# Performance Output Tracking for a One-Dimensional Wave-Heat Cascade System with Unmatched Disturbance

Authors: Yulong Liu
Journal: Complexity
 (2019)

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## Abstract

The objective of this paper is to solve the performance output regulation problem for a wave-heat cascade system with unmatched disturbance. Applying the series expansion, the auxiliary trajectory for the cascade system is constructed and the unmatched disturbance is rejected. Meanwhile, the controller and observer only based on error signal are designed, and the performance output regulation problem is solved. Under the control feedback, the performance output can track the reference signal, and the regulation error goes to zero asymptotically. Finally, some numerical simulations are presented for illustration.


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## Body

## 1. Introduction

In recent years, regulating the output of a given distributed parameter system is one of the central problems in control theory. The objective of the output regulation is to construct a controller such that the performance output of the given plant can track a reference signal. Comparing with the finite-dimensional system, there exist various so-called noncollocated problems in distributed parameter system. The system we consider is described by the one-dimensional wave-heat cascade system as follows:(1)v˙t=Gvt,t≥0,uttx,t=uxxx,t,x∈0,1,t>0,ux0,t=c1z0,t+c2u0,t,t≥0,ux1,t=-c0ut1,t,t≥0,ztx,t=zxxx,t,x∈0,1,t>0,zx0,t=dt,t≥0,z1,t=Ut,t≥0,ux,0=u0x,x∈0,1,utx,0=u1x,x∈0,1,zx,0=z0x,x∈0,1,Yout=u1,t,t≥0,where u0,u1, and z0 are the initial state, Yout is the performance output, d(t) is the disturbance, and U:[0,∞)→R is the control input. Considering the control plant (1) in the state space H=Cn×H1(0,1)×L2(0,1), we are going to design a control law so that the performance output Yout tracks the given reference signal Yref in the presence of the external disturbance d.Recently, the performance output tracking for a wave equation with harmonic disturbance is considered in [1] where the control and disturbance are unmatched. In [2], the same problem is also studied for a wave with a general boundary disturbance by the disturbance treatment technique which is firstly proposed in [3]. These results were extended to regulate the output of a Schrödinger equation in [4, 5]. However, the literature mentioned above is considered in a relative easy situation where the performance output is always collocated to the control actuation. Therefore, one of the objectives of this paper is to deal with the problem that the performance output is not collocated to the control actuation. The latest progress of this problem is considered in [6–9]. They use the method of backstepping, servo system, and the adaptive control approach to solve the performance output regulation problem in noncollocated case. The proposed method design can also be extended to the more general control system. Meanwhile, [10, 11] construct various controllers to solve the robust output regulation of distribution parameter systems by internal model principle. Different from the methods mentioned above, in this paper, a novel auxiliary trajectory and servomechanism is designed to cope with the noncollocated problem for system (1).In addition, the problem of the output regulation for cascaded system is seldom considered. Concerning the cascaded system, a lot of results have been obtained only for stabilization problem [12–15]. In view of this reason, another objective of this paper is to solve the problem of the output regulation for the cascaded system (1) by a novel method of auxiliary trajectory and servomechanism.For the disturbance and the reference signal, in system (1), both d(t) and Yref are supposed to be the harmonic signals of the following form:(2)dt=∑j=0mAjsin⁡ωjt+Bjcos⁡ωjt,m∈N,Yreft=∑k=0m~A~ksin⁡ω~kt+B~kcos⁡ω~kt,m~∈N,where A~k,B~k, Aj,Bj are unknown amplitudes and ωj,ω~k are known frequencies, j=1,2,…,m, k=1,2,…,m~. By a simple computation, both of them can be rewritten as an output of the following exosystem:(3)v˙t=Gvt,t≥0,Yref=Fvt,t≥0,dt=Qvt,t≥0,v0=v0,where G∈Rn×n is the system matrix; F and Q are known n-dimensional row vectors; and the initial state v0 depends on the amplitudes A~k,B~k, Aj,Bj and hence is unknown. Throughout this paper, we always assume that G is invertible and diagonalizable with σ(G)={λi}i=1n⊂iR/{0} and Re〈Gx,x〉Cn=0, ∀x∈Cn. Under this assumption, the general harmonic signal can be written as an output of the exosystem (3). With this assumption, the mathematical foundations of the output regulation problem in question can be found in [16, 17].Now, we need to design a controller such that the regulation errorYe=Yref-Yout (the only measurement for controller design) satisfies(4)limt→∞⁡Yet=limt→∞⁡Yreft-Youtt=0.We proceed as follows. In Section2, the servomechanism will be designed by the method of auxiliary trajectory and the negative impacts of the unmatched disturbance will be canceled. In Section 3, the controller will be designed and the performance output regulation problem will be solved under the control feedback. In Section 4, we design an observer based on error signal and prove the convergence of the observer by the Lyapunov functional method. In Section 5, the uniform boundedness of the loop system will be presented and proved. Section 6 presents some numerical simulations to illustrate the effectiveness of the control law, followed by concluding remarks in Section 7.


