Output Feedback Stabilization for One-Dimensional Wave Equation Subject to Boundary Disturbance

Transcription

1 8 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 6, NO. 3, MARCH 5 Output Feedback Stabilization for One-Dimensional Wave Equation Subject to Boundary Disturbance Bao-Zhu Guo and Feng-Fei Jin Abstract We consider boundary output feedback stabilization for a one-dimensional anti-stable wave equation subject to general control matched disturbance. The active disturbance rejection control (ADRC approach is adopted in investigation. Using the output of the system, we first design a variable structure unknown input type state observer which is shown to be exponentially convergent. The disturbance is estimated, in real time, through an extended state observer for an ODE reduced from the PDE observer. The disturbance is then canceled in the feedback loop by its approximated value. The stability of the resulting closed-loop system is proven. Simulation results are presented to validate the theoretical conclusions and to exhibit the peaking value reduction by time varying gain instead of constant high gain. Index Terms Disturbance rejection, output feedback, wave equation. I. INTRODUCTION The active disturbance rejection control (ADRC initiated by Han [5] is a new control technology that is used to deal with the systems which have large external disturbances or internal structure uncertainties. This control strategy is very different to traditional ones, by which the total disturbance of system coming from internal uncertainty and external disturbance is estimated, in real time, through an extended state observer and is canceled in the feedback loop. This reduces the control energy significantly in practice []. It is sharp contrast to other control strategies like sliding mode control where the high gain is used to suppress external disturbance in worst case. Very recently, we apply ADRC to stabilization for distributed parameter systems with external disturbance in [] where the state feedback control is used. The aim of this paper is to design an observer based boundary pointwise output feedback stabilizing control law for a wave equation subject to general external boundary disturbance considered in []. Naturally, the state feedback presented in [] is a starting point. Manuscript received December 5, 3; revised March,, May 7,, and June, ; accepted June,. Date of publication July 8, ; date of current version February 9, 5. Recommended by Associate Editor M. Opmeer. B-Z. Guo is with the School of Mathematical Sciences, Shanxi University, Taiyuan, China, the Academy of Mathematics and Systems Science, Academia Sinica, Beijing 9, China, and also with the School of Computational and Applied Mathematics, University of Witwatersrand, Johannesburg Wits 5, South Africa ( bzguo@iss.ac.cn. F-F. Jin is with the School of Mathematics Science, Qingdao University, China and also with the School of Computational and Applied Mathematics, University of Witwatersrand, Johannesburg Wits 5, South Africa ( jinfengfei@amss.ac.cn. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier.9/TAC The system that we are concerned with is an anti-stable wave equation governed by the following PDEs: u tt (x, t =u xx (x, t, x (,,t>, u x (,t= qu t (,t, t, u x (,t=u(t+d(t, t, ( u(x, = u (x,u t (x, = u (x, x [, ], y o (t ={u t (,t,u t (,t,u(,t},t where u is the state, U is the input (control; y o is the output (measurement, that is, the boundary pointwise signals u t (,t, u t (,t, and u(,t are measured; <q is a constant number. The unknown disturbance d is supposed to satisfy d Hloc (,, and bounded: d(t M for some M > and all t. System ( represents an anti-stable distributed parameter physical system explained in detail in [6]. It is noted that due to special boundary conditions, if there is no output signal u(,t, the zero dynamic of system may contain some nonzero constant state. We proceed as follows. In Section II, we design a variable structure unknown input type state observer. The convergence of the observer is proven by the Lyapunov functional method. In Section III, we design an output feedback control law for system ( based on an extended state observer that is designed for an ODE reduced from the observer. Some numerical simulations are presented in Section IV to illustrate the effect of the control law. II. STATE OBSERVER DESIGN In this section, we design a variable structure state observer for system (. This is, to the best of our knowledge, the first unknown input type observer for PDEs. The idea combines the work [8] for a one-dimensional heat equation and a direct variable structure output feedback control for a wave equation presented in [3] where only the vibrating energy can be controlled to be convergent to zero. To ensure the existence of the solution, we need to limit the symbolic function as a set-valued function as follows:, x >, sign(x = [, ], x =, (, x <. An unknown input type state observer for system ( is designed as follows: û tt (x, t =û xx (x, t, û x (,t=(c + q[û t (,t u t (,t] qû t (,t, û x (,t U(t c [û t (,t u t (,t] (3 c [û(,t u(,t] M sign (û t (,t u t (,t M sign (û(,t u(,t, û(x, = û (x, û t (x, = û (x where M >M + α, M >M + M + β,α>,β> and c, c,c are positive design parameters to be determined. Introduce the error IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 6, NO. 3, MARCH 5 85 variable (ũ, ũ t =(û u, û t u t.thenũ satisfies ũ tt (x, t =ũ xx (x, t ũ x (,t=cũ t (,t ũ x (,t c ũ t (,t c ũ(,t M sign (ũ t (,t M sign (ũ(,t d(t Let E(t = ρ(t = [ũ t (x, t+ũ x(x, t ] dx + c ũ (,t+m ũ(,t (x ũ t (x, tũ x (x, tdx V (t =E(t+δρ(t, <δ<. (5 It is obvious that ( δe(t V (t E(t. Let f : R R be a differentiable real function. Throughout the paper, we understand the derivative of the Lyapunov function V (f(t along function f is as (see, e.g., [, p.38] V (f(t =lim inf [V (f(t V (f(t ], t R. (6 t t t t Lemma : Suppose that f : R R is a continuously differentiable real function. Let V (f(t = f(t. Then, by definition (6 V (f(t = sign (f(t f (t for almost all t R. (7 Proof: Let ( N = {t f(t =,f (t }. (8 We claim that that N must be a countable set. Actually, for any t N,sincef (t, by the continuity of f, there exists a δ > such that f is monotone in (t δ,t + δ. In particular, f(t f(t =for all t (t δ,t + δ \{t }. In other words, there is only one point in (t δ,t + δ belonging to N. Since all the mutually disjoint open sets of R are countable, it follows that N must be countable. For any t R, there are three cases: a f(t. In this case, it is easily found that V (f(t = sign(f(t f (t ;bf(t =, f (t. In this case, t N; c f(t =f (t =. In this case, it is easily computed by definition that V (f(t = sign(f(t f (t =. Combining all these cases, we obtain (7. By Lemma and the definition of the Lyapunov derivative defined by (6, we differentiate functions E and ρ given by (5 formally along the solution of ( to obtain Ė(t ũ t (,tũ x (,t ũ t (,t+c ũ(,tũ t (,t + M sign (ũ(,t ũ t (,t = c ũ t (,t c ũ(,tũ t (,t M ũ t (,t M sign (ũ(,t ũ t (,t dũ t (,t cũ t (,t + c ũ(,tũ t (,t+m sign (ũ(,t ũ t (,t = c ũ t (,t M ũ t (,t dũ t (,t cũ t (,t (9 ρ(t = [ũ t (,t+ũ x(,t ] +ũ t (,t+ũ x(,t [ũ t (x, t+ũ x(x, t ] dx. ( From