boundary stabilization of a cascade of ode-wave systems...

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2017; 27:252–280Published online 31 May 2016 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.3572

Boundary stabilization of a cascade of ODE-wave systems subjectto boundary control matched disturbance

Jun-Jun Liu1 and Jun-Min Wang2,*,†

1College of Mathematics, Taiyuan University of Technology, Taiyuan, 030024 Shanxi, China2School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

SUMMARY

In this paper, we are concerned with a cascade of ODE-wave systems with the control actuator-matched dis-turbance at the boundary of the wave equation. We use the sliding mode control (SMC) technique and theactive disturbance rejection control method to overcome the disturbance, respectively. By the SMC approach,the disturbance is supposed to be bounded only. The existence and uniqueness of solution for the closed-loopvia SMC are proved, and the monotonicity of the ‘reaching condition’ is presented without the differentia-tion of the sliding mode function, for which it may not always exist for the weak solution of the closed-loopsystem. Considering that the SMC usually requires the large control gain and may exhibit chattering behav-ior, we then develop an active disturbance rejection control to attenuate the disturbance. The disturbanceis canceled in the feedback loop. The closed-loop systems with constant high gain and time-varying highgain are shown respectively to be practically stable and asymptotically stable. Then we continue to con-sider output feedback stabilization for this coupled ODE-wave system, and we design a variable structureunknown input-type state observer that is shown to be exponentially convergent. The disturbance is esti-mated through the extended state observer and then canceled in the feedback loop by its approximated value.These enable us to design an observer-based output feedback stabilizing control to this uncertain coupledsystem. Copyright © 2016 John Wiley & Sons, Ltd.

Received 20 June 2015; Revised 5 April 2016; Accepted 3 May 2016

KEY WORDS: boundary control; disturbance rejection; sliding mode control; active disturbance rejectioncontrol; output feedback

1. INTRODUCTION

In the past several decades, the boundary control of systems described by partial differentialequations (PDEs) has become an important research topic in the area of distributed parameter sys-tem control [1]. The contributions can be found in [2, 3] and the references therein. Traditionally,the system is controlled in the ideal operational environment with exact mathematical model and nointernal and external disturbances. This can be found in many researches [4–6] and so on.

However, when the external disturbance on boundary exists, the new approach is needed to dealwith the uncertainties. The general methods to reject the disturbance such as the internal modelprinciple for output regulation, the robust control for systems with uncertainties, the sliding modecontrol (SMC) in various situations [7, 8], and the adaptive control for systems with unknownparameters, to name just a few. In [9], based on the semigroup theory, the SMC is used to deal witha class of abstract infinite-dimensional systems where the control and disturbance are all assumed to

*Correspondence to: Jun-Min Wang, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing100081, China.

†E-mail: [email protected]

Copyright © 2016 John Wiley & Sons, Ltd.

BOUNDARY STABILIZATION OF A CASCADE OF ODE-WAVE SYSTEMS 253

be bounded (mainly in distributed control). The boundary stabilization for a one-dimensional heatequation with boundary disturbance is studied in [10], where the SMC is designed for the first-orderPDEs obtained through an integral transformation on the heat equation. Very recently, the slidingmode boundary stabilizer is designed for a one-dimensional unstable heat, wave, and Schrödingerequation in [2, 11–16] and [17], respectively. Another powerful method in dealing with the distur-bance is the active disturbance rejection control (ADRC) method. The ADRC, as an unconventionaldesign strategy, was first proposed by Han in the 1990s [18]. One of the remarkable features ofADRC is that the disturbance is first estimated in real time through an extended state observer [19]and it is then compensated (canceled) in the feedback loop. Because of this estimation/cancelationnature, the control energy can be significantly reduced [20] in the closed-loop system. Furthermore,it has been proved that the ADRC can deal with very complicated uncertainties and disturbances,including coupling of the external disturbance, the system unmodeled dynamics, and the superaddedunknown part of control input. It has been now acknowledged to be an effective control strategy forlumped parameter systems in the absence of proper models and in the presence of model uncertainty.Its convergence has been proved for lumped parameter systems in [19]. Other method in dealing withuncertainty includes the Lyapunov function-based method; see [21–23] and the references therein.

Controls of the ODE systems with infinite-dimensional actuator dynamics described by PDEshave attracted much attention over the last decades; see [24–31] and the references therein. Becauseproblems concerning coupled systems have been interesting areas for long, both exist in many prac-tical control systems such as electromagnetic coupling, mechanical coupling, and coupled chemicalreactions, and researchers have worked out fruitful results in both areas. However, the externaldisturbance is not considered in these works. An ODE-wave and an ODE-heat system, withoutconsidering disturbance, have been considered in [26] and [32], respectively.

In this paper, we are concerned with stabilization for the following ODE-wave cascade systemthrough Dirichlet interconnection: 8<

:PX.t/ D AX.t/C Bu.0; t/;ut t .x; t/ D uxx.x; t/;ux.0; t/ D 0;ux.1; t/ D U.t/C d.t/;

(1.1)

whereX.t/ 2 Rn�1 is the ODE state, U.t/ is the scalar input to the entire system, u.x; t/ is the stateof the PDE dynamics of the actuator governed by a wave equation, and A 2 Rn�n and B 2 Rn�1

are known constant matrices. It is supposed that†.A;B/ is stabilizable and both d and its derivativeare uniformly bounded, that is, jd.t/j 6M1 and j Pd.t/j 6M2 for some M1;M2 > 0 and all t > 0.

We first introduce a feedback-stabilizing mechanism to ODE part by the transformation .X; u/ 7!.X; v/ in the form [26]:

X.t/ D X.t/; (1.2)

v.x; t/ D u.x; t/ �

Z x

0

�.x � y/u.y; t/dy �

Z x

0

m.x � y/ut .y; t/dy � �.x/X.t/; (1.3)

where �.x/ D KM.x/ with K being chosen so that AC BK is Hurwitz and

m.x/ D

Z x

0

�.�/Bd�; �.x/ D

Z x

0

�.�/ABd�; M.x/ D ŒI 0�e

24 0 A2I 0

35x �I

0

�:

Moreover, differentiating 1.3 with respect to x at x D 1, we have

vx.1; t/ D ux.1; t/ �

Z 1

0

�x.1 � y/u.y; t/dy �

Z 1

0

mx.1 � y/ut .y; t/dy � �0.1/X.t/;

Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2017; 27:252–280DOI: 10.1002/rnc

254 J-J. LIU AND J-M. WANG

and thus, transformation 1.2–1.3 brings system 1.1 into the following intermediate cascade system:8<:PX.t/ D .AC BK/X.t/C Bv.0; t/;vt t .x; t/ D vxx.x; t/;vx.0; t/ D 0;

vx.1; t/ D U.t/C d.t/ �R 10 �x.1 � y/u.y; t/dy �

R 10 mx.1 � y/ut .y; t/dy � �

0.1/X.t/:(1.4)

The transformation 1.3 is invertible, that is,

u.x; t/ D v.x; t/ �

Z x

0

�.x � y/v.y; t/dy �

Z x

0

n.x � y/vt .y; t/dy � �.x/X.t/; (1.5)

where �.x/ D �KN.x/ and

�.s/ D

Z s

0

�.�/ABd�; n.s/ D

Z s

0

�.�/Bd�; N.x/ D ŒI 0�e

24 0 .AC BK/2I 0

35x �I

0

�:

Next, we design for system .X; v/ to .X;w/, by the following transformation [33]:

X.t/ D X.t/; (1.6)

w.x; t/ D v.x; t/C c0

Z x

0

vt .y; t/dy; (1.7)

where 0 < c0 ¤ 1 is the design parameter. Moreover, we can get the inverse of 1.6–1.7 from [2](we can take q D 0 in (4) of [2]):

X.t/ D X.t/;

v.x; t/ D w.x; t/Cc0

c20 � 1

Z x

0

wt .y; t/dy �c20

c20 � 1

Z x

0

wx.y; t/dy:(1.8)

Thus, transformation 1.6 brings the intermediate cascade system 1.4 into the following targetcascade system:8ˆ<ˆ:

PX.t/ D .AC BK/X.t/C Bw.0; t/;wt t .x; t/ D wxx.x; t/;wx.0; t/ D c0wt .0; t/;

wx.1; t/ D U.t/C d.t/ �R 10 �x.1 � y/u.y; t/dy �

R 10 mx.1 � y/ut .y; t/dy � �

0.1/X.t/

C c0ut .1; t/ � c0KBu.1; t/ � c0R 10 �.1 � y/ut .y; t/dy

� c0R 10mxx.1 � y/u.y; t/dy � c0�.1/AX.t/:

(1.9)Therefore, with two transformations 1.2–1.3 and 1.6–1.7, the original system 1.1 and the targetsystem 1.9 are equivalent. So we need to consider systems 1.1 and 1.9 in the real state space

H D Rn �H 1.0; 1/ � L2.0; 1/

with the inner product given by 8.X; f1; g1/>; .Y; f2; g2/> 2 H,

˝.X; f1; g1/

>; .Y; f2; g2/>˛D X>YC

Z 1

0

�f 01.x/f

02.x/C g1.x/g2.x/

�dxCf1.1/f2.1/: (1.10)

It is easy to see that for any a; b 2 H and � 2 R, the following assertions are held: (i) ha; bi Dhb; ai; (ii) h�a; bi D �hb; ai; (iii) h�aC b; ci D �ha; bC ci; and (iv) ha; ai > 0. Hence, 1.10 is aninner product for H.



The main contribution of this paper is to apply the SMC approach to reject the disturbance inthe stabilization of the system (1.9), the reaching condition and the existence and uniqueness ofthe solution for all states in the state space are established, and to apply the ADRC approach toestimate the disturbance in the stabilization of the system (1.9). The disturbance is canceled in thefeedback loop. The closed-loop systems with constant high gain and time-varying high gain areshown respectively to be practically stable and asymptotically stable. Due to the original system 1.1and the target system 1.9 are equivalent, we continue to consider output feedback stabilization forsystem 1.1; we design a variable structure unknown input-type state observer that is shown to beexponentially and estimate the disturbance in terms of the estimated state. These enable us to designan observer-based output feedback stabilizing control to this uncertain coupled system.

