Region-of-Attraction (ROA) Certificates for Dynamical Systems

Necessary Software: Matlab, and the @polynomial class which is included in the SOSTOOLS release. The software packages SOSTOOLS, Sedumi, Yalmip and PENBMI were used to compute the certificates in the files below, but are not required to manipulate the data.

Polynomial Certificates for Topcu/Packard/Seiler Automatica submission, Matlab .mat file (updated June 7, 2007).

Polynomial Certificates for Topcu/Packard IEEE TAC Special Issue submission, Matlab .mat file (updated July 2, 2007).

G. Chesi, A. Garulli, A. Tesi and A. Vicino, "LMI-based computation of optimal quadratic Lyapunov functions for odd polynomial systems," Int. J. Robust and Nonlinear Control, volume 15, pp. 35-49, 2005.

Example 1
Example 2. Increasing the degree of the multipliers in the Pointwise-max quadratic case yields excellent results.
Example 3, Pointwise-max of quadratics certifies global stability.
Example 4. Four results: quadratic; pointwise-max of quadratics; quartic; and pointwise-max of quartics. Each case improves on the preceding case.
Example 5, bootstrap works well here. Quadratic (with spherical shape factor for p) certifies normalized volume of 21.3. Using that Lyapunov function as a shape factor for the quartic yields a Lyapunov function which certifies a normalized volume of 32.8. Quartic, with spherical shape factor ends up only certifying normalized volume of 18.8.

J. Hauser and M. Lai, "Estimating Quadratic Stability Domains by Nonsmooth Optimization," American Control Conference, 571--576, 1992.

Example (pg 575). The paper uses a quadratic Lyapunov function which certifies a region-of-attraction with Volume=30.7. The Lyapunov functions in the .mat file are quadratic, pointwise-max of 2 quadratics, and quartic, certifying volumes 32.9, 33.9 and 70.4 respectively.

A. Vannelli and M. Vidyasagar, "Maximal Lyapunov Functions and Domains of Attraction for Autonomous Nonlinear Systems," Automatica, vol. 21, pp. 69-80, 1985.

Example 4 (pg 78). The difference between degree 6, and pointwise-max of degree 6 is minimal, though both are (generally) superior to the two quartic versions. Pointwise-max of quadratics was completely ineffective, and is not included.

T. Vincent and W. Grantham, Nonlinear and Optimal Control Systems, 1997, Wiley-Interscience.

System taken from text. Pointwise-Max functions appear to be ineffective, though the degree 6 polynomial Lyapunov function is significantly better than the quadratic Lyapunov function.

Van der Pol equations are analyzed extensively in many papers. Some certificates obtained using PENBMI are available.

Two simple 1-dimensional examples illustrate that systems with vector fields of even-degree can also be handled.