Abstract
As it is the case for linear systems, understanding the influence of interconnections on stability and asymptotic behavior is of paramount importance. In the case of nonlinear systems, a powerful concept in the analysis of interconnections is the notion of gain function of an input-to-state stable system. Using this concept, it is possible to develop a nonlinear version of the small-gain theorem, which is useful in the analysis as well as in the design of feedback laws. This chapter describes this theorem and how it can be used in the design of stabilizing feedback laws for nonlinear systems.
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Notes
- 1.
More precisely, if the \({\mathscr {L}}_2\) gains of the two component systems are upper-bounded by two numbers \(\bar{\gamma }_1\) and \( \bar{\gamma }_2\) satisfying \(\bar{\gamma }_1\bar{\gamma }_2 \le 1\).
- 2.
See [9] for an introduction to the property of input-to-state stability.
- 3.
A function \(\gamma : [0,\infty ) \rightarrow [0,\infty )\) satisfying \(\gamma (0)=0\) and \(\gamma (r)<r\) for all \(r>0\) is called a simple contraction. Observe that if \(\gamma _1\circ \gamma _2(\cdot )\) is a simple contraction, then also \(\gamma _2\circ \gamma _1(\cdot )\) is a simple contraction. In fact, let \(\gamma _1^{-1}(\cdot )\) denote the inverse of the function \(\gamma _1(\cdot )\), which is defined on an interval of the form \([0,r_1^*)\) where
$$ r_1^*= \lim _{r \rightarrow \infty }\gamma _1(r). $$If \(\gamma _1\circ \gamma _2(\cdot )\) is a simple contraction, then
$$ \gamma _2(r)< \gamma _1^{-1}(r) \;\;\mathrm{for}\;\;\mathrm{all}\;\;0<r<r_1^*, $$and this shows that
$$ \gamma _2(\gamma _1(r)) <r\;\;\mathrm{for}\;\;\mathrm{all}\;\;r>0, $$i.e., \(\gamma _2\circ \gamma _1(\cdot )\) is a simple contraction .
- 4.
- 5.
The reader should have no difficulty in checking that a function defined as
has the indicated property.
- 6.
See [1] for further results of this kind.
- 7.
Use the fact that \(x\alpha (x) = |x|\alpha (|x|).\)
- 8.
Note that the resulting system is a special case of the system in Fig. 8.2, namely the interconnection of
$$ \dot{x} = q(v_1,x) + b(v_2)u \quad z = x \quad y = x $$and
$$ \begin{array}{rcl} \dot{x}_\mathrm{p} &{}=&{} f_\mathrm{p}(x_\mathrm{p},z)\\ \dot{\mu }&{}=&{} 0\end{array} \quad \begin{array}{rcl} v_1 &{}=&{} x_\mathrm{p}\\ v_2 &{}=&{} \mu \end{array}$$with control \(u=h_\mathrm{c}(y)\).
- 9.
- 10.
See also [12].
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Isidori, A. (2017). The Small-Gain Theorem for Nonlinear Systems and Its Applications to Robust Stability. In: Lectures in Feedback Design for Multivariable Systems. Advanced Textbooks in Control and Signal Processing. Springer, Cham. https://doi.org/10.1007/978-3-319-42031-8_8
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