Minimum Energy Control Problem of Positive Fractional Discrete-time Systems

The minimum energy control problem of positive fractional-discrete time linear systems is addressed. Necessary and sufficient conditions for the reachability of the system are established. Sufficient conditions for the solvability of the minimum energy control of the positive fractional discrete-time systems are given. A procedure for computation of the optimal sequence of inputs minimizing the quadratic performance index is proposed. INTRODUCTION In positive systems inputs, state variables and outputs take only non-negative values. Examples of positive systems are industrial processes involving chemical reactors, heat exchangers and distillation columns, storage systems, compartmental systems, water and atmospheric pollution models. A variety of models having positive linear systems behaviour can be found in engineering, management science, economics, social sciences, biology and medicine, etc. Positive linear systems are defined on cones and not on linear spaces. Therefore, the theory of positive systems is more complicated and less advanced. An overview of state of the art in standard positive systems is given in the monographs (Farina and Rinaldi 2000; Kaczorek 2002). The realization problem for positive standard and singular continuous-time systems with delays was formulated and solved in (Kaczorek 2007c, 2007d). The reachability, controllability and minimum energy control of positive linear discrete-time systems with time-delays have been considered in (Busłowicz and Kaczorek 2004; Xie and Wang 2003). The realization problem for cone systems has been addressed in (Kaczorek 2006). The reachability and controllability to zero of positive fractional linear systems have been investigated in (Kaczorek 2008a, 2007a, 2007b, 2008b; Klamka 2002; Klamka 2005). Mathematical fundamentals of fractional calculus are given in the monographs (Miller and Ross 1993; Nishimoto 1984; Oldham and Spanier 1974; Oustalup 1993; Podlubny 1999). The fractional order controllers have been developed in (Oustalup 1993). A generalization of the Kalman filter for fractional order systems has been proposed in (Sierociuk and Dzieliński 2006). Some other applications of fractional order systems can be found in (Ferreira and Machado 2003; Moshrefi-Torbati and Hammond 1998; Ortigueira 1997; Ostalczyk 2000, 2004a, 2004b; Podlubny 2002; Samko et al. 1993; Vinagre et al. 2002; Vinagre and Feliu 2002; Gałkowski and Kummert 2005). The minimum energy control problem has been solved for different classes of linear systems in (Klamka 1991, 1976, 1983; Kaczorek and Klamka 1986). In this paper the minimum energy control problem will be addressed for positive fractional discrete-time linear systems. The paper is organized as follows. In section 2 the solution of the state equation and the necessary and sufficient conditions for the positivity of the fractional systems are recalled. Necessary and sufficient conditions for the reachability of the positive fractional systems are established in section 3. The main result of the paper is presented in section 4 in which the minimum energy control problem is formulated and solved. Concluding remarks are given in section 5. To the best knowledge of the author the minimum energy control problem for the positive fractional discrete-time linear systems have not been considered yet. POSITIVE FRACTIONAL SYSTEMS Let be the set of real matrices and The set of matrices with nonnegative entries will be denoted by and The set of nonnegative integers will be denoted by n m × R n m × 1 : n n× R = R . . m n × m n × + R 1 : n n× + + R = R Z+ and the identity matrix by n n × . n I In this paper definition of the fractional difference of the form (Kaczorek 2007a) 0 ( 1) , k j k j x j α α − = ⎛ ⎞ Δ = − ⎜ ⎟ ⎝ ⎠ ∑ k j x