Sampling-Based Polytopic Trees for Approximate Optimal Control of Piecewise Affine Systems

Abstract

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...

Linear Encodings for Polytope Containment Problems

“…We show applications of our results on several problems for which existing methods are not satisfactory. We provide alternative methods for computing Hausdorff distance between polytopes, zonotope order reduction [13], inner approximation of orthogonal projections, and built upon our prior work in [14] to compute explicit model predictive control (MPC) schemes for constrained hybrid systems.…”

“…In this section, we study a particular application of our results to formal synthesis of controllers for hybrid systems. The method is based on our framework in [14], where polytope-to-polytope control strategies were used to design feedback strategies for piecewise affine systems. The task is reaching a goal region, which itself is given as a union of polytopes in the state-space, while respecting state and control constraints.…”

“…We can also consider mixed-integer formulations to encode hybrid relations between (A t , B t , c t ) and (x t , u t ), but the details are omitted here. The full algorithm and its theoretical guarantees are reported in [14]. Here we only provide the essential details related to the contributions of this paper.…”

“…Using the results of this paper, we improve a crucial computational part of the algorithm in [14]. When adding a branch to the tree, we design a polytopic trajectory such that the final polytope is contained within one of the polytopes in the tree, i.e.…”

“…an instance of polytope containment problem. In [14], we explored efforts to compute the Hpolytope form of these AH-polytopes, either using Fourier-Motzkin elimination which was very slow and numerically unstable for our applications, or approximating the transformation matrix by a leftinvertible one which also caused numerical issues as fine approximations led to ill-conditioned matrices. Here we use our results on AH-polytope in AHpolytope containment, in particular zonotope containment, to present an alternative approach that does not require H-polytope forms of the polytopes in the tree.…”