Abstract
Stabilizing a system in a neighborhood of a steady state is one of the first goals of Control Theory. For this purpose, we cast a glow in § 4.2 on the notions of stability and of asymptotic stability of an equilibrium point for general dynamical systems as discussed in § 2.4. The case of linear dynamical systems is treated in § 4.3. For linear systems in the plane, we provide a detailed classification of the stability of the zero equilibrium in § 4.4. In § 4.5, we introduce the tangent linear system and we discuss to what extent the stability of an equilibrium point can be deduced from the stability properties of the tangent linear system. Stability theory in the sense of Lyapunov and, especially, Lyapunov functions are discussed in § 4.6. At last, we consider controlled nonlinear dynamical systems, and discuss how to stabilize them locally around an equilibrium point in § 4.7.
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Notes
- 1.
The symbol \(\mathrm o \) corresponds to the Small-o notation: \(f({z})=\mathrm o \big (g({z})\big )\) as \({z}\rightarrow {z}_{0}\) if and only if \(|f({z})| / |g({z})| \rightarrow ~0\) as \({z}\) goes to \({z}_{0}\).
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© 2013 Springer-Verlag Berlin Heidelberg
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d’Andréa-Novel, B., De Lara, M. (2013). Stability of an Equilibrium Point. In: Control Theory for Engineers. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34324-7_4
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DOI: https://doi.org/10.1007/978-3-642-34324-7_4
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34323-0
Online ISBN: 978-3-642-34324-7
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