Robust Linear Static Anti-Windup With Probabilistic Certificates

Abstract

Abstract-In this paper, we address robust static anti-windup compensator design and performance analysis for saturated linear closed loops in the presence of nonlinear probabilistic parameter uncertainties via randomized techniques. The proposed static anti-windup analysis and robust performance synthesis correspond to several optimization goals, ranging from minimization of the nonlinear input/output gain to maximization of the stability region or maximization of the domain of attraction. We also introduce a novel paradigm accounting for uncertainties in the energy of the disturbance inputs.Due to the special structure of linear static anti-windup design, wherein the design variables are decoupled from the Lyapunov certificates, we introduce a significant extension, called scenario with certificates (SwC), of the so-called scenario approach for uncertain optimization problems. This extension is of independent interest for similar robust synthesis problems involving parameter-dependent Lyapunov functions. We demonstrate that the scenario with certificates robust design formulation is appealing because it provides a way to implicitly design the parameter-dependent Lyapunov functions and to remove restrictive assumptions about convexity with respect to the uncertain parameters. Subsequently, to reduce the computational cost, we present a sequential randomized algorithm for iteratively solving this problem. The obtained results are illustrated by numerical examples.

Algorithms for OptimalACPower Flow in the Presence of Renewable Sources

“…In this case, although we cannot assure that the solution obtained satisfies the constraints, the risk of failure can be bounded a-priori. This risk is a function of the number of samples used and can be made as small as desired; see [45], [46]. An important feature of scenario approach is that the underlying distribution of the uncertainty does not need to be known.…”

“…In the case where some of the variables depend on the uncertainty, as it is the case in power networks, one needs an approach that can deal with so-called certificates. This problem was addressed in [45], [46] where the so-called Scenario with Certificates (SwC) approach was developed. Consider the robust optimization problem with certificates which is a generalization of the problem considered in this chapter…”

“…SwC-AC-OPF Design Procedure i) Given probabilistic levels ε, and β compute N SwC according to [45], [46] (see note 5 on randomized algorithms for robust optimization).…”

“…. , Q G, [N ] , W We remark that the results in [45], [46] provides an a-priori guarantee that the SwC-AC-OPF design procedure is such that the probabilistic constraints of the Chance-Constrained AC-OPF (22) are satisfied with very high confidence. Moreover, although the number of variables in SwC-AC-OPF is proportional to the number of samples N SwC (which can be high if one requires an extremely small risk of failure) the structure of the resulting optimization problem is amenable to parallelization and, hence, it can be efficiently solved even for large networks.…”

Robust Static Output Feedback Design with Deterministic and Probabilistic Certificates

“…variables whose existence is necessary to prove the existence of a SOF, but that are not involved in the controller construction. This paves the way to the use of recent results on probabilistic robust design, based on the so-called scenario-with-certificates (SwC) approach [12]. This approach represents one of the more recent findings in the area of randomized methods for systems and control [4,20], emerged in the last decade to synergize with the standard deterministic methods for control of systems with uncertainty.…”