Zero-Error Prescribed-Time Tracking Control for Switched Non-Square MIMO Nonlinear Systems Subject to Asymmetric Time-Varying Output Constraints

Abstract

Previous studies typically assume that output constraints are symmetric or time-invariant. However, effectively addressing asymmetric and time-varying output constraints remains an unsolved issue, especially in the case of switched non-square multi-input multi-output (MIMO) nonlinear systems. To tackle this challenge, this paper first establishes a prescribed-time (PT) Lyapunov criterion for switched nonlinear systems. Second, an asymmetric nonlinear mapping (ANM) method is proposed to handle asymmetric time-varying output constraints. Compared to the barrier Lyapunov function (BLF) approach, the ANM method relaxes the initial output conditions and the constraint functions are not necessarily required to be opposite in sign. Finally, a PT tracking controller is designed for a class of switched MIMO nonlinear systems with a non-square control coefficient matrix by using the time-varying gain technique. This controller achieves zero-error tracking within the prescribed time while ensuring that output constraints are satisfied. The effectiveness of the proposed control strategy is validated through numerical and simulation examples.

1. Introduction

Switched systems, comprising multiple subsystems governed by a deterministic switching law, are extensively utilized to model diverse real-world scenarios, including aircrafts, robotics, and network systems. As a result, the control and stabilization of these systems have garnered significant research interest, as presented in [1,2,3] and related studies. It is worth noting that many practical engineering systems are often modeled by more complex MIMO nonlinear systems, which have been extensively studied in [4,5,6,7]. Furthermore, nonlinear systems with a non-square control coefficient matrix find extensive applications across various engineering domains, including spacecraft attitude control, flexible robotic manipulators, and the swing-up control of gymnastic robots, as illustrated in [8,9,10]. However, control strategies for non-square switched MIMO nonlinear systems remain an open research problem. Therefore, exploring the tracking control problem for non-square switched MIMO nonlinear systems presents a challenging and significant research endeavor.

Recently, an increasing number of researchers are focusing on the finite-time (FT) control of nonlinear systems due to its advantages, such as faster convergence rates and enhanced robustness against disturbances. To achieve FT stability for nonlinear systems, various sophisticated and well-designed methods have been proposed, including the homogeneous approach, backstepping design, dynamic gain control, and adding a power integrator (see [11,12,13,14,15]). It is worth noting that the estimation of settling time in FT control heavily depends on initial conditions. To address this limitation, the concept of fixed-time (FxT) stability was introduced (see [16,17,18,19]). FxT stability is notable for its ability to estimate an upper bound of settling time that is independent of initial conditions. However, while FxT stability provides advantages in settling time estimation, the relationship between control parameters and the upper bound of the settling time is often complex, and the settling time is frequently overestimated. As a result, the concept of prescribed-time (PT) stability has been introduced, allowing users to arbitrarily specify the system’s convergence time based on practical requirements, regardless of the initial conditions. Specifically, PT stability has been successfully applied to various complex control problems, including uncertain nonlinear systems with time delays in [20]. Due to its distinct advantages, PT stability has garnered significant research interest in the control community (see [21,22,23]).

On the other hand, many real-world systems are often required to satisfy specific output constraints. As a result, the control of nonlinear systems with output constraints has attracted significant attention, as highlighted in studies such as [24,25,26,27]. The BLF has proven to be an effective tool for both controller design and stability analysis in such systems. As the system state variable approaches the predefined constraint boundary, the BLF value tends to infinity. Ensuring the boundedness of the BLF can effectively prevent constraint violations. However, as noted in [28], control algorithms based on BLFs often require transforming the output constraint problem into a tracking error constraint problem, which results in more conservative initial output conditions. By introducing a nonlinear transformation that is dependent on the output, the authors of [28] proposed an adaptive control method for single-input single-output systems with time-varying output constraints, which relaxes the initial output condition requirements. Nevertheless, this approach does not account for switching or MIMO characteristic in the modeling. Additionally, the time-varying output constraints considered in [28] are not arbitrary functions but must satisfy specific conditions. For instance, the output $y\left(t\right)$ is required to satisfy $-{\epsilon }_l\left(t\right)<y\left(t\right)<{\epsilon }_u\left(t\right)$, where ${\epsilon }_l\left(t\right)$ and ${\epsilon }_u\left(t\right)$ are assumed to be positive functions.

Motivated by these observations, we now formulate the following research question:

For switched MIMO systems with time-varying output constraints and non-square control coefficient matrices, how can we achieve prescribed-time trajectory tracking control?

In this paper, we aim to provide a comprehensive answer to this question. The contributions of this work are listed in the following three aspects:

(1)

The conservatism in initial conditions inherent to BLF-based control methods, as observed in [24,25,26,27], poses significant challenges for practical implementation. Although the integral BLF approach proposed in [29] alleviates this issue to some extent, it remains ineffective in addressing time-varying asymmetric output constraints. To address this limitation, we employ the ANM method, effectively alleviating the conservatism in the system’s initial output constraints.

(2)

The proposed ANM method is applicable to asymmetric time-varying output constraints, including unilateral constraints, unconstrained cases, and constant constraints, offering greater generality compared to the nonlinear transferred function approach discussed in [28]. Our approach permits the constraint functions to be arbitrary smooth time-varying functions, in contrast to [27,28,30], where they are required to have opposite signs.

(3)

Compared to the MIMO systems with square coefficient matrices studied in [4,5,6], this work proposes a unified design approach that is applicable not only to square MIMO systems but also effectively handles non-square MIMO systems, achieving zero-error tracking within a prescribed time.

The rest of this paper is organized as follows. Section 2 presents the system model and preliminaries. Section 3 details the proposed control strategy and conducts an in-depth stability analysis. Section 4 presents numerical and simulation examples with corresponding results to validate the theoretical findings. The paper concludes with some remarks in Section 5.

2. Problem Formulation

2.1. System and Assumptions

This work investigates the following switched MIMO system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_i=x_{i+1},i=1,2,\dots ,n-1\hfill \\ {\dot{x}}_n={\mathbf{H}}_{\sigma }\left(x_1\right)u+{\mathbf{F}}_{\sigma }(t,x),\hfill \\ y=x_1,\hfill \end{array}\right.\end{array}$$

(1)

where $x_i={[x_{i1},x_{i2},\dots ,x_{im}]}^T\in {\mathbb{R}}^m$, $i=1,2,\dots ,n$, $m\le n$, $x={[x_1,x_2,\dots ,x_n]}^T\in {\mathbb{R}}^{nm}$ and $u={[u_1,u_2,\dots ,u_q]}^T\in {\mathbb{R}}^q$ ($q\ge m$) are the state and control input vectors, respectively. Differently from the switching mechanism in [31], where the switching is governed by a stochastic Markov process, our switching signal, denoted as $\sigma :[t_0,+\infty )\to \mathcal{N}:=$$\{1,2,\dots ,N\}$, is a deterministic, piecewise constant and right continuous function. Throughout this paper, we assume that all the states of (1) remain continuous at each switching instant, and there is a finite number of switchings for the switching signal $\sigma \left(t\right)$ within any finite time interval. For any $\sigma \in \mathcal{N}$, the matrix ${\mathbf{H}}_{\sigma }\left(x_1\right)\in {\mathbb{R}}^{m\times q}$ denotes the control coefficient matrix, which is not necessarily symmetric, and the function ${\mathbf{F}}_{\sigma }={[f_{1,\sigma },\dots ,f_{m,\sigma }]}^T\in {\mathbb{R}}^m$ consists of unknown smooth nonlinear vectors, which represents non-vanishing disturbance.

Let us denote ${\epsilon }_l(t)={[{\epsilon }_{l,1}(t),\dots ,{\epsilon }_{l,m}(t)]}^T\in {\mathbb{R}}^m$ and ${\epsilon }_u(t)={[{\epsilon }_{u,1}(t),\dots ,{\epsilon }_{u,m}(t)]}^T\in {\mathbb{R}}^m$ as the constraint function vectors. The objective of this paper is to design a tracking controller $u\in {\mathbb{R}}^q$ that ensures that all state variables remain bounded, while enabling the output $y=x_1$ to accurately track a reference signal $y_r:=y_r\left(t\right)={[y_{r,1}\left(t\right),\dots ,y_{r,m}\left(t\right)]}^T$ within any prescribed time $T_{pre}$ without violating the predetermined output constraint

$$\begin{array}{c}\hfill {\epsilon }_{l,j}(t)<y_j<{\epsilon }_{u,j}(t),j=1,2,\dots ,m.\end{array}$$

(2)

Remark 1. 

We emphasize that the constraint functions ${\epsilon }_{l,j}\left(t\right)$ and ${\epsilon }_{u,j}(t)$, $j=1,2,\dots ,m$ can be any smooth time-varying functions. This distinguishes our approach from the requirement in [27,28,30], where the two constraint functions must have opposite signs. In practical applications, output constraint functions may both be positive, both negative, or differ in sign. Our method offers a more general and flexible framework, broadening its applicability to a wider range of constrained control problems.

Assumption 1. 

For any $\sigma \in \mathcal{N}$, there exist the unknown bounded function $c_{\sigma }(t)$ and the known smooth function $f\left(x\right)$ such that

$$\begin{array}{c}\hfill |{\mathbf{F}}_{\sigma }(t,x)|\le c_{\sigma }(t)f(x),\end{array}$$

(3)

where $c_{\sigma }\left(t\right)$ is bounded by an unknown constant ${\overline{c}}_{\sigma }$.

Assumption 2. 

For any $\sigma \in \mathcal{N}$, the control coefficient matrix ${\mathbf{H}}_{\sigma }$ can be decomposed as

$$\begin{array}{c}\hfill {\mathbf{H}}_{\sigma }(x_1)=\mathbf{P}{\mathbf{Q}}_{\sigma }(x_1),\end{array}$$

(4)

where $\mathbf{P}=(I_m,{\iota }_{m+1},\dots ,{\iota }_q)\in {\mathbb{R}}^{m\times q}$ is a known matrix, with ${\iota }_p\in {\mathbb{R}}^m$, for $p=m+1,\dots ,q$, being nonzero vectors, and ${\mathbf{Q}}_{\sigma }(x_1)\in {\mathbb{R}}^{q\times q}$ is a square matrix that is not necessarily symmetric. Moreover, the matrix $\mathbf{P}({\mathbf{Q}}_{\sigma }(x_1)+{\mathbf{Q}}_{\sigma }^T(x_1)){\mathbf{P}}^T$ is symmetric and positive-definite, i.e.,

$$\begin{array}{c}\hfill {\lambda }_{min}\left(\mathbf{P}({\mathbf{Q}}_{\sigma }(x_1)+{\mathbf{Q}}_{\sigma }^T(x_1)){\mathbf{P}}^T\right)\ge \underline{\lambda }>0,\end{array}$$

where $\underline{\lambda }$ is a known constant and ${\lambda }_{min}(•)$ denotes the minimum eigenvalue of the matrix •.

