A fast fixed-time backstepping control method for systems with mismatched disturbances

Abstract

In this paper, a fast fixed-time backstepping control approach is developed for systems with mismatched disturbances. A novel fast fixed-time disturbance observer (FFTDO), which is convergent within fixed time regardless of initial conditions, is designed to efficiently estimate and compensate the disturbances. On the basis of FFTDO, a novel fast fixed-time backstepping control scheme is proposed, which can guarantee fast fixed-time convergence and high steady-state precision. In addition, a novel fixed-time filter is used to circumvent the problem of “explosion and complexity,” and to ensure overall fixed-time convergence. The fixed-time stability of the closed-loop system under the proposed controller is proved by the Lyapunov stability theory. The simulation results are carried out to confirm the feasibility and superiority of the proposed control strategy.

Introduction

Mismatched disturbances widely exist in many practical systems, which may lead to performance degradation of the control systems.^1–4 To achieve high control performance and strong robustness, several classical control design tools have been extensively employed to nonlinear systems with mismatched disturbances, such as LMI-based method, ⁵ Riccati method, ⁶ adaptive method, ⁷ and backstepping method. ⁸ Among these methods, backstepping control is a simple and effective method by using variable virtual control laws, which can not only guarantee global stability but also achieve good tracking performance in practical implementation.

As known, conventional backstepping control has two main drawbacks. One is weak robustness. For this problem, combining backstepping control with other robust control approaches is a usual solution. In Ahmadi et al.’s ⁹ paper, a novel backstepping controller using adaptive control technique was designed for wheeled mobile robots. In Liu et al.’s ¹⁰ paper, a new robust backstepping control strategy combining sliding mode control and neural networks was presented for spacecraft attitude regulation. In Zhang et al.’s ¹¹ paper, a disturbance observer-based backstepping control was developed for the optoelectronic stabilized platform affected with disturbances, which could improve the system tracking performance. In Liu et al.’s ¹² paper, sliding mode control method and disturbance observer technique were introduced to backstepping controller to improve the system performance against uncertainties. The other is the “explosion of complexity” problem, which is caused by repeated differentiations of virtual control. The common method to address this problem is to adopt filters such as first-order filters ¹³ and command filters ¹⁴ to obtain indirect differentiation. The dynamic surface control was proposed by Swaroop et al., ¹⁵ which can circumvent the problem of “explosion of complexity” by using a first-order filter at every step of conventional backstepping control. The backstepping control with first-order filters at every step is also known as dynamic surface control, and it has been successfully employed to various applications, such as pure feedback nonlinear systems, ¹⁶ two-axis frame systems, ¹⁷ robots, ¹⁸ and aircrafts. ¹⁹ The command-filtered backstepping is another effective control method. Command filters are used to produce virtual signals and approximate the derivatives of virtual signals,^20–23 and the errors resulting from the command filters can be reduced by combining compensation signals. It is noted that the conventional backstepping control and the modified backstepping control can only guarantee the closed-loop system asymptotically stable.

Compared with asymptotic control, finite-time control has more superior performance in terms of fast response and high control precision. Nevertheless, finite-time control cannot guarantee system stabilization within bounded time regardless of initial conditions. ²⁴ For various systems, the unknown initial states may result in large convergence time, which hinders the practical applications. Thus, it is necessary to design a robust controller that can guarantee system stabilization within bounded time. The fixed-time stability was firstly put forward in Polyakov, ²⁵ where the convergence time was bounded and irrelevant to the initial states. Zuo et al. ²⁶ developed a robust fixed-time control for generic linear systems in the existence of matched and mismatched disturbances. Golestani et al. ²⁷ presented a novel fixed-time control for a flexible spacecraft. The results of literature^26,27 cannot solve the fixed-time tracking problem for the high-order nonlinear systems with mismatched disturbances. In Huang et al.’s ²⁸ paper, the fixed-time backstepping control strategy was employed to the power system. Qiang et al. ²⁹ proposed a fixed-time backstepping control scheme for a class of nonlinear system against mismatched disturbances. In Yuan et al.’s ³⁰ paper, a adaptive fixed-time control was presented for nonlinear systems with full-state constraints. Nevertheless, speed up the convergence speed of fixed-time control is of great importance.

Combining backstepping method with fixed-time control can improve the convergence property of the backstepping method. Moreover, we can utilize command filters to circumvent the problem of “explosion of complexity” and utilize disturbance observers to improve the robustness in backstepping control. However, several existing filters and disturbance observers may not guarantee overall fixed-time stability. Fixed-time control for n-order systems with mismatched and matched disturbances is still a challenging task. Furthermore, how to enhance the convergence rate and the control precision of systems is still valuable for further research and is of great importance.

Motivated by the above analysis, this paper focuses on the development of a fast fixed-time backstepping control for n-order systems with mismatched disturbances such that superior performance including fast fixed-time convergence and high steady-state precision are provided. The main contributions of this paper are outlined as follows: (1) A novel fast fixed-time disturbance observer (FFTDO) is designed to estimate matched and mismatched disturbances, whose convergence time is irrelevant to initial states. (2) During the process of control design, the “explosion of complexity” problem is handled by employing a novel fixed-time filter. (3) With the help of the FFTDO and fixed-time filter, the fast fixed-time backstepping control (FFTBC) is developed, which can guarantee the fixed-time stability of the overall system with fast response and high control precision in the presence of mismatched and matched disturbances. FFTBC combines the advantages of both fixed-time control and dynamic surface control.

Problem statement

Nonlinear dynamic systems

Consider a system with mismatched disturbances, defined by

$$\begin{array}{c}\frac{d{x}_i}{\mathrm{dt}}=a_i(x)+b_i(x)x_{i+1}+d_i,iis\mathrm{fro}m1ton-1,\hfill \\ \frac{d{x}_n}{\mathrm{dt}}=a_n(x)+b_n(x)u+d_n,\hfill \\ y=x_1,\hfill \end{array}$$

(1)

where $x_i$ $(i\mathrm{is}\mathrm{from}1ton)$ denotes the system state. $u$ denotes the control input. $d_i$ represents the unknown disturbance. $a_i$ and $b_i\ne 0$ are smooth nonlinear functions in terms of $x$ .

This paper attempts to develop a novel fast fixed-time backstepping control method for system (1) such that the system output can track the desired reference signal within fixed time based on the following assumptions.

• Assumption 1. The disturbance $d_i$ $(i\mathrm{is}\mathrm{from}1ton)$ in system (1) and its derivative $\frac{d\left(d_i\right)}{\mathrm{dt}}$ are bounded by $\left|d_i\right|\le {\mathrm{I}}_i$ , $\left|\frac{d(d_i)}{\mathrm{dt}}\right|\le L_i$ , where ${\mathrm{I}}_i$ and $L_i$ are positive constants.

• Assumption 2. The desired reference signal $x_d$ and its time derivative $\frac{d{x}_d}{\mathrm{dt}}$ are known and bounded.

• Assumption 3. The function $b_i(x)$ satisfies the condition of $0<|b_i(x)|\le B_i$ , $i\mathrm{is}\mathrm{from}1ton$ . Here, $B_i$ is positive constant.

Useful lemmas

The following lemmas will be used in the analysis.

