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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 18, NO. 6, DECEMBER 2010 1041

A Fuzzy Approach for Robust Reference-Tracking-Control Design of Nonlinear Distributed Parameter

Time-Delayed Systems and Its ApplicationYu-Te Chang and Bor-Sen Chen , Fellow, IEEE

Abstract—This paper addresses the robust reference-tracking-control problem for nonlinear distributed parameter systems(NDPSs) with time delays, external disturbances, and measure-ment noises. The NDPS is measured at several sensor locationsfor output-feedback tracking control. A fuzzy-spatial state-spacemodel derived via finite-difference approach is introduced to rep-resent the nonlinear distributed parameter time-delayed system.Thus, we use a fuzzy interpolation method with several local lin-ear systems to approximate the nonlinear system and employ thefinite-difference method to approximate the partial differential op-

erators in fuzzy-spatial state-space model. Based on this model, arobust fuzzy-observer-based reference-tracking controller is pro-posed to control the NDPS to track a desired reference trajec-tory. First, a 2-D H

∞ tracking performance in a spatiotemporaldomain is proposed for robust tracking design of nonlinear dis-tributed parameter time-delayed systems. Then, an equivalent 1-DH ∞ reference-tracking design is developed to simplify the design

procedure, and the linear-matrix-inequality (LMI) technique is ap-plied to solve the control gains and observer gains for the robustH ∞ tracking-design problem via a systematic control-design pro-

cedure. Finally, a tracking-control-design example for the nervoussystem is given to confirm the proposed reference-tracking-controlscheme of nonlinear distributed parameter time-delayed systems.

Index Terms—Finite-difference approach, fuzzy interpola-

tion method, nonlinear distributed parameter systems (NDPSs),reference-tracking control, robust observer-based tracking con-trol, spatial state-space model, time delay.

I. INTRODUCTION

MOST physical systems are inherently distributed in space

and time, e.g., chemical engineering [1], biodynam-

ics [2], [3], and mechanical systems related to heat flows, fluid

flow, elastic wave, or flexible structure [4], [5]. In the past,

most physical systems were modeled by ordinary differential

equations (ODEs) in order to simplify and systematically solve

control-design problems. However, it is not sufficient to modelthe physical systems, if we consider the variation of the system

that is depended on the space. In chemical engineering, many

Manuscript received December 14, 2009; revised April 12, 2010; acceptedJune 17, 2010. Date of publication July 15, 2010; date of current versionDecember 3, 2010. This work was supported by the National Science Council(NSC) under Contract NSC 98-2221-E-007-113-MY3.

The authors are with the Laboratory of Control and Systems Biology, Depart-ment of Electrical Engineering, National Tsing Hua University, Hsinchu 30013,Taiwan (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TFUZZ.2010.2058809

chemical processes are characterized by the presence of spa-

tial variations and time delays [1]. Thus, the reaction–diffusion

equation is introduced to represent the chemical process. In

recent years, interactions between the mathematical and bio-

logical sciences have been increasing rapidly [2], [3]. In biol-

ogy, mathematical tools can help provide systematic analysis,

e.g., the stability or the robustness of biological systems. How-

ever, the evolution of physiological behavior is dependent on

time and space. For example, the nonlinear partial differentialHodgkin–Huxley (H–H) model has been applied to model sig-

nal transmission in a nervous system [2], [6]. Therefore, the

distributed parameter system (DPS), which is described by par-

tial differential equation (PDE), is more suitable to model the

spatiotemporal dynamic systems in biology.

In general, the tasks of control systems are one of two cate-

gories: stabilization and tracking. The stabilization problem is

to design a controller so that the states of the closed-loop system

can converge to an equilibrium point. In the tracking problem, a

controller is designed to guarantee that the output of the closed-

loop system can track a desired reference trajectory. Many stud-

ies have investigated the stabilization design problem of linear

DPSs (LDPSs), for example, the stability analysis of the LDPSs

introduced in [7] and [8]. Similarly, a robust stabilization of the

LDPS with the external disturbances has been developed to at-

tenuate the effect of external disturbances from the H ∞ control

point of view [9]. However, the control-design problem of non-

linear DPSs (NDPSs) is more complex than the control design

of the LDPSs. Based on Galerkin’s method, controller-design

schemes have been proposed to stabilize the NDPS by a residual

model filter [10] or an inertial manifold model [11]. Over the

past 20 years, the fuzzy approach that uses several local linear

models to interpolate a nonlinear system has been widely ap-

plied to the analysis of the nonlinear systems in various fields,

which are described by ODE [12]–[29]. Recently, the fuzzyapproach was already applied to the field of PDEs. A new tech-

nique using the adaptive fuzzy algorithm is proposed to obtain

the solutions of PDEs [30]. According to the adaptive scheme of

fuzzy-logic systems, a fuzzysolution with adjustable parameters

for the PDE can be obtained successfully. On the other hand,

an H ∞ fuzzy-observer-based control design [31] is proposed

for a class of nonlinear parabolic PDE systems with control

constraints. In addition, a robust stabilization problem for the

NDPS with time delay is studied using a fuzzy-control approach

in [32]. Galerkin’s method is applied to derive a set of nonlinear

ODEs for the NDPS [31], [32]. A new fuzzy state-space model

is proposed in [33] to represent the NDPSs based on Galerkin’s

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1042 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 18, NO. 6, DECEMBER 2010

method, with the advantage of avoiding obtaining a complex

nonlinear ODE. The robust H ∞ stabilization is developed to

attenuate the effects of modeling errors, external disturbances,

and measurement noises [33]. However, in a tracking-design

case, the asymptotic tracking cannot be achieved due to the

changing reference signal via Galerkin’s method, because this

tracking error will affect the residual subsystem continuously.Therefore, this robust H ∞ control-design method in [33] can-

not be extended from the stabilization problem to the tracking

problem of NDPSs.

For the reference-tracking problem of DPSs, there is only a

study of the linear case [34]. Byrnes et al. proposed an extended

output-regulation method to control the LDPS to track a refer-

ence model [34]. For the nonlinear case, to the best of our knowl-

edge, no result of tracking design for NDPSs has been presented

because of the complex nonlinearity, distributed parameter, and

design difficulty. However, the tracking-control design is a more

important control problem in practical applications, because a

system needs to be controlled in order to track a reference signal.

Recently, the tracking-control problems of biological systemshave become a very important topic in the biomedical engi-

neering. However, biological systems are always nonlinear and

compartment-dependent with process delays. For example, the

control problem of the nervous system in [6], [35], and [36]

should be about how to design a controller to make the mem-

brane voltage of nervous system tracking a given membrane-

voltage reference time-course, e.g., short electrical pulses, and

not simply how to stabilize the nervous system. In a realistic

model, a space-clamped axon was described as the nonlinear

H–H model of the spatiotemporal dynamics [36]. In this situa-

tion, the tracking-control design of NDPSs has the potential to

create a real-time therapeutic regime for the undesired neural os-cillation caused by disturbances and environmental noises [35],

[37]–[39]. In addition, the problem of time delay commonly

emerges in practical systems. Therefore, this paper studies a

robust tracking-control-design problem for NDPSs with time

delays, external disturbances, and measurement noise.

Tseng et al. studied a robust tracking problem for a non-

linear ODE system using the fuzzy-tracking-control scheme,

where a Takagi–Sugeno (T–S) fuzzy model was initially used

to represent the nonlinear ODE system [12]. Then, a fuzzy-

observer-based controller was developed to reduce the tracking

error as much as possible via the H ∞ tracking performance.

In this study, a robust tracking-control scheme for NDPSs with

time delays, external disturbances, and measurement noises is

proposed based on a fuzzy-observer-based controller. A design

procedure is introduced as follows. First, a fuzzy DPS with time

delay is proposed to approximate the NDPS with time delay by

interpolating several linear distributed parameter time-delayed

systems. Unlike using the infinite-dimensional ODE system to

represent the PDE system [33], for the convenience of tracking-

control design, the partial differential operator on space in PDE

could be approximated by a finite-difference operator. Then,

we can obtain a set of fuzzy finite-difference systems with the

approximation errors and the truncation errors to represent the

NDPS. When all finite-difference grid points are represented by

a spatial vector, a set of the fuzzy finite-difference systems is

represented by an equivalent fuzzy-spatial state-space system

by the Kronecker product method. In the output-feedback con-

trol design, a fuzzy observer based on fuzzy-spatial state-space

system is developed to estimate the state of the NDPS from the

output measurements at several sensor locations. Finally, a ro-

bust fuzzy-observer-based tracking-control scheme is proposed

to control the NDPS with time delay to track a reference model.At the same time, it also attenuates the effects of the time delay,

the approximation errors, the truncation errors, the measure-

ment noises, and the external disturbances. In order to treat the

robust tracking problem, a 2-D H ∞ tracking performance in a

spatiotemporal domain is introduced for NDPSs to efficiently at-

tenuate the effects of the time delay, the approximation error, the

truncation error, the external disturbance, and the measurement

noise. For the convenience of tracking-control design, the 2-D

H ∞ tracking performance is transformed to an equivalent 1-D

H ∞ tracking performance, when all finite-difference grid points

are represented by a spatial vector. Based on H ∞ attenuation

theory [12], the proposed fuzzy-observer-based tracking con-

troller is proved to guarantee that the NDPS can robustly track a reference model by efficiently eliminating the effect of time

delay, the approximation error, the truncation error, the external

disturbance, and the measurement noise on the tracking error

below a prescribed level. The effect of using a finite-difference

operator to approximate the partial differential operator on the

H ∞ tracking performance is also discussed. We have found that

the effect due to finite-difference approximation is of the order

O(∆2x ), when ∆x is the distance of the neighboring grid points.

Therefore, if the grid points in a spatial domain are dense enough

or the truncation error O(∆2x ) is small enough, the H ∞ track-

ing performance of the fuzzy-spatial state-space system will

approach the H ∞ tracking performance of the NDPS.Generally, in the observer-based control design of both ODE

and PDE systems, how to solve coupling linear matrix inequali-

ties (LMIs) to obtaincontrol gain and observer gain is still a diffi-

cult problem. A two-step procedure [40]–[42] and a transformed

technique [33] have previously been proposed to conservatively

solve the coupling LMIs. Although this coupling problem could

be solved by a BMI optimal technique [43], the BMI problem

is not a convex problem. Especially for a complex multivariable

constraint case, BMI is not an efficient method to solve the cou-

pling LMIs of fuzzy-observer-based tracking design for NDPSs.

In [44], a single-step approach was proposed to solve the BMI

problem for matrix decoupling of the fuzzy-observer-based sta-

bilization design of the fuzzy time-delay systems. In this study,

the BMI problem is approximated to an LMI problem via the

Schur complement and several inequalities [45]. Therefore, the

coupling problem can be efficiently solved by the conventional

LMI technique in the design procedure. Finally, in order to

emerge the importance of practical application, an example of

the reference-tracking-control design for the H–H nervous sys-

tem [2], [35], [46]–[48] in biochemical engineering is given to

illustrate the design procedure and to confirm the robust track-

ing performance. This tracking-control design of the H–H ner-

vous system is useful to suppress oscillations and blockage of

action potential transmission for patients suffering from nervous

system dysfunction.



