Volume 48, Issue 9 - 9

In this contribution different approaches are introduced for the description of discrete-continuous (hybrid) systems. The approaches were formulated by different research groups within the program KONDISK supported by the German Research Council DFG. The objectives of the research program are modelling, analysis, verification and synthesis of hybrid systems. The single contributions have thereby a common foundation, they are based on Petri nets.

A unifying modeling method is presented that (1) extends the declarative, equation based, object oriented modeling approach by discontinuous and variable structure components which arise from abstractions in physical system models and (2) combines this with imperative, reactive, discrete event controllers based on the statechart and the sequential function chart formalisms.

In this article, the problem of unmeasureable state variables and model uncertainties in hybrid control systems is considered. First, a state reconstruction method for a class of discrete-continuous systems with piecewise affine dynamics and discrete measurements, developed at Universität Karlsruhe (TH), is presented. Then, a concept for compensation of discrete control errors between a desired and the actual hybrid process state, developed at Technische Universität München, is proposed. An example process is used to illustrate the combined application of the methods in a hybrid control architecture.

In this article a general class of hybrid optimal control problems with continuous and discrete state variables and control inputs is defined. After a brief review of conventional optimal control, major novel challenges resulting from the hybrid nature are discussed. Some application problems are comparatively easy to solve because of the fixed or known sequence of discrete events; however, if the number and the sequence of discrete phases is not known a priori, the solution must then be found among a combinatorial number of possible sequence candidates. The article presents several preliminary approaches to the (numerical) solution of hybrid optimal control problems by hybrid dynamic programming, by decomposition using branch-and-bound, or fixing transversality conditions to obtain suboptimal solutions. The last two methods rely on the capabilities of the direct collocation method DIRCOL to solving multi-phase optimal control problems robustly and efficiently. Results obtained by the proposed methods are presented in two examples: an underactuated robotic system with a holding brake as the discrete component, and a hybrid, motorized traveling salesman problem.

This contribution deals with the synthesis of discrete supervisory control for switched linear systems. Via two successive approximation steps, this hybrid control problem is transformed into a purely discrete problem. The latter can then be solved using a slightly modified version of P. J. Ramadge´s and W. M. Wonham´s "supervisory control theory". By embedding both approximation steps within J. C. Willems´ "behavioural systems theory", we can guarantee that any solution of the resulting discrete synthesis problem will also solve the underlying hybrid problem. All algorithms employed within the approximation and the synthesis procedure terminate after a finite number of steps. The approach is illustrated by means of a simple three-plate thermal system.