IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, …

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 7, JULY 2013 1631

Parameter Estimation and Non-Collocated AdaptiveStabilization for a Wave Equation Subject toGeneral Boundary Harmonic Disturbance

Wei Guo and Bao-Zhu Guo, Senior Member, IEEE

Abstract—This paper is concerned with the parameter estima-tion and asymptotic stabilization of a 1-D wave equation that issubject to general harmonic disturbances at the controlled end andsuffers from instability at the other end. First, we design an adap-tive observer in terms of measured position and velocity. We thenadopt the backstepping method for infinite-dimensional systemsto design an observer-based output feedback law. The resultingclosed-loop system is shown to be asymptotically stable. And theestimates of the parameters converge to the unknown parameters.

Index Terms—Backstepping, boundary control, distributed pa-rameter systems, harmonic disturbance rejection.

I. INTRODUCTION

I N the past several decades, collocated boundary feedbackcontrol has played a dominant role in the boundary feedback

stabilization of time invertible infinite-dimensional systems de-scribed by wave and flexible beam equations ([1], [2], [5], [6]).Most of those systems studied are assumed to be conservative,meaning that the system energy remains constant when there isno boundary control imposed. The main idea of feedback sta-bilization design is to introduce a boundary damping throughcontrol to make the system energy decay polynomially or expo-nentially to zero as time goes to infinity. The control design isusually based on passive principle and is hence straightforward,although the stability analysis is hard inmany cases, in large partdue to the PDE’s nature. This is also the case for the stabiliza-tion of many multi-dimensional partial differential control sys-tems [23], [25], [26], [28]. The early non-dissipative boundarycontrol is designed in [8], [27], and the generalization (to cer-tain extend) of this design to 2D can be found in [24]. Anotherinteresting example of non-dissipative stabilization example is

Manuscript received December 30, 2011; revised August 02, 2012; acceptedDecember 06, 2012. Date of publication January 11, 2013; date of current ver-sion June 19, 2013. This work was supported by the Program for InnovativeResearch Team in UIBE, the National Natural Science Foundation of China,the National Basic Research Program of China (2011CB808002), and the Na-tional Research Foundation of South Africa. Recommended by Associate EditorC. Prieur.W. Guo is with School of Statistics, University of International Business and

Economics, Beijing 100029, China (e-mail: [email protected]).B.-Z. Guo is with the Academy of Mathematics and Systems Science,

Academia Sinica, Beijing 100190, China and also with the School of Com-putational and Applied Mathematics, University of the Witwatersrand, SouthAfrica (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAC.2013.2239003

presented in [12], which is later proved to be dissipative in [9]under the state transformation.In the last few years, particular attentions have been paid

to the boundary feedback stabilization for unstable wave andbeam equations by means of non-collocated control and obser-vation. In this situation, the passive principle cannot be directlyapplied anymore. An observer-based compensator which expo-nentially stabilizes the string system with a non-collocated ac-tuator/sensor configuration is proposed in [10]. A major changehas taken place since the backstepping method is introduced inPDEs ([18]). In [16], the controller and observer are designedusing both the displacement and velocity measurement via thebackstepping method to exponentially stabilize a one-dimen-sional wave equation that contains destabilizing anti-stiffnessboundary condition at its free end. The destabilizing term in [16]is proportional to the displacement, which makes the systemunstable in the sense that the uncontrolled system has positivereal eigenvalue. An extension is later presented in [11] wherethe controller and observer for a non-collocated wave equa-tion are designed using the displacement measurement only. Abreakthrough was made recently in [19], where the anti-stablewave equation with an anti-damping term on the uncontrolledboundary (which is different to the destabilizing term in [16]) isstabilized through a novel backstepping transformation method.The stabilization of unstable shear beam equation can be foundin [17] where the non-collocated boundary stabilization is dis-cussed by using the backstepping method and observer-basedfeedback. The stabilization of the anti-damping term in internaldomain of the spacial variable is addressed in [33]. However,uncertainties in the boundary control (input) or boundary mea-surement (output) are not considered in the aforementioned con-troller and observer designs.The early efforts for the design of adaptive controller and

observer for partial differential equation control systems arepresented in [20]–[22]; particularly for parabolic PDEs withboundary control and unknown parameters that may cause insta-bility of the system and affect the interior of the domain. Adap-tive stabilization for the most challenging anti-wave equationsystem can be found in [15]. A recent progress is made in [13]where the adaptive observer and controller are designed for aone-dimensional wave equation with corrupted output distur-bances and non-collocated control.We also refer to a recent nicework [3] where the stabilization of a one-dimensional heat equa-tion with boundary control corrupted by bounded disturbance isconsidered by means of sliding mode control and backsteppingmethod.