Assumption 3. 

For any $j=1,2,\dots ,m$, there exist the constants ${\underline{M}}_0,{\overline{M}}_0,M_1$ such that the time-varying functions ${\epsilon }_{l,j}(t),{\epsilon }_{u,j}(t)$ and their time derivative ${\dot{\epsilon }}_{l,j}(t),{\dot{\epsilon }}_{u,j}(t)$ satisfy

$$\begin{array}{c}\hfill {\epsilon }_{l,j}(t)\ge {\underline{M}}_0,{\epsilon }_{u,j}(t)\le {\overline{M}}_0,\left|{\dot{\epsilon }}_{l,j}(t)\right|\le M_1,\left|{\dot{\epsilon }}_{u,j}(t)\right|\le M_1,\end{array}$$

(5)

where ${\dot{\epsilon }}_{l,j}(t),{\dot{\epsilon }}_{u,j}(t)$ denote the first-order time derivatives of ${\epsilon }_{l,j}(t)$ and ${\epsilon }_{u,j}(t)$, respectively, i.e., ${\dot{\epsilon }}_{l,j}(t)\triangleq d{\epsilon }_{l,j}(t)/dt$ and ${\dot{\epsilon }}_{u,j}\left(t\right)\triangleq d{\epsilon }_{u,j}\left(t\right)/dt$.

Assumption 4. 

For any $j=1,2,\dots ,m$, there exist the known time-varying functions $N_{l,j}(t),N_{u,j}(t)$ and positive constants $N_{i,j}$, $i=1,2,\dots ,n$, such that the desired trajectory $y_{r,j}$ satisfies ${\epsilon }_{l,j}(t)<N_{l,j}(t)<y_{r,j}<N_{u,j}(t)<{\epsilon }_{u,j}(t)$ and its i-th time derivatives satisfy $|y_{r,j}^{\left(i\right)}(t)|\le N_{i,j}$, $i=1,2,\dots ,n.$

Remark 2. 

Assumption 2 is introduced to address the non-invertibility issue of non-square coefficient matrices. The non-square nature of the system introduces significant challenges in controller design, as illustrated in inequality (37). Assumption 2 provides the necessary theoretical foundation for achieving prescribed-time tracking control. It is worth noting that similar conditions have been successfully applied in [9,32]. Moreover, this assumption effectively characterizes various practical engineering systems, including spacecraft attitude control, flexible robotic manipulators, and the swing-up control of gymnastic robots, with relevant applications discussed in [8,9,10].

2.2. Preliminaries

In this paper, ${\mathbb{R}}^{+}$ and ${\mathbb{R}}^n$ denote the set of nonnegative real numbers and the real n-dimensional space, respectively. For a given matrix A, $A^T$ denotes its transpose. For a given vector X, $\left|X\right|$ is the Euclidean norm. Define $\parallel A\parallel ={\left({\sum }_{i=1}^n{\sum }_{j=1}^m{a}_{ij}^2\right)}^{{\displaystyle \frac{1}{2}}}$ for a matrix $A\in {\mathbb{R}}^{n\times m}$. Let ${\mathcal{C}}^{1,2}([a,b]\times {\mathbb{R}}^n;{\mathbb{R}}^{+})$ denote all nonnegative functions $V(t,\zeta )$ on $[a,b]\times {\mathbb{R}}^n$, which are continuously differentiable in t and twice continuously differentiable in $\zeta$.

We introduce the following scaling function:

$$\begin{array}{c}\hfill \mu (t)={\left({\displaystyle \frac{T}{t_0+T-t}}\right)}^{\nu },\forall t\in [t_0,t_0+T),\end{array}$$

(6)

where $\nu \ge max\{n,2\}$ is an integer and $T=T_{pre}>0$ is a freely prescribed time. Obviously, $\mu \left(t\right)$ is a monotonically increasing function on $[t_0,t_0+T)$ with $\mu \left(t_0\right)=1$ and ${lim}_{t\to t_0+T}\mu \left(t\right)=+\infty$. Meanwhile, $\mu \left(t\right)$ is continuously differentiable with respect to t and

$$\begin{array}{c}\hfill \dot{\mu }(t)={\displaystyle \frac{\nu }{T}}{\left({\displaystyle \frac{T}{t_0+T-t}}\right)}^{\nu +1}.\end{array}$$

(7)

In the following part, we introduce two lemmas, which are useful for our subsequent controller design and stability analysis.

Theorem 1. 

Consider a switched nonlinear system $\dot{\zeta }=f_{\sigma }(t,\zeta )$ with $\sigma =\sigma \left(t\right)$ being defined as that in (1) and $f_{\sigma }:[t_0,t_0+T)\times {\mathbb{R}}^r\to {\mathbb{R}}^r$ being a smooth function. If there exist a non-negative function $V(t,\zeta )\in {\mathcal{C}}^{1,2}([t_0,t_0+T)\times {\mathbb{R}}^r;{\mathbb{R}}^{+})$ and the positive constants $c_0$, ${\theta }_0$ such that for all $\zeta \in {\mathbb{R}}^r$, $\sigma \in \mathcal{N}$,

$$\begin{array}{cc}\hfill & \dot{V}(t,\zeta )={\displaystyle \frac{\partial V}{\partial t}}+{\displaystyle \frac{\partial V}{\partial {\zeta }^T}}{f}_{\sigma }\le -c_0\mu (t)V(t,\zeta )+{\theta }_0\mu (t),\mathrm{for}\mathrm{all}t\in [t_0,t_0+T),\hfill \end{array}$$

(8)

then the function $V(t,\zeta )$ satisfies

$$\begin{array}{cc}\hfill & V(t,\zeta (t))\le e^{-{\int }_{t_0}^t{c}_0\mu \left(s\right)ds}V(t_0,\zeta (t_0))+c_0^{-1}{\theta }_0.\hfill \end{array}$$

(9)

Proof. 

Let $\left\{t_l\right|l=0,1,2,\dots \}$ be the switching time instants of the switching signal $\sigma =\sigma \left(t\right)$ in system (19). During the time interval $[t_l,t_{l+1})$ ($l=0,1,2,\dots$), the ${\sigma }_l$-th subsystem is activated. Then, there exists a time interval $[t_{n-1},t_n)$ such that $t_0+T\in [t_{n-1},t_n)$. For any $t\in [t_0,t_0+T)$, from (6) and (8), we have

$$\begin{array}{c}\hfill e^{{\int }_{t_0}^t{c}_0\mu \left(s\right)ds}\dot{V}(t,\zeta )\le -c_0\mu (t)V(t,\zeta )e^{{\int }_{t_0}^t{c}_0\mu (s)ds}+{\theta }_0\mu (t)e^{{\int }_{t_0}^t{c}_0\mu (s)ds},\end{array}$$

(10)

which implies that

$$\begin{array}{c}\hfill \sum _{m=1}^n{\int }_{t\wedge t_{m-1}}^{t\wedge t_m}(e^{{\int }_{t_0}^{\tau }{c}_0\mu (s)ds}\dot{V}(\tau ,\zeta )+c_0\mu (\tau )V(\tau ,\zeta )e^{{\int }_{t_0}^{\tau }{c}_0\mu \left(s\right)ds})d\tau \le {\int }_{t_0}^t{\theta }_0\mu (\tau )e^{{\int }_{t_0}^{\tau }{c}_0\mu \left(s\right)ds}d\tau .\end{array}$$

(11)

We compute the term on the left side of (11) to obtain

$$\begin{array}{cc}\hfill & \sum _{m=1}^n{\int }_{t\wedge t_{m-1}}^{t\wedge t_m}\left(e^{{\int }_{t_0}^{\tau }{c}_0\mu (s)ds}\dot{V}(\tau ,\zeta )+c_0\mu (\tau )V(\tau ,\zeta )e^{{\int }_{t_0}^{\tau }{c}_0\mu \left(s\right)ds}\right)d\tau \hfill \\ \hfill =& \sum _{m=1}^n{\int }_{t\wedge t_{m-1}}^{t\wedge t_m}d\left(V(\tau ,\zeta )e^{{\int }_{t_0}^{\tau }{c}_0\mu \left(s\right)ds}\right)\hfill \\ \hfill =& \sum _{m=1}^{n}V(\tau ,\zeta )e^{{\int }_{t_0}^{\tau }{c}_0\mu \left(s\right)ds}{|}_{t\wedge t_{m-1}}^{t\wedge t_m}\hfill \\ \hfill =& V(\tau ,\zeta )e^{{\int }_{t_0}^{\tau }{c}_0\mu \left(s\right)ds}{|}_{t_0}^t\hfill \\ \hfill =& V(t,\zeta )e^{{\int }_{t_0}^t{c}_0\mu \left(s\right)ds}-V(t_0,\zeta ).\hfill \end{array}$$

(12)

On the right-hand side of (11), it can be derived that

$$\begin{array}{cc}\hfill & {\int }_{t_0}^t{\theta }_0\mu (\tau )e^{{\int }_{t_0}^{\tau }{c}_0\mu \left(s\right)ds}d\tau \le {\displaystyle \frac{{\theta }_0}{c_0}}\left(e^{{\int }_{t_0}^{\tau }{c}_0\mu \left(s\right)ds}-1\right).\hfill \end{array}$$

(13)

Based on (12) and (13), we have

$$\begin{array}{c}\hfill V(t,\zeta )-V(t_0,\zeta )e^{-{\int }_{t_0}^t{c}_0\mu \left(s\right)ds}\le c_0^{-1}{\theta }_0\left(1-e^{-{\int }_{t_0}^t{c}_0\mu \left(s\right)ds}\right)\le c_0^{-1}{\theta }_0.\end{array}$$

(14)

