Approximation Of The Least Measure For Switched Linear Systems

Switched systems are a class of hybrid systems consisting of a family of subsystemsand a switching law that orchestrates the switching among the subsystems. For switchedsystems, switching plays a nontrivial role which not only makes the systems remain somedynamical performances of the subsystems, but also introduces strong complexities. Inthis manner, switched systems are used to model a variety of complex nonlinear systems.Due to the practical importance and theoretical challenging, switched systems have beenattracting much attention in the last several decades.For a dynamical system, the largest divergence rate is the worst-case convergence rateof the state trajectories, which characterizes the system performance. For linear systems,as we all know, the largest divergence rate is equal to spectral abscissa (maximal realpart of the eigenvalues) in continuous time and the logarithm of the spectral radius indiscrete time. During the last two decades, a considerable progress has been made in thestudy of the largest divergence rate of switched linear systems. For discrete-time switchedlinear systems, the rate coincides with the logarithm of the (joint/generalized) spectralradius. The spectral radius is known to be equal to least common norm of the subsystemmatrices among all the matrix norm. For continuous-time switched linear systems, there isno counterpart of the spectral radius, as spectral abscissa in linear case. Recently, Sun hasproved that the largest divergence rate is equal to the least common matrix measure of thesubsystem matrices. From the computational point of view, several approaches have beendeveloped to approximate the spectral radius, for instance, the ellipsoid norm approach,the semi-defnite lifting procedure, the sum-of-squares technique and so on. While, forthe continuous-time case, a few algorithms have been introduced to approximate the ratefor switched linear systems with special structures. However, there are few computationalalgorithms for general systems.The purpose of this dissertation is to develop algorithms that approximate the leastcommon matrices measure of switched linear systems. The main contents and contribu-tions of this dissertation are summarized in the following:1. For marginally stable switched linear systems, there exists a unit sphere defned by Barabanov norm, such that the worst-case trajectory will stay on the sphere ifthe initial state is on it. Especially, the worst-case trajectory is a closed periodicorbit for2nd-or3rd-order switched linear system with two subsystems. Thus,1is an eigenvalue of the transition matrix in half-period. Furthermore, all thereal eigenvalues of the transition matrix in half-period are greater than1if thesystem is stable, while there exists a real eigenvalue is less than1if the system isinstable. From this geometrical property, a bisection algorithm could be designedto approximate the least common matrices measure for switched linear systems inany accuracy. However, this algorithm is no longer applicable to the systems whentheir orders are higher than three for the worst-case trajectories that might not beclosed periodic orbits. Thus the algorithm is no longer applicable.2. For general switched linear systems, from the defnitions of matrix (set) measureand least measure, the matrix (set) measure is dependent on the vector norm andthe least measure is the least value of the matrix (set) measure for all possible vectornorms. As any vector norm could be bounded by a homogeneous sum-of-squarespolynomial, it will allow us to approximate the matrix (set) measure by sum-of-squares polynomials. This problem can be transformed into a series of linear matrixinequalities. Considering the set of homogeneous polynomials in Rnis a convex cone,thus we could search a sum-of-squares polynomial to approximate the least commonnorm. Based on this polynomial, a approximation of the least measure is derived.Through this procedure, the approximation problem is transformed into generaleigenvalue problem, which could be solved via the LMI toolbox in MATLAB. Theaccuracy of this algorithm is dependent on the system matrices. If the eigenvaluesof the system are far away from the imaginary axis and concentrated, the algorithmhas a high precision. Otherwise, the algorithm is relatively conservative.3. As the algorithm based on the sum-of-squares technique is conservative when theeigenvalues are closed to the imaginary axis. Thus, a new algorithm is needed forthis case. For any stable switched linear system, there exists a convex piecewiseLyapunov function. This Lyapunov function can induce a piecewise quadratic vectornorm. It is easy to calculate the matrix (set) measure under this piecewise quadraticvector norm. Thus the least measure of all possible vector norm could be relaxedas the least measure for all possible piecewise quadratic vector norms. By the S- procedure lemma, the relaxed problem can be transformed into a general eigenvalueproblem of bilinear matrix inequalities which could be solved by grid method. Forthe case of the two-term piecewise quadratic vector norm, the parameters could bereduced by half through a variable substitution. It efectively improves the efciencyand accuracy of the algorithm.For the aforementioned three algorithms, there are numerical examples to illustratethe efectiveness.Finally, the conclusion and the prospects of future research are given at the end ofthis dissertation.