The rest of this paper is organized as follows. Section 2 is devoted to the disturbance rejec-tion by the SMC approach. The SMC is designed, and the existence and uniqueness of solutionof the closed-loop system are proved. The finite time ‘reaching condition’ is presented rigorously.In Section 3, we design a disturbance estimator by ADRC approach. First, we design the constanthigh-gain disturbance to show that the closed-loop is practically stable. However, constant high-gaindisturbance estimator shears the simple tuning in practice and noise filtering function but causespeaking value problem in the initial stage. To overcome the peaking value problem, we design atime-varying disturbance estimator and obtain the asymptotic stability. In Section 4, we design anunknown input observer for system 1.1 and show the convergence of the observer. The exponen-tial stability of the closed-loop system is presented, and we design an output feedback control lawfor system 1.1 based on an extended state observer that is designed for an ODE reduced from theobserver. Some concluding remarks are presented in Section 5.

2. SLIDING MODE CONTROLLER

In this section, we will integrate the backstepping approach and the SMC method to design thecontrol U.t/ such that system 1.1 (as depicted in Figure 1) is stabilized.

2.1. Design of sliding surface

Design the sliding surface Sw for system 1.9 as a closed-subspace of H:

Sw D®.X; f; g/> 2 Hjf .1/ D 0

¯: (2.1)

It is obvious that Sw is an infinite-dimensional manifold of H, on which system 1.9 becomes8<:PX.t/ D .AC BK/X.t/C Bw.0; t/;wt t .x; t/ D wxx.x; t/;wx.0; t/ D c0wt .0; t/;w.1; t/ D 0:

(2.2)

Figure 1. Block diagram of a cascade of ODE-wave systems.



Theorem 2.1The cascade ODE-wave system 2.2 in the sliding surface Sw is exponentially stable, that is, thereare positive constants C1 and C2 > 0 such that

jX.t/j2 C kw.x; t/k2 C kwt .x; t/k2 6 C1e�C2t

�jX.0/j2 C kw.0; t/k2 C kwt .0; t/k

2�; (2.3)

where j � j and k � k denote the norms in Rn and L2.0; 1/, respectively.

ProofLet V.t/ be a Lyapunov function:

V.t/ D X.t/>PX.t/C aE.t/; (2.4)

where the matrix P D P> > 0 is the solution to the Lyapunov equation

P.AC BK/C .AC BK/>P D �Q

for some Q D Q> > 0, the parameter a > 0 is to be chosen later, and the function E.t/ is definedby the following:

E.t/ D1

2

Z 1

0

w2x.x; t/dx C1

2

Z 1

0

w2t .x; t/dx C ı

Z 1

0

.x � 1/wx.x; t/wt .x; t/dx; (2.5)

where ı > 0 is an analysis parameter to be chosen later. By taking a derivative of V.t/ along thesolutions of system 2.2, and the boundary condition of 2.2 w.1; t/ D 0, we can get

PV .t/ D PX.t/>PX.t/CX.t/>P PX.t/C a PE.t/

D �X.t/>QX.t/C 2X>PBw.0; t/C a

Z 1

0

wx.x; t/wxt .x; t/dx

C a

Z 1

0

wt .x; t/wxx.x; t/dx C aı

Z 1

0

.x � 1/wxt .x; t/wt .x; t/dx:

Let �min.Q/ be the minimum eigenvalue of Q. By means of Young inequality ab 6 �2a2 C 1

2�b2,

we get

PV .t/ 6 � �min.Q/

2jX j2 C

2jPBj2

�min.Q/w2.0; t/ � awt .0; t/wx.0; t/

Caı

2w2t .0; t/ �

aı

2

Z 1

0

w2t .x; t/dx Caı

2w2x.0; t/ �

aı

2

Z 1

0

w2x.x; t/dx

D ��min.Q/

2jX j2 C

2jPBj2

�min.Q/w2.0; t/ � ac0w

2t .0; t/

Caı

2w2t .0; t/ �

aı

2

Z 1

0

w2t .x; t/dx Caı

2c20w

2t .0; t/ �

aı

2

Z 1

0

w2x.x; t/dx

6 � �min.Q/

2jX j2 C

10jPBj2

�min.Q/

Z 1

0

w2x.x; t/dx � ac0w2t .0; t/

Caı

2w2t .0; t/ �

aı

2

Z 1

0

w2t .x; t/dx Caı

2c20w

2t .0; t/ �

aı

2

Z 1

0

w2x.x; t/dx

D ��min.Q/

2jX j2 � a

�c0 �

ı

2�ı

2c20

w2t .0; t/

�

�aı

2�10jPBj2

�min.Q/

Z 1

0

w2x.x; t/dx �aı

2

Z 1

0

w2t .x; t/dx:



Let ı be small enough and a sufficiently large so that�c0 �

ı

2�ı

2c20

> 0;

�aı

2�10jPBj2

�min.Q/

> 0:

Then there is a C2 > 0 such that PV .t/ 6 �C2V.t/. The proof is then complete. �

By using the transformations 1.6 and 1.3, we have the sliding function for system 1.1:

Sw.t/ , w.1; t/ D v.1; t/C c0Z 1

0

vt .y; t/dy

D u.1; t/ �

Z 1

0

�.1 � y/u.y; t/dy �

Z 1

0

m.1 � y/ut .y; t/dy � �.1/X.t/

C c0

Z 1

0

ut .y; t/dy � c0

Z 1

0

Z s

0

�.s � y/ut .y; t/dyds

C c0

Z 1

0

Z s

0

my.s � y/uy.y; t/dyds � c0

Z 1

0

�.y/ PX.t/dy:

(2.6)

Hence, the original system 1.1 on the sliding surface 2.1 becomes8ˆ<ˆ:

PX.t/ D AX.t/C Bu.0; t/;ut t .x; t/ D uxx.x; t/;ux.0; t/ D 0;

u.1; t/ DR 10 �.1 � y/u.y; t/dy C

R 10 m.1 � y/ut .y; t/dy C �.1/X.t/

� c0R 10ut .y; t/dy C c0

R 10

R s0�.s � y/ut .y; t/dyds

� c0R 10

R s0my.s � y/uy.y; t/dyds C c0

R 10 �.y/

PX.t/dy;

(2.7)

which is exponentially stable by Theorem 2.1 because of the equivalence between 2.2 and 2.7.

2.2. State feedback controller

To motivate the control design, we differentiate the sliding function Sw.t/ in 2.6 formally withrespect to t to obtain the following:

PSw.t/ D wt .1; t/

D c0

�ux.1; t/ �

Z 1

0

�x.1 � y/u.y; t/dy �

Z 1

0

mx.1 � y/ut .y; t/dy � �0.1/X.t/

C ut .1; t/ �

Z 1

0

�.1 � y/ut .y; t/dy C

Z 1

0

my.1 � y/uy.y; t/dy � �.1/ PX.t/

(2.8)and hence

PSw.t/ D wt .1; t/

D c0

�U.t/C d.t/ �

Z 1

0

�x.1 � y/u.y; t/dy �

Z 1

0

mx.1 � y/ut .y; t/dy � �0.1/X.t/

C ut .1; t/ �

Z 1

0

�.1 � y/ut .y; t/dy C

Z 1

0

my.1 � y/uy.y; t/dy � �.1/ PX.t/:

(2.9)Design the feedback controller

U.t/ D�1

c0

�ut .1; t/ �

Z 1

0

�.1 � y/ut .y; t/dy C

Z 1

0

my.1 � y/uy.y; t/dy � �.1/ PX.t/

C

Z 1

0

�x.1 � y/u.y; t/dy C

Z 1

0

mx.1 � y/ut .y; t/dy C �0.1/X.t/C U0.t/;

(2.10)



where U0 is a new control. Then PSw.t/ D c0.U0.t/C d.t//. Let U0.t/ D �.M C /sign.Sw.t//.Then we have

PSw.t/ D �c0.M C /sign.Sw.t//C c0d.t/ (2.11)

and hence,

Sw.t/ PSw.t/ D �c0.M C /sign.Sw.t//Sw.t/C c0d.t/Sw.t/ 6 �c0jSw.t/j; > 0: (2.12)

Note that 2.12 is just the well-known ‘reaching condition’ for system 1.9. The sliding modecontroller is designed as follows:

U.t/ D�1

c0

�ut .1; t/ �

Z 1

0

�.1 � y/ut .y; t/dy C

Z 1

0

my.1 � y/uy.y; t/dy � �.1/ PX.t/

C

Z 1

0

�x.1 � y/u.y; t/dy C

Z 1

0

mx.1 � y/ut .y; t/dy C �0.1/X.t/

� .M C /sign.Sw.t//:(2.13)

The closed-loop system of system 1.9 under the state feedback controller 2.13 is as follows:8<:PX.t/ D .AC BK/X.t/C Bw.0; t/;wt t .x; t/ D wxx.x; t/;wx.0; t/ D c0wt .0; t/;

wt .1; t/ D � c0.M C / sign .Sw.t//C c0d.t/ , Qd.t/:

(2.14)

The next subsection is going to show the existence and uniqueness of the solution to 2.14 and thefinite time ‘reaching condition’ to the sliding mode surface Sw .

2.3. Well-posedness for the target system 2.14

In this section, we consider the target system 2.14. Write system 2.14 as follows:

d

dtZ.�; t / D AZ.�; t /C B Qd.t/; (2.15)

where Z.t/ D .X;w;wt /, B D .0;�ı0.x � 1/C ı.x � 1/; 0/, and´A.X; f; g/ D ..AC BK/X C Bf .0/; g; f 00; /; 8.X; f; g/ 2 D.A/D.A/ D ¹.X; f; g/ 2 Rn �H 2.0; 1/ �H 1.0; 1/jf 0.0/ D c0g.0/; g.1/ D 0º:

(2.16)

Let A be defined by 2.16. For any .X; f; g/> 2 D.A/, we have˝A.X; f; g/>; .X; f; g/>

˛D X>.ACBK/>XCB>Xf .0/�c0jg.0/j

2 6Mk.X; f; g/k2: (2.17)

Then it follows from 2.24 later, which 1 2 �.A/, the resolvent set of A, and the spectrum ofA consists of the isolated eigenvalues only. Therefore, by 2.17, the bounded perturbation of thesemigroup, and the Lumer–Phillips theorem [34], we have that A generates a C0-semigroup of eAt

on H.A direct computation shows that the adjoint operator of A is given by the following:´

A�.Y; '; / D ..AC BK/>Y;� C 2B>Y � B>Yx;�'00/; 8.Y; g/ 2 D.A�/;D.A�/ D ¹.Y; '; / 2 Rn �H 2.0; 1/ �H 1.0; 1/j'0.0/ D �c0 .0/; .1/ D 0º:

(2.18)



The dual system of 2.15 is hence given by the following:8<:PX�.t/ D .AC BK/>X�.t/;w�t t .x; t/ D w

