Design of a Finite-Time Bounded Tracking Controller for Time-Delay Fractional-Order Systems Based on Output Feedback

Abstract

This paper focuses on a class of fractional-order systems with state delays and studies the design problem of the finite-time bounded tracking controller. The error system method in preview control theory is first used. By taking fractional-order derivatives of the state equations and error signals, a fractional-order error system is constructed. This transforms the tracking problem of the original system into an input–output finite=time stability problem of the error system. Based on the output equation of the original system and the error signal, an output equation for the error system is constructed, and a memory-based output feedback controller is designed by means of this equation. Using the input–output finite-time stability theory and linear matrix inequality (LMI) techniques, the output feedback gain matrix of the error system is derived by constructing a fractional-order Lyapunov–Krasovskii function. Then, a fractional-order integral of the input to the error system is performed to derive a finite-time bounded tracking controller for the original system. Finally, numerical simulations demonstrate the effectiveness of the proposed method.

1. Introduction

The study of precise models is key to accurately understanding the laws of operation of phenomena and, subsequently, predicting and transforming the world. In both nature and engineering practice, it has been observed that many control systems exhibit non-locality and long memory characteristics, which are difficult to describe accurately using traditional integer-order differential equations [1,2]. Fractional-order calculus provides a new tool that can more accurately capture and simulate the behavior of these systems [3,4]. Furthermore, due to measurement errors, component aging, and other factors, systems inevitably exhibit time-delay characteristics during the modeling process [5,6]. Therefore, studying fractional-order systems with time delays is of significant theoretical importance and practical value.

As a key research branch of time-delay fractional-order systems, the tracking control problem focuses on how to design a tracking controller that enables the system output to follow the reference signal [7,8,9]. To date, many research studies have been performed on this problem. For example, adaptive tracking control [10], tracking control based on the internal model principle [11], and sliding mode tracking control [12]. However, it is worth mentioning that current research on the tracking control of time-delay fractional-order systems mainly focuses on the system’s behavior over an infinite time interval. That is, it characterizes the steady-state performance of the system, but does not reflect the transient performance of the system. In many practical engineering problems, the primary concern is the system’s behavior within a fixed time interval, with the aim of ensuring that the system’s output stays within the domain defined by the reference signal over a finite period. For instance, it is desired that a robot moves along a planned path within a given time span. In response to this, the concept of finite-time bounded tracking control has been proposed in the academic community, and studies have been conducted on various types of systems [13,14,15]. In [13], based on the theory of partial state stability and finite-time boundedness, a design method of finite-time bounded tracking controllers for discrete linear time systems is presented. In [14], the concept of finite-time bounded tracking for discrete-time systems is extended to continuous-time delay systems, and the finite-time bounded tracking control problem for these systems is studied. In [15], a state feedback approach is used to design a finite-time bounded tracking controller for linear fractional-order systems without delays.

Compared to fractional-order systems without delays, the finite-time bounded tracking control problem for delayed fractional-order systems is more challenging. One of the main reasons is that the presence of time delays causes the system to be described by fractional-order differential-difference equations, which have infinite-dimensional characteristics [16]. This paper investigates the finite-time bounded tracking control problem for a class of delayed fractional-order systems. The key motivation of our work is to address the challenges posed by delayed fractional-order systems, which frequently arise in various real-world applications such as engineering, biology, and economics [5,6]. These systems exhibit memory and hereditary properties, making them more complex to analyze and control than integer-order systems. Extending the concept of finite-time bounded tracking control to delayed fractional-order systems enables more effective handling of time-delay effects and fractional dynamics within a finite-time framework. Thus, this provides a solid theoretical foundation and method for the application of finite-time bounded tracking control in time-delay fractional-order systems. In this paper, by utilizing the error system method from preview control theory, the finite-time bounded tracking problem is transformed into an input–output finite-time stability problem. Then, the output equation is constructed, and an output feedback controller for the error system is designed using this equation. By designing a fractional-order Lyapunov–Krasovskii function for the closed-loop error system and applying linear matrix inequality (LMI) methods, the output feedback controller for the error system is derived. Subsequently, sufficient conditions and the corresponding controller are derived to achieve finite-time bounded tracking of the original system’s output with respect to the reference signal. The main contributions of this paper are as follows:

(a)

The concept of finite-time bounded tracking control is first extended to delayed fractional-order systems, and a finite-time bounded tracking controller is designed for them. This expansion enriches the theoretical system of fractional-order systems and offers new control possibilities.

(b)

Considering the difficulty in measuring system states, an output feedback method is adopted. The output equation is modified to utilize both output and error signal information, and a memory output feedback controller is designed.

(c)

To deal with state time-delay in fractional-order systems, a fractional-order Lyapunov–Krasovskii function is constructed. By leveraging the relationship between Caputo and Riemann–Liouville fractional-order derivatives, the controller’s existence conditions are deduced.

The research plan of this article is as follows: Some basic notions and properties for fractional calculus will be reviewed in Section 2. Section 3 will propose the problem of finite-time bounded tracking control for time-delay fractional-order systems, and give some basic assumptions. To design the finite-time bounded tracking controller for such systems, a fractional error system will be constructed, and by designing a fractional-order Lyapunov–Krasovskii function for the error system and applying LMI methods, the output feedback controller will be derived in Section 4. Section 5 will demonstrate the effectiveness of the proposed method through numerical simulations. Conclusions will be drawn in Section 6.

Notations: 

The notations in this paper are standard. For any matrix $Q$, $Q\in {ℝ}^n$ and $Q\in {ℝ}^{m\times n}$ denote an n-dimensional matrix and an $m\times n$ matrix, respectively. $I$ is the identity matrix. $Q>0 (\ge 0)$ indicates that $Q$ is positive definite (semi-definite), and $Q^T$ denotes the transpositions of $Q$. The symbol $*$ in a matrix indicates symmetric entries.

2. Preliminaries

This section introduces the basic definitions and properties of fractional calculus required in this paper. Because the Riemann–Liouville derivative is well-suited to mathematical analysis due to its intrinsic properties and widespread use in modeling dynamic systems with memory effects, and the Caputo derivative is particularly useful in control applications as it allows the initial conditions to be expressed in terms of integer-order derivatives, which are more physically interpretable [2], this article adopts Riemann–Liouville and Caputo fractional-order derivatives. Without loss of generality, it is assumed in this paper that the lower limits of fractional integrals and derivatives are zero.