• Lemma 1. For any positive real numbers $x$ , $y$ and $\theta \ge 1$ , the following inequality holds ³¹ :

$$(x-y{)}^{\theta }\le x^{\theta }-y^{\theta }$$

(2)

• Lemma 2. For any positive real numbers $a$ , $b$ , $c$ and positive constants $m$ , $n$ satisfying that $1/m+1/n=1$ , the following relation holds ³² :

$$\mathrm{ab}\le c^m\frac{a^m}{m}+c^{-n}\frac{b^n}{n}$$

(3)

• Lemma 3. For any nonnegative real numbers ${\zeta }_1,{\zeta }_2,\cdots ,{\zeta }_n$ , $\psi >1$ , and $0<\phi <1$ , the following two relations hold ³³ :

$$\sum _{i=1}^n{\zeta }_i^{\psi }\ge n^{1-\psi }{\left(\sum _{i=1}^n{\zeta }_i\right)}^{\psi }$$

$$\sum _{i=1}^n{\zeta }_i^{\phi }\ge {\left(\sum _{i=1}^n{\zeta }_i\right)}^{\phi }$$

(4)

• Lemma 4. Consider a positive Lyapunov function $V$ that satisfies the following equality

$$\frac{\mathrm{dV}}{\mathrm{dt}}+{\alpha }_1{V}^{{\theta }_1}+{\alpha }_2{V}^{{\theta }_2}+{\alpha }_{3}V=0$$

(5)

where ${\alpha }_1,{\alpha }_2,{\alpha }_3>0$ , ${\theta }_1>1$ , $0<{\theta }_2<1$ . The function $V$ can be stabilized to the origin with the convergence time $T$ bounded by

$$T<\frac{1}{{\alpha }_3({\theta }_1-1)}\ln (1+\frac{{\alpha }_3}{{\alpha }_1})+\frac{1}{{\alpha }_3(1-{\theta }_2)}\ln (1+\frac{{\alpha }_3}{{\alpha }_2})$$

(6)

Proof. Denote $\eta =V^{1-{\theta }_2}$ and $\beta =({\theta }_1-{\theta }_2)/(1-{\theta }_2)$ , applying (5), the time derivative of $\eta$ is

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d\eta }{\mathrm{dt}}=(1-{\theta }_2)V^{-{\theta }_2}\frac{\mathrm{dV}}{\mathrm{dt}}}\\ =-(1-{\theta }_2)({\alpha }_1{\eta }^{\beta }+{\alpha }_2+{\alpha }_3\eta )\hfill \end{array}$$

(7)

Solving (7), the upper bound of convergence time can be estimated as

$$\begin{array}{c}\multicolumn{1}{c}{\underset{{\eta }_0\to \infty }{lim}T({\eta }_0)=\underset{{\eta }_0\to \infty }{lim}\frac{1}{1-{\theta }_2}(\underset{1}{\overset{{\eta }_0}{\int }}\frac{1}{{\alpha }_1{\eta }^{\beta }+{\alpha }_2+{\alpha }_3\eta }d\eta +\underset{0}{\overset{1}{\int }}\frac{1}{{\alpha }_1{\eta }^{\beta }+{\alpha }_2+{\alpha }_3\eta }d\eta )}\\ <\underset{{\eta }_0\to \infty }{lim}\frac{1}{1-{\theta }_2}(\underset{1}{\overset{{\eta }_0}{\int }}\frac{1}{{\alpha }_1{\eta }^{\beta }+{\alpha }_3\eta }d\eta +\underset{0}{\overset{1}{\int }}\frac{1}{{\alpha }_2+{\alpha }_3\eta }d\eta )\hfill \\ <\underset{{\eta }_0\to \infty }{lim}\frac{1}{1-{\theta }_2}\underset{1}{\overset{{\eta }_0}{\int }}\frac{1}{{\alpha }_1{\eta }^{\beta }+{\alpha }_3\eta }d\eta +\frac{1}{{\alpha }_3(1-{\theta }_2)}\ln (1+\frac{{\alpha }_3}{{\alpha }_2})\hfill \end{array}$$

(8)

Define $\varpi =1+\ln \eta$ and $\chi =e^{(\beta -1)(\varpi -1)}$ , and denote ${\varpi }_0$ and ${\chi }_0$ as the initial value of $\varpi$ and $\chi$ , we obtain

$$\begin{array}{c}\underset{{\eta }_0\to \infty }{lim}\frac{1}{1-{\theta }_2}\underset{1}{\overset{{\eta }_0}{\int }}\frac{1}{{\alpha }_1{\eta }^{\beta }+{\alpha }_3\eta }d\eta \hfill \\ =\frac{1}{1-{\theta }_2}\underset{{\varpi }_0\to \infty }{lim}\underset{1}{\overset{{\varpi }_0}{\int }}\frac{1}{{\alpha }_1{e}^{(\beta -1)(\varpi -1)}+{\alpha }_3}d\varpi \hfill \\ =\frac{1}{(1-{\theta }_2)(\beta -1)}\underset{{\chi }_0\to \infty }{lim}\underset{1}{\overset{{\chi }_0}{\int }}\frac{1}{\chi ({\alpha }_1\chi +{\alpha }_3)}d\chi \hfill \\ =\frac{1}{{\alpha }_3({\theta }_1-1)}\ln (1+\frac{{\alpha }_3}{{\alpha }_1})\hfill \end{array}$$

(9)

Then,

$$T<\frac{1}{{\alpha }_3({\theta }_1-1)}\ln (1+\frac{{\alpha }_3}{{\alpha }_1})+\frac{1}{{\alpha }_3(1-{\theta }_2)}\ln (1+\frac{{\alpha }_3}{{\alpha }_2})$$

(10)

The proof of Lemma 4 is completed.

• Lemma 5. Consider the dynamic system

$$\frac{\mathrm{dx}}{\mathrm{dt}}=f(x),x(0)=x_0$$

(11)

where $x\in R^n$ , and $f(x):D\to R^n$ is continuous on an open neighborhood $D\subseteq R^n$ of the origin, and $f(0)=0$ . Suppose there exists a positive definite and radially unbounded function $V(x):R^n\to R$ , and an arbitrarily small positive real number $\rho$ , ${\theta }_1>1$ , and $0<{\theta }_2<1$ such that $\frac{\mathrm{dV}}{\mathrm{dt}}\le -{\alpha }_1{V}^{{\theta }_1}-{\alpha }_2{V}^{{\theta }_2}-{\alpha }_{3}V+\rho$ . Then, the system will converge into an arbitrarily small region near the origin, that is, $V\le 2\upsilon$ with ${\alpha }_1\mathrm{sig}(\upsilon {)}^{{\theta }_1}+{\alpha }_2\mathrm{sig}(\upsilon {)}^{{\theta }_2}+{\alpha }_3\upsilon =\rho$ in fixed time given by

$$T<\frac{1}{{\alpha }_3({\theta }_1-1)}\ln (1+\frac{{\alpha }_3}{{\alpha }_1})+\frac{1}{{\alpha }_3(1-{\theta }_2)}\ln (1+\frac{{\alpha }_3}{(2^{{\theta }_2}-1){\alpha }_2})$$

(12)

Proof. The dynamics of function $V$ can be expressed as

$$\frac{\mathrm{dV}}{\mathrm{dt}}\le -{\alpha }_1{V}^{{\theta }_1}-{\alpha }_2{V}^{{\theta }_2}-{\alpha }_{3}V+{\alpha }_1{\upsilon }^{{\theta }_1}+{\alpha }_2{\upsilon }^{{\theta }_2}+{\alpha }_3\upsilon$$

(13)

For ${\theta }_1>1$ and in accordance with Lemma 1, one obtains $V^{{\theta }_1}-{\upsilon }^{{\theta }_1}\ge (V-\upsilon {)}^{{\theta }_1}$ . For $0<{\theta }_2<1$ , if $V\ge 2\upsilon$ , one obtains $V^{{\theta }_2}-{\upsilon }^{{\theta }_2}\ge (2^{{\theta }_2}-1)(V-\upsilon {)}^{{\theta }_2}$ . ³⁴ Then, we have

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{dV}}{\mathrm{dt}}\le -{\alpha }_1{V}^{{\theta }_1}-{\alpha }_2{V}^{{\theta }_2}-{\alpha }_{3}V+{\alpha }_1{\upsilon }^{{\theta }_1}+{\alpha }_2{\upsilon }^{{\theta }_2}+{\alpha }_3\upsilon }\\ \le -{\alpha }_1(V-\upsilon {)}^{{\theta }_1}-(2^{{\theta }_2}-1){\alpha }_2(V-\upsilon {)}^{{\theta }_2}-{\alpha }_3(V-\upsilon )\hfill \end{array}$$

(14)

Denote $W=V-\upsilon$ , there is

$$\frac{\mathrm{dW}}{\mathrm{dt}}\le -{\alpha }_1{W}^{{\theta }_1}-(2^{{\theta }_2}-1){\alpha }_2{W}^{{\theta }_2}-{\alpha }_{3}W$$

(15)

On the basis of Lemma 4, the convergence time can be estimated as

$$T<\frac{1}{{\alpha }_3({\theta }_1-1)}\ln (1+\frac{{\alpha }_3}{{\alpha }_1})+\frac{1}{{\alpha }_3(1-{\theta }_2)}\ln (1+\frac{{\alpha }_3}{(2^{{\theta }_2}-1){\alpha }_2})$$

(16)

It is concluded that the function $V$ can converge to arbitrarily small region near the origin, that is, $V\le 2\upsilon$ with ${\alpha }_1\mathrm{sig}(\upsilon {)}^{{\theta }_1}+{\alpha }_2\mathrm{sig}(\upsilon {)}^{{\theta }_2}+{\alpha }_3\upsilon =\rho$ , within fixed time regardless of initial states. It is easy to demonstrate that when system trajectory enters into the ultimate bound, the relation $\frac{\mathrm{dV}}{\mathrm{dt}}<0$ holds. Thus, any system trajectory entering into the ultimate bound will persistently remain in it.

