Analysis and synthesis of positive systems with periodicity

Abstract

This thesis is concerned with the problems of analysis and synthesis for positive systems with periodicity, including standard positive periodic systems and periodic piecewise positive systems.

For the standard positive periodic systems, the input-output gains of both discrete-time and continuous-time positive periodic systems are investigated. For the discrete-time case, the lifting technique is applied to analyze the input-output gains of the systems. The results show that the ell_1- and ell_infty-gains can be characterized by linear inequalities. For the continuous-time case, the L_1- and L_infty-gains characterization problem turns into the existential problem of a positive periodic vector function. Then the duality relation between the ell_1- (L_1-) gain and ell_infty- (L_infty-) gain is further investigated. By taking advantage of the duality property, a dual system of the positive periodic system is given for calculating the input-output gain.

For periodic piecewise positive systems, their positivity, stability, and performance are analyzed. The work examines the following three aspects: 1) The monotonicity of linear periodic piecewise positive systems is first studied, and a time-varying co-positive Lyapunov function for periodic piecewise positive systems is employed and a sufficient condition for the asymptotic stability of the system is established. Based on the provided co-positive Lyapunov function and the sufficient stability condition, an unweighted upper bound of L_1-gain of the system is given; 2) By constructing a time-scheduled co-positive Lyapunov function with a new interpolation function, an equivalent stability condition for periodic piecewise positive systems is established, and the asymptotic stability of the system can be determined via linear programming. Based on the asymptotic stability condition, the spectral radius of the state transition matrix is characterized by linear inequalities. It is shown that the estimated spectral radius converges to the true value from above when the number of linear inequalities approaches infinity. The relation between the spectral radius of the state transition matrix and the convergent rate of the system is also revealed. Furthermore, an iterative algorithm is developed to stabilize the system by decreasing the spectral radius of the state transition matrix; 3) The lambda-stability and L_1-gain of the periodic piecewise positive systems with delays are studied. A co-positive Lyapunov-Krasovskii function is used, and the sufficient stability condition shows that the time-delay will affect the convergent speed of the state. Both continuous and discontinuous Lyapunov-Krasovskii function is given to characterize the unweighted L_1-gain of the systems. By taking advantage of the positivity property of the state, less conservative conditions are given with linear inequalities.