Polynomial Constraints for Robustness Analysis of Nonlinear Systems

Abstract

This paper presents a framework for abstracting uncertain or non-polynomial components of dynamical systems using polynomial constraints. This enables the application of polynomial-based analysis tools, such as sum-of-squares programming, to a broader class of non-polynomial systems. A numerical method for constructing these constraints is proposed. The relationship between polynomial constraints and existing integral quadratic constraints (IQCs) is investigated, providing transformations of IQCs into polynomial constraints. The effectiveness of polynomial constraints in characterizing nonlinearities is validated via numerical examples to compute inner estimates of the region of attraction for two systems.

Figures (5)

• Figure 3: Simple local polynomial constraints on two nonlinearities. Shaded regions indicate nonnegativity. The quadratic-based constraint is a 4th degree polynomial and the cubic-based and Padé approximant-based constraints are 6th degree. The Padé approximant-based constraints are both computed using \ref{['eq:pade-approx-constraint']} with $\epsilon_1 = \epsilon_2 = 0.1$, $k = 1$, and the $[3/2]$ Padé approximant.
• Figure 4: Triple Integrator: Local constraints on $\tanh(v) - v$. Shaded regions indicate nonnegativity. The numerically-constructed polynomial constraint requires the additional constraint that $w \in [-3, 3]$ for the constraint to be valid for $v \in [-4, 4]$ to exclude regions where $|w| \gg |v|$. The Padé approximation-based constraint is valid for $v \in [-4, 4]$, and the sector constraint is valid for $[-2.1, 2.1]$. The polynomial constraints give tighter bounds on $\tanh(v) - v$ than does the sector constraint, while also being valid for a wider region.
• Figure 5: Triple Integrator: Regions of attraction, synthesized using different constraints on $\Delta$, projected onto planes. The ROA computed using the polynomial constraints has a larger volume. Trajectories converging to the origin plotted in green.
• Figure 6: System with Exponential: Local constraints on $e^v - v - 1$. Shaded regions indicate nonnegativity. The sector constraint is valid for $v \in [-1, 0.53]$. The polynomial constraint is valid for $v \in [-4, 2]$ with the additional constraint that $w \in [0, 3.1]$ to exclude regions where $|w| \gg |v|$. The polynomial constraints give tighter bounds than does the sector constraint, in particular in the upper bound where $v > 0$, which is important for stability analysis of \ref{['eq:exp-sys']}.
• Figure 7: System with Exponential: Inner estimates of region of attraction of origin, computed using two different characterizations of $\Delta$. The green lines denote trajectories converging to the origin and the red lines denote trajectories diverging from the origin. The ROAs computed using the polynomial constraint have larger areas, with additional computation leading to larger ROAs.

Theorems & Definitions (11)

• Definition 1
• Theorem 1
• proof
• Remark 1
• Theorem 2
• proof
• Lemma 1
• proof
• Corollary 1
• Corollary 2