As well known, switched systems consist of a family of subsystems described by differential or difference equations, along with a switching law that orchestrates the transitions between these subsystems. In the few decades, switched systems have garnered considerable attention from researchers due to their powerful potential in various practical applications, such as stirred tank reactors [1], automobile control systems [2], and other fields. Numerous theoretical results related to switched systems have been provided, especially concerning the Lyapunov stability [3,4,5,6].

It is important to note that the previous research results have mainly focused on asymptotic stability and exponential stability, which describe the system’s behavior over an infinite-time interval. However, in practical industrial processes such as flight control [7] and mobile robots [8], the operating time may be specific and finite. This has sparked significant attention to the issue of finite-time stability (FTS) ever since it was first introduced in [9]. In some cases, only the output, not the state, needs to be constrained within a bound. As a special case of FTS, input-output finite-time stability (IO-FTS) examines the effects of output constraints on system performance [10,11]. A system is considered IO-FTS if, for a given class of norm bounded input signals over a specified time interval T, the outputs of the system do not exceed a predetermined threshold during T. There has been significant research on the problem of IO-FTS, covering various system types, such as singular systems [12,13], fuzzy systems [14], Markovian systems [15], and so on. More recently, several interesting results have been investigated concerning IO-FTS for switched systems [16,17,18]. In [16], through the construction of several linear copositive Lyapunov functions and the utilization of the mode-dependent average dwell time (MDADT) methodology, sufficient conditions are derived to ensure the IO-FTS of the resultant closed-loop switched system. A suitable state feedback controller is designed for switched singular continuous-time systems in [17].

In an ideal scenario, the switching of the controller is perfectly synchronized with the switching of the system. However, in many real-world situations, there can be a delay in the switching signal available to the controller compared to the system’s switching, resulting in asynchronous switching in the closed-loop system. This asynchrony arises due to the time required for communication between the system and the controller through a communication channel. As a result, there are inevitable instances of asynchronous switchings between the system mode and the controller. As well known, sliding mode control (SMC) has proven to be an effective robust control approach against uncertainties and external disturbances, with successful applications in a wide variety of complex systems and engineering domains. The SMC has been extended to switched systems [19,20,21,22]. However, it’s worth noting that all the aforementioned works considered system dynamic behaviors over a sufficiently long (in principle infinite) time interval without any constraint on transient dynamics. To address finite-time SMC, Zhao et al. [23] explored the finite-time bound (FTB) problem of SMC for a class of switched systems with unmeasured states but still under the synchronous switching assumption. However, there is a lack of results regarding the IO-FTS problem of SMC for a class of switched systems under asynchronous switching.

In this work, we will investigate the IO-FTS problem of SMC for a class of uncertain switched systems under asynchronous switching. Firstly, the IO-FTS concept is introduced for the relevant switched system and a suitable SMC law is designed to ensure the reachability of the mode-dependent sliding surface within a finite-time interval. Utilizing ADT and MLF techniques, the corresponding IO-FTS for both the reaching phase and sliding motion phase are attained under asynchronous switching. Subsequently, we present a partitioning strategy that effectively analyzes the IO-FTS characteristics of the switched system over the whole finite-time interval.

Notations. Throughout this paper, the symbol $\left|\right|·\left|\right|$ denotes a real vector or induced matrix norm for vectors in the Euclidean space. ${\mathbb{R}}^{+}$ denotes the set of nonnegative real numbers. ${\mathbb{R}}^n$ is the n-dimensional vector space and $\mathbb{N}$ is the set of nonnegative integers. The notation $P>0(<0)$ means that P is real symmetric and positive definite (negative definite) matrix and I is used to represent an identity matrix of appropriate dimensions. For any symmetric matrix P, ${\lambda }_{max}\left(P\right)$ and ${\lambda }_{min}\left(P\right)$ denote the maximum and minimum eigenvalues of matrix P, respectively, and `*’ represents an ellipsis for terms induced for symmetry. And we define $\mathrm{He}\left\{P\right\}\triangleq P+P^T$.