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1632 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 7, JULY 2013

In this paper, we are concerned with the parameter estima-tion and asymptotic stabilization of a one-dimensional waveequation that is subject to general harmonic disturbances at thecontrolled end and suffers from instability at the other end. Anadaptive observer is designed in terms of the measured end po-sition and velocity to estimate the unknown parameters and thestate simultaneously. An observer-based output feedback lawis designed through the backstepping method to stabilize thesystem in the time of rejecting the disturbance. The resultingclosed-loop system is shown to be asymptotically stable.The paper is organized as follows. We give the main results

in Section II. The proofs of these results are then provided inSection III. In Section IV, we presents some illustrative simula-tion results. Conclusions are made in Section V.

II. PROBLEM FORMULATION AND MAIN RESULTS

In this paper, we are concerned with the following one-di-mensional wave equation:

(1)

where (or ) denotes the derivative of with respect to ,(or ) the derivative with respect to , is the boundary con-

trol (input) at the right end, is the boundary measurement(output), and is the initial value; parameter is a realconstant, represents the general harmonic disturbance whichhas the following form:

with , being the frequencies and ,, , the amplitudes. Note that the amplitudes

are assumed to be unknown in this paper. Such kind of distur-bance is more general than the finite sum of periodic harmonicdisturbance that can be considered as the approximation of theperiodic disturbance signal in terms of Fourier expansion ([4],[32]).The physical motivation of problem (1) has been well ex-

plained in [15] where the unstable wave equation models theclassical Rijke tube control experiment in combustion dynamicswith control of a loudspeaker, and a pressure sensor (micro-phone) near the speaker. Both the actuator and the sensor areplaced as far as possible from the flame front, for their thermalprotection. The unstable damping that we include in our modelof wave dynamics represents the injection of energy in propor-tion to the amplitude field, see Fig. 1 below that comes from [15,Figure 1] (velocity field is replaced by amplitude field). The dif-ference between our model (1) and that in [15] is that we havethe external disturbance at the control end, because the oppositeend of the flame front of Rijke tube is exposed to the air fromwhich the external disturbance is imposed to the boundary. Forother control strategies of such process, we refer to [15] and thereferences therein. Moreover, our study for system (1) have thefollowing engineering applications:

Fig. 1. Control of a thermoacoustic instability and disturbance in a Rijke tube(a duct-type combustion chamber).

• Knock sensor in measure of knocking in combustionprocess of automotive engine. The observer design suchas the one in our stabilization of system (1), is of signifi-cance by itself. As illustrated in the interesting work [32]and a more recent paper [4] for combustion process ofautomotive engine. The knocking measurement needs tobe realized by estimation of a wave equation driven by aperiodic unknown signal through the knock sensor. In [4],the periodic unknown signal is assumed to admit a finitesum of periodic harmonic series. This is based on the factthat any periodic signal can be expanded through Fourierdecomposition. Also, the period disturbance rejection isthe objective of the repetitive control ([34], [35]). Herewe remove the limitation of the periodicity of the signalwhich covers the problem of [4] as its special case in theobserver design.

• Estimation of the harmonic signal. The observer-basedfeedback stabilization for system (1) studied in this paperis an online estimation/cacellation strategy by which theunknown harmonic disturbance is estimated. The onlineestimation of the unknown harmonic signal which is thesum of a finite number of sinusoids has been addressed bymany authors, see [30] and the references therein.

Note that in the absence of disturbance and when ,system (1) is unstable in the sense that its open-loop systemhas positive real eigenvalues in the open right half complexplane. This instability also occurs in combustion dynamics. Anice physical explanation is illustrated in [15].Also note that system (1) is a non-minimum phase system

when . Indeed, the transfer function from to outputis found to be

which has at least one real pole . Moreover, the control(at ) and the observation (at ) is non-collocated.An adaptive regulator for the unstable system (1) with the

disturbance in the form of is designed understate feedback in [14] to achieve both parameter estimation andstabilization. Only output feedback is used in this paper to tacklethe general harmonic disturbance.The main result of this paper states that for any given fre-

quencies , , one can always construct anobserver-based adaptive scheme to achieve both parameter esti-mation and stabilization. Unlike the stabilization by disturbancerejection only as that in [3] for the heat equation, the objective

