Lyapunov Functions can Exactly Quantify Rate Performance of Nonlinear Differential Equations

TL;DR

This paper develops a unified framework for quantifying rate performance in nonlinear ODEs by introducing beta-stability, a normalized, time-invariant generalization of exponential, rational, and finite-time stability. It proves a necessary-and-sufficient Lyapunov characterization: beta-stability with rate $k$ on a forward-invariant set is equivalent to the existence of a Lyapunov function $V$ satisfying $M^{-1}\alpha(x)\le V(x)\le \alpha(x)$ and $\dot V(x)\le k\rho(V(x))$ (when differentiable), with two-measure extensions. The authors then translate these conditions into SOS programs for exponential, rational, and finite-time stability, enabling computation of tight lower bounds on rate performance and regions of performance; they also demonstrate a conservatism scaling rule in the rational case. Numerical experiments on Lorenz and van der Pol systems show that SOS-derived bounds closely track simulation and reveal regions where rate performance can be certified. Overall, the work provides a practical, computationally tractable approach to quantify and certify rate and gain performance in nonlinear dynamics.

Abstract

Pointwise-in-time stability notions for Ordinary Differential Equations (ODEs) provide quantitative metrics for system performance by establishing bounds on the rate of decay of the system state in terms of initial condition -- allowing stability to be quantified by e.g. the maximum provable decay rate. Such bounds may be obtained by finding suitable Lyapunov functions using, e.g. Sum-of-Squares (SOS) optimization. While Lyapunov tests have been proposed for numerous pointwise-in-time stability notions, including exponential, rational, and finite-time stability, it is unclear whether these characterizations are able to provide accurate bounds on system performance. In this paper, we start by proposing a generalized notion of rate performance -- with exponential, rational, and finite-time decay rates being special cases. Then, for any such notion and rate, we associate a Lyapunov condition which is shown to be necessary and sufficient for a system to achieve that rate. Finally, we show how the proposed conditions can be enforced using SOS programming in the case of exponential, rational, and finite-time stability. Numerical examples in each case demonstrate that the corresponding SOS test can achieve tight bounds on the rate performance with accurate inner bounds on the associated regions of performance.

Figures (8)

• Figure 1: Rational stability bound $\beta_{\textnormal{r}}(y,t):=\frac{y}{1+yt}$ for $y=2$ and $y=1$. Since $\beta_{\textnormal{r}}(2,0.5)=\beta_{\textnormal{r}}(1,0)$, time-invariance of $\beta_{\textnormal{r}}$ implies that also $\beta_{\textnormal{r}}(2,t+0.5)=\beta_{\textnormal{r}}(1,t)$ for all $t\geq 0$.
• Figure 2: Illustration of the set $\mathcal{W}(x):=\{y\in\mathbb{R}_{+}\mid \alpha_{1}(\phi_{f}(x,t))\leq\beta(y,t),~\forall t\in\mathbb{R}_{+}\}$ (in blue) for a particular curve $\alpha_{1}(\phi_{f}(x,t))$. The converse Lyapunov function in the proof of Thm. \ref{['thm:LF_necessity']} is then given by $V(x):=\inf_{y\in\mathcal{W}(x)}y$. Note that if $\beta(y,-t)$ is well-defined and $\beta(\beta(y,t),-t)=y$, then $\beta(\alpha_{1}(\phi_{f}(x,t)),-t)\leq y$ for any $y\in\mathcal{W}(x)$ and $t\geq 0$. It follows that, in this case, $V(x):=\sup_{t\in[0,\infty)}\beta(\alpha_{1}(\phi_{f}(x,t)),-t)$.
• Figure 3: Norm $\left\lVert{\phi_{f}(x,t)}\right\rVert_{2}$ of simulated solution to the Lorenz system in \ref{['eq:Lorenz_ODE']} on logarithmic scale, starting with $x=\textnormal{e}_{2}=(0,1,0)^T$. The exponential bounds $M_{d}e^{-0.4688t}$ for $M_{d}$ as in Tab. \ref{['tab:exponential_stability_Lorenz']} are also plotted for $d\in\{1,2,4,8\}$.
• Figure 4: Greatest lower bound on rate $k_R$ and associated least upper bound on gain $M_R$ for exponential stability of the van der Pol system in \ref{['eq:vdP_ODE']} as a function of radius, $R$, of the centered circular domain on which the conditions were verified as computed by SOSP \ref{['eq:exponential_stability_SOSP']} with $d=8$.
• Figure 5: Forward-invariant domains of the reverse-time van der Pol Equation \ref{['eq:vdP_ODE']} in which exponential stability can be verified with several rates $k$ by solving the SOSP \ref{['eq:exponential_stability_SOSP']}, along with the simulated region of asymptotic attraction in black.