# RobustH∞ Feedback Compensator Design for Linear Parabolic DPSs with Pointwise/Piecewise Control and Pointwise/Piecewise Measurement

Authors: Liu Yaqiang; Ren Zhigang; Jin Zengwang
Journal: Complexity
 (2021)

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## Abstract

In this paper, a robustH∞ control problem of a class of linear parabolic distributed parameter systems (DPSs) with pointwise/piecewise control and pointwise/piecewise measurement has been investigated via the robust H∞ feedback compensator design approach. A unified Lyapunov direct approach is proposed in consideration of the pointwise/piecewise control and point/piecewise measurement based on the distributions of the actuators and sensors. A new type of Luenberger observer is developed on the continuous interval of space domain to track the state of the system, and an H∞ performance constraint with prescribed H∞ attenuation levels is proposed in this paper. By utilizing Lyapunov technique, mathematical inequalities, and integration theory, a sufficient condition based on LMI for the exponential stability of the corresponding closed-loop coupled system under an H∞ performance constraint is presented. Finally, the effectiveness of the proposed design method is verified by numerical simulation results.


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## Body

## 1. Introduction

Distributed parameter systems (DPSs) are infinite-dimensional in nature and are generally modeled by partial differential equations (PDEs). DPSs are widely used in engineering systems [1–4], such as thermodynamics, chemical engineering, missile, aerospace, aviation, and nuclear fission and fusion. Control problem of DPSs has attracted extensive attention due to the important applications in engineering systems, such as the vibration control of flexible structures that the vibration process can be described by Euler–Bernoulli equations, the diffusion control of oil spill that the diffusion phenomena can be described by diffusion equations, and the temperature control of heating furnace that the thermal conduction process can be described by heat equations. In recent decades, fruitful achievements in the design of DPSs control have been achieved from many scholars all over the world [5–17].Generally, control forms of DPSs can be divided into boundary control and in-domain control based on the actuators’ location. Fruitful achievements on boundary control of DPSs that the actuators are located at the boundary area have been published already. For example, the boundary control problem of flexible robot manipulators has been developed to solve the DPSs with flexible structures [18–20]. This technique has been extended to the boundary antidisturbance control and boundary adaptive robust control for flexible DPSs [21–23]. Boundary control of 1D nonlinear parabolic DPSs has been studied in [24], in which the continuum backstepping method is utilized. Boundary feedback control of DPSs has been addressed in [25], and a novel combination of feedback idea and backstepping approach is presented in [26]. Sampled-data boundary control and sliding mode boundary control of DPSs have been studied in [27, 28], where a sampled-data strategy for the boundary control problem is proposed. Fuzzy boundary control based on the T-S fuzzy DPS model is shown in [29, 30]. H∞ boundary control has been proposed in [31] that a linear matrix inequality (LMI) approach has been utilized. Meanwhile, there are also some achievements on in-domain control of DPSs. For example, pointwise control of DPSs with T-S fuzzy DPS model has been developed in [32], where a fuzzy state feedback controller is designed. Furthermore, this technique has been extended to the [33, 34]. Robust sampled-data control has been proposed in [35, 36], where the sampled-data pointwise controller method is applied. Mobile piecewise control of 1D DPSs has been studied in [37] that a mobile actuator-plus-sensor network is developed, and this technique has been extended to solve the 2D DPSs in [38]. More recently, collocated control and noncollocated control of in-domain control in DPSs have been studied deeply. Static collocated feedback control has been presented in [39, 40] that the static collocated pointwise and piecewise feedback controller has been designed for parabolic DPSs. For the noncollocated control that the actuators and sensors can never be placed at the same location exactly, the static feedback control has been studied in [32, 36, 41, 42], and