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].
In this paper we present a new approach of using input-output linearization to control a single input, single output, input-affine nonlinear non-minimum phase system. We will show that, if the linearized system is stabilizable, we can redefine the output of the system such that the input-output linearized system is locally asymptotically stable. Furthermore we develop an LQR technique for designing the redefined output, which assures stabilization of the zerodynamics. Simulations of a physical system show that the resulting controller, which in a way fuses LQR techniques with input-output linearization, outperforms a regular LQR feedback controller and demonstrates a big region of attraction. The presented technique can be used to regulate the system around an equilibrium and to achieve tracking for certain trajectories. Conditions are established that allow the asymptotic regulation and tracking of desired trajectories for the original output. We will demonstrate the control design on a two-dimensional Segway model.
We show that given a desired closed-loop response for a system, there exists an affine subspace of controllers that achieve this response. By leveraging the existence of this subspace, we are able to separate controller design from closed-loop design by first synthesizing the desired closed-loop response and then synthesizing a controller that achieves the desired response. This is a useful extension to the recently introduced System Level Synthesis framework, in which the controller and closed-loop response are jointly synthesized and we cannot enforce controller-specific constraints without subjecting the closed-loop map to the same constraints. We demonstrate the importance of separating controller design from closed-loop design with an example in which communication delay and locality constraints cause standard SLS to be infeasible. Using our new two-step procedure, we are able to synthesize a controller that obeys the constraints while only incurring a 3% increase in LQR cost compared to the optimal LQR controller.
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