## 2. Trajectory Planning for the Disturbance

In this section, we are going to cancel the negative impacts of the disturbance by trajectory planning. We first consider the unmatched disturbanceQv(t) in (1). Inspired by [18], we need to find the transformation to convert system (1) into the target system that is a disturbance free system, and at the same, both the controller and the output keep invariant. For this purpose, we suppose that the auxiliary trajectory satisfies the following system:(5)ϕtx,t=ϕxxx,t,x∈0,1,t>0,ϕx0,t=Qvt,t≥0,ϕ0,t=P1vt,t≥0,where P1 is an n-dimensional row vector such that ϕ(1,t)=0. Inspired by [18] and [19, Chapter 12], again, we try to find a special solution of system (5) in the following form:(6)ϕx,t=∑n=0∞αntxnn!,x∈0,1,t≥0.Inserting (6) into system (5), we have(7)α˙nt=αn+2t,n=0,1,2⋯α0t=P1vt,α1t=Qvt,which leads to(8)ϕx,t=∑n=0∞P1vntx2n2n!+∑n=0∞Qvntx2n+12n+!=P1∑n=0∞Gnx2n2n!vt+Q∑n=0∞Gnx2n+12n+1!vt=P1cosh⁡xG1/2+QG-1/2sinh⁡xG1/2vt.Since ϕ(1,t)=0, we get(9)ϕ1,t=P1cosh⁡G1/2+QG-1/2sinh⁡G1/2vt=0or equivalently(10)P1=-QG-1/2tanh⁡G1/2.We suppose(11)ψttx,t=ψxxx,t,x∈0,1,t>0,ψx1,t=c0ψt1,t,t≥0,ψ1,t=P2vt,t≥0,where P2 is an n-dimensional row vector such that ψx(0,t)=c1ϕ(0,t)-c2ψ(0,t). We try to find a special solution of system (11) in the following form:(12)ψx,t=∑n=0∞βntx-1nn!,x∈0,1,t≥0.Inserting (11) into system (12), we have(13)β¨nt=βn+2t,n=0,1,2…β0t=P2vt,β1t=c0P2Gvt,which leads to(14)ψx,t=P2cosh⁡x-1Gvt+c0P2sinh⁡x-1Gvt.Sinceψx(0,t)=c1ϕ(0,t)-c2ψ(0,t), we get(15)ψx0,t=P2sinh⁡-GGvt+c0P2cosh⁡-GGvt=c1P1vt-c2P2cosh⁡-Gvt+c0P2sinh⁡-Gvt,or equivalently(16)P2=c1P1·H-1,where(17)H=sinh⁡-GG+c0cosh⁡-GG+c2cosh-G+c2c0sinh⁡-G.If we let(18)ρx,t=ux,t-ψx,t,x∈0,1,t≥0,φx,t=zx,t-ϕx,t,x∈0,1,t≥0,then, by (1), (5), and (11)(19)ρttx,t=ρxxx,t,x∈0,1,t>0,ρx1,t=c0ρt1,t,t≥0,ρx0,t=c1φ0,t-c2ρ0,t,t≥0,φtx,t=φxxx,t,x∈0,1,t>0,φx0,t=0,t≥0,φ1,t=Ut,t≥0,v˙t=Gvt,t≥0,and(20)Yet=Yreft-ψ1,t-ρ1,t=F-P2vt-ρ1,t.Comparing system (1) with system (19), one can find that the disturbance Qv(t) in system (1) has been canceled in system (19). The next objective is to stabilize the regulation error Ye. To this end, we will apply the trajectory planning again to bring the external signal F-P2v(t) into the control channel such that the regulation error Ye translates into a state of the target system. Naturally, we only need to stabilize the target system to achieve our goal.We suppose that the trajectory satisfies the following system:(21)ηttx,t=ηxxx,t,x∈0,1,t>0,ηx1,t=-c0ηt1,t,t≥0,ηx0,t=c1μ0,t-c2η0,t,t≥0,μtx,t=μxxx,t,x∈0,1,t>0,μx0,t=0,t≥0,μ1,t=P3vt,t≥0,where P3 is an n-dimensional row vector that will be determined later.Suppose that the “ρ-part" of (21) satisfies(22)ηttx,t=ηxxx,t,x∈0,1,t>0,ηx1,t=-c0ηt1,t,t≥0,η1,t=F-P2vt,t≥0.Taking (5), (6), (7)n and (8) into account, system (22) admits a special solution(23)ηx,t=F-P2cosh⁡x-1Gvt+c0F-P2sinh⁡x-1Gvt.It implies that(24)ηx0,t=F-P2sinh⁡-GGvt+c0F-P2cosh⁡-GGvt,ηx0,t=c1μ0,t-c2η0,t.From (24), we have(25)μ0,t=1c1F-P2·H·vt,where H is defined by (17).Suppose that the “μ-part” of (21) satisfies(26)μtx,t=μxxx,t,x∈0,1,t>0,μx0,t=0,t≥0,μ0,t=1c1F-P2·H·vt,t≥0.We will find a special solution for “μ-part" of (26) that takes the form(27)μx,t=∑n=0∞ξntxnn!,x∈0,1,t≥0.Hence,(28)μx,t=1c1F-P2·H·cosh⁡xG1/2vt.More specifically,(29)μ1,t=P3vt=1c1F-P2·H·cosh⁡G1/2vt.Or equivalently(30)P3=1c1F-P2·H·cosh⁡G1/2.If we let(31)εx,t=ηx,t-ρx,t,γx,t=μx,t-φx,t,then, by (19) and (21),(32)εttx,t=εxxx,t,x∈0,1,t>0,εx1,t=-c0εt1,t,t≥0,εx0,t=c1γ0,t+c2ε0,t,t≥0,γtx,t=γxxx,t,x∈0,1,t>0,γx0,t=0,t≥0,γ1,t=P3vt-Ut,t≥0,and(33)Yet=F-P2vt-ρ1,t=η1,t-φ1,t=ε1,t.


