Time-varying Projected Dynamical Systems with Applications to Feedback Optimization of Power Systems

Abstract

This paper is concerned with the study of continuous-time, non-smooth dynamical systems which arise in the context of time-varying non-convex optimization problems, as for example the feedback-based optimization of power systems. We generalize the notion of projected dynamical systems to time-varying, possibly non-regular, domains and derive conditions for the existence of so-called Krasovskii solutions. The key insight is that for trajectories to exist, informally, the time-varying domain can only contract at a bounded rate whereas it may expand discontinuously. This condition is met, in particular, by feasible sets delimited via piecewise differentiable functions under appropriate constraint qualifications. To illustrate the necessity and usefulness of such a general framework, we consider a simple yet insightful power system example, and we discuss the implications of the proposed conditions for the design of feedback optimization schemes.Index Terms-Non-smooth analysis, nonlinear dynamical systems, power systems. arXiv:1809.07288v2 [math.OC]

Temporal viability regulation for control affine systems with applications to mobile vehicle coordination under time-varying motion constraints

“…A detailed calculation of the contingent cone and the proof is omitted here. The case with active constraint (the set under (8)) has also been discussed in [10]. If a subset of the inequality constraint of the vector function g becomes active, then the formula (8) in Lemma 1 applies to only the subset of active inequality constraints.…”

“…Remark 1: A non-empty temporal contingent cone T t F (x) for all time t is a necessary condition to ensure the existence of the control input u(t) associated with the time-varying vector field f (x(t), u(t), t). A sufficient condition to guarantee non-empty T t F (x) along the solution x(t) and time t is the forward Lipschitz continuity of the set F (t) with respect to time t (see [10,Theorem 1]). According to [10,Proposition 4], a sufficient condition to ensure the forward Lipschitz continuity of the set F (t) is (i) the gradient vector ∇ x g(x(t), t) has full rank, and (ii) the time-varying function g(x(t, t)) is Lipschitz continuous in t. In this paper we may further suppose g(x(t), t) is a C 1 function of both the state x(t) and time t, 3 which automatically guarantees the second condition.…”

“…We remark that available results in the literature on controlled invariance and viability regulation mostly deal with time-invariant constraints or only state-dependent viable sets. Time-varying constraints have attracted some recent attention for some particular control systems, such as [4], [10]. In this paper, we aim to develop general theories for time-varying viability regulation for general dynamical control systems, and in particular for control affine systems.…”

“…The main contributions of this paper include a development of temporal viability theory (with motivations of temporal contingent cone in a recent paper [10]), temporal viability regulation, and control law design for controlled temporal invariance of control affine systems. To illustrate their applications, we will consider two typical examples in mobile vehicle coordination control with time-varying motion constraints (in terms of distance and visibility maintenance), and justify their real-time performance guarantees with the developed temporal viability control.…”

A Feedback-Based Regularized Primal-Dual Gradient Method for Time-Varying Nonconvex Optimization

“…It can be seen that under the initial condition (x 1 (0), x 2 (0)) = (0, 0), (4.5) admits the solution x 1 (t) = t, x 2 (t) = 0, but (4.6a) and (4.6b) do not have solutions. In [22], the authors introduced a formulation similar to (4.6a) based on the notion of temporal tangent cones, which is a generalization of tangent cones in time-varying situations. Next we study the tracking performance of the system of differential inclusions (4.4), and present the following theorem which is the continous-time counterpart of Theorem 3.2.…”