Finite-Time Stability of Hybrid Systems: A Multiple Generalized Lyapunov Functions Approach

TL;DR

This paper introduces a less conservative method for ensuring finite-time stability in hybrid systems using multiple generalized Lyapunov functions that can increase during flows and jumps, broadening applicability.

Contribution

It proposes a novel approach allowing Lyapunov functions to increase, expanding the class of hybrid systems for which finite-time stability can be guaranteed.

Findings

Finite-time stability can be achieved even with increasing Lyapunov functions.

The method is less conservative than previous approaches.

Numerical example confirms the effectiveness of the proposed approach.

Abstract

This paper studies finite-time stability of a class of hybrid systems. We present sufficient conditions in terms of multiple generalized Lyapunov functions for the origin of the hybrid system to be finite-time stable. More specifically, we show that even if the value of the generalized Lyapunov functions increase between consecutive switches, finite-time stability can be guaranteed if the finite-time convergent mode is active long enough. In contrast to earlier work where the Lyapunov functions are required to be decreasing during the continuous flows and non-increasing at the discrete jumps, we allow the generalized Lyapunov functions to increase \emph{both} during the continuous flows and the discrete jumps. As thus, the derived stability results are less conservative compared to the related literature. Numerical example demonstrates the efficacy of the proposed methods.

Equations

$$\begin{array}{r}\dot{y}(t)=f(y(t)),\end{array}$$

$$\begin{array}{r}\dot{V}(y)+cV(y{)}^{\alpha }\le 0,y\in \mathcal{V}\setminus \{0\},\end{array}$$

$$\begin{array}{rl}\dot{x}(t)& =f_{{\sigma }_f}(x(t))x(t)\in C,\\ x(t^{+})& =g_{{\sigma }_g}(x(t))x(t)\in D,\end{array}$$

$$\begin{array}{r}\sum _{k=1}^p|V_F(x(t_{F_{k+1}}))-V_F(x(t_{F_k+1}))|\le \alpha (\Vert x_0\Vert ),\end{array}$$

$$\displaystyle\sum\limits_{k=1}^{p}\Big{(}V_{F}(x(t_{F_{k+1}}))^{1-\beta}-V_{F}(x(t_{F_{k}+1}))^{1-\beta}\Big{)}\leq\bar{\alpha}(\|x_{0}\|).$$

$$\begin{split}\sum\limits_{k=0}^{p}\Big{(}V_{i^{k+1}}(x(t_{{k+1}}))-&V_{i^{k}}(x(t_{{k+1}}))\Big{)}\leq\alpha_{1}(\|x_{0}\|),\end{split}$$

$$\begin{split}\sum\limits_{k=0}^{p}\Big{(}V_{i^{k}}(x(t_{{k+1}}))-&V_{i^{k}}(x(t_{k}))\Big{)}\leq\alpha_{2}(\|x_{0}\|),\end{split}$$

$$\begin{split}\sum\limits_{t\in J_{i}}\Big{(}V_{i}(x^{+}(t))-V_{i}(x(t))\Big{)}&\leq\alpha_{3}(\|x_{0}\|);\end{split}$$

$${\dot{V}}_F\le -c{V}_F^{\beta },$$

$$\begin{array}{r}\sum _{k=1}^p|{\bar{V}}_{F_{k+1}}-{\bar{V}}_{F_k+1}|\le {\alpha }_4(\Vert x_0\Vert );\end{array}$$

$$V_{i^p}(x(t_p))$$

$$\displaystyle\quad+\sum\limits_{k=0}^{p-1}\Big{(}V_{i^{k}}(x(t_{{k+1}}))-V_{i^{k}}(x(t_{k}))\Big{)}$$

$$\displaystyle\quad+\sum\limits_{k=0}^{p}\sum_{t\in J_{k}\bigcap[t_{k},t_{k+1})}\Big{(}V_{i^{k}}(x(t^{+}))-V_{i^{k}}(x(t))\Big{)}$$

$$\stackrel{\text{eqref}Hybcond1,\text{eqref}Hybcond2,\text{eqref}Hybcond3}{\le }{\alpha }_0(\Vert x_0\Vert )+{\alpha }_1(\Vert x_0\Vert )$$

$$+{\alpha }_2(\Vert x_0\Vert )+N{\alpha }_3(\Vert x_0\Vert ),$$

$$V_F(x(t_{F_i}))\le \alpha (\Vert x_0\Vert ).$$

$$\begin{split}&|\bar{T}_{F_{k}}|\leq\frac{\bar{V}_{F_{k}}^{1-\beta}}{c((1-\beta)}-\frac{\bar{V}_{F_{k}+1}^{1-\beta}}{c(1-\beta)}\\ \implies\sum\limits_{k=1}^{M}&|\bar{T}_{F_{k}}|\leq\sum\limits_{k=1}^{M}\Big{(}\frac{\bar{V}_{F_{k}}^{1-\beta}}{c((1-\beta)}-\frac{\bar{V}_{F_{k}+1}^{1-\beta}}{c(1-\beta)}\Big{)}\\ &=\frac{\bar{V}_{F_{1}}^{1-\beta}}{c(1-\beta)}+\sum\limits_{i=1}^{M-1}\frac{\bar{V}_{F_{i+1}}^{1-\beta}-\bar{V}_{F_{i}+1}^{1-\beta}}{c(1-\beta)}-\frac{\bar{V}_{F_{M}+1}^{1-\beta}}{c(1-\beta)}.\end{split}$$