the boundary condition at x = in (, we can obtain the following estimation: ũ x(,t [c ũ t (,t+c ũ(,t+m sign (ũ t (,t +M sign (ũ(,t + d(t] =[c ũ t (,t+c ũ(,t] +[M sign (ũ t (,t +M sign (ũ(,t + d(t] +[M sign (ũ t (,t + M sign (ũ(,t + d(t] [c ũ t (,t+c ũ(,t] = c ũ t (,t+c ũ (,t+c c ũ t (,tũ(,t +c M ũ t (,t +c M sign (ũ(,t ũ t (,t +c d(tũ t (,t+c M sign (ũ t (,t ũ(,t +c M ũ(,t +c d(tũ(,t +[M sign (ũ t (,t + M sign (ũ(,t + d(t] c ũ t (,t+c ũ (,t+c c ũ t (,tũ(,t +c M sign (ũ(,t ũ t (,t+c β ũ(,t +[M sign (ũ t (,t + M sign (ũ(,t + d(t] ( where we considered a + S b + S for any set S R if a b. Choose δ sufficiently small, to obtain V (t =Ė(t+δ ρ(t [ c δ( + c ] ũ t (,t (c + δ c δc c ũ t (,t δ c ũ (,t c δβ ũ(,t [M + d(t+δc M ] ũ t (,t δ [ũ t (x, t+ũ x(x, t ] dx γv (t, t >a.e. for γ = { min δ, c δ, c } δβ >. ( M That is, V decays exponentially if all the computations from (9 to ( make sense. For the last point, we need to show the existence of the partial differential inclusion solution to ( (very similar to the Filippov solution caused by the discontinuity of the symbolic function []. The uniqueness is not expected for even a simple ODE like ẋ(t sign(x(t,x( = with symbolic function, there are at least three solutions x(t =t, x(t = t,andx(t =. Theorem : For any initial value (ũ, ũ H (, H (, satisfying compatible conditions ũ ( = cũ (, ũ ( c ũ ( c ũ ( M sign (ũ ( M sign (ũ ( d( (3 and measurable disturbance d Hloc (,, ( admits at least one partial differential inclusion solution (ũ, ũ t C(, ; H (, H (,. Moreover, any partial differential inclusion solution (ũ, ũ t C(, ; H (, H (, is exponentially stable E(t δ E(e γt ( where E is defined in (5; γ comes from (; and <δ</ is specified below (5. This shows that the state observer (3 is exponentially convergent. Proof: We need only show the existence of the partial differential inclusion solution to ( since the exponential stability has been proved by ( and the Sobolev imbedding theorem that H (, L (, : There exists a constant C>such that as t s ũ t (,t ũ t (,s C ũ t (,t ũ t (,s H (,. (5

3 86 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 6, NO. 3, MARCH 5 It is noted that when sign(ũ t (,t and sign(ũ(,t in some interval, ( admits unique classical solution. It follows from [, p.8] that for linear system ( by considering U (t =sign(ũ t (,t and U (t =sign(ũ(,t, we only need to consider its sliding mode solution in some finite interval. We may suppose without loss of generality that this interval is just (,θ with <θ</. Thereare three different cases according to the signs of u(,t and u t (,t on interval (,θ. Case. u(,t on [,θ. In this case, u t (,t on [,θ as well. Finding the sliding mode solution amounts to finding functions U eq i (t [, ] for t [,θ and U eq i H (,θ, i=,, such that ũ tt (x, t =ũ xx (x, t, x (,, t [,θ, a.e. ũ x (,t=cũ t (,t ũ x (,t= M U eq (t M U eq (t d(t ũ(x, = ũ (x ũ t (x, = ũ (x (6 and ũ(,t=ũ t (,t= on [,θ [7, p.6]. Now we try to find U eq i. Since on the discontinuous surfaces, u(,t on [,θ, wesolve ũ tt (x, t =ũ xx (x, t, x (,, t [,θ ũ x (,t=cũ t (,t (7 ũ(,t= The classical solution of (7 can be obtained by the characteristic line method as follows: x+t ũ(x, t = [ũ (x + t+ũ (x t] + ũ (xdx, x t x t and x + t. (8 For x t and x + t, the solution is ũ(x, t = [ũ ( + ũ (x + t] + c (c + x+t t x ũ (xdx [ũ (x+ũ (x] dx (9 where we used the left boundary condition: ũ x (,t=cũ t (,t.when x t and x+t, the right boundary condition of (7 gives ũ(x, t = [ũ (x t ũ ( x t] + ũ (xdx Hence x t x t ũ (xdx. ( ũ x (,t=ũ ( t ũ ( t = M U eq (t M U eq (t d(t ( which amounts to ũ ( t ũ ( t+d(t = M U eq (t M U eq (t. ( If for all t [,θ, ũ ( t ũ ( t+d(t >M + M, there is no sliding mode solution and the solution exists in classical sense. However, when ũ ( t ũ ( t+d(t M + M for all t [,θ, we can find a pair of equivalent controls U eq eq (t =U (t = [ũ M + M ( t ũ ( t+d(t] H (,θ. (3 The H property of the equivalent control makes system (6 posses classical solution (see Case 3 below. By (8, (9, and ( that (ũ, ũ t C(,θ; H (, H (,. Case. ũ(,t ũ(, = ũ ( on [,θ. In this case, u t (,t on [,θ as well. Assume ũ ( > without generality. Finding the sliding mode solution amounts to finding function U eq (t [, ] for t [,θ and U eq H (,θ such that ũ tt (x, t =ũ xx (x, t, x (,, t [,θ, a.e. ũ x (,t=cũ t (,t ( ũ x (,t= c ũ ( M U eq (t M d(t Now we try to find U eq.sinceu(,t ũ ( on [,θ, wesolve ũ tt (x, t =ũ xx (x, t, x (,, t [,θ ũ x (,t=cũ t (,t (5 ũ(,t=ũ ( The classical solution to (5 are the same as (8 for x t and x + t ; and as (9 for x t and x + t. Whenx t and x + t, the right boundary condition of (5 gives ũ(x, t = [ũ (x t ũ ( x t] + ũ (xdx ũ (xdx +ũ (. (6 Hence which amounts to x t x t ũ x (,t=ũ ( t ũ ( t = c ũ ( M U eq (t M d(t (7 M U eq (t = ũ ( t+ũ ( t M d(t. (8 If for all t [,θ, ũ ( t ũ ( t+m + d(t >M, there is no sliding mode solution and the solution exists in classical sense. However, when ũ ( t ũ ( t+m + d(t M for all t [,θ, we can find the equivalent control U eq (t = M [ũ ( t ũ ( t+m + d(t] H (,θ. (9 Once again it follows from (8, (9, and (6 that (ũ, ũ t C(,θ; H (, H (,. Case 3. ũ(,t on (,θ and Cases and do not occur. In this situation, the classical solution of ( exists and is unique in H (, H (,. For instance, when ũ(,t >, ũ t (,t >, system ( can be written as ũ tt (x, t =ũ xx (x, t, x (,, t [,θ, a.e., ũ x (,t=cũ t (,t, (3 ũ x (,t= c ũ t (,t c ũ(,t M M d(t, ũ(x, = ũ (x, ũ t (x, = ũ (x and its solution in H (, H (, can be obtained by ( ũ ũ t (ũ t = e At + e A(t s B[ M M d(s]ds (3 ũ

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 6, NO. 3, MARCH 5 87 where the operators A and B are defined by A(f,g =(g, f, (f,g D(A D(A = { (f,g H (, H (, A(f,g H (, L (, f ( = cg( f ( = c g( c f(} ( B =. (3 δ(x This is because A generates a C -semigroup e At on H (, L (, and B is admissible for e At ([]. The solution is classical solution if the initial value (ũ, ũ C(, ; H (, H (, and d Hloc (, satisfy the compatible condition (3 (see, e.g., Proposition..5 of [9, p. 8]. III. OUTPUT FEEDBACK CONTROL LAW VIA ADRC In this section, we design an observer based output feedback control law for system (. In [] where a state feedback is designed, the following function: y(t = u t (x, tdx qu(,t. (33 plays an important role. Since we have obtained an approximation of the state by observer (3, it is naturally to replace u in (33 by û. That is, in what follows, by y, we simply mean that: y(t Δ = û t (x, tdx qû(,t (3 which is determined completely by the output of system (. Find the derivative of y in (3 to obtain ẏ(t = d u t (x, tdx qu(,t d + ũ t (x, tdx qũ(,t = u x (,t+qu t (,t qu t (,t+ d ũ t (x, tdx qũ(,t = U(t+d(t+f (t (35 where f(t = ũt(x, tdx qũ(,t satisfies f(t N e γt, t for some N > and γ>specified in ( (36 in terms of ( and ũ is the solution to (. To estimate the disturbance d, we design, inspired from the ADRC to lumped parameter system presented in [] where the constant high gain is used, the following extended state observer with time varying high gain for system (35 as follows: { ŷ(t =U(t+ ˆd(t r(t[ŷ(t y(t] (37 ˆd(t = r (t[ŷ(t y(t] where r C( R +, R + is a time varying gain satisfying r (t >, lim t + r(t =+ ṙ(t r(t M 3 for some M 3 >, lim t + lim t + r(te γt =for γ>in (. d(t r(t = (38 The existence of gain function r in (38 depends on the disturbance which has growth rate less than γ. This relaxes the condition in [] where d is assumed to be uniformly bounded. It is known from [] that in (37, ˆd is used to estimate d, which is confirmed by the following Lemma. This is a remarkable feature of ADRC. Lemma : Let (ŷ, ˆd be the solution of (37. Then lim [ ŷ(t y(t + ˆd(t d(t ] =. (39 Proof: Set ỹ(t =r(t[ŷ(t y(t], d(t = ˆd(t d(t. ( Then (ỹ, d satisfies { ỹ(t = r(t [ ỹ(t d(t ] + ṙ(t (t r(t d(t = r(tỹ(t d(t. ( Introduce variable z(t =ỹ(t+r(tf(t. Then ( is equivalent to { z(t = r(t [ z(t d(t ] + ṙ(t r(t z(t+r (tf(t d(t = r(t z(t+r (tf(t d(t. ( The existence of the local classical solution to ( is guaranteed by the local Lipschitz condition of the right side of (. The global solution is ensured by the Lyapunov function presented below. Actually, define a Lyapunov function for system ( as follows: V (x,x =x + 3 x x x. (3 Then V is positive definite owing to the following inequality: V (x,x x + x V (x,x, x,x R. ( Differentiate V along the solution of ( to obtain V ( z(t, d(t [ ] ṙ(t = z(t r(t z(t+r(t d(t+ r(t z(t+r (tf(t where +3 d(t[ r(t z(t+r (tf(t d(t] z(t [ r(t z(t+r (tf(t d(t ] [ ] d(t ṙ(t r(t z(t+r(t d(t+ r(t z(t+r (tf(t [ = r(t+ ṙ(t ] z (t r(t r(t d (t ṙ(t r(t d(t z(t + d(t [ z(t 3 d(t ] + r (tf(t [ d(t+ z(t ] φ(tv ( z(t, d(t + [ d(t +3r (t f(t ] ( z(t, d(t φ(tv ( z(t, d(t + [ d(t +3r (t f(t ] V ( z(t, d(t (5 By assumption (38 φ(t =r(t Reorganizing (5 gives d V ( z(t, d(t φ(t + sup t [, 3ṙ(t r(t. (6 lim φ(t =+. (7 V ( z(t, d(t [ d(t +3r (t f(t ], t. (8

5 88 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 6, NO. 3, MARCH 5 Hence d V ( z(t, d(t e t φ(τdτ + V φ(t ( z(t, d(t e t φ(τdτ [ d(t +3r (t f(t ] e t φ(τdτ, t (9 or ( d V ( z(t, d(t e t φ(τdτ [ d(t +3r (t f(t ] e t φ(τdτ, t. (5 Integrate both sides of (5 from to t with respect to t to obtain V ( z(t, d(t e t φ(τdτ V ( z(, d( + [ d(s +3r (s f(s ] e Therefore, for all t, wehave V ( z(t, d(t V ( z(, d( e + [ t ( t s φ(τdτ ds, t. (5 φ(sds d(s +3r (s f(s ] e e φ(sds s φ(τdτ ds. (5 Equation (5 shows that the local solution never blows up, so the global solution of ( exists. It is obvious that the first term on the right side of (5 converges t to zero as t owing to the fact e (/ φ(sds (t t owing to (7. Since e (/ φ(sds as t, we can apply the L Hospital rule to the second term of the right side of (5, to obtain lim = lim [ ( d(s +3r (s f(s ] e φ(sds e [ ( d(t +3r (t f(t ] e e t φ(sds s φ(τdτ ds _ t φ(τdτ φ(t 8 d(t 6 r (t f(t = lim + lim φ(t φ(t 8 d(t = lim r(t r(t φ(t + lim r(t φ(t 6 r(t f(t = (53 where in the last equality of (53, we used (36, (38, and (6. By (5 and (53, we have lim V ( z(t, d(t = which implies [ ] lim z(t + d(t =. (5 Furthermore, since ỹ(t = z(t r(tf(t, it follows from (38 that lim ỹ(t =. However, since ŷ(t y(t =ỹ(t/r(t, wefinally obtain lim ŷ(t y(t =. (55 Equation (39 then follows from (5 and (55. Remark : If we set the time varying gain in (3.5 as constant high gain r(t =/ε, then under the more strict assumption that d is uniformly bounded, it was proved in [] the following practical stability that there exists a t ε > such that ŷ(t y(t + ˆd(t d(t C ε, t t ε (56 where C > is an ε-independent constant. The constant high gain is useful because by the time varying gain in (37, the output is sensitive to the high frequency noise entering the system through the control channel. But the time varying gain can reduce significantly the peaking value. Recommended control strategy is to use the time varying gain in (37 first to reduce the peaking value