the observer-based dynamic feedback control has been designed in [43–47]. The estimation problems in controller design of DPSs have been studied in [48–53], and for some DPSs with unknown parameters, parameter estimation methods have been applied in [54–58].The design and analysis methods have also been extended to switched control systems and filtering technique in [59–62]. Although there have been many promising efforts, there are still many control problems of DPSs need to be studied in the future.In general, disturbance problems of DPSs are unavoidable because of the errors from model calculations and equipments. Thus, an approach of robustH∞ control is proposed to deal with the control problem of DPSs with external disturbances. The robust H∞ control has attracted much attention from many scholars over the past few decades. For example, an H∞ static output feedback boundary controller for semilinear parabolic and hyperbolic DPSs is proposed in [31]. This idea has extended to solve the sampled-data distributed H∞ control problem for a class of parabolic DPSs in [35]. An H∞ fuzzy observer-based controller is proposed for a class of nonlinear parabolic DPSs in [63], and this technique has developed to the observer-based H∞ sampled-data fuzzy control design in [46, 64] and mixed H2/H∞ fuzzy observer-based feedback control design in [65]. In this paper, we will extend the works in [66, 67] to design the H∞ output feedback compensator for linear parabolic DPSs with external disturbances by using a unified Lyapunov approach. A sufficient condition for the static H∞ feedback compensator can stabilize the DPSs under an H∞ performance constraint with the collocated observation case which is first proposed in terms of standard linear matrix inequalities (LMIs); then, another sufficient condition for the observer-based dynamic H∞ feedback compensator can stabilize the DPSs under an H∞ performance constraint with the noncollocated observation case which is developed by using the Lyapunov direct method, Poincaré–Wirtinger inequality’s variants, Cauchy–Schwartz inequality, integration by parts, and first mean value theorem for definite integrals.The main contributions and novelty of this paper compared with the existing works before are summarized as follows:(i)
Different from the results in [32–34, 43, 44] that the pointwise/piecewise control and collocated (or noncollocated) pointwise/piecewise measurement are discussed separately, all the forms of control and observation are considered in this paper by a unified Lyapunov direct approach.(ii)
In contrast to the works in [66] that a unified Lyapunov-based compensator design for linear parabolic DPSs with free disturbance, this paper presents the static and dynamic robust H∞ output feedback compensator design for linear DPSs with external disturbances. Meanwhile, a new type of Luenberger observer is designed for noncollocated observation in space with continuous observation functions.(iii)
AnH∞ performance constraint in the sense of ⋅2 is proposed to deal with the external disturbance of the model and measurement disturbance in the measurement output.The organizational structure of the remaining parts of this paper is arranged as follows: first, the problem formulation of this paper and some preliminary knowledge are presented in Section2. Then, the static output feedback compensator design and observer-based dynamic output feedback compensator design in terms of collocated and noncollocated observation in space satisfying the H∞ performance constraint are shown in Section 3. Section 4 provides some numerical simulation results of the corresponding closed-loop systems to show the effectiveness of the proposed design method. Finally, brief conclusions are followed in Section 5.Notation:ℜ, ℜm, and ℜm×n denote the set of all real numbers, m-dimensional Euclidean space, and the set of all m×n matrices, respectively. ℋ≜ℒ20,L is a real Hilbert space of square integrable functions with the inner product induced norm ⋅2. ℒ20,∞;ℋ is a real Hilbert space of square integrable functions ϖ⋅,t with ϖℒ2≜∫0∞ϖ⋅,t2dt. ℒ20,∞;ℜn is a real Hilbert space of square integrable functions ζt with ζℒ2≜∫0∞ζtdt. utz,t stands for the partial derivative of uz,t with respect to t, i.e., utz,t=∂uz,t/∂t, uzz,t, and uzzz,t stands for the first-order and second-order partial derivative of uz,t with respect to z, i.e., uzz,t=∂uz,t/∂z, uzzz,t=∂2uz,t/∂z2, respectively. ℳ and N denote two sets of natural numbers, i.e., ℳ≜1,2,…,m, N≜1,2,…,n.