$$\sum _{i=1}^{M-1}\frac{{\bar{V}}_{F_{i+1}}^{1-\beta }-{\bar{V}}_{F_i+1}^{1-\beta }}{c(1-\beta )}\le \frac{\bar{\alpha }(\Vert x_0\Vert )}{c(1-\beta )}.$$

$$\begin{array}{rl}{\bar{T}}_F+\frac{{\bar{V}}_{F_M+1}^{1-\beta }}{c(1-\beta )}& \le \frac{{\bar{V}}_{F_1}^{1-\beta }}{c(1-\beta )}+\sum _{i=1}^{M-1}\frac{{\bar{V}}_{F_{i+1}}^{1-\beta }-{\bar{V}}_{F_i+1}^{1-\beta }}{c(1-\beta )}\\ & \le \gamma (\Vert x_0\Vert ).\end{array}$$

$$\begin{array}{r}{\bar{T}}_F+\frac{{\bar{V}}_{F_M+1}^{1-\beta }}{c(1-\beta )}\le \gamma (\Vert x_0\Vert )={\bar{T}}_F,\end{array}$$

$${\bar{t}}_{F_M+1}-{\bar{t}}_{F_M}$$

$$f_1=\left[\begin{array}{c}0.01x_1^2+x_2\\ -0.01x_1^3+x_2\end{array}\right],f_2=\left[\begin{array}{c}0.01x_1-x_2\\ -x_1^2+0.01x_2\end{array}\right],$$

$$f_3=\left[\begin{array}{c}-x_1-x_2\\ x_1-x_2\end{array}\right],f_4=\left[\begin{array}{c}0.01x_1^2+0.01x_1{x}_2\\ -0.01x_1^3+x_2^2\end{array}\right],$$

$$f_5=\left[\begin{array}{c}{x}_2-20\text{sign}(x_1)|x_1{|}^{\alpha }\\ -10\text{sign}(x_1)|x_1{|}^{2-2\alpha }\end{array}\right].$$

$${\left(\sum _{i=1}^M{z}_i\right)}^r\le \sum _{i=1}^M{z}_i^r\le M^{1-r}{\left(\sum _{i=1}^M{z}_i\right)}^r.$$

$$\begin{array}{r}{a}^r-b^r\le (a-b{)}^r.\end{array}$$

$$\begin{array}{rl}\sum _{i=1}^k(a_i^r-b_i^r)& \le \sum _{i\in I_1}(a_i^r-b_i^r)\le \sum _{i\in I_1}(a_i-b_i{)}^r\\ & \le m^{1-r}{\left(\sum _{i\in I_1}(|a_i-b_i|)\right)}^r.\end{array}$$

$$\begin{array}{rl}& \sum _{i=1}^k(V_F(x(t_{F_{i+1}}){)}^r-V_F(x(t_{F_i+1}){)}^r)\\ \le & m^{1-r}{\left(\sum _{i\in I_1}(|V_F(x(t_{F_{i+1}}))-V_F(x(t_{F_i+1}))|)\right)}^r\\ \le & m^{1-r}{\left(\sum _{i=1}^k|V_F(x(t_{F_{i+1}}))-V_F(x(t_{F_i+1}))|\right)}^r\\ \le & m^{1-r}\alpha (\Vert x(0)\Vert {)}^r\triangleq \bar{\alpha }(\Vert x(0)\Vert )\end{array}$$

Full text

I Introduction

Hybrid systems exhibit continuous state evolution and discrete state jumps, and therefore can capture the behavior of complex dynamical systems. The introductory paper [1] provides an overview and merits of hybrid systems. Stability of switched and hybrid systems has been studied extensively; for a detailed presentation of the solution concepts and the notion of stability for hybrid systems, the interested readers are referred to [2, 3].

