Symbolic Optimal Control

Abstract

We present novel results on the solution of a class of leavable, undiscounted optimal control problems in the minimax sense for nonlinear, continuous-state, discrete-time plants. The problem class includes entry-(exit-)time problems as well as minimum time, pursuit-evasion and reach-avoid games as special cases. We utilize auxiliary optimal control problems ("abstractions") to compute both upper bounds of the value function, i.e., of the achievable closed-loop performance, and symbolic feedback controllers realizing those bounds. The abstractions are obtained from discretizing the problem data, and we prove that the computed bounds and the performance of the symbolic controllers converge to the value function as the discretization parameters approach zero. In particular, if the optimal control problem is solvable on some compact subset of the state space, and if the discretization parameters are sufficiently small, then we obtain a symbolic feedback controller solving the problem on that subset. These results do not assume the continuity of the value function or any problem data, and they fully apply in the presence of hard state and control constraints.

Least-violating symbolic controller synthesis for safety, reachability and attractivity specifications

“…It is important to mention that the dynamic programming fixed point (3.5), (3.6) can be seen as a special case of the one considered in the work [30,39]. Some of the results below can be obtained using results of [30].…”

“…Symbolic control has been used to tackle a number of optimal control problems involving either cumulative costs such as minimal-time [21,14], entry-time problems [9], finite [22] or infinite [17] horizon problems, or average costs [32]. The research that is the most closely related to the present work are [7] and [30,39]. In [7], the authors study dynamic programming formulations that are similar to those characterizing safety and uniform reachability controllability measures.…”

“…However, in this work, the characterization of the level sets of the value function in terms of controllability measures is not established, and the synthesis of controllers is not discussed. In [30] and [39], the authors study a large class of optimal control problems, which can capture our dynamic programming formulation to synthesize least-violating controllers for uniform reachability, but not for safety. Finally, uniform attractivity specifications are not covered by these papers.…”

Guaranteed Optimal Reachability Control of Reaction-Diffusion Equations Using One-Sided Lipschitz Constants and Model Reduction

“…In [16], the authors focus on (periodically) sampled systems, and perform reachability analysis using convex polytopes as state set representations. In [27,37,19,46,47], the authors construct an over-approximation of the set of trajectories using a growth bound (bounding the distance of neighboring trajectories) exploiting the notion of one-sided Lipschitz constant (also called "logarithmic norm" or "matrix norm"). The notion of "one-sided Lipschitz (OSL) constant" has been introduced independently by Dahlquist [17] and Lozinskii [36] in order to derive error bounds in initial value problems (see survey in [51]).…”

“…We used ourselves OSL constants in the context of symbolic optimal control in [14]. The main difference with previous work [27,37,19,46,47] is that our method makes use of explicit Euler's algorithm for ODE integration (cf. [32,33]) instead of sophisticated algorithms such as Lohner's algorithm [27] or interval Taylor series methods [44].…”