## 2. Problem Formulation and Preliminaries

In this paper, we consider a class of one-dimensional linear parabolic DPSs with external disturbances of the following form:(1)utz,t=uzzz,t+ηuz,t+dz,t+hTzUt,z∈0,L,yt=∫0Lszuz,tdz+ωt,subject to the Robin boundary conditions in one dimension,(2)uzz,t|z=0=a1u0,t,uzz,t|z=L=−a2uL,t,and the initial condition,(3)uz,0=u0z,z∈0,L,where z∈0,L⊂ℜ denotes the spatial position between 0,L, and t≥0 denotes time, respectively. u⋅,t≜uz,t,z∈0,L∈ℋ, denotes the state variable. η>0 is a known constant. d⋅,⋅≜dz,t,z∈0,L∈ℒ20,∞,ℋ is an unknown bounded external disturbance and satisfies ∫0∞d⋅,t22dt<∞. hz≜hzh2z⋯hmzT∈ℜm is a known integrable vector function of z, and the element hiz describes the distribution of i-th actuator on the spatial domain 0,L. Ut is a set of control inputs from m actuators, expressed as Ut≜u1tu2t⋯umtT∈ℜm. yt is a set of measurement outputs from n sensors, expressed as yt≜y1ty2t⋯yntT∈ℜn. sz≜s1zs2z⋯snzT∈ℜn is a known integrable vector function of z, and the element sjz describes the distribution of j-th sensor on the spatial domain 0,L. ω⋅≜ω1tω2t⋯ωntT∈ℒ20,∞;ℜn is the n-dimensional bounded measurement disturbance vector. It should be pointed out that when η>0.25π2L−2, the one-dimensional linear parabolic DPSs is unstable.Remark 1.
It is worth noting that equation (1) is equivalent to the following general form [68]:(4)u¯tz,t=u¯zzz,t+2fzz,tu¯zz,t+η+fz2z,t+fzzz,t−ftz,tu¯z,t+dz,t+h¯TzU¯t,z∈0,L,through the conversion of the following state variables and control variables:(5)fz,t=f1z+f2t,u¯z,t=exp−fz,tuz,t,h¯Tz=exp−f1zhTz,U¯t=exp−f2tUt,where f⋅,t≜fz,t∈ℋ20,L is a known scalar function and continuously differentiable in time t.In practical applications of DPSs, the number of actuators and sensors is usually limited and actives at specified point or part thereof in the spatial domain, respectively. Therefore, the in-domain control forms of DPSs are generally divided into pointwise control and local piecewise control according to the distribution of actuators. In this paper, two forms of in-domain control are both considered; the actuators’ spatial distribution functionshiz are described as follows:(6)hiz≜δz−z¯i,i∈ℳpointwise control case,(7)hiz≜1z∈z¯i−,z¯i+0elsewhere,i∈ℳlocal piecewise control case,where δ⋅ is the Dirac delta function [69]. The points z¯i,i∈ℳ, and local subdomains z¯i−,z¯i+,i∈ℳ, satisfy 0<z¯1<z¯2<⋯<z¯m<L and 0<z¯1−<z¯1+<z¯2−<z¯2+<⋯<z¯m−<z¯m+<L, which imply the chosen functions giz,i∈ℳ, produce pointwise control at the points z¯i and local piecewise uniform control over z¯i−,z¯i+, respectively. Meanwhile, the spatial domain 0,L can be divided into m parts by a spatial domain decomposition approach that 0=z1<z2⋯<zm+1=L. The