The survey paper [4] studies Lyapunov stability (LS), Lagrange stability and asymptotic stability (AS) for stochastic hybrid systems (SHS), and provides Lyapunov conditions for stability in probability. The paper also presents open problems on converse results on the stability in probability of SHS. The review paper [5] discusses sufficient conditions for stability of three classes of hybrid systems, namely, (i) systems modeled on a time scale, (ii) impulsive hereditary systems, and (iii) weakly coupled systems defined on Banach spaces. In [6], the authors study hybrid systems exhibiting delay phenomena such as memory. They establish sufficient conditions for AS using Lyapunov-Razumikhin functions and Lyapunov-Krasovskii functionals. More recently, pointwise AS of hybrid systems is studied in [7], where the notion of set-valued Lyapunov functions is used to establish sufficient conditions for AS of a closed set. In [8], the authors impose an average dwell-time for the discrete jumps and devise Lyapunov-based sufficient conditions for exponential stability of closed sets; see [7, 8] for details on the notion of stability of closed sets.

In contrast to AS, which pertains to convergence as time goes to infinity, finite-time stability (FTS)111With slight abuse of notation, we use FTS to denote the phrase ”finite-time stability” or ”finite-time stable”, depending on the context. is a concept that requires convergence of the solution in finite time [9]. FTS of switched and hybrid systems has gained popularity in the last few years. The authors in [10] consider the problem of designing a controller for a linear switched system under delay and external disturbance with finite- and fixed-time convergence. In [11], the authors design a hybrid observer and show finite-time convergence in the presence of unknown, constant bias. In [12], the authors study FTS of nonlinear impulsive dynamical systems, and present sufficient conditions to guarantee FTS. The work in [11, 12] considers discrete jumps in the system states in a continuously evolving system, i.e., they consider one model for the continuous dynamics, and one model for the discrete dynamics. In this work, we consider the general case with $N_{f}$ continuous flows and $N_{g}$ discrete jump dynamics, where $N_{f},N_{g}$ can be any positive integers. The authors in [13] present conditions in terms of a common Lyapunov function for FTS of hybrid systems. They require the value of the Lyapunov function to be decreasing during the continuous flow and non-increasing at the discrete jumps. In [14], the authors consider a switched system whose subsystems possess a homogeneous Lyapunov function and are of constant switching intervals. In [15], the authors introduce the concept of locally homogeneous system and show FTS of switched systems with uniformly bounded uncertainties. More recently, [16] studies FTS of homogeneous switched systems by introducing the concept of hybrid homogeneous degree, and relating negative homogeneity with FTS. In [17], the authors consider systems in strict-feedback form with positive powers and design a controller as well as a switching law so that the closed-loop system is FTS. In this work, we do not assume that the subsystems of the hybrid system are homogeneous or in strict feedback form, and present conditions for FTS in terms of multiple generalized Lyapunov functions.

In this paper, we consider a general class of hybrid systems under arbitrary switching, and develop sufficient conditions for FTS of its equilibrium. We develop sufficient conditions for FTS of the origin of the hybrid system in terms of multiple generalized Lyapunov functions. We relax the requirement in [13, 18] that the Lyapunov function is non-increasing at the discrete jumps and strictly decreasing during the continuous flow. More specifically, we use generalized Lyapunov functions for each mode, and allow these functions to increase both during the continuous flow and at the discrete jumps, and only require that these increments are bounded. In other words, we allow the hybrid system to have unstable modes. We show that if the origin is stable and if there exists an FTS mode that is active for a sufficiently long cumulative time, then the origin is FTS under arbitrary switching. In our earlier work [19], we have presented conditions for FTS of switched systems. The current paper extends these results to a general class of hybrid systems. To the best of authors’ knowledge, this is a first work on FTS of hybrid systems using multiple generalized Lyapunov functions. We summarize in Remark 2 how earlier results on FTS of continuous-time, switched and hybrid systems are special cases of the results presented in this paper.

The paper is organized as follows: In Section II, we present an overview of FTS followed by conditions for FTS of hybrid systems under arbitrary switching. Section III evaluates the performance of the proposed method via simulation results. Our conclusions and thoughts on future work are summarized in Section IV.

II FTS of Hybrid Systems

II-A Notations

We denote by $\|\cdot\|$ the Euclidean norm of vector $(\cdot)$, $|\cdot|$ the absolute value if $(\cdot)$ is scalar and the length if $(\cdot)$ is a time interval. The set of non-negative reals is denoted by $\mathbb{R}_{+}=[0,\infty)$, set of non-negative integers by $\mathbb{Z}_{+}$ and set of positive integers by $\mathbb{N}$. We denote by $\textrm{int}(S)$ the interior of the set $S$, and by $t^{-}$ and $t^{+}$ the time just before and after the time instant $t$, respectively.

