Certification of Linear Inclusions for Nonlinear Systems

TL;DR

The paper tackles certifying that a given set of vertex linear systems forms a polytopic Linear Difference Inclusion (LDI) for a nonlinear system around an equilibrium inside a region $\mathcal{Z}$. It introduces a Farkas' lemma–based optimization problem that yields a Yes/No certificate: a negative optimum proves the candidate vertex set is not an LDI, while a nonnegative global optimum (for all $c>0$) confirms it. The framework analyzes deviations $\delta x$ and $\delta u$, enforcing nonnegative weights $\alpha(x,u)$ with sum 1 to realize the inclusion, and can reduce conservatism in LDI enclosures even when the equilibrium is not at the origin. Because the resulting problem is non-convex, it relies on global solvers or multi-start strategies, and the paper demonstrates both limitations and improvements through illustrative examples, suggesting directions for more reliable and tighter LDI determinations.

Abstract

In this work, we propose novel method for certifying if a given set of vertex linear systems constitute a linear difference inclusion for a nonlinear system. The method relies on formulating the verification of the inclusion as an optimization problem in a novel manner. The result is a Yes/No certificate. We illustrate how the method can be useful in obtaining less conservative linear enclosures for nonlinear systems.

Figures (3)

• Figure 1: Example \ref{['Example_1']}. The blue triangle is the state $x$. The red cross is the actual successor state $x^+$ that results from the nonlinear system equations. The green circles are $\bar{A}_ix$, $i=\{1,2,3,4\}$. The successor state $x^+\notin Co(\bar{A}_ix, i=\{1,2,3,4\})$. The optimal value corresponding to this solution is $-0.385$.
• Figure 2: Example \ref{['Example_1']} (another optimal solution). The blue triangle is the state $x$. The red cross is the actual successor state $x^+$ that results from the nonlinear system equations. The green circles are $\bar{A}_ix$, $i=\{1,2,3,4\}$. The successor state $x^+\notin Co(\bar{A}_ix, i=\{1,2,3,4\})$. The optimal value corresponding to this solution is $-0.385$.
• Figure 3: Example \ref{['Example_2']} with $\Bar{A}^n_1$ and $\Bar{A}^n_2$. The blue triangle is the state $x$ which is $-2-0.20001^T$. The red cross is the actual successor state $x^+=0.04999-2.40002^T$ that results from the nonlinear system equations. The green circles are $\bar{A}^n_ix$, $i=\{1,2\}$. The right green circle is at $-10^{-5}-2.40002^T$. The succesor state $x^+\notin Co(\bar{A}_ix, i=\{1,2\})$. The optimal value corresponding to this solution is $-0.0499$.

Theorems & Definitions (8)

• Remark 1
• Lemma 1
• Theorem 1
• proof
• Remark 2
• Remark 3
• Example 1
• Example 2