GUO AND GUO: PARAMETER ESTIMATION AND NON-COLLOCATED ADAPTIVE STABILIZATION 1633

of our design is twofold, i.e., estimation of the unknown param-eters and rejection of the harmonic disturbance.We design the following adaptive observer for system (1)

(2)

where and henceforth , , , , ,are positive design parameters. Note that here and in the rest ofthe paper, we omit the (obvious) domains for and .Let be the error between system

(1) and observer (2). It is easy to see that is governed by

(3)

where

are the parameter errors.Define a Lyapunov function for system (3) as follows:

(4)

A formal calculation shows that

(5)

This can be considered as a motivation of designing observer(2) and the update laws for the parameter errors , ,

given in (3).Let with being an operator defined

in as follows:

(6)

Theorem 2.1: Suppose that, and they satisfy the following compat-

ible condition:

(7)

and

(8)

Then system (3) admits a unique classical solution . That is tosay, for any time

By the Sobolev embedding theorem ([31, p. 85]), it follows that.

Remark 2.1: In Theorem 2.1, condition 7 is the natural com-patible condition for the classical solution of (3), and condition(8) is for the existence of the more smoother solution that weshall need in the proof of Theorem 2.2 in Section III.Definition 2.1: For any initial data

, the weak solution of (3)is defined as the limit of any convergent subsequence of

in the space whereis the classical solution (ensured

by Theorem 2.1) with the initial condition (for all )

which satisfies

By (4) and (5), the above weak solution is well defined,since it does not depend on the choice of initial sequence

.


Theorem 2.2: Suppose that

(9)

in observer (2). Then for any initial value, the solution

of system (3) is asymptotically stable in the sense that

We propose the following observer-based feedbackcontroller:

(10)

Controller (10) contains two parts: one part is used to overcomethe instability caused by the term on the left end of(1), and the another [last term in controller (10)] is used to cancelthe effect of the disturbance. Simply speaking, it is obtained bythe applying the backstepping method in PDEs which transfersthe “ ” part of system (2) to the “ ” part of (14) later for whichthe stability is almost straightforward. Moreover, if we compareoutput feedback controller (10) and the state feedback controller(1.3) of [14], we see that our controller here is just to replace thestate in [14] by . In this sense, our result is something of thewell-known separation principle.The closed-loop of system (2) corresponding to controller

(10) becomes

(11)

Consider the invertible ible change of variable

(12)

where is a Volterra transformation [16]. The inverseis given by

(13)

The transformation (12) converts system (11) into

(14)

where is given by (3) and

(15)

The recommended choices of the control gains are as follows:, , and satisfies (9).

Now we turn to transformed system (14) without dynamicequations for and , , since they havebeen determined by the error system (3) already. The systemnow reads

(16)

We consider system (16) in the energy space. The norm of is induced by the inner product

for any . Define the operatoras follows:

(17)

Theorem 2.3: For each initial value , there ex-ists a unique solution to (16); and for all

, there exists a (depending on only) such that

where

(18)

Moreover, for each andwith

and satisfying the compatible conditions (7) and (8),there exists a classical solution to (16).Theorem 2.4: The transformed system (16) is asymptotically

stable. That is, for any , the (weak) solution of(16) justified by Theorem 2.3 satisfies


We go back to the closed-loop system of (1) and (2) under thefeedback (10)

(19)

Let us consider system (19) in the state space.

Theorem 2.5: For any initial value, there exists a unique (weak)

solution to (19) such that. Moreover, the

closed-loop solution of(19) is asymptotically stable in the sense that

And

PROOF OF THE MAIN RESULTS

Proof of Theorem 2.1

Since system (3) is non-autonomous, we introduce someadditional variables to make it time invariant. For thispurpose, observer that the harmonic disturbance functionvector is a solu-tion to the following autonomous equation with initial value

:

(20)

where

......

......

Let be the usual Hilbert space with the inner productand inner product induced norm . We consider system

(3) and (20) in the energy state space with theinner product

where is defined in Definition 2.1.Define the operator as follows:

(21)

Then system (3) and (20) can be written as a nonlinear evolutionequation

(22)

where

Equation (22) is a nonlinear autonomous revolution system.However, same as [14], it seems hard to use nonlinear semi-group to prove its well-posedness due to the lack of dissipa-tivity of defined by (21) (or for any constant ).Hence we invoke the Galerkin method to establish the existenceand the uniqueness of the solution of (3). To do this, we need abasis to construct the Galerkin approximation, which can be re-alized by the operator defined by (6). It is obvious that isunbounded self-adjoint positive definite in with com-pact resolvent. A simple computation shows that the eigenpairs

are

(23)where satisfies

Since defined by (23) is approximately normalized(i.e., for some constants ,


independent of ), it forms an orthogonal Riesz basis for.

