. This paper summarizes the definitions and several of the main results of an approach to hybrid systems, which combines finite automata and linear systems, developed by the author in the early 1980s. Some related more recent results are briefly mentioned as well. 1 Introduction - The Need for Hybrid Systems Linear control theory is well-developed and highly sophisticated, and is widely applied in areas ranging from aerospace to automotive control. Linear systems provide highly accurate models of many physical systems; furthermore, the use of linear systems as "robust" controllers often allows the tolerance of even severe model nonlinearities and uncertainties. However, it remains a fact that many continuous physical processes cannot be satisfactorily modeled linearly, nor can be adequately regulated by means of linear controllers alone. It has long been recognized that the control of more complex systems than those handled by the linear theory will require switching mechanisms (disco...

Abstract. The input to state stability (ISS) paradigm is motivated as a generalization of classical linear systems concepts under coordinate changes. A summary is provided of the main theoretical results concerning ISS and related notions of input/output stability and detectability. A bibliography is also included, listing extensions, applications, and other current work. 1

The analysis and design of nonlinear feedback systems has recently undergone an exceptionally rich period of progress and maturation, fueled, to a great extent, by (1) the discovery of certain basic conceptual notions, and (2) the identification of classes of systems for which systematic decomposition approaches can result in effective

This paper deals with the design of linear systems with saturating actuators where the actuator limitations have to be incorporated a priori into control design. We take a semi-global approach to solve some of the central control problems for such systems. These problems include: stabilization, input-additive disturbance rejection and robust stabilization in the presence of matched nonlinear uncertainties. We develop further the semi-global design technology which was initiated in our earlier work [7] and utilize it to deal with these control problems.

This paper discusses various continuity and incremental-gain properties for neutrally stable linear systems under linear feedback subject to actuator saturation. The results complement our previous ones, which applied to the same class of problems and provided finite-gain stability. Keywords: saturated-input linear systems, operator stability, finite incremental gain 1 Introduction

The low-and-high gain design technique, which was initiated in [10] for a chain of integrators and completed in [13] for general linear asymptotically null controllable with bounded controls systems, was conceived for semi-global control problems beyond stabilization and was related to the performance issues such as semi-global stabilization with enhanced utilization of the available control capacity and semi-global disturbance rejection. Although the low-and-high gain design technique as initiated in [10] is based on explicit eigenstructure assignment, its full development in [13] is based on the solution of an algebraic Riccati equation (ARE) which is parameterized in an arbitrarily small scalar, called low gain parameter. In this paper, we develop a new complete lowand -high gain design procedure which is based on explicit eigenstructure assignment. The new design approach avoids the solution of the numerically stiff parameterized ARE. Submitted to International Journal of Robust an...

The purpose of this paper is to examine the problem of controlling a linear time-invariant system subject to input saturation in order to have its output track (or reject) a family of reference (or disturbance) signals produced by some external generator. It is shown that a semi-global framework for this problem is a natural one. Within this framework, a set of solvability conditions are given and feedback laws which solve the problem are constructed. The theory developed in this paper incorporates earlier results established for linear systems without input saturation.

For the problem of stabilization of mildly nonlinear systems with saturated control, a simple and computationally attractive method of designing a stabilizing gain scheduled output feedback is proposed. The resulting controller is characterized by an efficient use of the actuator capacity ("agressive" control), and is supplied with rigorously proven input/output performance bounds. In particular, for the case of stabilization of a linear time invariant system without exponentially unstable modes, the closedloop map from the plant disturbances and state estimation noise to the state and control variables is shown to be L2-bounded . Key Words: saturated control, gain scheduling, stability, input - output performance. 1 Introduction The presence of saturation limits on the controller gain and gain rate complicates the control law design even in the case of a model that is otherwise linear and time-invariant. Saturation of control actuators is recognized as a "headache" for the control ...

analysis, switching surfaces. In this paper we develop an entirely new constructive global analysis methodology for a class of hybrid systems known as Piecewise Linear Systems (PLS). This methodology consists in inferring global properties of PLS solely by studying their behavior at switching surfaces associated with PLS. The main idea is to analyze impact maps, i.e., maps from one switching surface to the next switching surface, by constructing quadratic Lyapunov functions on switching surfaces. We found that an impact map induced by an LTI flow between two switching surfaces can be represented as a linear transformation analytically parameterized by a scalar function of the state. This representation of impact maps allows the search for quadratic surface Lyapunov functions to be done by simply solving a set of LMIs. Global asymptotic stability, robustness, and performance of limit cycles and equilibrium points of PLS can this way be efficiently checked. These new results were successfully applied to certain classes of PLS. Although this analysis methodology yields only sufficient criteria of stability, it has shown to be very successful in globally analyzing a large number of examples with a locally stable limit cycle or equilibrium point. In fact, it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology. Examples analyzed include systems of relative degree larger than one and of high dimension, for which no other analysis methodology could be applied. 1

This note addresses the problem of satisfying pointwise-in-time input and/or state hard constraints in nonlinear control systems. The approach is based on conceptual tools of predictive control and consists of adding to a primal compensated nonlinear system a Reference Governor (RG). This is a discrete-time device which on-line handles the reference to be tracked, taking into account the current value of the state, in order to satisfy the prescribed constraints. The resulting hybrid system is proved to fulfill the constraints, as well as stability and tracking requirements. I. Introduction In recent years the field of feedback control of dynamic systems with input and/or state-related constraints has received considerable attention [1], [2]. Most of this research has addressed regulation problems for systems subject to input saturation. More recently, moving horizon optimal control [3], [4], [5] and model predictive control [6], [7] have been proved to be effective tools to deal with ...