By A.Stephen Morse

A logic-based switching controller is one whose subsystems contain not just widespread dynamical parts similar to integrators, summers, profits and so on. yet event-driven good judgment and linked switches to boot. In any such procedure the predominantly logical part is the manager, mode changer, and so forth. there was growing to be curiosity in recent times in making a choice on what might be won from employing "hybrid" controllers of this sort. To this finish a workshop used to be hung on Block Island with the purpose of bringing jointly participants to debate the learn and customary curiosity within the box. This quantity not just comprises contributions from those that have been current at Block Island but in addition extra fabric from those that weren't. themes lined contain: hybrid dynamical platforms, regulate of hard-bound restricted and nonlinear structures, automobile difficulties related to switching keep an eye on and procedure keep watch over within the face of large-scale modeling blunders.

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T h e systems chosen were the traction control and Anti-Lock Braking systems used in cars. These systems were selected since the controllers developed by the Industry were based on extensive testing and simulation coupled with engineering intuition to obtain an acceptable rulebased controller. There was (and still is) a desire by the companies (in our case, Lucas Automotive Ltd) to see if more a rigorous analysis of these systems was possible. The main reason was to obtain better methods of assessing the safety of these systems, rather t h a n control, since the failure of such a system could lead to expensive litigation.

If the equations are continuous and the time step is chosen sufficiently small, Xk+l is close to a:k and the Newton-Raphson iteration has better chances to converge using zk as initial guess for ~:k+1. In this case, the accuracy with which the trajectory is computed is of no interest. What is important is to reach the equilibrium point with the minimum amount of computation. The time-step is then controlled by the number of iterations needed to compute zk+l and not by the error criterion. 5 Validation An important consideration in the design of embedded controllers is related to system validation: it is common knowledge that validation has to be performed with much more attention than in the past.

The first step is to take each cell or polytope generated from the model and map the movement of the trajectories between the polytope boundaries as dictated by the linear system defined for that polytope's interior. The trajectory tangents will have The Analysis of Piecewise Linear Dynamical Systems 51 two principle behaviours approaching the interior boundaries of the cell: either they will point out indicating an 'exit' from the cell; or they point in, indicating an 'entrance' to the cell.

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