Measures and LMIs for optimal control of piecewise-affine systems

Abstract

This paper considers the class of deterministic continuous-time optimal control problems (OCPs) with piecewise-affine (PWA) vector field, polynomial Lagrangian and semialgebraic input and state constraints. The OCP is first relaxed as an infinite-dimensional linear program (LP) over a space of occupation measures. This LP, a particular instance of the generalized moment problem, is then approached by an asymptotically converging hierarchy of linear matrix inequality (LMI) relaxations. The relaxed dual of the original LP returns a polynomial approximation of the value function that solves the Hamilton-Jacobi-Bellman (HJB) equation of the OCP. Based on this polynomial approximation, a suboptimal policy is developed to construct a state feedback in a sample-and-hold manner. The results show that the suboptimal policy succeeds in providing a stabilizing suboptimal state feedback law that drives the system relatively close to the optimal trajectories and respects the given constraints.

Measures and LMIs for Lateral F-16 MRAC Validation

“…until they reach the final state belonging to a set x(T ) ∈ {J ≤ 3 × 10 −3 }, then the control law is validated. The main results of this section rely heavily on theoretical background discussed extensively in [9,11,10]. The procedure consists of writing our validation problem as a piecewise polynomial dynamical optimization problem…”

“…Since x r (t) is both linear and autonomous, we can go directly to (17) by using (9), piecewise polynomial dynamical optimization [10] to global occupation measure µ…”

Measures and LMIs for Lateral F-16 MRAC Validation

“…The main results of this section rely heavily on theoretical background discussed extensively in [9,11,10]. The procedure consists of writing our validation problem as a piecewise polynomial dynamical optimization problem…”

Occupation measure methods for modelling and analysis of biological hybrid systems

“…From this infinite-dimensional LP, Lasserre et al (2008) defines hierarchies of Linear Matrix Inequalities (LMI) relaxations, to synthesise a sequence of polynomial controls converging to the solutions of the optimal control problem. In Abdalmoaty et al (2013) the authors propose an extension to piecewise affine systems. Our underlying idea of constructing a suboptimal control with an iterative algorithm is similar to (Abdalmoaty et al, 2013, Section 4).…”