We can then follow the steps as those in [14] to construct aGalerkin scheme to prove the existence and uniqueness for theclassical solution to error system (3). The details are omitted.


By density argument, we may assume without loss of gen-erality that the initial value be-longs to and satisfies compatible conditions (7)and (8). Construct the Lyapunov functional for system (22)as follows:

(24)

where , , . Thetime derivative of along the solution of system (22) is foundto be

This shows that , hence

(25)

In particular, one has

(26)

Similarly, let

It is found that the time derivative of along the solution of (3)can be found as

(27)

Integrating over on both sides of (27) gives

(28)

From (28), one has

(29)

Take in (29) to give


(30)

It is found from (25), (26) and (30) that

which implies that the trajectory of system (22)

is precompact in . In light of Lasalle’s invariance principle([36, p. 1618]), any solution of system (22) tends to the maximalinvariant set of the following:

Now, since , it follows thatand , . Thus the solution reduces to

(31)

We now show that (31) admits zero solution only. To this end,we first consider the equation

(32)

Introduce a Hilbert space with theinner product

Define a linear operator associated to system (32)

(33)

We now claim that is skew-symmetric in . In fact, for any

(34)

Thus all eigenvalues of are located on the imaginary axis.Next we claim that each eigenvalue of is geometrical

simple and hence algebraically simple from general functionalanalysis theory. To see this, we solve the eigenvalue problem

for any . The solution is withsatisfying

(35)

Solve (35) in the case of to give

(36)

When

(37)

with

(38)

or

(39)

Hence is geometrically simple. From (36)–(38) we see that. Therefore for any eigenfunction of .

If , , then. Thus , is

not solution of (39). For given , , , it iseasy to follow that is a solution to (39) if only if satisfies

(40)

In fact, assume , and is a solution to(39). Then

Hence

(41)

From (41), one has


If , then

Thus

which implies that is a solution to (39). Hence, if we choosesatisfying condition (9)

, then

(42)

Finally, we claim that the spectrum of consists of isolatedeigenvalues only. In fact, for a given and ,

, solve , i.e.

to give

(43)

where , satisfy the following algebra equations:

(44)

A simple computation together with (38) shows that the deter-minant of coefficient of (44) ,which implies that

and hence

for some constant . By the Sobolev embedding theorem,is compact on . That is, is a skew-adjoint oper-

ator with compact resolvent on . Consequently, the spectrumof consists of isolated eigenvalues only.Furthermore, from (37) and (38), we can obtain the following

asymptotic expressions of eigen-pairs of :

(45)

Define

(46)

By general theory of functional analysis, forms anorthogonal Riesz basis for . Therefore, the solution of (32)can be represented as

where the constants are determined by the initial con-dition. That is

Hence

Therefore

(47)

We now show that , for all . Since other-wise, if there exists such that, then due to the fact for all . Fur-thermore, the smoothness of the initial value guarantees that

, which implies that there ex-ists an integer such that

(48)

Since for any , , and, , one has, for


(49)

Integrating both sides of (49) and using (42), (48), and the fact, we obtain

Since the right side of the above inequality has an upper boundfor all , we get that , which is a contradiction.By (47), and ,

.We have thus proved that , that is

The proof is complete.


It is well-known that the operator defined by (17) generatesa -semigroup of contractions on ([7]). From (17), it isreadily found that its adjoint operator is

(50)

Denote by , as that given in (18), and the Dirac distri-bution. Then (16) is equivalent to

(51)

or in operator form

(52)

where . By the equivalence between (16) and(51), one can prove Theorem 2.3 as that in [11]. the proof iscomplete.Remark 3.1: In the above setting, we understand the solution

of (52) in by identifying by its extension definedby

where is the dual space of with the pivot space.


