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An Exact, Finite Dimensional Representation for Full-Block, Circle Criterion Multipliers

Felix Biertümpfel, Bin Hu, Geir Dullerud, Peter Seiler

TL;DR

The paper introduces a finite-dimensional, copositivity-based representation for the complete set of full-block circle-criterion multipliers in the interconnection of a discrete-time LTI system with a non-repeated, sector-bounded nonlinearity. It establishes that the set of incremental input/output pairs of a specially constructed piecewise-linear function coincides with the set of input/output pairs of all non-repeated nonlinearities, enabling a finite characterization of the multipliers via copositivity constraints. The authors prove an exact equivalence between the newly defined $\mathcal{M}_{\text{inc}}$ and the traditional full-block set $\mathcal{M}_{fb}$, with exact results available for input/output dimension $m\le4$, and demonstrate improved conservatism over standard diagonal or vertex relaxations through a numerical example. The approach yields a practical SDP framework to certify internal stability and induced $\ell_2$-gain with reduced computational burden for small-scale nonlinearities, while highlighting the need for copositive checks or relaxations in higher dimensions.

Abstract

This paper provides the first finite-dimensional characterization for the complete set of full-block, circle criterion multipliers. We consider the interconnection of a discrete-time, linear time-invariant system in feedback with a non-repeated, sector-bounded nonlinearity. Sufficient conditions for stability and performance can be derived using: (i) dissipation inequalities, and (ii) Quadratic Constraints (QCs) that bound the input/output pairs of the nonlinearity. Larger classes of QCs (or multipliers) reduce the conservatism of the conditions. Full-block, circle criterion multipliers define the complete set of all possible QCs for non-repeated, sector-bounded nonlinearities. These provide the least conservative conditions. However, full-block multipliers are defined by an uncountably infinite number of constraints and hence do not lead to computationally tractable solutions if left in this raw form. This paper provides a new finite-dimensional characterization for the set of full-block, circle criterion multipliers. The key theoretical insight is: the set of all input/output pairs of non-repeated sector-bounded nonlinearities is equal to the set of all incremental pairs for an appropriately constructed piecewise linear function. Our new description for the complete set of multipliers only requires a finite number of matrix copositivity constraints. These conditions have an exact, computationally tractable implementation for problems where the nonlinearity has small input/output dimensions $(\le 4)$. We illustrate the use of our new characterization via a simple example.

An Exact, Finite Dimensional Representation for Full-Block, Circle Criterion Multipliers

TL;DR

The paper introduces a finite-dimensional, copositivity-based representation for the complete set of full-block circle-criterion multipliers in the interconnection of a discrete-time LTI system with a non-repeated, sector-bounded nonlinearity. It establishes that the set of incremental input/output pairs of a specially constructed piecewise-linear function coincides with the set of input/output pairs of all non-repeated nonlinearities, enabling a finite characterization of the multipliers via copositivity constraints. The authors prove an exact equivalence between the newly defined and the traditional full-block set , with exact results available for input/output dimension , and demonstrate improved conservatism over standard diagonal or vertex relaxations through a numerical example. The approach yields a practical SDP framework to certify internal stability and induced -gain with reduced computational burden for small-scale nonlinearities, while highlighting the need for copositive checks or relaxations in higher dimensions.

Abstract

This paper provides the first finite-dimensional characterization for the complete set of full-block, circle criterion multipliers. We consider the interconnection of a discrete-time, linear time-invariant system in feedback with a non-repeated, sector-bounded nonlinearity. Sufficient conditions for stability and performance can be derived using: (i) dissipation inequalities, and (ii) Quadratic Constraints (QCs) that bound the input/output pairs of the nonlinearity. Larger classes of QCs (or multipliers) reduce the conservatism of the conditions. Full-block, circle criterion multipliers define the complete set of all possible QCs for non-repeated, sector-bounded nonlinearities. These provide the least conservative conditions. However, full-block multipliers are defined by an uncountably infinite number of constraints and hence do not lead to computationally tractable solutions if left in this raw form. This paper provides a new finite-dimensional characterization for the set of full-block, circle criterion multipliers. The key theoretical insight is: the set of all input/output pairs of non-repeated sector-bounded nonlinearities is equal to the set of all incremental pairs for an appropriately constructed piecewise linear function. Our new description for the complete set of multipliers only requires a finite number of matrix copositivity constraints. These conditions have an exact, computationally tractable implementation for problems where the nonlinearity has small input/output dimensions . We illustrate the use of our new characterization via a simple example.

Paper Structure

This paper contains 13 sections, 8 theorems, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Background
  4. Quadratic Constraints
  5. Stability Condition
  6. QCs for Sector-Bounded Nonlinearities
  7. Comment on Repeated Nonlinearities
  8. Main Result
  9. Incremental Graph for a Piecewise Linear Function
  10. New Characterization for Full-Block Multipliers
  11. Computational Characteristics of Multipliers
  12. Numerical Example
  13. Conclusion

Key Result

Theorem 1

Consider the interconnection $F_u(G,\Delta_\Phi)$ with the LTI system $G$ defined in eq:LTINom. Assume the following: Given these assumptions, if $\Phi: \mathbb{R}^{m} \to \mathbb{R}^{m}$ is any non-repeated nonlinearity in the sector $[\alpha,\beta]$ then $F_u(G,\Phi)$ is well-posed and internally stable with $\|F_u(G,\Delta_\Phi)\|_{2\to 2} < \gamma$.

Figures (3)

  • Figure 1: Feedback interconnection $F_u(G,\Delta_\Phi)$ of an LTI system $G$ and perturbation $\Delta_\Phi$ defined by a repeated nonlinearity $\Phi$.
  • Figure 2: The left plot shows $f_{\alpha\beta}(v)$ vs. $v$ for $\alpha=1$ and $\beta=3$. The right plot shows the sector bound (shaded blue) for this choice of $[\alpha,\beta]$. The points $(\bar{v},\bar{w}) = (1.5,4.5)$ and $(\hat{v},\hat{w}) = (-2,-2)$ are also shown on the left along with the increment on the right, $(\bar{v}-\hat{v}, \bar{w}-\hat{w}) = (3.5,6.5)$ (red dots).
  • Figure 3: Achieved gain bound over sector bound $\beta$: $\mathcal{M}_{\text{d}}$ (\ref{['pl:Md']}), $\mathcal{M}_{\text{c}}$ (\ref{['pl:Mfc']}), $\mathcal{M}_\text{inc}$ (\ref{['pl:Mcmp']})

Theorems & Definitions (21)

  • Definition 1: Sector Bound
  • Definition 2: Well-Posedness
  • Definition 3: Internal Stability
  • Definition 4: Induced $\ell_2$ Gain
  • Definition 5
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 11 more