Peter Giesl

Meshless Collocation: Error Estimates with Application to Dynamical Systems

In this paper, we derive error estimates for generalized interpolation, in particular collocation, in Sobolev spaces. We employ our estimates in collocation problems using radial basis functions and extend and improve previously known results for elliptic problems. Finally, we use meshless collocation to approximate Lyapunov functions for dynamical systems.

On the determination of the basin of attraction of discrete dynamical systems

Consider a discrete dynamical system given by the iteration with exponentially asymptotically stable fixed point . In this paper, we seek to study its basin of attraction using sublevel sets of Lyapunov functions. We prove the existence of a smooth Lyapunov function. Moreover, we present an approximation method of this Lyapunov function using radial basis functions. Error estimates show that one can determine every connected and bounded subset of the basin of attraction with this method. Examples include an application to the region of convergence of Newtons method.

Computation of Lyapunov functions for nonlinear discrete time systems by linear programming

Given an autonomous discrete time system with an equilibrium at the origin and a hypercube containing the origin, we state a linear programming problem, of which any feasible solution parameterizes a continuous and piecewise affine (CPA) Lyapunov function for the system. The linear programming problem depends on a triangulation of the hypercube. We prove that if the equilibrium at the origin is exponentially stable, the hypercube is a subset of its basin of attraction, and the triangulation fulfils certain properties, then such a linear programming problem possesses a feasible solution. We present an algorithm that generates such linear programming problems for a system, using more and more refined triangulations of the hypercube. In each step the algorithm checks the feasibility of the linear programming problem. This results in an algorithm that is always able to compute a Lyapunov function for a discrete time system with an exponentially stable equilibrium. The domain of the Lyapunov function is only limited by the size of the equilibriums basin of attraction. The system is assumed to have a right-hand side, but is otherwise arbitrary. Especially, it is not assumed to be of any specific algebraic type such as linear, piecewise affine and so on. Our approach is a non-trivial adaptation of the CPA method to compute Lyapunov functions for continuous time systems to discrete time systems.

Computation of continuous and piecewise affine Lyapunov functions by numerical approximations of the Massera construction

The numerical construction of Lyapunov functions provides useful information on system behavior. In the Continuous and Piecewise Affine (CPA) method, linear programming is used to compute a CPA Lyapunov function for continuous nonlinear systems. This method is relatively slow due to the linear program that has to be solved. A recent proposal was to compute the CPA Lyapunov function based on a Lyapunov function in a converse Lyapunov theorem by Yoshizawa. In this paper we propose computing CPA Lyapunov functions using a Lyapunov function construction in a classic converse Lyapunov theorem by Massera. We provide the theory for such a computation and present several examples to illustrate the utility of this approach.

On the basin of attraction of limit cycles in periodic differential equations

We consider a general system of ordinary differential equations (x) over dot = f (t, x), where x is an element of R-n, and f (t + T, x) = f (t, x) for all (t, x) is an element of R x R-n is a periodic function. We give a sufficient and necessary condition for the existence and uniqueness of an exponentially asymptotically stable periodic orbit. Moreover, this condition is sufficient and necessary to prove that a subset belongs to the basin of attraction of the periodic orbit. The condition uses a Riemannian metric, and we present methods to construct such a metric explicitly.

Computation and Verification of Lyapunov Functions

Lyapunov functions are an important tool to determine the basin of attraction of equilibria in Dynamical Systems through their sublevel sets. Recently, several numerical construction methods for Lyapunov functions have been proposed, among them the RBF (Radial Basis Function) and CPA (Continuous Piecewise Affine) methods. While the first method lacks a verification that the constructed function is a valid Lyapunov function, the second method is rigorous, but computationally much more demanding. In this paper, we propose a combination of these two methods, using their respective strengths: we use the RBF method to compute a potential Lyapunov function. Then we interpolate this function by a CPA function. Checking a finite number of inequalities, we are able to verify that this interpolation is a Lyapunov function. Moreover, sublevel sets are arbitrarily close to the basin of attraction. We show that this combined method always succeeds in computing and verifying a Lyapunov function, as well as in determini...

MUSCULOSKELETAL STABILIZATION OF THE ELBOW — COMPLEX OR REAL

Both sensory information and mechanical properties of the musculoskeletal system are necessary for fast and appropriate reactions of humans and animals to environmental perturbations. In this paper, we focus on the musculoskeletal system and study the stability of a human elbow in an equilibrium state. We derive a biomechanical model of the human elbow, including an antagonistic pair of muscles, and investigate the stability analytically based on the theory of Ljapunov. Depending on the elbow angle and the level of coactivation, we obtain the following three qualitatively different behaviors: unstable, stable with real eigenvalues, and stable with complex eigenvalues. If the eigenvalues are real, then the system is critically damped; for complex eigenvalues, solutions near the equilibrium are oscillating. Based on experimental data, we found that in principle real and complex behaviors may occur in human arm movements. The experiments support the analytical predictions. Furthermore, in agreement with the simulations, we found differences in the experimental results among the subjects. The results of this study support the assumption that arm movements around an equilibrium point may be self-stabilized without sensory feedback or motor control, based only on mechanical properties of musculoskeletal systems.

Approximation of domains of attraction and lyapunov functions using radial basis functions

Abstract Consider a general system of autonomous ordinary differential equations and a given asymptotically stable, hyperbolic equilibrium. We present a method to construct an explicit Lyapunov function for this equilibrium with which we can determine a positively invariant subset of its domain of attraction. We first prove the existence of three Lyapunov functions with certain orbital derivatives. Then we approximate them by using radial basis function methods. We prove that, using a sufficiently fine grid, we thus obtain a function with negative orbital derivative.

Implementation of a fan-like triangulation for the CPA method to compute Lyapunov functions

An integral part of the CPA method to compute Continuous and Piecewise Affine Lyapunov functions for nonlinear systems is the generation of a suitable triangulation. Recently, the CPA method was revised by using more advanced triangulations and it was proved that it can compute a CPA Lyapunov function for any nonlinear system possessing an exponentially stable equilibrium. This more advanced triangulation scheme includes a simplicial fan close to the equilibrium of the system, for which the Lyapunov function is computed. In this paper we prove a result that allows for a simpler, more general and more efficient generation of this simplicial fan and thus improves the implementation of the CPA method. Moreover, the simplicial fan subdivision at the origin is also of great importance in methods relying on a conic decomposition of the state-space.

Stability Optimization of Juggling

Biological systems like humans or animals have remarkable stability properties allowing them to perform fast motions which are unparalleled by corresponding robot configurations. The stability of a system can be improved if all characteristic parameters, like masses, geometric properties, springs, dampers etc. as well as torques and forces driving the motion are carefully adjusted and selected exploiting the inherent dynamic properties of the mechanical system. Biological systems exhibit another possible source of self-stability which are the intrinsic mechanical properties in the muscles leading to the generation of muscle forces. These effects can be included in a mathematical model of the full system taking into account the dependencies of the muscle force on muscle length, contraction speed and activation level. As an example for a biological motion powered by muscles, we present periodic single-arm self-stabilizing juggling motions involving three muscles that have been produced by numerical optimization. The stability of a periodic motion can be measured in terms of the spectral radius of the monodromy matrix. We optimize this stability criterion using special purpose optimization methods and leaving all model parameters, control variables, trajectory start values and cycle time free to be determined by the optimization. As a result we found a self-stable solution of the juggling problem.