This completes the proof of Theorem 1. □

3. Control Design and Stability Analysis

3.1. State-Constrained Transformation

Let us define the tracking error $z={[z_1,\dots ,z_m]}^T$, where $z_j=y_j-y_{r,j}=x_{1j}-y_{r,j}$ for $j=1,2,\dots ,m$. In light of the output constraint (2), it can be verified that $z_j$ ($j=1,2,\dots ,m$) must satisfy the constraint condition:

$$\begin{array}{c}\hfill -E_{l,j}\left(t\right):={\epsilon }_{l,j}\left(t\right)-y_{r,j}\left(t\right)<z_j<{\epsilon }_{u,j}\left(t\right)-y_{r,j}\left(t\right):=E_{u,j}\left(t\right).\end{array}$$

(15)

To tackle the issue of output constraint (15),we introduce the following asymmetric nonlinear mapping (ANM):

$$\begin{array}{c}\hfill {\zeta }_j={\mathcal{I}}_{\{z_j\ge 0\}}{\displaystyle \frac{2}{\pi }}{E}_{u,j}tan\left({\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{z_j}{E_{u,j}}}\right)+{\mathcal{I}}_{\{z_j<0\}}{\displaystyle \frac{2}{\pi }}{E}_{l,j}tan\left({\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{z_j}{E_{l,j}}}\right),j=1,2,\dots ,m,\end{array}$$

(16)

where ${\mathcal{I}}_A={\mathcal{I}}_A(w)$ is an indicator function defined by

$$\begin{array}{c}\hfill {\mathcal{I}}_A(w)=\left\{\begin{array}{cc}1,\hfill & w\in A,\hfill \\ 0,\hfill & w\notin A.\hfill \end{array}\right.\end{array}$$

The aforementioned transformation is well-defined on the condition that the error variables $z_j$, $j=1,\dots ,m$ adhere to the constraint condition (15). Consequently, if ${sup}_{t\ge t_0}{\left|\zeta \left(t\right)\right|}^2={sup}_{t\ge t_0}\left(|{\zeta }_1{\left(t\right)|}^2+\dots +{\left|{\zeta }_m\left(t\right)\right|}^2\right)<+\infty$ is satisfied, then the variables $z_j$, $j=1,2,\dots ,m$ will indeed comply with the constraint condition (15). In fact, the ANM method is designed to transform the output-constrained problem into a bounded tracking error problem by introducing a nonlinear mapping function (15). We see that the output constraints will not be violated as long as ${\zeta }_j$ is bounded. Moreover, according to Assumptions 3 and 4, we know that both $E_{l,j}(t)$ and $E_{u,j}(t)$ are positive and bounded, i.e., there exist positive constants $M_{l,j}$ and $M_{u,j}$ such that $0<E_{l,j}(t)\le M_{l,j}$ and $0<E_{u,j}(t)\le M_{u,j}$. Thus, if $z_j$, $j=1,2,\dots ,m$ satisfies (15), then it follows from (16) that

$$\begin{array}{c}\hfill z_j={\mathcal{I}}_{\{{\zeta }_j\ge 0\}}{\displaystyle \frac{2}{\pi }}{E}_{u,j}arctan\left({\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{{\zeta }_j}{E_{u,j}}}\right)+{\mathcal{I}}_{\{{\zeta }_j<0\}}{\displaystyle \frac{2}{\pi }}{E}_{l,j}arctan\left({\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{{\zeta }_j}{E_{l,j}}}\right),j=1,2,\dots ,m.\end{array}$$

(17)

Noting that $|arctanx|\le \left|x\right|$ for $x\in \mathbb{R}$, we further know that $|z_j|\le |{\zeta }_j|$ holds for all $j=1,2,\dots ,m$. In addition, it can be inferred that $z_j=z_j({\zeta }_j,E_{u,j},E_{l,j})$ is continuously differentiable with respect to its arguments and

$$\begin{array}{c}\hfill {\dot{z}}_j={\displaystyle \frac{\partial z_j}{\partial {\zeta }_j}}{\dot{\zeta }}_j+{\displaystyle \frac{\partial z_j}{\partial E_{u,j}}}{\dot{E}}_{u,j}+{\displaystyle \frac{\partial z_j}{\partial E_{l,j}}}{\dot{E}}_{l,j}.\end{array}$$

(18)

It is easy to compute that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial z_j}{\partial E_{u,j}}}={\mathcal{I}}_{\{{\zeta }_j\ge 0\}}({\displaystyle \frac{2}{\pi }}arctan({\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{{\zeta }_j}{E_{u,j}}})-{\displaystyle \frac{{\zeta }_j}{E_{u,j}(1+{(\frac{\pi }{2}·\frac{{\zeta }_j}{E_{u,j}})}^2)}}),\hfill \\ \hfill & {\displaystyle \frac{\partial z_j}{\partial E_{l,j}}}={\mathcal{I}}_{\{{\zeta }_j<0\}}({\displaystyle \frac{2}{\pi }}arctan({\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{{\zeta }_j}{E_{l,j}}})-{\displaystyle \frac{{\zeta }_j}{E_{l,j}(1+{(\frac{\pi }{2}·\frac{{\zeta }_j}{E_{l,j}})}^2)}}).\hfill \end{array}$$

Due to the fact that

$$\begin{array}{c}\hfill \underset{ϵ\to 0^{+}}{lim}{\displaystyle \frac{z_j(0+ϵ,E_{u,j},E_{l,j})-z_j(0,E_{u,j},E_{l,j})}{ϵ}}=\underset{ϵ\to 0^{-}}{lim}{\displaystyle \frac{z_j(0+ϵ,E_{u,j},E_{l,j})-z_j(0,E_{u,j},E_{l,j})}{ϵ}}=1,\end{array}$$

we can obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial z_j}{\partial {\zeta }_j}}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{1+{\left(\frac{\pi }{2}·\frac{{\zeta }_j}{E_{u,j}}\right)}^2}},\hfill & \mathrm{if}{\zeta }_j>0,\hfill \\ 1,\hfill & \mathrm{if}{\zeta }_j=0,\hfill \\ {\displaystyle \frac{1}{1+{\left(\frac{\pi }{2}·\frac{{\zeta }_j}{E_{l,j}}\right)}^2}},\hfill & \mathrm{if}{\zeta }_j<0.\hfill \end{array}\right.\end{array}$$

Using the transformation (16), together with the switched MIMO system (1), yields

$$\begin{array}{c}\hfill {\dot{\zeta }}_j={\psi }_j(x_{2j}-{\dot{y}}_{r,j})+{\mathcal{I}}_{\{{\zeta }_j\ge 0\}}{s}_{u,j}{\dot{E}}_{u,j}+{\mathcal{I}}_{\{{\zeta }_j<0\}}{s}_{l,j}{\dot{E}}_{l,j},j=1,2,\dots ,m,\end{array}$$

(19)

where ${\psi }_j$, $s_{u,j}$ and $s_{l,j}$ are defined by ${\psi }_j={\mathcal{I}}_{\{{\zeta }_j\ge 0\}}{\mathrm{\Delta }}_{u,j}+{\mathcal{I}}_{\{{\zeta }_j<0\}}{\mathrm{\Delta }}_{l,j}$, $s_{u,j}={\displaystyle \frac{{\zeta }_j}{E_{u,j}}}-{\displaystyle \frac{2}{\pi }}{\mathrm{\Delta }}_{u,j}arctan({\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{{\zeta }_j}{E_{u,j}}})$, $s_{l,j}={\displaystyle \frac{{\zeta }_j}{E_{l,j}}}-{\displaystyle \frac{2}{\pi }}{\mathrm{\Delta }}_{l,j}arctan({\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{{\zeta }_j}{E_{l,j}}})$ with ${\mathrm{\Delta }}_{u,j}=1+{\left({\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{{\zeta }_j}{E_{u,j}}}\right)}^2$, ${\mathrm{\Delta }}_{l,j}=1+{({\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{{\zeta }_j}{E_{l,j}}})}^2$.

Remark 3. 

Compared with the existing results, such as [24,25], the ANM method offers a broader range of applicability and can be categorized into the following cases:

Case 1 [Symmetric time-varying constraints, i.e., ${\epsilon }_{l,j}\left(t\right)=-{\epsilon }_j\left(t\right)$ and ${\epsilon }_{u,j}\left(t\right)={\epsilon }_j\left(t\right)$, $j=1,\dots ,m$]. Under the assumption of symmetric and time-varying output constraints, the ANM (16) still works with

$$\begin{array}{c}\hfill E_{l,j}\left(t\right)=y_{r,j}\left(t\right)-{\epsilon }_j\left(t\right),E_{u,j}\left(t\right)={\epsilon }_j\left(t\right)-y_{r,j}\left(t\right).\end{array}$$

(20)

Case 2 [Symmetric constant constraints, i.e., ${\epsilon }_{l,j}\left(t\right)=-\epsilon$ and ${\epsilon }_{u,j}\left(t\right)=\epsilon$, $j=1,\dots ,m$]. In this case, the output constraints are assumed to be positive numbers. Then, the system (1) degenerates into the case studied in [24,25]. Response: The underline is necessary to clearly present the classification and should be retained in all highlighted parts.

Case 3 [Unilateral constraints or non-constraints, i.e., ${\epsilon }_{l,j}\left(t\right)=-\infty$ or ${\epsilon }_{u,j}\left(t\right)=+\infty$, $j=1,\dots ,m$]. In the unilateral constraint case, the output variable $y\left(t\right)$ is required to be $-\infty <y_j\left(t\right)<{\epsilon }_{u,j}\left(t\right)$ or ${\epsilon }_{l,j}\left(t\right)<y_j\left(t\right)<+\infty$ for all $t\ge 0$. Taking the first case as an example, the ANM (16) is changed into

$$\begin{array}{c}\hfill {\zeta }_j={\mathcal{I}}_{\{z_j\ge 0\}}{\displaystyle \frac{2}{\pi }}{E}_{u,j}tan\left({\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{z_j}{E_{u,j}}}\right)+{\mathcal{I}}_{\{z_j<0\}}{z}_j,j=1,2,\dots ,m.\end{array}$$

On the other hand, our results can be extended to the case where no constraints are imposed on the output $y\left(t\right)$. In this case, the ANM (16) can reduce into a trivial transformation ${\zeta }_j=z_j$, $j=1,2,\dots ,m$.

