An improved closed-loop stability related measure for finite-precision digital controller realizations

Abstract

Abstract-The pole-sensitivity approach is employed to investigate the stability issue of the discrete-time control system, where a digital controller, implemented with finite word length (FWL), is used. A new stability related measure is derived, which is more accurate in estimating the closed-loop stability robustness of an FWL implemented controller than some existing measures for the pole-sensitivity analysis. This improved stability measure thus provides a better criterion to find the optimal realizations for a generic controller structure that includes output-feedback and observer-based controllers. A numerical example is used to verify the theoretical analysis and to illustrate the design procedure.Index Terms-Closed-loop stability, digital controller, finite word length, optimization.

A Unifying Framework for Finite Wordlength Realizations

“…It can be used to evaluate the overall sensitivity. From the properties of the -norm, we have (30) where is the Frobenius norm. Definition 8: The MIMO transfer function sensitivity is defined by (31) Next, we introduce a new operator that simplifies the subsequent expressions for and the transfer function sensitivity matrix.…”

Toward tools and methodology for the fixed-point implementation OF linear filters

“…Let (λ k ) 1 k n denote the poles and zeros of the system (i.e. the eigenvalues of A).Since the FWL error that can cause a stable system to become unstable is determined by how close the pole are to 1 and how sensitive they are to the parameter perturbations, the following measure is classically used [5]:…”

“…Generally ω k = 1 1−|λ k | to give more weight for the poles closed to the unit circle [5]. The pole sensitivity measure is also used in closed-loop context, in some stability-related measures [2].…”

Solving Finite Word Length Realization Problems in the Framework of Structured Singular Value

“…However, different realizations possess different degrees of stability robustness to FWL errors. An FWL design is to select optimal realizations for the given filter/control law by optimizing some FWL stability measures, such as the Frobenius-norm pole sensitivity measure υ f [4], the l 1 -based stability measure υ l [5], the 1-norm pole sensitivity measure υ 1 [6], [7], the stability radius measure υ r [8] and the pole sensitivity sum measure υ s [9]. In fact, the FWL stability measure υ proposed in [10] quantifies the FWL stability characteristics of a realization best.…”