Definition 1

([17], Left Riemann–Liouville (R-L) Fractional Integral). Let the function $x(t)$  be integrable, and  $\alpha$  be any given positive real number. The left Riemann–Liouville fractional integral of order  $\alpha$ is defined as

$${}_{0}I_t^{\alpha }x(t)={\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha )}}{\displaystyle {\int }_0^t{(t-\tau )}^{\alpha -1}x(\tau )d\tau },$$

where ${}_{0}I_t^{\alpha }$ represents the $\alpha$-order integral operator in the interval $[0,t]$, $\mathsf{\Gamma }(\alpha )={\displaystyle {\int }_0^{\infty }{e}^{-t}{t}^{\alpha -1}dt}$.

Based on Definition 1, the following definitions of the Riemann–Liouville fractional derivative and the Caputo fractional derivative can be obtained.

Definition 2

([17], Left Riemann–Liouville Fractional Derivative). Let the function $x(t)$ be integrable, $\alpha >0$ , and n be the smallest integer greater than  $\alpha$, i.e.,  $n-1<\alpha <n$. The left Riemann–Liouville fractional derivative of order  $\alpha$  is defined as

$$\begin{array}{l}{}_0{}^{RL}D_t^{\alpha }x(t)=D^n({}_{0}I_t^{(n-\alpha )}x(t))\\ ={\displaystyle \frac{1}{\mathsf{\Gamma }(n-\alpha )}}{\displaystyle \frac{{\mathrm{d}}^n}{{\mathrm{dt}}^n}}\left({\displaystyle {\int }_{t_0}^t{(t-\tau )}^{n-\alpha -1}x(\tau )d\tau }\right)\end{array},$$

where $D^n$ is the integer-order derivative operator.

Definition 3

([17], Left Caputo Fractional Derivative). Let the function $x(t)$ be differentiable, $\alpha >0$, and n be the smallest integer greater than  $\alpha$, i.e.,  $n-1<\alpha <n$. The left Caputo fractional derivative of order  $\alpha$  is defined as

$$\begin{array}{l}{}_0{}^{C}D_t^{\alpha }x(t)={}_{0}I_t^{n-\alpha }(D^{n}x(t))\\ ={\displaystyle \frac{1}{\mathsf{\Gamma }(n-\alpha )}}{\displaystyle {\int }_0^t{(t-\tau )}^{n-\alpha -1}{x}^{(n)}(\tau )d\tau }\end{array}.$$

Remark 1. 

From Definitions 2 and 3, it can be seen that the $\alpha$-order left Riemann–Liouville fractional derivative of  $x(t)$  is essentially obtained by first performing an  $(n-\alpha )$-order integral on  $x(t)$, followed by an  $n$-order derivative. In contrast, the  $\alpha$-order left Caputo fractional derivative is obtained by first taking the  $n$-order derivative, followed by an  $(n-\alpha )$-order integral. Furthermore, in general, the left Riemann–Liouville fractional derivative and the left Caputo fractional derivative are not equivalent. The relationship between them is given by

$${}_0{}^{RL}D{}_t{}^{\alpha }x(t)={}_0{}^{C}D{}_t{}^{\alpha }x(t)+{\displaystyle \sum _{k=0}^{n-1}\frac{t^{k-\alpha }}{\mathsf{\Gamma }(k-\alpha +1)}}{x}^{(k)}(0).$$

where $x^{(0)}(t)=x(t)$, but under the condition $x^{(k)}(0)=0$$(k=0,1,\cdots ,n-1)$, they are equivalent [17].

For simplicity, we abbreviate ${}_{0}I_t^{\alpha }x(t)$ as $I^{\alpha }x(t)$, ${}_0{}^{RL}D{}_t{}^{\alpha }x(t)$ as ${}^{RL}D{}^{\alpha }x(t)$, and ${}_0{}^{C}D{}_t{}^{\alpha }x(t)$ as ${}^{C}D{}^{\alpha }x(t)$.

In the process of designing the controller, we need to use the following lemmas.

Lemma 1

([18]). Let $0<\alpha <1$, and let  $x(t)\in R^n$ be a continuously differentiable function. For $t\ge t_0$, we have

$${}^{C}D{}^{\alpha }\left[x^T(t)Px(t)\right]\le x^T(t)P\left({}^{C}D{}^{\alpha }x(t)\right)+{\left({}^{C}D{}^{\alpha }x(t)\right)}^{T}Px(t),$$

where $P\in R^{n\times n}$ and $P>0$.

Lemma 2

([19]). Suppose $x(t)$ is a non-negative continuous function. Then, for any $t\in [0,T]$, we have

$$I^{\alpha }x(t)={\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha )}}{\displaystyle {\int }_0^t{(t-s)}^{\alpha -1}}x(s)ds\le a\iff x(t)\le {\displaystyle \frac{\mathsf{\Gamma }(1+\alpha )}{T^{\alpha }}}a,$$

where $a>0$ is the upper bound of $I^{\alpha }x(t)$.

Lemma 3

([20]). Let $\alpha >0$, $n-1<\alpha <n$, and $x(t)$ be an $n$-order differentiable function. The composite formula for the fractional integral and the Caputo-type fractional derivative of the same order is given by

$$I^{\alpha }({}^{C}D{}^{\alpha }x(t))=x(t)-{\displaystyle \sum _{k=0}^{n-1}\frac{t^k}{k!}{x}^{(k)}(0)}.$$

Lemma 4

(Schur’s Complement [21]). Consider the matrix $S=\left[\begin{array}{cc}{S}_{11}& S_{12}\\ S_{12}^T& S_{22}\end{array}\right]$, where $S_{11}$ and $S_{22}$ are both symmetric and invertible. The following three inequalities are equivalent:

(1) 

$S<0$;

(2) 

$S_{11}<0,S_{22}-S_{12}^T{S}_{11}^{-1}{S}_{12}<0$;

(3) 

$S_{22}<0,S_{11}-S_{12}{S}_{22}^{-1}{S}_{12}^T<0$.

3. Problem Statement and Basic Assumptions

Consider the following linear fractional-order system with state delays:

$$\left\{\begin{array}{l}{}^{C}D{}^{\alpha }x(t)=Ax(t)+A_{1}x(t-\gamma )+Bu(t)+Hw(t)\\ y(t)=Cx(t)\\ x(t)=\eta (t), t\in \left[-\gamma ,0\right]\end{array}\right.,$$

(1)

where $0<\alpha <1$; $x(t)\in {ℝ}^n$ is the state variable, $u(t)\in {ℝ}^m$ is the control input, $w(t)\in {ℝ}^q$ is the disturbance, and $y(t)\in {ℝ}^p$ is the output; $\gamma >0$ is the state delay; $A\in {ℝ}^{n\times n}$, $A_1\in {ℝ}^{n\times n}$, $B\in {ℝ}^{n\times m}$, $H\in {ℝ}^{n\times q}$, and $C\in {ℝ}^{p\times n}$ are constant matrices; and $\eta (t)$ is the initial function of the state vector.