�xx.x; t/;

w�x.0; t/ D c0w�t .0; t/ � 2c0B

>X�; w�t .1; t/ D B>X�

y.t/ D B�0�X�; w�.x; t/; w�t .x; t/

�D w�x.1; t/C w

�.1; t/:

(2.19)

Let

E�.t/ D .X�/>PX� C1

2

Z 1

0

�w�2x .x; t/C .w

�t .x; t/ � B

>X�/2�dx C

1

2w�2.1; t/: (2.20)

Because A generates a C0-semigroup on H, and so does for A�, there exist constants !;M! > 0

such that E�.t/ 6 M!e!tE�.0/ for all t > 0. Let ��.t/ D

R 10 xw

�x.x; t/.w

�t .x; t/ � B

>X�/dx.Then we have

d

dt��.t/ D

Z 1

0

xw�xt .x; t/w�t .x; t/dx C

Z 1

0

xw�x.x; t/w�t t .x; t/dx � B

>X�Z 1

0

xw�xt .x; t/dx

D

Z 1

0

xw�t .x; t/dw�t C

Z 1

0

xw�x.x; t/dw�x � B

>X�Z 1

0

xdw�t

D1

2w�2t .1; t/ �

1

2

Z 1

0

w�2t .x; t/dx C1

2w�2x .1; t/ �

1

2

Z 1

0

w�2x .x; t/dx

� B>X�w�t .1; t/C B>X�

Z 1

0

w�t .x; t/dx

D �1

2.B>Y /2 �

1

2

Z 1

0

w�2t .x; t/dx C1

2w�2x .1; t/

�1

2

Z 1

0

w�2x .x; t/dx C B>X�

Z 1

0

w�t .x; t/dx

D1

2w�2x .1; t/ �

1

2

Z 1

0

�w�2x .x; t/C .w

�t .x; t/ � B

>X�/2�dx

(2.21)from which we haveZ T

0

1

2w�2x .1; t/dt 6 ��.T / � ��.0/C

Z T

0

E�.t/dt 6 .T C 2/E�.0/:

Because

1

2

Z T

0

w�2.1; t/dt 6Z T

0

E�.t/dt 6 TE�.t/ 6 TN!e!TE�.0/; (2.22)

we haveZ T

0

Œw�x.1; t/C w�.1; t/�2dt 6 2

Z T

0

Œw�2x .1; t/C w�2.1; t/�dt 6 DTE�.0/: (2.23)

Furthermore, a direct computation shows that 1 2 �.A�/ and

.I �A�/�10@Xfg

1A D0@ .I � .AC BK/>/�1X

C1ex C C2e

�x CM.x/

f � C1ex � C2e

�x �M.x/C 2B>X � B>Xx

1A ; (2.24)



where 8<:M.x/ D

R x0 sinh.x � s/

�g.s/ � f .s/ � 2B>X C B>Xs

�ds;

C1 D e�1�f � c2e

�1 �M.1/C B>X�;

C2 De�1.1�c0/.f �M.1/CB>X/�c0.M.0/�f �2B>X/CM 0.0/

1Cc0Ce�2�c0e�2:

Hence, .I � A�/�1 exists and is compact on H by the Sobolev embedding theorem [34], so as forthe A. Moreover,

B�.I �A�/�10@Xfg

1A D 2C1e C Z 1

0

e1�s.g.s/ � f .s/ � 2B>X C B>Xs/ds: (2.25)

Hence, B�.I �A�/�1 is bounded on H. This together with 2.23 shows that B� is admissible for theC0-semigroup eA

�t generated by A�. Therefore, system 2.15 admits a unique solution. Moreover,for any T > 0, there exists a constant CT > 0 such that

k.X.t/; w.�; t /; wt .�; t //>kH 6 CT

hk.X.0/; w.�; 0/; wt .�; 0//

T kH C k QdkL2.0;T /

i; 8t 2 Œ0; T �:

(2.26)We may suppose without loss of generality that w.1; 0/ > 0, because w.1; 0/ < 0 is similar. In thiscase, it follows from 2.14 that

Qd.t/ D �c0.M C /C c0d.t/: (2.27)

Now, for Qd defined by 2.27, because H 10 .0; T / is dense in L2.0; T /, take Qdn 2 H 1

0 .0; T / such that

limn!1

k Qdn � QdkL2.0;T / D 0:

Let .Xn.t/; wn.�; t /; wnt .�; t //> be the solution of 2.15 corresponding to Qdn, and the initial value.X.0/; wn.�; 0/; wnt .�; 0//

> 2 D.A/ where

limn!1

k.Xn.0/; wn.�; 0/; wnt .�; 0//> � .X.0/; w.�; 0/; wt .�; 0//

>kH D 0:

It follows from 2.26 that

limn!1

k.Xn.0/; wn.�; 0/; wnt .�; 0//> � .X.0/; w.�; 0/; wt .�; 0//

>kH D 0

uniformly in t 2 Œ0; T �. By Proposition 4.2.1 of [35] p.120, we know that .Xn.t/, wn.�; t /,wnt .�; t //

> is the classical solution of 2.15 (or 2.14). Consequently,

wnt .1; t/ D Qdn.t/ or wn.1; t/ D wn.1; 0/C

Z t

0

Qdn./d:

Passing to the limit as n!1 in the aforementioned equality, we obtain

w.1; t/ D w.1; 0/C

Z t

0

Qd./d; 8t 2 Œ0; T �: (2.28)

Because T is arbitrary, we see that Sw.t/ D w.1; t/ to 2.14 with Qd defined by is continuous inŒ0;1/ for any initial value in the state space. Furthermore, owing to 2.28, it has



Sw.1; t/ D w.1; t/ D w.1; 0/C

Z t

0

Qd./d D w.1; 0/C

Z t

0

Œ�c0.M C /C c0d.t/�

6 w.1; 0/ � c0t:(2.29)

It is seen that Sw.t/ is decreasing in t . Because w.1; 0/ > 0, there exists some t0 > 0 such thatw.1; t/ > 0 for t 2 Œ0; t0/ and w.1; t/ D 0 for all t > t0.

Remark 2.2When w.1; 0/ < 0, 2.29 becomes

Sw.1; t/ D w.1; t/ D w.1; 0/C

Z t

0

Qd./d D w.1; 0/C

Z t

0

Œc0.M C /C c0d.t/�

> w.1; 0/C c0t:(2.30)

In this case, Sw.t/ is a continuous increasing function, and hence, there exists some t0 > 0 suchthat w.1; t/ < 0 for t 2 Œ0; t0/ and w.1; t/ D 0 for all t > t0.

Returning to the original system by the inverse transformation, we obtain the first main result ofthis paper.

Theorem 2.3Suppose that d is measurable and jd.t/j 6M0 for all t > 0, and let SU be the sliding mode functiongiven by the following:

SU .t/ D u.1; t/�

Z 1

0

�.1� y/u.y; t/dy �

Z 1

0

m.1� y/ut .y; t/dy � �.1/X.t/C c0

Z 1

0

v.y; t/dy:

(2.31)Then for any .u.�; 0/; ut .�; 0// 2 H, SU .0/ ¤ 0, there exists a tmax > 0 such that the closed-loopsystem of 1.1 under the feedback control 2.13 is as follows:8

ˆ<ˆ:


ux.1; t/ DR 10�x.1 � y/u.y; t/dy C

R 10mx.1 � y/ut .y; t/dy C �

0.1/X.t/

� 1c0

ut .1; t/ �

R 10�.1 � y/u.y; t/dy �

R 10my.1 � y/ut .y; t/dy � �.1/ PX.t/

�� .M C /sign.SU .t//C d.t/;

(2.32)

which admits a unique solution u 2 C.0; tmaxIH/ and SU .t/ D 0 for all t > tmax . On the slidingmode surface SU .t/ D 0, system 1.1 becomes8

ˆ<ˆ:


u.1; t/ DR 10�.1 � y/u.y; t/dy C

R 10m.1 � y/ut .y; t/dy

C �.1/X.t/ � c0R 10 ut .y; t/dy C c0

R 10

R s0�.s � y/ut .y; t/dyds

� c0R 10

R s0my.s � y/uy.y; t/dyds C c0

R 10 r.y/

PX.t/dy;

(2.33)

which is equivalent to 2.2 and hence is exponentially stable in H with the decay rate �c.

It is remarked that systems 2.14 and 2.32 are equivalent with the two step transformations 1.2–1.3and 1.6–1.7.

3. ACTIVE DISTURBANCE REJECTION CONTROL

It is well known that the so-called chattering behavior is associated with the SMC, due to disconti-nuity of control. In this section, we shall use a direct approach to attenuate rather than to reject the



disturbance. This is the key to the ADRC method in finite-dimensional systems. The idea is to firstestimate the disturbance and then to cancel the estimate in the feedback loop. Unlike the SMC thatusually uses high gain control, the control effort in ADRC is found to be moderate.

3.1. Constant high-gain estimator-based feedback

In estimating the disturbance, we assume that the derivative of the disturbance is bounded: j Pd.t/j 6M2 for some M2 > 0 and all t > 0. The objective now is to design a continuous U.t/ that canstabilize system 1.9 in the presence of disturbance. In view of 2.13, this controller is designedas follows:

U.t/ D U1.t/ CU2.t/ D1

1�c20

wx.1; t/CR 10 �x.1 � y/u.y; t/dy

CR 10mx.1 � y/ut .y; t/dy C �

0.1/X.t/C U2.t/;(3.1)

where U2.t/, also continuous, is to be designed in what follows. Under control 3.1, the closed-loopof system 1.9 becomes 8<

ˆ:PX.t/ D .AC BK/X.t/C Bw.0; t/;wt t .x; t/ D wxx.x; t/;wx.0; t/ D c0wt .0; t/;

wt .1; t/ D �1�c2

0

c0d.t/ �

1�c20

c0U2.t/:

(3.2)

Introduce a variable y.t/ D w.1; t/. Then the boundary condition at x D 1 in 3.2 gives that

Py.t/ D �1 � c20c0

d.t/ �1 � c20c0

U2.t/: (3.3)

It is seen that 3.3 is an ODE with state y and control U2. Now, we are able to design an extendedstate observer to estimate both y and d as follows:8<: POy".t/ D �

1�c20

c0Od".t/ �

1�c20

c0U2.t/ �

1". Oy".t/ � y.t//;

POd".t/ D1"2

c01�c2

0

. Oy".t/ � y.t//;(3.4)

where " is the tuning small parameter. Then errors Qy".t/ D Oy".t/ � y.t/ and Qd".t/ D Od".t/ � d.t/satisfy

