Analysis and synthesis of positive systems under l₁ and L₁ performance

Abstract

This thesis is concerned with the analysis and synthesis of positive systems under ℓ1 and L1 performance. Two classes of systems are considered: positive linear systems and positive Takagi-Sugeno (T-S) fuzzy systems. For positive linear systems, the controller, state-bounding observer and filter design problems are considered. Due to the special structures and unique features of positive systems, some previous approach used for general systems, such as similarity transformation, are no longer applicable to positive systems. First, the stabilization problem for positive linear systems is studied. In detail, analytical formulae to compute the exact values of ℓ1-induced and L1-induced norms are presented. Moreover, it is shown how the necessary and sufficient conditions can be constructed such that the closed-loop system is stable and satisfies a prescribed L1-induced performance. For single-input multiple-output (SIMO) positive systems, analytical solutions are established to show how the optimal ℓ1-induced and L1-induced controllers are designed. In addition, the L1-induced sparse state-feedback controller is investigated for continuous-time interval positive systems. Then, to estimate the state of positive systems at all times, the problem of positive state-bounding observers for interval positive systems is studied under the L1-induced performance. Necessary and sufficient conditions are presented to design a pair of state-bounding positive observers. Finally, the positive filtering problem is addressed for positive systems under the L1-induced performance. A pair of positive filters with error-bounding feature is designed to estimate the output of positive systems and the obtained results are expressed in terms of linear programming problems. For positive T-S fuzzy systems, the controller and filter design problems are investigated under the ℓ1-induced performance. First, novel performance characterization of positive fuzzy systems is established. Sufficient conditions are further established for the existence of state-feedback controller. An iterative convex optimization algorithm is developed to solve the design conditions. Furthermore, a pair of error-bounding positive filters are constructed to estimate the output of positive T-S fuzzy systems. A new performance characterization is first established to guarantee the asymptotic stability of the filtering error system with the ℓ1-induced performance. Then, sufficient conditions expressed by linear programming problems are derived to design the required filters.