Fixed-time stability super-twisting sliding mode control with mismatched disturbances

Abstract

In practice, many systems are affected by mismatched disturbances, which do not satisfy the match condition. This study focuses on a variable gain super-twisting sliding mode controller that achieves global robust fixed-time stabilization of second-order systems with matched and mismatched disturbances without disturbances observers. The proposed controller and the sliding variables adopt a state-dependent variable exponent coefficient, which improves the robustness properties of a sliding surface and switching control law. At the same time, the control strategy alleviates the chattering phenomenon by using the variable-gain super-twisting algorithm. The settling time is calculated a priori by the Lyapunov method. Simulation results show that, compared with recently published controllers, our proposed control strategy can suppress the mismatched disturbance more rapidly, simply, and effectively.

Introduction

In engineering practice, control systems suffer unknown disturbances between real-time plant dynamics and their mathematical model, which deteriorate closed-loop control performance and even destroy system stability. The main unknown disturbances in the system are unmodeled dynamics, parameter perturbations, and exogenous disturbances. According to whether the disturbances enjoy the matching condition or not, they can be divided into matched and mismatched disturbances. Matched disturbances^1,2 are the most common disturbances encountered in control applications and are handled in the state equation by a control input. Conversely, the mismatched disturbances^3,4 are difficult to eliminate because they are not present in access channels to cross-coupling and interconnection states. Our challenge is designing robust controllers to eliminate mismatched disturbances in systems.

To achieve the desired closed-loop performance, we need to design a robust control algorithm for mismatched systems. The authors solve the problem of estimated and compensated mismatched disturbances by using observers; for instance, disturbance observers,^5–7 and extended state observers.^8–10 However, it is difficult to design a low-pass filter Q(s) in the disturbance observer, and the extended state observer only handles the slowly varying disturbances. The above approach also increases the complexity of controllers and computational costs. Research groups take advantage of the robust properties of variable structure control to reject mismatched disturbances, such as backstepping methods, ¹¹ and methods based on sliding mode control (SMC).^12–14 Backstepping control possesses excellent performance for high-order nonlinear systems with disturbances, together with adaptive control, ¹⁵ observer,^16,17 which increases the complexity of the design. To enhance the robustness properties of the controller, the shorter settling time (i.e. the time that dynamical systems reach their equilibrium point.) is a must. In recent years, finite-time stabilization,^18–21 and fixed-time stabilization,^22–25 using a terminal sliding mode control (TSMC) provided by the nonlinear sliding surface have received much attention. Fixed-time stability pre-sets an explicit upper bound on the settling time that is independent of the initial conditions. Compare to asymptotic stabilization and finite-time stabilization, fixed-time stabilization provides a faster convergence rate, higher accuracy, and a stronger anti-disturbance ability for a variety of systems. A new second-order fixed-time sliding mode controller without disturbance observers has been used to reject mismatched disturbance. However, these methods are all based on SMC and inevitably produce high-frequency oscillations (i.e. so-called chattering), which is an inherent drawback of SMC. ¹⁴

The main reason for the chattering phenomenon of sliding mode control is the discontinuous control input, and a higher-order SMC approach can solve this problem effectively. Levant has proposed a continuous second-order sliding mode super-twisting algorithm (STA) to suppress chattering. But the perturbation bound is known in advance and the gain is constant. ²⁶ Moreno et al. have developed a time-varying gain super-twisting algorithm (VGSTA) with gradient-bound gains adjusted. ²⁷ The VGSTA precisely compensates the disturbance with variable gain following, when the disturbance prior is unknown and allowed to grow as a function of time and the state variables. VGSTA improves the robustness of the system and realizes the dynamic control input, which is desired for engineering applications. A practical fractional-order VGSTA has been proposed to improve the tracking performance of wafer stages for semiconductor manufacturing. ²⁸ A finite-time convergence TSMC with VGSTA as the reaching law of an active fault-tolerant scheme has been developed for multi-robotics systems in the presence of actuator faults. ²⁹ But the current methods, the STA and VGSTA are finite-time stable and have never been used to suppress mismatch interference. ³⁰

