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Exact Characterization of Admissible Bandwidths for Linear Disturbance Observers
Hamin Chang, Hyuntae Kim
TL;DR
The paper asks for the entire set of Q-filter time constants that ensure robust internal stability under plant uncertainty. It formulates the τ-dependent conditions using bivariate polynomials and constructs an exact procedure for recovering the admissible intervals. The set is open and semialgebraic with finitely many interval components, and the example has a disconnected admissible set.
Problem
The paper addresses finding the entire set of positive Q-filter time constants for which the DOB closed loop is robustly internally stable.
Source: §2 · p. 3Method
It reformulates the τ-dependent stability conditions as bivariate-polynomial feasibility tests and uses exact algebraic decomposition with membership tests to recover admissible components.
Source: §1 · p. 1, §3.2 · pp. 4–5Results
The admissible set is an open semialgebraic subset of (0, ∞), hence a finite union of open intervals, and the example’s admissible set is disconnected.
Source: §3.2 · p. 4, §4 · p. 5Takeaways & Limitations
The characterization recovers every stabilizing interval, including disconnected and degenerate components, rather than restricting bandwidth selection to small τ.
Source: §1 · p. 1, §4 · p. 5The characterization applies under the stated Q-filter assumptions, including a Hurwitz denominator, low-pass condition l − k ≥ ν, proper implementation without unstable pole-zero cancellations, and unit DC gain.
Source: §2 · p. 2
Abstract
from arXiv · showhide
This paper characterizes the entire set of admissible Q-filter bandwidths, equivalently parameterized by the time constant $\tau$, that guarantee robust internal stability of a linear disturbance observer (DOB) under parametric plant uncertainties. Classical small-$\tau$ stability results provide a rigorous foundation for Q-filter bandwidth design. Building on this foundation, we formulate the $\tau$-dependent robust internal-stability problem as a semialgebraic feasibility problem in bivariate polynomials and prove that the complete admissible set is a finite union of open intervals under the standing assumptions. A constructive procedure with exact membership tests recovers all stabilizing intervals. A numerical example with an uncertain plant family containing non-minimum-phase members illustrates a disconnected admissible set.
1 Introduction
The paper seeks the complete admissible Q-filter time-constant set for robust DOB stability, extending small-τ sufficient conditions and an existing exact range framework. It represents the τ-dependent conditions with bivariate polynomials and constructs a finite algebraic decomposition that recovers every admissible component.
1 Introduction
- Small-τ sufficient conditions provide a rigorous foundation, while Chang et al. (2020) established exact vertex and pairwise conditions whose admissible-range intersection identifies robustly stabilizing time constants.Source: §1 · p. 1
- The method classifies finite stability-boundary mechanisms and constructs an exact algebraic decomposition that recovers every admissible component, including disconnected and degenerate cases.Source: §1 · p. 1
2 Problem Formulation
The problem is to find the entire set of positive Q-filter time constants that makes the DOB closed loop robustly internally stable. The formulation specifies plant and filter assumptions, recalls small-τ and characteristic-polynomial tests, and uses corner and pairwise-segment conditions to characterize stability.
2 Problem Formulation
- Plants have fixed numerator and denominator degrees and common relative degree; the nominal model belongs to the uncertainty family, and the Q-filter is low-pass with a Hurwitz denominator and unit DC gain.The assumptions also require a proper implemented block without unstable pole-zero cancellations; the nominal numerator must have no zeros outside the open left half-plane when the admissible set is nonempty.Source: §2 · p. 2
- The DOB interconnection is represented by a transfer matrix from reference and disturbance inputs to tracking error and plant input.Source: §2 · pp. 1–2
- Internal stability requires a well-posed implementation with proper, stable transfers from independent internal inputs to all internal signals, and robust stability requires this for every plant.Source: §2 · p. 2
- For sufficiently small positive τ, a recalled sufficient condition requires a stable nominal feedback interconnection, minimum-phase plants, and a Hurwitz polynomial for every plant.Source: §2 · p. 2
- The robust-stability test is equivalent to requiring the plant-dependent characteristic polynomial to be Hurwitz for every uncertain plant.Source: §2 · p. 2
- The characteristic polynomial depends affinely on uncertain coefficients, so its convex hull is represented by finitely many distinct corner families with a common degree and positive leading sign.Identical corner families are removed before enumeration.Source: §2 · pp. 2–3
- Robust stability holds exactly when every corner polynomial and every segment between each pair of corner polynomials is Hurwitz for all segment parameters.Testing all pairs is exact even when some segments are not exposed edges of the polynomial polytope.Source: §2 · p. 3
- The objective is to find all τ > 0 for which the closed-loop system is robustly internally stable for every uncertain plant.Source: §2 · p. 3
3 Main Result
The paper recasts fixed-τ segment stability as bivariate polynomial conditions and proves that the admissible set is a finite union of open intervals. It gives algebraic boundary conditions and an exact procedure for recovering those intervals.
3.1 Reformulation of Stability Condition
- For fixed τ with Hurwitz endpoint polynomials, a segment fails the stability test exactly when some η > 0 satisfies Gij(η; τ) = 0 and Fij(η; τ) ≤ 0.The bivariate condition captures the imaginary-axis crossing that makes an interpolated polynomial non-Hurwitz.Source: §3.1 · p. 3
3 Main Result
- The construction first intersects corner Hurwitz conditions, expressed through leading principal minors, with the pairwise segment conditions.If a relevant minor is identically zero, the corresponding Vi and thus R are empty; the construction otherwise considers nonzero minors.Source: §3 · p. 3, §3.2 · p. 4
3.2 Construction of the Set of Admissible τ
- The admissible set R is open and semialgebraic, hence a finite union of possibly unbounded open intervals.This describes the full set of τ values satisfying the robust-stability tests.Source: §3.2 · p. 4
- At a positive endpoint of an admissible component, a corner Hurwitz condition reaches its boundary or a pairwise segment condition has a boundary witness.The pairwise witness includes Gij = 0, Fij ≤ 0, and ∂ηGij = 0.Source: §3.2 · p. 4
- Real quantifier elimination converts the stability conditions into a quantifier-free formula in τ, enabling exact membership tests and interval recovery.The procedure partitions positive τ at roots of the formula's polynomials, tests the resulting cells, and merges connected admissible cells.Source: §3.2 · pp. 4–5
4 Numerical Example
The example applies the classification to an uncertain plant family containing a non-minimum-phase member and finds a disconnected admissible set with a component away from the origin.
4 Numerical Example
- The example specifies an outer-loop controller and a Q-filter parameterized by τ.Their displayed transfer functions define the numerical setup.Source: §4 · p. 5
- The uncertain plant family has β1 ∈ [−0.2, 2], includes a non-minimum-phase member, and has two degree-eight corner polynomials.The nominal model is the family member at β1 = 1.Source: §4 · p. 5
- Exact duplicate removal leaves 14 boundary candidates, and cell tests resolve the extraneous roots.The boundary-containing set combines corner Hurwitz-minor roots with resultant roots.Source: §4 · p. 5
- The admissible set consists of two open intervals, including a component away from the origin.This demonstrates that stabilizing bandwidths need not be confined to a local small-τ range.Source: §4 · p. 5