CHANG AND CHEN: FUZZY APPROACH FOR ROBUST REFERENCE-TRACKING-CONTROL DESIGN 1043

The main contribution of this paper is given as follows.

1) The difficulty of the reference-tracking-control design for

an NDPS with time delay, external disturbances, and mea-

surement noises was overcome by the proposed fuzzy-

spatial state-space model via a finite-difference method.

2) A 2-D H ∞ tracking performance in a spatiotemporal do-

main is proposed for robust tracking design of NDPS andis then transformed into an equivalent 1-D H ∞ tracking

performance for a fuzzy-spatial state-space system.

3) A fuzzy-observer-based tracking controller is success-

fully developed to control the NDPSs to efficiently track

a desired reference model based on the H ∞ tracking

performance.

This robust H ∞ reference-tracking-control scheme can be

systematically designed through output feedback via the help of

the MATLAB LMI toolbox.

Notations: For the convenience of problem description and

control design, we define the separable Hilbert space in the

following.

1) y(t)2 =

ni= 1 |yi (t)|2 , where y(t) = [y1 (t), . . . ,

yn (t)]T .

2) L2 (R+ ;Rn ) is the space of the n-dimensional measurable

functions z(t) ∈ Rn defined on t ∈ R+ = [0,∞) such

that z(t)2L2 (R+ ;Rn )

= ∞

0 z(t)2 dt < ∞.

II. REFERENCE-TRACKING-CONTROL-PROBLEM

FORMULATION FOR NONLINEAR DISTRIBUTED

PARAMETER TIME-DELAYED SYSTEMS

In this section, the reference-tracking control problems for

the NDPS are formulated. Time delays, external disturbances,

and measurement noise frequently appear in practical physicalsystems. Therefore, we consider the following NDPS with time

delay, external disturbance, and measurement noise:

∂y(x, t)

∂t = Ay(x, t) + f (y(x, t), y(x, tτ ))

+ g(x)u(t) + gd (x)d(t) (1)

z(t) = h(x)y(x, t) + Dn n(t) (2)

for x = [x1 , x2 ]T ∈ U ⊂ R2+ and t > 0, where y(x, t)

=

[y1 (x, t), . . . , yn (x, t)]T ∈ Rn is the state variable, x and t are

the space and time variables, respectively, f (y(x, t), y(x, tτ )) ∈Rn is a nonlinear function satisfying f (0, 0) = 0, tτ denotes

the delayed time, i.e., tτ = t− τ , τ > 0, the distribution of

the control force u(t) is provided by p-point force actuators,

i.e., u(t) ∈ R p is the applied force to be designed as u(t) =[u1 (t), . . . , u p (t)]T , and the influence function g(x) is an n × p-matrix form whose elements are of delta function δ (x− pi ).

For example, n = 1, g(x) = [g1 δ (x− p1 ), . . . , g p δ (x − p p )],

where pi are control-force locations, and gi δ (x− pi ) = gi

for x = pi or gi δ (x− pi ) = 0 for x = pi . The measured out-

put z(t) may be interpreted as observations or as parts of

the system whose behavior we wish to influence, and the

output z(t) ∈ Rq is a vector, i.e., z(t) = [z1 (t), . . . , zq (t)]T ,

where q is the number of observations and the observation

influence function h(x) ∈ Rq ×n is a matrix function. For ex-

ample, n = 1, h(x) = [δ (x− q 1 ), . . . , δ (x− q q )]T . Therefore,

in the case that is free from measurement noise, we have

zi (t) = δ (x− q i )y(x, t) = y(q i , t), with q i as the ith sensor

locations. gd (x) ∈ Rn× pd is an interactive location matrix of

the external disturbance, d(t) ∈ L2 (R+ ;R p d ) is the vector of

the external disturbance, Dn ∈ Rq is the influence matrix of the

measurement noise, and n(t) ∈ L2 (R

+ ;R

) is the measurementnoise at the locations of observation. The differential operator

A in L2 (U ;Rn ) is defined as follows [7]:

Ay(x, t) = κ1

∂ 2

∂x21

y(x, t) + κ2∂ 2

∂x22

y(x, t)

where (∂ 2 /∂x2k )y(x, t)

= [(∂ 2 /∂x2

k )y1 (x, t), . . . , (∂ 2 /∂x2k )

yn (x, t)]T ∈ Rn , for k = 1 and 2, in the 2-D case, κ1 and

κ2 are the diagonal matrices, y(x, t) ∈ D(A) = y(x, t) ∈L2 (U ;Rn )| y(x, t), (∂y(x, t)/∂x1 ), (∂y(x, t)/∂x2 ) are ab-

solutely continuous, and κ1 (∂ 2 y(x, t)/∂x21 ) + κ2 (∂ 2 y(x, t)/

∂x22 ) ∈ L2 (U ;Rn ). The initial value is given by y(x, 0) =

y0 (x). The boundary condition is given by the Dirichlet bound-ary condition, i.e., y(x, t) = 0 on ∂U , or the Neumann boundary

condition, i.e., ∂y(x, t)/∂x = 0 on ∂U . Remark 1: In this paper, we address the robust tracking-

control-design problem for NDPSs via the proposed fuzzy ap-

proach. Therefore, we studied the constant delay for NDPSs. In

general, the cases of time-varying delay or multiple time delays

can be studied by combining the proposed fuzzy approach and

the delay-dependent approach for ODE systems [21]–[24].

A desired reference trajectory yR (x, t) ∈ Rn is generated by

the following linear distributed parameter reference model:

∂yR (x, t)

∂t = AR y

R

(x, t) + AR yR

(x, t) + gR (x)r(t) (3)

whereAR is a specified linear differential operator, AR isa spec-

ified matrix, gR (x) is a specified influence function, and r(t) is a

bounded reference input. The linear distributed parameter refer-

ence model in (3) can be designed similar to the reference model

in [12] and [34]. First, we design the matrix AR and the refer-

ence input r(t) to decide the behavior of the time evolution [34].

Then, we choose the appropriate diffusion coefficients in differ-

ential operator AR according to the practical-application case.

The task of tracking control is to make y(x, t) in (1) and (2)

track the desired trajectory yR (x, t) generated by (3).

The external disturbance d(t) and themeasurement noise n(t)

are uncertain, and the reference input r(t) could be arbitrarilyassigned by users, which can be all considered as disturbancesof

the tracking system. Therefore, the robust 2-D tracking-control

design should be specified so that the effect of the external

disturbance d(t), the measurement noise n(t), and the reference

input r(t) on the tracking error in the spatiotemporal domain

must be below a prescribed level ρ as follows: tf

0

U yR (x, t)− y(x, t)2 dxdt tf

0 v(t)2 dt≤ ρ2 (4)

or

tf

0

U yR (x, t)− y(x, t)

2

dxdt ≤ ρ

2 tf

0 v(t)

2

dt (5)




where v(t) = [r(t)T , d(t)T , n(t)T ]T is considered as a vector

of disturbances. If the initial condition is considered, then tf

0

U

yR (x, t)− y(x, t)2 dxdt≤V (y0 ) + ρ2

tf

0

v(t)2 dt

(6)

for some positive function V (·) > 0. The inequality in (4) or (5)

is called the 2-D H ∞ tracking performance on the spatiotempo-ral domain. Its physical meaning is that the effect of all distur-

bances on the tracking error at the total space U × [0, tf ] must

be attenuated below a prescribed level ρ from the energy per-

spective. Note that the norm tf

0

U yR (x, t)− y(x, t)2 dx dt

describes the energy of tracking error for all positions in the

spatiotemporal space U × [0, tf ], which is different from the

conventional H ∞ tracking performance that is only in the time

domain.

The robust H ∞ tracking-control problem is formulated as

follows: Given a prescribed disturbance attenuation level ρ, a

robust controller u(t) is designed to attenuate the effect of the

external disturbances d(t) and measurement noises n(t) on thetracking error yR (x, t) − y(x, t) below ρ from the view point

of total energy on the space U × [0, tf ] to robustly track the

desired reference trajectory yR (x, t). In other words, the H ∞tracking performance in (5) is achieved.

Remark 2: The fuzzy-tracking-control design for nonlinear

ODE systems in [12] has previously been used to control the

states of the nonlinear dynamic system to track a desired state

trajectory; however, in this paper, it is more difficult to design

the tracking control of the NDPS in (1) because the NDPS is

a spatiotemporal dynamic system, and the H ∞ tracking per-

formance (4) or (6) should be achieved in the spatiotemporal

domain. An output regulation for an LDPS developed previ-

ously [34] is designed to track a reference output generated by

an exogenous system. This reference output is limited to some

specific trajectories, e.g., stable trajectory or sinusoidal trajec-

tory. However, in this paper, the tracking problem of NDPSs is

to track any desired spatiotemporal trajectory, which could be

generated in (3). Therefore, this tracking-control-design scheme

is more general than the output-regulation design in [34].

Remark 3: We previouslyused Galerkin’s method to represent

a PDE by an infinite-dimensional ODE to solve a stabilization

problem for the NDPS [33]. However, the effect of residual

subsystem on the tracking error cannot be neglected because

the asymptotic tracking cannot be achieved in NDPS due to

the change of yR (x, t). Moreover, Galerkin’s method is diffi-cult to extend into the multispace variable case for the state

y(x, t). Therefore, in this paper, we applied the finite-difference

approach to solve the tracking-control problem for the NDPS.

III. SYSTEM REPRESENTATION BY FUZZY-SPATIAL

STATE-SPACE MODEL

In the control-design problems of NDPSs, the main prob-

lem is to obtain a suitable state-space model to represent the

NDPSs. At first, a T–S fuzzy DPS with time delay is proposed

to approximate the NDPS with time delay as follows [33]:

Rule i : IF y1 (x, t) is F 1i , and . . . , and, yn (x, t) is F ni

THEN∂y(x, t)

∂t = Ay(x, t) + Ai y(x, t) + Aτ ,i y(x, tτ )

+ g(x)u(t) + gd (x)d(t)

where F j i is the grade of the membership of y j (x, t), and Ai ∈

R

n×n

and Aτ ,i ∈ R

n×n

are the system parameters without timedelay and with time delay τ in local LDPSs, respectively. The

overall fuzzy DPS can be formulated as follows [12], [33], [49],

[50]:

∂y(x, t)

∂t =

M i= 1

µi (y(x, t))Ay(x, t) + Ai y(x, t)

+ Aτ ,i y(x, tτ )

+ g(x)u(t) + gd (x)d(t) + ε(x, t)

(7)

where µi (y(x, t)) = (

n j = 1 F j i (y j (x, t)))/(

M i= 1

n j =1 F j i

(y j (x, t))). F j i (y j (x, t)) is the grade of the membership of

y j (x, t) or the possibility function of y j (x, t). The denomina-

tors of µi (y(x, t)) are only for normalization so that the total

sum of the fuzzy basesM

i= 1 µi (y(x, t)) = 1.