Remark 2. 

The model (1) represents a time-delay fractional-order system, which is widely used to describe processes with memory and hereditary properties, such as those in a viscoelastic mechanical system [22] and power system [6]. The inclusion of a time delay reflects the practical phenomenon where system states depend not only on the current input but also on past states. The condition $0<\alpha <1$  arises because fractional-order derivatives with orders in this range are particularly effective for modeling such memory-dependent behaviors [5,6].

Let $r(t)$ be the reference signal of the system (1), and let $r(t)=\phi (t)$ for $t\in [-\gamma ,0]$, where $\phi (t)$ represents the initial function. Additionally, denote the error signal $e(t)$ as the difference between $r(t)$ and the output signal $y(t)$, namely,

$$e(t)=r(t)-y(t).$$

(2)

Extend $y(t)$ to the interval $\left[-\gamma ,0\right]$ while maintaining $y(t)=Cx(t)$. Then, the initial condition of the error signal can be obtained as $e(t)=C\eta (t)-\phi (t)$, $t\in \left[-\gamma ,0\right]$.

The reference signal $r(t)$ and the disturbance signal $w(t)$ of system (1) satisfy the following basic assumptions.

Assumption 1.

Suppose the reference signal $r(t)$ is a piecewise continuously differentiable function and satisfies

$${\left({\displaystyle \frac{T^{1-\alpha }}{\mathsf{\Gamma }(2-\alpha )}}\right)}^2\underset{t\in [0,T]}{\sup }{\dot{r}}^T(t)Q_1\dot{r}(t)\le c_{11}.$$

where  $Q_1\in {ℝ}^{p\times p}$  is a given positive definite matrix, and  $c_{11}$  is a given positive constant.

Assumption 2.

Suppose the disturbance signal $w(t)$ is a piecewise continuously differentiable function and satisfies

$${\left({\displaystyle \frac{T^{1-\alpha }}{\mathsf{\Gamma }(2-\alpha )}}\right)}^2\underset{t\in [0,T]}{\sup }{\dot{w}}^T(t)Q_2\dot{w}(t)\le c_{22}.$$

where  $Q_2\in {ℝ}^{q\times q}$  is a given positive definite matrix, and  $c_{22}$  is a given positive constant.

We extend the definition of finite-time bounded tracking in reference [15] to linear time-delay fractional order systems (1). Thus, the following definition is obtained.

Definition 4.

Given scalars $c_1>0$, $c_2>0$, $T>0$, and matrices $Q>0$ and $\mathsf{\Phi }>0$, under zero initial conditions (i.e., $\eta (t)=0 (t\in \left[-\gamma ,0\right])$, when$\underset{t\in [0,T]}{\sup }{w}^T(t)Qw(t)\le c_1$, if for all $t\in [0,T]$ and $\gamma >0$, we have $e^T(t)\mathsf{\Phi }e(t)<c_2$, then the delayed fractional-order system (1) achieves finite-time bounded tracking of the reference signal $r(t)$ with respect to $(c_1,c_2,Q,\mathsf{\Phi },T)$.

The purpose of this paper is to design an output feedback controller for the fractional-order linear system (1), such that the output $y(t)$ of the closed-loop system achieves finite-time bounded tracking of the reference signal $r(t)$ with respect to $(c_1,c_2,Q,\mathsf{\Phi },T)$.

4. Controller Design

To design the tracking controller for system (1), we first construct an error system that includes the error signal.

Take the $\alpha$-order Caputo fractional derivative of both sides of Equation (2), and based on the output equation of system (1), we obtain

$${}^{C}D{}^{\alpha }e(t)={}^{C}D{}^{\alpha }r(t)-{}^{C}D{}^{\alpha }y(t)={}^{C}D{}^{\alpha }r(t)-C\left({}^{C}D{}^{\alpha }x(t)\right)$$

(3)

Next, taking ${}^{C}D^{\alpha }$ on both sides of the state equation of system (1), we obtain

$$\begin{gathered} {}^{C}D{}^{\alpha }({}^{C}D{}^{\alpha }x(t))={}^{C}D{}^{\alpha }\left(Ax(t)\right)+{}^{C}D{}^{\alpha }\left(A_{1}x(t-\gamma )\right)+{}^{C}D{}^{\alpha }\left(Bu(t)\right)+{}^{C}D{}^{\alpha }\left(Hw(t)\right) \\ =A\left({}^{C}D{}^{\alpha }x(t)\right)+A_1\left({}^{C}D{}^{\alpha }x(t-\gamma )\right)+B\left({}^{C}D{}^{\alpha }u(t)\right)+H\left({}^{C}D{}^{\alpha }w(t)\right). \end{gathered}$$

(4)

Introduce a new state vector in the form of

$$\overline{x}(t)=\left[\begin{array}{c}{}^{C}D{}^{\alpha }x(t)\\ e(t)\end{array}\right]\in {ℝ}^{p+n}.$$

Thus, combining Equations (3) and (4), we obtain the following:

$${}^{C}D{}^{\alpha }\overline{x}(t)=\overline{A}\overline{x}(t)+{\overline{A}}_1\overline{x}(t-\gamma )+\overline{B}\left({}^{C}D{}^{\alpha }u(t)\right)+\overline{H}\overline{w}(t)$$

(5)

where

$$\begin{gathered} \overline{A}=\left[\begin{array}{cc}A& 0\\ -C& 0\end{array}\right], {\overline{A}}_1=\left[\begin{array}{cc}{A}_1& 0\\ 0& 0\end{array}\right], \overline{B}=\left[\begin{array}{c}B\\ 0\end{array}\right], \\ \overline{H}=\left[\begin{array}{cc}H& 0\\ 0& I\end{array}\right], \overline{w}(t)=\left[\begin{array}{c}{}^{C}D{}^{\alpha }w(t)\\ {}^{C}D{}^{\alpha }r(t)\end{array}\right]. \end{gathered}$$

System (5) is the desired error system.

From the output equation $y(t)=Cx(t)$ of system (1), we know that $y(t)$ is measurable, and the reference signal $r(t)$ is known. Therefore, $e(t)$ can also be obtained online. To fully utilize this information, we introduce the following observation equation for the error system (5):

$$\overline{y}(t)=\overline{C}\overline{x}(t)$$

(6)

where $\overline{C}=\left[\begin{array}{cc}C& 0\\ 0& I\end{array}\right]$.

