Article Contents
Article Contents

Contraction metric computation using numerical Integration and quadrature

• We present a novel method to compute contraction metrics for general nonlinear dynamical systems with exponentially stable equilibria. Such a contraction metric delivers information on the long term behaviour of the system and is robust with respect to perturbations of the dynamics, even perturbations that shift the equilibrium. We prove that our method is always able to deliver a contraction metric in any compact subset of the basin of attraction of an exponentially stable equilibrium. Further, we demonstrate the applicability of the method by computing contraction metrics for two three-dimensional systems from the literature.

Mathematics Subject Classification: Primary: 65L07, 37B25; Secondary: 65L06, 34D20.

 Citation:

• Figure 1.  Example 3.1: the blue surface (left) is the boundary between the area where the verification condition (VP1) is satisfied and where it is not satisfied, while the green surface (right) describes the verification condition (VP2) suggested by the Numerical Integration-CPA method. Hence, $P$ is a contraction metric within the intersection of the areas bounded by both surfaces

Figure 2.  Example 3.1: The yellow dots indicate the failing points of the Lyapunov-like function while the red closed shape is a level set of it. The green surface is the boundary of the area where verification conditions (VP2) are satisfied; the area where the constraints (VP1) are satisfied is a superset of this set, hence both (VP1) and (VP2) are satisfied within the area bounded by the green surface. From the results we can conclude that there is exactly one equilibrium in the set bounded by the red surface, that it is exponentially stable, and that the red set is a subset of its basin of attraction

Figure 3.  Example 3.2. Left: The magenta dots are the equilibria. The blue surface is the boundary between the area where the verification condition (VP1) is satisfied and where it is not satisfied. Middle: The green surface is the boundary between the area where the verification condition (VP2) is satisfied and where it is not satisfied; hence, $P$ is a contraction metric within the intersection of the areas bounded by the blue and green sets. Clearly, in this example the blue set is far behind the green set and does not have any effect on limiting the area. Right: This plot shows a level set of a Lyapunov-like function in red and the failing points of that function in yellow. The set bounded by the red surface is positively invariant subset of the basin of attraction of an exponentially stable equilibrium in its interior

Figure 4.  System (33): The green surface indicates the boundary of the area where (VP2) is satisfied. The yellow points indicate where the Lyapunov-like function fails to satisfy $(V_P)'_+( {\bf x})<0$, and the red set is a level set of $V_P$. From left to right, the first plot is for the small perturbation $\varepsilon = 0.03$, the second plot shows the problem with the large perturbation $\varepsilon = 0.1$, and the last plot displays a new Lyapunov-like function computed for the large perturbation $\varepsilon = 0.1$. Hence, the set bounded by the red surface is a positively invariant subset of the basin of attraction of an exponentially stable equilibrium in the interior of the set for the perturbed system (33) with $\varepsilon = 0.03$ (left) and $\varepsilon = 0.1$ (right)

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