In this paper we study the effects of finite-precision controller implementation on the stability and performance of sampled-data systems. A framework is presented in which these effects can be analyzed from a statistical point of view. We analyze the behavior of finite-precision controllers at fast sampling rates and consider the problem of 2 -induced norm minimization, subject to a wordlength constraint.
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In this paper we consider the typical sampled-data feedback configuration shown in Fig. I, where P denotes a continuous-time finite-dimensional linear plant, K,! a discrete-time finite-dimensional linear controller, H,, and S,, synchronized hold and sampler at a sampling period h. and W an antialiasing (stable and strictly proper) filter. We study the effect of finite wordlength implementation of the digital controller on stability and LCZp erformance of the feedback system, and the robustness of the feedback system to uncertainty due to finite precision arithmetic. In particular, we propose a framework to address the following type of problems: i) Given P. Kd. W. h estimate the wordlength needed to ensure stability of the feedback system. ii) For K,! = ShKHlr being a digital implementation of a fixed stabilizing analog controller K. study the behavior of the wordlength needed to ensure stability as h -+ 0. iii) Given P. Kd. W. h estimate the wordlength necessary to ensure that the &-performance of the feedback system lies within a prespecified tolerance. iv) Once again, for K,! = ShKHh. study the behavior of this estimate as h -+ 0. v) Given P. W. h and a fixed wordlength for the controller implementation, minimize the induced &-performance of the feedback system. The exact solution to the above problems is not known. We assume that the errors due to finite arithmetic in the controller implementation are random in nature and proceed to assess their effect from a statistical point of view. We provide sufficient conditions which guarantee robustness of stability and performance, albeit with a certain probability. A fixed point arithmetic format is assumed throughout. Some related work has appeared in [ 1 11 where the authors analyze the degradation of an LQG cost function from a statistical point of view. A statistical approach to the coefficient wordlength problem for digital filters may be found in . For more recent work on finite wordlength implementation and design see , [lo],  and the references therein. The organization of the paper is as follows. In Section I1 we consider the problem of stability robustness, in the face of finite precision controller implementation and develop a statistical estimate of the wordlength required to ensure stability. A similar development Manuscript received October 26, 1993; revised February 18, 1994. This work was supported by Grant NSF/ECS-9016050 and AFOSR Grant AFlF49620-92-J-024 1. The authors are with the Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455 USA. IEEE Log Number 9405663.