II-B Overview of FTS

Consider the system:

[TABLE]

where $y\in\mathbb{R}^{n}$, $f:D\rightarrow\mathbb{R}^{n}$ is continuous on an open neighborhood $D\subseteq\mathbb{R}^{n}$ of the origin and $f(0)=0$. The origin is said to be an FTS equilibrium of (1) if it is Lyapunov stable and finite-time convergent, i.e., for all $y(0)\in\mathcal{N}\setminus\{0\}$, where $\mathcal{N}$ is some open neighborhood of the origin, $\lim_{t\to T}y(t)=0$, where $T=T(y(0))<\infty$ [9]. The authors also presented Lyapunov conditions for FTS of (1):

Theorem 1** ([9]).**

*Suppose there exists a continuous function $V$: $D\rightarrow\mathbb{R}$ such that the following hold:

(i) $V$ is positive definite

(ii) There exist real numbers $c>0$ and $\alpha\in(0,1)$ , and an open neighborhood $\mathcal{V}\subseteq D$ of the origin such that*

[TABLE]

where $\dot{V}(y)\triangleq(D^{+}(V\circ\psi^{x}))(0)$ is the upper-right Dini derivative of $V(y)$. Then the origin is a FTS for (1).

II-C Main result

In this section, we consider the class of hybrid systems $\mathcal{H}=\{\mathcal{F},\mathcal{G},C,D\}$ described as

[TABLE]

where $x\in\mathbb{R}^{n}$ is the state vector with $x(t_{0})=x_{0}$, $f_{i}(x)\in\mathcal{F}\triangleq\{f_{i}\}$ for $i\in\Sigma_{f}\triangleq\{1,2,\cdots,N_{f}\}$ is the continuous flow for the system which are allowed on the subset of the state space $C\subset\mathbb{R}^{n}$ and $g_{j}(x)\in\mathcal{G}\triangleq\{g_{j}\}$ for $j\in\Sigma_{g}\triangleq\{1,2,\cdots,N_{g}\}$ defines the discrete behavior, which is allowed on the subset $D\subset\mathbb{R}^{n}$. Define $x^{+}(t)\triangleq x(t^{+})$. The switching signals $\sigma_{f}$ and $\sigma_{g}$ are assumed to be piecewise constant, right-continuous signals, which can depend upon both state and time. Details about definitions and solution concepts of the hybrid system (3) can be found in [3].

Denote by $T_{i_{k}}=[t_{i_{k}},t_{i_{k}+1})$ the interval in which the flow $f_{i}$ is active for the $k-$th time for $i\in\Sigma_{f}$ and $k\in\mathbb{N}$, and $t^{d}_{j_{m}}$ as the time when discrete jump $x^{+}=g_{j}(x)$ takes place for the $m-$th time for $j\in\Sigma_{g}$ and $m\in\mathbb{N}$. Define $J_{i}=\{t^{d}_{j_{m}}\;|t^{d}_{j_{m}}\in T_{i_{k}},j\in\Sigma_{g},m\in\mathbb{N}\}$ as the set of all time instances when a discrete jump takes place when the continuous flow $f_{i}$ is active. We assume that the solution $x(t)$ to (3) exists and is maximal. For each interval $T_{i_{k}}$, define the largest connected sub-interval $\bar{T}_{i_{k}}\subset T_{i_{k}}$, such that there is no discrete jump in $\bar{T}_{i_{k}}$, i.e., $\textrm{int}(\bar{T}_{i_{k}})\bigcap J_{i_{k}}=\emptyset$. For example, if $T_{i_{1}}=[0,1)$ and $J_{i}=\{0.2,0.4,0.75\}$, then $\bar{T}_{i_{1}}=[0.4,0.75)$. We make the following assumptions:

Assumption 1**.**

The origin is the only equilibrium point for all the continuous flows and discrete jumps, i.e., $f_{i}(0)=g_{j}(0)=0$ for all $f_{i}\in\mathcal{F}$ and $g_{j}\in\mathcal{G}$.

In this work, per Assumption 1, we restrict our focus on the case when there is a unique equilibrium of the hybrid system (3). The case when there exists some $g_{j}\in\mathcal{G}$ and a set $\bar{D}\neq\{0\}$ such that $g_{j}(x)=0$ for all $x\in\bar{D}\subset D$, can be treated by studying the stability property of set $\bar{D}$; see [13, 18].

Assumption 2**.**

The time interval $\bar{T}_{F_{k}}$ for the FTS mode $F\in\Sigma$ is such that its length satisfies $|\bar{T}_{F_{k}}|\geq t_{d}>0$ for all $k\in\mathbb{N}$.

Assumption 2 implies that for the FTS flow $f_{F}$, in each interval $T_{F_{k}}$ when the system (3) evolves along the flow $f_{F}$, there exists a sub-interval $\bar{T}_{F_{k}}$ of non-zero length $t_{d}$ such that there is no discrete jump in the system state during $\bar{T}_{F_{k}}$. Let $\bar{T}_{F_{k}}=[\bar{t}_{F_{k}},\bar{t}_{F_{k}+1})$ with $\bar{t}_{F_{k}+1}-\bar{t}_{F_{k}}\geq t_{d}$, and $\{\bar{V}_{F_{1}},\bar{V}_{F_{2}},\cdots,\bar{V}_{F_{p}}\}$ and $\{\bar{V}_{F_{1}+1},\bar{V}_{F_{2}+1},\cdots,\bar{V}_{F_{p}+1}\}$ be the sequence of the values of the generalized Lyapunov function $V_{F}$ at the beginning and end of the intervals $\bar{T}_{i_{k}}$, respectively. Let $i^{0},i^{1},\cdots,i^{l},\cdots\in\Sigma_{f}$ be the sequence of modes active on the intervals $[t_{0},t_{1}),[t_{1},t_{2}),\cdots,[t_{l},t_{l+1}),\cdots$, respectively. The conditions for FTS of the origin of (3) in terms of a common Lyapunov function are already given in [13, 18]. In this work, we direct our focus on conditions in terms of multiple generalized Lyapunov functions defined as follows.