The proof of Lemma 5 is completed.

Fast fixed-time backstepping control design based on fixed-time disturbance observer

Fixed-time disturbance observer design

Motivated by the work of Yan et al., ³⁵ a fast fixed-time disturbance observer (FFTDO) is constructed as

$$\{\begin{array}{c}\frac{d{\hat{x}}_i}{\mathrm{dt}}=a_i(x)+b_i(x)x_{i+1}-{\kappa }_i\mathrm{sig}(x_i\prime {)}^{{\sigma }_1}-{\kappa }_i\prime \mathrm{sig}(x_i\prime {)}^{{\sigma }_2}-{\kappa }_i″x_i\prime +z_i\hfill \\ \frac{d{z}_i}{\mathrm{dt}}=-l_i\mathrm{sign}(x_i\prime )\hfill \\ \frac{d{\hat{x}}_n}{\mathrm{dt}}=a_n(x)+b_n(x)u-{\kappa }_n\mathrm{sig}({x\prime }_n{)}^{{\sigma }_1}-{\kappa \prime }_n\mathrm{sig}({x\prime }_n{)}^{{\sigma }_2}-{\kappa ″}_n{x\prime }_n+z_n\hfill \\ \frac{d{z}_n}{\mathrm{dt}}=-l_n\mathrm{sign}({x\prime }_n)\hfill \end{array}$$

(17)

where ${\hat{x}}_i$ is the estimation of $x_i$ , $x{\prime }_i={\hat{x}}_i-x_i$ . ${\kappa }_i,\kappa {\prime }_i,{\kappa ″}_i,l_i>0$ when $i$ is from 1 to $n$ . ${\sigma }_1>1$ and $0<{\sigma }_2<1$ .

The estimated disturbance can be expressed as

$$\{\begin{array}{c}{\hat{d}}_i=-{\kappa }_i\mathrm{sig}(x_i\prime {)}^{{\sigma }_1}-{\kappa }_i\prime \mathrm{sig}(x_i\prime {)}^{{\sigma }_2}-{\kappa }_i″x_i\prime +z_i\hfill \\ \frac{d{z}_i}{\mathrm{dt}}=-l_i\mathrm{sign}(x_i\prime )\hfill \end{array}$$

(18)

where ${\hat{d}}_i$ is the estimation of $d_i$ , $i$ is from 1 to $n$ .

• Theorem 1. For system (1), the disturbance observer is designed as (17). Then, the disturbance observer error of the FFTDO can converge to the zero uniformly in fixed time, given by

$$\begin{array}{c}\multicolumn{1}{c}{T_i\le (\frac{1}{{\kappa }_i\prime ({\sigma }_1-1)}\ln (1+\frac{{\kappa }_i\prime }{{\kappa }_i})+\frac{1}{{\kappa }_i″(1-{\sigma }_2)}\ln (1+\frac{{\kappa }_i″}{{\kappa }_i\prime })).}\\ (1+\frac{1}{m_i(1/M_i-h({\kappa }_i\prime )/{\kappa }_i\prime )})\hfill \end{array}$$

(19)

where $M_i=l_i+L_i$ , $m_i=l_i-L_i$ , $h(\kappa {\prime }_i)=1/\kappa {\prime }_i+(2e/m_i\kappa {\prime }_i{)}^{1/3}$ , and $e$ represents the base of the natural logarithms.

Proof. Denote $d{\prime }_i={\hat{d}}_i-d_i$ . Taking the time derivative of $x_i$ and combining (1) and (17), we obtain

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d{x}_i\prime }{\mathrm{dt}}=\frac{d{\hat{x}}_i}{\mathrm{dt}}-\frac{d{x}_i}{\mathrm{dt}}}\\ =a_i+b_i{x}_{i+1}+{\hat{d}}_i-(a_i+b_i{x}_{i+1}+d_i)\hfill \\ =d_i\prime \hfill \end{array}$$

(20)

The estimate error dynamics can be expressed as

$$\{\begin{array}{c}\frac{d{x}_i\prime }{\mathrm{dt}}=-{\kappa }_i\mathrm{sig}(x_i\prime {)}^{{\sigma }_1}-{\kappa }_i\prime \mathrm{sig}(x_i\prime {)}^{{\sigma }_2}-{\kappa }_i″x_i\prime +{\mathrm{\Gamma }}_i\hfill \\ \frac{d{\mathrm{\Gamma }}_i}{\mathrm{dt}}=-l_i\mathrm{sign}(x_i\prime )-\frac{d(d_i)}{\mathrm{dt}}\hfill \end{array}$$

(21)

where ${\mathrm{\Gamma }}_i$ is an auxiliary variable.

For (21), $\left|\frac{d\left(d_i\right)}{\mathrm{dt}}\right|\le L_i$ . when $l_i>L_i$ , it can be obtained

$$\frac{d{x}_i\prime }{\mathrm{dt}}\le -{\kappa }_i\mathrm{sig}(x{\prime }_i{)}^{{\sigma }_1}-\kappa {\prime }_i\mathrm{sig}(x{\prime }_i{)}^{{\sigma }_2}-{\kappa ″}_{i}x{\prime }_i$$

(22)

In accordance with Lemma 4 and (22), $x{\prime }_i$ can converge to equilibrium in a fixed time

$$T_{1i}\le \frac{1}{{\kappa }_i\prime ({\sigma }_1-1)}\ln (1+\frac{{\kappa }_i\prime }{{\kappa }_i})+\frac{1}{{\kappa }_i″(1-{\sigma }_2)}\ln (1+\frac{{\kappa }_i″}{{\kappa }_i\prime })$$

(23)

Considering the second equation of (18), by the end of $T_{1i}$ , ${\mathrm{\Gamma }}_i$ is upper bounded by

$$|{\mathrm{\Gamma }}_i(T_{1i})|\le M_i{T}_{1i}$$

(24)

The trajectory of the system (18) starts at $(0,{\mathrm{\Gamma }}_i(T_{1i}))$ for $t>T_{i1}$ . According to Wang et al.’s ²⁸ paper, the latter trajectory converges to the equilibrium for a time

$$T_{2i}\le \frac{\sum _j^{n}d{x}_i\prime (t_j)/\mathrm{dt}}{m_i}$$

(25)

in which $t_j(jis\mathrm{fro}m1ton)$ represents the time interval between the consecutive intersection of the trajectory with the axis $x_i=0$ , such that $x_i(t_j)=0$ and $\frac{d{x}_i\prime (t_j)}{\mathrm{dt}}={\mathrm{\Gamma }}_i(t_j)$ . According to Wang et al.’s ²⁸ paper, $|\frac{d{x}_i\prime (t_{j+1})}{\mathrm{dt}}/\frac{d{x}_i\prime (t_j)}{\mathrm{dt}}|\le M_{i}h(\kappa {\prime }_i)/\kappa {\prime }_i$ and the mandatory condition for convergence of the time series (25) is $M_{i}h(\kappa {\prime }_i)/\kappa {\prime }_i<1$ . Then, $T_{2i}$ can be given by

$$T_{2i}\le \frac{d{x}_i\prime (t_1)/\mathrm{dt}}{m_i(1-M_{i}h({\kappa }_i\prime )/{\kappa }_i\prime )}=\frac{{\mathrm{\Gamma }}_i(T_{1i})}{m_i(1-M_{i}h({\kappa }_i\prime )/{\kappa }_i\prime )}$$