locations of the actuators for pointwise control and local piecewise control satisfy the relationships z¯i∈zi,zi+1,i∈ℳ, and z¯i−,z¯i+⊂zi,zi+1,i∈ℳ.Similar to the actuators’ distribution, the in-domain observation forms are generally divided into pointwise measurement and local piecewise measurement; the sensors’ spatial distribution functionssjz,j∈N are described in this paper as follows:(8)sjz≜δz−z˜j,j∈Npointwise measurement case,(9)sjz≜1z∈z˜j−,z˜j+0elsewhere,j∈Nlocal piecewise measurement case.The pointsz˜j,j∈N, and local subdomains z˜j−,z˜j+,j∈N satisfy 0<z˜1<z˜2<⋯<z˜n<L and 0<z˜1−<z˜1+<z˜2−<z˜2+<⋯<z˜n−<z˜n+<L, which imply the chosen functions cjz,j∈N produce pointwise observation at the points z˜j and local piecewise uniform observation over z˜j−,z˜j+, respectively. At the same time, the spatial domain 0,L can be divided into n parts by a spatial domain decomposition approach that 0=z^1<z^2⋯<z^n+1=L. The locations of the sensors for pointwise measurement and local piecewise measurement satisfy the relationships z˜j∈zj,zj+1,j∈N and z˜j−,z˜j+⊂z^j,z^j+1,j∈N.For the linear parabolic DPS (1)–(3), the following H∞ performance constraint is proposed under the zero initial condition u0⋅=0:(10)∫0∞u⋅,t22dt≤γ12∫0∞d⋅,t22dt+γ22∫0∞ωt2dt,where γ1>0 and γ2>0 are the prescribed H∞ attenuation levels.For the development of stability analysis in this paper, two exponential stability definitions in the sense of⋅2 of the linear parabolic DPS (1)–(3) are defined.Definition 1.
The linear parabolic DPS (1)–(3) with Ut=0 and d⋅,t=0 is exponentially stable in the sense of ⋅2, when there exist two constants ɛ1≥1 and ɛ2>0 satisfying the expression u⋅,t2≤ɛ1u0⋅2exp−ɛ2t for any t≥0.Definition 2.
The linear parabolic DPS (1)–(3) with the designed output feedback compensator is exponentially stable in the sense of ⋅2 under an H∞ performance constraint, when the corresponding closed-loop DPS with d⋅,t=0 and ωt=0 is exponentially stable in the sense of ⋅2; meanwhile, the H∞ performance constraint in (10) is ensured when the initial value of uz,t is zero (u0z=0) and all d⋅,⋅∈ℒ20,∞;ℋ,ω⋅∈ℒ20,∞;ℜn.The following lemmas are very useful for the development of the robustH∞ compensator design in this paper.Lemma 1 (Poincaré–Wirtinger inequality’s variants).
For a scalar functionu∈ℋ10,L, we have(11)∫0Lus−uz2ds≤4τ1π−2∫0Ldusds2ds,∫0Lus−ul2ds≤4τ2π−2∫0Ldusds2ds,where z∈0,L, τ1≜maxz2,L−z2, ul≜z2−z1−1∫z1z2uzdz, z1,z2⊂0,L, and τ2≜maxz22,L−z12. For more detailed details, please refer to the lemma in [66].Lemma 2.
(Cauchy–Schwartz inequality [70]). For any two scalar functions ϑ1∈ℋ10,L and ϑ2∈ℋ10,L, there exists constant τ3>0 which makes the following inequality hold:(12)∫0Lϑ1zϑ2zds≤ϑ1zϑ2z≤τ32ϑ1z2+12τ3ϑ2z2.


