Nonlinear physical systems often include phenomena that occur at multiple time scales. To simplify behavior analysis, fast time scale phenomena along with associated small, parasitic effects are commonly abstracted away and system behavior is described as multiple piecewise continuous modes of operation. System models incorporate a meta-level control model operating on top of the data flow model to select active model fragments [6, 10]. However, systems with mixed continuous and discrete components, the so-called hybrid systems, can exhibit extremely complex behaviors due to nonlinearities and discontinuities [7]. Examples of hybrid systems include traffic control systems, electric power circuits, and economic models.
Numerical analysis of these systems is often hampered by the steep
gradient near discontinuities. When the system state reaches or
exceeds a threshold value, the discrete model is invoked, mode switches
occur, and a new model fragment is selected that governs behavior in the active
mode of operation. The threshold function specifies switching surfaces
in phase space along which discontinuous changes in the system may
occur.
Traditional integration
schemes, such as the Runge-Kutta method, are very sensitive to the
steep gradients that occur at these
discontinuities and may perform poorly when a fixed step numerical
approximation is applied.
To permit efficient computational analysis of hybrid systems, we need a well-defined semantics for modeling behaviors at discontinuities, and simulation schemes that can seamlessly combine continuous behavior generation with discrete mode switches. This paper presents a model semantics for a class of hybrid systems operating in the so-called sliding regimes, a region where a system chatters between two different modes such as in the anti-lock braking system, and describes a method for simulating sliding mode behavior efficiently. The switching surfaces in the physical system behavior description arise from modeling artifacts that abstract the hysteresis effects of small, unmodeled parameters. Our simulation algorithm is based on Filippov's construction of equivalence dynamics in sliding regimes. Simulation results on several examples have shown little error when using a large time step along the switching surface.
The sliding mode simulation algorithm relies on the fact that the equivalence dynamics on a sliding surface is defined as the limiting behavior when switching tends to be infinitely fast. We have developed an adaptive algorithm elsewhere that can accurately follow a sliding trajectory and generate control signals at discrete times by exploiting the equivalence in control signals [12]. In contrast, the algorithm described in this paper exploits the equivalence in dynamics for sliding mode systems and presents an alternative method for adaptively following trajectories at discontinuous boundaries.