Design of Linear Control Laws for Minimum Uniform Quantization Tracking Error

The uniform quantization effects present in the implementation of numeric regulators introduce undesired tracking errors, even though their continuous-time counterparts can ensure ideal steady-state response. Without perturbing the transient response, the state realization of the regulator can be scaled to reduce the influence of the steady-state artifacts. A main theoretical contribution is thus proposed, based on two complementary aspects. The starting point is given by an analytical bound of the quantization error. On one hand, this guaranteeable bound is minimized by the existence of an optimally scaled similarity matrix for the Jordan form of the closed-loop state matrix. On the other hand, a balancing scheme for the numeric regulator further reduces the quantization effects for a predefined hardware configuration. Mathematical guarantees to enforce said properties are then presented, developing sufficient conditions. Finally, the proposed method is illustrated on a case study which demonstrates the non-conservative nature of the optimized bound in comparison to the default value from the characterization theorem.