In (7), we use the fuzzy interpolation via M local LDPSs to

approximate the NDPS in (1). Theapproximation error ε(x, t) ∈R

n is defined as follows:

ε(x, t) = f (y(x, t), y(x, tτ ))

−M

i= 1

µi (y(x, t))[Ai y(x, t) + Aτ ,i y(x, tτ )]. (8)

The bound of ε(x, t) could be estimated according to the fol-

lowing theorem.

Theorem 1 [33]: Supposing f (y(x, t), y(x, tτ )) is a con-

tinuous function defined on a compact set U ⊂ Rn , i.e.,

y(x, t), y(x, tτ ) ∈ U, then for two arbitrary constants σ >

0, and στ > 0, the fuzzy function f (y(x, t), y(x, tτ )) =

M i= 1 µi (y(x, t))[Ai y(x, t) + Aτ ,i y(x, tτ )] could be con-

structed to approximate the nonlinear function f (y(x, t),

y(x, tτ )), and the approximation error is bounded by σ andστ , i.e., ε(x, t)2 ≤ σ2y(x, t)2 + σ2

τ y(x, tτ )2 .

The finite-difference scheme [51], [52] has been widely ap-

plied to obtain numerical solutions of PDEs. In this paper, we

use the finite-difference method to represent the NDPS. Con-

sider a typical grid mesh, as shown in Fig. 1. The state y(x, t)is represented by yk, l (t) ∈ Rn at the grid node xk, l (x1 = k∆x ,

x2 = l∆x ), where k = 0, . . . , N 1 + 1, and l = 0, . . . , N 2 + 1,

i.e., y(x, t)|x= xk , l

= yk, l (t). Note that the grid nodes k = 0,

k = N 1 + 1, l = 0, or l = N 2 + 1 are the grid nodes at the

boundary. At the interior points of grid, i.e., 0 < k < N 1 + 1,and 0 < l < N 2 + 1, the central-difference approximation for

the linear differential operator can be written as follows




Fig. 1. Finite-difference grids on the spatiodomain.

[51], [52]:

(Ay(x, t))x=x k , l

= κ1

∂ 2 y(x, t)

∂x21

x= xk , l

+ κ2

∂ 2 y(x, t)

∂x22

x=x k , l

= κ1yk +1 ,l (t) + yk−1,l (t)− 2yk ,l (t)

∆2x

+ κ2yk ,l + 1 (t) + yk ,l−1 (t)− 2yk, l (t)

∆2x

+Ok ,l (∆2x ). (9)

The remainder term Ok ,l (∆2x ) ∈ R

n is called the local trunca-

tion error.

A fuzzy finite-difference model can be constructed to repre-

sent the state yk ,l (t) of the NDPS at x = xk ,l in (7) as follows:

yk ,l (t) =

M

i=1

µi (yk, l )

Ai yk ,l (t) + Aτ ,i yk ,l (tτ )

+ Bk ,l u(t) + Bd,k,ld(t) + εk, l (t)

+ 1

∆2x

κ1 yk + 1,l (t) + 1

∆2x

κ1 yk−1,l (t)− 2

∆2x

κ1 yk ,l (t)

+ 1

∆2x

κ2 yk, l+1 (t) + 1

∆2x

κ2 yk, l−1 (t)− 2

∆2x

κ2 yk ,l (t)

+Ok ,l (∆2x ). (10)

The elements of matrix Bk ,l = [Bk,l,1 , . . . , Bk,l,p ] ∈ Rn× p are

defined as Bk,l,i = gi for xk, l = pi , or Bk,l,i = 0 for xk, l = pi ,

where gi is the influence function at the location pi and is

defined in (1). The definition of Bd,k,l is similar to Bk ,l . The

approximation error is given by εk ,l (t) = ε(x, t)|x= xk , l .We defined a spatial state vector y(t) to collect the states

yk ,l (t) ∈ Rn at all grid nodes in Fig. 1. For Dirichlet boundary

conditions [52], the values of yk, l (t) at boundary are fixed,

for example, y(x, t) = 0 on ∂U . We have yk, l (t) = 0 at k =0, N 1 + 1, or l = 0, N 2 + 1. Therefore, the spatial state vector

y(t) ∈ Rn N is defined as follows:

y(t) = [yT 1,1 (t), . . . , yT

1,l (t), . . . , yT 1,N 2

(t), . . . , yT k, l (t)

. . . , yT N 1 ,1 (t), . . . , yT

N 1 ,l (t), . . . , yT N 1 ,N 2

(t)]T (11)

where N = N 1 × N 2 . Note that n is the dimension of the

vector yk ,l (t) for each grid node, and N 1 × N 2 is the number

of grid nodes. For example, letting N 1 = 2, and N 2 = 2, then

we have y(t) = [yT 1,1 (t), yT

1,2 (t), yT 2,1 (t), yT

2 ,2 (t)]T . For Neu-

mann boundary conditions [52], i.e., ∂ y(x, t)/∂x = 0 on ∂ U ,the boundary condition is given as (∂y(x, t)/∂x)x= xk , l

= 0 at

k = 0, N 1 + 1, l = 0, N 2 + 1. Therefore, if the grid nodes at

boundary are also considered in the spatial state vector y(t),

then y(t) ∈Rn N

in (11) should be modified as follows:y(t) = [yT

0,0 (t), . . . , yT 0,l (t), . . . , yT

0,N 2 +1 (t), . . . , yT k, l (t)

. . . , yT N 1 + 1,0 (t), . . . , yT

N 1 +1 ,l (t), . . . , yT N 1 +1 ,N 2 +1 (t)]T

(12)

where N = (N 1 + 2)× (N 2 + 2).

In order to simplify the index of the node yk, l (t) ∈ Rn in

the spatial state vector y(t) ∈ RnN , we denote the symbol

y j (t) ∈ Rn to replace yk, l (t). Note that the index j is from 1

to N , i.e., y1 (t) = y1,1 (t), y2 (t)

= y1,2 (t), . . ., y j (t)

= yk, l (t),

. . ., yN (t) = yN 1 ,N 2 (t), where j = (k − 1)N 1 + l in (11). The

fuzzy finite-difference model of two indices in (10) could berepresented with only one index as follows:

y j (t) =M

i= 1

µi (y j )

Ai y j (t) + Aτ ,i y j (tτ )

+ B j u(t)

+ Bd, j d(t) + ε j (t) + T j y(t) + O j (∆2x ) (13)

where finite-difference matrix T j ∈ Rn×nN expresses the inter-

action from the other grid nodes to the node y j (t) = yk ,l (t) as

follows:

T j y(t) =

κ1

∆2x

yk + 1,l (t) + κ1

∆2x

yk−1,l (t) − 2κ1

∆2x

yk, l (t)

+ κ2

∆2x

yk ,l +1 (t) + κ2

∆2x

yk ,l−1 (t)− 2κ2

∆2x

yk ,l (t).

The measurement output z(t) in (2) can be represented as

follows:

z(t) =

N j = 1

C j y j (t) + Dn n(t) (14)

where the matrix C j is defined as C j= C k, l = [C k,l,1 , . . . ,

C k,l,q ]T ∈ R

q ×n , in which the element is given as C k,l,i = I nfor xk, l = q i , or C k,l,i = 0 for xk ,l = q i . Note that q i is the ith

sensor location, as defined in (2). The matrix Dn is also defined

in (2).We collect all states y j (t) of grid nodes in (13) and (2) to

the state y(t) in (11) or (12). Then, a diagonal fuzzy weighting

matrix µi (y) ∈ RN ×N is defined as µi (y) = diag(µi (y1 ), . . . ,

µi (yN )) and the n × n identity matrix by I n . The Kronecker

product can be used to simplify the representation. Some prop-

erties for µi (y) can be obtained as follows:

Lemma 1: Using the properties of Kronecker product, we

have the following properties.

1) µi (y) ⊗ Ai = (µi (y) ⊗ I n )(I N ⊗ Ai ) ∈ RnN ×nN

where Ai ∈ Rn×n .

2) (µi (y) ⊗ I n )(µ j (y) ⊗ I n ) = (µi (y)µ j (y) ⊗ I n ) ∈

RnN ×nN .




3)M

i= 1 µi (y) = I N ,M

i= 1 (µi (y)⊗ I n ) = I nN andM

i= 1M j = 1 (µi (y)µ j (y)⊗ I n ) = I nN .

Proof: Theproperties 1) and 2) are the fundamental properties

for the Kronecker product [53]. The property 3) can be proven

via the fuzzy fundamental propertyM

i=1 µi (y j (t)) = 1 for j =1, . . . , N .

Using the Kronecker product, the systems in (13) and (14)can be written as the following fuzzy-spatial state-space system:

y(t) =

M i=1

(µi (y)⊗Ai )y(t) + (µi (y)⊗Aτ ,i )y(tτ )

+ T y(t) + Bu(t) + Bd d(t) + ε(t) + O(∆2

x )

=M

i=1

µi (y)

(I N ⊗Ai )y(t) + (I N ⊗Aτ ,i )y(tτ )

(15)

+ T y(t) + Bu(t) + Bd d(t) + ε(t) + O(∆2x )

z(t) = C y(t) + Dn n(t) (16)

where we define µi (y) = µi (y)⊗ I n , and the spatial state vector

y(t) ∈ Rn N is denoted in (11) or (12) to represent y(x, t) at all

finite-difference grid points on the spatial domain in Fig. 1. The

corresponding matrices are defined as T = [T T 1 , . . . , T T

N ]T ∈

RnN ×nN , B = [BT 1 , . . . , BT

N ]T ∈ RnN × p , Bd = [BT

d, 1 , . . . ,

BT d, N ]

T ∈ Rn N × p d , and C = [C 1 , . . . , C N ] ∈ Rq ×n N . The

matrix Dn is defined in (2). The approximation error ε(t) is

defined as ε(t) = [ε1 (t)T , . . . , εN (t)T ]T .

The physical meaning of (15) and (16) is that the NDPS in

(1) and (2), at all finite-difference grid points on the spatial

domain in Fig. 1, can be represented by the fuzzy-spatial state-

space system in (15) and (16). In Theorem 1, the bound of approximate error ε(x, t) can be proved to be less than two

arbitrary constants σ and στ . The bound for approximate error

ε(t) can be obtained by the following corollary.

Corollary 2: If the bounds of ε(x, t) are provided with σ and

στ , then the bound of ε(t) could also be estimated as ε(t)2 ≤σ2y(t)2 + σ2

τ y(tτ )2 .

Proof: By Theorem 1, we have ε j (t)2 ≤ σ2y j (t)2 +σ2

τ y j (tτ )2 . Thus

ε(t)2 =N

j =1

ε j (t)2 ≤N

j = 1

σ2y j (t)2 + σ2τ y j (tτ )

2

= σ 2y(t)2 + σ2τ y(tτ )

2 .

Remark 4: Similarly, the reference model in (3) could be

transformed into a linear spatial state-space reference model by

the finite-difference method, i.e., the desired reference trajectory

yR (t) at all grid points can be generated by the following linear

spatial state-space reference model:

yR (t) = (I N ⊗AR )yR (t) + T R yR (t) + BR r(t) + OR (∆2x )

(17)

where AR is a specified system matrix, BR is the specified

influence matrix, T R is the finite-difference matrix, andOR (∆2x )

is the truncation error.

Fig. 2. Block diagram of the tracking-control scheme for the NDPS in (1) and(2).