3.2. Controller Design

Let us define

$$\begin{array}{cc}\hfill & {\eta }_{1j}=\mu (t){\xi }_{1j},{\xi }_{1j}={\zeta }_j,j=1,2,\dots ,m,\hfill \\ \hfill & {\eta }_i=\mu (t){\xi }_i,{\xi }_i=x_i-{\varphi }_{i-1},i=2,\dots ,n,\hfill \end{array}$$

(21)

where ${\varphi }_1,\dots ,{\varphi }_{n-1}\in {\mathbb{R}}^m$ are the virtual controllers to be specified later. In the following, we focus on the control design of the switched MIMO system (1). The design process is carried out by using the backstepping methodology.

Step 1. Let us consider a candidate Lyapunov function $V_1={\displaystyle \frac{1}{2}}{\sum }_{j=1}^m{\left|{\eta }_{1j}\right|}^2$. Then, the dynamics of ${\eta }_{1j}$, $j=1,2,\dots ,m$, are

$$\begin{array}{cc}\hfill {\dot{\eta }}_{1j}& =\mu (t){\dot{\xi }}_{1j}+\dot{\mu }(t){\xi }_{1j}\hfill \\ \hfill & =\mu (t){\psi }_j(x_{2j}-{\dot{y}}_{r,j})+\mu (t){\mathcal{I}}_{\{{\zeta }_j\ge 0\}}{s}_{u,j}{\dot{E}}_{u,j}+\mu (t){\mathcal{I}}_{\{{\zeta }_j<0\}}{s}_{l,j}{\dot{E}}_{l,j}+\dot{\mu }(t){\xi }_{1j}.\hfill \end{array}$$

(22)

Combining with (19) yields

$$\begin{array}{cc}\hfill {\dot{V}}_1& =\sum _{j=1}^m{\eta }_{1j}\left(\mu (t){\dot{\xi }}_{1j}+\dot{\mu }(t){\xi }_{1j}\right)\hfill \\ \hfill & =\mu (t)\sum _{j=1}^m{\eta }_{1j}{\psi }_j(x_{2j}-{\dot{y}}_{r,j})+\mu (t)\sum _{j=1}^m{\eta }_{1j}{\mathcal{I}}_{\{{\zeta }_j\ge 0\}}{s}_{u,j}{\dot{E}}_{u,j}\hfill \\ \hfill & +\mu (t)\sum _{j=1}^m{\eta }_{1j}{\mathcal{I}}_{\{{\zeta }_j<0\}}{s}_{l,j}{\dot{E}}_{l,j}+\sum _{j=1}^m{\eta }_{1j}\dot{\mu }(t){\xi }_{1j}.\hfill \end{array}$$

(23)

For any $j=1,2,\dots ,m$, in accordance with Assumptions 3 and 4, there exist the positive constants ${\overline{M}}_{u,j}$ and ${\overline{M}}_{l,j}$ such that $\left|{\displaystyle \frac{{\dot{E}}_{u,j}}{E_{u,j}}}\right|\le {\overline{M}}_{u,j}$ and $\left|{\displaystyle \frac{{\dot{E}}_{l,j}}{E_{l,j}}}\right|\le {\overline{M}}_{l,j}$. It is noteworthy that the relationship ${\eta }_{1j}=\mu \left(t\right){\xi }_{1j}=\mu \left(t\right){\zeta }_j$ holds true for $j=1,2,\dots ,m$. Therefore, by using the fact $|{\sum }_{j=1}^m{x}_j{y}_j|\le \sqrt{{\sum }_{j=1}^m{x}_j^2}·\sqrt{{\sum }_{j=1}^m{y}_j^2}$ for $x_j,y_j\in \mathbb{R}$ and Young’s inequality, we can derive

$$\begin{array}{cc}\hfill \left|\mu \left(t\right)\sum _{j=1}^m{\eta }_{1j}{\mathcal{I}}_{\{{\zeta }_j\ge 0\}}{s}_{u,j}{\dot{E}}_{u,j}\right|& \le \mu \left(t\right)\sum _{j=1}^m(1+{\mathrm{\Delta }}_{u,j}){\overline{M}}_{u,j}|{\eta }_{1j}\left|\right|{\zeta }_j|\hfill \\ \hfill & \le \sqrt{\sum _{j=1}^m{(1+{\mathrm{\Delta }}_{u,j})}^2{\overline{M}}_{u,j}^2}\sqrt{\sum _{j=1}^m{\left|{\eta }_{1j}\right|}^4}\hfill \\ \hfill & \le D_u\sum _{j=1}^m{\left|{\eta }_{1j}\right|}^2,\hfill \end{array}$$

(24)

where $D_u={\sum }_{j=1}^m(1+{\mathrm{\Delta }}_{u,j}){\overline{M}}_{u,j}$. Similarly,

$$\begin{array}{cc}\hfill \left|\mu \left(t\right)\sum _{j=1}^m{\eta }_{1j}{\mathcal{I}}_{\{{\zeta }_j<0\}}{s}_{l,j}{\dot{E}}_{l,j}\right|& \le D_l\sum _{j=1}^m{\left|{\eta }_{1j}\right|}^2,\hfill \end{array}$$

(25)

where $D_l={\sum }_{j=1}^m(1+{\mathrm{\Delta }}_{l,j}){\overline{M}}_{l,j}$. Denote ${\eta }_1={[{\eta }_{11},\dots ,{\eta }_{1m}]}^T$, ${\xi }_1={[{\xi }_{11},\dots ,{\xi }_{1m}]}^T$, ${\dot{y}}_r={[{\dot{y}}_{r,1},\dots ,{\dot{y}}_{r,m}]}^T$ and $\mathbf{\Psi }=\mathrm{diag}\{{\psi }_1,\dots ,{\psi }_m\}$. Obviously, the matrix $\mathbf{\Psi }$ is invertible due to the positiveness of ${\psi }_j$, $j=1,2,\dots ,m$. It follows from (23)–(25) that

$$\begin{array}{cc}\hfill {\dot{V}}_1& \le \mu (t){\eta }_1^T\mathbf{\Psi }(x_2-{\dot{y}}_r)+D_1{\left|{\eta }_1\right|}^2+\dot{\mu }(t){\eta }_1^T{\xi }_1\hfill \\ \hfill & =\mu (t){\eta }_1^T\mathbf{\Psi }(x_2-{\varphi }_1)+\mu (t){\eta }_1^T\mathbf{\Psi }({\varphi }_1-{\dot{y}}_r)+D_1{\left|{\eta }_1\right|}^2+\dot{\mu }(t){\eta }_1^T{\xi }_1,\hfill \end{array}$$

(26)

where $D_1=D_u+D_l$. Due to the fact that ${\eta }_1=\mu (t){\xi }_1$, the last term ${\eta }_1^T\dot{\mu }(t){\xi }_1$ in the above equation can be rewritten as $\dot{\mu }(t){\mu }^{-1}(t){\left|{\eta }_1\right|}^2$. Noting that ${\psi }_j\ge 1$ for $j=1,2,\dots ,m$, we have ${\eta }_1^T\mathbf{\Psi }{\eta }_1\ge {\left|{\eta }_1\right|}^2$. By taking

$$\begin{array}{c}\hfill {\varphi }_1=-\left(k_1{\eta }_1+{\displaystyle \frac{1}{2}}{\xi }_1+D_1{\xi }_1+\dot{\mu }(t){\mu }^{-1}(t){\xi }_1\right)+{\dot{y}}_r\end{array}$$

(27)

with $k_1>0$ being a real number, then it follows that

$$\begin{array}{cc}\hfill & \mu (t){\eta }_1^T\mathbf{\Psi }({\varphi }_1-{\dot{y}}_r)\hfill \\ \hfill & =-\mu (t){\eta }_1^T\mathbf{\Psi }\left(k_1{\eta }_1+{\displaystyle \frac{1}{2}}{\xi }_1+D_1{\xi }_1+\dot{\mu }(t){\mu }^{-1}(t){\xi }_1\right)\hfill \\ \hfill & =-\mu (t)k_1{\eta }_1^T\mathbf{\Psi }{\eta }_1-{\displaystyle \frac{1}{2}}{\eta }_1^T\mathbf{\Psi }{\eta }_1-D_1{\eta }_1^T\mathbf{\Psi }{\eta }_1-\dot{\mu }(t){\mu }^{-1}(t){\eta }_1^T\mathbf{\Psi }{\eta }_1\hfill \\ \hfill & \le -\mu (t)k_1|{\eta }_1{|}^2-{\displaystyle \frac{1}{2}}|{\eta }_1{|}^2-D_1|{\eta }_1{|}^2-\dot{\mu }(t){\mu }^{-1}(t){|{\eta }_1|}^2.\hfill \end{array}$$

(28)

Hence, we know from (26) that ${\dot{V}}_1\le {\eta }_1^T\mathbf{\Psi }{\eta }_2-\mu (t)k_1|{\eta }_1{|}^2-{\displaystyle \frac{1}{2}}{\left|{\eta }_1\right|}^2$.

Step 2. Let us consider a candidate Lyapunov function $V_2=V_1+{\displaystyle \frac{1}{2}}{\left|{\eta }_2\right|}^2$. The dynamic of ${\eta }_2$ is

$$\begin{array}{c}\hfill {\dot{\eta }}_2=\mu \left(t\right)\left({\xi }_3+{\varphi }_2\right)-\mu \left(t\right){\dot{\varphi }}_1+\dot{\mu }\left(t\right){\mu }^{-1}\left(t\right){\eta }_2.\end{array}$$

(29)

Then, along with the solution of (29), it follows that

$$\begin{array}{cc}\hfill {\dot{V}}_2& ={\dot{V}}_1+{\eta }_2^T{\dot{\eta }}_2\hfill \\ \hfill & \le {\eta }_1^T\mathbf{\Psi }{\eta }_2-\mu (t)k_1|{\eta }_1{|}^2-{\displaystyle \frac{1}{2}}{|{\eta }_1|}^2\hfill \\ \hfill & +{\eta }_2^T{\eta }_3+\mu (t){\eta }_2^T{\varphi }_2-\mu (t){\eta }_2^T{\dot{\varphi }}_1+\dot{\mu }(t){\mu }^{-1}(t){\left|{\eta }_2\right|}^2.\hfill \end{array}$$

(30)

Note that ${\parallel \mathbf{\Psi }\parallel }^2={\sum }_{j=1}^m{\left|{\psi }_j\right|}^2$. By using the Young inequality, we have

$$\begin{array}{cc}\hfill & {\eta }_1^T\mathbf{\Psi }{\eta }_2\le {\displaystyle \frac{1}{2}}\sum _{j=1}^m{|{\psi }_j|}^2|{\eta }_2{|}^2+{\displaystyle \frac{1}{2}}{\left|{\eta }_1\right|}^2.\hfill \end{array}$$

By taking

$$\begin{array}{c}\hfill {\varphi }_2=-k_2{\eta }_2-{\displaystyle \frac{1}{2}}{\xi }_2-{\displaystyle \frac{1}{2}}\sum _{j=1}^m{\left|{\psi }_j\right|}^2{\xi }_2-\dot{\mu }\left(t\right){\mu }^{-1}\left(t\right){\xi }_2+{\dot{\varphi }}_1\end{array}$$

(31)

with $k_2>0$ being a real number, we have ${\dot{V}}_2\le {\eta }_2^T{\eta }_3-\mu (t)k_1|{\eta }_1{|}^2-\mu (t)k_2|{\eta }_2{|}^2-{\displaystyle \frac{1}{2}}{\left|{\eta }_2\right|}^2$.

