Stabilization of Multivariable Nonlinear Systems: Part I
This chapter addresses the problem of asymptotically stabilizing a multivariable system having vector relative degree. To this end, a multivariable version of the property of being strongly minimum phase is introduced and it is shown that, if a system has
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Stabilization of Multivariable Nonlinear Systems: Part I
10.1 The Hypothesis of Strong Minimum-Phase In this and in the following chapter we consider nonlinear systems having m inputs and m outputs, modeled by equations of the form (9.1) and we suppose that Assumption 9.1 holds. This assumption guarantees the existence of a globally defined change of coordinates x˜ = Φ(x) in which x˜ can be split in two subsets z ∈ Rn−d and ξ ∈ Rd . The set ξ —in turn—is split in subsets, each one of which consists of a string of ri of components ξi,1 , ξi,2 , . . . , ξi,ri which are seen to satisfy equations of the form (9.34).1 The collection of all (9.34), for i = 1, . . . , , is a set of m 1r1 + m 2 r2 + · · · + m r = d equations that characterizes what has been called a partial normal form. A full normal form is obtained by adding the Eq. (9.40) that models the flow of the complementary set z of new coordinates. It should be observed, in this respect, that a strict normal form, namely a normal form in which (9.40) takes the simpler structure (9.41), exists only under additional (and strong) assumptions (see Remark 9.6). The problem addressed in the present and in the following chapter is the design of feedback laws yielding global stability or stability with guaranteed domain of attraction. To this end it is appropriate to consider a property that can be viewed as an extension, to multivariable systems, of the property of strong minimum phase considered in Definition 6.1 and that in fact reduces to such property in the case of a single-input single-output system. Definition 10.1 Consider a system of the form (9.1), with f (0) = 0 and h(0) = 0. Suppose the system satisfies Assumption 9.1 so that a normal form can be globally
1 Recall
that ξi j ∈ Rm i and that m 1 + m 2 + · · · + m = m.
© Springer International Publishing Switzerland 2017 A. Isidori, Lectures in Feedback Design for Multivariable Systems, Advanced Textbooks in Control and Signal Processing, DOI 10.1007/978-3-319-42031-8_10
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10 Stabilization of Multivariable Nonlinear Systems: Part I
defined. The system is strongly minimum-phase if there exist a class K L function β(·, ·) and a class K function γ (·) such that, for any x(0) ∈ Rn and any admissible input function u(·) : [0, ∞) → Rm , so long as x(t) is defined the estimate z(t) ≤ β(z(0), t) + γ (ξ(·)[0,t] )
(10.1)
holds. It is seen from the previous definition that, if a strict normal form exists, the role of the input u(·) in the previous definition is irrelevant and the indicated property reduces to the property that system (9.41), viewed as a system with state z and input ξ , is input-to-state stable. As in Chap. 6, it will be useful to consider also the stronger version of the property in question in which γ (·) and β(·) are bounded as in (6.18). Definition 10.2 A system is strongly—and also locally exponentially—minimumphase if, for any x(0) ∈ Rn and any admissible input function u(·) : [0, ∞) → Rm , so long as x(t) is defined an estimate of the form (10.1) holds, where β(·, ·) a
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