Abstract
A class of differential inclusion systems – the linear polytope systems – is discussed in this chapter. This kind of system can be viewed as another extension of the linear control systems to the set-valued mappings. This chapter contains four sections. In the first section, we present the definition of the linear polytope system and the motivation of investigation. Section 4.2 deals with the convex hull Lyapunov function which is the main tool in this chapter. Then Sect. 4.3 considers the control of the linear polytope system. We apply the conclusions of Sect. 4.3 to deal with saturated control at the last section of this chapter.
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Notes
- 1.
Recall the definition of Moore-Penrose inverse, assume that A is an \( n\times n \) symmetric matrix, \( n\times n \) matrix G is a Moore-Penrose inverse of A, such that \( AGA=G \), \( GAG=A \), \( {(AG)}^T=AG \), \( {(GA)}^T=GA \) hold.
- 2.
In accordance with the general notation, \( \frac{\partial {V}_c\left({x}_0\right)}{\partial x} \) represents \( {\left.\frac{\partial {V}_c(x)}{\partial x}\right|}_{x={x}_0} \).
References
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© 2016 Shanghai Jiao Tong University Press, Shanghai and Springer-Verlag Berlin Heidelberg
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Han, Z., Cai, X., Huang, J. (2016). Linear Polytope Control 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_4
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DOI: https://doi.org/10.1007/978-3-662-49245-1_4
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