Controller Synthesis for Discrete-Time Polynomial Systems via Occupation Measures

Abstract

In this paper, we design nonlinear state feedback controllers for discrete-time polynomial dynamical systems via the occupation measure approach. We propose the discrete-time controlled Liouville equation, and use it to formulate the controller synthesis problem as an infinite-dimensional linear programming problem on measures, which is then relaxed as finitedimensional semidefinite programming problems on moments of measures and their duals on sums-of-squares polynomials. Nonlinear controllers can be extracted from the solutions to the relaxed problems. The advantage of the occupation measure approach is that we solve convex problems instead of generally non-convex problems, and the computational complexity is polynomial in the state and input dimensions, and hence the approach is more scalable. In addition, we show that the approach can be applied to over-approximating the backward reachable set of discrete-time autonomous polynomial systems and the controllable set of discrete-time polynomial systems under known state feedback control laws. We illustrate our approach on several dynamical systems.

Learning control for polynomial systems using sum of squares relaxations

“…In this case, the matrix X 1,T contains the derivatives of the states at the sampling times when the measurements are taken (see [6,Remark 2]). In this paper, we will focus on continuous-time polynomial systems because this allows us to adopt the tools from [7], [10], [11], while for discrete-time polynomial systems less results are available [12], [13].…”

Learning control for polynomial systems using sum of squares relaxations

“…In this case, the matrix X 1,T contains the derivatives of the states at the sampling times when the measurements are taken (see [6,Remark 2]). In this paper we will focus on continuous-time polynomial systems because this allows us to adopt the tools from [7], [11], [12] for this class of systems, while for discrete-time polynomial systems less results are available [13], [14].…”

Controller Synthesis for Discrete-time Hybrid Polynomial Systems via Occupation Measures

“…The procedure of extracting a controller for mode i from the moments of the measure ν i is the same as in Section IV.C. of [48].…”

“…In this example, we linearize the dynamics and work with the PWA system as in [22]. (We could have approximated the system with higher order polynomials, synthesized the controller, and run on the real system as in Example E in [48], but we are more interested in the comparison with the traditional model predictive control (MPC) approach. )…”