## 3. Controller Design

Equation (33) implies that we only need to stabilize system (32) to achieve output regulation (1) without input delay. In this way, the controller with τ=0 can be designed easily(34)Ut=P3vt=1c1F-P2·H·cosh⁡G1/2vt,under which, we get the closed-loop system of (34)(35)v˙t=Gvt,t≥0,uttx,t=uxxx,t,x∈0,1,t>0,ux1,t=-c0ut1,t,t≥0,ux0,t=c1z0,t+c2u0,t,t≥0,ztx,t=zxxx,t,x∈0,1,t>0,zx0,t=dt,t≥0,z1,t=1c1F-P2·H·cosh⁡G1/2vt,t≥0,ux,0=u0x,x∈0,1,utx,0=u1x,x∈0,1,zx,0=z0x,x∈0,1.Theorem 1.
For any initial state(v(0),u(·,0),z(·,0))∈H, system (35) has unique solution (v(t),u(·,t),z(·,t))∈C([0,∞);H) such that, for any t0>0,(36)Yet≤L1e-ω1t,∀t≥t0,where L1 and ω1 are two positive constants. Moreover, the state of the closed-loop system (35) is uniformly bounded(37)supt∈0,∞⁡vt,u·,t,z·,tH<+∞.Proof.
We first consider the following transformed system:(38)εttx,t=εxxx,t,x∈0,1,t>0,εx1,t=-c0εt1,t,t≥0,εx0,t=c1γ0,t+c2ε0,t,t≥0,γtx,t=γxxx,t,x∈0,1,t>0,γx0,t=0,t≥0,γ1,t=0,t≥0,with the initial state(39)εx,0=ψx,0+ηx,0-ux,0,x∈0,1,γx,0=μx,0+ϕx,0-zx,0,x∈0,1,where ψ, η, μ, and ϕ are defined by (14), (23), (28), and (8), respectively. As system (38) is a cascade of the heat equation and the wave equation and the “γ-subsystem" of (38) is independent of the “ε-subsystem", it is well known that there exists a unique solution (ε(·,t),γ(·,t))∈C([0,∞);H1(0,1)×L2(0,1)) to system (38).
Consider the Lyapunov function(40)V=12∫01εx2dx+∫01εt2dx+c2ε20+δ∫011+xεxxεtxdx+12∫01γ2xdx.Using the Cauchy-Schwarz and Yong’s inequalities, there exist m1,m2 such that(41)m1Π≤V≤m2Π,where(42)Π=∫01εx2dx+∫01εt2dx+ε20+∫01γ2xdx.Therefore V is positive definite.
The derivative ofV along the solution of system (38) is(43)dVdt=∫01εxεtxdx+∫01εtεttdx+c2ε0εt0+δ∫011+xεxtεt+εxεttdx+∫01γγtdx=∫01εxεtxdx+∫01εtεxxdx+εx0,t-c1γ0,tεt0+δ∫011+xεxtεt+εxεxxdx+∫01γγxxdx=∫01εxεtxdx-∫01εxtεxdx+εx1εt1-c1γ0,tεt0+δ∫011+xεxtεt+εxεxxdx-∫01γx2dx≤-c0εt21-c1γ0,tεt0+δ21+xεt2+εx201-δ2∫01εt2+εx2dx-∫01γx2dx≤-c0-δ-δc02εt21-δ2-c12εt20-δ2∫01εt2+εx2dx-1-c12-δc1c2∫01γ2dx-δc22-2δc1c2ε20which is negative definite for(44)0<c1<12,2c1<c2<1,c1<δ<c01+c02.It follows from (40) and (41) that(45)Πt≤Me-t/MΠ0for some possibly large M, which proves the exponential stability of the “ε-γ” system.
On the other hand, since the “v-subsystem” of (35) is independent of the other subsystems, it admits a unique solution v(t)∈C[0,∞);Cn.
We define(46)ux,t=ψx,t+ηx,t-εx,t,x∈0,1,t≥0,zx,t=μx,t+ϕx,t-γx,t,x∈0,1,t≥0,where ψ, η, μ, and ϕ are defined by (14), (23), (28), and (8), respectively. Now, it is easy to verify that such a defined (v(t),u(·,t),z(·,t))∈C([0,∞);H) is a solution of system (35). Moreover, the uniformly bounded (37) holds due to (45), (46), and the fact that G is dissipative.
The proof is complete.


