Stability Analysis And Control Design Of Piecewise-linear Systems

In the fields of theoretical study and engineering practice, there exist large numbersof control systems that behave piecewise-linear dynamics. Moreover, the piecewise-lineartheory provides a significant means of analysis and design for nonlinear control systems.As a consequence, it is important to improve and develop the methods to analyze anddesign the piecewise-linear systems from the viewpoint of control theory and controlengineering.This dissertation discusses the motion analysis, stability analysis and control designof piecewise-linear systems, and extends the piecewise-linear theory to the control designof a class of nonlinear systems. The main contributions are listed as follows:The motion properties are explored for both of the discrete-time and continuous-timepiecewise-linear systems. First, based on the transition region and transition direction,the state space partitions of two classes of systems are further refined. Then, for therefined state space partitions, the piecewise uniqueness property of transition region andthe piecewise uniqueness property of transition direction are established for the discrete-time and continuous-time piecewise-linear systems, respectively.Consider the discrete-time piecewise-linear systems without afne items, by intro-ducing the concept of generating function, a new stability analysis method is presented.First, by constructing the generating function and analyzing its properties including ho-mogeneity, piecewise-sub-additivity and piecewise-convexity, a necessary and sufcientstability criterion is proposed. Then, an approach to compute the exponential growth rateof the systems is proposed via the radius of convergence of generating function. Finally,the algorithm for computing the generating function is developed and the error analysis iscarried out.For the continuous-time piecewise-linear systems without afne items, by introduc-ing the integral function, a new stability analysis method is presented. First, the primaryproperties of the integral function are investigated based on the piecewise unique propertyof transition direction, such that the equivalent relationship between system stability andconvergence of integral function can be established. Then, an approach to estimate theexponential growth rate of the systems is proposed via the radius of convergence of the integral function. Finally, the algorithm for computing the integral function is developedand illustrated through examples.For the H_∞control design problem of piecewise-linear systems, this dissertation firstproposes a generalized diferential based piecewise Lyapunov function method, which candeal with the case that the sliding motion occurs on the boundary of polyhedral regions.By the proposed piecewise Lyapunov function method, the H_∞control problem is formu-lated as the feasibility of a set of linear matrix inequalities, which is numerically tractable.Furthermore, by constructing the appropriate piecewise adaptation law, an adaptive H_∞control design approach is developed for uncertain piecewise-linear systems based on theproposed piecewise Lyapunov function method.Considering a class of applications, the dissertation extends the H_∞control designmethod to the nonlinear systems with linear time-invariant input channel. First, a difer-ential inclusion based piecewise linearization method is proposed such that the nonlinearsystems can be formulated as a class of piecewise-linear diferential inclusions. Then,applying the piecewise Lyapunov function method, the H_∞controllers are designed forthe piecewise-linear diferential inclusions. Finally, the closed-loop stability and perfor-mance of the original nonlinear systems are analyzed and the design method is illustratedthrough examples.