Definition 1**.**

(Generalized Lyapunov Function): A continuous, positive definite function $V_{j}:\mathbb{R}^{n}\rightarrow\mathbb{R}_{+}$ is called a generalized Lyapunov function if there exists a continuous function $\phi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ satisfying $\phi(0)=0$, such that $V_{j}(x(t))\leq\phi(V_{j}(x(t_{j_{i}})))$ for all $t\in[t_{j_{i}},t_{j_{i}+1})$.

Definition 2**.**

(Class-$\mathcal{GK}$ function): A function $\alpha:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is called a class-$\mathcal{GK}$ function if it is increasing, i.e., for all $x>y\geq 0$, $\alpha(x)>\alpha(y)$, and right continuous at the origin with $\alpha(0)=0$.

Definition 3**.**

(Class-$\mathcal{GK}_{\infty}$ function): A function $\alpha:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is called a class-$\mathcal{GK}_{\infty}$ function if it is a class-$\mathcal{GK}$ function, and $\lim_{r\to\infty}\alpha(r)=\infty$.

Note that the class-$\mathcal{GK}$ functions have similar composition properties as those of class-$\mathcal{K}$ functions, i.e., for $\alpha_{1},\alpha_{2}\in\mathcal{GK}$ and $\alpha\in\mathcal{K}$, we have:

• •

$\alpha_{1}\circ\alpha_{2}\in\mathcal{GK}$ and $\alpha_{1}+\alpha_{2}\in\mathcal{GK}$;

• •

$\alpha\circ\alpha_{1}\in\mathcal{GK}$, $\alpha_{1}\circ\alpha\in\mathcal{GK}$ and $\alpha_{1}+\alpha\in\mathcal{GK}$.

Lemma 1**.**

If a generalized Lyapunov function $V_{F}$ satisfies

[TABLE]

for all $p\in\mathbb{N}$ where $\alpha\in\mathcal{GK}$, then there exists $\bar{\alpha}\in\mathcal{GK}$ such that the following holds for all $0\leq\beta<1$ and all $p\in\mathbb{N}$:

[TABLE]

Proof.

See Appendix A. ∎

We now present our main result for FTS of hybrid systems.

Theorem 2**.**

The origin of (3) is FTS if there exist generalized Lyapunov functions $V_{i}(x)$ for each $i\in\Sigma_{f}$ and the following hold:

• (i)

There exists $\alpha_{1}\in\mathcal{GK}$, such that

[TABLE]

holds for all $p\in\mathbb{Z}_{+}$;

• (ii)

There exists $\alpha_{2}\in\mathcal{GK}$ such that

[TABLE]

holds for all $p\geq 0$;

• (iii)

There exists $\alpha_{3}\in\mathcal{GK}$ such that for all $i\in\Sigma_{f}$,

[TABLE]

• (iv)

There exists $F\in\Sigma_{f}$ such that $\dot{y}=f_{F}(y)$ is FTS, and there exist a positive definite generalized Lyapunov function $V_{F}$ and constants $c>0$, $0<\beta<1$ such that

[TABLE]

for all $t\in\bigcup[t_{F_{k}},t_{F_{k}+1})\setminus J_{F}$;

• (v)

There exists $\alpha_{4}\in\mathcal{GK}$ such that for all $p\geq 1$,

[TABLE]

• (vi)

The accumulated duration of activation for mode $F$ without any discrete state jump satisfy $\bar{T}_{F}=\sum_{i}|\bar{T}_{i_{k}}|=\gamma(\|x_{0}\|)$ where $\gamma\in\mathcal{GK}$ where $|\bar{T}_{i_{k}}|$ is the length of the interval $T_{i_{k}}$.

Moreover, if all the conditions hold globally, the functions $V_{i}$ are radially unbounded for all $i\in\Sigma_{f}$, and $\alpha_{l}\in\mathcal{GK}_{\infty}$ for $l\in\{1,2,3,4\}$, then the origin of (3) is globally FTS (GFTS).

Proof.

