Robustness of Finite State Automata

The classical robust control deals with systems which can be approximated by nite order linear time-invariant (LTI) models, uses integral constraints, such as induced gain bounds, to assess robustness with respect to the error of such approximation, and employs H-In nity optimization to design robust linear controllers. In this paper, a parallel approach is developed, in which nite state stochastic automata play the role of LTI models. Analogs of the KalmanYakubovich-Popov Lemma, the S-procedure losslessness theorem, and H-In nity design are derived. Introduction Robustness analysis and optimization is a major source of eÆcient design tools for the modern control engineer. The classical robust control deals with systems which can be approximated by LTI models. The di ernece between such approximations and the true system dynamics is described by integral constraints, such as induced gain bounds or Integral Quadratic Constraints [1]. Constructively veri able conditions of stability and performance, such as the small gain theorem, are used to assess stability and performance of systems de ned by nominal LTI dynamics and integral constraints. Ultimately, the task of robust LTI feedback design is reduced to induced gain minimization, such as H-In nity optimization, which employs extensively quadratic Lyapunov functions. While being the dominant tool for computer-aided design and analysis of systems modeled by near-linear di erential equations, this framework fails to provide adequate treatment in the case of hybrid systems, i.e. systems which combine continuous and discrete state dynamics. A major objective of the paper is creation of an alternative robust control framework in which nite state stochastic automata serve as a basic system model. Systems under consideration are represented as interconnections of \nominal" controlled nite state automata and the \uncertain feedback" systems described by integral constraints representing modeling error. Lyapunov functions are used for analysis and design. The theorems presented in this paper are quite elementary, and can be viewed as simpli ed versions of the standard results of dynamic programming [2]. However, they highlight a potentially powerful framework for nonlinear 1 to appear in the Proceedings of Mohammed Dahleh memorial symphosium. Email: [email protected]. This work was supported by NSF, AFOSR, and DARPA feedback design. In this framework, one has to start with nding a reduced model of the original system. 1 System Models In this section, basic principles of system modeling and design using nite state stochastic automata and integral constraints are introduced. 1.1 Finite Alphabet Feedback Design This subsection contains motivation for using the uncertain nite state automata models as de ned later in the paper. Observer-Based Feedback. Our ultimate goal is to develop tools for optimizing the controller K in the feedback loop shown on Figure 1, where P