Parameter-Dependent Lyapunov Functions and Stability Analysis of Linear Parameter-Dependent Dynamical Systems

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Zhang, Xiping
Tsiotras, Panagiotis
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The purpose of this thesis is to develop new stability conditions for several linear dynamic systems, including linear parameter-varying (LPV), time-delay systems (LPVTD), slow LPV systems, and parameter-dependent linear time invariant (LTI) systems. These stability conditions are less conservative and/or computationally easier to apply than existing ones. This dissertation is composed of four parts. In the first part of this thesis, the complete stability domain for LTI parameter-dependent (LTIPD) systems is synthesized by extending existing results in the literature. This domain is calculated through a guardian map which involves the determinant of the Kronecker sum of a matrix with itself. The stability domain is synthesized for both single- and multi-parameter dependent LTI systems. The single-parameter case is easily computable, whereas the multi-parameter case is more involved. The determinant of the bialternate sum of a matrix with itself is also exploited to reduce the computational complexity. In the second part of the thesis, a class of parameter-dependent Lyapunov functions is proposed, which can be used to assess the stability properties of single-parameter LTIPD systems in a non-conservative manner. It is shown that stability of LTIPD systems is equivalent to the existence of a Lyapunov function of a polynomial type (in terms of the parameter) of known, bounded degree satisfying two matrix inequalities. The bound of polynomial degree of the Lyapunov functions is then reduced by taking advantage of the fact that the Lyapunov matrices are symmetric. If the matrix multiplying the parameter is not full rank, the polynomial order can be reduced even further. It is also shown that checking the feasibility of these matrix inequalities over a compact set can be cast as a convex optimization problem. Such Lyapunov functions and stability conditions for affine single-parameter LTIPD systems are then generalized to single-parameter polynomially-dependent LTIPD systems and affine multi-parameter LTIPD systems. The third part of the thesis provides one of the first attempts to derive computationally tractable criteria for analyzing the stability of LPV time-delayed systems. It presents both delay-independent and delay-dependent stability conditions, which are derived using appropriately selected Lyapunov-Krasovskii functionals. According to the system parameter dependence, these functionals can be selected to obtain increasingly non-conservative results. Gridding techniques may be used to cast these tests as Linear Matrix Inequalities (LMI's). In cases when the system matrices depend affinely or quadratically on the parameter, gridding may be avoided. These LMI's can be solved efficiently using available software. A numerical example of a time-delayed system motivated by a metal removal process is used to demonstrate the theoretical results. In the last part of the thesis, topics for future investigation are proposed. Among the most interesting avenues for research in this context, it is proposed to extend the existing stability analysis results to controller synthesis, which will be based on the same Lyapunov functions used to derive the nonconservative stability conditions. While designing the dynamic ontroller for linear and parameter-dependent systems, it is desired to take the advantage of the rank deficiency of the system matrix multiplying the parameter such that the controller is of lower dimension, or rank deficient without sacrificing the performance of closed-loop systems.
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