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Local Stability and Stabilization of Quadratic-Bilinear Systems using Petersen's Lemma

Amir Enayati Kafshgarkolaei, Maziar S. Hemati

TL;DR

The paper addresses scalable local stability analysis and stabilization of quadratic-bilinear (QB) systems by exploiting Petersen's Lemma to convert nonconvex sign-definiteness conditions into tractable LMIs. The authors derive a main synthesis result that yields a static state-feedback gain $K = Y P^{-1}$ ensuring stability inside an ellipsoid $\hat{\mathcal{E}} = \{ x : x^T P^{-1} x \le 1 \}$ with a Lyapunov function $V(x)= x^T P^{-1} x$, and a corollary provides ellipsoidal ROA estimates for autonomous QB systems. The approach scales quadratically with the state dimension and is demonstrated on three benchmark problems, showing competitive accuracy to existing tools while enabling analysis of systems with hundreds of states. This framework offers a practical path to apply QB models across domains by providing scalable stability certificates and stabilizing controllers that were previously limited by computational complexity.

Abstract

Quadratic-bilinear (QB) systems arise in many areas of science and engineering. In this paper, we present a scalable approach for designing locally stabilizing state-feedback control laws and certifying the local stability of QB systems. Sufficient conditions are established for local stability and stabilization based on quadratic Lyapunov functions, which also provide ellipsoidal inner-estimates for the region of attraction and region of stabilizability of an equilibrium point. Our formulation exploits Petersen's Lemma to convert the problem of certifying the sign-definiteness of the Lyapunov condition into a line search over a single scalar parameter. The resulting linear matrix inequality (LMI) conditions scale quadratically with the state dimension for both stability analysis and control synthesis, thus enabling analysis and control of QB systems with hundreds of state variables without resorting to specialized implementations. We demonstrate the approach on three benchmark problems from the existing literature. In all cases, we find our formulation yields comparable approximations of stability domains as determined by other established tools that are otherwise restricted to systems with up to tens of state variables.

Local Stability and Stabilization of Quadratic-Bilinear Systems using Petersen's Lemma

TL;DR

The paper addresses scalable local stability analysis and stabilization of quadratic-bilinear (QB) systems by exploiting Petersen's Lemma to convert nonconvex sign-definiteness conditions into tractable LMIs. The authors derive a main synthesis result that yields a static state-feedback gain ensuring stability inside an ellipsoid with a Lyapunov function , and a corollary provides ellipsoidal ROA estimates for autonomous QB systems. The approach scales quadratically with the state dimension and is demonstrated on three benchmark problems, showing competitive accuracy to existing tools while enabling analysis of systems with hundreds of states. This framework offers a practical path to apply QB models across domains by providing scalable stability certificates and stabilizing controllers that were previously limited by computational complexity.

Abstract

Quadratic-bilinear (QB) systems arise in many areas of science and engineering. In this paper, we present a scalable approach for designing locally stabilizing state-feedback control laws and certifying the local stability of QB systems. Sufficient conditions are established for local stability and stabilization based on quadratic Lyapunov functions, which also provide ellipsoidal inner-estimates for the region of attraction and region of stabilizability of an equilibrium point. Our formulation exploits Petersen's Lemma to convert the problem of certifying the sign-definiteness of the Lyapunov condition into a line search over a single scalar parameter. The resulting linear matrix inequality (LMI) conditions scale quadratically with the state dimension for both stability analysis and control synthesis, thus enabling analysis and control of QB systems with hundreds of state variables without resorting to specialized implementations. We demonstrate the approach on three benchmark problems from the existing literature. In all cases, we find our formulation yields comparable approximations of stability domains as determined by other established tools that are otherwise restricted to systems with up to tens of state variables.

Paper Structure

This paper contains 10 sections, 2 theorems, 35 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background and Preliminaries
  3. Quadratic-Bilinear Systems
  4. Local Stability of an Equilibrium Point
  5. Local Stability Analysis and Stabilizing Controller Synthesis for Quadratic-Bilinear Systems
  6. Illustrative Examples
  7. Local Stability Analysis: Two-State Quadratic System from amato2006region
  8. Local Stability Analysis: 9-State Model of a Sinusoidally-Forced Shear Flow Moehlis2004
  9. Stabilizing Controller Synthesis: Three-state Quadratic-Bilinear System from Amato2009
  10. Conclusions

Key Result

Theorem 1

Let $\varepsilon>0$ be given. If $P=P^\mathsf{T}\succ0$ and $Y\in\mathbb{R}^{m \times n}$ satisfy the LMI then the linear state feedback gain stabilizes the QB system eq:qb inside the ellipsoid and the quadratic form is a Lyapunov function for the closed-loop system inside the stabilizability ellipsoid $\hat{\mathcal{E}}$.

Figures (4)

  • Figure 1: ROA estimates for the 2-state example from amato2006region. (a) ROA estimates are overlaid on the system's phase portrait. Ellipsoidal estimates from \ref{['cor:trace']} are plotted in grayscale, and the ROA estimate resulting from their union in blue. The polytope and associated ellipsoidal ROA estimate from amato2006region are plotted in yellow and green, respectively). (b) Quantitative results obtained from \ref{['cor:trace']} gridded on $\varepsilon$. The red ellipsoid and the red marker in (a) and (b), respectively, correspond to the maximum trace ellipsoid over the set computed.
  • Figure 2: An empirical comparison of computation time versus state dimension $n$ for \ref{['cor:trace']}, the quadratic constraint method of Kalur et al. KalurEtAl21, and the polytopic estimation method of Amato et al. amato2006region based on "stacking" the 2-state example from amato2006region.
  • Figure 3: ROA estimates for the 9-state model from Moehlis2004 versus Reynolds number ($Re$). (a) Results from applying \ref{['cor:trace']} over a grid of $\varepsilon$ for different $Re$. (b) Comparison of the maximum ellipsoidal ROA estimate from \ref{['cor:trace']} determined via bisection (blue) is compared with ellipsoidal ROA estimates based on the quadratic constraint method of Kalur et al. KalurEtAl21 (red).
  • Figure 4: ROS estimates for the 3-state example from Amato2009. (a) Results from applying \ref{['cor:largest_stabilizability_ellipsoid']} over a grid of $\varepsilon$, with the maximum estimate highlighted in red. (b)–(d) Cross-sections of ellipsoidal estimates from \ref{['cor:largest_stabilizability_ellipsoid']} (grayscale), the ROS estimate determined from their union (blue), and the largest ellipsoidal estimate based on the polytopic ROS method of Amato2009.

Theorems & Definitions (4)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • Remark 2