(26)

It should be pointed out that the condition $l_i>L_i$ in

Theorem 1 is the necessary condition for the convergence of the FFTDO. When $(x{\prime }_i,z_i)$ converges to the origin, the observer error of the FFTDO converges to zero:

$$d{\prime }_i=-{\kappa }_i\mathrm{sig}(x{\prime }_i{)}^{{\sigma }_1}-\kappa {\prime }_i\mathrm{sig}(x{\prime }_i{)}^{{\sigma }_2}-{\kappa ″}_{i}x{\prime }_i+{\mathrm{\Gamma }}_i=0$$

(27)

Thus, the convergence time is estimated as

$$\begin{array}{c}\multicolumn{1}{c}{T_i=T_{i1}+T_{i2}\le (\frac{1}{{\kappa }_i\prime ({\sigma }_1-1)}\ln (1+\frac{{\kappa }_i\prime }{{\kappa }_i})+\frac{1}{{\kappa }_i″(1-{\sigma }_2)}\ln (1+\frac{{\kappa }_i″}{{\kappa }_i\prime })).}\\ (1+\frac{1}{m_i(1/M_i-h({\kappa }_i\prime ))/{\kappa }_i\prime })\hfill \end{array}$$

(28)

The proof of Theorem 1 is completed.

Fixed-time filter design

A novel fixed-time filter is introduced as

$$\tau \frac{d{x}_c}{\mathrm{dt}}+\mathrm{sig}(x_c-x_d{)}^{p/q}+\mathrm{sig}(x_c-x_d{)}^{q/p}+x_c-x_d=0$$

(29)

where $\tau >0$ , $p>q>1$ . $x_d$ and $x_c$ represent the input and output values of the filter, respectively.

• Remark 1. Denote the filter tracking error $\epsilon =x_c-x_d$ . The time derivative of the filter tracking error $\epsilon$ can be derived that $\frac{d\epsilon }{\mathrm{dt}}=-\frac{1}{\tau }(\mathrm{sig}{(\epsilon )}^{p/q}+\mathrm{sig}{(\epsilon )}^{q/p}+\epsilon )-\frac{d{x}_d}{\mathrm{dt}}$ . Suppose $\left|\frac{d{x}_d}{\mathrm{dt}}\right|\le {\iota }_1$ is satisfied and there exists a sufficiently small time constant $\tau (b_1,{\iota }_1)$ for any given positive number $b_1$ . Then, according to Lemma 5, it is easy to prove that $\epsilon$ will converge to the region $\left|\epsilon \right|\le \tau (b_1,{\iota }_1){\iota }_1$ in fixed time. Next, the approximation performance of first-order differential of the filter tracking error is considered. Define the first-order differential of the filter tracking error as $\xi =\frac{d{x}_c}{\mathrm{dt}}-\frac{d{x}_d}{\mathrm{dt}}$ . Then $\frac{d\xi }{\mathrm{dt}}=\frac{d^2\epsilon }{d{t}^2}=-\frac{1}{\tau }(\frac{p}{q}{\left|\epsilon \right|}^{p/q-1}+\frac{q}{p}{\left|\epsilon \right|}^{q/p-1}+1)\xi -\frac{d^2{x}_d}{d{t}^2}$ . If $\left|\frac{d^2{x}_d}{d{t}^2}\right|\le {\iota }_2$ is satisfied and there is a sufficiently small time constant $\tau (b_2,{\iota }_2)$ for any given positive number $b_2$ , then $\xi$ will exponentially converge to the region $\left|\xi \right|\le \tau (b_2,{\iota }_2){\iota }_2$ .

• Remark 2. In the existing literature, the widely-used first-order filter is $\tau \frac{d{x}_c}{\mathrm{dt}}+x_c=x_d$ . Here, $\tau$ is a small time constant. To guarantee tracking accuracy, the parameter $\tau$ needs to be enough small. Nevertheless, in the proposed fixed-time filter, the parameters $\tau$ , $p$ and $q$ all play the important role in determining the convergence speed and tracking precision, thus the demand for parameter $\tau$ is relaxed.

Fast fixed-time backstepping controller design

In this subsection, a fast fixed-time backstepping controller based on FFTDO is developed. The structure diagram of the proposed control strategy is shown in Figure 1. The design procedure contains $n$ steps. At each step, a virtual control law is constructed to obtain fixed-time stability for each state variable. Finally, the control law is designed at step $n$ .

• Step 1: Consider the first equation of system (1). Define the tracking error of $x_1$ as $e_1=x_1-x_d$ , taking the derivative of $e_1$ yields

$$\frac{d{e}_1}{\mathrm{dt}}=\frac{d{x}_1}{\mathrm{dt}}-\frac{d{x}_d}{\mathrm{dt}}=a_1(x)+b_1(x)x_2+d_1-\frac{d{x}_d}{\mathrm{dt}}$$

(30)

Figure 1.

Structure diagram of the proposed control strategy.

To obtain fixed-time convergence for error variable $e_1$ , the virtual control law ${\overline{x}}_{2c}$ can be derived as

$${\overline{x}}_{2c}=-b_1(x{)}^{-1}(k_1\mathrm{sig}(e_1{)}^{\frac{p}{q}}+k{\prime }_1\mathrm{sig}(e_1{)}^{\frac{q}{p}}+{k″}_1{e}_1+a_1(x)+{\hat{d}}_1-\frac{d{x}_d}{\mathrm{dt}})$$

(31)

where $k_1,k{\prime }_1,{k″}_1>0$ , $p>q>1$ , and ${\hat{d}}_1$ is the estimation of $d_1$ by the FFTDO (17).

Similar to the dynamic surface control, a new state variable $x_{2c}$ is introduced to avoid the direct differentiation of the virtual control law ${\overline{x}}_{2c}$ . Moreover, to ensure overall fast fixed-time convergence, the proposed fixed-time filter is employed

$$\tau \frac{d{x}_{2c}}{\mathrm{dt}}+\mathrm{sig}(x_{2c}-{\overline{x}}_{2c}{)}^{p/q}+\mathrm{sig}(x_{2c}-{\overline{x}}_{2c}{)}^{q/p}+x_{2c}-{\overline{x}}_{2c}=0$$

(32)

Define the filter error ${\delta }_2=x_{2c}-{\overline{x}}_{2c}$ . Differentiating ${\delta }_2$ yields

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d{\delta }_2}{\mathrm{dt}}=\frac{d{x}_{2c}}{\mathrm{dt}}-\frac{d{\overline{x}}_{2c}}{\mathrm{dt}}}\\ =-\frac{1}{\tau }(\mathrm{sig}({\delta }_2{)}^{p/q}+\mathrm{sig}({\delta }_2{)}^{q/p}+{\delta }_2)-\frac{d{\overline{x}}_{2c}}{\mathrm{dt}}\hfill \end{array}$$

(33)

Substituting (31)–(32) into (30), we obtain

$$\frac{d{e}_1}{\mathrm{dt}}=-k_1\mathrm{sig}(e_1{)}^{\frac{p}{q}}-k{\prime }_1\mathrm{sig}(e_1{)}^{\frac{q}{p}}-{k″}_1{e}_1+b_1(x)(e_2+{\delta }_2)+d{\prime }_1$$

(34)

Step $i$ : Consider the $\mathrm{ith}$ $(i\mathrm{is}\mathrm{from}2\mathrm{to}n-1)$ equation of system (1). Define the tracking error of $x_i$ as $e_i=x_i-x_{\mathrm{ic}}$ , the time derivative of $e_i$ is

$$\frac{d{e}_i}{\mathrm{dt}}=\frac{d{x}_i}{\mathrm{dt}}-\frac{d{x}_{\mathrm{ic}}}{\mathrm{dt}}=a_i(x)+b_i(x)x_{i+1}+d_i-\frac{d{x}_{\mathrm{ic}}}{\mathrm{dt}}$$