PQy.t/ D �1 � c20c0

Qd.t/ �1

"Qy.t/;

PQd.t/ D1

"2c0

1 � c20Qy.t/ � Pd.t/; (3.5)

which can be rewritten as follows:

d

dt

�Qy".t/Qd".t/

D A

�Qy".t/Qd".t/

C B Pd.t/; (3.6)

where

A D

0@ �1"�1�c2

0

c01"2

c01�c2

0

0

1A ; B D

�0

�1

:

A direct computation gives the eigenvalues of A:

� D �1

2"˙

p3

2"j: (3.7)



Hence, we have

eAt D

0B@ �1�2��1

e�1t ��2

�2 � �1e�2t �1�2

C".�2��1/.e�2t � e�1t /

C"�2��1

.e�1t � e�2t / � �2�2��1

e�1t C�1

�2 � �1e�2t

1CAand

eAtB D

0@ �1�2C".�2��1/

.e�2t � e�1t /

� �2�2��1

e�1t C�1

�2 � �1e�2t

1A ; C" D1

"2c0

1 � c20:

Because �Qy.t/Qd.t/

D eAt

�Qy.0/Qd.0/

CR t0eA.t�s/B Pd.s/ds; (3.8)

the first term previously can be arbitrarily small as t ! 1 by the exponential stability of eAt ,and the second term can also be arbitrarily small as " ! 0 because of boundedness of Pd and theexpression of C" and eAtB . As a result, . Qy.t/; Qd.t// can be arbitrarily small as t !1 and "! 0.

The state feedback controller to 3.2 is designed as follows:

U2.t/ D � satOd".t/

�CM

c0

1 � c20w.1; t/; (3.9)

where

sat .x/ D

8<:M1; x >M1 C 1;�M1; x 6 �M1 � 1;x; x 2 .�M1 � 1;M1 C 1/:

(3.10)

It is clearly seen from 3.9 that this controller is just used to cancel the disturbance d because A gen-erates an exponentially stable C0-semigroup. This estimation/cancelation strategy 3.9 is obviouslyan economic strategy. Under the feedback 3.9, the closed-loop system of 3.2 becomes8

ˆˆ<ˆˆ:


wt .1; t/ D1�c2

0

c0.sat. Od".t// � d.t// �Mw.1; t/;

w.1; t/ D y.t/;

Py.t/ D1�c2

0

c0.sat. Od".t// � d.t// �Mw.1; t/;

POy".t/ D �1". Oy".t/ � y.t// �Mw.1; t/;

POd".t/ D1"2

c01�c2

0

. Oy".t/ � y.t//:

(3.11)

Proposition 3.1Assume that jd j 6 M1 and j Pd j 6 M2 are measurable. Then for any initial value .X.0/, w.�; 0/,wt .�; 0/, Oy".0/, and Od".0// 2 H � R2, the closed-loop system 3.11 admits a unique solution .X , ´,Oy", Od" /> 2 C.0;1IH �R2/, and

limt!1"!0

��.X.t/; ´.�; t /; Oy".t/; Od".t/ � d.t//��H�R2

D 0:



ProofUsing the error dynamics 3.5, we see that 3.11 is equivalent to the following:8

ˆ<ˆˆ:

PX.t/ D .AC BK/X.t/C Bw.0; t/;

wt t .x; t/ D wxx.x; t/;

wx.0; t/ D c0wt .0; t/;

wt .1; t/ D1�c2

0

c0

satQd".t/C d.t/

�� d.t/

��Mw.1; t/;

PQy".t/ D �1"Qy".t/ �

1�c20

c0Qd".t/;

PQd".t/ D1"2

c01�c2

0

Qy".t/ � Pd.t/:

(3.12)

It is seen that in 3.12, the variable . Qy; Qd/ is independent of the ‘w part’ and can be made as smallas desired as t ! 1 and " ! 0. Thus, we need to consider only the ‘w part’ that is re-writtenas follows: 8<

ˆ:PX.t/ D .AC BK/X.t/C Bw.0; t/;wt t .x; t/ D wxx.x; t/;wx.0; t/ D c0wt .0; t/;

wt .1; t/ D c�10

�1 � c20

� sat. Qd".t/C d.t/

�� d.t// �Mw.1; t/:

(3.13)

System 3.13 will be considered in the state Hilbert space H D H 2.0; 1/ � L2.0; 1/. Define²A.X; f; g/ D ..AC BK/X C Bf .0/; g; f 00; /; 8.X; f; g/ 2 D.A/D.A/ D ¹.X; f; g/ 2 Rn �H 2.0; 1/ �H 1.0; 1/jf 0.0/ D c0g.0/; g.1/ D �Mf.1/º:

(3.14)

Similar to 2.15, system 3.13 can be rewritten as an evolution equation in H as follows:

d

dtZ.�; t / D AZ.�; t /C c�10

�1 � c20

�B

sat. Qd.t/C d.t/�� d.t//; (3.15)

where Z.t/ D .X;w;wt / and B D .0; �ı0.x � 1/C ı.x � 1/; 0/. We can show that A generatesan exponentially stable C0-semigroup on H and B is admissible to the semigroup eAt [2]. Then forŒX.0/; w.�; 0/; wt .�; 0/� 2 H, there exists a unique solution ŒX;w;wt � 2 C.0;1IH/:

Z.�; t / D eAtZ.�; 0/C c�10�1 � c20

� Z t

0

eA.t�s/B

sat. Qd".s/C d.s// � d.s/�ds: (3.16)

By 3.8, for any given "0 > 0, there exist t0 > 0 and "1 > 0 such thatˇ.sat. Qd".s/C d.s// � d.s//

ˇ< "0 for all t > t0 and 0 < " < "1:

We rewrite solution of 3.16 as follows:

Z.�; t / D eAtZ.�; 0/C c�10�1 � c20

�eA.t�t0/

Z t0

0

eA.t0�s/B

satQd".s/C d.s/

�� d.s/

�ds

C c�10�1 � c20

� Z t

t0

eAD.t�s/B

satQd".s/C d.s/

�� d.s/

�ds:

(3.17)The admissibility of B implies that 8 Qd 2 L1.0;1/,��Z t0

0

eA.t�s/B

satQd".s/C d.s/

�� d.s/

�ds

��2H6 Ct0 j

satQd".s/C d.s/

�� d.s/

�k2L2loc.0;t/

6 t0Ct0.2M1 C 1/2;

(3.18)



for some constant Ct0 that is independent of Qd . Because eAt is exponentially stable, it follows fromProposition 2.5 of [36] that��R tt0 eA.t�s/BŒsat. Qd".s/C d.s// � d.s/�ds

��D

��R t0 eA.t�s/B.0Þt0 hsat. Qd".s/C d.s// � d.s/ids

��6 L

��Œsat. Qd" C d/ � d��L1.0;1/

6 L"0;

(3.19)

where L is a constant that is independent of Qd and

.d1Þ�d2/.t/ D

²d1.t/; 0 6 t 6 ;d2.t � /; t > ;

(3.20)

where the left-hand side of 3.20 denotes the -concatenation of d1 and d2. Suppose that

keAtk 6 L0e�!t for some L0; ! > 0:

By 3.17, 3.18, and 3.19, we have

kZ.�; t /k 6 L0e�!t kZ.�; 0/k C L0Ct0c�10�1 � c20

�e�!.t�t0/.2M1 C 1/

2 C L"0: (3.21)

The first two terms of 3.21 tend to 0 as t !1. The result is then proved by the arbitrariness of "0.�

Theorem 3.2Suppose jd j 6M1 and j Pd j 6M2 are measurable. Then for ŒX.0/; u.�; 0/; ut .�; 0/� 2 H, the closed-loop of system 1.18

ˆ<ˆˆ:

PX.t/ D AX.t/C Bu.0; t/; t > 0;ut t .x; t/ D uxx.x; t/; x 2 .0; 1/; t > 0;ux.0; t/ D 0;

ux.1; t/ D1

c20�1wx.1; t/C

R 10 �x.1 � y/u.y; t/dy C

R 10 mx.1 � y/ut .y; t/dy

C� 0.1/X.t/CM c01�c2

0

w.1; t/ � Od.t/C d.t/;

POy".t/ D �1�c2

0

c0Od".t/ �

1�c20

c0U2.t/ �

1". Oy".t/ � y.t//;

POd".t/ D1"2

c01�c2

0

. Oy".t/ � y.t//;

(3.22)

admits a unique solution .X.�/; u.x; t/; ut .x; t//> 2 C.0;1IH/, and the solution of system 3.22tends to any arbitrary given vicinity of 0 as t !1 and "! 0.

3.2. Time-varying high-gain estimator-based feedback

In the previous subsection, we estimate the disturbance d by constant high gain. This brings thenotorious peaking value problem in estimator. In this section, we propose a novel disturbance esti-mator by time-varying high gain. This improves the performance through four aspects: (i) Thepractical stability claimed by Theorem 3.2 becomes the asymptotic stability; (ii) the boundedness ofderivative of disturbance is relaxed in much extent; (iii) the peaking value is reduced significantly;and (iv) the possible non-smooth control 3.9 becomes smooth. The possible trouble brought by thisapproach is the high-frequency noise sensitivity.

Now, we design the following extended state observer with time-varying high gain for system 3.3as follows: 8<

:POy.t/ D

c20�1

c0U2.t/C

c20�1

c0Od.t/ � g.t/Œ Oy.t/ � y.t/�;

POd.t/ D �g2.t/

²c0c20�1Œ Oy.t/ � y.t/�

³;

(3.23)



where g 2 C 1Œ0;1/ is a time-varying gain function satisfying the following:²g.t/ > 0; Pg.t/ > 0; 8 t > 0:g.t/!1 as t !1; supt2Œ0;1/ j Pg.t/=g.t/j <1:

(3.24)

In addition, we assume that the disturbance d 2 H 1loc.0;1/ satisfies

limt!1

j Pd.t/j C jd.t/j

g.t/D 0: (3.25)

By condition 3.25, both d and Pd are allowed (at least mathematically) to grow exponentially at anyrate by choosing properly the gain function g. This relaxes the condition in the previous sectionwhere d and Pd are assumed to be uniformly bounded.