Repeating this step, we can obtain the dynamic of ${\eta }_i$

$$\begin{array}{c}\hfill {\dot{\eta }}_i=\mu (t)\left({\xi }_{i+1}+{\varphi }_i\right)-\mu (t){\dot{\varphi }}_{i-1}+\dot{\mu }(t){\mu }^{-1}(t){\eta }_i\end{array}$$

(32)

and find a ${\varphi }_i$ ($i=3,\dots ,n-1$) in the form of

$$\begin{array}{c}\hfill {\varphi }_i=-k_i{\eta }_i-{\xi }_i-\dot{\mu }(t){\mu }^{-1}(t){\xi }_i+{\dot{\varphi }}_{i-1}\end{array}$$

(33)

with $k_i>0$ being a real number, such that

$$\begin{array}{c}\hfill {\dot{V}}_i\le {\eta }_i^T{\eta }_{i+1}-\mu (t)\sum _{l=1}^i{k}_l|{\eta }_l{|}^2-{\displaystyle \frac{1}{2}}{|{\eta }_i|}^2,\end{array}$$

(34)

where $V_i=V_{i-1}+{\displaystyle \frac{1}{2}}{\left|{\eta }_i\right|}^2$, $i=3,\dots ,n-1$, and $k_3,\dots ,k_{n-1}$ are positive real numbers.

Stepn. Consider a Lyapunov function $V_n=V_{n-1}+{\displaystyle \frac{1}{2}}{\left|{\eta }_n\right|}^2$. The dynamic of ${\eta }_n$ is

$$\begin{array}{c}\hfill {\dot{\eta }}_n=\mu (t){\mathbf{H}}_{\sigma }u+\mu (t){\mathbf{F}}_{\sigma }-\mu (t){\dot{\varphi }}_{n-1}+\dot{\mu }(t){\mu }^{-1}(t){\eta }_n.\end{array}$$

(35)

Then, along with the solution of (35), it follows that

$$\begin{array}{cc}\hfill {\dot{V}}_n& ={\dot{V}}_{n-1}+{\eta }_n^T{\dot{\eta }}_n\hfill \\ \hfill & \le {\eta }_{n-1}^T{\eta }_n-\mu (t)\sum _{l=1}^{n-1}{k}_l|{\eta }_l{|}^2-{\displaystyle \frac{1}{2}}{\left|{\eta }_{n-1}\right|}^2+\mu (t){\eta }_n^T{\mathbf{H}}_{\sigma }u\hfill \\ \hfill & +\mu (t){\eta }_n^T{\mathbf{F}}_{\sigma }-\mu (t){\eta }_n^T{\dot{\varphi }}_{n-1}+\dot{\mu }(t){\mu }^{-1}(t){\left|{\eta }_n\right|}^2.\hfill \end{array}$$

(36)

By using the Young inequality, we have

$$\begin{array}{cc}\hfill & {\eta }_{n-1}^T{\eta }_n\le {\displaystyle \frac{1}{2}}|{\eta }_{n-1}{|}^2+{\displaystyle \frac{1}{2}}{|{\eta }_n|}^2,\hfill \\ \hfill & \mu (t){\eta }_n^T\left({\mathbf{F}}_{\sigma }-{\dot{\varphi }}_{n-1}\right)\le \mu (t){\left(2\theta \right)}^{-1}\left(f^2+{\left|{\dot{\varphi }}_{n-1}\right|}^2\right){\left|{\eta }_n\right|}^2+\mu \left(t\right)\theta M,\hfill \end{array}$$

where $M=max\{1,{\overline{c}}_1^2,\dots ,{\overline{c}}_N^2\}$, $\theta >0$ is a control parameter. Note, that

$$\begin{array}{cc}\hfill {\eta }_n^T{\mathbf{H}}_{\sigma }{\mathbf{P}}^T{\eta }_n& ={\displaystyle \frac{1}{2}}{\eta }_n^T\mathbf{P}({\mathbf{Q}}_{\sigma }+{\mathbf{Q}}_{\sigma }^T){\mathbf{P}}^T{\eta }_n+{\displaystyle \frac{1}{2}}{\eta }_n^T\mathbf{P}({\mathbf{Q}}_{\sigma }-{\mathbf{Q}}_{\sigma }^T){\mathbf{P}}^T{\eta }_n\hfill \\ \hfill & \ge {\displaystyle \frac{1}{2}}\underline{\lambda }{\left|{\eta }_n\right|}^2.\hfill \end{array}$$

(37)

Hence, by taking

$$\begin{array}{c}\hfill u=-2{\underline{\lambda }}^{-1}\left(k_n+{\displaystyle \frac{1}{2}}{\mu }^{-1}(t)+{\left(2\theta \right)}^{-1}\left(f^2+{\left|{\dot{\varphi }}_{n-1}\right|}^2\right)+\dot{\mu }(t){\mu }^{-2}(t)\right){\mathbf{P}}^T{\eta }_n\end{array}$$

(38)

with $k_n>0$ being a real number, we have

$$\begin{array}{cc}\hfill & \mu (t){\eta }_n^T{\mathbf{H}}_{\sigma }u\hfill \\ \hfill & =-2{\underline{\lambda }}^{-1}\mu (t)k_n{\eta }_n^T{\mathbf{H}}_{\sigma }{\mathbf{P}}^T{\eta }_n-{\underline{\lambda }}^{-1}{\eta }_n^T{\mathbf{H}}_{\sigma }{\mathbf{P}}^T{\eta }_n\hfill \\ \hfill & -{\underline{\lambda }}^{-1}\mu (t){\theta }^{-1}\left(f^2+{\left|{\dot{\varphi }}_{n-1}\right|}^2\right){\eta }_n^T{\mathbf{H}}_{\sigma }{\mathbf{P}}^T{\eta }_n-2{\underline{\lambda }}^{-1}\dot{\mu }(t){\mu }^{-1}(t){\eta }_n^T{\mathbf{H}}_{\sigma }{\mathbf{P}}^T{\eta }_n\hfill \\ \hfill & \le -\mu (t)k_n|{\eta }_n{|}^2-{\displaystyle \frac{1}{2}}|{\eta }_n{|}^2-\mu (t){\left(2\theta \right)}^{-1}\left(f^2+{|{\dot{\varphi }}_{n-1}|}^2\right)|{\eta }_n{|}^2-\dot{\mu }(t){\mu }^{-1}(t){|{\eta }_n|}^2.\hfill \end{array}$$

Therefore, we obtain

$$\begin{array}{c}\hfill {\dot{V}}_n\le -\mu (t)\sum _{l=1}^n{k}_l{\left|{\eta }_l\right|}^2+\mu (t)\mathsf{\Theta }\le -2\mu (t)k{V}_n+\mu (t)\mathsf{\Theta },\end{array}$$

(39)

where $k=min\{k_1,k_2,\dots ,k_n\}$ and $\mathsf{\Theta }=\theta M$ are positive constants.

Remark 4. 

Since ${\mathbf{H}}_{\sigma }$ in system (1) is not necessarily a square matrix, directly applying the inverse of ${\mathbf{H}}_{\sigma }$ for controller design, as in [5,6,24], is unfeasible. To address this, we decompose ${\mathbf{H}}_{\sigma }$ into a non-square matrix $\mathbf{P}$ and a square matrix ${\mathbf{Q}}_{\sigma }$, as stated in Assumption 2. This enables controller design even when ${\mathbf{H}}_{\sigma }$ is non-invertible. However, since ${\mathbf{Q}}_{\sigma }$ is unknown and cannot be directly utilized, constructing a control input based on ${\mathbf{H}}_{\sigma }$ is challenging. To resolve this, we incorporate inequality (37) and design a prescribed-time tracking controller relying solely on $\mathbf{P}$, eliminating the dependence on ${\mathbf{Q}}_{\sigma }$ and ensuring convergence within the prescribed time.

3.3. Stability Analysis

Theorem 2. 

Under Assumptions 1–4, there exists a tracking controller u in the form of (38) for system (1) such that for any initial data $y\left(t_0\right)$ satisfying ${\epsilon }_{l,j}(t_0)<y_j(t_0)<{\epsilon }_{u,j}(t_0)$, $j=1,2\dots ,m$,

(1) 

the prescribed output constraint is not violated, i.e., the output $y(t)={[y_1(t),\dots ,y_m(t)]}^T$ satisfies

$$\begin{array}{c}\hfill {\epsilon }_{l,j}(t)<y_j(t)<{\epsilon }_{u,j}(t),j=1,2\dots ,m,\forall t\in [t_0,t_0+T);\end{array}$$

(40)

(2) 

all signals of the closed-loop system are bounded on $[t_0,t_0+T)$;

(3) 

the tracking error $z(t)=y(t)-y_r(t)$ possesses the following property:

$$\begin{array}{c}\hfill \underset{t\to t_0+T}{lim}\left[|z(t)|+|\dot{z}(t)|\right]=0,\end{array}$$

(41)

where $\dot{z}(t)=\dot{y}(t)-{\dot{y}}_r(t)$ represents the first time derivative of $z\left(t\right)$.

Proof. 