(35)

To obtain fixed-time convergence for error variable $e_i$ , the virtual control law ${\overline{x}}_{i+1c}$ can be constructed as

$$\begin{array}{c}\multicolumn{1}{c}{{\overline{x}}_{i+1c}=-b_i(x{)}^{-1}(k_i\mathrm{sig}(e_i{)}^{\frac{p}{q}}+k{\prime }_i\mathrm{sig}(e_i{)}^{\frac{q}{p}}+{k″}_i{e}_i+a_i(x)+{\hat{d}}_i-\frac{d{x}_{\mathrm{ic}}}{\mathrm{dt}})}\end{array}$$

(36)

The proposed fixed-time filter is designed as

$$\begin{array}{c}\multicolumn{1}{c}{\tau \frac{d{x}_{i+1c}}{\mathrm{dt}}+\mathrm{sig}(x_{i+1c}-{\overline{x}}_{i+1c}{)}^{p/q}+\mathrm{sig}(x_{i+1c}-{\overline{x}}_{i+1c}{)}^{q/p}}\\ +x_{i+1c}-{\overline{x}}_{i+1c}=0\hfill \end{array}$$

(37)

The filter error is expressed as ${\delta }_{i+1}=x_{i+1c}-{\overline{x}}_{i+1c}$ . Taking the derivative of ${\delta }_{i+1}$ results in

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d{\delta }_{i+1}}{\mathrm{dt}}=\frac{d{x}_{i+1c}}{\mathrm{dt}}-\frac{d{\overline{x}}_{i+1c}}{\mathrm{dt}}}\\ =-\frac{1}{\tau }(\mathrm{sig}({\delta }_{i+1}{)}^{p/q}+\mathrm{sig}({\delta }_{i+1}{)}^{q/p}+{\delta }_{i+1})-\frac{d{\overline{x}}_{i+1c}}{\mathrm{dt}}\hfill \end{array}$$

(38)

Substituting (36)–(37) into (35) leads to

$$\frac{d{e}_i}{\mathrm{dt}}=-k_i\mathrm{sig}(e_i{)}^{\frac{p}{q}}-k{\prime }_i\mathrm{sig}(e_i{)}^{\frac{q}{p}}-{k″}_i{e}_i+b_i(x)(e_{i+1}+{\delta }_{i+1})+d{\prime }_i$$

(39)

Step $n$ : Consider the final equation of system (1). Define the tracking error of $x_n$ as $e_n=x_n-x_{\mathrm{nc}}$ , the time derivative of $e_n$ is

$$\frac{d{e}_n}{\mathrm{dt}}=\frac{d{x}_n}{\mathrm{dt}}-\frac{d{x}_{\mathrm{nc}}}{\mathrm{dt}}=a_n(x)+b_n(x)u+d_n-\frac{d{x}_{\mathrm{nc}}}{\mathrm{dt}}$$

(40)

Then, the control law can be established as

$$\begin{array}{c}u=-b_n(x{)}^{-1}\hfill \\ (k_n\mathrm{sig}(e_n{)}^{\frac{p}{q}}+k{\prime }_n\mathrm{sig}(e_n{)}^{\frac{q}{p}}+{k″}_n{e}_n+a_n(x)+{\hat{d}}_n-\frac{d{x}_{\mathrm{nc}}}{\mathrm{dt}})\hfill \end{array}$$

(41)

Substituting (41) into (40), we get

$$\frac{d{e}_n}{\mathrm{dt}}=-k_n\mathrm{sig}(e_n{)}^{\frac{p}{q}}-k{\prime }_n\mathrm{sig}(e_n{)}^{\frac{q}{p}}-{k″}_n{e}_n+d{\prime }_n$$

(42)

Stability analysis

This section presents the fixed-time stability analysis of the system (1) under the proposed control method.

• Assumption 4. There is a positive constant $\vartheta$ so that the initial condition satisfies the inequality $\sum _{i=1}^n{e}_i^2+\sum _{i=2}^n{\delta }_i^2\le 2\vartheta$ .

• Remark 3. It is worth noticing that Assumption 4 is not really restrictive because $\vartheta$ can be designed large enough.

• Theorem 2. For the system (1) under abovementioned assumptions, the disturbance observer is designed as (17) and the control law is designed as (41). Then, the system output will track the desired reference signal $x_d$ within fixed time.

Proof. We consider a Lyapunov function in the following form

$$V=0.5\sum _{i=1}^n{e}_i^2+0.5\sum _{i=2}^n{\delta }_i^2+0.5\sum _{i=1}^{n}d{\prime }_i^2$$

(43)

It can be deduced from Theorem 1 that $d{\prime }_i$ can converge to zero within fixed time $T_i$ . It means that $d{\prime }_i=0$ for all $t>t_1=\max (T_i)$ , $(i\mathrm{is}\mathrm{from}1ton)$ . Then, equation (43) can be rewritten as

$$V=0.5\sum _{i=1}^n{e}_i^2+0.5\sum _{i=2}^n{\delta }_i^2$$

(44)

Combining (33)–(34), (38)–(39), and (42), the time derivative of $V$ can be obtained

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{dV}}{\mathrm{dt}}=\sum _{i=1}^n{e}_i\frac{d{e}_i}{\mathrm{dt}}+\sum _{i=2}^n{\delta }_i\frac{d{\delta }_i}{\mathrm{dt}}}\\ =e_1[-k_1\mathrm{sig}(e_1{)}^{p/q}-{k\prime }_1\mathrm{sig}(e_1{)}^{q/p}-{k″}_1{e}_1+b_1(x)(e_2+{\delta }_2)]\hfill \\ +\sum _{i=2}^{n-1}{e}_i[(-k_i\mathrm{sig}{(e_i)}^{p/q}-k_i\prime \mathrm{sig}(e_i{)}^{q/p}-k_i″e_i)+b_i(x)(e_{i+1}+{\delta }_{i+1})]\hfill \\ -e_n(k_n\mathrm{sig}(e_n{)}^{p/q}+{k\prime }_n\mathrm{sig}(e_n{)}^{q/p}+{k″}_n{e}_n)\hfill \\ -\sum _{i=2}^n(\frac{{\left|{\delta }_i\right|}^{p/q+1}+{\left|{\delta }_i\right|}^{q/p+1}+{\delta }_i^2}{\tau }+{\delta }_i\frac{d{\overline{x}}_{\mathrm{ic}}}{\mathrm{dt}})\hfill \end{array}$$

(45)

In accordance with the control objective $e_1=0$ and Assumption 4, there are positive constants $A$ and $\vartheta$ such that $\Omega =\{x_d^2+{(\frac{d{x}_d}{\mathrm{dt}})}^2+{(\frac{d^2{x}_d}{d{t}^2})}^2\le A\}$ and ${\Omega }_i=\{\sum _{i=1}^n{e}_i^2+\sum _{i=2}^n{\delta }_i^2\le 2\vartheta \}$ are compact sets in $R^3$ and $R^{2i-1}$ respectively. $\Omega \times {\Omega }_i$ is a compact set in $R^{2i+2}$ . ³⁴ Thus, $\left|\frac{d{\overline{x}}_{\mathrm{ic}}}{\mathrm{dt}}\right|$ is upper bounded satisfying that $\left|\frac{d{\overline{x}}_{\mathrm{ic}}}{\mathrm{dt}}\right|\le {\Upsilon }_i$ on $\Omega \times {\Omega }_i$ .

