Invariant Manifolds, Zero Dynamics and Stability
Part of the Progress in Systems and Control Theory book series (PSCT, volume 9)
We consider ordinary differential equations which can be written as coupled pairs
with x ∈ ℝn and y ∈ ℝm. It will be tacitly assumed throughout this paper that g, h and all mappings — as s and w — which appear in the sequel are everywhere defined smooth CN-functions of their respective variables for some appropriate integer N ≥ 1. That solutions of an ordinary differential equation exist on a given (finite) time interval will also be taken for granted. Concerning (1.1) our basic assumption is
$$\dot x = g(t,x,y),\dot y = h(t,x,y)$$
so that y = 0 represents a global invariant manifold for the system (1.1). The differential equation
$$h(t,x,0) = 0$$
Then describes the dynamics which prevail within this basic invariant man-ifold. For shortness we refer to (1.3) as to the differential equation of the “zero dynamics” for (1.1) with (1.2).
$$\dot x = g(t,x,0)$$
KeywordsOrdinary Differential Equation Invariant Manifold Polynomial System Background Material Respective Variable
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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© Springer Science+Business Media New York 1991