Then, for system (5) and its observation Equation (6), we design the following memory-based output feedback controller:

$${}^{C}D{}^{\alpha }u(t)=F_1\overline{y}(t)+F_2\overline{y}(t-\gamma ).$$

(7)

where $F_1$ and $F_2$ are matrices to be determined. Under the action of controller (7), the closed-loop system

$${}^{C}D{}^{\alpha }\overline{x}(t)=\left(\overline{A}+\overline{B}{F}_1\overline{C}\right)\overline{x}(t)+\left({\overline{A}}_1+\overline{B}{F}_2\overline{C}\right)\overline{x}(t-\gamma )+\overline{H}\overline{w}(t)$$

(8)

is obtained.

In the following, we solve for the output feedback gain matrices $F_1$ and $F_2$ in system (8) and obtain the following theorem.

Theorem 1. 

Under Assumptions 1 and 2, if there are positive definite matrices $P>0$ and $M>0$  such that

$$\left[\begin{array}{ccc}P\left(\overline{A}+\overline{B}{F}_1\overline{C}\right)+{\left(\overline{A}+\overline{B}{F}_1\overline{C}\right)}^{T}P+M& P\left({\overline{A}}_1+\overline{B}{F}_2\overline{C}\right)& P\overline{H}\\ *& -M& 0\\ *& *& -Q\end{array}\right]<0$$

(9)

and

$${\tilde{C}}^T\mathsf{\Phi }\tilde{C}-{\displaystyle \frac{c_2{T}^{1-\alpha }}{c_1\mathsf{\Gamma }(2-\alpha )T}}P<0,$$

(10)

where $\tilde{C}=\left[\begin{array}{cc}0& I\end{array}\right]$, then system (8) can achieve finite-time bounded tracking of the reference signal $r(t)$ with respect to $(c_1,c_2,Q,\mathsf{\Phi },T)$, where $c_1\ge c_{11}+c_{22}$ and $Q=\left[\begin{array}{cc}{Q}_2& 0\\ 0& Q_1\end{array}\right]$.

Proof of Theorem 1.

Using positive definite matrices $P$  and $M$, which satisfy the matrix inequalities (9) and (10), we construct the following fractional Lyapunov–Krasovskii function:

$$V(t)=I^{1-\alpha }\left({\overline{x}}^T(t)P\overline{x}(t)\right)+{\displaystyle {\int }_{t-\gamma }^t{\overline{x}}^T(s)M\overline{x}(s)ds}.$$

(11)

taking the derivative of $V(t)$ in (11) along the trajectory of system (8). Based on the definition of the Riemann–Liouville fractional derivative, as well as the relationship between the Caputo fractional derivative and the Riemann–Liouville fractional derivative, we obtain

$$\begin{gathered} \dot{V}(z(t))={}^{RL}D{}^{\alpha }({\overline{x}}^T(t)P\overline{x}(t))+{\overline{x}}^T(t)M\overline{x}(t)-{\overline{x}}^T(t-\gamma )M\overline{x}(t-\gamma ) \\ ={}^{C}D{}^{\alpha }({\overline{x}}^T(t)P\overline{x}(t))+{\displaystyle \frac{t^{-\alpha }}{\mathsf{\Gamma }(1-\alpha )}}{\overline{x}}^T(0)P\overline{x}(0)+{\overline{x}}^T(t)M\overline{x}(t)-{\overline{x}}^T(t-\gamma )M\overline{x}(t-\gamma ). \end{gathered}$$

(12)

Since $\overline{x}(t)=\left[\begin{array}{c}{}^{C}D{}^{\alpha }x(t)\\ e(t)\end{array}\right]=0$, when $t\in \left[-\gamma ,0\right]$, we have

$${\displaystyle \frac{t^{-\alpha }}{\mathsf{\Gamma }(1-\alpha )}}{\overline{x}}^T(0)P\overline{x}(0)=0.$$

(13)

Applying Lemma 1 to Equation (12) and incorporating (13) and system (8), we obtain

$$\begin{gathered} \dot{V}(t)\le {\overline{x}}^T(t)P\left({}^{C}D{}^{\alpha }\overline{x}(t)\right)+{\left({}^{C}D{}^{\alpha }\overline{x}(t)\right)}^{T}P\overline{x}(t)+{\overline{x}}^T(t)M\overline{x}(t)-{\overline{x}}^T(t-\gamma )M\overline{x}(t-\gamma ) \\ ={\overline{x}}^T(t)P\left[\left(\overline{A}+\overline{B}{F}_1\overline{C}\right)\overline{x}(t)+\left({\overline{A}}_1+\overline{B}{F}_2\overline{C}\right)\overline{x}(t-\gamma )+\overline{H}\overline{w}(t)\right] \\ +{\left[\left(\overline{A}+\overline{B}{F}_1\overline{C}\right)\overline{x}(t)+\left({\overline{A}}_1+\overline{B}{F}_2\overline{C}\right)\overline{x}(t-\gamma )+\overline{H}\overline{w}(t)\right]}^{T}P\overline{x}(t) \\ +{\overline{x}}^T(t)M\overline{x}(t)-{\overline{x}}^T(t-\gamma )M\overline{x}(t-\gamma ) \\ ={\overline{x}}^T(t)\left[P\left(\overline{A}+\overline{B}{F}_1\overline{C}\right)+{\left(\overline{A}+\overline{B}{F}_1\overline{C}\right)}^{T}P+M\right]\overline{x}(t)-{\overline{x}}^T(t-\gamma )M\overline{x}(t-\gamma ) \\ +{\overline{x}}^T(t)P\left({\overline{A}}_1+\overline{B}{F}_2\overline{C}\right)\overline{x}(t-\gamma )+{\overline{x}}^T(t-\gamma ){\left({\overline{A}}_1+\overline{B}{F}_2\overline{C}\right)}^{T}P\overline{x}(t) \end{gathered}$$

(14)

Letting $\zeta (t)=\left[\begin{array}{c}\overline{x}(t)\\ \overline{x}(t-\gamma )\\ \overline{w}(t)\end{array}\right]$, then the right side of Equation (14) can be written as a quadratic form in terms of $\zeta (t)$, that is,

$$\begin{gathered} \dot{V}(t)\le {\zeta }^T(t)\left[\begin{array}{ccc}P\left(\overline{A}+\overline{B}{F}_1\overline{C}\right)+{\left(\overline{A}+\overline{B}{F}_1\overline{C}\right)}^{T}P+M& P\left({\overline{A}}_1+\overline{B}{F}_2\overline{C}\right)& P\overline{H}\\ *& -M& 0\\ *& *& 0\end{array}\right]\zeta (t) \\ ={\zeta }^T(t)\left[\begin{array}{ccc}P\left(\overline{A}+\overline{B}{F}_1\overline{C}\right)+{\left(\overline{A}+\overline{B}{F}_1\overline{C}\right)}^{T}P+M& P\left({\overline{A}}_1+\overline{B}{F}_2\overline{C}\right)& P\overline{H}\\ *& -M& 0\\ *& *& -Q\end{array}\right]\zeta (t) \\ +{\overline{w}}^T(t)Q\overline{w}(t). \end{gathered}$$