Remark 5: The state variable y(x, t) in (1) depends on the

space x and time t. Based on the finite-difference scheme [51],

[52], the spatial state vector y(t) in (11) or (12) was used to

represent the y(x, t) at all grid points. Similarly, the state vari-

ables yR (x, t), at all grid points, can be represented by spa-

tial state vector yR (t). In this situation, the t

f 0

U yR (x, t) −y(x, t)2 dxdt in the 2-D H ∞ tracking performance (4) or (6)

could be modified by the 1-D form tf

0 yR (t)− y(t)2 ∆2x dt

for the finite-difference systems (15) and (17) to simplify the

design procedure. As ∆x → 0, the 1-D systems in (15) and (17)

will approach 2-D systems (7) and (3), respectively, and the 1-D

integration form will approach the 2-D integration form.

IV. ROBUST FUZZY-OBSERVER-BASED

TRACKING-CONTROL DESIGN

In Section III, a fuzzy DPS in (7) was proposed to ap-

proximate the NDPS in (1). Then, according to the finite-

difference method, the fuzzy-spatial state-space model in (15)and (16) can be constructed for NDPSs. In this section, a fuzzy-

observer-based tracking-control scheme is developed to solve

the tracking-control-design problem of NPDSs. Then, a set of

the complex matrix inequalities is derived to guarantee the ro-

bust H ∞ tracking performance. Finally, the design procedure is

given to simplify the robust tracking-design problem of NDPSs

by solving a set of LMIs to obtain the controller gains and

observer gains.

A. Robust H ∞ Fuzzy-Observer-Based

Tracking-Control Design

A fuzzy-observer-based tracking controller based on the

fuzzy-spatial state-space system in (15) and (16) is proposed

to estimate the state of fuzzy-spatial state-space system and

then using a state feedback scheme to control the NDPS in (1)

and (2) to thereby robustly track a desired trajectory yR (x, t)generated by the reference model in (3). The fuzzy-observer-

based tracking controller is proposed as the following form (see

Fig. 2):

˙y(t) =

M i=1

µi (y)

(I N ⊗Ai )y(t) + (I N ⊗Aτ ,i )y(tτ )

+ T y(t) + Bu(t) + Gi (C y(t)− z(t))

(18)




u(t) =M

i=1

N j = 1

µi (y j )K i, j [y j (t) − yR ,j (t)] (19)

=M

i=1

K i µi (y)[y(t)− yR (t)] (20)

where Gi ∈ RnN ×q and K i

= [K i, 1 , . . . , K i, N ] ∈ R p×n N are,

respectively, the observer gains and the control gains to be

designed. The fuzzy weighting matrix µi (y) is defined as

µi (y) = (µi (y)⊗ I n ). In the observer equation (18), the ob-

server gains Gi are designed so that the spatial state-estimation

error y(t)− y(t) is as small as possible. In (19), the N

fuzzy tracking controllers u j (t) = M

i=1 µi ( y j )K i, j [y j (t) −yR ,j (t)] ∈ R

p are designed to make the tracking errors y j (t) −yR ,j (t) as small as possible in spite of the fuzzy approximation

error, the truncation error, the time delay, the external distur-

bance, and the measurement noise, i.e., to achieve the robust

H ∞ tracking design. The overall controller u(t) = N

j = 1

u j (t)can be formulated in (20).

The dynamic of the estimated error e(t) = y(t)− y(t) is

given by

e(t) =M

i=1

µi (y)[(I N ⊗Ai )y(t) + (I N ⊗Aτ ,i )y(tτ )]

+ T y(t) + Bd d(t) + ε(t) + O(∆2x )

−

M i= 1

µi (y)[(I N ⊗Ai )y(t) + (I N ⊗Aτ ,i )y(tτ )

+ T y(t) + Gi (C y(t) − z(t))]

=

M i1 =1

M i2 =1

µi1(y)µi2

(y)[((I N ⊗Ai2) + Gi2

C )e(t)

+ (I N ⊗Aτ ,i 2)e(tτ ) + T e(t) + (I N ⊗ (Ai1

−Ai2))y(t)

+ (I N ⊗ (Aτ ,i 1 −Aτ ,i 2

))y(tτ ) + Bd d(t) + Gi2Dn n(t)]

+ ε(t) + O(∆2x ). (21)

Combining the fuzzy system in (15) with the observer-

based tracking controller, the closed-loop fuzzy-tracking-control system could be described by the following augmented

system:

˙y(t) = A(µ, µ)y(t) + Aτ (µ, µ)y(tτ ) + Bv (µ)v(t)

+ ε(t) + O(∆2x ) (22)

where y(t) = [yR (t)T , y(t)T , e(t)T ]T , ε(t) = [ 0, ε(t)T ,ε(t)T ]T , and O(∆2

x ) = [OR (∆2x )T , O(∆2

x )T ,O(∆2x )T ]T . The

other notations are defined as follows:

A(µ, µ) =

M

i1 =1

M

i2 =1

Ψ11 0 0

Ψ21 Ψ22 Ψ23

0 Ψ32 Ψ33

Aτ (µ, µ)

=

M i1 =1

M i2 =1

0 0 0

0 µi1(y)(I N ⊗Aτ ,i 1

) 0

0 0 µi2(y)(I N ⊗Aτ ,i 2

)

Bv (µ) =

M i2 =1

B

R 0 0

0 Bd 0

0 Bd µi2(y)Gi2

Dn

where Ψ11

= I N ⊗AR + T R , Ψ21

= −BK i2

µi2(y), Ψ22

=

µi1(y)(I N ⊗Ai1

) + T + BK i2 µi2

(y), Ψ23= −BK i2

µi2(y),

Ψ32= µi1

(y)µi2(y)(I N ⊗ (Ai1

−Ai2)), and Ψ33

= µi2

(y)((I N ⊗Ai2

) + T + Gi2C ). Note that the augmented system in

(22) includes the reference model in (17), the fuzzy-spatial state-

space system in (15) with the fuzzy controller in (20), and the

estimated-error dynamic in (21).

The robust H ∞ fuzzy-observer-based tracking controller is

specified to guarantee that the effect of the time delay, the ap-proximation error, the truncation error, the external disturbance,

and the measurement noise on the tracking error yR (t)− y(t)and the estimation error e(t) are attenuated below a prescribed

level ρ from the energy perspective. This design problem is

called the H ∞ observer-based tracking-control-design problem.

Based on the augmented system in (22) and Remark 5, the H ∞observer-based tracking-control performance should be modi-

fied to include the state-estimation error e(t) and the effect of

truncation error O(∆2x ) as follows:

tf

0 R1 (yR (t)− y(t))2 + R2 e(t)2 dt

tf

0 v(t)2

+ ¯O(∆

2x )

2

dt

≤ ρ2 (23)

where the weighting matrices R1= diag(∆x R1 , . . . , ∆x R1 ) ∈

RnN ×nN , and R2

= diag(∆x R2 , . . . , ∆x R2 ) ∈ R

n N ×n N , with

R1 ∈ Rn×n and R2 ∈ R

n×n , i.e., the effect of disturbances,

measurement noise, and truncation error on the tracking error

and the estimation error, should be below the attenuation level ρ.

The weighting matrices R1 and R2 denote the tradeoff between

the tracking error and the estimation error. Since the trunca-

tion error due to finite-difference approach appears in (22), its

effect should be included in the H ∞ tracking performance in

the design procedure. According to (6), the H ∞ observer-based

tracking performance in (23) can be represented by the follow-ing inequality: tf

0

Ry(t)2 dt ≤ V (y(0)) + ρ2

tf

0

v(t)2 + O(∆2x )2 dt

(24)

for some positive function V (y(0)) when the initial condition

y(0) is also considered, where

R =

R1 −R1 0

0 0 R2

and y(t) is the state vector of the augmented system in (22).

Therefore, the H ∞ observer-based tracking-control design

for NDPSs is based on how to specify the control gains and




observer gains of the fuzzy-observer-based controller in (18)

and (20) so that the H ∞ observer-based tracking performance

in (23) or (24) could be achieved.

Remark 6: The term Ry(t)2 can be represented as

Ry(t)2 = N

j =1 R1 (yR ,j (t)− y j (t))2 ∆2x + R2 (y j (t)−

y j (t))2 ∆2x , where ∆2

x is the area of the gridded rectangle, and

the term Ry(t)2

, which is called the Riemann sum [54], canbe used to approximate the integration of the 2-D H ∞ tracking

performance in a spatiotemporal domain. Therefore, if the grid-

ded spacing ∆x of the finite-difference approach is sufficiently

small so thatO(∆2x ) → 0, then the H ∞ observer-based tracking

performance in (23) will approach the 2-D H ∞ observer-based

tracking performance in the spatiotemporal domain tf

0

U R1 (yR (x, t) − y(x, t))2 + R2 e(x, t)2 dx dt tf

0 v(t)2 dt≤ ρ2

(25)

where e(x, t) = y(x, t)− y(x, t).

Lemma 2 [55]: X T P Y + Y T P X ≤ ξX T P X + 1

ξ Y T P Y

for any positive constant ξ , a symmetric matrix P = P T ≥ 0,and two vectors X and Y with appropriate dimensions.

To solve the H ∞ tracking problem in (25), let us choose a

Lyapunov function V (y(t)) for the augmented system in (22)

as

V (y(t)) = y(t)T P y(t) +

t

tτ

y(s)T Qy(s)ds (26)

where P = P T > 0, and Q = QT > 0. Based on (26), we can

obtain the following result for the robust H ∞ tracking problem

in (24).

Theorem 3: For the augmented system (22) with a prescribed

disturbance attenuation level ρ in (24), if there exist two sym-

metric positive-definite matrices P and Q, the control gains K i ,

and the observer gains Gi , for i = 1, . . . , M , such that

Π(µ, µ) =

Θ11 (µ, µ) P Aτ (µ, µ) P Bv (µ)

AT τ (µ, µ)P ξ Στ −Q 0

BT v (µ)P 0 −ρ2 I

< 0 (27)

where Θ11 (µ, µ) = A(µ, µ)T P + P A(µ, µ) + ( 1

ρ2 + 1ξ )P P +

ξ Σ + RT R + Q, Σ = diag(0, 2σ2 I, 0), and Στ

= diag(0,

2σ2τ I, 0), then the H ∞ observer-based tracking performance

for NDPSs in (24) is guaranteed by the fuzzy-observer-based

controller in (18)–(20).

Proof: See Appendix A.

Remark 7: 1) Recently, some approaches based on the fuzzy

Lyapunov function [56] or piecewise Lyapunov function [57]

have been developed to relax the conservativeness of stability

andstabilization problems.The purpose of this paper is to extend

the fuzzy H ∞ tracking-control approach to the field of NDPSs.

Therefore, we use the Lyapunov function V (y(t)) with com-

mon P and Q to simplify the reference-tracking-control-design

procedure of NDPSs with time delay. For the conservative anal-

ysis, see [17], [21], [22], and [56]–[59]. 2) When v (t) ≡ 0 in

(22), it is seen from (A7) in Appendix A that y(t) → 0, and

Ry(t) = [( R1 yR (t) − R1 y(t))T , ( R2 e(t))T ]T → 0 as t →∞

[12], i.e., the inequality in (27) guarantees the asymptotical sta-

bility and asymptotical tracking of (22) simultaneously.

