Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions
Goncalves J. M. and Megretski A. and Dahleh M. A.
IEEE Transactions on Automatic Control, Volume 48, Number 12, Pages 2089-2106, December 2003Abstract
In this paper we develop an entirely new constructive global analysis methodology for a class of hybrid systems known as Piecewise Linear Systems (PLS). This method- ology consists of inferring global properties of PLS solely by studying their behavior at switching surfaces associated with PLS. The main idea is to analyze impact maps, i.e., maps from one switching surface to the next switching surface. These maps are proven globally stable by constructing quadratic Lyapunov functions on switching sur- faces. Impact maps are known to be \unfriendly" maps in the sense that they are highly nonlinear, multivalued, and not continuous. We found, however, that an impact map induced by an LTI ow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex subsets of linear manifolds. This representation of impact maps allows the search for quadratic surface Lyapunov functions (SuLF) to be done by simply solving a set of LMIs. Global asymptotic stability, robustness, and performance of limit cycles and equilibrium points of PLS can this way be e ciently checked. These new results were successfully applied to certain classes of PLS: relay feedback, on/o and saturation systems. Although this analysis methodology yields only su cient criteria of stability, it has shown to be very successful in globally analyz- ing a large number of examples with a locally stable limit cycle or equilibrium point. In fact, it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology. Examples analyzed include systems of relative degree larger than one and of high dimension, for which no other analysis methodology could be applied. This success in globally analyzing certain classes of PLS has shown the power of this new methodology, and suggests its potential toward the analysis of larger and more complex PLS.
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- @Article{,
- author = {Goncalves J. M. and Megretski A. and Dahleh M. A.},
- journal = {IEEE Transactions on Automatic Control},
- title = {Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions},
- year = {2003},
- month = {December},
- number = {12},
- pages = {2089-2106},
- volume = {48}
- }
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