A piecewise linear system consists of a set of linear time-invariant (LTI) subsystems, with a switching sequence specifying an active subsystem at each time instant. This paper studies the adaptive control problem of single-input, single-output (SISO) piecewise linear systems. By employing the knowledge of the time instant indicator functions of system parameter switches, a new controller structure parametrization is proposed for the development of a stable adaptive control scheme with reduced modeling error in the estimation error signal used for parameter adaptive laws. This key feature is achieved by the new control scheme's ability to avoid a major parameter swapping term in the error model, with the help of indicator functions whose knowledge is available in many applications. A direct state feedback model reference adaptive control (MRAC) scheme is presented for such systems to achieve closed-loop signal boundedness and small output tracking error in the mean square sense, under the usual slow system parameter switching condition. Simulation results on linearized NASA GTM models are presented to demonstrate the effectiveness of the proposed scheme.

This paper studies the adaptive output feedback control problem for single-input, single-output piecewise linear systems with switched parameters. By employing the knowledge of the time instant indicator functions of system parameter switching, a new controller structure parametrization is proposed for the development of a stable adaptive control scheme which is capable of reducing the modeling error in the estimation error signal used for parameter adaptive laws. This key feature is achieved by the new control scheme's ability to avoid a major parameter swapping term in the error model, with the help of indicator functions whose knowledge is available in many applications. It is shown that under the usual slow system parameter switching condition, closed-loop stability (signal boundedness) and small output tracking error in the mean square sense are achieved. Simulation results on linearized NASA GTM models are presented, demonstrating the effectiveness and performance improvement of the proposed adaptive control scheme. I. INTRODUCTIONAdaptive control of plants that are modeled or approximated by linear time-invariant (LTI) systems has been studied extensively in the literature [4], [12]. Yet in many practical applications, an LTI system model may be insufficient to describe an actual plant. Adaptive control of linear timevarying (LTV) plants had been a focus of research, see, for example, [5], [7], [14]. The adaptive control designs typically require that the plant parameters are smooth and their time variations are sufficiently slow. To relax the slow variation requirement, a new model reference controller structure was proposed in [15], which allows arbitrary plant parameter variations, as long as the structured part of controller parameters are properly incorporated in the adaptive control design and the unstructured controller parameters vary slowly. However, the assumption of the smoothness of plant parameters excludes their applicability to a wide class of controlled plants, the socalled piecewise linear systems. By a piecewise system (also called a "switched system" in the literature by many researchers), we mean a dynamical system whose dynamics switches among a set of continuoustime subsystems according to certain switching criteria. The motivation to study piecewise linear systems is two-fold. On the one hand, as shown in [10], [11], a nonlinear system may be modeled as a piecewise linear system for control design, which is expected to be capable of expanding the system operating range. On the other hand, many practical systems are of a hybrid nature and require several dynamical subsystems to describe their behavior [6].

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As an effort to establish a stability and performance metric for adaptive control systems and an attempt to expand nonlinear system operating range with linearizationbased designs in the presence of uncertainties, this dissertation focuses on the gain margin (GM) of adaptive control systems and the development of novel adaptive control schemes for piecewise linear systems. The contributions are a systematic gain margin analysis for a variety of adaptive control systems, and a framework of solutions to the open problems in adaptive control of uncertain piecewise linear systems. The gain margin of adaptive control systems is defined as the specification of the parameter range of a control gain matrix in a designed adaptive control system for maintaining the desired closed-loop signal boundedness and asymptotic tracking performance. A systematic gain margin analysis is conducted for continuous-and discrete-time, direct and indirect model reference adaptive control (MRAC) systems, I gratefully acknowledge the financial support provided by NSF under grant EC-S0601475 and by NASA Langley Research Center under grant NNX08AB99A. I want to thank my wife Ran Zheng, my son Brandon, my parents and my parentsin-law, without whom this dissertation would not have been possible and meaningful. This dissertation is dedicated to my family.

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