in the initial stage to a reasonable level and then apply the constant high gain. The constant high gain has the advantage of filtering the high frequency noise. In [], the state feedback control law is designed as U(t = +qk u q+k t(,t+m q S q+k U (t ˆd(t S U (t = +qk q(q+k u(,t+ u(,t q q (57 q+k q u t (x, tdx. In terms of Theorem and Lemma, û is an approximation of u and ˆd is an approximation of d, we design naturally an observer based output feedback control law as follows: U(t = +qk u q+k t(,t+m q q+k ŜU (t ˆd(t Ŝ U (t = +qk q(q+k û(,t+ û(,t (58 q q q+k q ût(x, tdx. It is seen that the control law (58 is simply replacing S U by ŜU in (57. The closed-loop system then becomes u tt (x, t =u xx (x, t u x (,t= qu t (,t u x (,t= +qk u q+k t(,t+m q q+k ŜU (t ˆd(t+d(t û tt (x, t =û xx (x, t û x (,t=(c + q[û t (,t u t (,t] qû t (,t û x (,t +qk u q+k t(,t+m q q+k ŜU (t ˆd(t c (û t (,t u t (,t c (û(,t u(,t (59 M sign(û t (,t u t (,t M sign (û(,t u(,t y(t = û t (x, tdx qû(,t ŷ(t = +qk u q+k t(,t+m q q+k ŜU (t r(t(ŷ(t y(t ˆd(t = r (t(ŷ(t y(t. Theorem : With the output feedback control (58, for any initial value (u(,,u t (, H (, H (,, (û(,, û t (, H (, H (,, ŷ( R, ˆd( R, any partial differential inclusion solution of closed loop system (59 satisfies where F (t = + lim F (t = [ u t (x, t+u x(x, t ] dx + c u (,t [û t (x, t+û x(x, t ] dx+ c û (,t+ ŷ(t + ˆd(t d(t. (6

6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 6, NO. 3, MARCH 5 89 Fig.. Displacements of closed-loop system (59 with disturbance d(t =. (a Displacement u(x, t; (b displacement û(x, t; (c disturbance d(t and its estimation ˆd. Proof: By error variables ũ =û u in ( and (ỹ, d =(r(ŷ y, ˆd d in (, we have the following equivalent system (6 for system (59, which reads: u tt (x, t =u xx (x, t u x (,t= qu t (,t u x (,t= +qk u q+k t(,t+m q S q+k U (t +M q S q+k U (t d(t ũ tt (x, t =ũ xx (x, t,x (, ũ x (,t=cũ t (,t ũ x (,t c ũ t (,t c ũ(,t M sign (ũ t (,t M sign (ũ(,t d(t ỹ(t = r(t ( ỹ(t+ d(t + ṙ(t r(t ỹ(t r(tf (t d(t = r (tỹ(t d(t (6 where S U (t =ŜU (t S U (t. The ũ part of (6 is just ( and the ODE part of (6 is just (. By Theorem and Lemma [ũ t (x, t+ũ x(x, t] dx + c ũ (,t +M ũ(,t as t (6 ŷ(t y(t, d(t = ˆd(t d(t as t. Now, we show the convergence of the u part of (6. To this end, we rewrite u part as follows: u tt (x, t =u xx (x, t u x (,t= qu t (,t u x (,t= +qk q+k u t(,t+m q S q+k U (t (63 + M q S q+k U (t d(t. However, (63 is exactly the same as that in the closed-loop system (6 of [] by replacing d ε (t with M((q /(q + k S U (t+ d(t. Since we have ũ(,t=ũ(,t ũ(,t ũ(,t + ũ x (y, tdy ũ x(y, tdy. By (6, ũ(,t as t. This together with (57 and (58, and (6, by noting ũt(x, tdx ũ t (x, tdx, shows that S U (t =ŜU (t S U (t = q(q+k q+k ũ(,t q q ũ t (x, tdx as. Therefore, M((q /(q + k S U (t+ d(t. In terms of (5, (, and (6, we obtain ([] [ u x (x, t+u t (x, t ] dx + u (,t as t. (6 Since f( = f( f (xdx for any f H (,, wehave û(,t ũ(,t + u(,t ũ x(x, tdx This together with (6 and (6 leads to By (3, (6, and (6, we have ( + ũ(,t + u x(x, tdx + u(,t. û(,t as t. (65 y(t as t. (66 The result then follows from (6, (6, and (66. IV. NUMERICAL SIMULATIONS In this section, we present some simulation results to illustrate the effect of the control law. The simulation results are for the closedloop system (59. Here, we simply specify sign( =. The other parameters are chosen as and the initial values are q =,k =.,M =,c=.8,c =., c =,M =5,M =,r(t =t + (67 u(x, = x, u t (x, = x, û(x, = cos x, û t (x, = x, ŷ( =, ˆd( =. (68 The disturbances are chosen as d =and d =sin(t+e t respectively. The finite difference method is adopted in computation of the displacements. The time and space step are chosen as. and., respectively. Fig. (a and (b show the displacements u and û in system (59 respectively with d =.Fig.