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

Liu Yaqiang,Jin Zengwang,Ren Zhigang

doi:10.1155/2021/6660857

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

Liu Yaqiang, Jin Zengwang + Show 1 more

Open Access

https://doi.org/10.1155/2021/6660857

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Abstract

In this paper, a robust H ∞ 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.

Highlights

Distributed parameter systems (DPSs) are infinite-dimensional in nature and are generally modeled by partial differential equations (PDEs)
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
Performance constraint with the collocated observation case which is first proposed in terms of standard linear matrix inequalities (LMIs); 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, Poincare–Wirtinger inequality’s variants, Cauchy–Schwartz inequality, integration by parts, and first mean value theorem for definite integrals

Summary

Introduction

Distributed parameter systems (DPSs) are infinite-dimensional in nature and are generally modeled by partial differential equations (PDEs). Pointwise control of DPSs with T-S fuzzy DPS model has been developed in [32], where a fuzzy state feedback controller is designed This technique has been extended to the [33, 34]. 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. Performance constraint with the collocated observation case which is first proposed in terms of standard linear matrix inequalities (LMIs); 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, Poincare–Wirtinger inequality’s variants, Cauchy–Schwartz inequality, integration by parts, and first mean value theorem for definite integrals. M and N denote two sets of natural numbers, i.e., M ≜ {1, 2, . . . , m}, N ≜ {1, 2, . . . , n}

Problem Formulation and Preliminaries

Robust H‘ Feedback Compensator Design

Numerical Simulation

Conclusions