An idempotent-analytic ISS small gain theorem with applications to complex process models



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In this dissertation a general nonlinear input-to-state stability small gain theory is developed using idempotent analytic techniques. The small gain theorem presented may be applied to system complexes, such as those arising in process modelling, and allows for the determination of a practical compact attractor in the system’s state space. Thus, application of the theorem reduces the analysis of the system to one semi-local in nature. In particular, physically practical bounds on the region of operation of a complex system may be deduced. The theorem is proved within the context of the idempotent semiring K ⊂ End⊕ 0 (R≥0). We also show that particular to linear and power law input-to-state disturbance gain functions the deduction of the resulting sufficient condition for input-to-state stability may be performed efficiently, using any suitable dynamic programming algorithm. We indicate, through examples, how an analysis of the (weighted, directed) graph of the system complex gives a computable means to delimit (in an easily understood form) robust input-to-state stability bounds. Applications of the theory to practical chemical engineering systems yielding novel results round out the work and conclude the main body of the dissertation.