(15)

If (9) holds, then combing (15), we have

$$\dot{V}(t)<{\overline{w}}^T(t)Q\overline{w}(t).$$

(16)

According to the zero initial condition, we know

$$V(0)=I^{1-\alpha }\left({\overline{x}}^T(0)P\overline{x}(0)\right)+{\displaystyle {\int }_{-\gamma }^0{\overline{x}}^T(s)M\overline{x}(s)ds}=0.$$

(17)

Integrating both sides of (16) from 0 to $t$, and incorporating (17), we obtain

$$V(t)<{\displaystyle {\int }_0^t{\overline{w}}^T(s)Q\overline{w}(s)}ds.$$

(18)

Then, we estimate the upper bound of $V(t)$. Since $\overline{w}(t)=\left[\begin{array}{c}{}^{C}D{}^{\alpha }w(t)\\ {}^{C}D{}^{\alpha }r(t)\end{array}\right]$ and $Q=\left[\begin{array}{cc}{Q}_2& 0\\ 0& Q_1\end{array}\right]$, we have

$${\overline{w}}^T(t)Q\overline{w}(t)={\left({}^{C}D{}^{\alpha }r(t)\right)}^T{Q}_1\left({}^{C}D{}^{\alpha }r(t)\right)+{\left({}^{C}D{}^{\alpha }w(t)\right)}^T{Q}_2\left({}^{C}D{}^{\alpha }w(t)\right).$$

(19)

For the first term of (19), according to Assumption 1, $\dot{r}(t)$ has only the first kind of discontinuity at most, and thus is a bounded function. Additionally, the function ${(t-s)}^{-\alpha }$ with respect to $s$ retains its sign on the interval $[0,t]$, so by the mean value theorem for integrals [23], there exists $\xi \in [0,t]$ such that

$${\displaystyle {\int }_0^t{(t-s)}^{-\alpha }\dot{r}(s)}ds=\left[{\displaystyle {\int }_0^t{(t-s)}^{-\alpha }}ds\right]\dot{r}(\xi )={\displaystyle \frac{1}{1-\alpha }}{t}^{1-\alpha }\dot{r}(\xi ).$$

Therefore, we obtain

$$\begin{array}{cc}{}^{C}D{}^{\alpha }r(t)\hfill & ={\displaystyle \frac{1}{\mathsf{\Gamma }(1-\alpha )}}{\displaystyle {\int }_0^t{(t-s)}^{-\alpha }\dot{r}(s)}ds\hfill \\ \hfill & ={\displaystyle \frac{1}{(1-\alpha )\mathsf{\Gamma }(1-\alpha )}}{t}^{1-\alpha }\dot{r}(\xi )\hfill \\ \hfill & ={\displaystyle \frac{t^{1-\alpha }\dot{r}(\xi )}{\mathsf{\Gamma }(2-\alpha )}}\hfill \end{array}$$

Thus, we have

$$\begin{array}{cc}{\left({}^{C}D{}^{\alpha }r(t)\right)}^T{Q}_1\left({}^{C}D{}^{\alpha }r(t)\right)\hfill & ={\left({\displaystyle \frac{t^{1-\alpha }\dot{r}(\xi )}{\mathsf{\Gamma }(2-\alpha )}}\right)}^T{Q}_1\left({\displaystyle \frac{t^{1-\alpha }\dot{r}(\xi )}{\mathsf{\Gamma }(2-\alpha )}}\right)\hfill \\ & \le {\left({\displaystyle \frac{t^{1-\alpha }}{\mathsf{\Gamma }(2-\alpha )}}\right)}^2\underset{\xi \in [0,t]}{\max }{\dot{r}}^T(\xi )Q_1\dot{r}(\xi )\hfill \\ & \le {\left({\displaystyle \frac{T^{1-\alpha }}{\mathsf{\Gamma }(2-\alpha )}}\right)}^2\underset{t\in [0,T]}{\sup }{\dot{r}}^T(t)Q_1\dot{r}(t).\hfill \end{array}$$

Furthermore, according to Assumption 1, we have

$${\left({}^{C}D{}^{\alpha }r(t)\right)}^T{Q}_1\left({}^{C}D{}^{\alpha }r(t)\right)\le c_{11}.$$

(20)

Similarly, under the conditions of Assumption 2, the second term on the right side of (19) satisfies

$${\left({}^{C}D{}^{\alpha }w(t)\right)}^T{Q}_2\left({}^{C}D{}^{\alpha }w(t)\right)\le c_{22}.$$

(21)

By combining Equations (19)–(21), we obtain

$${\overline{w}}^T(t)Q\overline{w}(t)\le c_{11}+c_{22}\le c_1.$$

(22)

Substituting Equation (22) into Equation (18), we obtain an upper bound for $V(t)$ as

$$V(t)<{\displaystyle {\int }_0^t{c}_1}d\tau \le {\displaystyle {\int }_0^T{c}_1}d\tau =c_{1}T.$$

(23)

Given that

$$V(t)=I^{1-\alpha }\left({\overline{x}}^T(t)P\overline{x}(t)\right)+{\displaystyle {\int }_{t-\gamma }^t{\overline{x}}^T(s)M\overline{x}(s)ds}$$

and ${\displaystyle {\int }_{t-\gamma }^t{\overline{x}}^T(s)M\overline{x}(s)ds}\ge 0 (t\in \left[0,T\right])$, along with Equation (23), we can conclude that

$$I^{1-\alpha }\left({\overline{x}}^T(t)P\overline{x}(t)\right)<c_{1}T.$$

(24)

Since $c_{1}T>0$, applying Lemma 2 to (24), we obtain

$${\overline{x}}^T(t)P\overline{x}(t)<{\displaystyle \frac{c_1\mathsf{\Gamma }(2-\alpha )T}{T^{1-\alpha }}}.$$

(25)

On the other hand, $e(t)=\left[\begin{array}{cc}0& I\end{array}\right]\overline{x}(t)=\tilde{C}\overline{x}(t)$. According to (10) and (25), when $t\in \left[0,T\right]$, we obtain

$$e^T(t)\mathsf{\Phi }e(t)={\overline{x}}^T(t){\tilde{C}}^T\mathsf{\Phi }\tilde{C}\overline{x}(t)<{\displaystyle \frac{c_2{T}^{1-\alpha }}{c_1\mathsf{\Gamma }(2-\alpha )T}}{\overline{x}}^T(t)P\overline{x}(t)={\displaystyle \frac{c_2{T}^{1-\alpha }}{c_1\mathsf{\Gamma }(2-\alpha )T}}V(t)<c_2,$$

which completes the proof of Theorem 1. □

Note that the matrix inequality (9) in Theorem 1 contains the terms $P\left(\overline{A}+\overline{B}{F}_1\overline{C}\right)$ and $P\left({\overline{A}}_1+\overline{B}{F}_2\overline{C}\right)$, where $P$, $F_1$, and $F_2$ are all unknown. This means the matrix inequality (9) is not an LMI. In order to facilitate the solution of $P$, $F_1$, and $F_2$, we need to convert the matrix inequality in Theorem 1 into an LMI. Then, we obtain Theorem 2.