## 3. RobustH∞ Feedback Compensator Design

Based on the distributions of actuators and sensors thathiz,i∈ℳ, in (6) (or (7)) and sjz,j∈N, in (8) (or (9)), the observation obtained from sensors can be divided into collocated observation in space (i.e., hz=sz) and noncollocated observation in space (i.e., hz≠sz). In other words, the collocated observation in space is a special case of noncollocated observation. Meanwhile, the noncollocated observation in space (i.e., hz≠sz) consists of the following cases: pointwise control and noncollocated pointwise observation case, pointwise control and noncollocated piecewise observation case, piecewise control and noncollocated pointwise observation case, and piecewise control and noncollocated piecewise observation case. In this section, all the noncollocated observation cases will be considered to study the robust H∞ dynamic output feedback compensator design for the DPS (1)–(3).A new type of Luenberger-type observer for the DPS (1)–(3) is constructed as follows:(13)u^tz,t=u^zzz,t+ηu^z,t+h^TzUt+s^TzL^yt−y^t,t>0,z∈0,L,u^zz,t|z=0=a1u^0,t,u^zz,t|z=L=−a2u^L,t,u^z,0=u^0z,z∈0,L,y^t=∫0Lszu^z,tdz,t≥0,where u^z,t denotes the state of the observer; 0<L^≜diagl1,l2,…,ln∈ℜn×n is the observer gain to be determined.The observation functionss^z≜s^1zs^2z⋯s^nzT are choosen as(14)s^jz≜1z∈z^j,z^j+10elsewhere,j∈N,such that 0=z^1<z^2<⋯<z^n<z^n+1=L.Then, we design an observer-based dynamic output feedback compensator of the following form:(15)Ut=−K^∫0Lh^zu^z,tdz,t≥0,where 0<K^≜diagk1,k2,…,km∈ℜm×m is the compensator gain in the form of m×m diagonal matrix, and the compensator functions h^z≜h^1zh^2z⋯h^mzT∈ℜm are choosen as(16)h^iz≜1z∈zi,zi+10elsewhere,i∈ℳ,such that 0=z1<z2<⋯<zm<zm+1=L.The estimation error state is defined as(17)ez,t≜uz,t−u^z,t,z∈0,L.From formulas (1)–(3) and (13)–(17), the estimation error system is represented as(18)etz,t=ezzz,t+ηez,t+dz,t−s^TzL^∫0Lszez,tdz−s^TzL^ωt,z∈0,L,ezz,t|z=0=a1e0,t,ezz,t|z=L=−a2eL,t,ez,0=e0z,z∈0,L,where the initial value e0z≜u0z−u^0z.Substituting the designed dynamic feedback compensator (15) and the estimation error state (17) into the DPS (1)–(3), the following closed-loop system is obtained as follows:(19)utz,t=uzzz,t+ηuz,t+dz,t−hTzK^∫0Lh^zuz,tdz+hTzK^∫0Lh^zez,tdz,z∈0,L,uzz,t|z=0=a1u0,t,uzz,t|z=L=−a2uL,t,uz,0=u0z,z∈0,L.Hence, the closed-loop coupled DPS is represented by the estimation error system (18) and the closed-loop system (19) with expressions (6) (or (7)) and (8) (or (9)). The objective of this subsection is to seek an effective method to design an observer-based dynamic output feedback compensator such that the resulting closed-loop coupled DPS is exponentially stable under an H∞ performance constraint in the sense of ⋅2 with prescribed H∞ attenuation levels γ1 and γ2.The following theorem provides a sufficient condition for the exponential stability of the closed-loop coupled DPS (18), (19), (6) (or (7)), and (8) (or (9)) with d⋅,t=0,ωt=0 in the sense of ⋅2.Theorem 1.
