Control synthesis from temporal logic specifications has gained popularity in recent years. In this paper, we use a model predictive approach to control discrete time linear systems with additive bounded disturbances subject to constraints given as formulas of signal temporal logic (STL). We introduce a (conservative) computationally efficient framework to synthesize control strategies based on mixed integer programs. The designed controllers satisfy the temporal logic requirements, are robust to all possible realizations of the disturbances, and optimal with respect to a cost function. In case the temporal logic constraint is infeasible, the controller satisfies a relaxed, minimally violating constraint. An illustrative case study is included.
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.
The polytope containment problem is deciding whether a polytope is a contained within another polytope. This problem is rooted in computational convexity, and arises in applications such as verification and control of dynamical systems. The complexity heavily depends on how the polytopes are represented. Describing polytopes by their hyperplanes (H-polytopes) is a popular representation. In many applications we use affine transformations of H-polytopes, which we refer to as AH-polytopes. Zonotopes, orthogonal projections of Hpolytopes, and convex hulls/Minkowski sums of multiple Hpolytopes can be efficiently represented as AH-polytopes. While there exists efficient necessary and sufficient conditions for AH-polytope in H-polytope containment, the case of AH-polytope in AH-polytope is known to be NPcomplete. In this paper, we provide a sufficient condition for this problem that is cast as a linear program with size that grows linearly with the number of hyperplanes of each polytope. Special cases on zonotopes, Minkowski sums, convex hulls, and disjunctions of H-polytopes are studied. These efficient encodings enable us to designate certain components of polytopes as decision variables, and incorporate them into a convex optimization problem. We present examples on the zonotope containment problem, polytopic Hausdorff distances, zonotope order reduction, inner approximations of orthogonal projections, and demonstrate the usefulness of our results on formal controller verification and synthesis for hybrid systems.
Piecewise affine (PWA) systems are widely used to model highly nonlinear behaviors such as contact dynamics in robot locomotion and manipulation. Existing control techniques for PWA systems have computational drawbacks, both in offline design and online implementation. In this paper, we introduce a method to obtain feedback control policies and a corresponding set of admissible initial conditions for discrete-time PWA systems such that all the closed-loop trajectories reach a goal polytope, while a cost function is optimized. The idea is conceptually similar to LQR-trees [1], which consists of 3 steps:(1) open-loop trajectory optimization, (2) feedback control for computation of "funnels" of states around trajectories, and (3) repeating (1) and (2) in a way that the funnels are grown backward from the goal in a tree fashion and fill the state-space as much as possible. We show PWA dynamics can be exploited to combine step (1) and (2) into a single step that is tackled using mixed-integer convex programming, which makes the method suitable for dealing with hard constraints. Illustrative examples on contact-based dynamics are presented.benefit of capturing contact dynamics [8,9]. PWA models are also popular in traffic networks [10] and gene circuits [11].Controlling PWA systems is difficult since the controller has to determine both the temporal order of modes and the inputs applied at each one. Completeness is important -not finding a solution when one exists is undesirable. Given an initial condition and a goal, the (optimal) trajectory that steers the state to the goal while respecting the dynamics and state/control constraints can be obtained using mixed-integer convex programming (MICP). Time is discretized to obtain a finite number of decision variables. Given a discrete-time PWA model and a fixed time horizon (steps required to get into the goal), MICP approaches are sound and complete -they find optimal solutions if they exist. However, they come at a large computational price. Their unreliable computation time, even for obtaining a feasible solution instead of the optimal one, hinders online implementation for robotic tasks with fast dynamics. While there is a great deal of research on improving the runtime of MICP solvers, they are still orders of magnitude too slow for most robot control problems.An alternative is to move much of the computational burden to offline phase so the real-time implantation is a lookup table of simple control laws. However, existing approaches are not efficient even for relatively small systems. (Nondeterministic) finite-state abstractions [12] require state/control discretization, which scales poorly in high dimensions. Synthesizing PWA control laws corresponding to stabilizing piecewise quadratic (PWQ) Lyapunov functions was studied in [13,14]. But the state-control partition producing the control laws was assumed to be the same as those of PWA dynamics, which is very restrictive and the method may fail while other solutions exist [14].Multi-parametric programming [15] for mo...
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