A Sums-of-Squares Extension of Policy Iterations

In order to address the imprecision often introduced by widening operators in static analysis, policy iteration based on min-computations amounts to considering the characterization of reachable value set of a program as an iterative computation of policies, starting from a post-fixpoint. Computing each policy and the associated invariant relies on a sequence of numerical optimizations. While the early research efforts relied on linear programming (LP) to address linear properties of linear programs, the current state of the art is still limited to the analysis of linear programs with at most quadratic invariants, relying on semidefinite programming (SDP) solvers to compute policies, and LP solvers to refine invariants. We propose here to extend the class of programs considered through the use of Sums-of-Squares (SOS) based optimization. Our approach enables the precise analysis of switched systems with polynomial updates and guards. The analysis presented has been implemented in Matlab and applied on existing programs coming from the system control literature, improving both the range of analyzable systems and the precision of previously handled ones.Comment: 29 pages, 4 figure

A new solution approach to polynomial LPV system analysis and synthesis

Based on sum-of-squares (SOS) decomposition, we propose a new solution approach for polynomial LPV system analysis and control synthesis problems. Instead of solving matrix variables over a positive definite cone, the SOS approach tries to find a suitable decomposition to verify the positiveness of given polynomials. The complexity of the SOS-based numerical method is polynomial of the problem size. This approach also leads to more accurate solutions to LPV systems than most existing relaxation methods. Several examples have been used to demonstrate benefits of the SOS-based solution approach

Strong Stationarity Conditions for Optimal Control of Hybrid Systems

We present necessary and sufficient optimality conditions for finite time optimal control problems for a class of hybrid systems described by linear complementarity models. Although these optimal control problems are difficult in general due to the presence of complementarity constraints, we provide a set of structural assumptions ensuring that the tangent cone of the constraints possesses geometric regularity properties. These imply that the classical Karush-Kuhn-Tucker conditions of nonlinear programming theory are both necessary and sufficient for local optimality, which is not the case for general mathematical programs with complementarity constraints. We also present sufficient conditions for global optimality. We proceed to show that the dynamics of every continuous piecewise affine system can be written as the optimizer of a mathematical program which results in a linear complementarity model satisfying our structural assumptions. Hence, our stationarity results apply to a large class of hybrid systems with piecewise affine dynamics. We present simulation results showing the substantial benefits possible from using a nonlinear programming approach to the optimal control problem with complementarity constraints instead of a more traditional mixed-integer formulation.Comment: 30 pages, 4 figure