(cshowsthat ˆd tracks the disturbance d very satisfactorily. Fig. displays the corresponding counterparts of Fig. 3 with d =sint + e t. Fig. 3 demonstrates the displacements of system (59 with constant high gain r(t =8, the maximal value of r specified in (67 in the time span [,8] ([]. The peaking value phenomenon for disturbance tracking is clearly observed in Fig. 3(c. Compared with Fig., we see additional advantage of the time varying gain that it reduces

7 83 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 6, NO. 3, MARCH 5 Fig.. Displacements of closed-loop system (59 with disturbance d(t =sint + e t. (a The displacement u(x, t; (b the displacement û(x, t; (c disturbance d(t and its estimation ˆd. Fig. 3. Displacements of closed-loop system (59 with constant high gain r(t =8. (a Displacement u(x, t; (b displacement û(x, t; (c disturbance d(t and its estimation ˆd(t. Fig.. Displacements of closed-loop system (59 with time varying gain and constant gain. (a Displacement u(x, t; (b displacement û(x, t; (c disturbance d(t and its estimation ˆd(t. dramatically the peaking value in the initial stage no matter for the displacement or for the disturbance tracking. The price paid by time varying gain is that the convergence is slightly slowly, which can be seen by comparison of Figs. 3 and. Fig. displays the same parts as that in Fig. 3 but we choose the varying gain r as { t +, t+< 8, r(t = (69 8, t+ 8 to verify the strategy suggested by Remark. This means that we use the time varying gain in the initial stage and then change to the constant high gain afterwards. Figs. (a and (b are similar to Figs. 3(a and 3(b, respectively, but in Fig. (c, we can see clearly the peaking reduction compared with Fig. 3(c. REFERENCES [] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides. Dordrecht, The Netherlands: Kluwer Academic Publishers, 988. [] B. Z. Guo and F.-F. Jin, Sliding mode and active disturbance rejection control to stabilization of one-dimensional anti-stable wave equations subject to disturbance in boundary input, IEEE Trans. Autom. Control, vol. 58, no. 5, pp. 69 7, 3. [3] B. Z. Guo and W. Kang, The Lyapunov approach to boundary stabilization of an anti-stable one-dimensional wave equation with boundary disturbance, Int. J. Robust Nonlin. Control, vol., no., pp. 5 69,. [] B. Z. Guo and Z. L. Zhao, On the convergence of extended state observer for nonlinear systems with uncertainty, Syst. Control Lett.,vol.6,no.6, pp. 3,. [5] J. Q. Han, From PID to active disturbance rejection control, IEEE Trans. Ind. Electron., vol. 56, no. 3, pp. 9 96, 9. [6] M. Krstic, Adaptive control of an anti-stable wave PDE, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., vol. 7, no. 6, pp ,. [7] Y. Orlov, Discontinuous Systems Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions. Berlin, Germany: Springer-Verlag, 9. [8] A. Pisano and Y. Orlov, Boundary second-order sliding-mode control of an uncertain heat process with unbounded matched perturbation, Automatica, vol. 8, no. 8, pp ,. [9] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups. Basel, Switzerland: Birkhäuser, 9. [] J. A. Walker, Dynamical Systems and Evolution Equations: Theory and Applications. New York: Plenum Press, 98. [] Q. Zheng and Z. Gao, An energy saving, factory-validated disturbance decoupling control design for extrusion processes, in Proc. th World Congress Intell. Control Autom.,, pp