Section: New Results

Local Analysis of Lurie Systems

Participants : Elena Panteley [L2S,CNRS] , Stephen Duncan [University of Oxford] , Thomas Lathuiliere [University of Oxford] , Giorgio Valmorbida.

An important aspect of nonlinear systems is the fact that stability might only be a local property. This means that associated to a stable equilibrium point or periodic trajectory, there is a region of attraction. Such a region is formed by points of trajectories converging to the stable sets. An important task of practical interest is then to estimate these regions via numerical methods that rely on the model of the system. As an illustration, it might be of interest to know the region of safe operation of an electric motor in order to preserve its integrity or, in the case of an autonomous vehicle, limit the operating condition for safety purposes.

For the particular class of Lurie systems, namely systems defined by the interconnection of a linear system and a static nonlinearity, it is possible to compute estimates based on sector inequalities characterizing the nonlinearities in the system. If further information, such as the slope of the nonlinearity is available, one can better characterize local properties such as regions of stability, and input-output relations such as reachable sets and local nonlinear gains.

To obtain these characterizations we rely on numerical methods based on convex optimization. These methods are based on the solution of Lyapunov inequalities yielding Lyapunov functions that are quadratic on both the states and the nonlinearity and has an integral term on the nonlinearity [39].

Moreover, whenever a more precise characterization of the nonlinearity is at hand as for instance nonlinearities having rational Jacobian, one can generalize the local analysis methods using polynomial optimization. This includes the case of standard Lurie systems by considering the interconnection of a polynomial system with static sector nonlinearities that have rational Jacobian. In this setting we have proposed conditions that relax the requirement on the candidate Lyapunov function [17], which serve as stability certificates, from being sum-of-squares of polynomial with respect to the nonlinearities and the Lurie-Postnikov terms from being non-negative.

Further to the stability analysis we were interested in another important phenomenon and its analysis through numerical methods : the existence of limit cycles on nonlinear systems. Such a phenomenon is relevant since it can be used as a method to design stable oscillators with known amplitude and frequency but also to evaluate and suppress undesirable oscillations in engineered systems. In order to proceed with this analysis we have limited our attention to a particular class of systems defined by a Liénard systems and formulate sufficient conditions for existence and uniqueness of limit cycles for systems with a non-differentiable vector field. As an application we consider the example of a linear system with saturation [24]. Moreover, for planar saturating systems we present sufficient conditions for the existence of periodic orbits and we characterize inner and outer sets bounding the periodic orbits. A method to build these bounds, based on the solution to a convex optimization problem is proposed and numerical examples optimizing the region bounding the limit cycle illustrate the technique [25].