Safe Non-Stochastic Control of Linear Dynamical Systems

Abstract

We study the problem of safe control of linear dynamical systems corrupted with non-stochastic noise, and provide an algorithm that guarantees (i) zero constraint violation of convex time-varying constraints, and (ii) bounded dynamic regret, i.e., bounded suboptimality against an optimal clairvoyant controller that knows the future noise a priori. The constraints bound the values of the state and of the control input such as to ensure collision avoidance and bounded control effort. We are motivated by the future of autonomy where robots will safely perform complex tasks despite real- world unpredictable disturbances such as wind and wake disturbances. To develop the algorithm, we capture our problem as a sequential game between a linear feedback controller and an adversary, assuming a known upper bound on the noise's magnitude. Particularly, at each step <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t=1, \ldots, T$</tex> , first the controller chooses a linear feedback control gain <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K_{t}\in \mathcal{K}_{t}$</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{K}_{t}$</tex> is constructed such that it guarantees that the safety constraints will be satisfied; then, the adversary reveals the current noise <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$w_{t}$</tex> and the controller suffers a loss <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$f_{t}(K_{t})$</tex> - e.g., <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$f_{t}$</tex> represents the system's tracking error at upon the realization of the noise. The controller aims to minimize its cumulative loss, despite knowing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$w_{t}$</tex> only after <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K_{t}$</tex> has been chosen. We validate our algorithm in simulated scenarios of safe control of linear dynamical systems in the presence of bounded noise.