Based on Assumption 3 and Lemma 2, the following inequalities can be obtained

$$\begin{array}{c}\multicolumn{1}{c}{e_1{b}_1(x)e_2\le B_1\left|e_1\right|\left|e_2\right|}\\ \le B_1(\frac{c_1^{q/p+1}}{q/p+1}{\left|e_1\right|}^{q/p+1}+\frac{c_1^{-(p/q+1)}}{p/q+1}{\left|e_2\right|}^{p/q+1})\hfill \end{array}$$

(46)

$$\begin{array}{c}\multicolumn{1}{c}{e_1{b}_1(x){\delta }_2\le B_1\left|e_1\right|\left|{\delta }_2\right|}\\ \le B_1(\frac{{\gamma }_1^{p/q+1}}{p/q+1}{\left|e_1\right|}^{p/q+1}+\frac{{\gamma }_1^{-(q/p+1)}}{q/p+1}{\left|{\delta }_2\right|}^{q/p+1})\hfill \end{array}$$

(47)

$$\begin{array}{c}\multicolumn{1}{c}{e_i{b}_i(x)e_{i+1}\le B_i\left|e_i\right|\left|e_{i+1}\right|}\\ \le B_i(\frac{c_i^{q/p+1}}{q/p+1}{\left|e_i\right|}^{q/p+1}+\frac{c_i^{-(p/q+1)}}{p/q+1}{\left|e_{i+1}\right|}^{p/q+1})\hfill \end{array}$$

(48)

$$\begin{array}{c}\multicolumn{1}{c}{e_i{b}_i(x){\delta }_{i+1}\le B_i\left|e_i\right|\left|{\delta }_{i+1}\right|}\\ \le B_i(\frac{{\gamma }_i^{p/q+1}}{p/q+1}{\left|e_i\right|}^{p/q+1}+\frac{{\gamma }_i^{-(q/p+1)}}{q/p+1}{\left|{\delta }_{i+1}\right|}^{q/p+1})\hfill \end{array}$$

(49)

$$-{\delta }_i\frac{d{\overline{x}}_{\mathrm{ic}}}{\mathrm{dt}}\le \left|{\delta }_i\right|\left|{\Upsilon }_i\right|\le \frac{{\beta }_i^{p/q+1}}{p/q+1}{\left|{\delta }_i\right|}^{p/q+1}+\frac{{\beta }_i^{-(q/p+1)}}{1+q/p}{\left|{\Upsilon }_i\right|}^{q/p+1}$$

(50)

Substituting (46)–(50) into (45), it can be derived that

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{dV}}{\mathrm{dt}}\le -(k_1-\frac{{\gamma }_1^{p/q+1}}{p/q+1}{B}_1){\left|e_1\right|}^{p/q+1}}\\ -({k\prime }_1-\frac{c_1^{q/p+1}}{q/p+1}{B}_1){\left|e_1\right|}^{q/p+1}-{k″}_1{e}_1^2\hfill \\ -\sum _{i=2}^{n-1}[(k_i-\frac{{\gamma }_i^{p/q+1}}{p/q+1}{B}_i-\frac{c_{i-1}^{-(p/q+1)}}{p/q+1}{B}_i){\left|e_i\right|}^{p/q+1}\hfill \\ +(k_i\prime -\frac{c_i^{q/p+1}}{q/p+1}{B}_i)·{\left|e_i\right|}^{q/p+1}+k_i″e_i^2]\hfill \\ -(k_n-\frac{c_{n-1}^{-(p/q+1)}}{p/q+1}{B}_n){\left|e_n\right|}^{p/q+1}-{k\prime }_n{\left|e_n\right|}^{q/p+1}\hfill \\ -{k″}_n{e}_n^2+\sum _{i=2}^n\frac{{\gamma }_{i-1}^{-(q/p+1)}}{q/p+1}{B}_i{\left|{\delta }_i\right|}^{q/p+1}\hfill \\ -\sum _{i=2}^n\frac{1}{\tau }({\left|{\delta }_i\right|}^{p/q+1}+{\left|{\delta }_i\right|}^{q/p+1}+{\delta }_i^2)\hfill \\ +\sum _{i=2}^n(\frac{{\beta }_i^{p/q+1}}{p/q+1}{\left|{\delta }_i\right|}^{p/q+1}+\frac{{\beta }_i^{-(q/p+1)}}{1+q/p}{\left|{\Upsilon }_i\right|}^{q/p+1})\hfill \end{array}$$

$$\begin{array}{c}=-\left(k_1-\frac{{\gamma }_1^{p/q+1}}{p/q+1}{B}_1\right){\left|e_1\right|}^{p/q+1}-\left({k^{\prime }}_1-\frac{c_1^{q/p+1}}{q/p+1}{B}_1\right){\left|e_1\right|}^{q/p+1}-{k^{″}}_1{e}_1^2\\ -\sum _{i=2}^{n-1}[\left(k_i-\frac{{\gamma }_i^{p/q+1}}{p/q+1}{B}_i-\frac{c_{i-1}^{-(p/q+1)}}{p/q+1}{B}_i\right){\left|e_i\right|}^{p/q+1}+\left({k^{\prime }}_i-\frac{c_i^{q/p+1}}{q/p+1}{B}_i\right)\cdot \\ {\left|e_i\right|}^{q/p+1}+{k^{″}}_i{e}_i^2]-\left(k_n-\frac{c_{n-1}^{-(p/q+1)}}{p/q+1}{B}_n\right){\left|e_n\right|}^{p/q+1}-{k^{\prime }}_n{\left|e_n\right|}^{q/p+1}-{k^{″}}_n{e}_n^2\\ -\sum _{i=2}^n\left(\frac{1}{\tau }-\frac{{\beta }_i^{p/q+1}}{p/q+1}\right){\left|{\delta }_i\right|}^{p/q+1}-\sum _{i=2}^n\left(\frac{1}{\tau }-\frac{{\gamma }_{i-1}^{-(q/p+1)}}{q/p+1}{B}_i\right){\left|{\delta }_i\right|}^{q/p+1}\\ -\sum _{i=2}^n\frac{1}{\tau }{\delta }_i^2+\sum _{i=2}^n\frac{{\beta }_i^{-(q/p+1)}}{1+q/p}{\left|{\Upsilon }_i\right|}^{q/p+1}\end{array}$$

(51)

where $k_1>(B_1{\gamma }_1^{p/q+1})/(p/q+1)$ , ${k^{\prime }}_1>(B_1{c}_1^{q/p+1})/(q/p+1)$ , $k_i>B_i({\gamma }_i^{p/q+1}+c_{i-1}^{-(p/q+1)})/(p/q+1)$ , ${k^{\prime }}_i>(B_i{c}_i^{q/p+1})/(q/p+1)$ , $1/\tau >\max [{\beta }_i^{p/q+1}/(p/q+1),(B_i{\gamma }_{i-1}^{q/p+1})/(q/p+1)]$ , $k_n>(B_n{c}_{n-1}^{-(p/q+1)})/(p/q+1)$ .

For convenience, we define $K_1$ , $K_2$ , $K_3$ , and $\mathrm{\Delta }$ as follows

$$\begin{array}{c}\multicolumn{1}{c}{K_1=min\{k_1-\frac{{\gamma }_1^{p/q+1}}{p/q+1}{B}_1,k_i-\frac{{\gamma }_i^{p/q+1}}{p/q+1}{B}_i-\frac{c_i^{-(p/q+1)}}{p/q+1}{B}_i,}\\ k_n-\frac{c_{n-1}^{-(p/q+1)}}{p/q+1}{B}_n,\frac{1}{\tau }-\frac{{\beta }_i^{p/q+1}}{p/q+1}\}\hfill \end{array},$$

$$K_2=min\{{k\prime }_1-\frac{c_1^{q/p+1}}{q/p+1}{B}_1,k_i\prime -\frac{c_i^{q/p+1}}{q/p+1}{B}_i,{k\prime }_n,\frac{1}{\tau }-\frac{{\gamma }_{i-1}^{-(q/p+1)}}{q/p+1}{B}_i\}$$

$$K_3=min\{k_i″\},i\mathrm{is}\mathrm{from}1\mathrm{to}n,$$

$$\mathrm{\Delta }=\sum _{i=2}^n\frac{{\beta }_i^{-(q/p+1)}}{1+q/p}{\left|{\Upsilon }_i\right|}^{q/p+1}$$

(52)

