It is preferable to develop general purpose algorithms for analysis. This has meant solving partial differential inequalities (PDI's) by the use of finite difference methods over R^n. The study of PDI's has seen a resurgence of interest, mainly in the form of nonlinear H_\infty control.
In this paper, an alternative means of determining the region of stability of systems with piecewise linear dynamics is presented. The method uses a convex polytope of a regular form to approximate sets which are invariant under the system dynamics. It is believed that this method will prove to have algorithmic advantages over finite difference methods.
For ease of presentation, the results are given for the planar case (R^2). This allows geometric explanations to be given for proofs, and a good deal of the imposing mathematics associated with the higher order cases is eliminated. The higher order cases are to be developed elsewhere, but understanding this case is very useful in order to develop an intuition with regards to the mathematics.
Some related work is that which appears in a series of papers [1],[2],[3],[4],[5] (amongst others, see the further references therein). The results in the listed papers are concerned with constrained input systems, which form an important subset of the piecewise linear systems studied herein. The previous work has been concerned with determining the polyhedral sets which are invariant and respect input constraints - i.e., lie within the the linear operating region. The present work is aimed at determining the largest invariant sets within the full range of operating conditions; the reduction of conservatism is at the cost of a more involved set of computations.
The form of the paper will now be summarised. A nonlinear transformation on a polytope is developed which preserves the geometric connection information - an isomorphism. This allows a standard polytope to be distorted in a controlled fashion, allowing it to be used as an approximant. This general isomorphism is then applied to the problem of approximating the region of attraction of the origin, by the tactic of ensuring that the polytope remains invariant under the flow at all times. This generic algorithm on piecewise linear systems is applied to a the case of a linear system with a saturating feedback, and graphical results shown. There then follow a series of comments on issues raised by its development, and brief concluding remarks are given. Finally, the proofs to statements in the body of the paper are deferred to an appendix.
[2] G. Bitsoris and M. Vassilaki. Constrained regulation of linear systems. Automatica, 31(2):223-227, 1995.
[3] P.-O. Gutman and M. Cwikel. An algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states. IEEE Trans. Automat. Contr., 32(3):251 - 254, 1987.
[4] S. Tarboureich and C. Burgat. Positively invariant sets for constrained continuous-time systems with cone properties. IEEE Trans. Automat. Contr., AC-39(2):401-405, 1994.
[5] M. Vassilaki and G. Bitsoris. Constrained regulation of linear continuous-time dynamical systems. Systems and Control Letters, 13:247 - 252, 1989.