We first assume that andwith

and satisfying the compatible conditions (7) and (8).Now, Theorem 2.3 assures that the classical solution exists. Let

Then there exists an such that

(53)

The time derivatives of and along the solution of (16)are, respectively


and

Let

Then

Performing two completions of square gives

(54)

where

In view of (53), we have

(55)

Take to get

(56)

where . Now, by virtue of (55), we have

(57)

Let and apply the Gronwall’s inequalityto (56) to conclude

(58)

By (24) and Theorem 2.2, one has

Given , we choose such that for ,. Then

Choosing large enough, the first term on the right handside above is less than and thusfor large enough, which implies that

(59)

Combining (58) and (59), we have . Thistogether with (55) gives

Finally, since is dense in, and is dense in , for any

and, we can take and

suchthat


The result then follows from the density argument and the con-clusion just justified for the classical solution.


For any initial value

from (3) and (15), it is easy to verify that

and , which implies that there existsa unique solution to (3) and (16), respectively. Let

Then a direct computation shows that such a definedsatisfies (19) with initial value . This solutionis unique by the following invertible transformation

......

where

......

......

......

......

and by the uniqueness of solution to (3) and (16). The asymp-totic stability follows from (12), Theorems 2.2 and 2.4.

III. NUMERICAL SIMULATION

In this section, we present some numerical simula-tions to illustrate the main results. Notice that there isan invertible transform between the closed-loop system

and the systemgiven by

.........

......

......

...

Fig. 2. Amplitude (left) and error (right).

We only need to give the numerical simulation results for thesystem : which is describedby

(60)

The values of the parameters are: , , ,, , , ,

, , , , , ,, , and . The initial values are taken

as

,otherwise,


Fig. 3. Parameters estimation , , and .

Fig. 4. Time trace of control .

In the simulation, the second order in time equations arefirst converted into a system of two first order equations, andthen Backward Euler Method in time and Chebyshev spectralmethod in space are used. The grid size is taken as andtime step . The code was programmed in Matlab([29]).The numerical results for and are presented in

Fig. 2. We see that system (60) is indeed asymptotically stable.Fig. 3 shows the approximation of the parameters. It

is seen that the estimates , , andwith initial values , , and

approximate, respectively, the system parametersand reasonably well.

Fig. 4 shows that the control is quite reasonable, and since itis the output feedback, it is stabilized after some time.

IV. CONCLUSION

We consider the boundary output feedback stabilization ofa one-dimensional wave equation subject to boundary distur-bance. Two features of the system make the problem difficult:a) the boundary control and observation are non-collocated; andb) the boundary input is subject to harmonic disturbance with ar-bitrary frequencies and unknown magnitude. We design an infi-nite-dimensional observer and an adaptive update laws to esti-

mate online the unknown parameters. An output feedback con-troller is then designed by the backstepping method for infinite-dimensional systems. It is shown that the resulting closed-loopsystem is asymptotically stable, and that the estimates of the pa-rameters converge to the true values of the unknown parameters.The idea used in this paper can be potentially applied to solve theanti-stable problem discussed in [19] and [33]. Another problemleft is that the method here cannot be used to copy with the con-stant disturbance, which needs the separate discussion.

ACKNOWLEDGMENT

The authors would like to acknowledge the suggestions andcomments by anonymous referees and editors which improvethe manuscript substantially.

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Wei Guo received the Ph.D. degree from theAcademy of Mathematics and Systems Science, theChinese Academy of Science, Beijing, China, in2004.He worked in the School of Computational and

Applied Mathematics, University of the Witwater-srand, South Africa in 2007 as a Postdoctoral Fellow.He is currently a Professor with the University ofInternational Business and Economics, Beijing.His research interests include control of systemsdescribed by partial differential equations, and the

adaptive control of infinite-dimensional systems.

Bao-Zhu Guo (SM’09) received the Ph.D. degree inapplied mathematics from the Chinese University ofHong Kong in 1991.From 1985 to 1987, he was a Research Assistant

at the Beijing Institute of Information and Control,China. During the period 1993 to 2000, he was withthe Beijing Institute of Technology, first as an Asso-ciate Professor (1993–1998) and subsequently a Pro-fessor (1998–2000). Since 2000, he has been withthe Academy of Mathematics and Systems Science,the Chinese Academy of Sciences, where he is a Re-

search Professor in mathematical system theory. He is the author or co-author ofover 130 international peer/refereed journal papers, and three books includingStability and Stabilization of Infinite Dimensional Systems with Applications(Springer-Verlag, 1999). His research interests include the theory of control andapplication of infinite-dimensional systems.Dr. Guo received the One Hundred Talent Program Award from the Chinese

Academy of Sciences (1999), and the National Science Fund for DistinguishedYoung Scholars (2003).

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