Formal Methods for Control Synthesis: An Optimization Perspective

Abstract

In control theory, complicated dynamics such as systems of (nonlinear) differential equations are controlled mostly to achieve stability. This fundamental property, which can be with respect to a desired operating point or a prescribed trajectory, is often linked with optimality, which requires minimizing a certain cost along the trajectories of a stable system. In formal verification (model checking), simple systems, such as finite-state transition graphs that model computer programs or digital circuits, are checked against rich specifications given as formulas of temporal logics. The formal synthesis problem, in which the goal is to synthesize or control a finite system from a temporal logic specification, has recently received increased interest. In this article, we review some recent results on the connection between optimal control and formal synthesis. Specifically, we focus on the following problem: Given a cost and a correctness temporal logic specification for a dynamical system, generate an optimal control strategy that satisfies the specification. We first provide a short overview of automata-based methods, in which the dynamics of the system are mapped to a finite abstraction that is then controlled using an automaton corresponding to the specification. We then provide a detailed overview of a class of methods that rely on mapping the specification and the dynamics to constraints of an optimization problem. We discuss advantages and limitations of these two types of approaches and suggest directions for future research.

Control Barrier Functions for Multi-Agent Systems Under Conflicting Local Signal Temporal Logic Tasks

“…are given by the quadratic program (10), which is additionally subject to (11) andû i ∈ U i , and wherex i,c is the stacked vector of the states of the agents in V i ∪ j∈Ni(t) ∪ l∈Li,j (t) V l j . Note that CR i (t) introduces discontinuities that, however, do not affect the existence of solutions since CR i (t) is piecewise continuous.…”

“…where r i is maximized, while x i (t) / ∈ O i for all t ≥ 0. Proof: The quadratic program (10), additionally subject to (11)…”

“…Signal temporal logic (STL) [6], on the other hand, allows to impose tasks with strict deadlines and offers a closer connection to the physical system by the introduction of robust semantics [7], [8], hence offering the benefit of not necessarily requiring an abstraction of the system. Control methods for STL then consider discrete-time systems and result, even for single-agent systems, in computationally costly mixed integer linear programs [9]- [11]. We recently proposed an alternative approach for continuous-time systems by using time-varying feedback control strategies [12]- [14], which are computationally efficient and inherently robust due its feedback nature.…”

Formal Controller Synthesis via Genetic Programming

“…Nevertheless, in recent years, tools have been developed for formal control synthesis for hybrid systems with temporal logic specifications. Most of these methods fit into one of three main paradigms: synthesis by means of 1) finite (bi-)simulation abstractions [2,3], 2) control Lyapunov and/or barrier functions [4,5] and 3) online optimization-based methods [6]. The first paradigm relies on discretization of the state space and therefore suffers from the curse of dimensionality, resulting in controllers taking the form of enormous look-up tables, which complicates their implementation [7].…”

“…Optimization-based methods typically employ model-predictive control to optimize a cost function related to the temporal logic specification, see e.g. [26,27] and the survey [6].…”