## 4. Observer Design

In this section, one will design the observer forv(t). According to the ideal of [18], the state observer can be designed as follows:(47)v^˙t=Gv^t+c3F-P2∗-y˙et-ρ^t1,t+F-P2Gv^t,t≥0,ρ^ttx,t=ρ^xxx,t,x∈0,1,t>0,ρ^x1,t=-c0ρ^t1,t-c3yet-c3ρ^1,t+c3F-P2v^t,t≥0,ρ^x0,t=c1φ^0,t+c2ρ^0,t,t≥0,φ^tx,t=φ^xxx,t,x∈0,1,t>0,φ^x0,t=0,t≥0,φ^1,t=Ut,t≥0,where 0<c3<1 is the tuning parameter and K=F-P1 is the conjugate transpose of K∗.Let(48)v~x,t=vx,t-v^x,t,ρ~x,t=ρx,t-ρ^x,t,φ~x,t=φx,t-φ^x,t.We have the error system as follows:(49)v~˙t=Gv~t-c3K∗ρ~t1,t+c3K∗KGv~t,t≥0,ρ~ttx,t=ρ~xxx,t,x∈0,1,t>0,ρ~x1,t=-c0ρ~t1,t-c3ρ~1,t-c3Kv~t,t≥0,ρ~x0,t=c1φ~0,t+c2ρ~0,t,t≥0,φ~tx,t=φ~xx,x∈0,1,t>0,φ~x0,t=0,t≥0,φ~1,t=0,t≥0,where c3 is the tuning parameter and K=F-P2 is the conjugate transpose of K∗.Moreover, we have the following theorem.Theorem 2.
Suppose thatG+c3K∗KG is Hurwitz. Then solution of system (49) is asymptotically stable.Proof.
SinceG+c3K∗K is Hurwitz, there exists a positive constant a1 such that(50)G+c3K∗Kv~2≤-a1v~Cn2.
Consider the following Lyapunov function.(51)V1=12v~2+12∫01ρ~x2dx+∫01ρ~t2dx+c2ρ~20+c3ρ~21+12∫01φ~2xdx
The derivative ofV1 along the solution of system (49) is(52)dV1dt=v~v~˙+∫01ρ~xρ~txdx+∫01ρ~tρ~ttdx+c2ρ~0ρ~t0+∫01φ~φ~tdx+c3ρ~1ρ~t1=G+c3K∗Kv~2t+c3K∗ρ~t1,tv~t+∫01ρ~xρ~txdx+∫01ρ~tρ~xxdx+c2ρ~0ρ~t0+∫01φ~φ~tdx+c3ρ~1ρ~t1=G+c3K∗Kv~2t+c3K∗ρ~t1,tv~t+∫01ρ~xρ~txdx+ρ~tρ~x01-∫01ρ~xtρ~xdx+ρ~x0-c1φ~0ρ~t0+∫01φ~φ~xxdx+c3ρ~1ρ~t1≤G+c3K∗Kv~2t+c3K∗ρ~t1v~t-c0ρ~t12-c3ρ~1ρ~t1-c3Kv~tρ~t1-c1φ~0ρ~t0-∫01φ~x2dx+c3ρ~1ρ~t1≤-a1vCn2-c0ρ~t12+c12φ~02+ρ~t02-∫01φ~x2dx≤-a1vCn2-c0-c12ρ~t12-1-c12∫01φ~x2dxwhich is negative definite for(53)c1<2,c0>c12.This completes the proof of the theorem.Theorem 3.
Suppose thatG+c3K∗K is Hurwitz. Then, for any initial state (v(0),ρ(·,0),φ(·,0),v^(0),φ^(·,0),z^(·,0))∈H2 and U∈Hloc1(0,∞), system (14)-(47) admits a unique solution(54)vt,ρ·,t,φ·,t,v^t,ρ^·,t,φ^·,t∈C0,∞;H2such that(55)vt-v^t,ρ·,t-ρ^·,t,φ·,t-φ^·,tH→0ast→∞.Proof.
It is well known that, for any(v(0),ρ(·,0),φ(·,0))∈H and U∈Hloc1(0,∞), system (19) admits a unique solution(56)vt,ρ·,t,φ·,t∈C0,∞;H.From Theorem 2, we know that system (49) admits a unique solution (v~(t),ρ~(·,t),φ~(·,t))∈C([0,∞);H) with initial state (v~(0),ρ~(·,0),φ~(·,0))=(v(0)-v^(0),ρ(·,0)-ρ^(·,0),φ(·,0)-φ^(·,0),) such that(57)v~t,φ~·,t,z~·,tH→0ast→∞.Then, we define(58)v^t=vt-v~t,x∈0,1,ρ^x,t=ρx,t-ρ~x,t,x∈0,1,φ^x,t=φx,t-φ~x,t,x∈0,1,Hence, (v(t),ρ(·,t),φ(·,t),v^(t),ρ^(·,t),φ^(·,t))∈C([0,∞);H2) is a solution of system (19)-(47). Moreover, by (57) and (58), we can see (55) holds. The proof is complete.


