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2017 | OriginalPaper | Buchkapitel

6. Stabilization of Nonlinear Systems via State Feedback

verfasst von : Alberto Isidori

Erschienen in: Lectures in Feedback Design for Multivariable Systems

Verlag: Springer International Publishing

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Abstract

This chapter considers single-input single-output nonlinear systems that can be brought, by change of coordinates, to a special form having essentially the same structure as the normal form of a linear system. For such systems, it is also possible to define concepts and properties which identify a class of systems that can be seen as a nonlinear analogue of the class of linear system having all zeros with negative real part. This makes it possible to systematically develop stabilization methods that, in various forms, extend to nonlinear systems the stabilization methods presented in Chap. 2.

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Vorheriges Kapitel Coordination and Consensus of Linear Systems

Nächstes Kapitel Nonlinear Observers and Separation Principle

By “smooth map” we mean a $C^\infty $ map, i.e., a map for which partial derivatives of any order are defined and continuous. In what follows, f(x) and g(x) will be sometimes regarded as smooth vector fields of $\mathbb R^n$.

In fact, the right-hand side of the upper equation for each x is an affine function of u.

Some of the topics presented in this chapter are covered in various textbooks on nonlinear systems, such as [1‐6]. The approach here follows that of [2]. For further reading, see also [14‐16].

The first of the two properties is clearly needed in order to have the possibility of reversing the transformation and recovering the original state vector as $x =\varPhi ^{-1}(\tilde{x})$, while the second one guarantees that the description of the system in the new coordinates is still a smooth one.

Let $\lambda $ be real-valued function and f a vector field, both defined on a subset U of $\mathbb R^n$. The function $L_f\lambda $ is the real-valued function defined as

$$L_f\lambda (x)=\sum _{i=1}^n{\partial \lambda \over \partial x_i}f_i(x):= {\partial \lambda \over \partial x}f(x).$$

This function is sometimes called the derivative of $\lambda $ along f. If g is another vector field, the notation $L_gL_f\lambda (x)$ stands for the derivative of the real-valued function $L_f\lambda $ along g and the notation $L_f^k\lambda (x)$ stands for the derivative of the real-valued function $L_f^{k-1}\lambda $ along f.

See [2, p. 140–142] for a proof of Propositions 6.1 and 6.2.

Let $\lambda $ be a real-valued function defined on a subset U of $\mathbb R^n$. Its differential, denoted $d\lambda (x)$, is the row vector

$$ d\lambda (x) = \left( \begin{matrix} \displaystyle {\partial \lambda \over \partial x_1} \displaystyle {\partial \lambda \over \partial x_2} \cdots \displaystyle {\partial \lambda \over \partial x_n}\end{matrix}\right) := {\partial \lambda \over \partial x}.$$

See, e.g., [2].

See [7] and also [2] for the proof.

Let f and g be vector fields of $\mathbb R^n$. Their Lie bracket, denoted [f, g], is the vector field of $\mathbb R^n$ defined as

$$ [f,g](x) = {\partial g \over \partial x}f(x) -{\partial f \over \partial x}g(x). $$

The vector field $ad_{f}^{k}g$ is recursively defined as follows

$$ ad_{f}^{0}g=g \qquad ad_{f}^{k}g = [f,ad_{f}^{k-1}g].$$

If $\tau $ is a vector field of $\mathbb R^n$, the flow of $\tau $ is the map $\varPhi _t^{\tau }(x)$, in which $t\in \mathbb R$ and $x\in \mathbb R^n$, defined by the following properties: $\varPhi _0^{\tau }(x)=x$ and

$$ {\mathrm{d} \varPhi _t^{\tau }(x) \over \mathrm{d}\,t} = \tau (\varPhi _t^{\tau }(x)). $$

In other words, $\varPhi _t^{\tau }(x)$ is the value at time $t\in \mathbb R$ of the integral curve of the o.d.e. $\dot{x} = \tau (x)$ passing through $x=0$ at time $t=0$. The vector field $\tau $ is said to be complete if $\varPhi _t^{\tau }(x)$ is defined for all $(t,x) \in \mathbb R\times \mathbb R^n$.

For further reading, see [8] and the references cited thererein.

We have tacitly assumed, as in Sect. 2.1, that the triplet A, B, C is a minimal realization of its transfer function.

If we deal with a locally defined normal form, all functions of time are to be seen as functions defined in a neighborhood of $t=0$. Otherwise, if the normal form is globally defined, such functions are defined for all t for which the solution of (6.12) is defined.

In the second part of the definition, of the property of input-of-state stability is invoked. For a summary of the main characterizations of such property and a number of related results, see Sect. B.2 of Appendix B. For further reading about the property of input-to-state stability, see references [10, 11, 14] of Appendix B. For further reading about the notion of a minimum-phase system, see also [8, 9].

See Sect. B.2 in Appendix B.

In what follows, property (6.16) will be sometime expressed in the equivalent form: V(x) is positive definite and proper .

This analogy is the motivation for the introduction of the term “minimum-phase” to indicate the asymptotic properties considered in Definition 6.1.

For additional reading on such design procedure, as well as on its use in problems of adaptive control, see [3, 10].

To check that this is always possible, observe that the difference

$$ \bar{f}(z,\xi ) = f(z,\xi ) - f(z,0) $$

is a smooth function vanishing at $\xi =0$, and express $\bar{f}(z,\xi )$ as

$$ \bar{f}(z,\xi ) = \int _0^1{\partial \bar{f}(z,s\xi ) \over \partial s}ds = \int _0^1\Bigl [{\partial \bar{f}(z,\zeta ) \over \partial \zeta }\Bigr ]_{\zeta = s\xi }\xi ds. $$

With a mild abuse of notation, we use here x instead of $\tilde{x}$, to denote the vector of coordinates that characterize the normal form of the system.

The arguments used in this proof are essentially those originally proposed in [11] and frequently reused in the literature.

In fact, let x be a point of $S_d^c$ for which $\xi =0$. Then $x\in S_0$, which implies $x\in S^{\,\prime }$. Any of such x cannot be in $S^{\,\prime \prime }$.

See Sect. B.1 of Appendix B, in this respect.

Note that such T only depends on the choice of ${\mathscr {C}}$ and $\varepsilon $ and not on the value of k.

For further reading, see also [12].

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