Abstract
In a system consisting of the interconnection of several component subsystems, some of which could be only poorly modeled, stability analysis and feedback design might not be easy tasks. Thus, methods allowing to understand the influence of interconnections on stability and asymptotic behavior are important. The methods in question are based on the use of a concept of gain, which can take alternative forms and can be evaluated by means of a number of alternative methods. This chapter describes the various alternative forms of such concept of gain, and shows why this is useful in the analysis of stability of interconnected systems. A major consequence is the development of a systematic method for stabilization in the presence of (general class of) model uncertainties.
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Notes
- 1.
- 2.
The concept of dissipation inequality was introduced by J.C. Willems in [13], to which the reader is referred for more details.
- 3.
If, in the actual physical system, the components of the input u(t) are voltages (or currents), the quantity \(\Vert u(t)\Vert ^2\) can be seen as instantaneous power, at time t, associated with such input and its integral over a time interval \([t_0,t_1]\) as energy associated with such input over this time interval.
- 4.
See Sect. A.5 in Appendix A. This is the case if \(x(0)=\varPi _1\), in which \(\varPi _1\) is the first column of the solution \(\varPi \) of the Sylvester equation (A.29).
- 5.
Recall that the norm of a matrix \(T\in \mathbb R^{p\times m}\) is defined as
$$ \Vert T\Vert = \sup _{\Vert u\Vert \ne 0}{\Vert Tu\Vert \over \Vert u\Vert } = \max _{\Vert u\Vert =1}\Vert Tu\Vert . $$ - 6.
A matrix (of real numbers) with this structure is called an Hamiltonian matrix and has the property that its spectrum is symmetric with respect to the imaginary axis (see Lemma A.5 in Appendix A).
- 7.
See Proposition A.1 in Appendix A.
- 8.
See Proposition A.2 in Appendix A.
- 9.
Note that an equivalent condition is that the matrix \( I - D_1D_2\) is nonsingular.
- 10.
- 11.
Since \(\alpha \) is a small angle, it makes sense to take
$$ C_0 = \left( \begin{matrix}- \frac{2\ell _0 g}{J_0}&{} 0 &{} 0 &{} 0\\ \end{matrix}\right) ,$$where \(\ell _0\) and \(J_0\) are the nominal values of \(\ell \) and J, in which case
$$ D_\mathrm{p}= \mathrm{row}\left( 2g\left( \frac{ \ell _0}{J_0}- \frac{ \ell }{J}(\cos \alpha )\right) , \frac{2}{M}(\sin \alpha )\right) .$$ - 12.
- 13.
The inequality (3.56), for each fixed \({\mathscr {X}}\) is a linear matrix inequality in \(\mathbf{K}\) and, for each fixed \(\mathbf{K}\) is a linear matrix inequality in \({\mathscr {X}}\).
- 14.
- 15.
Observe that, since \(\mathrm{Ker}(P)\) and \(\mathrm{Ker}(Q)\) are subspaces of \(\mathbb R^m\), then \(W_P\) and \(W_Q\) are matrices having m rows.
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Isidori, A. (2017). The Small-Gain Theorem for Linear 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_3
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