Remark 3.3It is worth mentioning that, in theory, the time-varying gain Extended State Observer (ESO)degrades the ability of ESO to filter high-frequency noise, while the constant gain ESO does not. Inpractical applications, we can use time-varying gain g.t/ as follows: (i) given a small initial valueg.0/ > 0; (ii) from the constant high gain, we obtain the convergent high-gain value r that can alsobe obtained by trial and error experiment for practical systems; and (iii) our gain function is usuallysupposed to grow continuously from g.0/ > 0 to r . For instance, we take g satisfying Pg.t/ D ag.t/,a > 0. In this case, we can compute the switching time as a�1ln.r=g.0//, where a is used to con-trol the convergent speed and the peaking value. The larger a is, the faster convergence but largerpeaking, while the smaller a is, the low convergence speed and smaller peaking.

Once again, Od is used to estimate d , which is confirmed by the following Lemma 3.4.

Lemma 3.4Let . Oy; Od/ be the solution of 3.23. Then

limt!1j Oy.t/ � y.t/j D 0; lim

t!1j Od.t/ � d.t/j D 0: (3.26)

ProofSet

Qy.t/ D g.t/Œ Oy.t/ � y.t/�; Qd.t/ Dc20 � 1

c0. Od.t/ � d.t//: (3.27)

Then the error . Qy; Qd/ is governed by the following:8<: PQy.t/ D �g.t/hQy.t/ � Qd.t/

iC Pg.t/

g.t/Qy.t/;

PQd.t/ D �g.t/ Qy.t/ �c20�1

c0Pd.t/:

(3.28)

The existence of the local classical solution to 3.28 is guaranteed by the local Lipschitz conditionof the right side of 3.28. The global solution will be ensured by the following Lyapunov functionargument. Define

V.t/ D Qy2.t/C3

2Qd2.t/ � Qy.t/ Qd.t/: (3.29)

Then

1

2V.t/ 6 Qy2.t/C Qd2.t/ 6 2V.t/:



Differentiate V along the solution of 3.28 to obtain the following:

PV .t/ D 2 Qy.t/ PQy.t/C 3 Qd.t/PQd.t/ � PQy.t/ Qd.t/ �

PQd.t/ Qy.t/

D 2 Qy.t/

²�g.t/

hQy.t/ � Qd.t/

iCPg.t/

g.t/Qy.t/

³C 3 Qd.t/

²�g.t/ Qy.t/ �

c20 � 1

c0Pd.t/

³� Qd.t/

²�g.t/

hQy.t/ � Qd.t/

iCPg.t/

g.t/Qy.t/

³� Qy.t/

²�g.t/ Qy.t/ �

c20 � 1

c0Pd.t/

³D

��g.t/C

2 Pg.t/

g.t/

�Qy2.t/ � g.t/ Qd2.t/ �

Pg.t/

g.t/Qy.t/ Qd.t/C

c20 � 1

c0Pd.t/

hQy.t/ � 3 Qd.t/

i6 � 1

2�.t/V .t/C 4

ˇc20 � 1

c0

ˇj Pd.t/jk. Qy.t/; Qd.t//k

6 � 12�.t/V .t/C

p2

�4

ˇc20 � 1

c0

ˇj Pd.t/j C 3g.t/jy2.t/j

�pV.t/;

(3.30)where by assumptions 3.24 and 3.25 ,

�.t/ D g.t/ � supt2Œ0;1/

ˇ3 Pg.t/

g.t/

ˇ!1 as t !1; (3.31)

and hence, there exists a t0 > 0 such that �.t/ > 0, for each t > t0. This, together with 3.30,shows thatp

V.t/

dt6 �1

4�.t/

pV.t/C

p2

2

�4

ˇc20 � 1

c0

ˇ ˇPd.t/

ˇC 3g.t/jy2.t/j

�;8 t > 0:

It then follows that

pV.t/ 6

pV.0/e�

14

R t0�.s/dsC

Z t

0

�p2

�4

ˇc20�1

c0

ˇ ˇPd.s/

ˇC 3g.t/jy2.t/j

�e14

R s0 �.�/d�ds

2e14

R t0�.s/ds

: (3.32)

The first term on the right-hand side of 3.32 is obviously convergent to 0 as t ! 1 owingto 3.31. Apply the L’Hospital rule to the second term on the right-hand side of 3.32 to obtainlimt!1

pV.t/ D 0, which amounts to the following:

limt!1

hQy2.t/C Qd2.t/

iD 0:

This leads to the following:

limt!1

hj Qy.t/j C j Qd.t/j

iD 0:

The proof is complete. �

By Lemma 3.4, we design naturally the feedback control

U2.t/ D � Od.t/CMc0

1 � c20w.1; t/; (3.33)



under which the closed-loop of system 3.2 is as follows:8ˆ<ˆ:


wt .1; t/ D �c20�1

c0. Od.t/ � d.t// �Mw.1; t/;

POy.t/ Dc20�1

c0U2.t/C

c20�1


POd.t/ D � g2.t/

²c0c20�1Œ Oy.t/ � y.t/�

³:

(3.34)

Proposition 3.5Assume that the time-varying gain g 2 C 1Œ0;1/ satisfies 3.24 and the disturbance d 2 H 1

loc.0;1/

satisfies 3.25. Then for any initial value .X.0/; w.0/; wt .0/; Oy.0/; Od.0// 2 H � R2, there exists aunique solution .X; ´; Oy; Od/ 2 C.0;1IH � R2/ to system 3.34. Moreover, 3.34 is asymptoticallystable in the sense that

limt!1k.X.t/; w.�; t /; wt .�; t /; Oy.t/; Od.t/ � d.t//kH�R2 D 0:

ProofUsing the error variables . Qy; Qd/ defined in 3.27, we can write the equivalent system of 3.34as follows: 8

ˆ<ˆ:


wt .1; t/ D � Qd.t/ �Mw.1; t/;

PQy.t/ D � g.t/hQy.t/ � Qd.t/

iC Pg.t/

g.t/Qy.t/;

PQd.t/ D � g.t/ Qy.t/ �c20�1

c0Pd.t/:

(3.35)

The ‘ODE part’ of 3.35 is just the system 3.28, which is shown to be convergent by Lemma 3.4.The ‘.X;w;wt / part’ of 3.35 is similar to 3.13, and the proof hence becomes similar to the proof ofProposition 3.1. The details are omitted. �

Returning back to system 1.1 by the inverse transformations 1.5 and 1.8, and feedback control3.1 and 3.33, we have proved, from Proposition 3.5, the following Theorem 3.6.

Theorem 3.6Assume that the time-varying gain g 2 C 1Œ0;1/ satisfies 3.24 and the disturbance d 2 H 1

loc.0;1/

satisfies 3.25. Then for any initial value .X.0/; u.�; 0/; ut .�; 0/; Oy.0/; Od.0// 2 H � R2, the closed-loop of system 1.1 following8

ˆ<ˆˆ:

PX.t/ D AX.t/C Bu.0; t/; t > 0;ut t .x; t/ D uxx.x; t/; x 2 .0; 1/; t > 0;ux.0; t/ D 0;

ux.1; t/ D�1

c20�1wx.1; t/C

R 10 �x.1 � y/u.y; t/dy C

R 10 mx.1 � y/ut .y; t/dy

C� 0.1/X.t/CM c01�c2

0

w.1; t/ � Od.t/C d.t/;

POy.t/ Dc20�1

c0U2.t/C

c20�1


POd.t/ D � g2.t/

²c0c20�1Œ Oy.t/ � y.t/�

³(3.36)

admits a unique solution .X; u; ut ; Oy; Od/> 2 C.0;1IH �R2/, and 3.36 is asymptotically stable:

limt!1k.X.t/; u.�; t /; ut .�; t /; Oy.t/; Od.t/ � d.t//kH�R2 D 0;



where y.t/ D w.1; t/, U2.t/ D � Od.t/ C Mc0

1 � c20w.1; t/, and w; v are given by 1.7 and

1.3, respectively.

4. OUTPUT FEEDBACK STABILIZATION FOR SYSTEM 1.1

In this section, we are concerned with the boundary output feedback stabilization for the cascadedODE-wave system 1.1. For convenience, we restate system 1.1 as follows:8

<ˆ:PX.t/ D AX.t/C Bu.0; t/;ut t .x; t/ D uxx.x; t/;ux.0; t/ D 0; ux.1; t/ D U.t/C d.t/;X.0/ D X0; u.x; 0/ D u0.x/; ut .x; 0/ D u1.x/;Y0.t/ D ¹CX.t/; ut .0; t/; ut .1; t/; u.1; t/º ;

(4.1)

where .C;A/ is supposed to be observable and Y0.t/ is the output measurement, that is to say, theboundary signals CX.t/, ut .0; t/, ut .1; t/, and u.1; t/ are measured.

Now, we design an unknown input-type state observer for system 4.1; to ensure the existence ofthe solution, we need to set the multi-valued function as follows:

sign .x/ D

8<: 1; x > 0;

Œ�1; 1� ; x D 0;�1; x < 0:

Then the state observer for system 4.1 is as follows:8ˆ<ˆ:

POX.t/ D A OX.t/C LC. OX �X/C B Ou.0; t/;Out t .x; t/ D Ouxx.x; t/;Oux.0; t/ D cŒ Out .0; t/ � ut .0; t/�;Oux.1; t/ 2 U.t/ � c1Œ Out .1; t/ � ut .1; t/� � c2Œ Ou.1; t/ � u.1; t/�

�M3 sign . Out .1; t/ � ut .1; t// �M4 sign . Ou.1; t/ � u.1; t//;OX.0/ D OX0; Ou.x; 0/ D Ou0.x/; Out .x; 0/ D Ou1.x/;

(4.2)

where M3 > M1 C ˛, M4 > M1 C M3 C ˇ, ˛ > 0, ˇ > 0, c, c1, and c2 are positive designparameters to be determined, and L is chosen so that AC LC is Hurwitz. Set the error variable�

QX.t/; Qu.x; t/; Qut .x; t/�DOX.t/ �X.t/; Ou.x; t/ � u.x; t/; Out .x; t/ � ut .x; t/

�: (4.3)

Then the error system becomes8<ˆ:PQX.t/ D .AC LC/ QX.t/C B Qu.0; t/;Qut t .x; t/ D Quxx.x; t/;Qux.0; t/ D c Qut .0; t/;Qux.1; t/ 2 �c1 Qut .1; t/ � c2 Qu.1; t/ �M3 sign . Qut .1; t// �M4sign. Qu.1; t// � d.t/;QX.0/ D QX0; Qu.x; 0/ D Qu0.x/; Qut .x; 0/ D Qu1.x/:

(4.4)

The next Theorem 4.1 shows that error system 4.4 is exponentially convergent.

