Robust Static Output Feedback Design with Deterministic and Probabilistic Certificates

Abstract

Static output feedback design for linear plants is well known to be a challenging non-convex problem. The presence of plant uncertainty makes this challenge even harder. In this chapter, we propose a new BMI formulation with Svariables which includes an interesting link between state feedback, output injection, state injection and static output feedback gains in a unified framework. Based on this formulation, the robust design problem is suitably addressed by iterative optimization procedures with either deterministic or probabilistic viewpoints exploiting the fact that Lyapunov certificates are separated from the control gain design variables. The deterministic approach is for affine polytopic systems. The probabilistic approach requires no assumption on the uncertain system and is based on the Scenario with Certificates (SwC) method which was recently proposed to address certain static anti-windup design problems. Numerical results illustrate the effectiveness of the approach in both deterministic and stochastic cases.

Static Output Feedback Stabilization of Discrete-Time Linear Systems With Stochastic Dynamics Determined by an Independent Identically Distributed Process

“…This paper is concerned with the former theme (i.e., output feedback synthesis). In the case of deterministic systems, complete linearization of the inequality condition for static output feedback stabilization controller synthesis is known to be difficult [7]. Hence, a similar difficulty finally remains also in the present arguments for stochastic systems.…”

“…Since C is generally not a nonsingular square matrix in the present output feedback control problem, we cannot apply the change of variables to the product KCX in the above inequality. This is the issue in the output feedback synthesis, which does not occur in the state feedback synthesis (see, e.g., [7] for the associated discussions on deterministic systems). Even if we try to solve the bilinear matrix inequality (BMI) (20) iteratively by fixing a part of decision variables (i.e., K or X) at each iteration, the way of selecting the initial value for the variables becomes a problem; we have no reasonable way of the selection for K and X in (20).…”

“…From this lemma, it is meaningless in (27) to consider F that does not satisfy (9), which in turn implies that selecting a stabilizing state feedback gain as the variable F is reasonable. Although such selection of F does not necessarily lead to a globally optimal (or near) solution, a similar idea is empirically known to be effective in the case of deterministic systems [7]. Hence, this paper also exploits this idea and proposes the following two-step synthesis method for static output feedback stabilization controller synthesis.…”