Let $x_{0}\in D$, where $D$ is some open neighborhood of the origin. For all $p\in\mathbb{Z}_{+}$, we have that

[TABLE]

where $\alpha_{0}(r)=\max\limits_{i\in\Sigma,\|x\|\leq r}V_{i}(x)$. Since $V_{i}$ are positive definite, $\alpha_{0}$ is positive definite. Hence, there exists $\alpha\in\mathcal{GK}$ such that $\alpha_{0}(\|x_{0}\|)+\alpha_{1}(\|x_{0}\|)+\alpha_{2}(\|x_{0}\|)+N\alpha_{3}(\|x_{0}\|)\leq\alpha(\|x_{0}\|)$ for all $x_{0}\in D$. Consequently:

[TABLE]

We also know that during $\bar{T}_{F_{k}}$, there is no discrete jump for all $k\in\mathbb{N}$. As per condition (iv), $\dot{V}_{F}\leq-cV_{F}^{\beta}$ for all $t\in\bigcup\bar{T}_{F_{k}}$. Using this and the Comparison lemma, we obtain:

[TABLE]

Using (11), we obtain that $\frac{V_{F_{1}}^{1-\beta}}{c(1-\beta)}\leq\frac{\alpha(\|x_{0}\|)^{1-\beta}}{c(1-\beta)}$. Define $\gamma_{1}(\|x_{0}\|)\triangleq\frac{\alpha(\|x_{0}\|)^{1-\beta}}{c(1-\beta)}$ so that $\gamma_{1}\in\mathcal{GK}$. From condition (v) and Lemma 1, there exists $\bar{\alpha}\in\mathcal{GK}$ such that:

[TABLE]

Define $\gamma(\|x_{0}\|)\triangleq\gamma_{1}(\|x_{0}\|)+\frac{\bar{\alpha}(\|x_{0}\|)}{c(1-\beta)}$ so that we obtain:

[TABLE]

Clearly, $\gamma\in\mathcal{GK}$. Now, with $\bar{T}_{F}=\gamma(\|x_{0}\|)$, we obtain

[TABLE]

which implies $\frac{\bar{V}_{F_{M}+1}^{1-\beta}}{c(1-\beta)}\leq 0$. But $\bar{V}_{F}\geq 0$, which implies $\bar{V}_{F_{M}+1}=0$. Hence, if mode $F$ is active for the accumulated time $\bar{T}_{F}$ without any discrete jump in the system state, the value of the function $V_{F}$ converges to [math] as $t\rightarrow\bar{t}_{F_{M}+1}$.

From Assumption 2, $|\bar{T}_{F_{k}}|\geq t_{d}$ for all $k\in\mathbb{N}$, and hence $Mt_{d}\leq\sum\limits_{i=1}^{M}|\bar{T}_{F_{k}}|=\gamma(\|x_{0}\|)$, which implies that $M\leq\frac{\gamma(\|x_{0}\|)}{t_{d}}<\infty$. Now we show that the time of convergence is finite, i.e., $\bar{t}_{F_{M}+1}<\infty$. From the above analysis, we have that if $\sum\limits_{i=1}^{M}|\bar{T}_{F_{i}}|=\bar{T}_{F}$, then there exists an interval $[\bar{t}_{F_{M}},\bar{t}_{F_{M}+1})$ such that $\bar{V}_{F_{M}+1}=0$. Since $\bar{V}_{F_{i}}\leq\bar{\alpha}(\|x_{0}\|)<\infty$, we obtain that

[TABLE]

for all $x_{0}\in D$. Now, there are two cases possible. If $\bar{t}_{F_{M}}<\infty$, then, we obtain that $\bar{t}_{F_{M}+1}\leq\bar{t}_{F_{M}}+\frac{\bar{\alpha}(\|x_{0}\|)^{1-\beta}}{c(1-\beta)}<\infty$ for all $x_{0}\in D$. If $\bar{t}_{F_{M}}=\infty$, we obtain that the time of activation for mode $F$ reads $\bar{T}_{F}=\sum\limits_{i=1}^{M-1}|\bar{T}_{i}|<\gamma(\|x_{0}\|)$, which contradicts condition (vi). Hence, for condition (vi) to hold, it is required that $\bar{t}_{F_{M}}<\infty$ and hence, we obtain that $\bar{t}_{F_{M}+1}<\infty$. Hence, the trajectories of (3) reach the origin with finite number of active intervals of the continuous flow $f_{F}$.

