By Mikael K.-J. Johansson
This ebook offers a computational method of the research of nonlinear and unsure platforms. the focus is platforms with piecewise linear dynamics. the category of piecewise linear structures tested has nonlinear, in all likelihood discontinuous dynamics, and permits switching principles that contain reminiscence and common sense. those platforms may well convey astonishingly advanced behaviors. a few features of the profitable conception of linear structures and quadratic standards are prolonged the following to piecewise linear platforms and piecewise quadratic standards. The booklet additionally describes numerical techniques for assessing balance, computing brought on profits, and fixing optimum regulate difficulties for piecewise linear platforms. those advancements let researchers to research a wide and essentially vital classification of regulate structures that aren't simply handled while utilizing different recommendations.
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Extra info for Piecewise Linear Control Systems: A Computational Approach
Example text
6 Piecewise Quadratic Lyapunov Functions The previous section has laid the grounds for analysis of piecewise lin ear systems using piecewise quadratic Lyapunov functions. 3 we showed how it is possible to express the condition that the function be positive on the partition using linear" matrix inequalities. 24) + 2q~z + ri for z E Xi, E I1. We have the following result. 55 Chapter 4. 3) with u = 0 for all t _> 0 tends to zero exponentially. 2. 24) is a L y a p u n o v function for the system. If the partition X is pos itively invariant then every trajectory that starts in X tends to zero expo nentially.
These approaches are substantially more powerful than analysis based on quadratic Lyapunov functions, yet allow the analysis to be be carried out using efficient numerical computations. The application to system analysis and design of optimal control laws is described in the subsequent chapters. As always, it is a good idea to use "simple things first". After determin ing the equilibrium points of a piecewise linear system it is advisable to vet ify local stability properties first. For locally stable equilibrium points, one may invoke the tools developed in this chapter to try to extend the domain of analysis and in many cases establish global results.
I ............ i 1 y s T I / 00875871 .............................. /" ¢ ........... / 1 75[ ........... t ................................................................ 9 Analysis of smooth nonlinear system (left) via piecewise linear sector bounds on the nonlinearity (right). 9. In other words, assume that there exist vectors 1,, and ai, describing upper and lower bounds respectively, such that The closed loop system can then be described by the piecewise linear differ ential inclusion with 0 1 0 0 Notice that the integrator system could be replaced with a general piecewise linear system, and all results would follow similarly.
