Abstract
In the former chapter, control problems of linear polytope systems are considered by using the convex hull Lyapunov function. The linear polytope system is a convex combination of a finite set of finite linear systems. Involved set-valued mapping in the differential inclusion is the convex combination; hence, convex theory can be applied to deal with these control problems. In this chapter, the Luré differential inclusion system and its relative control problems are considered. This kind of differential inclusions is different from linear convex hulls; the set-valued mapping satisfies so-called sector condition, i.e., the image of the set-valued mapping is in a cone. Hence, it is a naturally nonlinear mapping.
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Notes
- 1.
The rank of rational matrix is defined on the field of rational functions. Denote that \( q=\mathrm{rank}W(s) \), then there exists at least one \( q\times q \) sub-matrix of W(s) whose determinant is not equal to 0.
- 2.
In fact, it only needs the condition that λ(s) is Hurwitz polynomial with first coefficient one. This assumption is convenient to the proof.
- 3.
\( {U}_{\perp } \) is an \( \left(n-m\right)\times n \) matrix, such that \( \left[\begin{array}{c}\hfill U\hfill \\ {}\hfill {U}_{\perp}\hfill \end{array}\right] \) is invertible and \( U{U}_{\perp}^T=0 \).
- 4.
In order to be simple, since C is full of row rank, \( {C}_{\perp } \) is \( \left(n-r\right)\times n \) matrix, and \( \left[\begin{array}{c}\hfill C\hfill \\ {}\hfill {C}_{\perp}\hfill \end{array}\right] \) is invertible with \( C{C}_{\perp}^T=0 \).
- 5.
When we define the norm of matrix, in order to be consistent with the norm of induced operator, the condition \( \left\Vert AB\right\Vert \le \left\Vert A\right\Vert \left\Vert B\right\Vert \) that is always additional considered, but it is not necessary for a vector.
- 6.
Barbalet Lemma: If f(t) is uniformly continuous, and the integral \( {\displaystyle \underset{0}{\overset{\infty }{\int }}f(t)dt} \) exists, then \( {f(t)\to 0}\;\left(t\to \infty \right) \). Please see Rochafellar (1970).
- 7.
The meaning of \( A\left|G\left(j\omega \right)\right| \sin \left(\omega t+\phi \right) \) is that if \( G(s)={\left[\begin{array}{ccc}\hfill {g}_1(s)\hfill & \hfill \cdots \hfill & \hfill {g}_n(s)\hfill \end{array}\right]}^T \), then \( A\left|G\left(\mathrm{j}\omega \right)\right| \sin \left(\omega t+\phi \right)=A{\left[\begin{array}{ccc}\hfill \left|{g}_1\left(\mathrm{j}\omega \right)\right| \sin \left(\omega t+\angle {g}_1\left(\mathrm{j}\omega \right)\right)\hfill & \hfill \cdots \hfill & \hfill \left|{g}_n\left(\mathrm{j}\omega \right)\right| \sin \left(\omega t+\angle {g}_n\left(j\omega \right)\right)\hfill \end{array}\right]}^T \).
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Han, Z., Cai, X., Huang, J. (2016). Luré Differential Inclusion Systems. In: Theory of Control Systems Described by Differential Inclusions. Springer Tracts in Mechanical Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49245-1_5
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DOI: https://doi.org/10.1007/978-3-662-49245-1_5
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