Since it is still very difficult to solve the matrix inequal-

ity in (27) to find control gains K i and observer gains Gi for

H ∞ observer-based tracking design of NDPSs, a simplifica-

tion procedure is given below to improve the solution of matrix

inequality in (27). We can define the Lyapunov function forthe j th fuzzy finite-difference model in (13), the j th reference

finite-difference model, and the j th estimated-error dynamic as

follows:

V j (yR ,j (t), y j (t), e j (t))

= ψ j (t)T

P 11 ,j P 12 ,j 0

P 12 ,j P 11 ,j 0

0 0 P 33 ,j

ψ j (t)

+ t

tτ

ψ j (s)T Q11 ,j Q12 ,j 0

Q12 ,j Q11 ,j 0

0 0 Q33 ,j

ψ j (s)ds

where ψ j (t) = [yR ,j (t)T , y j (t)T , e j (t)T ]T . Then, the Lyapunov

function V (y(t)) in (26) for the augmented system (22) can

be formulated as V (y(t)) = N

j = 1 V j (yR, j (t), y j (t), e j (t)).

Therefore, the following forms for the matrices P and Q in

(26) can be easily obtained as

P =

P 11 P 12 0

P 12 P 11 0

0 0 P 33

> 0

Q =

Q11 Q12 0

Q12 Q11 0

0 0 Q33

> 0 (28)

where P ij = diag(P ij ,1 , . . . , P ij,N ) ∈ RnN ×nN , and Qij =diag(Qij ,1 , . . . , Qij,N ) ∈ RnN ×nN .

Remark 8: For simplicity, we define

P (11)=

P 11 P 12

P 12 P 11

, Q(11)

=

Q11 Q12

Q12 Q11

, P (22)

= P 33

and Q(22)= Q33 for the matrices P and Q in (28).

The following lemma gives a transformation technique from

the summation of matrices into a large matrix form to simplify

the design procedure of the H ∞ observer-based tracking con-

trol in Theorem 3. The details of the derivation are given in

Appendix B. Thus, we can obtain the following results for

A(µ, µ), Aτ (µ, µ), and Bv (µ).

Lemma 3: The system matrix A(µ, µ) in (22) can be repre-

sented as follows:

A(µ, µ) = MT [ A + Ξ] M (29)




where

M =

M(11) 0

0 M(22)

, Ξ =

Ξ(11) 0

0 Ξ(22)

A =

A(11) + B K (11)

B K (12)

A(21) A(22) + G(22)

C

.

Moreover, for the matrices Aτ (µ, µ) and Bv (µ), we have

Aτ (µ, µ) = MT Aτ , Bv (µ) = MT Bv (30)

where

Aτ =

Aτ (11) 0

0 Aτ (22)

Bv =

H T

(11)Bv (11) 0

H T (22)Bv (21)

Gv (22) Bv (22)

H = H (11) 0

0 H (22)

=

I 2nN 0 . . . 0 0 0 . . . 0

0 0 . . . 0 I nN 0 . . . 0

∈ R

3n×3n N

in which the dimension number N is defined as N = N × M ,with M = 1 + (M × M ).

Proof: See Appendix B.

The related matrices in Lemma 3 are defined in Appendix B.

Since the matrix A(k ) is singular, we add the relaxed matrices

Ξ(k ) to adjust the relationship between the matrices A(k ) , K (k ) ,

and G(k ) .

Lemma 4: Since µi1(y)µi2

(y) is the diagonal matrix and the

matrix P has the form, as given in (28), it is easy to prove

the relationship P (11) M

(11)i1 i2

= M(11)i1 i2

P (11) , and P (22) M

(22)i1 i2

=

M(22)i1 i2

P (22). Moreover, the matrix MP can be represented

as MP = P M, where P = diag( P (11), P (22) ) = diag(I M ⊗

P (11), I M ⊗ P (22) ) ∈ R

3n N ×3n N .

In Theorem 3, the sufficient condition of the H ∞ tracking

control in (27) includes the fuzzy weighting matrices µ(y) and

µ(y). Based on Lemma 3, we obtain the following main result.

Theorem 4: For the augmented system (22) with a prescribed

disturbance attenuation level ρ, if there exist two symmetric

positive-definite matrices P and Q in (28), the relaxed matrix

Ξ, the control gains K i , and the observer gains Gi , for i =1, . . . , M , in (18)–(20) such that

Π =

Π11 P Aτ

P Bv

AT τ

P ξ Στ −Q 0

BT v

P 0 −ρ2 I

< 0 (31)

then the H ∞ tracking-control performance in (24) is guaranteed

for a prescribed disturbance-attenuation level ρ by the fuzzy-

observer-based controller in (18)–(20). In (31), we have

Π11= ( A + Ξ)T P + P ( A + Ξ) + H T

ξ Σ + RT R + Q

H

+ 1

ρ2 +

1

ξ P H T H P.

Proof: Using the equality MP = P M in Lemma 4, we have

A(µ, µ)T P + P A(µ, µ) = ( MT A M)T P + P ( MT A M)

= MT AT P M+ MT P A M.

Based on Lemma 3, Π(µ, µ) in (27) can be reformulated as

follows:

Π(µ, µ) =

M 0 0

0 I 0

0 0 I

T

×

Π11 P Aτ

P Bv

AT τ

P ξ Στ −Q 0

BT v

P 0 −ρ2 I

M 0 0

0 I 0

0 0 I

. (32)

Finally, if Π < 0 in (31), then Π(µ, µ) < 0.

Remark 9: The control gains K i and the observer gains Gi ,

for i = 1, . . . , M , are included in the matrix A. The matrix P

can be obtained by the matrix P in Lemma 4.The H ∞ tracking-control-design problem is to specify the

observer gains Gi and the control gains K i , for i = 1, . . . , M ,to satisfy the inequality in (31). In the observer-based control-

design problem, the observer gains Gi and the control gains K iare always coupled with the matrix P [41]. In this situation,

the matrix inequality in (31) will be a complex bilinear matrix

inequality (BMI). A systematic design procedure is developed

to solve this problem in the next section.

B. Solving Robust H ∞ Tracking-Control Problem Via

the Linear-Matrix Inequality

Some algorithms to solve local optimal BMI solutions havebeen proposed via the augmented Lagrangian method [43] and

the iteration method [31], [40]–[42]. However, because the BMI

problem is nonconvex, these algorithms are still inefficient to

solve BMI problems with multiple variables. A fuzzy-observer-

based H ∞ control design was studied in [44] for a T–S fuzzy

time-delay system without the measurement noise. In this paper,

the measurement noise at the measured output is considered in

the NDPS. For the robust H ∞ tracking problem of the NDPS,

the 2-D H ∞ tracking performance is addressed. Therefore, we

proposed a different method to overcome a more complex prob-

lem. Note that the inequality in (31) is still BMI, even if the

matrices P and Q were chosen as (28). The following lemma

is introduced to reduce a BMI to an LMI that can be efficientlysolved with the conventional LMI technique.

Lemma 5 [45]: Given a positive symmetric matrix X of ap-

propriate dimension, if the inequalityΩ11 − 2ςX ςI

ςI Ω22

< 0 (33)

holds, then we have Ω11 + X Ω22 X < 0. In other words,

the inequalities Ω22 < 0 and −2ςX − ς 2 Ω−122 < 0 imply that

X Ω22 X < −2ςX − ς 2 Ω−122 < 0.

Proof: First, by the Schur complement, the inequality in (33)

is equivalent to Ω22 < 0 and Ω11 − 2ςX − ς 2 Ω−122 < 0. From

Ω22 < 0, we have (X + ς Ω−122 )T Ω22 (X + ς Ω−122 ) < 0, which is




equivalent to X Ω22 X < −2ςX − ς 2 Ω−122 . Then, we can obtain

the inequality Ω11 + X Ω22 X < 0. The lemma is completely

proven.

Let us define the matrix X as follows:

X = diag( X (11) , X (22) ) = diag(I M ⊗

X (11), I M ⊗ X (22) )

(34)

where

X (11)=

X 11 X 12

X 12 X 11

=

P 11 P 12

P 12 P 11

−1

and X (22)= X 11 − X 12 . We also define some symbols with

respect to the matrix Q in (28) as follows:

S (11) = X (11) Q(11)

X (11) and

S (11) = X (11) H T (11)

Q(11)H (11) X (11) = H T

(11) S (11)H (11) .

(35)

Using the matrices P and Q in (28), the sufficient conditions

for robust H ∞ tracking control in Theorem 3 can be derived as

follows.

Theorem 5: For the augmented system in (22) with a pre-

scribed disturbance attenuation level ρ, suppose there exist the

symmetric matrices X (11) > 0, S (11) > 0, P (22) > 0, Q(22) >

0, the matrices Y i and Z i , i = 1, . . . , M , Ξ(11), Ξ(22) , and the

scalars ξ > 0, ξ 2 > 0 such that

Ω11 Ω12 Ω13 Ω14

Ω21 Ω22 Ω23 0Ω31 Ω32 Ω33 0Ω41 0 0 Ω44

< 0. (36)

Then, the H ∞ tracking performance in (24) is guaranteed by the

fuzzy-observer-based tracking controller in (18)–(20), i.e., theapproximation error and the time delay can be tolerated, and the

effect of the truncation error, the external disturbance, and the

measurement noise on the tracking error canbe attenuated below

a prescribed level ρ by the fuzzy-observer-based controller with

observer gains Gi = P −133 Z i and control gains K i = Y i (X 11 −

X 12 )−1 , i = 1, . . . , M . In (36)

Ω11 =

Θ11 BY (12) ςI 0

Y T (12)BT −2ς X (22) 0 ςI

ςI 0 −2I + ξ 2 I AT (21) P (22)

0 ςI P (22) A(21)

Θ22

Θ11

= Θ11 + S (11) − 2ς X (11)

Θ11= X (11)

AT (11) + A(11)

X (11) + BY (11) + Y T (11)

BT

+ ΞT (11) + Ξ(11)

Y (11)= K (11)

X (11), Y (12)= K (12)

X (22)

Ξ(11)= Ξ(11)

X (11)

Θ22= Θ22 + H T

(22) Q(22) H (22)

Θ22= AT

(22) P (22) + P (22)

A(22) + Z (22) C + C T Z T

(22)

+ ΞT (22) + Ξ(22)

Z (22)= P (22)

G(22), Z v (22)= P (22)

Gv (22)

Ξ(22)= P (22) Ξ(22)

Ω21 =

X (11)

AT τ (11) 0 0 0

0 0 0 AT τ (22)P (22)

Ω22 =

−S (11) 0

0 −Q(22)

, Ω12 = ΩT

21

Ω31 =

Σ1/2(11)H T

(11)X (11), 0, 0, 0

, Ω13 = ΩT 31

Ω32 =

Σ1/2τ (11), 0

, Ω23 = ΩT

32 , Ω33 = −2I + ξI

Ω14 ,1 =

X (11)H T (11)RT

(11), X (11) , H T (11) ,

H T (11) , H T

(11) Bv (11)

, Ω41 ,1 = ΩT

14 ,1

Ω14 ,2 =

H T (22)RT

(22), P (22) H T (22), P (22)H T

(22) ,

P (22)H T (22) Bv (21), Z v (22)

Bv (22) , Ω41 ,2 = ΩT 14 ,2

Ω44 ,1 = diag(−I,−ξ 2 I,−ρ2 I,−ξI ,−ρ2 /2I )

Ω44 ,2 = diag(−I,−ρ2 I,−ξI ,−ρ2 /2I,−ρ2 I )

Ω41 =

Ω41 ,1 0 0 0

0 0 0 Ω41 ,2

, Ω14 = ΩT

41

Ω44 =

Ω44 ,1 0

0 Ω44 ,2

where Σ(11)= diag(0, 2σ2 I ), Στ ,(11)

= diag(0, 2σ2

τ I ),

R(11)= [ R1 , −R1 ], and R(22)

= R2 .

