Simultaneous Coefficient Assignment of Discrete-Time Multi-Input Multi-Output Linear Time-Varying System: A New Approach for Compensator Design

Abstract

In this paper, a linear time-varying input-output system is considered and its realization as a linear time-varying autoregressive moving average system (ARMA) is studied. A time-varying z-transform is also introduced and its properties are studied. Furthermore a time-varying version of the coefficient assignment problem well known in time invariant system theory as the pole placement problem is posed and analyzed. A r-tuple of discrete time, linear time-varying plants with m inputs and p outputs are considered together with a single p input m output linear time-varying compensator. The design objective is to construct a single compensator that “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 $p \times m$ 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 an algorithm is obtained that is nonrecursive and involves solving for each time instant a system of linear equations with time-varying coefficients. The associated time-varying matrix has the “Sylvester matrix structure.” Such a structure is well-known in pole placement of time-invariant systems by dynamic compensation. Additionally the problem of coefficient assignment of the autoregressive part of the ARMA system is considered in the closed loop, without splitting up into a cascade of two subsystems as before. A new recursive algorithm to analyze this problem has been introduced. The proposed algorithm has no counterpart in the time-invariant system design and thus represents a new design procedure. A special case of this algorithm for the single-input single-output system has been described in detail. An interesting feature of the proposed recursive algorithm is that one obtains a nonlinear recursion on the compensator parameters that would assign a prespecified sequence of coefficients for the closed-loop system. For a specific design problem it is shown that the dynamics of this nonlinear recursion is chaotic.