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Abstract: Complex systems typically possess a hierarchical structure, characterized by continuous-variable dynamics at the lowest level and logical decision-making at the highest. Virtually all control systems today perform computer-coded checks and issue logical as well as continuous-variable control commands. Such are ``hybrid'' systems.
Traditionally, the hybrid nature of these systems is suppressed by converting them into either purely discrete or continuous entities. Motivated by real-world problems, we introduce ``hybrid systems'' as interacting collections of dynamical systems, evolving on continuous-variable state spaces, and subject to continuous controls and discrete phenomena.
We identify the discrete phenomena that arise in hybrid systems and review previously proposed models. We propose a hybrid control model, coupling differential equations and automata, that encompasses them. Our unified model is natural for posing and solving hybrid analysis and control problems.
We discuss topological issues that arise in hybrid systems analysis. Then we compare the computational capabilities of analog, digital, and hybrid machines by proposing intuitive notions of analog machines simulating digital ones. We show that simple continuous systems possess the power of universal computation. Hybrid systems have further simulation capabilities. For instance, we settle the famous asynchronous arbiter problem in both continuous and hybrid settings. Further, we develop analysis tools for limit cycle existence, perturbation robustness, and stability. We analyze a hybrid control system, typically used in aircraft, that logically switches between two conventional controllers. Stability of such systems has previously only been tested using extensive simulation; we prove global asymptotic stability for a realistic set of cases. Our tools demonstrate robustness of this stability with respect to ``continuation'' of the logical function.
We systematize the notion of a hybrid system governed by a hybrid controller using an optimal control framework. We prove theoretical results that lead us to algorithms for synthesizing such hybrid controllers. In particular, we prove existence of optimal and near optimal controls and derive ``generalized quasi-variational inequalities'' that the associated value function satisfies. We outline algorithms for solving these inequalities, based on a generalized Bellman equation, impulse control algorithms, and linear programming. Several illustrative examples are solved. The synthesized optimal hybrid controllers verify engineering intuition.
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Phone: 617-253-2161
FAX: 617-258-8553
E-mail: chris@lids.mit.edu
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Abstract: We propose a very general framework for hybrid control problems which encompasses several types of hybrid phenomena considered in the literature. A specific control problem is studied in this framework, leading to an existence result for optimal controls. The ``value function'' associated with this problem is expected to satisfy a set of ``generalized quasi-variational inequalities'' which are formally derived.
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Abstract: We explore the simulation and computational capabilities of hybrid and continuous dynamical systems. The continuous dynamical systems considered are ordinary differential equations (ODEs). For hybrid systems we concentrate on models that combine ODEs and discrete dynamics (e.g., finite automata). We review and compare four such models from the literature. Notions of simulation of a discrete dynamical system by a continuous one are developed. We show that hybrid systems whose equations can describe a precise binary timing pulse (exact clock) can simulate arbitrary reversible discrete dynamical systems defined on closed subsets of $\R^n$. The simulations require continuous ODEs in $\R^{2n}$ with the exact clock as input. All four hybrid systems models studied here can implement exact clocks. We also prove that any discrete dynamical system in $\Z^n$ can be simulated by continuous ODEs in $\R^{2n+1}$. We use this to show that smooth ODEs in $\R^3$ can simulate arbitrary Turing machines, and hence possess the power of universal computation. We use the famous asynchronous arbiter problem to distinguish between hybrid and continuous dynamical systems. We prove that one cannot build an arbiter with devices described by a system of Lipschitz ODEs. On the other hand, all four hybrid systems models considered can implement arbiters even if their ODEs are Lipschitz.
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Abstract: This paper outlines work on the stability analysis of hybrid systems. Particularly, we concentrate on the continuous dynamics and model the finite dynamics as switching among finitely many continuous systems. We introduce multiple Lyapunov functions as a tool for analyzing Lyapunov stability. We use IFS theory as a tool for Lagrange stability. By enforcing the conditions of our theorems, one can also synthesize hybrid systems with desired stability properties.
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Abstract: This paper outlines work on the stability analysis of hybrid systems. Particularly, we concentrate on the continuous dynamics and model the finite dynamics as switching among finitely many continuous systems. We introduce multiple Lyapunov functions as a tool for analyzing Lyapunov stability. We use IFS theory as a tool for Lagrange stability. By enforcing the conditions of our theorems, one can also synthesize hybrid systems with desired stability properties.
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Abstract: We discuss topological issues that arise when differential equations and finite automata interact (hybrid systems). In particular, we examine topologies for achieving continuity of maps from a set of measurements of continuous dynamics to a finite set of input symbols and from a finite set of output symbols into the control space for those continuous dynamics. Finding some anomalies in completing this loop, we discuss a new view of hybrid systems that may broach them and is more in line with traditional control systems. In fact, the most widely used fuzzy control system is related to this new view and does not possess these anomalies. Indeed, we show that fuzzy control leads to continuous maps (from measurements to controls) and that all such continuous maps may be implemented via fuzzy control.
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