A System Level Approach to Discrete-Time Nonlinear Systems

Abstract

We will show that there is a universal connection between the achievable closed-loop dynamics and the corresponding feedback controller that produces it. This connection shows promise to lead to new methods for robust nonlinear control in discrete-time. We will show that, given a causal nonlinear discrete-time system and controller, the resulting closed-loop is a solution to a nonlinear operator equation. Conversely, any causal solution to the nonlinear operator equation is a closedloop that can be achieved by some causal controller. Moreover, solutions can be substituted into a simple dynamic controller structure, which we will refer to as a system level controller, to obtain an implementation of the unique corresponding feedback controller. System level controllers could be an attractive approach for robust nonlinear control, as we will show that even when they are parametrized with approximate solutions to the operator equation, they can still produce robustly stable closed loops. We will provide theoretical results that state how grade of approximation and robust stability of the closed loop are related. Additionally, we will explore some first applications of our results. Using the cart-pole system as an illustrative example, we will derive how to design robust discrete-time trajectory tracking controllers for continuous-time nonlinear systems. Secondly, we will introduce a particular class of system level controller that shows to be particularly useful for linear systems with actuator saturation and state constraints; The special structure of the controller allows for simple stability and performance analysis of the closed-loop in presence of disturbances. The structure also offers simple ways to do antiwindup compensation, and provides a new nonlinear approach to the constrained LQG problem. A particular application to large-scale systems with actuator saturation and safety constraints is presented in our companion paper [1].

Achieving Performance and Safety in Large Scale Systems with Saturation using a Nonlinear System Level Synthesis Approach

“…Theorem III.3 (Characterization of CLMs [9]). A composite operator Ψ = (Ψ x , Ψ u ) ∈ C( n , n × m ) are realizable CLMs of the open loop (2a) if and only if they satisfy the operator equation…”

“…Theorem III.5 (Internal Stability of Closed Loop [9]). If f is uniformly continuous and the operator (Ψ x , Ψ u ) is…”

“…III.4) can implement arbitrary CLMs for nonlinear systems of the form (1). The results in [9] motivate a new approach for nonlinear control synthesis: Searching for stable operators Ψ x , Ψ u that satisfy (3) and constructing a corresponding system level controller SL(Ψ x , Ψ u ) by (Def. III.4).…”

“…Our work is based on [9], which describes the systemlevel characterization of the closed loops of general nonlinear discrete-time systems. Moreover, [9] introduces a simple universal control structure, called a system level controller, that has the capacity to stabilize any nonlinear system if parametrized with the according closed loop maps. In this paper, we will show that just using a very special case of the framework presented in [9] provides new promising tools for control design.…”

“…Moreover, [9] introduces a simple universal control structure, called a system level controller, that has the capacity to stabilize any nonlinear system if parametrized with the according closed loop maps. In this paper, we will show that just using a very special case of the framework presented in [9] provides new promising tools for control design. In particular, we will illustrate how a simple projection nonlinearity can become a powerful tool for solving the problems described above.…”