Consider a class of linear parabolic DPSs (1)–(3) with d⋅,t=0,ωt=0 for noncollocated observation (i.e., hz≠sz). If there exist compensator gains ki, i∈ℳ, and scalars l^j, j∈N ,satisfying the following LMI constraints:(20)Π^i≜η−π24ϕiπ2σi4ϕi−ki20π2σi4ϕi−ki2−π2σi24ϕiki0ki−2ε<0,i∈ℳ,(21)Ψ^j≜qη−qπ24φj+ɛ2qπ2σ^j4φj−l^j2qπ2σ^j4φj−l^j2−qπ2σ^j24φj<0,j∈N,in which(22)σ^j≜1,j∈Npointwise observation case,1z˜jR−z˜jL,j∈Nlocal piecewise observation case,φj≜maxz^j+1−z˜j2,z˜j−z^j2,j∈Npointwise measurement case,maxz^j+1−z˜jL2,z˜jR−z^j2,j∈Npiecewise measurement case,and then there exists an observer-based dynamic output feedback compensator (15) ensuring the exponential stability of the closed-loop coupled DPS (18), (19), (6) (or (7)), and (8) (or (9)) with d⋅,t=0,ωt=0 in the sense of ⋅2.Proof.
Assume the LMIs (20) and (21) are fulfilled. Consider the following Lyapunov function candidate:(23)Vt=V1t+V2t,in which(24)V2t=0.5q∫0Le2z,tdz,where q>0 is a Lyapunov parameter to be determined.
The time derivative ofV1t for the closed-loop system (19) is(25)V˙1t≤−∫0Luz2z,tdz+η∫0Lu2z,tdz−∫0Luz,thTzdzK^∫0Lh^zuz,tdz+∫0Luz,thTzdzK^∫0Lh^zez,tdz=−∫0Luz2z,tdz+η∫0Lu2z,tdz−∑i=1mkiuzc,t∫zizi+1uz,tdz+∑i=1mkiuzc,t∫zizi+1ez,tdz.
Based on the Cauchy–Schwartz inequality, the following inequality is fulfilled for any scalarɛ>0:(26)∫zizi+1kiuzc,tez,tdz≤12ɛ−1∫zizi+1ki2u2zc,tdz+ɛ∫zizi+1e2z,tdz.
Due toz¯i∈zi,zi+1 and z¯i−,z¯i+⊂zi,zi+1,i∈ℳ, by Poincaré–Wirtinger inequality’s variants in Lemma 1, the following inequality is fulfilled for each z∈zi,zi+1,i∈ℳ:(27)∫zizi+1uz2z,tdz≥π24ϕi∫zizi+1uz,t−σiuzc,t2dz.
Substituting the inequalities (27) and (26) into (25), we get(28)V˙1t≤−∑i=1m∫zizi+1π24ϕiuz,t−σiuzc,t2dz+η∑i=1m∫zizi+1u2z,tdz−∑i=1m∫zizi+1kiuzc,tuz,tdz+12ɛ∑i=1m∫zizi+1ki2u2zc,tdz+ɛ2∑i=1m∫zizi+1e2z,tdz=∑i=1m∫zizi+1uiTz,tΠiuiz,tdz+ɛ2∫0Le2z,tdz,where(29)Πi≜η−π24ϕiπ2σi4ϕi−ki2π2σi4ϕi−ki2−π2σi24ϕi+ki22ɛ<0,i∈ℳ.
The time derivative ofV2t is(30)V˙2t≤−q∫0Lez2z,tdz+qη∫0Le2z,tdz−q∫0Lez,ts^TzL^∫0Lszez,tdz=−q∫0Lez2z,tdz+qη∫0Le2z,tdz−∑j=1nl^jez^s,t∫z^jz^j+1ez,tdz,where l^j=qlj,ez^s,t=∫0Lsjzez,tdz,j∈N. The following expression is utilized by applying integration by parts and considering the boundary conditions in (18):(31)∫0Lez,tezzz,tdz=ez,tezz,t|z=0z=L−∫0Lez2z,tdz=−a2e2L,t−a1e20,t−∫0Lez2z,tdz≤−∫0Lez2z,tdz.
Asz˜j∈z^j,z^j+1,z˜j−,z˜j+⊂z^j,z^j+1,j∈N, and 0=z^1<z^2<⋯<z^n<z^n+1=L. By Lemma 1, we get for each z∈z^j,z^j+1,j∈N,(32)∫z^jz^j+1ez2z,tdz≥π24φj∫z^jz^j+1ez,t−σ^jez^s,t2dz.
Thus, the expression (30) can be rewritten as(33)V˙2t≤−qπ24φj∑j=1n∫z^jz^j+1ez,t−σ^jez^s,t2dz+qη∑j=1n∫z^jz^j+1e2z,tdz−∑j=1n∫z^jz^j+1l^jez^s,tez,tdz=∑j=1n∫z^jz^j+1ejTz,tΨjejz,tdz,where ejz,t≜ez,tez^s,tT,j∈N, and(34)Ψj≜qη−qπ24φjqπ2σ^j4φj−l^j2qπ2σ^j4φj−l^j2−qπ2σ^j24φj<0,j∈N.
From (28) and (33), the time derivative of Vt defined in (23) is given as(35)V˙t=V˙1t+V˙2t≤∑i=1m∫zizi+1uiTz,tΠiuiz,tdz+ɛ2∫0Le2z,tdz+∑j=1n∫z^jz^j+1ejTz,tΨjejz,tdz≤∑i=1m∫zizi+1uiTz,tΠiuiz,tdz+∑j=1n∫z^jz^j+1ejTz,tΨ^jejz,tdz.
By the Schur complement and the LMI constraint (20), we have(36)Πi<0,i∈ℳ.
It can be deduced from the LMI (21) and the inequality (36) that there exists an appropriate constant α2>0, satisfying the following inequalities:(37)Πi+0.5α2I≤0,i∈ℳ,Ψ^j+0.5α2I≤0,j∈N.