Consider the uncertain switched system as follows: where $x\left(t\right)\in {\mathbb{R}}^n$ represents the state; $u\left(t\right)\in {\mathbb{R}}^m$ is the control input; $y\left(t\right)\in {\mathbb{R}}^q$ is the measured output; $w\left(t\right)\in {\mathbb{R}}^r$ is the external disturbance; $f_{\sigma \left(t\right)}(x\left(t\right),t))\in {\mathbb{R}}^m$ is a nonlinear function satisfying $\parallel f_{\sigma \left(t\right)}(x\left(t\right),t))\parallel \le {ϵ}_{\sigma \left(t\right)}\parallel x\left(t\right)\parallel$ with ${ϵ}_{\sigma \left(t\right)}>0$ a known constant. The matrices $\{A_{\sigma \left(t\right)},B_{\sigma \left(t\right)},C_{\sigma \left(t\right)},D_{\sigma \left(t\right)}:\sigma \left(t\right)\in {\mathcal{N}}_c\}$ is a family of known matrices depending on an index set ${\mathcal{N}}_c=\{1,2,\dots ,s\}$, and $\sigma \left(t\right):{\mathbb{R}}^{+}\to {\mathcal{N}}_c$ specifies the index of the active subsystem at each time instant t. The switching signal of subsystems is given by $\sigma \left(t\right):\{(i_0,t_0),\dots ,(i_k,t_k),\dots |i_k\in {\mathcal{N}}_c,k\in \mathbb{N}\}$, which means that the $i_k$-th subsystem is activated when $t\in [t_k,t_{k+1})$. Furthermore, $\sigma \left(t\right)=i$ indicates the activation of the i-th subsystem. Due to the asynchronous switching, the practical switching instant of the controller may differ from that of the system. For convenience, ${\sigma }^{\prime }\left(t\right)$ is introduced to denote the practical switching time of the controller. ${\sigma }^{\prime }\left(t\right):\{(i_0,t_0+{\Delta }_0),\dots ,(i_k,t_k+{\Delta }_k),\dots |i_k\in {\mathcal{N}}_c,k\in \mathbb{N}\}$ implies that the $i_k$-th controller operates within the interval $t\in [t_k+{\Delta }_k,t_{k+1}+{\Delta }_{k+1})$, where $|{\Delta }_k|<{inf}_{k\ge 0}(t_{k+1}-t_k)$, ${\Delta }_k>0$ is said to be the mismatched period between the controller and the system.

For each switching signal $\sigma \left(t\right)=i,i\in {\mathcal{N}}_c$, we denote the system associated with the i-th subsystem by Thus, system (1) can be rewritten as

Next, we will proceed to establish the definition of input-output finite-time stability on the switched system (1). Additionally, the definition of the average dwell time is also recalled for the development of the main results in this work.

In this work, we utilize integral sliding mode control to effectively stabilize a switched system (1) under asynchronous switching. In general, there are two periods for SMC design. The first period is the reaching phase, that is, the state trajectories are driven onto the established sliding mode surface. During this period, the state is controlled by the designed sliding mode controller. The second period is the sliding motion, that is, the state trajectories move along the sliding mode surface. During this period, the state may be taken under the enforcement of equivalent control law $u_{eq}\left(t\right)$.

For the purpose of presentation clarity, it is convenient to denote the mismatched period as ${\Omega }_1$ and the matched period as ${\Omega }_2$.

The integral sliding surface function is designed as follows: where matrix G is chosen such that $GB$ is nonsingular and $K_j$ will be designed later.

Within the designated finite interval $[0,T]$, a suitable sliding mode controller is designed to drive the trajectories of the estimated state onto the specified sliding surface $s\left(x\right(t),j)=0$ in a finite time $T^{*}$ with $T^{*}<T$ and then are maintained there for the remaining time interval $[T^{*},T]$. In order to achieve this objective, the SMC law is constructed as where the robust term $\eta \left(t\right)$ is given as with ${\delta }_1\triangleq {max}_{i\in {\mathcal{N}}_c}\left\{{\delta }_{1i}\right\}$, ${\delta }_2\triangleq {max}_{i\in {\mathcal{N}}_c}\left\{{\delta }_{2i}\right\}$, ${\delta }_{1i}\triangleq \parallel G{A}_i\parallel +\parallel G{E}_i\parallel \parallel H_i\parallel +{ϵ}_i\parallel GB\parallel$, ${\delta }_{2i}\triangleq \delta \parallel G{D}_i\parallel$. The gain $K_j$ will be obtained in Theorem 4 and $\varrho$ is the adjustable parameter to be further described in Theorem 1.

In the forthcoming sections, the aforementioned SMC law will be firstly proven to ensure the reachability of the sliding surface within the interval $[0,T^{*}]$. Subsequently, the derived sufficient conditions for IO-FTS of the closed-loop switched system during both the reaching phase $[0,T^{*}]$ and the sliding motion phase $[T^{*},T]$ will be presented.

The analysis of the reachability of sliding surface $s\left(x\right(t),j)=0$ will be analyzed in the subsequent theorem.

In this section, IO-FTS problems of the switched systems (1) during the reaching phase and the sliding motion phase will be separately analyzed using the ADT method, respectively. Subsequently, the sliding mode gains for the whole phase will be provided.

During $t\in {\Omega }_1$, the SMC law mismatches with the subsystem, that is, $\sigma \left(t\right)=i\ne {\sigma }^{\prime }\left(t\right)=j$. Then, the SMC law is formulated as

Substituting the above SMC law into the switched system (1), the closed-loop switched system can be obtained as follows: with ${\overline{A}}_{ij}=A_i+\Delta A_i+\overline{B}{K}_j$, $\overline{B}=B{\left(GB\right)}^{-1}$, ${\eta }_s(t,j)=(\varrho +\eta \left(t\right))\mathrm{sgn}\left(s(x\left(t\right),j)\right)$.

When $t\in {\Omega }_2$, the matched SMC law can be described as

By substituting (20) into the switched system (1), we can get the closed-loop system with ${\overline{A}}_i=A_i+\Delta A_i+\overline{B}{K}_i$, ${\eta }_s(t,i)=(\varrho +\eta \left(t\right))\mathrm{sgn}\left(s(x\left(t\right),i)\right)$.