Theorem 2.

Under Assumptions 1 and 2, there are appropriate matrices $L>0$, $Z>0$, $Y_1$, and $Y_2$, such that

$$\left[\begin{array}{ccc}L{\overline{A}}^T+\overline{A}L+Y_1^T{\overline{B}}^T+\overline{B}{Y}_1+Z& {\overline{A}}_{1}L+\overline{B}{Y}_2& \overline{H}\\ *& -Z& 0\\ *& *& -Q\end{array}\right]<0$$

(26)

and

$$\left[\begin{array}{cc}-{\displaystyle \frac{c_2{T}^{1-\alpha }}{c_1\mathsf{\Gamma }(2-\alpha )T}}L& L{\tilde{C}}^T\\ *& -{\mathsf{\Phi }}^{-1}\end{array}\right]<0,$$

(27)

then system (8) can achieve finite-time bounded tracking of the reference signal $r(t)$ with respect to $(c_1,c_2,Q,\mathsf{\Phi },T)$, where the output feedback gain matrices are $F_1=Y_1{L}^{-1}{\overline{C}}^T{\left(\overline{C}{\overline{C}}^T\right)}^{-1}$and $F_2=Y_2{L}^{-1}{\overline{C}}^T{\left(\overline{C}{\overline{C}}^T\right)}^{-1}$.

Proof of Theorem 2.

It is sufficient to prove that the conditions of Theorem 1 are equivalent to the conditions of this theorem.

By multiplying the matrix $diag(P^{-1},I,I)$  on both sides of (9), we obtain

$$\left[\begin{array}{ccc}{P}^{-1}{\overline{A}}^T+\overline{A}{P}^{-1}+P^{-1}{\overline{C}}^T{F}_1^T{\overline{B}}^T+\overline{B}{F}_1{\overline{C}}_1{P}^{-1}+P^{-1}M{P}^{-1}& {\overline{A}}_1{P}^{-1}+\overline{B}{F}_2{\overline{C}}_1{P}^{-1}& \overline{H}\\ *& -P^{-1}M{P}^{-1}& 0\\ *& *& -Q\end{array}\right]<0.$$

(28)

When letting $L=P^{-1}$, $Z=P^{-1}M{P}^{-1}$, $Y_1=F_1\overline{C}{P}^{-1}$, and $Y_2=F_2\overline{C}{P}^{-1}$ in (28), then we have (26).

Furthermore, by multiplying $L$ on both sides of (10), we can obtain

$$L{\tilde{C}}^T\mathsf{\Phi }\tilde{C}L-{\displaystyle \frac{c_2{T}^{1-\alpha }}{c_1\mathsf{\Gamma }(2-\alpha )T}}LPL<0$$

(29)

Since $L=P^{-1}$ and $\mathsf{\Phi }>0$, (29) can be written as

$$-{\displaystyle \frac{c_2{T}^{1-\alpha }}{c_1\mathsf{\Gamma }(2-\alpha )T}}L-L{\overline{C}}^T{(-{\mathsf{\Phi }}^{-1})}^{-1}\overline{C}L<0$$

(30)

According to Lemma 2, (30) can be equivalently transformed into (27).

Under $Y_1=F_1\overline{C}{P}^{-1}$, $Y_2=F_2\overline{C}{P}^{-1}$, and $L=P^{-1}$, it is obtained that

$$F_1\overline{C}=Y_1{L}^{-1}$$

(31)

and

$$F_2\overline{C}=Y_2{L}^{-1}.$$

(32)

In addition, the matrix $\overline{C}=\left[\begin{array}{cc}C& 0\\ 0& I\end{array}\right]$ is of full rank, so the matrix ${\left(\overline{C}{\overline{C}}^T\right)}^{-1}$ exits. When multiplying ${\overline{C}}^T{\left(\overline{C}{\overline{C}}^T\right)}^{-1}$ on both sides of (31) and (32), it follows that $F_1=Y_1{L}^{-1}{\overline{C}}^T{\left(\overline{C}{\overline{C}}^T\right)}^{-1}$ and $F_2=Y_2{L}^{-1}{\overline{C}}^T{\left(\overline{C}{\overline{C}}^T\right)}^{-1}$. This completes the proof. □

Finally, by solving for $u(t)$ from the control input ${}^{C}D{}^{\alpha }u(t)=F_1\overline{y}(t)+F_2\overline{y}(t-\gamma )$ of system (8), we can obtain the control input for system (1). To clarify the controller structure, we appropriately block the output feedback gain matrices $F_1$ and $F_2$, yielding

$$F_1=\left[\begin{array}{cc}{F}_{1y}& F_{1e}\end{array}\right]\mathrm{and} F_2=\left[\begin{array}{cc}{F}_{2y}& F_{2e}\end{array}\right],$$

where $F_{1y}\in {ℝ}^{m\times n}$, $F_{1e}\in {ℝ}^{m\times p}$, $F_{2y}\in {ℝ}^{m\times n}$, and $F_{2e}\in {ℝ}^{m\times p}$. In addition, based on $\overline{y}(t)=\overline{C}\overline{x}(t)$, $\overline{C}=\left[\begin{array}{cc}C& 0\\ 0& I\end{array}\right]$, $\overline{x}(t)=\left[\begin{array}{c}{}^{C}D{}^{\alpha }x(t)\\ e(t)\end{array}\right]$, and $y(t)=Cx(t)$, we can derive that

$$\overline{y}(t)=\left[\begin{array}{cc}C& 0\\ 0& I\end{array}\right]\left[\begin{array}{c}{}^{C}D{}^{\alpha }x(t)\\ e(t)\end{array}\right]=\left[\begin{array}{c}{}^{C}D{}^{\alpha }\left(Cx(t)\right)\\ e(t)\end{array}\right]=\left[\begin{array}{c}{}^{C}D{}^{\alpha }y(t)\\ e(t)\end{array}\right],$$

and

$$\overline{y}(t-\gamma )=\left[\begin{array}{cc}C& 0\\ 0& I\end{array}\right]\left[\begin{array}{c}{}^{C}D{}^{\alpha }x(t-\gamma )\\ e(t)\end{array}\right]=\left[\begin{array}{c}{}^{C}D{}^{\alpha }\left(Cx(t-\gamma )\right)\\ e(t)\end{array}\right]=\left[\begin{array}{c}{}^{C}D{}^{\alpha }y(t-\gamma )\\ e(t)\end{array}\right].$$

