Abstract
AbstractIt is claimed in [1] that the notion of the relative degree in nonlinear control theory is closely related to that of the differentiation index for nonlinear differential‐algebraic equations (DAEs). In this paper, we give more insights on this claim via a recent proposed concept (see [2]) called the explicitation of DAEs. The explicitation attaches a class of control systems to a given DAE, we show that the relative degree of the systems in the explicitation class is invariant in some sense and that the differentiation index of the original DAE coincides with the maximum of the relative degree of the explicitation systems.
Highlights
We study nonlinear differential-algebraic equations (DAEs) of the following form
The notion of the differentiation index plays a significant role in the numerical solution theory of nonlinear DAEs [1,5,6], which is the least number of differentiations such that the differential array of (3) could recover x as a function of x and t only
Apart from the similarities in the definitions and initiatives of the differentiation index and the relative degree, we show one simple case that the two notions are related: For a SISO control system Σ of form (2), by setting the output y = 0, we get a DAE with the generalized state (x, v) ∈ Rn+1
Summary
We study nonlinear differential-algebraic equations (DAEs) of the following form. We study nonlinear control systems of form (2), the relative degree (see Definition 3.1 below) of (2) is, roughly speaking, the number of differentiations of the outputs y such that the inputs u can be recovered. Apart from the similarities in the definitions and initiatives of the differentiation index and the relative degree, we show one simple case that the two notions are related: For a SISO control system Σ of form (2), by setting the output y = 0, we get a DAE with the generalized state (x, v) ∈ Rn+1. We use a recent proposed method called the explicitation which attaches a class of control systems to a DAE, we study the relations of the relative degree of the control systems in the explicitation class and the differentiation index of the DAE
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