minus004

minus004

minus002

minus002

yb

(m)

xb (m)

I-PD controlyb

(m)

xb (m)



0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

Time (s)

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)

Time (s)Time (s)



xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)


0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)Time (s)

Time (s)Time (s)

xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)






7 Conclusions



References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 002 004

0

002

004

006

Reference

minus006minus006

minus004

minus004

minus002

minus002 0 002 004

0

002

004

006

minus006minus006

minus004

minus004

minus002

minus002

yb

(m)

xb (m)

I-PD controlyb

(m)

xb (m)



0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

Time (s)

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)

Time (s)Time (s)



xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)


0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0005

01

02040

minus01minus005

minus40minus20

02040

minus40minus20

0005

01

minus01minus005

Time (s)Time (s)

Time (s)Time (s)

xb

(m)

120572(d

eg)

yb

(m)

120573(d

eg)






7 Conclusions



References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











7 Conclusions



References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article generalized kalman-yakubovich-popov...

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