Substituting the inequalities (37) to (35), we obtain(38)V˙t≤−0.5α2∑i=1m∫zizi+1uiTz,tuiz,tdz−0.5α2∑j=1n∫z^jz^j+1ejTz,tejz,tdz≤−α2V1t+q−1V2t≤−α2κVt,where κ≜min1,q−1. Integrating from 0 to t for the inequality (38), we can get V1t≤V10exp−α1t, which implies u⋅,t2≤u0⋅2exp−0.5α1t, t≥0. Thus, we can obtain u⋅,t22+e⋅,t22≤max1,q/min1,qu0⋅22+e0⋅22exp−0.5α2κt. In other words, the closed-loop coupled DPS (18), (19), (6) (or (7)), and (8) (or (9)) with d⋅,t=0,ωt=0 is exponentially stable in the sense of ⋅2 . The proof is complete.Next, theH∞ performance analysis is developed for the closed-loop coupled DPS (18), (19), (6) (or (7)), and (8) (or (9)) with the initial value u0⋅=0 and all d∈ℒ20,∞;ℋ,ω∈ℒ20,∞;ℜn. The following theorem provides a sufficient condition for exponential stability in the sense of ⋅2 of the closed-loop coupled DPS (18), (19), (6) (or (7)), and (8) (or (9)) under a prescribed H∞ performance constraint (10) based on the LMI conditions.Theorem 2.
Consider a class of linear parabolic DPSs (1)–(3), given H∞ attenuation levels γ1>0,γ2>0 and constant 0<r<1, if there exist compensator gains ki, i∈ℳ and scalars l^j, j∈N, satisfying the following LMI constraints:(39)Γ^i≜piqi00.5qi−π2σi24ϕiki00ki−2ɛ00.500−rγ12Δzi<0,i∈ℳ,(40)Θj≜hjvj0.5q−l^j2vj−qπ2σ^j24φj000.5q0−1−rγ12Δz^j0−l^j200−γ22Δz^j<0,j∈N,in which(41)pi≜η−π24ϕi+1,qi≜π2σi4ϕi−ki2,hj≜qη−qπ24φj+ɛ2,vj≜qπ2σ^j4φj−l^j2,Δz^j≜z^j+1−z^j,j∈N,and then there exists an observer-based dynamic output feedback compensator (15) ensuring the exponential stability of the closed-loop coupled DPS (18), (19), (6) (or (7)), and (8) (or (9)) under a prescribed H∞ performance constraint (10).Proof.
From expressions (18), (19), and (35), the time derivative of Vt is rewritten as(42)V˙t≤∑i=1m∫zizi+1uiTz,tΠiuiz,tdz+∑j=1n∫z^jz^j+1ejTz,tΨ^jejz,tdz+∫0Luz,tdz,tdz+q∫0Lez,tdz,tdz−∑j=1n∫z^jz^j+1l^jωjtez,tdz.
Considering theH∞ performance constraint in (10) and constant 0<r<1, we have(43)V˙t+u⋅,t22−γ12d⋅,t22−γ22ωt2≤∑i=1m∫zizi+1uiTz,tΠiuiz,tdz+∑j=1n∫z^jz^j+1ejTz,tΨ^jejz,tdz+∫0Luz,tdz,tdz+q∫0Ldz,tez,tdz−∑j=1n∫z^jz^j+1l^jωjtez,tdz+∑i=1m∫zizi+1u2z,tdz−∑i=1mrγ12Δzi∫zizi+1d2z,tdz−∑j=1n1−rγ12Δz^j∫z^jz^j+1d2z,tdz−∑j=1nγ22Δz^j∫z^jz^j+1ωj2tdz≤∑i=1m∫zizi+1u^iTz,tΓiu^iz,tdz+∑j=1n∫z^jz^j+1e^jTz,tΘje^jz,tdz,where u^iz,t≜uz,tuzc,tdz,tT,i∈ℳ, e^jz,t≜ez,tez^s,tdz,tωjtT,j∈N, and(44)Γi≜η−π24ϕi+1π2σi4ϕi−ki20.5π2σi4ϕi−ki2−π2σi24ϕi+ki22ɛ00.50−rγ12Δzi,i∈ℳ.
By the Schur complement and the LMI constraint (40), we have(45)Γi<0,i∈ℳ.
It is easily obtained from (43) in consideration of the inequality (45) and the LMI (40) that(46)V˙t+u⋅,t22−γ12d⋅,t22−γ22ωt2≤0,∀t≥0.
Intergrating (46) from 0 to ∞ and considering the zero initial value u0⋅=0, we get the expression (10). In other words, the H∞ performance constraint (10) with the given prescribed H∞ attenuation levels γ1>0 and γ2>0 is guaranteed for the closed-loop coupled DPSs (18), (19), (6) (or (7)), and (8) (or (9)) with the initial value u0⋅=0 and all d∈ℒ20,∞;ℋ,ω∈ℒ20,∞;ℜn. Meanwhile, the LMI constraints (20) and (21) in Theorem 1 can be derived from the LMI constraints (39) and (40) in Theorem 2. Therefore, the closed-loop coupled DPS (18), (19), (6) (or (7)), and (8) (or (9)) is exponential stable in the sense of ⋅2 under a prescribed H∞ performance constraint (10). The proof is complete.