From (51)–(52) and using Lemmas 3 and 4, we get

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{dV}}{\mathrm{dt}}\le -K_1(\sum _{i=1}^n{\left|e_i\right|}^{p/q+1}+\sum _{i=2}^n{\left|{\delta }_i\right|}^{p/q+1})}\\ -K_2(\sum _{i=1}^n{\left|e_i\right|}^{q/p+1}+\sum _{i=2}^n{\left|{\delta }_i\right|}^{q/p+1})\hfill \\ -K_3(\sum _{i=1}^n{e}_i^2+\sum _{i=2}^n{\delta }_i^2)+\mathrm{\Delta }\hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\le -{(2n-1)}^{\frac{1-p/q}{2}}{K}_1{(\sum _{i=1}^n{e}_i^2+\sum _{i=2}^n{\delta }_i^2)}^{\frac{p/q+1}{2}}}\\ -K_2{(\sum _{i=1}^n{e}_i^2+\sum _{i=2}^n{\delta }_i^2)}^{\frac{q/p+1}{2}}\hfill \\ -K_3(\sum _{i=1}^n{e}_i^2+\sum _{i=2}^n{\delta }_i^2)+\mathrm{\Delta }\hfill \\ =-{\mathrm{\Gamma }}_1{V}^{\frac{p/q+1}{2}}-{\mathrm{\Gamma }}_2{V}^{\frac{q/p+1}{2}}-{\mathrm{\Gamma }}_{3}V+\mathrm{\Delta }\hfill \end{array}$$

(53)

where ${\mathrm{\Gamma }}_1=2^{(p/q+1)/2}(2n-1{)}^{(1-p/q)/2}{K}_1$ , ${\mathrm{\Gamma }}_2=2^{(q/p+1)/2}{K}_2$ , ${\mathrm{\Gamma }}_2=2K_3$ .

The parameters can be chosen satisfying the inequality $-{\mathrm{\Gamma }}_1{\vartheta }^{(p/q+1)/2}-{\mathrm{\Gamma }}_2{\vartheta }^{(q/p+1)/2}-{\mathrm{\Gamma }}_3\vartheta +\mathrm{\Delta }\le 0$ . When the initial condition satisfies Assumption 4, it can be obtained that $\frac{\mathrm{dV}}{\mathrm{dt}}\le 0$ . Thus, $\Xi =\{V\le \vartheta \}$ is an invariant set, that is, if $V(0)\le \vartheta$ , then it can be known that $V(t)\le \vartheta$ for all $t\ge 0$ . The ultimate bound of the overall system can be derived from the following equation

$$-{\mathrm{\Gamma }}_1{V}^{\frac{p/q+1}{2}}-{\mathrm{\Gamma }}_2{V}^{\frac{q/p+1}{2}}-{\mathrm{\Gamma }}_{3}V+\mathrm{\Delta }=0$$

(54)

It is noted that (54) is a polynomial algebraic equation with fractional exponent. The analytical solution of this equation is difficult to be obtained. Nevertheless, the ultimate bound of the overall system can be roughly estimated by solving the following equations

$$-{\mathrm{\Gamma }}_1{V}^{\frac{p/q+1}{2}}+\mathrm{\Delta }=0$$

$$-{\mathrm{\Gamma }}_2{V}^{\frac{q/p+1}{2}}+\mathrm{\Delta }=0$$

$$-{\mathrm{\Gamma }}_{3}V+\mathrm{\Delta }=0$$

(55)

Solving (55), the ultimate bound of the overall system is obtained as

$$\underset{t\to \infty }{lim}V(t)\le 3min\{{(\mathrm{\Delta }/{\mathrm{\Gamma }}_1)}^{2q/(p+q)},{(\mathrm{\Delta }/{\mathrm{\Gamma }}_2)}^{2p/(p+q)},\mathrm{\Delta }/{\mathrm{\Gamma }}_3\}$$

(56)

With appropriate control parameters and according to Lemma 5, the system output can track the desired reference signal $x_d$ within a finite time bounded by

$$\begin{array}{c}\multicolumn{1}{c}{t_2<\frac{2}{{\mathrm{\Gamma }}_3(p/q-1)}\ln (1+\frac{{\mathrm{\Gamma }}_3}{{\mathrm{\Gamma }}_1})+\frac{2}{{\mathrm{\Gamma }}_3(1-q/p)}}\\ \ln (1+\frac{{\mathrm{\Gamma }}_3}{(2^{q/p}-1){\mathrm{\Gamma }}_2})\hfill \end{array}$$

(57)

Considering the convergence time of FFTDO, the total convergence time of system (1) is $t\le t_1+t_2$ .

The proof of Theorem 2 is completed.

Simulation results and discussion

• Example 1. The advantages of the proposed filter (29) will be shown in comparison with several present filters. The desired reference signal is set as $x_d=\sin (5t)$ . The initial values are set as $x_c(0)=x_d(0)=0$ , $\frac{d{x}_c(0)}{\mathrm{dt}}=\frac{d{x}_d(0)}{\mathrm{dt}}=0$ .

Different filters can be expressed as: (1) The proposed filter: $\tau \frac{d{x}_c}{\mathrm{dt}}+\mathrm{sig}(x_c-x_d{)}^{p/q}+\mathrm{sig}(x_c-x_d{)}^{q/p}+x_c-x_d=0$ , (2) The first-order filter ¹³ : $\tau \frac{d{x}_c}{\mathrm{dt}}+x_c=x_d$ , (3) The finite-time filter ³⁶ : $\tau \frac{d{x}_c}{\mathrm{dt}}+\mathrm{sig}(x_c-x_d{)}^{q/p}=0$ , (4) The fixed-time filter ³⁴ : $\tau \frac{d{x}_c}{\mathrm{dt}}+\mathrm{sig}(x_c-x_d{)}^{p/q}+\mathrm{sig}(x_c-x_d{)}^{q/p}=0$ . Define $e=x_c-x_d$ and $e_d=\frac{d{x}_c}{\mathrm{dt}}-\frac{d{x}_d}{\mathrm{dt}}$ .

For fair comparison, the parameters of these filters are set as: $\tau =0.02$ , $p=13$ , and $q=11$ . Comparison simulation results are displayed in Figures 2 and 3. Figure 2 shows the performance of state tracking error under these four filters, and Figure 3 illustrates the approximation of first-order differential of the state tracking error under these four filters. It is obvious that the proposed fixed-time filter has superior approximation performance over the other three existing filters.

• Example 2. Consider a single-link manipulator system including actuator dynamics, the system mathematical model can be expressed as^37,38

$$\begin{array}{c}M\frac{d^{2}q}{d{t}^2}+B\frac{\mathrm{dq}}{\mathrm{dt}}+N\sin (q)=I+{\rho }_2,\hfill \\ D\frac{\mathrm{dI}}{\mathrm{dt}}+\mathrm{HI}=V-K_m\frac{\mathrm{dq}}{\mathrm{dt}}+{\rho }_3,\hfill \end{array}$$

(58)

where $q,\frac{\mathrm{dq}}{\mathrm{dt}},\frac{d^{2}q}{d{t}^2}$ stand for the link angular position, velocity and acceleration, respectively. $M$ is the lumped inertia, $B$ is the friction coefficient, $N$ is the lumped load term, $D$ is the rotor inductance, $I$ is the motor current, $H$ is the rotor resistance, $V$ is the voltage, $K_m$ is the back-emf coefficient. ${\rho }_2$ and ${\rho }_3$ denote external disturbances. Here, the parameters are given as $M=1\mathrm{kg}{\mathrm{m}}^2/\mathrm{rad}$ , $B=1$ , $N=10\mathrm{kgA}$ , $D=0.05\mathrm{H}$ , $H=0.5\Omega$ , and ${\mathrm{K}}_{\mathrm{m}}=10\mathrm{Nm}/\mathrm{A}$ .

Figure 2.

State tracking error of filters.

Figure 3.

First-order differential of state tracking error of filters.

