Sub-optimal boundary control of semilinear pdes using a dyadic perturbation observer

Abstract

Abstract-In this paper, we present a sub-optimal controller for semilinear partial differential equations, with partially known nonlinearities, in the dyadic perturbation observer (DPO) framework. The dyadic perturbation observer uses a two-stage perturbation observer to isolate the control input from the nonlinearities, and to predict the unknown parameters of the nonlinearities. This allows us to apply well established tools from linear optimal control theory to the controlled stage of the DPO. The small gain theorem is used to derive a condition for the robustness of the closed loop system.

Robust adaptive boundary control of semilinear PDE systems using a dyadic controller

“…In particular, the choice of the filter H(s) is arbitrary within the bounds of the small-gain condition. Some initial results on using linear quadratic regulator to design the control law have been presented by the authors in their previous work 26 , but further development is needed along those lines. Other avenues for extending this work include further experimental validation and demonstration, understanding how to use output feedback in place of full-state feedback, and determining fundamental restrictions, if any, on the ability of this technique to accommodate unstable linear dynamics.…”

“…The control design method can be used for systems of the formẇ = g w + (t, w), where g need not be stable. An extension of the DAC to such systems is examined rigorously in our previous work 26 and also illustrated (without a formal proof) via an example in Section 6.…”

“…On the one hand, it opens up the possibility of using well-understood tools from linear control such as the linear quadratic regulator as demonstrated in our previous work. 26 This contrasts with the existing approaches to solving optimal control problems for semilinear systems. 10,24 On the other hand, it allows us to guarantee robustness rigorously using the small gain theorem as shown in this paper.…”

Transient and Asymptotic Properties of Robust Adaptive Controllers in the Presence of Non-Coercive Lyapunov Functions

“…In the next section, we will derive an observer for estimating the states; for now, we use ( 9) and (10) to investigate tracking. If we could choose σ(t) = r(t) − y p (t), we would get that the tracking error y(t) − r(t) = y h (t) − σ(t); therefore, σ(t) can serve as the reference signal for y h (t).…”

“…The operators A, B and C are the state, control, and output operators, respectively. Coercive Lyapunov functions feature prominently in the derivation of adaptive laws, and help ensure appropriate bounds on the tracking error [9,10,11]. Our objective is to determine how the guaranteed bounds change in the absence of a coercive solution to the usual, unmodified Lyapunov equation.…”