This study proposes a global robust fixed-time stability super-twisting sliding mode control (FXSTSMC) method for second-order systems with matched and mismatched disturbances that without disturbance observers. By adopting the SMC strategy, the state-dependent variable exponent coefficients were used in the sliding surface and the controllers, and the system obtained robust global fixed-time stability of the state variable error with robust fixed-time stability of the sliding variable $s$ . To the best of the author’s knowledge, it is the first time that mismatched disturbances are rejected by combining the robustness properties of the sliding surface and a variable-gain super-twisting control input. According to the simulation results, the effectiveness of the new control algorithm is verified, it could apply to high-performance control or develop a fixed-time differentiator. The main research contributions of this paper are as follows:

• (1) The proposed new super-twisting sliding mode controller of the mismatched disturbances system is robust globally fixed-time stabilization and has a shorter convergence time than other similar methods. Compared with the literature, ¹⁴ the proposed method converges to the origin about 0.2 s faster.

• (2) The variable-gain super-twisting switching control law suppress chattering more effectively. In comparison to the literature, ¹⁴ the proposed control method cuts the input magnitude and frequency in half.

The rest of this paper is organized as follows: the preliminaries of the theory used in the paper in Section 2; a dynamic model of the motor drive system with uncertainties in Section 3; the fixed-time stability super-twisting sliding control design in Section 4, the simulation in Section 5 to verify the effectiveness of a fixed-time sliding mode and super-twisting control; the conclusion is presented in Section 6.

Preliminaries

Consider the following common differential equation

$$\{\begin{array}{c}\stackrel{·}{x}\left(t\right)=f\left(x\left(t\right)\right),x\left(t\right)\in R^n\hfill \\ \multicolumn{1}{c}{x\left(0\right)=x_0,}\end{array}$$

(1)

with $x(t)$ as the state of the system, the initial state is $x_0,f$ is a locally bounded continuous function.

• Definition 1 ²² : The system (1) is globally fixed-time stable, the following two conditions must be met:

1. System (1) is globally finite-time stable;

2. The setting-time $T$ is bounded, $T>0,T\left(x_0\right)\le T$ , and does not depend on the initial state $x_0$ .

• Lemma 1 ¹⁴ : The function $A\left(x\right)=\sigma \left|x\right|\left(\frac{2\ln \left|x\right|+1+\mu x^2}{{(1+\mu x^2)}^2}\right)$ ${\left|x\right|}^{\frac{\sigma x^2}{1+\mu x^2}}$ , for $x\ge 0$ , we have $A\left(x\right)>0$ when $x>1$ and $\left|A\left(x\right)\right|\le 2\sigma (1+\mu )$ when $0\le \mathrm{x}\le 1$ .

Notation: Introduce the following denotation $E$ is the natural constant down to $\ln \left(E\right)=1$ ; ${\left[x\right]}^k={\left|x\right|}^k\mathrm{sign}\left(x\right)$ ; The $\mathrm{sign}(x)$ denotes the symbolic function.

Dynamic model of the motor drive system with uncertainties

Consider the dynamic model of the drive system described by following second-order differential, the motion equation can be expressed as follows:

$$J\stackrel{··}{\theta }+T_f+T_L=\mathrm{Qu},$$

(2)

where $\stackrel{··}{\theta }$ is the mechanical angular acceleration of the rotor; $J$ is the inertia of the motor and load; $T_L$ is the load torque; $Q$ is the torque constant of the motor, $u$ is the control input. $T_f$ is the friction from the motor.

The friction model is the approximate piecewise linear friction model that can be written as follows, ³¹ $\alpha ,\beta$ is obtained by using linear fitting through field data.

so combine the load, the $T_f$ satisfies the following equation:

$$F_f=J^{-1}(T_f+T_L)=\alpha \stackrel{·}{\theta }+\beta ,$$

(3)

Substituting (3) into (2), the system dynamics become

$$\stackrel{·}{\omega }=\mathrm{Ku}-F_f,$$

(4)

with $K=\frac{Q}{J},\omega =\stackrel{·}{\theta }$ . $\omega$ is the mechanical angular speed.

