The Existence Of A Lyapunov Function For Several Special Classes Of Switched Positive Systems

This dissertation focuses on the existence of a Lyapunov function for severalspecial classes of switched positive systems (SPSs). For such classes of SPSs, based onthe topology, matrix, algebra, and control theory, we shall establish some checkableexistence criteria of common linear copositive Lyapunov functions (CLCLFs),common diagonal quadratic Lyapunov functions (CDQLFs), and switched linearcopositive Lyapunov functions (SLCLFs), respectively. The main topics of thisdissertation are consisted of the following aspects:1. We focus on the CLCLF existence for SPSs. This work follows two main ideas:linear programming approach and convex cone approach.Based on the linear programming approach, we shall deal with the case when thesubsystems of SPSs are discrete-time linear time-invariant (LTI) systems and when theSPSs composed of continuous-and discrete-time linear LTI subsystems, respectively.For the former case, firstly, the SPSs composed of pairs of second order discrete-timelinear LTI subsystems are considered. By simple geometrical discussion, a determinantcondition of the existence of a CLCLF shall be derived. Then this result is naturallyextended to the case of sets of second order discrete-time linear LTI subsystems.Secondly, two special classes of high order discrete-time linear LTI SPSs areinvestigated: the case when the system matrices are upper/lower triangular form andthe case of when the systems matrices are same block upper/lower triangular form. Weshall show that the CLCLF for upper/lower triangular form must exist and present theapproach of defining a CLCLF. For same block upper/lower triangular form, we shallshow that the existence of a CLCLF for such systems is equivalent to the existence ofa CLCLF for the SPSs composed of the same diagonal block matrices; Based on thesame formulation, a series of results of the former case shall be extended to the casewhen the SPSs composed of continuous-time and discrete-time linear LTI subsystems.On the other hand, applying the convex cone approach, we first investigate thegeneral SPSs composed of a finite of discrete-time linear LTI subsystems. Someequivalent conditions of the CLCLF existence for such systems shall be presented. An advantage is that these results are easy to be checked by algebraic approach. Moreover,for some special SPSs, we can construct a CLCLF according to the proof process.Secondly, we extend these results to the discrete-time linear SPSs with multiple timedelays.2. We study the CDQLFs existence for continuous-time SPSs composed of sets ofsubsystems. Firstly, the case of linear LTI subsystems is considered. Under theassumptions of the irreducibility of systems matrices and the nonexistence of theCDQLF, through the approximation thought, we refute the existence of hyperplanebetween a finite of convex cones, then this will contradict the assumptions. That is, theintersection between a finite of convex cones always exists, i.e., the SPSs admit aCDQLF under the former assumptions. Secondly, we deal with the case when thesubsystems of SPSs are cooperative and nonlinear. By introducing the homogeneity,irreducibility, and cooperativity of nonlinear vector fields, a necessary criterion for theexistence of a CDQLF is derived by the essentially same discussion as the LTI case.3. We address the SLCLF existence problem for the linear SPSs with multiple timedelays. Obviously, the existence of such a Lyapunov function can guarantee theexponential stability of SPSs. We first transform the SPSs to polytopic systems with theparticularity that the allowable values for the dynamical matrix are those correspondingto the vertices of the polytope. Furthermore, we formulate, by linear programmingapproach, some equivalent conditions for the existence of a SLCLF, such conditions arecharacteristic as linear programming and linear matrix inequality problems.