Stability of Uncertainty Piecewise Affine Time-Delay Systems with Application to All Wheel Drive Clutch Control.

Abstract

Piecewise affine (PWA) systems provide good flexibility and traceability for modeling a variety of nonlinear systems. The stability of PWA systems is an important but challenging problem since the stability of the sub-systems does not directly imply the stability of the global system. Meanwhile, time delays and uncertainty exist in many practical systems in engineering and introduce various complex behaviors such as oscillation, instability and poor performance. To ensure the stability of the practical control systems developed via the PWA system framework, the stability of uncertain PWA time-delay systems is investigated. In addition, a quantitative description of asymptotic behavior for time-delay systems is also studied. First, the stability problem for uncertain piecewise affine time-delay systems is investigated. It is assumed that there exists a constant time delay in the system and the uncertainly is norm-bounded. Sufficient conditions for the stability of nominal systems and the stability of systems subject to uncertainty are derived using the Lyapunov-Krasovskii functional with a triple integration term. This approach handles switching based on the delayed states (in addition to the states) for a PWA time-delay system, considers structured as well as unstructured uncertainty, and reduces the conservativeness of previous approaches. Second, an application of the PWA system framework to the modeling and control of an automotive all wheel drive clutch system is presented. The open-loop system is modeled as a PWA system, followed by the design of a piecewise linear feedback controller. The stability of the closed-loop system is examined using the proposed stability method. Finally, a new Lambert W function based approach for estimation of the decay function for time-delay systems is presented. Using this solution form, a decay function estimate, which is less conservative than existing methods, is obtained.