As a result, it is acquired that

$$\begin{array}{c}{}^{C}D{}^{\alpha }u(t)=\left[\begin{array}{cc}{F}_{1y}& F_{1e}\end{array}\right]\left[\begin{array}{c}{}^{C}D{}^{\alpha }y(t)\\ e(t)\end{array}\right]+\left[\begin{array}{cc}{F}_{2y}& F_{2e}\end{array}\right]\left[\begin{array}{c}{}^{C}D{}^{\alpha }y(t-\gamma )\\ e(t-\gamma )\end{array}\right]\hfill \\ =F_{1y}\left({}^{C}D{}^{\alpha }y(t)\right)+F_{1e}e(t)+F_{2y}\left({}^{C}D{}^{\alpha }y(t-\gamma )\right)+F_{2e}e(t-\gamma )\hfill \end{array}.$$

(33)

By performing (33) with $\alpha$-order integration from 0 to $t$ on both sides, and applying Lemma 3, we obtain

$$\begin{array}{c}u(t)-u(0)=F_{1y}{I}^{\alpha }\left({}^{C}D{}^{\alpha }y(t)\right)+F_{1e}{I}^{\alpha }\left(e(t)\right)+F_{2y}{I}^{\alpha }\left({}^{C}D{}^{\alpha }y(t-\gamma )\right)+F_{2e}{I}^{\alpha }\left(e(t-\gamma )\right)\hfill \\ =F_{1y}\left(y(t)-y(0)\right)+F_{1e}{I}^{\alpha }\left(e(t)\right)+F_{2y}\left(y(t-\gamma )-y(-\gamma )\right)+F_{2e}{I}^{\alpha }\left(e(t-\gamma )\right)\hfill \end{array}$$

Considering the zero initial condition, we have

$$u(t)=F_{1y}y(t)+F_{1e}{I}^{\alpha }\left(e(t)\right)+F_{2y}y(t-\gamma )+F_{2e}{I}^{\alpha }\left(e(t-\gamma )\right).$$

(34)

Based on Theorem 2 and the above analysis, we can derive Theorem 3.

Theorem 3. 

If Assumption 1 and Assumption 2 hold, and there are appropriate matrices $L>0$, $Z>0$, $Y_1$, and $Y_2$such that Equations (26) and (27) hold, then the control input for system (1) can be taken as Equation (34). Under (34), the output $y(t)$of system (1) can achieve finite-time bounded tracking of the reference signal $r(t)$ with respect to $(c_1,c_2,Q,\mathsf{\Phi },T)$.

Remark 3.

This paper studies the fractional-order model of the control system for the case $0<\alpha <1$ and  $x(0)=0$. Since, in this case, the left Riemann–Liouville fractional derivative is equivalent to the left Caputo fractional derivative, the controller designed in this paper is applicable to fractional-order models under both definitions.

Remark 4.

The merit of our finite-time bounded tracking controller lies in its ability to guarantee the output within a given threshold of the desired tracking signal in a finite time. This characteristic is particularly advantageous for systems requiring precise and time-critical operations, as it ensures both stability and a rapid response. Additionally, finite-time control methods often exhibit a better transient performance compared to asymptotic control approaches, making them suitable for practical applications where time efficiency and bounded-state behavior are essential [13,14]. The effectiveness of the controller will be demonstrated through numerical simulations.

5. Numerical Simulation

Consider the fractional-order linear system (1), where

$$\begin{gathered} \alpha =0.8, A=\left[\begin{array}{ccc}0& -1.5& 1\\ 2& 0& -0.6\\ 1& -1.5& 1\end{array}\right], A_1=\left[\begin{array}{ccc}0& 0.6& 0.18\\ -0.12& 0.6& 0\\ 0.03& 0& 0\end{array}\right], \\ B=\left[\begin{array}{c}-1\\ 1.5\\ -3\end{array}\right], H=\left[\begin{array}{c}-2\\ 5\\ -4\end{array}\right]\mathrm{and} C=\left[\begin{array}{ccc}1& 0& 1\end{array}\right], \end{gathered}$$

with initial conditions $x(t)=\eta (t)=0, t\in \left[-0.1,0\right]$.

Let $\mathsf{\Phi }=0.1I$, $c_1=0.26$, $c_2=1$, and $T=15$. The controller is designed for two different reference signals and disturbance signals, followed by numerical simulations.

(a) The reference signal is taken as

$$r(t)=0.1\cos (2t).$$

(35)

The weight matrix is chosen as $Q_1=1$, and then we can obtain

$${\left({\displaystyle \frac{T^{1-\alpha }}{\mathsf{\Gamma }(2-\alpha )}}\right)}^2\underset{t\in [0,T]}{\sup }{\dot{r}}^T(t)Q_1\dot{r}(t)\approx 0.094<0.16=c_{11}.$$

The disturbance signal is taken as

$$w(t)=0.03.$$

Let the weight matrix $Q_2=0.6$. After calculation, we obtain

$${\left({\displaystyle \frac{T^{1-\alpha }}{\mathsf{\Gamma }(2-\alpha )}}\right)}^2\underset{t\in [0,T]}{\sup }{\dot{w}}^T(t)Q_2\dot{w}(t)=0<0.1=c_{22}.$$

At this point, $Q=\left[\begin{array}{cc}{Q}_1& 0\\ 0& Q_2\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 0.6\end{array}\right]$ and $c_{11}+c_{22}=0.26\le c_1$.

