Robust H‘ Feedback Compensator Design for Linear Parabolic DPSs with Pointwise/Piecewise Control and Pointwise/ Piecewise Measurement

School of Automation and Guangdong-HongKong-Macao Joint Laboratory for Smart Discrete Manufacturing, Guangdong University of Technology, Guangzhou, Guangdong 510006, China School of Automation and Guangdong Discrete Manufacturing KnowledgeAutomation Engineering Technology Research Center, Guangdong University of Technology, Guangzhou, Guangdong 510006, China School of Cybersecurity, Northwestern Polytechnical University, Xi’an, Shanxi 710072, China National Engineering Laboratory for Integrated Aero-Space-Ground-Ocean Big Data Application Technology, Xi’an, Shanxi 710072, China


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][2][3][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][6][7][8][9][10][11][12][13][14][15][16][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][19][20].
is technique has been extended to the boundary antidisturbance control and boundary adaptive robust control for flexible DPSs [21][22][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 indomain 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][44][45][46][47]. e estimation problems in controller design of DPSs have been studied in [48][49][50][51][52][53], and for some DPSs with unknown parameters, parameter estimation methods have been applied in [54][55][56][57][58]. e design and analysis methods have also been extended to switched control systems and filtering technique in [59][60][61][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.
us, an approach of robust H ∞ control is proposed to deal with the control problem of DPSs with external disturbances. e 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].
is 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 observerbased H ∞ sampled-data fuzzy control design in [46,64] and mixed H 2 /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.
e 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]  (iii) An H ∞ 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.
e 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 Section 2. en, 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.

Problem Formulation and Preliminaries
In this paper, we consider a class of one-dimensional linear parabolic DPSs with external disturbances of the following form: subject to the Robin boundary conditions in one dimension, and the initial condition, where z ∈ [0, L] ⊂ R denotes the spatial position between is a set of measurement outputs from n sensors, expressed as y(t) ≜ y 1 (t) y 2 (t) · · · y n (t) T ∈ R n . s(z) ≜ s 1 (z) s 2 (z) · · · s n (z)] T ∈ R n is a known integrable vector function of z, and the element s j (z) describes the distribution of j-th sensor on the spatial domain [0, L].
It should be pointed out that when η > 0.25π 2 L − 2 , the onedimensional linear parabolic DPSs is unstable.

Remark 1.
It is worth noting that equation (1) is equivalent to the following general form [68]: through the conversion of the following state variables and control variables: 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. erefore, 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 functions h i (z) are described as follows: where δ(·) is the Dirac delta function [69]. e points which imply the chosen functions g i (z), 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 � z 1 < z 2 · · · < z m+1 � L. e locations of the actuators for pointwise control and local piecewise control satisfy the Similar to the actuators' distribution, the in-domain observation forms are generally divided into pointwise measurement and local piecewise measurement; the sensors' spatial distribution functions s j (z), j ∈ N are described in this paper as follows: Complexity 3 e points z j , j ∈ N, and local subdomains which imply the chosen functions c j (z), 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. e locations of the sensors for pointwise measurement and local piecewise measurement satisfy the relationships For the linear parabolic DPS (1)-(3), the following H ∞ performance constraint is proposed under the zero initial condition u 0 (·) � 0: where c 1 > 0 and c 2 > 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.
e 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 ω(t) � 0 is exponentially stable in the sense of | · | 2 ; meanwhile, the H ∞ performance constraint in (10) is ensured when the initial value of u(z, t) is zero (u 0 (z) � 0) and all d(·, ·) ∈ L 2 (0, ∞; H), ω(·) ∈ L 2 (0, ∞; R n ). e following lemmas are very useful for the development of the robust H ∞ compensator design in this paper.

Robust H ' Feedback Compensator Design
Based on the distributions of actuators and sensors that h i (z), i ∈ M, in (6) (or (7)) and s j (z), j ∈ N, in (8) (or (9)), the observation obtained from sensors can be divided into collocated observation in space (i.e., h(z) � s(z)) and noncollocated observation in space (i.e., h(z) ≠ s(z)). In other words, the collocated observation in space is a special case of noncollocated observation. Meanwhile, the noncollocated observation in space (i.e., h(z) ≠ s(z)) 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: where u(z, t) denotes the state of the observer; 0 < L ≜ diag l 1 , l 2 , . . . , l n ∈ R n×n is the observer gain to be determined. e observation functions s(z) ≜ s 1 (z) s 2 (z) · · · s n (z) T are choosen as such that 0 � z 1 < z 2 < · · · < z n < z n+1 � L. en, we design an observer-based dynamic output feedback compensator of the following form: where 0 < K ≜ diag k 1 , k 2 , . . . , k m ∈ R m×m is the compensator gain in the form of m × m diagonal matrix, and the compensator functions h(z) ≜ h 1 (z) h 2 (z) · · · h m (z)] T ∈ R m are choosen as 4 Complexity such that 0 � z 1 < z 2 < · · · < z m < z m+1 � L. e estimation error state is defined as From formulas (1)- (3) and (13)- (17), the estimation error system is represented as where the initial value e 0 (z) ≜ u 0 (z) − u 0 (z).
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: 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)). e 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 c 1 and c 2 .

Conclusions
In this paper, the robust H ∞ 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   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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Conflicts of Interest
e authors declare that they have no conflicts of interest.