(1) According to Theorem 1 and (39), we know that $V_n$ remains bounded over the interval $[t_0,t_0+T)$. This, in turn, implies that the variables ${\eta }_i$, $i=1,\dots ,n$, are also bounded within the same interval. Observing (21), we deduce that ${\xi }_{1j}={\mu }^{-1}(t){\eta }_{1j}$, $j=1,2,\dots ,m$ are also bounded, as ${\zeta }_j$. Consequently, by referring to (16), we can assert that $-E_{l,j}(t)<z_j(t)<E_{u,j}(t)$, $j=1,2,\dots ,m$ hold for all $t\in [t_0,t_0+T)$. Given the relationship $y_j(t)=z_j(t)+y_{r,j}(t)$, $j=1,2,\dots ,m$, we can further obtain

$$\begin{array}{c}\hfill -E_{l,j}(t)+y_{r,j}(t)<y_j(t)<E_{u,j}(t)+y_{r,j}(t).\end{array}$$

(42)

Considering the definitions of $E_{l,j}(t)$ and $E_{u,j}(t)$ in (15), we conclude that the output $y\left(t\right)$ will obey the pre-assigned output constraint ${\epsilon }_{l,j}(t)<y_j(t)<{\epsilon }_{u,j}(t)$ provided that the initial value $y\left(t_0\right)$ satisfies ${\epsilon }_{l,j}(t_0)<y_j(t_0)<{\epsilon }_{u,j}(t_0)$.

(2) Based on (21) and (39), we know that ${\eta }_i$ and ${\xi }_i$, $i=1,\dots ,n$, are bounded over the interval $[t_0,t_0+T)$. Moreover, combining with ${\xi }_i={\mu }^{-1}\left(t\right){\eta }_i$, $i=1,2,\dots ,n$, it can be inferred that ${lim}_{t\to t_0+T}\left|{\xi }_i\right|=0$.

Next, we need to show the boundedness of the controller u. First, we consider the boundedness of the first virtual controller ${\varphi }_1$. Based on Assumption 4, it follows that $y_r$ is bounded. Noting that $\dot{\mu }\left(t\right){\mu }^{-1}\left(t\right)={\displaystyle \frac{\nu }{T}}\left({\displaystyle \frac{T}{t_0+T-t}}\right)$, we have ${lim}_{t\to t_0+T}\dot{\mu }\left(t\right){\mu }^{-1}\left(t\right){\xi }_1={\displaystyle \frac{\nu }{T}}{lim}_{t\to t_0+T}{\left({\displaystyle \frac{t_0+T-t}{T}}\right)}^{\nu -1}{\eta }_1=0$. Considering that ${\eta }_1$ and ${\xi }_1$ are both bounded on the interval $[t_0,t_0+T)$, we conclude that the virtual controller ${\varphi }_1=-\left(k_1{\eta }_1+{\displaystyle \frac{1}{2}}{\xi }_1+D_1{\xi }_1+\dot{\mu }\left(t\right){\mu }^{-1}\left(t\right){\xi }_1\right)+{\dot{y}}_r$ is bounded on $[t_0,t_0+T)$.

Second, we consider the boundedness of the second virtual controller ${\varphi }_2$. Recall that

$$\begin{array}{c}\hfill {\varphi }_2=-k_2{\eta }_2-{\displaystyle \frac{1}{2}}{\xi }_2-{\displaystyle \frac{1}{2}}\sum _{j=1}^m{\left|{\psi }_j\right|}^2{\xi }_2-\dot{\mu }(t){\mu }^{-1}(t){\xi }_2+{\dot{\varphi }}_1.\end{array}$$

For any $j=1,2,\dots ,m$, according to the definition of ${\psi }_j$ and the boundedness of $E_{l,j}$ and $E_{u,j}$, we have

$$\underset{t\to t_0+T}{lim}{\psi }_j=\underset{{\zeta }_j\to 0^{+}}{lim}\left(1+{({\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{{\zeta }_j}{E_{u,j}}})}^2\right)=\underset{{\zeta }_j\to 0^{-}}{lim}\left(1+{({\displaystyle \frac{\pi }{2}}·{\displaystyle \frac{{\zeta }_j}{E_{l,j}}})}^2\right)=1.$$

(43)

Given that ${\eta }_2$, ${\xi }_2$ are bounded, and ${lim}_{t\to t_0+T}\dot{\mu }(t){\mu }^{-1}(t){\xi }_2=0$, it is sufficient to ensure that ${\varphi }_2$ remains bounded, provided that ${\dot{\varphi }}_1$ is also bounded. Let us denote ${\varphi }_1={[{\varphi }_{11},\dots ,{\varphi }_{1m}]}^T$. Based on the definition of ${\varphi }_1$, we can derive that

$$\begin{array}{cc}\hfill {\dot{\varphi }}_{1j}& =-\left(k_1{\dot{\eta }}_{1j}+{\displaystyle \frac{1}{2}}{\dot{\xi }}_{1j}+D_1{\dot{\xi }}_{1j}+{\dot{D}}_1{\xi }_{1j}+{\displaystyle \frac{d}{dt}}(\dot{\mu }(t){\mu }^{-1}(t){\xi }_{1j})\right)+{\ddot{y}}_{r,j}\hfill \\ \hfill & =-\left(k_1{\dot{\eta }}_{1j}+{\displaystyle \frac{1}{2}}{\dot{\xi }}_{1j}+D_1{\dot{\xi }}_{1j}+{\dot{D}}_1{\xi }_{1j}+{\displaystyle \frac{\nu }{{(t_0+T-t)}^2}}{\xi }_{1j}+{\displaystyle \frac{\nu }{t_0+T-t}}{\dot{\xi }}_{1j}\right)+{\ddot{y}}_{r,j},\hfill \\ \hfill & j=1,2,\dots ,m.\hfill \end{array}$$

(44)

Denote ${\eta }_2={[{\eta }_{21},\dots ,{\eta }_{2m}]}^T$. For any $j=1,2,\dots ,m$, it follows from (22) and (27) that

$$\begin{array}{cc}\hfill {\dot{\eta }}_{1j}=& {\psi }_j{\eta }_{2j}-\mu \left(t\right){\psi }_j\left(k_1{\eta }_{1j}+{\displaystyle \frac{1}{2}}{\xi }_{1j}+D_1{\xi }_{1j}+\dot{\mu }\left(t\right){\mu }^{-1}\left(t\right){\xi }_{1j}\right)\hfill \\ \hfill & +\mu \left(t\right){\mathcal{I}}_{\{{\zeta }_j<0\}}{s}_{l,j}{\dot{E}}_{l,j}+\mu \left(t\right){\mathcal{I}}_{\{{\zeta }_j\ge 0\}}{s}_{u,j}{\dot{E}}_{u,j}+\dot{\mu }\left(t\right){\xi }_{1j}.\hfill \end{array}$$

(45)

Note, that

$$\begin{array}{c}\hfill |{\mathcal{I}}_{\{{\zeta }_j\ge 0\}}{s}_{l,j}{\dot{E}}_{l,j}|\le D_l|{\xi }_{1j}|,|{\mathcal{I}}_{\{{\zeta }_j\ge 0\}}{s}_{u,j}{\dot{E}}_{u,j}|\le D_u|{\xi }_{1j}|,\end{array}$$

and ${lim}_{t\to t_0+T}\left|{\xi }_{1j}\right|=0$. Hence, due to the fact that $\nu \ge 2$, we have

$$\begin{array}{cc}\hfill \underset{t\to t_0+T}{lim}{\dot{\xi }}_{1j}& =\underset{t\to t_0+T}{lim}{\displaystyle \frac{d}{dt}}\left({\mu }^{-1}(t){\eta }_{1j}\right)\hfill \\ \hfill & =\underset{t\to t_0+T}{lim}\left(-{\mu }^{-2}(t)\dot{\mu }(t){\eta }_{1j}+{\mu }^{-1}(t){\dot{\eta }}_{1j}\right)\hfill \\ \hfill & =\underset{t\to t_0+T}{lim}\left(-{\displaystyle \frac{\nu }{T}}{\left({\displaystyle \frac{T}{t_0+T-t}}\right)}^{1-\nu }-{\psi }_j{k}_1\right){\eta }_{1j}\hfill \\ \hfill & =\underset{t\to t_0+T}{lim}(-k_1{\eta }_{1j}).\hfill \end{array}$$

(46)

Since ${\eta }_{1j}$ is bounded over the interval $[t_0,t_0+T)$, we know that there exists a finite constant $\omega$ such that

$$\begin{array}{c}\hfill \underset{t\to t_0+T}{lim}{\displaystyle \frac{T^{\nu }{\xi }_{1j}}{{(t_0+T-t)}^{\nu }}}=\omega .\end{array}$$

(47)

Applying the L’Hôpital’s rule, we have

$$\begin{array}{c}\hfill \underset{t\to t_0+T}{lim}{\displaystyle \frac{-T^{\nu }{\dot{\xi }}_{1j}}{\nu {(t_0+T-t)}^{\nu -1}}}=\omega .\end{array}$$

(48)

Combining with (46) and applying the L’Hôpital’s rule again yields

$$\begin{array}{c}\hfill \underset{t\to t_0+T}{lim}{\displaystyle \frac{T^{\nu }{k}_1{\eta }_{1j}}{\nu {(t_0+T-t)}^{\nu -1}}}=\underset{t\to t_0+T}{lim}{\displaystyle \frac{-T^{\nu }{k}_1{\dot{\eta }}_{1j}}{\nu (\nu -1){(t_0+T-t)}^{\nu -2}}}=\omega .\end{array}$$

(49)

Case 1 For $\nu =2$, it follows from (49) that ${lim}_{t\to t_0+T}{\dot{\eta }}_{1j}=-2k_1^{-1}{T}^{-2}\omega$, which indicates that ${\dot{\eta }}_{1j}$ is bounded on the interval $[t_0,t_0+T)$. Moreover, it further follows that ${\eta }_{1j}$ tends to zero at a rate not slower than $(t_0+T-t)$. Thus, we know that ${\xi }_{1j}$ tends to zero at a rate not slower than ${(t_0+T-t)}^3$ as t approaches $t_0+T$. Thus, we can ensure the boundedness of ${\dot{\varphi }}_{1j}$ on $[t_0,t_0+T)$ according to (44).