Theorem 4.1For any initial value . QX.0/; Qu0; Qu1/ 2 Rn �H 2.0; 1/ �H 1.0; 1/ satisfying compatible conditions

Qu00.0/ D c Qu1.0/;

Qu00.1/ 2 �c1 Qu1.1/ � c2 Qu0.1/ �M3 sign . Qu1.1// �M4 sign . Qu0.1// � d.0/;(4.5)



and measurable disturbance d.t/ 2 H 1loc.0;1/, the error system 4.4 admits at least one partial

differential inclusion solution . QX.t/; Qu.�; t /; Qut .�; t // 2 C.0;1IRn � H 2.0; 1/ � H 1.0; 1// isexponentially stable

E.t/ 6 C2E.0/e��t ; (4.6)

where C2 and � are two positive constants and E.t/ is given by the following:

E.t/ DˇQX.t/

ˇ2RnC1

2

Z 1

0

Œ Qu2t .x; t/C Qu2x.x; t/�dx C

c2

2Qu2.1; t/CM4j Qu.1; t/j: (4.7)

This shows that the error system 4.4 is exponentially convergent.

ProofIt is seen from 4.4 that the state Qu is independent of the ‘ QX’ part, by Theorem 1 of the [37];the ‘ Qu’ part in 4.4 has at least one partial differential inclusion solution . Qu.�; t /, Qut .�; t // 2C.0;1IH 2.0; 1/ �H 1.0; 1//, which satisfies

1

2

Z 1

0

�Qu2x.x; t/C Qu

2t .x; t/

�dx C

c2

2Qu2.1; t/CM4j Qu.1; t/j

6 2

1 � 4ıe��1t

�1

2

Z 1

0

Œ Qu020 .x/C Qu

21.x/�dx C

1

2Qu20.1/CM4ju0.1/j

�;

(4.8)

where 0 < ı < 1=4, �1 D 1=2min ¹ı; c2ı=2; c2ıˇ=M4º. Next, we prove the ‘ QX’ part is alsoconvergent. By the Agmons inequality and Poincar inequality [1], we can obtain

Qu2.0; t/ 6 C3�Qu2.1; t/C

Z 1

0

Qu2x.x; t/dx

�(4.9)

for some constant C3 > 0. By the constant variation rule, we can get the solution of ‘ QX’ part that is

QX.t/ D e.ACLC/t QX.0/C

Z t

0

e.ACLC/.t�s/B Qu.0; s/ds: (4.10)

Because AC LC is Hurwitz, there exists two positive constants C4 and �2 > 0 such that

ke.ACLC/tk 6 C4e��2t : (4.11)

Hence,

ˇQX.t/

ˇRn6ˇe.ACLC/t QX.0/

ˇRnC

ˇZ 1

0

e.ACLC/.t�s/B Qu.0; s/ds

ˇRn

6 C4e��2t j QX.0/jRn C C5kBk�Qu20.1/C

Z 1

0

Œ Qu020 .x/C Qu

21.x/�dx

�e��2t

Z 1

0

e.�2��/sds;

where C5 is a positive constant. This together with 4.8 yields 4.6. �

We are now in a position to design a stabilizing output feedback control for system 4.1. Becausewe have got the state estimation through the observer claimed by Theorem 4.1, we can design anoutput feedback control for 4.1 as two parts:

U.t/ D U3.t/C U4.t/; (4.12)



where U3.t/ will be designed later for the rejection of the disturbance and U4.t/ is proposed tostabilize the closed-loop system (4.35a) and (4.35b) without disturbance

U4.t/ D

Z 1

0

�x.1 � y/ Ou.y; t/dy C

Z 1

0

mx.1 � y/ Out .y; t/dy C �0.1/ OX.t/ � c0ut .1; t/

C c0KBu.1; t/C c0

Z 1

0

�.1 � y/ Out .y; t/dy C c0

Z 1

0

mxx.1 � y/ Ou.y; t/dy

C c0�.1/A OX.t/ � k

²u.1; t/ �

Z 1

0

�.1 � y/ Ou.y; t/dy �

Z 1

0

m.1 � y/ Out .y; t/dy

� �.1/ OX.t/ � c0

Z 1

0

�Z x

0

�.x � y/ Out .y; t/dy C

Z x

0

mxx.x � y/ Ou.y; t/dy

�dx

C c0

Z 1

0

Œ Out .x; t/ �KB Ou.x; t/� dx � c0

Z 1

0

�.x/dxA OX.t/

³(4.13)

with c0 > 0; c0 ¤ 1 being a positive constant. By the transformations 1.2–1.3 and 1.6–1.7, theerror system 4.3, and the feedback control 4.12, and system 4.1 can be equivalently translated in thefollowing target system (similarly system 1.9 with state feedback control):8<

:PX.t/ D .AC BK/X.t/C Bw.0; t/;wt t .x; t/ D wxx.x; t/;wx.0; t/ D c0wt .0; t/;wx.1; t/ D � kw.1; t/C U3.t/C de.t/;

(4.14)

where de.t/ D d.t/C f .t/ and f .t/ is given by the following:

f .t/ D

Z 1

0

�x.1 � y/ Qu.y; t/dy C

Z 1

0

mx.1 � y/ Qut .y; t/dy C �0.1/ QX.t/

C c0

Z 1

0

�.1 � y/ Qut .y; t/dy C c0

Z 1

0

mxx.1 � y/ Qu.y; t/dy C c0�.1/A QX.t/

C k

²Z 1

0

�.1 � y/ Qu.y; t/dy C

Z 1

0

m.1 � y/ Qut .y; t/dy C cA QX.t/

Z 1

0

�.x/dx C �.1/ QX.t/

� c0

Z 1

0

�Qut .x; t/ �KB Qu.x; t/ �

Z x

0

�.x�y/ Qut .y; t/dy�

Z x

0

mxx.x�y/ Qu.y; t/dy

�dx

³:

(4.15)

By Theorem 4.1, f .t/ has the estimate, for some positive constants Mf and �f :

jf .t/j 6Mf e��f t ; 8t > 0: (4.16)

Next, we are going to design the control U3.t/ to stabilize 4.14 in the presence of de.t/. System4.14 can be rewritten as an abstract evolution equation:

d

dt

0@ X.t/

w.�; t /wt .�; t /

1A D A0

0@ X.t/

w.�; t /wt .�; t /

1AC B0 .U3.t/C de.t// ; (4.17)

where B0 D .0; 0; ı.x�1//> and A0 is an unbounded linear operator defined in H by the following:²A0.X; f; g/> D ..AC BK/X C Bf .0/; g; f 00/>; 8.X; f; g/> 2 D.A0/;D.A0/ D

®.X; f; g/> 2 Rn �H 2.0; 1/ �H 1.0; 1/jf 0.0/ D cg.0/; f 0.1/ D �kf .1/:

(4.18)

A simple computation shows that A�0 , the adjoint operator of A0, is given by the following:8<:A�0.X; f; g/> D ..AC BK/X C Bf .0/;�g;�f 00/>;8.X; f; g/> 2 D.A�0/;

D.A0/ D².X; f; g/> 2 Rn �H 2.0; 1/ �H 1.0; 1/

ˇf 0.0/ D �cg.0/;

f 0.1/ D �kf .1/; B>X D 0:

(4.19)



It is well known that A0 generates a C0-semigroup eA0t of contractions on H and B0 is admissibleto eA0t . Hence, we have Proposition 4.2 concerning the well-posedness of system 4.17.

Proposition 4.2The operator A0 defined in 4.18 generates a C0-semigroup eA0t of contractions on H, and B0 isadmissible to eA0t . Therefore, for .X.0/; w.�; 0/; wt .�; 0//> 2 H, U3 2 L2loc.0;1/, and de.t/ 2L2loc.0;1/, system 4.17 admits a unique solution .X.t/; w.�; t /; wt .�; t //> 2 H.

By Proposition 4.2, the weak solution of 4.14 satisfies the equality:

d

dthˆ;‰iH D hˆ;A�0‰iH C ŒU3.t/C de.t/�B

�0‰; (4.20)

where ˆ D .X;w;wt /> 2 H and ‰ D .Y; f; g/ 2 D.A�0/ is an arbitrary test function. It then

follows from 4.20 that

d

dt

�X>Y C

Z 1

0

Œwx.x; t/f0.x/C wt .x; t/g.x/�dx C kw.1; t/f .1/

�D X>.AC BK/>Y

�

Z 1

0

Œwx.x; t/g0.x/C wt .x; t/f

0.x/�dx � kw.1; t/g.1/C g.1/ŒU3.t/C de.t/�:

(4.21)If we choose .X; f .x/; g.x// D .0; 0; x/ 2 D.A�/ in the aforementioned equation 4.21, thenwe have

d

dt

Z 1

0

wt .x; t/xdx D �.k C 1/w.1; t/C w.0; t/C U3.t/C de.t/: (4.22)

Let

´.t/ D

Z 1

0

Owt .x; t/xdx; ´1.t/ D

Z 1

0

Qwt .x; t/xdx:

Then

d

dt´.t/ D

d

dt

Z 1

0

wt .x; t/xdx C P1.t/

D �.k C 1/w.1; t/C w.0; t/C U3.t/C de.t/C P1.t/

D U3.t/C de.t/ � .k C 1/ Ow.1; t/C Ow.0; t/C .k C 1/ Qw.1; t/ � Qw.0; t/C P1.t/:

Define

´2.t/ D .k C 1/ Qw.1; t/ � Qw.0; t/; ´3.t/ D �.k C 1/ Ow.1; t/C Ow.0; t/:

Then ´.t/ satisfies the equation:

P.t/ D U3.t/C de.t/C P1.t/C ´2.t/C ´3.t/: (4.23)

By Theorem 4.1 and the estimate 4.16, there exists a constant M´ > 0 so that

j´1.t/j C j´2.t/j 6M´e��f t ; 8 t > 0: (4.24)

We now estimate the new disturbance de.t/. Similar to equations 3.23–3.25, we design the followingextended state observer with time-varying high gain for system 4.23 as follows:´

PO.t/ D U3.t/C Ode.t/C ´3.t/ � r.t/Œ O.t/ � ´.t/�;POde.t/ D �r

2.t/Œ O.t/ � ´.t/�;(4.25)



where r 2 C 1ŒRC; RC/ is a time-varying gain function satisfying:8<:r.t/ > 0; Pr.t/ > 0; 8 t > 0:r.t/!1 as t !1; supt2Œ0;1/ j Pr.t/=r.t/j <1:

limt!1Pde.t/r.t/D 0; limt!1 r.t/e

��f t D 0:

(4.26)

Lemma 4.3Let . O ; Od/ be the solution of 4.25. Then

limt!1j O.t/ � ´.t/j D 0; lim

t!1j Ode.t/ � de.t/j D 0: (4.27)

ProofSet

Q.t/ D r.t/Œ O.t/ � ´.t/�; Qde.t/ D Ode.t/ � de.t/: (4.28)

Then . Q.t/; Qde.t// satisfies´PQ.t/ D �r.t/ Q.t/C r.t/ Qde.t/C

Pr.t/r.t/Q.t/ � r.t/Œ P1.t/C ´2.t/�;

PQde.t/ D �r.t/ Q.t/ � Pde.t/:(4.29)

The next to prove limt!1 j Q.t/j D 0 and limt!1 j Qde.t/j D 0 is similar to the situation of the prooffor Lemma 3.4. We omit the proof here and leave it to the readers. �

We compensate the disturbance by designing

U3.t/ D � Ode.t/: (4.30)

Under the feedback control 4.30, the closed-loop of system 4.14 becomes8<:PX.t/ D .AC BK/X.t/C Bw.0; t/;wt t .x; t/ D wxx.x; t/;wx.0; t/ D c0wt .0; t/;

wx.1; t/ D � kw.1; t/ � Qde.t/;

(4.31)

where Qde.t/ D Ode.t/ � de.t/.