Based on Theorem 3, by applying the LMI toolbox in MATLAB to solve the LMI conditions (26) and (27), the matrix variables $L$, $Y_1$, and $Y_2$ are obtained. Furthermore, from $F_1=Y_1{L}^{-1}{\overline{C}}^T{\left(\overline{C}{\overline{C}}^T\right)}^{-1}$ and $F_2=Y_2{L}^{-1}{\overline{C}}^T{\left(\overline{C}{\overline{C}}^T\right)}^{-1}$, the gain matrix is computed as

$$\begin{gathered} F_1=\left[\begin{array}{cc}4.80426050240453& 36.0279538629814\end{array}\right], \\ {F}_2=\left[\begin{array}{cc}0.0536488668833622& -0.00259806136809582\end{array}\right]. \end{gathered}$$

At this point, according to Equation (34), the controller has the following form:

$$\begin{array}{c}u(t)=F_{1y}y(t)+F_{1e}\left(I^{0.8}e(t)\right)+F_{2y}y(t-0.1)+F_{2e}\left(I^{0.8}e(t-0.1)\right)\\ =F_{1y}y(t)+{\displaystyle \frac{F_{1e}}{\mathsf{\Gamma }(0.8)}}{\displaystyle {\int }_0^t{(t-\tau )}^{-0.2}e(\tau )d\tau }+F_{2y}y(t-0.1)\\ +{\displaystyle \frac{F_{2e}}{\mathsf{\Gamma }(0.8)}}{\displaystyle {\int }_0^t{(t-\tau )}^{-0.2}e(\tau -0.1)d\tau }\end{array}$$

where

$$\begin{gathered} F_{1y}=4.80426050240453, F_{1e}=36.0279538629814, \\ {F}_{2y}=0.0536488668833622 \mathrm{and} F_{2e}=-0.00259806136809582. \end{gathered}$$

By adopting the Adams–Bashforth–Moulton algorithm [20], and MATLAB (R2023a, MathWorks, Natick, MA, USA) software, we obtain the following figures.

From Figure 1, it can be seen that under the action of the designed controller, the closed-loop output signal of system (1) always stays within the given domain of the reference signal over the specified time interval. Additionally, from Figure 2 and Figure 3, it is evident that within the time interval $[0,15]$, the error signal also remains within the given bounds. This indicates that, under the action of the designed controller, the output of system (1) achieves finite-time bounded tracking of the target signal $r(t)$ with respect to $(0.26,1,Q,0.1I,15)$. Figure 4 shows the input of the system, and it can be observed from the figure that the input is bounded.

(b) The reference signal is taken as

$$r(t)=\left\{\begin{array}{cc}0,\hfill & t<4\\ 0.25(t-4),\hfill & 4\le t\le 8\\ 1,\hfill & 8<t\le 20\end{array}\right..$$

(36)

At this point, we have $\dot{r}(t)=\left\{\begin{array}{cc}0,\hfill & t<4\\ 0.25,\hfill & 4\le t\le 8\\ 1,\hfill & 8<t\le 20\end{array}\right.$. The weight matrix is chosen as $Q_1=0.5$, and then we can obtain

$${\left({\displaystyle \frac{T^{1-\alpha }}{\mathsf{\Gamma }(2-\alpha )}}\right)}^2\underset{t\in [0,T]}{\sup }{\dot{r}}^T(t)Q_1\dot{r}(t)\approx 0.148<0.15=c_{11}.$$

The disturbance signal is taken as

$$w(t)=0.1e^{-t/10}\sin (4t)+0.2.$$

Let the weight matrix $Q_2=0.5$. After calculation, we obtain

$${\left({\displaystyle \frac{T^{1-\alpha }}{\mathsf{\Gamma }(2-\alpha )}}\right)}^2\underset{t\in [0,T]}{\sup }{\dot{w}}^T(t)Q_2\dot{w}(t)\approx 0.105<0.11=c_{22}.$$

At this point, $Q=\left[\begin{array}{cc}{Q}_1& 0\\ 0& Q_2\end{array}\right]=\left[\begin{array}{cc}0.5& 0\\ 0& 0.5\end{array}\right]$ and $c_{11}+c_{22}=0.26\le c_1$.

Similarly, by applying the LMI toolbox in MATLAB to solve the LMI conditions (26), (27), the matrix variables $L$, $Y_1$, and $Y_2$ are obtained. Furthermore, from $F_1=Y_1{L}^{-1}{\overline{C}}^T{\left(\overline{C}{\overline{C}}^T\right)}^{-1}$ and $F_2=Y_2{L}^{-1}{\overline{C}}^T{\left(\overline{C}{\overline{C}}^T\right)}^{-1}$, the gain matrix is computed as

$$\begin{gathered} F_1=\left[\begin{array}{cc}4.78376603285385& 35.1922569105882\end{array}\right], \\ {F}_2=\left[\begin{array}{cc}0.0536605142245947& -0.00241725627559958\end{array}\right]. \end{gathered}$$

Furthermore, based on Equation (34), we can obtain the form of control function $u(t)$ as follows

$$\begin{array}{c}u(t)=F_{1y}y(t)+F_{1e}\left(I^{0.8}e(t)\right)+F_{2y}y(t-0.1)+F_{2e}\left(I^{0.8}e(t-0.1)\right)\\ =F_{1y}y(t)+{\displaystyle \frac{F_{1e}}{\mathsf{\Gamma }(0.8)}}{\displaystyle {\int }_0^t{(t-\tau )}^{-0.2}e(\tau )d\tau }+F_{2y}y(t-0.1)\\ +{\displaystyle \frac{F_{2e}}{\mathsf{\Gamma }(0.8)}}{\displaystyle {\int }_0^t{(t-\tau )}^{-0.2}e(\tau -0.1)d\tau }\end{array}$$

where

$$\begin{gathered} F_{1y}=4.78376603285385, F_{1e}=35.1922569105882, \\ {F}_{2y}=0.0536605142245947 \mathrm{and} F_{2e}=-0.00241725627559958. \end{gathered}$$

By adopting the Adams–Bashforth–Moulton algorithm, and MATLAB software, we obtain the following figures.

Figure 5 and Figure 6, respectively, depict the output response curve and the tracking error curve of system (1) when the reference the signal is given by Equation (36).

As shown in Figure 7, the closed-loop system has also achieved finite-time bounded tracking of the target signal $r(t)$ with respect to $(0.26,1,Q,0.1I,15)$. From Figure 8, it can be seen that the input of the system is bounded.

6. Conclusions

This paper studies the design problem of the finite-time bounded output feedback tracking controller for linear time-delay fractional-order systems. The method of the error system in preview control theory is adopted to derive the fractional-order error system, thus transforming the tracking control problem of the original system into the input–output finite-time stability control problem of the error system. Then, the output equation is modified, and a memory output feedback controller is designed by utilizing the modified output equation. Combining the research method of input–output finite-time stability with the linear matrix inequality technique, the design method of the output feedback controller gain matrix characterized by linear matrix inequalities is presented. Numerical simulations demonstrate the effectiveness of the proposed controller.