Optimal realizations of finite wordlength digital filters and controllers

The problem of finding state space realizations that minimize the sensitivity to quantization error of a finite wordlength implementation of a discrete time linear system is considered. Finite wordlength implementations which invoke quantization either before or after multiplication, and possibly include error feedback, are represented in a unified framework. This framework is used to pose and solve four different optimal realization problems. These problems are formulated in a closed loop context, which contains open loop applications (filtering/estimation) as special cases. Our results may be used to find optimal realizations for the implementation of multivariable feedback controllers, or for the implementation of multivariable filters/estimators. Optimal realizations are derived based on either an H₂ or H_∞ roundoff noise gain subject to either H₂ or H_∞ scaling constraints. The H ₂ noise gain measures the departure from the ideal closed loop response (no signal quantization) when the spectral properties of the quantization error are known. The H_∞ noise gain measures worst-case deviation from ideal response when the variance of the quantization error is bounded but the spectral properties are otherwise unknown. The H₂ scaling constraints limit the size of the quantized internal signals when the spectral properties of the exogenous input to the closed loop system are known, while H_∞ scaling restricts the maximum possible size of the internal signals when the spectral properties of the exogenous input are not precisely known. One of the optimization problems has a well-known analytical solution; the other three are reduced to the problem of minimizing a linear function subject to affine matrix inequality constraints, which is a convex optimization problem whose global optimum may be readily found. This solution, together with the unified framework for the analysis of several FWL implementations, constitutes the main contribution of this paper