Spatial domain decomposition approach to dynamic compensator design for linear space‐varying parabolic MIMO PDEs

Parameter‐dependent observer‐based feedback compensator design of a space‐time‐varying PDE with application to a class of steelmaking processes

“…Remark The observer‐based feedback compensator (13) with (14) is also applicable to the collocated measurement case as pointed out in Remark 1. On the other hand, different from the observer‐based feedback compensator in Reference 19, where only the function is involved in the compensator's structure, the proposed parameter‐dependent observer‐based feedback compensator (13) and (14) takes the time‐varying diffusion coefficient , the time‐varying advection coefficient , the space‐time‐varying reaction coefficient , and the space‐varying control distribution function into account and its gains directly depend on the diffusion coefficient , the advection coefficient , the reaction coefficient , and the function .…”

“…Feedback control of linear/nonlinear PDEs with space‐varying coefficients has been discussed in References 15‐19. In References 16,17, the infinite‐dimensional backstepping boundary control design 4 was extended to respectively solve boundary stabilization control for coupled reaction–diffusion systems and coupled parabolic partial integro‐differential equations (PIDEs) with space‐varying reaction coefficients.…”

“…Via the modal decomposition approach, a constructive method for finite‐dimensional observer‐based control has been recently developed in Reference 18 for parabolic PDEs with space‐varying diffusion and reaction coefficients. By resorting to the extreme value theorem and variants of Poincaré–Wirtinger inequality, a spatial decomposition approach has been recently proposed in Reference 19 for observer‐based feedback compensator design of a reaction–diffusion equation with space‐varying coefficients. More recently, the backstepping boundary control design in Reference 17 has been extended in Reference 20 to exponential stabilization of coupled linear reaction–diffusion equations with a space‐time‐varying reaction coefficient.…”

“…More recently, the backstepping boundary control design in Reference 17 has been extended in Reference 20 to exponential stabilization of coupled linear reaction–diffusion equations with a space‐time‐varying reaction coefficient. Despite the above progress, the proposed controller structure in References 15‐20 does not contain the reaction coefficient directly. This coefficient is very important and causes the PDE instability 4 .…”

“…In this article, on the basis of the authors' previous work, 19 we will further discuss the problem of parameter‐dependent observer‐based feedback compensator design of a linear parabolic PDE with time‐varying diffusion and advection coefficients, space‐time‐varying reaction coefficient, and non‐collocated measurement noise/disturbance. Different from the authors' previous work, 19 the coefficients are represented in parametric form under their boundedness assumptions and the parameter form is directly involved in the feedback compensator's structure. Via the Lyapunov technique with variants of Poincaré–Wirtinger's inequality, the parameter‐dependent observer‐based feedback compensator design is formulated as a feasibility problem subject to a set of convex constraints.…”