If all the conditions (i)-(vi) hold globally and the functions $V_{i}$ are radially unbounded, we obtain that $\alpha_{0}$ is also radially unbounded. Using the fact that $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\in\mathcal{GK}_{\infty}$, we have $\bar{\alpha}(\|x_{0}\|)<\infty$ for all $\|x_{0}\|<\infty$, and hence, we obtain $t_{F_{M}+1}\leq t_{F_{M}}+\frac{\bar{\alpha}(\|x_{0}\|)^{1-\beta}}{c(1-\beta)}<\infty$ for all $x_{0}$, which implies GFTS. ∎

Condition (i) means that at switching instants, the cumulative value of the differences between the consecutive Lyapunov functions is bounded by a class-$\mathcal{GK}$ function; condition (ii) means that the cumulative increment in the value of the individual Lyapunov functions when the respective modes are active, is bounded by a class-$\mathcal{GK}$ function (see Figure 1). Some authors use the time derivative condition, i.e., $\dot{V}_{i}\leq\lambda V_{i}$ with $\lambda>0$, in place of condition (ii) to allow growth of $V_{i}$, hence, requiring the function to be continuously differentiable (e.g. [20]). Our condition allows the use of non-differentiable generalized Lyapunov function; condition (iii) means that the cumulative increment in the value of the generalized Lyapunov function $V_{i}$ is bounded by a class-$\mathcal{GK}$ function during the discrete jumps; condition (iv) means that there exists an FTS mode $F\in\Sigma$; condition (v) means that the accumulated increment in the value of the Lyapunov function $V_{F}$ during the ”switched-off” periods is bounded (see Figure 2); and condition (vi) means that the FTS mode $F$ is active for a minimum accumulated time.

Remark 1**.**

In general, the inequality (12) can be difficult to obtain directly. Consider the case when we only know that the mode $F$ is homogeneous, with negative degree of homogeneity. From [21, Theorem 7.2], we know that in this case, condition (iii) of Theorem 2 holds for some $\beta$, but its exact value might not be known. In this case, it is not possible to bound the left-hand side (LHS) of (12). Lemma 1 allows one to bound this LHS with a class-$\mathcal{GK}$ function without explicitly knowing the value of $\beta$.

Compared to [13, 18], our method is less conservative since we allow generalized Lyapunov functions to increase during the continuous flow (per (7)) as well as at the discrete jump (per (8)). Also, during the continuous flow the generalized Lyapunov functions are allowed to grow when switching from one continuous flow to another (per (6)), whereas the aforementioned work imposes that the common Lyapunov function is always non-increasing.

We discuss the relation of Assumption 2 with the average dwell-time (ADT) for discrete jumps method that is often used in the literature [8]. Discrete jumps under ADT means that in any given interval $[t_{1},t_{2}]$, the number of discrete jumps of the states $N([t_{1},t_{2}])$ satisfies $N([t_{1},t_{2}])\leq N_{0}+\delta(t_{2}-t_{1})$, where $\delta\geq 0$ and $N_{0}\geq 1$. (see [8, 22]). We note that if the conditions of the ADT method hold for the mode $F$, instead of the minimum dwell-time condition of Assumption 2, then the FTS result still holds since the parameter $M$ used in the proof of Theorem 2 can be defined as $M=N_{0}+\frac{T_{F}}{\tau_{a}}$, where $T_{F}$ is the total time of activation of mode $F$.

So far we have considered the cases when only one of the modes of the system (3) is FTS. Next, we present a result for the case when all the continuous modes of the hybrid system are FTS.

Theorem 3**.**

The origin of (3) is FTS if the origin of $\dot{x}=f_{i}(x)$ is FTS for all $i\in\Sigma_{f}$ and there exist generalized Lyapunov functions $V_{i}$ for each $i\in\Sigma_{f}$ such that conditions (i), (iii) and (v) of Theorem 2 are satisfied.

Proof.

Let $T_{i}$ denote the total time duration for which mode $i\in\Sigma$ is active without any discrete jump in the state. As the generalized Lyapunov function $V_{i}$ satisfies condition (iv) of Theorem (2), it satisfies the condition (ii) also with $\alpha_{2}=0$. Let $T_{F}(\|x_{0}\|)=\max\limits_{i}\gamma_{i}(\|x_{0}\|)$. In any time interval $[0,T)$, at least one mode $i$ satisfies $T_{i}\geq\frac{T}{N}$. Hence, for $T\geq T_{M}\triangleq NT_{F}(\|x_{0}\|)$, we obtain that there exists at least one mode $i$ satisfying $T_{i}\geq\frac{T}{N}\geq\frac{T_{M}}{N}=T_{F}(\|x_{0}\|)\geq\gamma_{i}(\|x_{0}\|)$. This implies that all the conditions of Theorem 2 are satisfied. Therefore, we obtain that the origin of (3) is FTS. ∎

Remark 2**.**

For switched systems, i.e., systems with no discrete jump, Theorem 2 guarantees FTS of the origin with conditions (i), (ii), (iv), (v) and (vi), since with $D=\emptyset$, condition (iii) is not needed. Hence, the results presented in this paper are applicable to switched systems as well. Furthermore, if $N_{f}=1$, i.e., if the system (3) reduces to a continuous-time dynamical system, Theorem 2 reduces to Theorem 1. Thus, the continuous-time result for FTS is a special case of Theorem 2. Finally, some of the results presented in [13] are special cases of Theorem 2 with $V=V_{F}$ and $\alpha_{3}=0$.

