On the Problem of Coefficient Assignment of Discrete Time Multi-Input Multi-Output Linear Time Varying Systems

We consider a linear time varying autoregressive moving average (ARMA) system and pose and analyze a time varying version of the coefficient assignment problem well known in time invariant system theory as the pole placement problem. We consider a r-tuple of discrete time, linear time varying plants with m inputs and p outputs together with a single p input m output linear time varying compensator. The design objective is to construct a single compensator which "coefficient assigns," and hence "bounded input bounded output stabilizes" under suitable additional technical assumptions, the set of r plants simultaneously in the closed loop. Such a problem is useful in robust design of linear time-varying control systems in the closed loop. Among the results, it is shown that a generic r-tuple of p × m plants (in a suitable topology) is simultaneously coefficient assignable, provided that r ≪ m/p. The design procedure involves splitting the closed loop system into an ARMA system in cascade with a moving average system. The coefficient assignment problem consists of assigning the coefficients of the autoregressive part of the ARMA subsystem. Thereby we obtain an algorithm which is non-recursive and involves solving for each time instant, a system of linear equations with time-varying coefficients. Additionally we consider the problem of coefficient assignment of the autoregressive part of the ARMA system in the closed loop, without splitting up into a cascade of two subsystems as before. A new recursive algorithm to analyse this problem has been introduced.