Performance limitations in the robust servomechanism control problem for discrete and sampled data systems

Fundamental performance limitations for error tracking/regulation are obtained in the robust servomechanism problem (RSP) for sampled data systems and for discrete periodic systems. In studying these problems, the “cheap control” problem for a multi-input/multi-output discrete time system is initially studied, and explicit expressions for the limiting steady state solution of the associated algebraic Riccati equation (ARE), as the weight on the control energy tends to zero, are obtained. Application of these results is then made to the sampled system robust servomechanism problem, and the periodic discrete system robust servomechanism problem. In particular, explicit expressions for the limiting performance costs for error tracking/regulation for the RSP for sampled systems and discrete periodic systems are obtained. These limitations can be characterized completely by the number and location of the unstable transmission zeros. The results are then extended to the optimal linear filtering problem for discrete time systems, where the limiting variance of the estimation error for the optimal H2 filter, is obtained. In addition, an explicit expression for the lowest achievable mean square error is obtained when the measurement noise is slowly varying and has intensity which tends to zero.