III Simulations

We present a numerical example with number of modes $N=5$, where the fifth mode is FTS, i.e., $F=5$. The system modes are chosen as

[TABLE]

The generalized Lyapunov functions are chosen as $V_{i}(x)=x^{T}P_{i}x$ where matrices $P_{i}$ are chosen as $P_{1}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix},P_{2}=\begin{bmatrix}5&2\\ 2&4\end{bmatrix},P_{3}=\begin{bmatrix}1&0\\ 0&3\end{bmatrix},P_{4}=\begin{bmatrix}6&1\\ 2&3\end{bmatrix}$, and $V_{5}(x)=\frac{k_{2}}{2\alpha}|x_{1}|^{2\alpha}+\frac{1}{2}|x_{2}|^{2}$. The discrete jumps are defined by $g=\begin{bmatrix}-1.1x_{1}\\ -1.1x_{2}\end{bmatrix}$ so that the states $x_{1}$ and $x_{2}$ change their signs and increase in the magnitude at the discrete jumps. Note that this example is more general than the examples considered in [13], as we allow the dynamics to have unstable modes. In this case, we allowed the switches in the continuous flows after $0.2$ sec, i.e., $|T_{i_{k}}|=0.2$ sec and discrete jumps after $0.1$ sec, so that $t_{d}=0.1$ sec, i.e., $|\bar{T}_{i_{k}}|=0.1$, for all $i\in\{1,2,\cdots,5\}$ and $k\in\mathbb{Z}_{+}$ (see Assumption 2). Figure 5 shows the switching signal $\sigma_{f}(t)$. Figure 5 shows the states $x_{1}(t)$ and $x_{2}(t)$ for the first 10 seconds of the simulation. The states change sign at the discrete jumps. It can be seen in the figure that the system moves away from the origin while operating in the unstable modes. Figure 5 shows the norm of the state vector $x(t)$ on log scale. It can be seen that the system states reach the origin in finite time. Figure 6 shows the evolution of generalized Lyapunov functions $V_{i}$ with respect to time for first 10 seconds of the simulation. As can be seen from the figure, the generalized Lyapunov functions increase at the time of discrete jumps, the switching instants, as well as during the continuous flows along the unstable modes.

The provided example validates that the system can achieve FTS even if one or more modes are unstable, if the FTS mode is active for long enough.

IV Conclusions and Future Work

In this paper, we studied FTS of hybrid systems. We showed that under some mild conditions on the bounds on the difference of the values of generalized Lyapunov functions, if the FTS mode is active for a minimum required time, then FTS can be achieved. Our proposed method allows the individual generalized Lyapunov functions to increase both during the continuous flow as well as at the discrete jumps, i.e., it allows for the hybrid system to have unstable modes.

Traditionally, optimization-based formulations are used to find control inputs that satisfy the Lyapunov conditions. Our ongoing research focuses on incorporating input and state constraints in the hybrid framework for specifications involving both spatial and temporal requirements. More specifically, we are investigating how to incorporate prescribed-time, rather than merely finite-time, convergence for the system modes, so that the overall framework can be used in the synthesis and analysis of controllers with spatiotemporal specifications.

Appendix A Proof of Lemma 1

Proof.

Let the function $V_{F}$ satisfy (4) for all $k\mathbb{N}$, where $\alpha\in\mathcal{GK}$. Lemma 3.3, 3.4 of [23] establish the following set of inequalities for $z_{i}\geq 0$ and $0<r\leq 1$

[TABLE]

Hence, we have that for $a\geq b\geq 0$ and $0<r\leq 1$, $a^{r}=(b+(a-b))^{r}\leq b^{r}+(a-b)^{r}$, or equivalently,

[TABLE]

Define $a_{i}\triangleq V_{F}(x(t_{F_{i+1}}))$ and $b_{i}\triangleq V_{F}(x(t_{F_{i}+1}))$ so that whenever $a_{i}\geq b_{i}$, or, $V_{F}(x(t_{F_{i+1}}))\geq V_{F}(x(t_{F_{i}+1}))$, we have that $a_{i}^{r}-b_{i}^{r}\leq(a_{i}-b_{i})^{r}$. Denote $i\in I_{1}\triangleq\{i_{1},i_{2},\cdots,i_{m}\}$ for some $m\leq k$, for which $a_{i}\geq b_{i}$ and $i\in I_{2}\triangleq\{i_{m+1},i_{m+2},\cdots,i_{k}\}$ for which $a_{i}\leq b_{i}$ (or equivalently, $a_{i}^{r}-b_{i}^{r}\leq 0$). Hence, we have that for any $0<r\leq 1$,

[TABLE]

Using this, we obtain that

[TABLE]

where $\bar{\alpha}$ is clearly a class $\mathcal{GK}$ function. Hence, we obtain that (5) holds for all $0<\beta=1-r<1$. ∎