Define $x_1=q$ , $x_2=\frac{\mathrm{dq}}{\mathrm{dt}}$ , $x_3=I$ , and $u=V$ . The system model can be rewritten as

$$\{\begin{array}{c}\frac{d{x}_1}{\mathrm{dt}}=x_2\hfill \\ \frac{d{x}_2}{\mathrm{dt}}=a_2(x)+x_3+d_2\hfill \\ \frac{d{x}_3}{\mathrm{dt}}=a_3(x)+b_{3}u+d_3\hfill \end{array}$$

(59)

where $a_2(x)=-10\sin (x_1)-x_2$ , $a_3(x)=(-10x_2-0.5x_3)/0.05$ , $b_3=1/0.05$ , $d_2={\rho }_2$ , $d_3={\rho }_3/0.05$ .

The initial states are $[x_1(0),x_2(0),x_3(0)]=[0.2,0.2,0.1{]}^T$ . The disturbances are assumed as $d_2=10\sin (\pi t)$ for $t\le 16\mathrm{s}$ , $d_2=10\sin (\pi t/2)$ for $t>16\mathrm{s}$ , $d_3=-20\sin (\pi t)$ for $t\le 12\mathrm{s}$ , $d_3=-20\sin (\pi t/3)$ for $t>12\mathrm{s}$ .The reference signal 1 is set as $x_d=0.5\sin (0.5t)$ . The reference signal 2 is set as $x_d=0.5\sin (2t)$ .

Here, to verify the performance of FFTDO, a fixed-time disturbance observer (FTDO) used in Xia et al. ⁸ is selected for validation. Figures 4 and 5 show the tracking performance of the FFTDO and FTDO. It can be founded from figures that the proposed FFTDO can accurately estimate the mismatched and matched disturbances in fixed time, which is lined with Theorem 1. Moreover, the proposed FFTDO can achieve faster tracking performance compared to the FTDO. The parameters of FFTDO are set as ${\sigma }_1=1.5$ , ${\sigma }_2=0.5$ , ${\kappa }_2=\kappa {\prime }_2={\kappa ″}_2=8$ , $l_2=30$ , ${\kappa }_3=\kappa {\prime }_3={\kappa ″}_3=20$ , $l_3=80$ .

Figure 4.

(a) $d_2$ and its estimation ${\hat{d}}_2$ ; (b) the zoomed section of image (a).

Figure 5.

(a) $d_3$ and its estimation ${\hat{d}}_3$ ; (b) the zoomed section of image (a).

To illustrate the superiority of the proposed control method, Zhang’s controller in Zhang and Yang ³⁷ is selected for comparison. The design procedure of the proposed controller for system (59) are listed in Table 1. The parameters of the fast fixed-time backstepping controller are selected as $\tau =0.01$ , $p=7$ , $q=5$ , $k_1=k_2=k_3=2$ , $k{\prime }_1=k{\prime }_2=k{\prime }_3=2$ , ${k″}_1={k″}_2={k″}_3=2$ .

Table 1.

Design procedure of the proposed control method.

Sequence	Fast fixed-time backstepping control
Virtual law	${\overline{x}}_{2c}=-(k_1\mathrm{sig}(e_1{)}^{\frac{p}{q}}+k{\prime }_1\mathrm{sig}(e_1{)}^{\frac{q}{p}}+{k″}_1{e}_1-\frac{d{x}_d}{\mathrm{dt}})$
New error variable	${\delta }_2=x_{2c}-{\overline{x}}_{2c}$ $\tau \frac{d{x}_{2c}}{\mathrm{dt}}+\mathrm{sig}({\delta }_2{)}^{p/q}+\mathrm{sig}({\delta }_2{)}^{q/p}+{\delta }_2=0$
Error	$e_2=x_2-x_{2c}$
Virtual law	${\overline{x}}_{3c}=-(k_2\mathrm{sig}(e_2{)}^{\frac{p}{q}}+k{\prime }_2\mathrm{sig}(e_2{)}^{\frac{q}{p}}+{k″}_2{e}_2+a_2(x)+{\hat{d}}_2-\frac{d{x}_{2c}}{\mathrm{dt}})$
New error variable	${\delta }_3=x_{3c}-{\overline{x}}_{3c}$ $\tau \frac{d{x}_{3c}}{\mathrm{dt}}+\mathrm{sig}({\delta }_3{)}^{p/q}+\mathrm{sig}({\delta }_3{)}^{q/p}+{\delta }_3=0$
Error	$e_3=x_3-x_{3c}$
Control law	$u=-\frac{1}{b_3}(k_3\mathrm{sig}(e_3{)}^{\frac{p}{q}}+k{\prime }_3\mathrm{sig}(e_3{)}^{\frac{q}{p}}+{k″}_3{e}_3+a_3(x)+{\hat{d}}_3-\frac{d{x}_{3c}}{\mathrm{dt}})$

The performance comparisons of the angular position tracking trajectory under the reference signal 1 and 2 are shown in Figures 6 and 8, respectively. The corresponding position tracking errors are depicted in Figures 7 and 9, respectively. It is obviously seen from Figures 6 to 9 that the proposed method responds faster and takes shorter time for the position tracking error convergence to a small region compared to the other controller. Moreover, the proposed method obtains smaller steady-state error than that of the other method.

Figure 6.

(a) Angular position $x_1$ under the reference signal 1; (b) the zoomed section of image (a).

Figure 7.

(a) Angular position tracking error $e_1$ under the reference signal 1; (b) the zoomed section of image (a).

Figure 8.

(a) Angular position $x_1$ under the reference signal 2; (b) the zoomed section of image (a).

Figure 9.

(a) Angular position tracking error $e_1$ under the reference signal 2; (b) the zoomed section of image (a).

To make the comparisons more accurately, the root-mean-square error (RMSE) and the Maximum error (MAE) ³⁹ are calculated using the steady simulation experiment data. The calculation results are displayed in Table 2. As can be seen from Table 2, compared with Zhang’s controller, the RMSE and MAE under the proposed control strategy are smaller, which shows that the proposed controller has the better control performance.

Table 2.

Performance comparisons of two controllers.

Case	Controller	RMSE	MAE
The reference signal 1	Zhang’s	0.0117	0.0021
	The proposed	0.0094	0.0011
The reference signal 2	Zhang’s	0.0132	0.0032
	The proposed	0.0097	0.0014

To confirm the fixed-time convergence of the system under the proposed control method, different cases are considered as: (1) Case 1: $[x_1(0),x_2(0),x_3(0)]=[0.2,0.2,0.1{]}^T$ , (2) Case 2: $[x_1(0),x_2(0),x_3(0)]=[3,-0.1,0.1{]}^T$ , (3) Case 3: $[x_1(0),x_2(0),x_3(0)]=[-1.5,3,0.1{]}^T$ .

As shown in Figure 10, the convergence time of system (58) with three different initial conditions are so close, which implies that the total convergence time is weak dependence on the initial states. The tracking error signals are depicted in Figure 11. Observed by Figure 11, the total convergence time of these three cases are upper bounded by 2 s, which verifies the upper-bound estimation in Theorem 2. The simulation results confirm that the angular position tracking error can converge to the equilibrium within fixed time, which indicates that the total convergence time is irrelevant to initial conditions and can be acquired in prior by the designed controller parameters.

Figure 10.

(a) Angular position $x_1$ with different initial states; (b) the zoomed section of image (a).

Figure 11.

(a) Angular position tracking error $e_1$ with different initial states; (b) the zoomed section of image (a).

All in all, it can be acclaimed that the proposed observer-based fast fixed-time backstepping control approach is available to achieve satisfactory fixed-time stabilization for n-order systems with mismatched and matched disturbances. Furthermore, comparative simulation results verify that the proposed control scheme has more superior performance with respect to fast dynamic response and high control precision compared to the other control method.

Conclusion

In this paper, a fast fixed-time backstepping control strategy is developed for systems with mismatched disturbances. The FFTDO is designed to estimate and compensate the disturbances, which can effectively enhance the robustness of the system. The fixed-time filter is employed to eliminate the problem of “explosion of complexity.” A fast fixed-time virtual control law is designed for each step, which can guarantee the fixed-time convergence and high steady-state precision. The fixed-time stability of the system is analyzed by the Lyapunov theory. Corresponding simulation results confirm the effectiveness and superiority of the proposed control approach. Notable is, we do not consider the input constraint, which may result in the degradation of tracking performance. The fast fixed-time control problem of n-order systems with input constraints will be our further research work.