Continuous-time Piecewise Affine Systems Research Articles

Exponential Stability of Continuous-Time Piecewise Affine Systems

Exponential Stability of Continuous-Time Piecewise Affine Systems

Asymptotic Stability of Piecewise Affine Systems With Filippov Solutions via Discontinuous Piecewise Lyapunov Functions

Asymptotic stability of continuous-time piecewise affine systems defined over a polyhedral partition of the state space, with possible discontinuous vector field on the boundaries, is considered. In this article, the feasible Filippov solution concept is introduced by characterizing single-mode Caratheodory, sliding mode, and forward Zeno behaviors. Then, a global asymptotic stability result through a (possibly discontinuous) piecewise Lyapunov function is presented. The sufficient conditions are based on pointwise classifications of the trajectories which allow the identification of crossing, unreachable, and Caratheodory boundaries. It is shown that the sign and jump conditions of the stability theorem can be expressed in terms of linear matrix inequalities by particularizing to piecewise quadratic Lyapunov functions and using the cone-copositivity approach. Several examples illustrate the theoretical arguments and the effectiveness of the stability result.

Recursive estimation in piecewise affine systems using parameter identifiers and concurrent learning

ABSTRACTPiecewise affine systems constitute a popular framework for the approximation of non-linear systems and the modelling of hybrid systems. This paper addresses the recursive subsystem estimation in continuous-time piecewise affine systems. Parameter identifiers are extended from continuous-time state-space models to piecewise linear and piecewise affine systems. The convergence rate of the presented identifiers is improved further using concurrent learning, which makes concurrent use of current and recorded measurements. In concurrent learning, assumptions on persistence of excitation are replaced by the less restrictive linear independence of the recorded data. The introduction of memory, however, reduces the tracking ability of concurrent learning because errors in the recorded measurements prevent convergence to the true parameters. In order to overcome this limitation, an algorithm is proposed to detect and remove erroneous measurements at run-time and thereby restore the tracking ability. Detailed examples are included to validate the proposed methods numerically.

Reachability analysis of continuous-time piecewise affine systems

This paper proposes an algorithm for the characterization of reachable sets of states for continuous-time piecewise affine systems. Given a model of the system and a bounded set of possible initial states, the algorithm employs an LMI approach to compute both upper and lower bounds on reachable regions. Rather than performing computations in the state-space, this method uses impact maps to find the reachable sets on the switching surfaces of the system. This tool can then be used to deduce safety and performance results about the system.