A System Level Approach to Discrete-Time Nonlinear Systems

Abstract

In this paper, we will show that there is a universal connection between the closed-loop and the corresponding realizing controller in nonlinear discrete-time systems: Given an achievable stable closed-loop, we can follow a systematic procedure to construct an internally stable causal controller that realizes the desired closed-loop. In the linear system case, this relationship has been used as a key result in the recently developed System Level Synthesis (SLS), and this paper shows how this relationship finds its analogue in the general nonlinear discrete-time system case. Necessary and sufficient conditions are presented that characterize the entire space of closedloops that are achievable by some causal controller for a given system. Furthermore, we will show that constructing the same causal controller from maps that are not achievable closed-loops, still can stabilize the nonlinear system if they approximate the feasibility conditions well enough. Finally, we will discuss how this method opens up new ways towards robust nonlinear controller synthesis, by exploring two direct applications of this approach: design of trajectory tracking controllers for nonlinear systems using linear SLS controllers, and a method to stably "blend" multiple linear SLS controllers into one nonlinear controller that improves closed-loop performance. The companion paper [1] applies the latter approach to large-scale systems with actuator saturation and safety constraints.