Proof: See Appendix C.

Remark 10: The control gains K i and the observer gains

Gi , for i = 1, . . . , M , are included in the matrices K (11) ,

K (12) , G(22), and Gv (22), respectively. We define Y (11)

i3 i4

=

K (11)i3 i4

X (11) = [−Y i4, Y i4

], Y (12)

i3 i4

= K

(12)i3 i4

X (22) = −Y i4,

Z (22)i1 i2

= P (22)

G(22)i1 i2

= Z i2, and Z

(22)v ,i 1 i2

= P (22)

Z (22)v ,i 1 i2

= Z v ,i 2

in which Y i4

= K i4

(X 11 − X 12 ) and Z i2

= P 33 Gi2

. Then, we

have the following forms for Y (k ) , k = 11 and 12, Z (22), and

Z v (22):

Y (k )=

0 Y (k )11 . . . Y (k )

M M

0 0 . . . 0...

... . . .

...

0 0 . . . 0

Z (22)=

0 0 . . . 0

Z (22)11 0 . . . 0

......

. . . ...

Z (22)M M 0 . . . 0

and Z v (22)= [0, Z (22)T

v ,11 , Z (22)T v ,12 , . . . , Z (22)T

v , M M ]T .




Remark 11: In (36), we use the term −2 + ξ to replace the

term −ξ −1 with the fact, −ξ −1 ≤ −2 + ξ . From the inequality

−2 + ξ < 0, we know that ξ < 2. Alternatively, we can use

the term −2 + ξ −1 to replace the term −ξ by the fact −ξ ≤−2 + ξ −1 . For−2 + ξ −1 < 0, we have ξ > 1/2. Therefore, we

can choose either of two constraints on ξ to obtain the suitable

scalar ξ in the LMI (36). For ξ 2 , we have the same result.The H ∞ tracking-control-design problem reduces to how

to specify the observer gains Gi = P −133 Z i and control gains

K i = Y i (X 11 − X 12 )−1 , i = 1, . . . , M by solving the LMI in

(36) with some positive-definite matrices. Finally, in order to

achieve the optimal attenuation of truncation error, external dis-

turbances, and measurement noises on the tracking error and

estimation error, the optimal H ∞ tracking-control-design prob-

lem for the NDPS (1) and (2) can be solved by the following

constrained optimization problem:

ρ0 = minY i , Z i , i=1 ,...,M

ρ

subject to X (11) > 0, S (11) > 0, P 33 > 0

Q33 > 0, ξ > 0, ξ 2 > 0, and (36). (37)

This is called an eigenvalue problem (EVP) [55] and can be

easily solved by the LMI technique.

Based on the above analysis, the robust H ∞ observer-based

tracking-control-design procedure for NDPSs is summarized in

the following steps.

Design procedure

Step 1: Given a desired reference model in (3), generate the

reference output yR (x, t) and a prescribed distur-

bance attenuation level ρ.

Step 2: Select the fuzzy membership functions and fuzzyrules to establish a fuzzy DPS in (7) to approximate

the NDPS in (1).

Step 3: Give grid size ∆x and N to construct the fuzzy-spatial

state-space model in (15) and (16).

Step 4: Solve the observer gains Gi and the tracking-control

gains K i of the fuzzy-observer-based tracking con-

troller, for i = 1, . . . , M , in (18)–(20) by solving the

LMI problem in (36), or solve K i and Gi from the

optimal H ∞ tracking-control problem in (37).

Step 5: Construct a fuzzy-observer-based tracking controller

in (18)–(20) to control the NDPSs in (1) and (2), and

track the reference model in (3).

Remark 12: 1) Note that the different grid size ∆x can be cho-

senfor thedifferent space variables x. 2) Note thatthe dimension

N depends on ∆x . In theory, the grid size ∆x is chosen as small

as possible, i.e, ∆x → 0. However, in this situation, the dimen-

sion N will increase to infinity. The computational complexity

will also increase to solve the LMI problem in (36). Therefore,

how to choose the grid size is a tradeoff problem. Suppose the

initial-value problem of the NDPS in (1) is well-posed [51] and

that the finite-difference scheme (method of lines scheme [60])

in (9) is consistent, i.e., O(∆2x ) → 0 as ∆x → 0 [51]. Apply-

ing Lax–Richtmyer equivalence theorem [51], the fuzzy-spatial

state-space system with d(t) ≡ 0 in (15) is stable if and only if

the finite-difference scheme is a convergent scheme in which the

solution of the fuzzy-spatial state-space system with d(t) ≡ 0 in

(15) can converge to the solution of the NDPS with d(t) ≡ 0 in

(1). Based on 2) of Remark 7, the stability of the fuzzy-spatial

state-space system with d(t) ≡ 0 in (15) can be guaranteed by

the solvableconditions of Theorem 5. In general, as N increases,

since the stability of every finite-difference dynamic equation

within ∆x should be guaranteed simultaneously, it will lead tothe conservative of the stability criterion. Therefore, the grid

size ∆x is chosen such that the LMI in (36) is solvable.

V. APPLICATION TO TRACKING CONTROL OF

HODGKIN–HUXLEY NERVOUS SYSTEMS

The nervous system consists of highly interconnected nerve

cells, which communicate by generating and transmitting short

action potential (i.e., short electrical pulse). Action potentials

are stereotypical and all-or-none electrical transient deflections

of the membrane voltage from its resting value at electrochem-

ical equilibrium [35]. The phenomenological model of action-

potential generation in the nerve-cell dynamic is described bythe H–H model [6], [35], [36]. An action potential is generated

at the initial segment of the nerve cell’s axon and propagated

to the synaptic contacts at the end of the axon. In nerve cells,

separation of ionic charge along the cell membrane causes a dif-

ference in electrical potential across the cell membrane. From

the dendrites of other nerve cells, nerve cells receive electrical

input signals. Depending on the spatiotemporal distribution of

the input current to depolarize the membrane voltage, the fir-

ing threshold can be reached after sufficient membrane voltage

depolarization, and then, an action potential will be triggered.

The H–H dynamic equations [36] represent a phenomenolog-

ical model of action potential generation in a nerve cell as a

function of a given current stimulus [2]. Electrical stimulationof a nerve cell with rectangular pulses has a range of clinical

applications, for example, activation of muscles by stimulating

the motor nerve cell fibers innervating muscles or activation of

different sensor-motor areas in the brain or spinal cord, such as

deepbrainstimulationfor Parkinsonism patients [35],[37]–[39].

The H–H dynamic model is expressed by nonlinear PDEs that

describe the spatiotemporal evolution of the membrane voltage

yv= yv (x, t). In the H–H dynamic model, the total current

across nerve cell membrane is the sum of the capacitive current,

the ionic currents, and the external current. The H–H model is

described by the total currents, leading to an equivalent electric

ionic currents as follows [2], [6], [36]:

cm∂yv

∂t = κm

∂ 2 yv

∂x2 + f (yv ) + g(x)I in j(t) + gd (x)d(t) (38)

z(t) = h(x)yv + Dn n(t) (39)

where f (yv ) = gK φ

4n (V K − yv ) + gNa φ3

m φh (V Na − yv ) +gm (V leak − yv ). The φn , φm , and φh are defined as the potas-

sium activation, the sodium activation, and the sodium inac-

tivation, respectively. The current generated from the flow of

potassium ions is determined by a maximum potassium con-

ductance gK , an ionic equilibrium potential V K expressing

steady-state potassium ion separation, and potassium activation

φn . Similarly, sodium-ion current is modeled with a maximal




sodium conductance gNa , an ionic equilibrium potential V Na ,

and sodium activation φm and inactivation φh . The remaining

ion currents are collectively modeled by a leakage current with

conductance gm and ionic equilibrium potential V leak . The con-stant cm is themembranecapacity perunit area. Theconstantκm

is defined as κm= ra /2R2 , where ra is the radius of the fiber,

and R2 is the specific resistance of the axoplasm. The potas-sium activation φn , the sodium activation φm , and the sodium

inactivation φh vary, depending on the change of the membrane

potential yv , and are given by the following equations:

φn=

αn (yv )

αn (yv ) + β n (yv ), φm

=

αm (yv )

αm (yv ) + β m (yv )

φh=

αh (yv )

αh (yv ) + β h (yv ).

The specific functions αn (yv ), β n (yv ), αm (yv ), β m (yv ),

αh (yv ), and β h (yv ) are proposed by Hodgkin and Huxley [36]

as

αn (yv ) = 0.01 10− yv

exp((10− yv )/10)− 1

β n (yv ) = 0.125 exp

−yv

80

αm (yv ) = 0.1 25− yv

exp((25− yv )/10)− 1

β m (yv ) = 4 exp

−yv

18

αh (yv ) = 0.07exp

−yv

20

β h (yv ) = 1exp((30− yv )/10) + 1

.

The remaining constants are κm = 0.336, cm = 1 F/cm2 , gK =36 mS/cm2 , gNa = 120 mS/cm2 , and gm = 0.3 mS/cm2 with

equilibrium potentials V K = −12 mV, V Na = 115 mV, and

V leak = 10.613 mV [6]. I in j(t) is an externally injected cur-

rent in a spatially localized axonal compartment. The potential

yv is measured in units of millivolts, current density is in units

of microamperes per square centimeters, and the unit of time is

milliseconds. The initial distribution of the membrane voltage

is given as yv (x, 0) = 0. The boundary conditions are the Neu-

mann boundary condition, i.e., ∂yv (x, t)/∂x = 0 at x = 0 and

x = 1.

When the H–H nervous system suffers the effect of ex-

ternal disturbances, the influence function of external dis-

turbances can be defined as gd (x) = [δ (x− (5/10))]. The

observation influence function is defined as h(x) = [δ (x−(2/10)), δ (x− (4/10)), δ (x− (6/10)), δ (x− (8/10))]T , i.e.,

the sensors are located at x = 2/10, 4/10, 6/10, 8/10.

The influence function of measurement noise is de-

noted as Dn = [0, 1, 0, 1]T . The control influence function

is denoted as g(x) = [δ (x− (1/10)), δ (x− (3/10)), δ (x −(5/10)), δ (x− (7/10)), δ (x− (9/10))]. For the convenience

of simulation, the measurement noise and the external distur-

bance are assumed as n(t) = sin(t) and d(t) = sin(t).