Case 2 For $\nu >2$, we can deduce from (49) that ${\eta }_{1j}$ tends to zero at a rate not slower than ${(t_0+T-t)}^{\nu -1}$ and ${\dot{\eta }}_{1j}$ tends to zero at a rate not slower than ${(t_0+T-t)}^{\nu -2}$ as t approaches $t_0+T$. Therefore, in conjunction with (44), we can conclude that ${\dot{\varphi }}_{1j}$ is bounded over the interval $[t_0,t_0+T)$.

Consequently, the boundedness of ${\dot{\varphi }}_1$ can be guaranteed over the interval $[t_0,t_0+T)$, which in turn implies that ${\varphi }_2$ is also bounded.

By repeatedly applying the L’Hôpital’s rule to (49), we establish that ${\eta }_{1j}^{\left(l\right)}$, for $j=1,2,\dots ,m$ and $l=1,\dots ,n-1$, approaches zero at a rate of no less than ${(t_0+T-t)}^{\nu -l-1}$ as t nears $t_0+T$. By analogous reasoning applied to (44)–(46), we can similarly demonstrate that ${\eta }_i^{\left(l\right)}$, $i=2,\dots ,n-1$, $l=1,\dots ,n-i$ also tends to zero at a comparable rate as t approaches $t_0+T$. Consequently, it is sufficient to show that ${\varphi }_i^{\left(l\right)}$, for $i=1,2,\dots ,n-1$ and $l=1,\dots ,n-i$, remains bounded over the interval $[t_0,t_0+T)$. Observing that $x_i={\xi }_i+{\varphi }_{i-1}$, $i=2,\dots ,n$, we deduce that $x_i$, $i=1,2,\dots ,n$ is bounded. In conjunction with (38), we can conclude that u is bounded on the interval $[t_0,t_0+T)$.

(3) Given that ${lim}_{t\to t_0+T}\left|{\xi }_1\left(t\right)\right|=0$, we know that ${lim}_{t\to t_0+T}\left|{\zeta }_j\left(t\right)\right|=0$, $j=1,2,\dots ,m$. In light of (17), we can deduce that

$$\begin{array}{c}\hfill \underset{t\to t_0+T}{lim}|z_j(t)|\le \underset{t\to t_0+T}{lim}|{\zeta }_j(t)|=0,j=1,2,\dots ,m,\end{array}$$

which asserts that ${lim}_{t\to t_0+T}\left|z\left(t\right)\right|={lim}_{t\to t_0+T}\sqrt{|z_1{\left(t\right)|}^2+\dots +{\left|z_m\left(t\right)\right|}^2}=0$ holds. Furthermore, considering the expression for ${\dot{z}}_j$ as given in (18), we have

$$\begin{array}{cc}\hfill |{\dot{z}}_j|& \le \left|{\displaystyle \frac{\partial z_j}{\partial {\zeta }_j}}\right||{\dot{\zeta }}_j|+\left|{\displaystyle \frac{\partial z_j}{\partial E_{u,j}}}\right||{\dot{E}}_{u,j}|+\left|{\displaystyle \frac{\partial z_j}{\partial E_{l,j}}}\right||{\dot{E}}_{l,j}|\hfill \\ \hfill & \le |{\dot{\zeta }}_j|+|{\displaystyle \frac{{\dot{E}}_{u,j}}{E_{u,j}}}||{\zeta }_j|+|{\displaystyle \frac{{\dot{E}}_{l,j}}{E_{l,j}}}||{\zeta }_j|,\hfill \end{array}$$

where we use the facts that $|{\displaystyle \frac{\partial z_j}{\partial {\zeta }_j}}|\le 1$, $|{\displaystyle \frac{\partial z_j}{\partial E_{u,j}}}|\le |{\displaystyle \frac{{\zeta }_j}{E_{u,j}}}|$ and $|{\displaystyle \frac{\partial z_j}{\partial E_{l,j}}}|\le |{\displaystyle \frac{{\zeta }_j}{E_{l,j}}}|$. Therefore, in conjunction with ${lim}_{t\to t_0+T}|\dot{{\zeta }_j}(t)|=0$ for $j=1,2,\dots ,m$, it can be concluded that ${lim}_{t\to t_0+T}\left|\dot{z}\left(t\right)\right|=0$ holds. The proof is complete. □

Remark 5. 

As noted in [28], the BLF method often requires the initial output to be restricted within a smaller range than the actual constraints, whereas the ANM method effectively avoids this limitation. To illustrate this issue, we consider the following system with output constrained by a constant bound:

$$\begin{array}{c}\hfill {\dot{x}}_1(t)=u+x_2(t),y=x_1(t),\end{array}$$

(50)

where $x(0)=x_0$, $u\in \mathbb{R}$ is the control input, and $x_1,x_2\in \mathbb{R}$ are the system states. The system output y is subject to the following constraint: $-{\epsilon }_r<y(t)<{\epsilon }_r$, where ${\epsilon }_r$ is a known positive constant. Define the tracking error as $z_e=y-y_r$, where $y_r$ is a known reference signal. We consider the following logarithmic BLF:

$$\begin{array}{cc}\hfill V& ={\displaystyle \frac{1}{2}}log{\displaystyle \frac{{\epsilon }_b^2}{{\epsilon }_b^2-{z_e}^2}},\hfill \end{array}$$

where ${\epsilon }_b={\epsilon }_r-Y$, $|y_r|\le Y<{\epsilon }_r$, Y is a positive constant. The BLF approach guarantees compliance with the output constraint by restricting the tracking error within the bound: $-{\epsilon }_b<z_e<{\epsilon }_b$. Recalling the definition of $z_e$, we obtain $y_r-{\epsilon }_r+Y=y_r-{\epsilon }_b<y<{\epsilon }_b+y_r={\epsilon }_r-Y+y_r$. When $t=0$, which implies that:

$$\begin{array}{c}\hfill y_r\left(0\right)-{\epsilon }_r+Y<y\left(0\right)<{\epsilon }_r-Y+y_r\left(0\right).\end{array}$$

(51)

Since $|y_r|\le Y$, it follows that $-Y<y_r(0)<Y$, leading to a more restrictive initial range for the output signal $y\left(0\right)$ than the original constraint $-{\epsilon }_r<y(t)<{\epsilon }_r$. Thus, by adopting the ANM method, we effectively circumvent the conservativeness associated with the BLF-based control algorithm regarding the initial value of the system output.

Remark 6. 

It is noteworthy that this work centers on achieving PT tracking control, a goal that diverges from the findings presented in [4,7,9,22]. The pivotal difference resides in our employment of the ANM method to address output constraints, thereby recasting the output-constrained challenge into an investigation of the boundedness of ${\zeta }_j$, $j=1,2,\dots ,m$. Conversely, by using the time-varying gain technique and L’Hôpital’s rule, a zero-error tracking outcome is attainable. For analogous results, refer to [7], where a full-state tracking result is obtained for normal MIMO systems. Indeed, due to the presence of output constraints and the ANM (16), a nonlinear term ${\mathcal{I}}_{\{{\zeta }_j\ge 0\}}{s}_{u,j}{\dot{E}}_{u,j}+{\mathcal{I}}_{\{{\zeta }_j<0\}}{s}_{l,j}{\dot{E}}_{l,j}$ is incorporated into (19). It can be easily verified that this nonlinear term vanishes as ${\zeta }_j$, $j=1,2,\dots ,m$ approaches zero. Consequently, a good tracking result, ${lim}_{t\to t_0+T}\left[|z(t)|+|\dot{z}(t)|\right]=0$, is ensured. Given that numerous physical systems can be represented as two-dimensional nonlinear systems, this tracking outcome is adequately sufficient for a broad spectrum of practical applications.

4. Example

In this section, two examples are provided to validate our theoretical results.

Example 1. 

Consider the following three-dimensional switched MIMO system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_1=x_2,\hfill \\ {\dot{x}}_2={\mathbf{H}}_{\sigma }\left(x_1\right)u+{\mathbf{F}}_{\sigma }(t,x),\hfill \\ y=x_1,\hfill \end{array}\right.\end{array}$$

(52)

where $x_1={[x_{11},x_{12}]}^T\in {\mathbb{R}}^2$,$x_2={[x_{21},x_{22}]}^T\in {\mathbb{R}}^2$, $x={[x_1,x_2]}^T\in {\mathbb{R}}^4$ and $u={[u_1,u_2,u_3]}^T\in {\mathbb{R}}^3$ are the state and control input vectors, respectively. The switching signal $\sigma \left(t\right)$ takes values from the set $\mathcal{N}=\{1,2\}$. The system parameters are defined as follows:

$${\mathbf{H}}_{\mathbf{1}}=\left[\begin{array}{ccc}0.9& 0& 0.2\\ 0& 0.9& 0.5\end{array}\right],{\mathbf{H}}_{\mathbf{2}}=\left[\begin{array}{ccc}0.6& 0& 0.2\\ 0& 0.6& 0.3\end{array}\right],{\mathbf{F}}_{\mathbf{1}}=\left[\begin{array}{c}0.05x_{12}\\ 0.02x_{11}\end{array}\right],{\mathbf{F}}_{\mathbf{2}}=\left[\begin{array}{c}0.02x_{21}\\ 0.04x_{22}\end{array}\right].$$

(53)

Consider the output variable $y={[y_1,y_2]}^T$, which is required to satisfy the time-varying constraints ${\epsilon }_{l,j}(t)<y_j<{\epsilon }_{u,j}(t)$$(j=1,2)$ for all $t\ge 0$. The time-varying constraint function vectors are defined as:

$${\epsilon }_l(t)={\left[\begin{array}{c}3.5-e^{-2t},3-e^{-t}\end{array}\right]}^T,{\epsilon }_u(t)={\left[\begin{array}{c}6.5+e^{-2t},6.5+e^{-t}\end{array}\right]}^T.$$

(54)

Furthermore, the reference trajectories are given by

$$\begin{array}{cc}\hfill y_r& ={\left[\begin{array}{c}sin3t+5,cos3t+5\end{array}\right]}^T.\hfill \end{array}$$

(55)

Besides, it can be verified that Assumption 1 holds by choosing $f(x)=|x_1|+|x_2|$. By choosing the matrix

$$\begin{array}{c}\hfill \mathbf{P}=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 2\end{array}\right],\end{array}$$

it can be verified that Assumption 2 holds, and the minimum eigenvalue satisfies $\underline{\lambda }=110$. Clearly, Assumption 3 is satisfied with ${\underline{M}}_0=2,{\overline{M}}_0=7.5$, $M_1=2$.

