By Mikael K.-J. Johansson
This e-book provides a computational method of the research of nonlinear and unsure structures. the main target is platforms with piecewise linear dynamics. the category of piecewise linear structures tested has nonlinear, almost certainly discontinuous dynamics, and permits switching ideas that comprise reminiscence and common sense. those platforms may perhaps express astonishingly complicated behaviors. a few facets of the profitable idea of linear structures and quadratic standards are prolonged right here to piecewise linear structures and piecewise quadratic standards. The e-book additionally describes numerical approaches for assessing balance, computing caused earnings, and fixing optimum keep watch over difficulties for piecewise linear structures. those advancements let researchers to research a wide and essentially very important type of regulate structures that aren't simply handled whilst utilizing different concepts.
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Extra resources for Piecewise Linear Control Systems: A Computational Approach
Sample text
A~ constructed from piecewise linear functions g(~r) are continuous and piece wise quadratic. However, rather than tailoring the analysis to linear sys terns interconnected with scalar nonlinearities we will aim for results that are applicable to general piecewise linear systems. Motivated by the exam pies above, we will employ Lyapunov functions that are continuous and piecewise quadratic. 5 Interlude: Describing Partition Properties In order to generalize the ideas used in the motivating example we need to be able to construct continuous piecewise quadratic functions on gen eral polyhedral partitions and we need to be able to enforce positivi/y (and negativity) of such functions.
3 is known as the S procedure and is treated in more detail in Appendix A. The method is conservative in general, but appears to work very well in practice. There is a direct relationship between cell identifiers and cell boundings. The key difference is that cell boundings use linear forms for bounding cells that contain the origin. This will be critical for formulating analysis pro cedures using strict LMIs. In Appendix A we give a simple algorithm for translating cell identifiers to cell boundings and show that this step does not introduce any conservatism in the Lyapunov function search.
How ever, each such refinement comes at the price of increased memory require ments for the model representation and increased computational cost for the analysis. It is thus natural to look for the simplest model that gives a s u f ficiently accurate answer in the analysis. The following example illustrates this idea. 11 (left). , [120]). The exact descrip tion has many segments and results in relatively demanding analysis corn putations. It is therefore attractive to base the analysis on an approximate model that requires less computations.