Lemma 4.4For any initial value .X.0/; w.�; 0/; wt .�; 0/ 2 H. If Qde.t/! 0 as t !1, then 4.31 admits a uniquesolution .X;w;wt / 2 H satisfying

limt!1

�jX.t/j2Rn C w

2.1; t/C

Z 1

0

wx.x; t/C

Z 1

0

w2t .x; t/dx

�D 0: (4.32)

ProofThe existence and uniqueness of solution of system 4.31 have been claimed by Proposition 4.2.Because Qde.t/! 0 as t !1, for any given � > 0, we may suppose that j Qde.t/j 6 � for all t > t0for some t0 > 0. Now, we write the solution of 4.31 as follows:0@ X.t/

w.�; t /wt .�; t /

1A D eA0t0@ X.0/

w.�; 0/wt .�; 0/

1AC Z t

0

eA0.t�s/B0 Qde.t/ds

D eA0t

0@ X.0/

w.�; 0/wt .�; 0/

1AC eA0.t�t0/ Z t0

0

eA0.t0�s/B0 Qde.t/ds CZ t

t0

eA0.t�s/B0 Qde.t/ds:

(4.33)



The admissibility of B0 implies that��Z t

0

eA0.t�s/B0 Qde.t/ds��2H6 Ct

�� Qde��2L2loc.0;t/6 t2Ct

�� Qde��2L1.0;t/

;

for some constant Ct that is independent of Qde . Because eAt is exponentially stable, by using theProposition 2.5 of [38] again, we have��Z t

t0

eA.t�s/B Qdeds�� D ��Z t

0

eA.t�s/B.0Þt0

Qdeds

�� 6 L �� Qde��L1.0;1/

6 L�;

whereL is a constant that is independent of Qde . Suppose that��eA0t�� 6 L0e�!t for someL0; ! > 0.

We have��0@ X.t/

w.�; t /wt .�; t /

1A��H

6 L0e�!2t��0@ X.0/

w.�; 0/wt .�; 0/

1A��H

C L0Ct0 t20 e�!2.t�t0/

�� Qde��L1.0;t0/

C L�: (4.34)

As t !1, the first two terms of 4.34 tend to 0. The result is then proved by the arbitrariness of �.�

Remark 4.5The observer system 4.2 is based on the controller/observer separation principle. Actually, it isfound that the error system 4.4 is independent on the control U.t/. Moreover, we design the controlU.t/ D U3.t/C U4.t/ in 4.12, where U4.t/ given by 4.13 is designed to stabilize the system 4.31without disturbance, while U3.t/ D � Ode.t/ given by 4.30 is designed to cancel the disturbance. Thesystem 4.31 (or 4.14) is independent on the observer system (4.2).

The closed-loop system of 4.1 then becomes8<ˆ:

PX.t/ D AX.t/C Bu.0; t/;ut t .x; t/ D uxx.x; t/;ux.0; t/ D 0; ux.1; t/ D U3.t/C U4.t/C d.t/;PO.t/ D ´3.t/ � r.t/Œ O.t/ � ´.t/�;POde.t/ D �r

2.t/Œ O.t/ � ´.t/�

(4.35a)

and 8<ˆ:POX.t/ D A OX.t/C LC. OX �X/C B Ou.0; t/;Out t .x; t/ D Ouxx.x; t/;Oux.0; t/ D cŒ Out .0; t/ � ut .0; t/�;Oux.1; t/ 2 U.t/ � c1Œ Out .1; t/ � ut .1; t/� � c2Œ Ou.1; t/ � u.1; t/�

�M3sign. Out .1; t/ � ut .1; t// �M4 sign . Ou.1; t/ � u.1; t//;

(4.35b)

where U3.t/; U4.t/ are two controls given by 4.30 and 4.13, respectively, de.t/ D d.t/C f .t/, andf .t/ is given by 4.15.

Theorem 4.6For any initial value .X.0/; u.�; 0/; ut .�; 0// 2 Rn �H 2.0; 1/�H 1.0; 1/, . OX.0/; Ou.�; 0/; Out .�; 0// 2Rn � H 2.0; 1/ � H 1.0; 1/, and . O.0/; Ode.0// 2 R2, any partial differential inclusion solution ofclosed-loop system 4.35 satisfies

limt!1

F.t/ D 0; (4.36)



where

F.t/ D jX.t/j2Rn C1

2

Z 1

0

Œu2t .x; t/C u2x.x; t/�dx C

c2

2u2.1; t/C jO.t/j

CˇOX.t/

ˇ2RnC1

2

Z 1

0

�Ou2t .x; t/C Ou

2x.x; t/

�dx C

c2

2Ou2.1; t/C

ˇOde.t/ � de.t/

ˇ:

(4.37)

ProofBy the error variables . Q.t/; Qde.t// D .r.t/Œ O.t/ � ´.t/�; Ode.t/ � de.t// given by 4.28 and�

QX.t/; Qu.x; t/; Qut .x; t/�DOX.t/ �X.t/; Ou.x; t/ � u.x; t/; Out .x; t/ � ut .x; t/

�;

system 4.35 is equivalent to the following:8<ˆ:PX.t/ D AX.t/C Bu.0; t/;ut t .x; t/ D uxx.x; t/;ux.0; t/ D 0; ux.1; t/ D U.t/C d.t/;PQ.t/ D �r.t/ Q.t/C r.t/ Qde.t/C

Pr.t/r.t/Q.t/ � r.t/Œ P1.t/C ´2.t/�;

PQde.t/ D �r.t/ Q.t/ � Pde.t/

(4.38a)

and8<:PQX.t/ D .ACKC/ QX.t/C B Qu.0; t/;Qut t .x; t/ D Quxx.x; t/;Qux.0; t/ D c Qut .0; t/;Qux.1; t/ 2 �c1 Qut .1; t/ � c2 Qu.1; t/ �M3sign. Qut .1; t// �M4sign. Qu.1; t// � d.t/:

(4.38b)

The . QX; Qu/ part of 4.38 is just the 4.4, and the . Q ; Qd/ part of 4.38 is just the Lemma 4.3; thus, wehave, as t !1,

1

2

Z 1

0

Œ Qu2x.x; t/C Qu2t .x; t/�dx C

c2

2Qu2.1; t/C jO.t/ � ´.t/j C j Ode.t/ � de.t/j ! 0: (4.39)

Now, we show the convergence of the .X; u/ part of 4.38. To this end, we rewrite .X; u/ partas follows:8ˆˆ<ˆˆ:

PX.t/ D AX.t/C Bu.0; t/;

ut t .x; t/ D uxx.x; t/; ux.0; t/ D 0;

ux.1; t/ D � Qde.t/CR 10 �x.1 � y/ Ou.y; t/dy C

R 10 mx.1 � y/ Out .y; t/dy C �

0.1/ OX.t/

�c0ut .1; t/C c0KBu.1; t/C c0R 10 �.1 � y/ Out .y; t/dy

Cc0R 10 mxx.1 � y/ Ou.y; t/dy C c0�.1/A

OX.t/

�k°u.1; t/ �

R 10�.1 � y/ Ou.y; t/dy �

R 10m.1 � y/ Out .y; t/dy � �.1/ OX.t/

�c0R 10

�R x0 �.x � y/ Out .y; t/dy C

R x0 mxx.x � y/ Ou.y; t/dy

�dx

C c0R 10Œ Out .x; t/ �KB Ou.x; t/� dx � c0

R 10 �.x/dxA

OX.t/±:

(4.40)

(a) (b) (c)

Figure 2. The divergence of the open-loop system 1.1 with U.t/ D d.t/ D 0.



Because system 4.40 is exactly the same as that in the closed-loop system 4.31 under thetransformations 1.2–1.3 and 1.6–1.7, the convergence of 4.40 follows from 4.39 and Lemma 4.4. �

5. NUMERICAL SIMULATION

In this section, we present some numerical simulations for the system 1.1 to illustrate theeffectiveness of controller, respectively.

Figures 2 and 3 show the open-loop and closed-loop results, respectively, with A D 2, B D �2,and d.t/ D 4 cos

�12t�. The initial conditions are chosen as X.0/ D 1, u.x; 0/ D x3 � 2, and

ux.t; 0/ D �x. It can be seen that without control, both the ODE and wave PDE are not stabilized

(a) (b) (c)

Figure 3. The convergence of system 2.32 with control 2.13.

(a) (b)

(c) (d)

Figure 4. The convergence of the system and the tracking of the disturbance by the constant high gain.



Figure 5. The convergence of the system and the tracking of the disturbance by the time-varying gain.

in Figure 2, after the proposed sliding mode controller 2.13 is applied with the design parametersthat are chosen as c0 D 2, M D 1, and D 25; both the ODE and wave PDE are stabilized to 0when they reach the sliding surface after some time as shown in Figure 3.

