Appendix: Practical Stability Definitions
This part is devoted to introduce some concepts and results of practical stability properties using in terms of Lyapunov functions [59].
Theory of stability is the basis of the control systems study. Moreover, the concept of practical stability allows to study the properties of a nonlinear system when the state of this system is bringing close to a set instead of the equilibrium point. From a practical point of view, a system will be considered stable if the deviations remain bounded around the equilibrium point. This is clear that, in practice, asymptotic stability towards a domain whose the size has to be determined, for performance checking, is sufficient. If the behavior of the system can be bounded by certain bounds, the notion of practical stability becomes useful. For example, for AC machines the stability towards a domain with specified bounds during a fixed time interval is a concept equivalent to a finite time stability.
Now, we introduce definitions which are useful for to guarantee the practical stability, in terms of Lyapunov functions.
Define the following class of function:
$$\begin{aligned} \mathbf W&= \{d_{1}\in C[\mathfrak {R}^{+},\mathfrak {R}^{+}]:\\&\quad \,\, d_{1}(l) \text{ is } \text{ strictly } \text{ increasing } \text{ in } l \text { and }\\&\quad \,\, d_{1}(l)\rightarrow \infty \text { as } l\rightarrow \infty \}. \end{aligned}$$
Let \(B_{r}=\{e \in \mathfrak {R}^n:\left\| e\right\| \le r\}\). Consider the dynamical system
$$\begin{aligned} \dot{e}=f(t,e), e(t_{0})=e_{0}, t_{0}\ge 0, \end{aligned}$$
(3.114)
System (3.114) is said to be
Definition 3.8
UPS Uniformly practically stable if, given \((\hbar _{1}, \hbar _{2})\) with \(0<\hbar _{1}<\hbar _{2}\), one has
$$\begin{aligned} \left\| e_{0}\right\| \le \hbar _{1} \Rightarrow \left\| e(t)\right\| \le \hbar _{2}, \qquad \forall t\ge t_{0}. \end{aligned}$$
(3.115)
Definition 3.9
UPQS Uniformly practically quasi-stable if, given \(\hbar _{1}> 0\), \(\mathfrak {I}> 0\), \(T>0\) and \(\forall t_{0}\in \mathfrak {R}^{+}\), one has
$$\begin{aligned} \left\| e_{0}\right\| \le \hbar _{1} \Rightarrow \left\| e(t)\right\| \le \mathfrak {I}, \qquad t\ge t_{0}+T. \end{aligned}$$
(3.116)
Definition 3.10
SUPS Strongly uniformly practically stable, if (UPS) and (UPQS) hold together.
A result of the practical stability in terms of Lyapunov-like functions is presented
Theorem 3.8
[59] Assume that
-
(i)
\(\hbar _{1}\) and \( \hbar _{2}\) are given such that \(0<\hbar _{1}<\hbar _{2}\),
-
(ii)
\(V\in C[\mathfrak {R}^{+}\times \mathfrak {R}^n,{R}^{+}]\) and \(V(t,e)\) is locally Lipschitz in \(e\),
-
(iii)
for \((t,e)\in \mathfrak {R}^{+}\times B_{\hbar _{2}}, d_{1}(\left\| e\right\| )\le V(t,e)\le d_{2}(\left\| e\right\| )\) and
$$\begin{aligned} \dot{V}(t,e) \le \wp (t,V(t,e)) \end{aligned}$$
(3.117)
where \(d_{1}\), \(d_{2}\in \mathbf W \) and \(\wp \in C[\mathfrak {R}^{+,2},\mathfrak {R}]\),
-
(iv)
\(d_{2}(\hbar _{1}) < d_{1}(\hbar _{2})\) holds.
Consequently, the practical stability properties of
$$\begin{aligned} \dot{l} =\wp (t,l), l(t_{0})=l_{0} \ge 0, \end{aligned}$$
(3.118)
implies the corresponding practical stability properties of system (3.114).
From the above theorem, the following criteria can be established
Corollary 3.1
[59] In Theorem 3.8, if \(\wp (t,l) = - \alpha _{1} l + \alpha _{2}\), with \(\alpha _{1}\) and \(\alpha _{2} > 0\), it implies strong uniform practical stability (SUPS) of system (3.114).
We can see that the solution of equation
$$\begin{aligned} \dot{l}(t) = - \alpha _{1} l(t) + \alpha _{2} \end{aligned}$$
(3.119)
is of the form
$$\begin{aligned} l(t)= l_{0}e^{-\alpha _{1} t-t_{0}}+\frac{\alpha _{2}}{\alpha _{1}}\left[ 1-e^{-\alpha _{1} (t-t_{0})}\right] , t \ge t_{0} \end{aligned}$$
(3.120)
The strong uniform practical stability of (3.119) is obtained.