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This paper presents a method for optimal control of hybrid systems. An inequality of Bellman type is considered and every solution to this inequality gives a lower bound on the optimal value function. A discretization of this "hybrid Bellman inequality" leads to a convex optimization problem in terms of finitedimensional linear programming. From the solution of the discretized problem, a value function that preserves the lower bound property can be constructed. An approximation of the optimal feedback control law is given and tried on some examples.

The use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control. Lower bounds

This paper presents an approach for solving optimal control problems of switched systems. In general, in

Optimal feedback solutions to the infinite horizon LQR problem with state and input constraints based on receding horizon real-time quadratic programming are well known. In this paper we develop an explicit solution to the same problem, eliminating the need for realtime optimization. A suboptimal strategy, based on a suboptimal choice of a finite horizon and imposing additional limitations on the allowed switching between active constraint sets on the horizon, is suggested in order to address the computer memory and processing capacity requirements of the explicit solution. It is shown that the resulting feedback controller is piecewise linear, and the piecewise linear structure is explored and exploited for computational analysis of stability and performance of the suboptimal constrained LQR. The piecewise linear structure can also be exploited for efficient real-time implementation of the controller.

This paper presents an approach for solving optimal control problems for switched systems

In optimal control problems of switched systems, in general, one needs to nd both optimal continuous inputs and optimal switching sequences, since the system dynamics vary before and after every switching instant. In a previous paper, weproved that an optimal control problem can be posed as a two stage optimization problem under some additional assumptions. In general, the two stage optimization problem is still dicult to solve analytically. In this paper, we develop a search algorithm to explore the solution of the two stage optimization problem and nd useful suboptimal solutions. This algorithm is motivated by the idea of dynamic programming which studies the value functions. The algorithm is used to determine suboptimal solutions for general switched linear quadratic problems. 1 Introduction Aswitched system is a hybrid system that consists of several subsystems and a switching law indicating the active subsystem at each time instant. Examples of switched systems can be found in c...

Dither signals are commonly used for compensating nonlinearities in feedback systems in electronics and mechanics. The seminal works by Zames and Shneydor and more recently by Mossaheb present rigorous tools for systematic design of dithered systems. Their results rely however on a Lipschitz assumption on the nonlinearity and thus do not cover important applications with discontinuities. The aim of this thesis is to provide some ideas and tools on how to analyse and design dither in nonsmooth systems. In particular, it is shown that a dithered relay feedback system can be approximated by a smoothed system. Guidelines are given for tuning the amplitude and the period time of the dither signal, in order to stabilize the nonsmooth system. Stability results based on Popov-like and Zames-Falb criteria jointly with some Linear Matrix Inequalities are proposed. Moreover it is argued that in dithered relay feedback systems the shape of dither signals is relevant for stabilization. Some peculiar behaviours of relay feedback systems dithered with a particular class of dither signals are presented. When the dither signal is a square wave, the dithered system can exhibit an asymmetric periodic orbit, though the smoothed system is asymptotically stable. We even show an example in which, by using a trapezoidal dither signal, both systems have a stable oscillation, but the period time for the oscillation of the smoothed system is different from the one of the dithered system. Finally some engineering applications are presented in order to show the usefulness of techniques and results discussed in the thesis. Thesis Supervisor: Franco Garofalo, Professor of Automatic Control Thesis Supervisor: Francesco Vasca, Associate Professor of Automatic Control Acknowledgements Yes, ...it's fina...

In this paper, we consider the hierarchical control problem for a class of uncertain hybrid dynamical systems. The continuous dynamics of this class of uncertain hybrid systems are described by linear difference state equations, whose right side functions are unknown but lie within some convex hulls of known functions. Our control

In this paper, we consider the controller synthesis problem for a class of uncertain hybrid dynamical systems. The goal is for the closed loop system to exhibit desired behavior under dynamic uncertainty and exteriors disturbances. The main question is whether there exists a controller such that the closed loop system satisfies the specification. The notion of attainability is introduced to refer to the specified behavior that can be forced to the plant by a control mechanism. We give a method for attainability checking by employing the predecessor operator and backward reachability analysis, and a procedure for controller design by using finite automata and linear programming techniques.