Figures 4 and 5 show visually the effectiveness of the proposed controller 3.1 for the ODE-wavecascade system. We choose A D 2, B D �2, and the initial values as X.0/ D 1, u.x; 0/ D x3 � 2,and ut .x; 0/ D �x. The time-varying gain is taken as follows:

g.t/ D

²t C 1; t C 1 6 10;10; t C 1 > 10: (5.1)

In Figure 4(a), the ODE state X.t/ with control in system is demonstrated. It is seen that the con-trol effect is very satisfactorily. Figure 4(b) shows the displacement u.x; t/ of system, Figure 4(c)shows the velocity ut .x; t/ of system , and Figure 4(d) demonstrates the disturbance estimation Od.t/compared with the disturbance d.t/. The convergence of the system is shown to be fast. Figure 5demonstrates the state X.t/ (Figure 5(a)), and the displacement u.x; t/ (Figure 5(b)) of system,and the disturbance estimation Od.t/ compared with d.t/ (Figure 5(d)), with the constant high gaing.t/ D 10. The peaking value for disturbance tracking is clearly observed in Figure 5(d). Comparedwith Figure 5(d), it is seen from Figure 5(d) that the peaking value is dramatically reduced.

Figure 6 presents some simulation results for the closed-loop system 4.35 to illustrate the effectof the control law U3.t/CU4.t/ with U3.t/; U4.t/ being given by 4.30 and 4.13, respectively. Here,we design the function sign.0/ D 0. The other parameters are chosen as follows:

c D 0:8; c1 D 1:2; c2 D 4;M3 D 5;M4 D 12; r.t/ D t C 1;



Figure 6. The convergence of the system and the disturbances with the estimation.



and the initial values are as follows:

u.x; 0/ D 4x; ut .x; 0/ D �x; Ou D cos 4x; Out .x; 0/ D x; O.0/ D 1; Od.0/ D 2:

The finite difference method is adopted in computation of the displacements. The time and spacestep are chosen as 0.01 and 0.001, respectively. Figure 6 shows the ODE state X.t/, bX.t/, thedisplacement of u.x; t/, the displacement of Ou.x; t/, the velocity of ut .x; t/ , the velocity of Out .x; t/,and disturbance d.t/ and its estimation Od.t/ with control in system, respectively. It is seen that thecontrol effect is very satisfactorily. In Figure 6(h), we choose the varying gain r.t/ as follows:

r.t/ D

²t C 1; t C 1 6 10;10; t C 1 > 10: (5.2)

This means that we use the time-varying gain in the initial stage and then change to the constant highgain afterwards. In Figure 6(h), we can see clearly the peaking reduction compared with Figure 6(g).

6. CONCLUDING REMARKS

In this paper, we are concerned with a cascade of ODE-wave systems with the control actuator-matched disturbance at the boundary of the wave equation. We use the SMC technic and the ADRCmethod to overcome the disturbance, respectively.

We first apply the SMC approach to reject the disturbance. Because in SMC approach, we do notneed to estimate the disturbance, the disturbance is assumed to be bounded only. The sliding modesurface is found to be a closed subspace of the state space. The closed-loop system is shown to havea unique (weak) solution and can reach the sliding mode surface in finite time. On the sliding modesurface, the system is shown to be exponentially stable with arbitrary-prescribed decay rate.

Furthermore, we also use the ADRC to stabilize the cascade ODE-PDE system. Both disturbanceestimators with constant high gain and time-varying gain are designed respectively. The practicalstability for the closed-loop system with constant high gain and asymptotic stability with time-varying gain is proved. The constant gain is easily tuning in practice but produces peaking valueproblem. The time-varying gain reduces peaking value significantly, but it brings sensitivity forhigh-frequency noise. The last point comes from the fact of tracking noise with disturbance togetherinstead of filtering the noise. A recommended scheme is to apply the time-varying gain in the initialstage so that the peaking value can reach a reasonable area and then apply the high gain.

Moreover, we design an unknown input-type observer by variable structure control method. Basedon the estimated state, we design, by an idea of extended state observer, a disturbance estimator toestimate the external disturbance. The disturbance is then compensated in the feedback loop. Thisidea comes from the ADRC approach. The closed-loop system is shown to be asymptotically stable.The idea is potentially promising for treating other uncertain PDEs.

ACKNOWLEDGEMENTS

This work is supported by the National Natural Science Foundation of China and the School YoungFoundation of Taiyuan University of Technology (2015QN062).

REFERENCES

1. Krstic M, Smyshlyaev A. Boundary Control of PDEs: A Course on Backstepping Designs. SIAM: Philadelphia, PA,2009.

2. Guo BZ, Jin FF. Sliding mode and active disturbance rejection control to the stabilization of anti-stable one-dimensional wave equation subject to boundary input disturbance. IEEE Transactions on Automatic Control 2013;58:1269–1274.

3. Guo BZ, Xu CZ. The stabilization of a one dimensional wave equation by boundary feedback with noncollocatedobservation. IEEE Transactions on Automatic Control 2007; 52:371–377.

4. Chen G, Delfour MC, Krall AM, Payre G. Modeling, stabilization and control of serially connected beam. SIAMJournal on Control and Optimization 1987; 25:526–546.

5. Luo ZH, Guo BZ, Morgul O. Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer-Verlag: London, 1999.



6. Machtyngier E. Exact controllability for the Schrödinger equation. SIAM Journal on Control and Optimization 1994;32:24–34.

7. Orlov Y, Utkin VI. Use of sliding modes in distributed system control problems. Automation and Remote Control1982; 43:1127–1135.

8. Utkin VI. Sliding mode control: mathematical tools, design and applications. In Nonlinear and Optimal ControlTheory, vol. 1932, Lecture Notes in Math. Springer-Verlag: Berlin, 2008; 289-347.

9. Orlov Y, Utkin VI. Sliding mode control in infinite-dimensional systems. Automatica 1987; 23:753–757.10. Drakunov S, Barbieeri E, Silver DA. Sliding mode control of a heat equation with application to ARC welding. IEEE

International Conference on Control Applications 1996:668–672.11. Cheng MB, Radisavljevic V, Su WC. Sliding mode boundary control of a parabolic PDE system with parameter

variations and boundary uncertainties. Automatica 2011; 47:381–387.12. Garrido AJ, Garrido I, Amundarain M, Alberdi M, De la Sen M. Sliding-mode control of wave power generation

plants. IEEE Transactions on Industry Applications 2012; 48(6):2372–2381.13. Liu JJ, Wang JM. Active disturbance rejection control and sliding mode control of one-dimensional unstable heat

equation with boundary uncertainties. IMA Journal of Mathematical Control and Information 2013; 32(1):97–117.14. Orlov Y, Pisano A, Usai E. Distributed-parameter second-order sliding-mode control techniques for the uncertain

heat and wave equations. 18th IFAC World Congress, Milan, Italy, 2011; 762–767.15. Orlov Y, Pisano A, Usai E. Boundary control and observer design for an uncertain wave process by second-order

sliding-mode technique. 52nd IEEE Conference on Decision and Control, Florence, Italy, December 10–13, 2013;472–477.

16. Wahyudie A, Abdi Jama1 M, Assi A, Noura H. Sliding mode control for heaving wave energy converter. 2013IEEE International Conference on Control Applications (CCA) Part of 2013 IEEE Multi-Conference on Systems andControl, Hyderabad, India, August 28–30 2013; 989–994.

17. Guo BZ, Liu JJ. Sliding mode control and active disturbance rejection control to the stabilization of one-dimensionalSchrodinger equation subject to boundary control matched disturbance. International Journal of Robust andNonlinear Control 2014; 24:2194–2212.

18. Han JQ. From PID to active disturbance rejection control. IEEE Transactions on Industrial Electronics 2009; 56:900–906.

19. Guo BZ, Zhao ZL. On the convergence of extended state observer for nonlinear systems with uncertainty. Systems &Control Letters 2011; 60:420–430.

20. Zheng Q, Gao Z. An energy saving, factory-validated disturbance decoupling control design for extrusion processes.The 10th World Congress on Intelligent Control and Automation, Beijing, 2012; 2891–2896.

21. Guo BZ, Kang W. The Lyapunov approach to boundary stabilization of an anti-stable one-dimensional wave equationwith boundary disturbance. International Journal of Robust and Nonlinear Control 2014; 24:54–69.

22. Orlov YV. Application of Lyapunov method in distributed systems. Automation and Remote Control 1983; 44:426–430.

23. Pisano A, Orlov Y, Usai E. Tracking control of the uncertain heat and wave equation via power-fractional andsliding-mode techniques. SIAM Journal on Control and Optimization 2011; 49:363–382.

24. d’Andrea-Novel B, Boustany F, Conrad F, Rao BP. Feedback stabilization of a hybrid PDE-ODE systems:application to an overhead crane. Mathematics of Control, Signals, and Systems 1994; 7(1):1–22.

25. d’Andrea-Novel B, Coron JM. Exponential stabilization of an overhead crane with flexible cable via a backsteppingapproach. Automatica 2000; 36(4):587–593.

26. Krstic M. Compensating a string PDE in the actuation or sensing path of unstable ODE. IEEE Transactions onAutomatic Control 2009; 54(6):1362–1367.

27. Krstic M. Compensating actuator and sensor dynamics governed by diffusion PDEs,. Systems and Control Letters2009; 58(5):372–377.

28. Krstic M, Smyshlyaev A. Backstepping boundary control for first order hyperbolic PDEs and application to systemswith actuator and sensor delays. Systems and Control Letters 2008; 57(9):750–758.

29. Li J, Liu YG. Adaptive control of the ODE systems with uncertain diffusion-dominated actuator dynamics.International Journal of Control 2012; 85:868–879.

30. Pietri DB, Krstic M. Delay-adaptive predictor feedback for systems with unknown long actuator delay. IEEETransactions on Automatic Control 2010; 55(9):2106–2112.

31. Susto GA, Krstic M. Control of PDE-ODE cascades with Neumann interconnections. Journal of Franklin Institute2010; 347(1):284–314.

32. Tang S, Xie C. Stabilization for a coupled PDE-ODE control system. Journal of The Franklin Institute 2011;348:2142–2155.

33. Smyshlyaev A, Krstic M. Boundary control for an anti-stable wave equation with anti-damping on uncontrolledboundary. Systems & Control Letters 2009; 58:617–623.

34. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag: NewYork, 1983.

35. Tucsnak M, Weiss G. Observation and Control for Operator Semigroups. Basel: Birkhauser, 2000.36. Weiss G. Admissibility of unbounded control operators. SIAM Journal on Control and Optimization 1989; 27:

527–545.37. Guo BZ, Jin FF. Output feedback stabilization for one-dimensional wave equation subject to boundary disturbance.

IEEE Transactions on Automatic Control 2015; 60:824–830.38. Weiss G. Admissible observation operators for linear semigroups. Israel Journal of Mathematics 1989; 65:17–43.


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