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Iterative, Small-Signal L2 Stability Analysis of Nonlinear Constrained Systems

Reza Lavaei, Leila J. Bridgeman

Abstract

This paper provides a method to analyze the small-signal L2 gain of control-affine nonlinear systems on compact sets via iterative semi-definite programs. First, a continuous piecewise affine storage function and the corresponding upper bound on the L2 gain are found on a bounded, compact set's triangulation. Then, to ensure that the state does not escape this set, a barrier function is found that is robust to small-signal inputs. Small-signal L2 stability then holds inside each sublevel set of the barrier function inside the set where the storage function was found. The bound on the inputs is also found while searching for a barrier function. The method's effectiveness is shown in a numerical example.

Iterative, Small-Signal L2 Stability Analysis of Nonlinear Constrained Systems

Abstract

This paper provides a method to analyze the small-signal L2 gain of control-affine nonlinear systems on compact sets via iterative semi-definite programs. First, a continuous piecewise affine storage function and the corresponding upper bound on the L2 gain are found on a bounded, compact set's triangulation. Then, to ensure that the state does not escape this set, a barrier function is found that is robust to small-signal inputs. Small-signal L2 stability then holds inside each sublevel set of the barrier function inside the set where the storage function was found. The bound on the inputs is also found while searching for a barrier function. The method's effectiveness is shown in a numerical example.
Paper Structure (12 sections, 7 theorems, 24 equations, 2 figures, 1 algorithm)

This paper contains 12 sections, 7 theorems, 24 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Statement
  4. Main Results
  5. Efficient Algorithms
  6. Iterative SDPs for Theorem 2
  7. Iterative SDPs for Theorem 3
  8. Algorithm 1: The Combined Search for Gain and Small-Signal Bounds
  9. Triangulation Refinement
  10. Numerical Simulation
  11. Conclusion
  12. Acknowledgements

Key Result

Lemma 1

Consider the triangulation $\mathcal{T}=\{\sigma_i\}_{i=1}^{m_{\mathcal{T}}}$, where $\sigma_i=\textrm{co}(\{x_{i,j}\}_{j=0}^n)$, and a set $\mathbf{W}=\left\{ W_x \right\}_{ x\in \mathbb{E}_\mathcal{T} } \subset \mathbb{R}$. Let $X_i\in\mathbb{R}^{n\times n}$ be a matrix that has $x_{i,j}-x_{i,0}$

Figures (2)

  • Figure 1: Finding a storage function and the corresponding gain, $\sqrt{\gamma}$
  • Figure 2: Finding a robust barrier function, the corresponding $\hat{u}$, and the level-set $\mathcal{A}$

Theorems & Definitions (14)

  • Definition 1: $\mathcal{L}$ stabilityzames1966input
  • Definition 2: Small-signal $\mathcal{L}$ stabilitykhalil
  • Definition 3: Affine independencegieslRevCPA2013
  • Definition 4: $n$-simplex gieslRevCPA2013
  • Definition 5: Triangulation gieslRevCPA2013
  • Lemma 1: gieslRevCPA2013
  • Lemma 2: gieslRevCPA2013,giesl2012
  • Definition 6: Modified Barrier Function Lavaei2024Lyap
  • Definition 7: Robust Barrier Function
  • Theorem 1
  • ...and 4 more