## 5. The Uniform Boundedness of the Loop System

Replacingv with v^, one will obtain the following closed-system of (1):(59)v˙t=Gvt,t≥0,uttx,t=uxxx,t,x∈0,1,t>0,ux0,t=c1z0,t+c2u0,t,t≥0,ux1,t=-c0ut1,t,t≥0,ztx,t=zxxx,t,x∈0,1,t>0,zx0,t=dt,t≥0,z1,t=Ut,t≥0,v^˙t=Gv^t+c3F-P2∗-y˙et-ρ^t1,t+F-P2Gv^t,t≥0,ρ^ttx,t=ρ^xxx,t,x∈0,1,t>0,ρ^x1,t=-c0ρ^t1,t-c3yet-c3ρ^1,t+c3F-P2v^t,t≥0,ρ^x0,t=c1φ^0,t+c2ρ^0,t,t≥0,φ^tx,t=φ^xxx,t,x∈0,1,t>0,φ^x0,t=0,t≥0,φ^1,t=Ut,t≥0.Moreover, one has the following theorem.Theorem 4.
Suppose thatG+c3K∗K is Hurwitz. Then, for any initial state (v(0),u(·,0),z(·,0),v^(0),ρ^(·,0),φ^(·,0))∈H2, the closed-loop (59) admits a unique solution(60)vt,u·,t,z·,t,v^t,ρ^·,t,φ^·,t∈C0,∞;H2such that(61)dt-Qv^tCn+Yet→0ast→∞.If we assume further that G is dissipative, then the state of system is uniformly bounded(62)supt∈0,∞⁡vt,u·,t,z·,t,v^t,ρ^·,t,φ^·,tH2<∞.Proof.
According to Theorems1 and 3, system (59) admits a unique solution(63)vt,u·,t,z·,t,v^t,φ^·,t,z^·,t∈C0,∞;H2with initial state (v(0),u(·,0),z(·,0),v^(0),φ^(·,0),z^(·,0))∈H2.
Next, We define(64)ux,t=ψx,t+ηx,t-εx,t,x∈0,1,zx,t=μx,t+ϕx,t-γx,t,x∈0,1,φ^x,t=φx,t-φ~x,t,x∈0,1,ρ^x,t=ρx,t-ρ~x,t,x∈0,1,v^t=vt-v~t,x∈0,1,and, by Theorems 1 and 2, it is easy to check(65)supt∈0,∞⁡vt,u·,t,z·,t,v^t,ρ^·,t,φ^·,tH2<∞.By (36) and (56), we can see (61) holds. The proof is complete.


## 6. Numerical Simulation

In this section, one presents some numerical simulations to validate our theory results. We give the numerical simulation results for system(v,u,v^,φ^) which is governed by (59). The corresponding parameters are chosen as(66)G=01-40and(67)F=1,0,Q=0,1,c0=1.3.The initial states are selected as(68)ux,0=0.01cos⁡2πx,φ^x,0=sin⁡2πx,v=0,2,v^=0,-3.The time step and space step are taken as 0.001s and 0.05s.The solution of the closed-loop system (59) is plotted in Figure 1. The output tracking and the disturbance estimation are plotted in Figure 2. In Figures 1 and 2, we choose c1=0.1, c2=0.7, c3=0.8.Figure 1
The solution of closed-loop system.
(a)
u ( x , t ) (b)
z ( x , t ) (c)
ρ ^ ( x , t ) (d)
φ ^ ( x , t )Figure 2
Output tracking and disturbance estimation.
(a)
Output tracking (b)
Disturbance estimationFigure1 displays the numerical results for closed-loop system in the noncollocated error feedback output regulation case. Figures 1(a) and 1(b) display the states of the controlled system u(x,t), z(x,t). The states of the observer ρ^(x,t), φ^(x,t) are plotted in Figures 1(c) and 1(d). It is obvious that all states are bounded.Figure2(a) shows that the output signal u(1,t) tracks the given reference signal Yref(t) as time evolves for noncollocated case. Obviously, output signal u(1,t) can track the given reference signal asymptotically. The disturbance estimation is plotted in Figure 2(b). The observer reconstructs the corresponding disturbance. Both of them show that the convergence is very effective and smooth. More importantly, no peaking phenomenon takes place. Comparing Figure 2(a) with Figure 2(b), it can be seen that after the convergence of the observer, the output converges to zero verifying disturbance rejection.


## 7. Concluding Remarks

In this paper, we mainly solved the performance output regulation problem for a wave-heat cascade system with unmatched disturbance. Applying a novel auxiliary trajectory method and servomechanism design, the performance output can track the given signal asymptotically. Thus, the contribution of this paper is to solve the noncollocated performance output regulation problem of the cascade system despite unmatched disturbance. The idea is potentially promising for treating other PDE-PDE or PDE-ODE systems to solve the performance output regulation which will be our future work.



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