Optimal control of uncertain stochastic systems subject to total variation distance uncertainty
Date
2012Source
SIAM Journal on Control and OptimizationVolume
50Issue
5Pages
2683-2725Google Scholar check
Keyword(s):
Metadata
Show full item recordAbstract
This paper is concerned with optimization of uncertain stochastic systems, in which uncertainty is described by a total variation distance constraint between the measures induced by the uncertain systems and the measure induced by the nominal system, while the payoff is a linear functional of the uncertain measure. Robustness at the abstract setting is formulated as a minimax game, in which the control seeks to minimize the payoff over the admissible controls while the uncertainty aims at maximizing it over the total variation distance constraint. It is shown that the maximizing measure in the total variation distance constraint exists, while the resulting payoff is a linear combination of L 1 and L ∞ norms. Further, the maximizing measure is characterized by a linear combination of a tilted measure and the nominal measure, giving rise to a payoff which is a nonlinear functional on the space of measures to be minimized over the admissible controls. The abstract formulation and results are subsequently applied to continuous-time uncertain stochastic controlled systems, in which the control seeks to minimize the payoff while the uncertainty aims to maximize it over the total variation distance constraint. The minimization over the admissible controls of the nonlinear functional payoff is addressed by developing a generalized principle of optimality or dynamic programming equation satisfied by the value function. Subsequently, it is proved that the value function satisfies a Hamilton-Jacobi-Bellman (HJB) equation. It is also shown that the value function is also a viscosity solution of the HJB equation. Finally, the linear quadratic case is studied, and it is shown that the infinity norm of a quadratic payoff is well defined and finite. Throughout the paper the formulation and conclusions are related to previous work found in the literature. © 2012 Society for Industrial and Applied Mathematics.
Collections
Cite as
Related items
Showing items related by title, author, creator and subject.
-
Article
Multiple model adaptive control with mixing
Kuipers, M.; Ioannou, Petros A. (2010)Despite the remarkable theoretical accomplishments and successful applications of adaptive control, the field is not sufficiently mature to solve challenging control problems where strict performance and robustness guarantees ...
-
Article
Robust adaptive attenuation of unknown periodic disturbances in uncertain multi-input multi-output systems
Jafari, S.; Ioannou, Petros A. (2016)In high-performance high-accuracy systems, the attenuation of vibrational disturbances is essential. In this paper, we design and analyze a robust output-feedback adaptive control scheme to attenuate noise-corrupted ...
-
Conference Object
Robust adaptive sliding control of linearizable systems
Xu, H.; Mirmirani, M.; Ioannou, Petros A.; Boussalis, Helen R. (2001)A switching adaptive control algorithm based on a sliding mode method is proposed for a class of single-input, single-output nonlinear systems with unknown dynamics. The plant is assumed to be linear-in-the-control input ...