Fixed-Time Stability of Time-Varying Hybrid Systems with Time-Delay

Abstract

In this paper, fixed-time stability (FxTS) of time-varying (TV) hybrid systems with time-delay (TD) is investigated. Starting with TV nonlinear systems with TD, some sufficient conditions are given to verify the FxTS of systems by using TV Lyapunov–Krasovskii functional and integral technique. Then, the obtained results are extended to the case of hybrid systems, several FxTS conditions are provided by using the multiple TV Lyapunov–Krasovskii functional and the function-dependent minimum dwell time technique. Comparing with the existing results, the obtained results can provide the more accurate estimation of the convergence time of systems, and can be applied to a wider class of systems. Finally, two numerical examples are given to illustrate the theoretical results.

Appendix

Proof of Corollary 1

Fixed-time convergence We separately consider the cases when $V(x_{t_0 } )>c$ and $V(x_{t_0 } )\le c$.

Case 1 Assume that $V(x_{t_0 } )>c$. From condition (2) we have $\dot{V}(x_t )\le -q(t)V^b(x_t )$, and hence

$$\begin{aligned} \frac{\dot{V}(x_t )}{V^b(x_t )}\le -q(t) \end{aligned}$$

(57)

By integrating both side of (57) from $t_0$ to $t_0 +\bar{t}$, we obtain

$$\begin{aligned} \int _{t_0 }^{t_0 +\bar{t}} {\frac{\dot{V}(x_t )}{V^b(x_t )}} dt\le -\int _{t_0 }^{t_0 +\bar{t}} {q(t)} dt \end{aligned}$$

(58)

where $\bar{t}\ge 0$ and $t_0 +\bar{t}$ denote the first time when $V(x_t )$ reaches c, then we have

$$\begin{aligned} \int _{V(x_{t_0 } )}^c {\frac{dz}{z^b}} \le -\int _{t_0 }^{t_0 +\bar{t}} {q(s)ds} \end{aligned}$$

(59)

and hence

$$\begin{aligned} \int _{t_0 }^{t_0 +\bar{t}} {q(t)} { }dt\le \frac{1}{b-1}\left( {c^{1-b}-V^{1-b}(x_{t_0 } )} \right) \le \frac{c^{1-b}}{b-1} \end{aligned}$$

(60)

Combining with condition (3), we further have

$$\begin{aligned} \int _{t_0 }^{t_0 +\bar{t}} {q(t)} { }dt\le \int _{t_0 }^{t_0 +H_1 } {q(t){ }} dt \end{aligned}$$

(61)

which implies that $\bar{t}\le H_1$, and hence $V(x_t )\le 1$ for $t\ge t_0 +H_1$ from the definition of $\bar{t}$ and condition (2).

Case 2 Assume that $V(x_{t_0 } )\le c$. From condition (3) we have $V(x_t )\le c$ for all $t\ge t_0$.

Combining Cases 1 and 2, we have that $V(x_t )\le c$ for $t\ge t_0 +H_1$.

Furthermore, by using contradiction, we show that there exists a $t_0 +H_1 \le t^*<t_0 +H_2$ such that $V(x_t )=0$ for $t\ge t^*$. Suppose that $V(x_t )>0$ for all $t\in [t_0 +H_1,{ }t_0 +H_2 )$. Then from condition (3), we have that for any $t\in [t_0 +H_1,{ }t_0 +H_2 )$,

$$\begin{aligned} \frac{\dot{V}(x_t )}{V^a(x_t )}\le -p(t) \end{aligned}$$

(62)

By integrating both side of (62) from $t_0 +H_1$ to $t_0 +H_2$, we obtain

$$\begin{aligned} \int _{t_0 +H_1 }^{t_0 +H_2 } {\frac{\dot{V}(x_t )}{V^a(x_t )}} dt\le -\int _{t_0 +H_1 }^{t_0 +H_2 } {p(t)} dt \end{aligned}$$

i.e.,

$$\begin{aligned} \frac{1}{1-a}\left( {V^{1-a}(x_{t_0 +H_1 } )-V^{1-a}(x_{t_0 +H_2 } )} \right) \ge \int _{t_0 +H_1 }^{t_0 +H_2 } {p(t)} dt \end{aligned}$$

(63)

Note that $V(x_{t_0 +H_1 } )\le c$ and condition (4), we further have

$$\begin{aligned} \frac{1}{1-a}\left( {c^{1-a}-V^{1-a}(x_{t_0 +H_2 } )} \right) \ge \int _{t_0 +H_1 }^{t_0 +H_2 } {p(t)} dt\ge \frac{c^{1-a}}{1-a} \end{aligned}$$

(64)

which is a contradiction. Therefore, the assumption does not hold, i.e., there exists a $t_0 +H_1 \le t^*<t_0 +H_2$ such that $V(x_{t^*} )=0$, and hence $V(x_t )=0$ for $t\ge t^*$ from condition (3). i.e., the system (1) is fixed-time convergent and the convergent time $T(\phi )\le T_{\max } =H_2$.

Global asymptotic stability from the above analysis we have that for any given $x_0 \in {{\mathbb {R}}}^n$, $V(x_t )$ is decreasing and $V(x_t )\rightarrow 0$ as $t\rightarrow \infty$, which means that the system (1) is globally asymptotically stable.

In a word, the system (1) is FxTS. This completes the proof. $\square$