Fig. 3. (a) Spatiotemporal profiles of the reference model in (40). (b) Spa-tiotemporal profiles of the H–H nervous system in (38). (c) Spatiotemporalprofiles of the tracking error y

R(x, t) − y

v(x, t). (d) Time profiles of the esti-

mated error e(t).

For communication in the nervous system, the nerve cells

should transmit a desired signal. Therefore, we control the ner-

vous system to track a reference signal. Suppose the desired

response of a H–H nervous system is specified by the following

reference model:

∂yR (x, t)

∂t = AR yR (x, t) + AR yR (x, t) + gR (x)r(t) (40)

where the differential operator is defined as AR yR (x, t) =

0.5∂ 2

yR (x, t)/∂x2

, AR = −1, and gR (x) = [200δ (x −(3/10)), 200δ (x− (7/10))]. The reference input r(t) is given

to generate the impulse-response signals to simulate a nervous

system as r(t) = exp(−0.5(t − 5)2 ) + exp(−0.5(t− 15)2 ).

The spatiotemporal profile of the reference model is shown

in Fig. 3(a). The control target is to design the control input

I in j (t) in (38) so that the state y (x, t) could track the desired

trajectory yR (x, t) in (40) as good as possible in spite of the

measurement noise n(t) and the external disturbance d(t),

i.e., the control input I in j(t) is designed such that the tracking

error yR (x, t)− yv (x, t) must be as small as possible under

the influence of the measurement noise n(t) and the external

disturbance d(t).

First, we establish a T–S fuzzy DPS as (7) with the trape-

zoidal membership functions. The range of the state is given as

yv (x, t) ∈ [−5, 30]. The operation points of the fuzzy DPS are

given at yv (x, t) = −5 and yv (x, t) = 30. The number of fuzzy

rules is M = 2. The parameters in the fuzzy DPS (7) are ob-

tained as A1 = −0.8089 and A2 = −2.6473. We can obtain the

boundsof theapproximateerror σ = 1.67× 10−2 and στ = 0 in

Theorem 1. Obviously, the proposed fuzzy model can approach

the nonlinear partial system accurately. We give the grid space

∆x = 0.1111 and N = 10; then, the finite-difference operator

can be constructed. Following the proposed design procedure in

the above section, the optimal H ∞ fuzzy-observer-based track-

ing controller could be obtained easily with ρ0 = 0.0440 by




solving the constrained optimization problem in (37). The spa-

tiotemporal profile of the controlled H–H nervous system is

shown in Fig. 3(b). The spatiotime profiles of the tracking error

yR (x, t)− y(x, t) between the nonlinear distributed parameter

system and the reference model are shown in Fig. 3(c).The time

profiles of the estimated error e(t) between the nonlinear dis-

tributed parameter system and the fuzzy observer are shownin Fig. 3(d). The simulation results show that the proposed ro-

bust H ∞ fuzzy-observer-based tracking controllerobviously can

control the trajectory of the H–H nervous system to track a de-

sired trajectory by efficiently attenuating the truncation error,

the external disturbances, and measurement noises. The H ∞tracking performance can be computed as follows:

300 y(t)T RT Ry(t)dt 30

0 v(t)T v(t)dt≈ 0.01712 < ρ2

0 = 0.04402 .

This conservative result is due to the conservative wayof solving

LMIs in the H ∞ control-tracking-design procedure. Therefore,the simulation example has shown the feasibility of the pro-

posed robust H ∞ fuzzy-tracking-control design of the NDPS

for potential practical applications. The effects of the external

disturbance and the measurement noise in the NDPS could be ef-

ficiently attenuated by the proposed robust H ∞ observer-based

tracking-control design.

VI. CONCLUSION

This paper proposed a 2-D H ∞ tracking performance in the

spatiotemporal domain for robust model reference-tracking con-

trol of NDPSs under time delay, external disturbances, and mea-surement noises. The robust model reference-tracking-control

problem for the NDPSs with time delay, external disturbances,

and measurement noise was successfully solved by the fuzzy-

spatial state-space system based on the finite-difference model.

A fuzzy-observer-based tracking controller is proposed to at-

tenuate the effect of the truncation error, external disturbances,

and measurement noise on the desired tracking performance be-

low a prescribed level to achieve robust tracking-control design

of NDPSs. In order to simplify the design procedure, the pro-

posed H ∞ fuzzy-observer-based tracking-control scheme for

NDPSs can be transformed from solving a BMI problem to

solving an LMI problem. Therefore, determining the observer

gains and the controller gains for the optimal H ∞ observer-

based tracking control becomes an LMI-based optimization

problem, which can be efficiently solved by the LMI Toolbox in

MATLAB. An H–H nervous tracking-control problem in bi-

ology engineering is provided to illustrate the practical appli-

cation of the H ∞ tracking-control scheme to NDPSs and to

confirm its robust tracking performance. The proposed robust

tracking-design method can also be applied to many fields, e.g.,

heat flows, elastic wave, flexible structures, chemical engineer-

ing, biodynamic systems, etc. Therefore, the proposed design

method was the potential for the robust tracking control of

NDPSs with time delay, external disturbances, and measure-

ment noises.

APPENDIX

A. Proof of Theorem 3

Proof: First, we differentiate the function V (y(t)) in

(26). Then, adding and subtracting the term ρ2 v(t)T v(t) −y(t)T RT Ry(t), we get

V (y(t)) = ˙y(t)T P y(t) + y(t)T P ˙y(t)

+ y(t)T Qy(t)− y(tτ )T Qy(tτ )

= ( A(µ, µ)y(t) + Aτ (µ, µ)y(tτ ) + Bv (µ)v(t)

+ ε(t) + O(∆2x ))T P y(t) + y(t)T P ( A(µ, µ)y(t)

+ Aτ (µ, µ)y(tτ ) + Bv (µ)v(t) + ε(t) + O(∆2x ))

+ y(t)T Qy(t)− y(tτ )T Qy(tτ )

− (ρ2 v(t)T v(t)− y(t)T RT Ry(t))

+ (ρ2 v(t)T v(t)− y(t)T RT Ry(t)). (A1)

By Lemma 2, we have

O(∆2x )T P y(t) + y(t)T P O(∆2

x )

≤ ρ2 O(∆2x )T O(∆2

x ) + 1

ρ2 y(t)T P P y(t) (A2)

ε(t)T P y(t) + y(t)T P ε(t)

≤ ξ ε(t)T ε(t) + 1

ξ y(t)T P P y(t) (A3)

where ξ is any positive constant. According to the bound of ε(t)in Corollary 2, we have

ε(t)T ε(t) = 2ε(t)T ε(t) ≤ y(t)T Σy(t) + y(tτ )T Στ y(tτ )(A4)

where Σ and Στ are defined as Σ = diag(0, 2σ2 I, 0), and

Στ = diag(0, 2σ2

τ I, 0). Based on the inequalities in (A2)–(A4),

from (A1), we obtain

V (y(t)) ≤ η(t)T Π(µ, µ)η(t) + ρ2 v(t)T v(t)

− y(t)T RT Ry(t) + ρ2 O(∆2x )T O(∆2

x ) (A5)

where η(t) = [y(t), y(tτ ), v(t)], and Π(µ, µ) is definedin (27).

Suppose the inequality Π(µ, µ) < 0 holds, then we have the

following inequality:

V 1 (y(t))≤ ρ2 v(t)T v(t)− y(t)T RT Ry(t) + ρ2 O(∆2x)T O(∆2

x).(A6)

Integrating (A6) from t = 0 to t = tf yields

V (y(tf )) ≤ V (y(0)) +

tf

0

ρ2v(t)2 − Ry(t)2

+ ρ2O(∆2x )2 dt. (A7)

Since V (y(tf )) > 0, we have the H ∞ tracking-performance

inequality in (24). Therefore, if the inequality in (27)

holds, then the H ∞ tracking performance in (24) could be

guaranteed.




B. Proof of Lemma 3

Since µi1(y) and µi2

(y) are in matrix form and have the

propertyM

i1 =1

M i2 =1 µi1

(y) µi2(y) = I , by Lemma 1, the

term A(µ, µ) in (22) can be represented as follows:

A(µ, µ) =

A(11)

0 00 0

+

M i1 =1

M i2 = 1

¯M

(11)

i1 i2 00 M

(22)i1 i2

×

A

(11)i1 i2

0

A(21)i1 i2

A(22)i1 i2

+ G(22)i1 i2

C

+

M i3 =1

M i4 =1

B K

(11)i3 i4

B K (12)i3 i4

0 0

M

(11)i3 i4

0

0 M(22)i3 i4

(B1)

where M(11)

i1 i2

= diag(µ

i1

(y)µi2

(y), µi1

(y)µi2

(y)), M(22)

i1 i2

=

µi1(y)µi2

(y), A(11)0

= diag(I N ⊗AR + T R , 0), A

(11)i1 i2

=

diag(0, (I N ⊗Ai1) + T ), A

(21)i1 i2

= [0, I N ⊗ (Ai1

−Ai2)],

A(22)i1 i2

= (I N ⊗Ai2

) + T , B = [0, BT ]T , K

(11)i3 i4

= [−K i4

,

K i4], K

(12)i3 i4

= −K i4

, and G(22)i1 i2

= Gi2

. Similarly, for Aτ (µ, µ)

and Bv (µ), we have

Aτ (µ, µ) =

M i1 =1

M i2 = 1

M

(11)i1 i2

0

0 M(22)i1 i2

A

(11)τ ,i 1 i2

0

0 A(22)τ ,i 1 i2

(B2)

Bv (µ) =

Bv (11) 0

Bv (21) 0

+

M i1 =1

M i2 =1

M

(11)i1 i2

0

0 M(22)i1 i2

×

0 0

0 G(22)i1 i2

Bv (22)

(B3)

where A(11)τ ,i 1 i2

= diag(0, I N ⊗Aτ ,i 1

), A(22)τ ,i 1 i2

= I N ⊗Aτ ,i 2

,

G(22)v ,i 1 i2

= Gi2

, Bv (11)= diag(BR , Bd ), Bv (21)

= [0, Bd ], and

Bv (22)= Dn .

Lemma 6: Given the matrices Ξ0

, Γi1

, Ξ1,i 1

, and Ξ2,i 1 i2

with

appropriate dimensions for i1 = 1, . . . , M , and i2 = 1, . . . , M ,we have the following equality:

Ξ0 +M

i1 =1

Γi1ΞT

1,i 1 +

M i2 =1

Ξ1,i 2Γi2

+M

i1 =1

M i2 =1

Γi1Ξ2,i 1 i2

Γi2

=

I

Γ1

...

ΓM

T

Ξ0 Ξ1,1 . . . Ξ1,M

ΞT 1,1 Ξ2,11 . . . Ξ2,1M

......

. . . ...

Ξ1,M Ξ2,M 1 . . . Ξ2,M M

I

Γ1

...

ΓM

. (B4)

Before further simplification, we define some notations as

B = I M ⊗

B, C = I M ⊗ C , and the following matrices:

M(k )=

I

M(k )11

...

