Abstract
In this chapter, we consider the class of CT-LTV systems with finite state jumps, which are linear continuous-time systems whose states undergo finite jump discontinuities at discrete instants of time. Such systems, namely Impulsive Dynamical Linear Systems (IDLSs) (Haddad et al., Impulsive and Hybrid Dynamical Systems, Princeton University Press, Princeton, 2006), can be regarded as a special class of hybrid systems, and they can be either time-dependent (TD-IDLSs) if the state jumps are time-driven or state-dependent (SD-IDLSs) if the state jumps occur when the trajectory reaches an assigned subset of the state space, the so-called resetting set. TD-IDLSs can also be seen as a special case of switching linear systems (Liberzon, Switching in Systems and Control, Springer, Berlin, 2003). Lyapunov stability and stabilization of hybrid systems have been thoroughly discussed in the literature (see, for instance, the monographs Liberzon, Switching in Systems and Control, Springer, Berlin, 2003; Haddad et al., Impulsive and Hybrid Dynamical Systems, Princeton University Press, Princeton, 2006; Pettersson, Analysis and Design of Hybrid Systems. Ph.D. Thesis, 1999, and references therein). In this chapter, we propose some necessary and sufficient conditions for the FTS of TD-IDLSs, while only a sufficient condition will be provided for SD-IDLSs.
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Notes
- 1.
r is the number of resetting times in [t 0,t 0+T], i.e., is the cardinality of \(\mathcal{T}\).
- 2.
The Chebyshev center of a set \({\mathcal{D}} \subseteq \mathbb{R}^{n}\) is defined as \(0_{\mathcal{D}}~:=~\arg\min_{x \in \mathbb{R}^{n}} ( \max_{\theta\in\mathcal{S}} \| x- \theta\| _{\infty})\).
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Amato, F., Ambrosino, R., Ariola, M., Cosentino, C., De Tommasi, G. (2014). FTS of IDLSs. In: Finite-Time Stability and Control. Lecture Notes in Control and Information Sciences, vol 453. Springer, London. https://doi.org/10.1007/978-1-4471-5664-2_7
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