Optimal realizations of finite wordlength digital controllers via affine matrix inequalities

The problem of finding state-space realizations that minimize the closed loop sensitivity to quantization noise of a finite wordlength digital controller, subject to scaling of the controller internal signals, is considered. Finite wordlength implementations which invoke quantization either before or after multiplication, and possibly include integer residue feedback, are represented in a unified framework. This framework is used to pose and solve four different optimal realization problems. These problems yield realizations that minimize the closed loop output roundoff noise gain subject to overflow or scaling constraints. Optimal realizations are derived based on either an H₂ or H_∞, roundoff noise gain subject to either H₂ or H_∞, scaling constraints. The H_∞ noise gain measures the departure from the ideal closed loop response, i.e. no signal quantization, when the quantization residual has fixed and known spectral characteristics, and the H_∞ noise gain measures worst-case deviation from ideal response when this latter assumption does not hold. The H₂ scaling constraints limit the power of the controller internal signals when the spectral properties of the exogenous input to the closed loop system are known, while H_∞, scaling restricts the maximum possible power 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.