Robust Stability of Discrete-time Disturbance Observers: Understanding Interplay of Sampling, Model Uncertainty and Discrete-time Designs

TL;DR

This work addresses robust stability of uncertain sampled-data systems controlled by a discrete-time disturbance observer (DT-DOB). Departing from conventional small-gain analyses, it analyzes the limiting root locations of the closed-loop characteristic polynomial as the sampling period ${\Delta}$ becomes small, emphasizing the influence of sampling zeros and discretization choices. A key result establishes that, under fast sampling, robust internal stability for all plants ${P(s)\in{\mathcal P}}$ is equivalent to (i) a CT nominal loop being internally stable, (ii) the plant being minimum phase, and (iii) the fast-dynamics polynomial ${\Psi^{\mathsf d}_{\sf fast}(z)}$ being Schur; a degenerative exception exists. The paper then offers a systematic design guideline for DT-DOB controllers and demonstrates through simulations on a two-mass-spring benchmark that careful discretization and a well-chosen DT-Q filter yield superior disturbance rejection and robust stability, while naive discretizations or overly wide Q-filters can destabilize the loop.

Abstract

In this paper, we address the problem of robust stability for uncertain sampled-data systems controlled by a discrete-time disturbance observer (DT-DOB). Unlike most of previous works that rely on the small-gain theorem, our approach is to investigate the location of the roots of the characteristic polynomial when the sampling is performed sufficiently fast. This approach provides a generalized framework for the stability analysis in the sense that (i) many popular discretization methods are taken into account; (ii) under fast sampling, the obtained robust stability condition is necessary and sufficient except in a degenerative case; and (iii) systems of arbitrary order and of large uncertainty can be dealt with. The relation between sampling zeros---discrete-time zeros that newly appear due to the sampling---and robust stability is highlighted, and it is explicitly revealed that the sampling zeros can hamper stability of the overall system when the Q-filter and/or the nominal model are carelessly selected in discrete time. Finally, a design guideline for the Q-filter and the nominal model in the discrete-time domain is proposed for robust stabilization under the sampling against the arbitrarily large (but bounded) parametric uncertainty of the plant.

Figures (6)

• Figure 1: Overall configuration of sampled-data system controlled by DT-DOB (dotted block)
• Figure 2: Another realization of DT-DOB scheme
• Figure 3: Step response $y(t)$ with different designs of Q-filters: CT nominal closed-loop system (dashed green); CT-DOB controlled systems with $\tau = 0.05$ (dash-dotted red) and $\tau = 0.025$ (solid black); and DT-DOB controlled systems with $\mathrm{Q}^\mathsf{d}_{\sf prop}(z;\Delta)$ (solid blue), $\mathrm{Q}^\mathsf{d}_{\sf ind, LBW}(z;\Delta)$ (solid black), and $\mathrm{Q}^\mathsf{d}_{\sf ind, SBW}(z;\Delta)$ (dash-dotted red)
• Figure 4: Frequency responses of DT Q-filters and sensitivity functions of DT-DOB controlled systems with $\mathrm{Q}^\mathsf{d}_{\sf prop}(z;\Delta)$ (solid blue), $\mathrm{Q}^\mathsf{d}_{\sf ind, LBW}(z;\Delta)$ (solid black), and $\mathrm{Q}^\mathsf{d}_{\sf ind, SBW}(z;\Delta)$ (dash-dotted red)
• Figure 5: Root contours of characteristic polynomial of DT-DOB controlled system under variation of $\Delta\in [0.001,0.3]$: The smaller $\Delta$ becomes, the brighter the color is.