Stability analysis and control synthesis for positive discrete-time linear systems by means of linear matrix inequalities

Author(s)

Amanda Spagolla

• Document type: Master's Dissertation
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Faculdade de Engenharia Elétrica e de Computação
• Defense date: 2019-03-25

Examining board members

Pedro Luis Dias Peres; Matheus Souza; Cristiano Marcos Agulhari

• Advisor: Cecília de Freitas Morais; Pedro Luis Dias Peres

Abstract

Linear positive systems are investigated in this dissertation, that is, linear systems whose state and output variables are non-negative for any non-negative input and non-negative initial condition. More precisely, the problem of control synthesis for linear parameter-varying (LPV) discrete-time positive systems is addressed by means of linear matrix inequality (LMI) based methodologies. To this aim, performance criteria such as the H-infinity and H-2 norms are taken into account. The static state and output feedback control design is investigated. As first contribution, an iterative procedure based in parameter-dependent LMIs is proposed to solve the problems of stabilization and H-infinity and H-2 control of LPV discrete-time positive systems. Among the main features of the iterative method, one can highlight the use of the control gain directly as optimization variable, instead of adopting the usual change of variables. This particularity of gain synthesis is very advantageous in the context of positive systems, since it eliminates the need of diagonal structures in the variables related to the gain recovery, required to satisfy the positiveness constraint applied to the closed-loop system. Another benefit is that the proposed method can be particularized to deal with time-invariant polytopic systems and switched systems (with arbitrary switching rule), also allowing the employment of particular structures in the gain, such as decentralization or module limitation of the gain entries, without constraining other variables of the problem. Further interesting properties of the proposed technique are listed as follows: the employment of relaxations applied jointly to the stability and positivity conditions to reduce the conservativeness of the iterative procedure; the guarantee of local convergence of the synthesis algorithm; the proposition of appropriate initial conditions that assure the existence of feasible solution at each iteration. As a second contribution, this dissertation presents an LMI condition combined with a scalar parameter search, with the scalar parameter confined to a well-defined interval, for the design of state-feedback controllers for discrete-time positive switched systems, that is capable of providing mode-dependent and mode-independent controllers. Finally, numerical examples illustrate the applicability and flexibility of the proposed methods, as well as the efficiency of the proposed approach when compared with other techniques from the literature (AU)

• FAPESP's process: 16/25273-8 - Stability analysis and control synthesis for positive discrete-time linear systems by means of linear matrix inequalities
• Grantee: Amanda Spagolla
• Support Opportunities: Scholarships in Brazil - Master