## 4. Numerical Simulation

In this section, we will provide some numerical simulations to demonstrate the effectiveness of the proposed design strategy. For the linear parabolic DPS (1)–(3), given the constants η=3,L=1,a1=0.5, and a2=0, set the initial value u0z=0.5sin0.5L−1πx+0.5π,z∈0,L, dz,t=cosπzexp−t, the open-loop profile of evolution of uz,t, and the open-loop trajectory of u⋅,t2, as shown in Figure 1. It is easily seen from Figure 1 that the linear parabolic DPS (1)–(3) with Ut=0 is unstable.Figure 1
Simulation results for the open-loop linear parabolic DPS (1)–(3). (a) Open-loop profile of evolution of uz,t. (b) Open-loop trajectory of u⋅,t2.
(a)(b)The numerical simulations of the corresponding closed-loop coupled DPS (18), (19), (7), and (9) for local piecewise control and noncollocated local piecewise measurement case in space are then shown. Assume three actuators and two sensors are applied for implementation of the proposed design method, that is, m=3,n=2, and ℳ=1,2,3,N=1,2. Three actuators are respectively distributed over 0.1L,0.3L,0.4L,0.6L, and 0.7L,0.9L over the spatial domain 0,L, and two sensors are respectively distributed over 0.2L,0.4L,0.6L,0.8L over the spatial domain 0,L, i.e., z¯1−=0.1L, z¯1+=0.3L, z¯2−=0.4L, z¯2+=0.6L, z¯3−=0.7L, z¯3+=0.9L and z˜1−=0.2L,z˜1+=0.4L,z˜2−=0.6L,z˜2+=0.8L, respectively. Set z1=0, z2=L/3, z3=2L/3, and z4=L and z^1=0,z^2=0.5L,andz^3=L. Through numerical calculations, we can get Δz1=Δz2=Δz3=L/3, Δz^1=Δz^2=0.5L, σ1=σ2=σ2=5L−1, σ^1=σ^2=5L−1, ϕ1=0.09L2,ϕ2=0.0711L2,ϕ3=0.09L2, and φ1=0.16L2,φ2=0.2178L2. Set ɛ=10,r=0.5; by solving the LMIs (39) and (40), we get the optimised H∞ attenuation levels as γ1=20.6102,γ2=18.6307, compensator gains k1=52.1015, k2=71.7887, and k3=52.1015, and observer gains l1=77.0853,l2=69.8957. Applying the designed observer-based dynamic output feedback compensator (15) with the calculated compensator gains and observer gains, the closed-loop profile of evolution of uz,t, the closed-loop trajectory of u⋅,t2, e⋅,t2, and the trajectory of the observer-based dynamic output feedback compensator Ut of the form (15) with dz,t=0, ωt=0 are shown in Figure 2. The simulation results indicate that the designed observer-based dynamic output feedback compensator (15) can stabilize the MIMO PDE (1)–(3) with d⋅,t=0,ωt=0 in the sense of ⋅2.Figure 2
Closed-loop coupled DPS (18), (19), (7), and (9) with dz,t=0 and ωt=0 for local piecewise control and noncollocated local piecewise measurement in space. (a) Closed-loop profile of evolution of uz,t. (b) Closed-loop trajectory of u⋅,t2. (c) Closed-loop trajectory of e⋅,t2. (d) Trajectory of dynamic output feedback compensator Ut.
(a)(b)(c)(d)Figure3 presents the closed-loop profile of evolution of uz,t, the closed-loop trajectory of u⋅,t2, the observer-based dynamic output feedback compensator Ut in (14), and the trajectory of γt for the closed-loop coupled DPS (18), (19), (7), and (9) with u0⋅=0 and dz,t=cosπxexp−t, ωt≜exp−0.1texp−0.3t. It is easily seen from Figure 3 that γt<minγ1,γ2, which implies ∫0∞u⋅,t22dt≤γ12∫0∞d⋅,t22dt+γ22∫0∞ωt2dt; the H∞ performance constraint (10) is guaranteed. The simulation results indicate that the designed observer-based dynamic output feedback compensator (15) can stabilize the linear parabolic DPS (1)–(3) with u0⋅=0 and dz,t=cosπxexp−t, ωt≜exp−0.1texp−0.3t, in the sense of ⋅2.Figure 3
Closed-loop coupled DPS (18), (19), (7), and (9) with u0⋅=0 and dz,t=cosπxexp−t, ωt≜exp−0.1texp−0.3t, for local piecewise control and noncollocated local piecewise measurement in space. (a) Closed-loop profile of evolution of uz,t. (b) Closed-loop trajectory of u⋅,t2. (c) Trajectory of observer-based dynamic output feedback compensator Ut. (d) The trajectory of γt.
(a)(b)(c)(d)


## 5. Conclusions

In this paper, the robustH∞ feedback compensator for a class of linear parabolic DPSs with external disturbances has been investigated in consideration of the pointwise/piecewise control and pointwise/piecewise measurement based on the distributions of the actuators and sensors. A new type of Luenberger observer is designed to solve the difficulty caused by noncollocated observation and track the state of the PDEs. It is different from the previous observer design method that all the cases of the pointwise/piecewise control and pointwise/piecewise measurement are considered via a defined unified distribution function. An observer-based dynamic output feedback compensator is designed and an H∞ performance constraint is proposed under the zero initial condition. By utilizing Poincaré–Wirtinger inequality’s variants, Cauchy–Schwartz inequality, integration by parts, and first mean value theorem for definite integrals, sufficient conditions on the exponential stability of the corresponding closed-loop system under an H∞ performance constraint in the sense of ⋅2 are presented in terms of LMI constraints. Finally, numerical simulation results of the resulting closed-loop systems are provided to illustrate the effectiveness of the proposed design strategy.



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*Source: 6660857-2021-04-12.xml*