Based on the control framework introduced in Section 3, we design a tracking controller as follows:

$$\begin{array}{c}\hfill u=-2{\underline{\lambda }}^{-1}\left(k_2+{\displaystyle \frac{1}{2}}{\mu }^{-1}(t)+{\left(2\theta \right)}^{-1}\left(f^2+{\left|{\dot{\varphi }}_1\right|}^2\right)+\dot{\mu }(t){\mu }^{-2}(t)\right){\mathbf{P}}^T{\eta }_2.\end{array}$$

(56)

where ${\zeta }_1$, ${\eta }_1$ and ${\eta }_2$ are defined in (21) with ${\varphi }_1=-\left(k_1{\eta }_1+{\displaystyle \frac{1}{2}}{\xi }_1+D_1{\xi }_1+\dot{\mu }(t){\mu }^{-1}(t){\xi }_1\right)+{\dot{y}}_r$. For the simulation, the initial condition is given as $x_1(0)={[4,4]}^T$, $x_2\left(0\right)={[4,4]}^T$, and the design parameters are set as $\theta =2.5$, $k_1=0.2$, $k_2=1$, $\nu =2$, and $T_{pre}=6$.

The simulation results are given in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7. Figure 2 illustrates that the output tracking errors $z_j(j=1,2)$ converge to zero at the prescribed time, while the norm of the derivative of the tracking error also converges to zero. Figure 3 demonstrates that the time-varying asymmetric output constraints are strictly satisfied, and the control signal u remains bounded. To better illustrate the advantages of the ANM method, we present the control performance and output tracking under different constraint settings (see Figure 4, Figure 5, Figure 6 and Figure 7), including unilateral constraints, constant constraints, and the unconstrained case. This confirms that the ANM method is applicable to all these scenarios, as stated in Remark 3. Compared to the unconstrained case, the control input u exhibits greater variation under time-varying asymmetric output constraints. This is because the system must exert more control effort to ensure that the state remains within the prescribed constraint boundaries.

Figure 1. The switching law $\sigma \left(t\right)$ in Example 1.

Figure 2. The output tracking error in Example 1.

Figure 3. Output and control input under asymmetric time-varying constraints in Example 1.

Figure 4. Output and control input under one-sided time-varying constraints ${\epsilon }_u\left(t\right)$ in Example 1.

Figure 5. Output and control input under one-sided time-varying constraints ${\epsilon }_l\left(t\right)$ in Example 1.

Figure 6. Output and control input under asymmetric constant constraints in Example 1.

Figure 7. Unconstrained output and control input in Example 1.

Example 2. 

Consider a spacecraft system driven by four reaction wheels as mentioned in [8], where the nonlinear dynamics of the system are given by

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\dot{x}}_1& =x_2,\hfill \\ \hfill {\dot{x}}_2& ={\mathbf{H}}_{\sigma }\left(x_1\right)u+{\mathbf{F}}_{\sigma }(t,x),\hfill \end{array}\right.\end{array}$$

where $x_1={[x_{11},x_{12},x_{13}]}^T\in {\mathbb{R}}^3$ represents the attitude angles of the spacecraft, $x_2={[x_{21},x_{22},x_{23}]}^T\in {\mathbb{R}}^3$ represents the angular velocity of the spacecraft, and $u={[u_1,u_2,u_3,u_4]}^T\in {\mathbb{R}}^4$ represents the control torques imposed on the four reaction wheels. ${\mathbf{H}}_{\sigma }(x_1)=J_{\sigma }^{-1}D$ is an extended inertia matrix with $J_{\sigma }$ being the inertia matrix, and ${\mathbf{F}}_{\sigma }(t,x)=-J_{\sigma }^{-1}S(x_2)J_{\sigma }{x}_2$ represents the nonlinear disturbance term with $S\left(x_2\right)$ being a skew-symmetric matrix of $x_2$ defined by

$$\begin{array}{c}\hfill S(x_2)=\left[\begin{array}{ccc}0& -x_{23}& x_{22}\\ x_{23}& 0& -x_{21}\\ -x_{22}& x_{21}& 0\end{array}\right].\end{array}$$

(57)

In this example, switching dynamics are incorporated into system modeling to capture the variation in outer surroundings, i.e., $\sigma \left(t\right)$ represents the switching signal taking values from the set $\mathcal{N}=\{1,2\}$. We select the matrices in the above model as

$$J_1=\left[\begin{array}{ccc}40& 0& 0\\ 0& 17& 0\\ 0& 0& 15\end{array}\right]\mathrm{kg}·{\mathrm{m}}^2,J_2=\left[\begin{array}{ccc}25& 0& 0\\ 0& 20& 0\\ 0& 0& 15\end{array}\right]\mathrm{kg}·{\mathrm{m}}^2,D=\left[\begin{array}{cccc}1& 0& 0& 1/\sqrt{3}\\ 0& 1& 0& 1/\sqrt{3}\\ 0& 0& 1& 1/\sqrt{3}\end{array}\right].$$

(58)

Assume that the output variable $y=x_1$, and $y={[y_1,y_2,y_3]}^T$ must fulfill ${\epsilon }_{l,j}(t)<y_j<{\epsilon }_{u,j}(t)$ ($j=1,2,3$) for all $t\ge 0$. The time-varying constraint function vectors ${\epsilon }_l(t)={[{\epsilon }_{l,1}(t),{\epsilon }_{l,2}(t),{\epsilon }_{l,3}(t)]}^T$ and ${\epsilon }_u(t)={[{\epsilon }_{u,1}(t),{\epsilon }_{u,2}(t),{\epsilon }_{u,3}(t)]}^T$ are set as

$$\begin{array}{cc}\hfill {\epsilon }_l(t)& ={\left[\begin{array}{c}0.25,0.2,0.3\end{array}\right]}^T,\hfill \\ \hfill {\epsilon }_u(t)& ={\left[\begin{array}{c}0.4+0.2e^{-6t},0.35+0.15e^{-6t},0.45+0.1e^{-6t}\end{array}\right]}^T.\hfill \end{array}$$

(59)

The desired trajectories of attitude angles are chosen as

$$\begin{array}{cc}\hfill y_r& ={\left[\begin{array}{c}0.3+0.05e^{-6t},0.25+0.04e^{-6t},0.35+0.03e^{-6t}\end{array}\right]}^T.\hfill \end{array}$$

(60)

It is easy to show that $|{\mathbf{F}}_{\sigma }(t,x)|\le c_{\sigma }(t)f(x)$, with $c_{\sigma }\left(t\right)$ being some unknown disturbance and $f\left(x\right)=|x_2{|}^2$(which is computable and not unique). Hence, it shows that Assumption 1 holds. Based on (58), it can be computed that the matrix $\mathbf{P}$ is setting as

$$\begin{array}{c}\hfill \mathbf{P}=\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 0& 1\\ 0& 0& 1& 1\end{array}\right],\end{array}$$

(61)

which implies that Assumption 2 holds with $\underline{\lambda }=0.055$. According to (2), Assumption 3 is satisfied with ${\underline{M}}_0=0.2,{\overline{M}}_0=0.6$, $M_1=1.2$.

Based on the control framework introduced in Section 3, we design a tracking controller as follows:

$$\begin{array}{c}\hfill u=-2{\underline{\lambda }}^{-1}\left(k_2+{\displaystyle \frac{1}{2}}{\mu }^{-1}(t)+{\left(2\theta \right)}^{-1}\left(f^2+{\left|{\dot{\varphi }}_1\right|}^2\right)+\dot{\mu }(t){\mu }^{-2}(t)\right){\mathbf{P}}^T{\eta }_2.\end{array}$$

(62)

where ${\zeta }_1$, ${\eta }_1$ and ${\eta }_2$ are defined in (21) with ${\varphi }_1=-\left(k_1{\eta }_1+{\displaystyle \frac{1}{2}}{\xi }_1+D_1{\xi }_1+\dot{\mu }(t){\mu }^{-1}(t){\xi }_1\right)+{\dot{y}}_r$. For the simulation, the initial condition is given as $x_1(0)={\left[\begin{array}{c}0.5,0.4,0.45\end{array}\right]}^T$, $x_2(0)={\left[\begin{array}{c}0,0,0\end{array}\right]}^T$, and the design parameters are set as $\theta =5$, $k_1=k_2=1$, $\nu =2$, and $T_{pre}=3$. The simulation results are given in Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12. Figure 9 shows that the output tracking errors $z_j$ ($j=1,2,3$) converge to zero at the prescribed time, verifying the effectiveness of the proposed control strategy in achieving zero-error trajectory tracking. Figure 10 illustrates that the control input u remains bounded throughout the tracking process. From Figure 11, we can infer that the time-varying asymmetric output constraints are satisfied, confirming that the ANM approach successfully prevents constraint violations. Figure 12 presents the variation of the tracking error under different prescribed times, further validating the flexibility of the proposed method in achieving prescribed-time convergence with different time settings.

Figure 8. The switching law $\sigma \left(t\right)$ in Example 2.

Figure 9. The output tracking error in Example 2.

Figure 10. The control input in Example 2.

Figure 11. The output response under asymmetric time-varying constraints in Example 2.

Figure 12. Comparison of tracking error under different prescribed times in Example 2.

Remark 7. 

Although the boundedness of all signals has been theoretically proven, numerical simulations often encounter issues due to the unbounded nature of the time-varying gain $\mu \left(t\right)$ as t approaches $T_{pre}$. To circumvent this problem and enable practical simulation, we introduce a small positive parameter $ϵ>0$ to modify $\mu \left(t\right)$, resulting in ${\tilde{T}}_{pre}=T_{pre}+ϵ$. This method is consistent with the techniques employed in [33,34].

5. Conclusions

A PT tracking control method is proposed for a class of switched MIMO nonlinear systems with asymmetric time-varying output constraints. By combining the tangent-type ANM and the backstepping recursive design strategy, the proposed controller effectively achieves zero-error tracking while ensuring that all closed-loop signals remain bounded and satisfy the specified output constraints. Simulation results confirm that the method successfully achieves the desired tracking performance. In future work, we will investigate prescribed-time tracking control for switched stochastic MIMO systems, considering both system uncertainties and external disturbances. Furthermore, we will develop an adaptive controller based on the backstepping method to achieve prescribed-time tracking control.