M(k )M M

, A(k )

=

A(k )0 0 . . . 0

A(k )11 0 . . . 0

..

.

..

.

. . . ..

.A

(k )M M 0 . . . 0

Ξ(k )=

−Ξ(k )0 − Ξ

(k )T 0 + Ξ

(k )3 Ξ

(k )0 − Ξ

(k )1,11

Ξ(k )T 0 − Ξ

(k )T 2,11 Ξ

(k )1,11 + Ξ

(k )T 2,11 − Ξ

(k )3

......

Ξ(k )T 0 − Ξ

(k )T 2,M M Ξ

(k )1,11 + Ξ

(k )T 2,M M

. . . . . . Ξ(k )0 − Ξ

(k )1 ,M M

. . . . . . Ξ(k )1,M M + Ξ

(k )T 2,11

. . . . . .

...

. . . . . . Ξ(k )1,M M + Ξ

(k )T 2,M M − Ξ

(k )3

K (k )=

0 K (k )11 . . . K

(k )M M

0 0 . . . 0

0...

. . . ...

0 0 . . . 0

G(22)

=

0 0 0 0

G(22)11 0 . . . 0

0...

. . . ...

G(22)M M 0 . . . 0

Aτ (k )= [0, A

(k )T τ ,11 , . . . , A

(k )T τ , M M ]

T

Gv (22)= [0, G

(22)T v ,11 , . . . , G

(22)T v , M M ]

T

where for M(k ) , Ξ(k ) , and Aτ (k ) , we have k = 11 and 22; for

A(k ) , k = 11, 21, and 22; for K (k ) , k = 11, and 12; the matrices

Ξ(11)0 ∈ R

2n N ×2nN , Ξ(22)0 ∈ R

nN ×nN , Ξ(11)1 ,i 1 i2

∈ R2n N ×2nN ,

Ξ(11)2,i 1 i2

∈ R2nN ×2nN , Ξ

(22)1 ,i 1 i2

∈ Rn N ×n N , Ξ

(22)2,i 1 i2

∈ RnN ×nN ,

Ξ(11)3 ∈ R

2n N ×2nN , and Ξ(22)3 ∈ R

n N ×nN are some relaxed

matrices applied in the following lemma, and other matrices

are defined in (B1)–(B3).

Remark 13: In the matrices A(k ) , Aτ (k ) , K (k ) , G(k ) ,

Gv (k ) , and Ξ(k ) , it should be noted that the subscript in-

dex i1 i2 of submatrices is defined as i1 i2 = 11, 12, . . . ,M M, for i1 = 1, . . . , M , and i2 = 1, . . . , M , i.e., (i1 , i2 ) =(1, 1), (1, 2), (1, 3), . . . , (M, M ).

Proof of Lemma 3

Proof: By the equality −I + M i1 = 1 M

i2 =1 M(k )i1 i2

= 0, for

k = 11 and 22, we can add the following equalities to relax the




equation given in (29):−I +

M i1 =1

M i2 =1

M(k )i1 i2

Ξ

(k )T 0 = 0

Ξ(k )0 −I +

M

i3 =1

M

i4 =1

M(k )i3 i4 = 0

−I +

M i1 =1

M i2 =1

M(k )i1 i2

M i3 =1

M i4 =1

Ξ(k )1 ,i 3 i4

M(k )i3 i4

= 0

M

i1 = 1

M i2 =1

M(k )i1 i2

Ξ(k )T 2,i 1 i2

−I +

M i3 =1

M i4 =1

M(k )i3 i4

= 0

Ξ

(k )3 −

M i1 =1

M i2 =1

M i3 =1

M i4 =1

M(k )i1 i2

Ξ(k )3

M(k )i3 i4

= 0.

Adding the above three equalities into (B1) and using Lemma

6, we can obtain the equation given in (29). Moreover, it is easyto prove the equation given in (30) by Lemma 6.

C. Proof of Theorem 5

Proof: First, the fact that the inequality in (36) implies the

inequality in (31) is proven in the following. Because the fact

(1− ξ )ξ −1 (1− ξ ) ≥ 0 implies the inequality −ξ −1 ≤ −2 + ξ ,

we have Ω33= − ξ −1 I ≤ Ω33 in (36). Similarly, we

have −ξ −12 ≤ −2 + ξ 2 . Using the aforementioned inequality

and Schur complement, the inequality in (36) implies the

inequality, Ω11 − Ω13 Ω−1

33 Ω31 − Ω12

Ω22 − Ω23

Ω−133 Ω32

−1

Ω21 − Ω14 Ω−144 Ω41 < 0, which can be written as follows:

Θ11 BY (12) ςI 0

Y T (12) BT −2ς X (22) 0 ςI

ςI 0 −ξ −12 I AT

(21) P (22)

0 ςI P (22) A(21) Θ22 + Υ22

< 0

(C1)

where

Θ11= Θ11 + Υ11 − 2ς X (11)

Υ11= X (11) H T

(11) Φ(11) H (11) X (11) + Aτ (11) Φ−1

τ (11) AT τ (11)

+ 1ξ 2

X (11) X (11) +

1ρ2

+ 1ξ

H T

(11) H (11)

+ 1

ρ2 2H T

(11) Bv (11)

BT v (11)H (11)

Υ22= H T

(22) Φ(22) H (22) + P (22) Aτ (22) Φ−1

τ (22) AT τ (22)

P (22)

+

1

ρ2 +

1

ξ

P (22) H T

(22) H (22) P (22)

+ 1

ρ2 2 P (22) H T

(22) Bv (21)

BT v (21)H (22)

P (22)

+ 1

ρ2 Z v (22) Bv (22) BT v (22) Z

T v (22)

in which Φ(k )= (ξ Σ(k ) + RT

(k ) R(k ) + Q(k ) ), and Φτ (k )=

(ξ Στ (k ) − Q(k ) ), k = 11 and 22. Note that Σ(22) = 0 and that

Στ (22) = 0. By Lemma 5, the inequality in (C1) implies the

following inequality:

Ω =

Θ11 Θ12

ΘT 12 X (22)

Θ22 + Υ22

X (22) < 0 (C2)

where Θ11= Θ11 + Υ11 − ξ −1

2 X (11)

X (11), and Θ12=

BY (12) + X (11) A(21)T

P (22) X (22).

On the other hand, by the Schur complement, the inequality

in (31) is equivalent to the following form:

( A + Ξ)T P + P ( A + Ξ) + H T (ξ Σ + RT R + Q)H

+ P Aτ (ξ Στ −Q)−1 AT τ

P +

1

ρ2 +

1

ξ

P H T H P

+ 1

ρ2P Bv

BT v

P < 0. (C3)

Note that Σ = diag(Σ(11), Σ(22) ), RT R = diag( RT (11) R(11),

RT (22)R(22) ), Q = diag( Q(11), Q(22) ), Στ = diag(Στ ,(11),

Στ (22)), and (ξ Στ −Q)−1 = diag

(ξ Στ (11) − Q(11) )−1 ,

(ξ Στ (22) − Q(22) )−1

.

Pre- and postmultiplying the above inequality by the matrix

X in (34), and substituting the definitions of P , A, Aτ , Bv , and

H in (29) and (30) into this inequality, we obtain

Ω =

Θ11

Θ12

ΘT 12

X (22) Θ22 X (22)

+

Ψ11 0

0 Ψ22

+ Ψτ ,11 0

0 Ψτ ,22

+

1ρ2 + 1ξ

×

H T

(11)H (11) 0

0 X (22) P (22)H T

(22)H (22) P (22)

X (22)

+ 1

ρ2

ΨT

v ,11 Ψv ,11

ΨT v ,11

Ψv ,22

ΨT v ,22

Ψv ,11 ΨT

v ,22 Ψv ,22 + ΨT

n, 22 Ψn ,22

< 0

(C4)

where Z v (22), Θ11 , and Θ22 are defined in (36), and Θ12 is

defined in (C2). The other notations are defined as follows:

Ψ11= X (11) H T

(11)Φ(11)H (11) X (11)

Ψ22= X (22) H T

(22)Φ(22)H (22) X (22)

Ψτ ,11= Aτ (11) Φ−1

τ (11) AT τ (11)

Ψτ ,22= X (22)

P (22) Aτ (22) Φ−1

τ (22) AT τ (22)

P (22) X (22)

Ψv ,11= BT

v (11) H (11)

Ψv ,22= BT

v (21) H (22) P (22)

X (22)

Ψn, 22

= BT v (22) Z v (22) X (22) .




By Lemma 2, we can obtain the following inequality: 0 ΨT

v ,11 Ψv ,22

ΨT v ,22

Ψv ,11 0

≤

ΨT

v ,11 Ψv ,11 0

0 ΨT v ,22

Ψv ,22

.

(C5)

Using the above inequality, we can get the inequality Ω ≤ Ω,in which Ω and Ω are defined in (C2) and (C4), respectively.

Therefore, the inequality Ω < 0 implies the inequality Ω < 0.

Since theinequality in (C4) is equivalent to theinequality in (31),

and the inequality (36) implies the inequality in (C2), we know

the inequality in (36) implies the inequality in (31). Finally,

using the result of Theorem 3, we can prove that if the inequality

in (36) holds, then the H ∞ tracking-control performance in

(24) can be guaranteed by the fuzzy-observer-based tracking

controller in (18)–(20).

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Yu-Te Chang received the B.S. and M.S. degrees inelectrical engineering from Chung Hua University,Hsinchu, Taiwan, in 2002 and 2004, respectively. Heis currently working toward the Ph.D. degree in elec-trical engineering with the National Tsing Hua Uni-versity, Hsinchu.

His current research interests include robust con-trol, fuzzy control and nonlinear systems, and partialdifferential equations.

Bor-Sen Chen (F’01) received the B.S. degree fromTatung Institute of Technology, Taipei, Taiwan, in1970,the M.S. degree from theNationalCentral Uni-versity, Chungli, Taiwan, in 1973, and the Ph.D. de-gree from the University of Southern California, LosAngeles, in 1982.

From 1973 to 1987, he was a Lecturer, an Asso-

ciate Professor, and a Professor with Tatung Instituteof Technology. He is currently a Professor of electri-cal engineering and computer science with the Labo-ratory of Control and Systems Biology, Department

of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan. Heis an Editor of the Asian Journal of Control. He is a member of the EditorialAdvisory Board of the International Journal of Control, Automation, and Sys-tems. He is a member of Editorial Board of Fuzzy Sets and Systems. He hadbeen the Editor-in-Chief of the International Journal of Fuzzy Systems from2006 to 2008. He is currently the Editor-in-Chief of the International Journalof Systems and Synthetic Biology. His current research interests include controlengineering, signal processing, and systems biology.

Prof. Chen was the recipient of the Distinguished Research Award from theNational Science Council of Taiwan four times. He was also the recipient of the Automatic Control Medal from the Automatic Control Society of Taiwanin 2001. He was an Associate Editor of the IEEE TRANSACTIONS ON FUZZY

SYSTEMS from 2001 to 2006. He is a Research Fellow with the National Sci-ence Council of Taiwan and is the holder of the excellent Scholar Chair inengineering.

a fuzzy approach for robust reference-tracking of ball bearing systems

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