Considering model uncertainties of friction parameters in different states $\Delta \alpha ,\Delta \beta$ (i.e. static, Stribeck, or sliding), each term can be expressed as deviations as in the equation. For example, a viscous friction with an unknown viscous coefficient grows linearly in the state and does not satisfy condition (4).

$$F_f=({\alpha }_N+\Delta \alpha )\stackrel{·}{\theta }+({\beta }_N+\Delta \beta ),K=K_N+\Delta K,$$

(5)

where ${\alpha }_N,{\beta }_N,K_N$ and $\Delta K,\Delta \alpha ,\Delta \beta$ are the nominal and uncertain parts of $K,\alpha ,\beta$ , respectively. The disturbance $\Delta K,\Delta \alpha ,\Delta \beta$ are assumed bounded.

the mechanical dynamics can be rewritten as

$$\stackrel{··}{\theta }+({\alpha }_N+\Delta \alpha )\stackrel{·}{\theta }+({\beta }_N+\Delta \beta )=(K+\Delta K)u.$$

(6)

Consider the second-order drive system with $\Delta \omega$ mismatched uncertainties given,

$$\{\begin{array}{c}\stackrel{·}{\theta }=\omega +\Delta \omega \hfill \\ \multicolumn{1}{c}{\stackrel{·}{\omega }=(K+\Delta K)u-({\alpha }_N+\Delta \alpha )\stackrel{·}{\theta }-({\beta }_N+\Delta \beta ).}\end{array}$$

(7)

If we choose the state variable as $x={\left[\begin{array}{cc}{x}_1\hfill & x_2\hfill \end{array}\right]}^T={\left[\begin{array}{cc}\theta \hfill & \stackrel{·}{\theta }\hfill \end{array}\right]}^T\in R^2$ , then the mechanical dynamics (7) can be rewritten in the state space form as

$$\{\begin{array}{c}{\stackrel{·}{x}}_1=x_2+d_1\left(t\right)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2=-{\alpha }_N{x}_2-{\beta }_N+(g\left(x\right)+\Delta g)u+d_2\left(t\right)+d_3(x_2,t),}\end{array}$$

(8)

where $d_1\left(t\right)=\Delta x_2$ , $d_2\left(t\right)=-{\alpha }_N\Delta x_2-\Delta \alpha \Delta x_2-\Delta \beta$ , $d_3(x_2,t)=-\Delta \alpha x_2$ . The disturbances satisfies:

• Assumption 1: Consider that $d_1\left(t\right)$ is a continuous function of mismatched disturbance is bounded for $t>0$ , there exists a positive constant $D_1^{+}$ , and $\left|d_1\left(t\right)\right|<D_1^{+}$ . ¹⁴

• Assumption 2: The uncertainties in the inertia, friction, and disturbance are all bounded in practical applications. ⁹ $\Delta \alpha ,\Delta \beta ,\Delta g$ are bounded.

FXSTSMC design

In this section, a FXSTSMC method is proposed for the dynamic system (8). The system state could track the desired trajectory quickly and accurately, and the convergence time is bounded by a constant regardless of the initial system conditions.

Define $x_d$ as the desired position trajectory, ${\stackrel{·}{x}}_d$ as desired velocity trajectory, and ${\stackrel{··}{x}}_d$ as desired accelerated velocity, all of them are continuous functions. Let $x_1$ faster track a desired trajectory $x_d$ , the $x_2$ trajectory of a system ${\stackrel{·}{x}}_1-d_1\left(t\right)$ . The tracking error $e=x_1-x_d$ .

The controller design

The main objective of this subsection is to design a fixed-time stability super twisting sliding mode controller with the sliding surface, equivalent control, and variable super-twisting switching control law shown in Figure 1.

Figure 1.

Block diagram of the proposed controller.

Consider the sliding surface by using the track error variable exponent coefficient given.

$$s=\stackrel{·}{e}+c_1{\left[e\right]}^{\frac{{\sigma }_1{e}^2}{1+{\mu }_1{e}^2}}+c_{2}e,$$

(9)

with $\frac{{\sigma }_1}{1+{\mu }_1}=\eta >1,c_1>0$ , $c_2>2{\sigma }_1{c}_1(1+{\mu }_1)>0$ .

As shown in Figure 1. we designed the sliding mode control law written as

$$u\left(x\right)=-\frac{1}{g\left(x\right)}(u_{\mathrm{eq}}+u_{\mathrm{sc}}),$$

(10)

where $u_{\mathrm{eq}}$ represents the equivalent control law and $u_{\mathrm{sc}}$ represents the switching control with the variable gain super-twisting algorithm.

The equivalent control $u_{\mathrm{eq}}$ is implemented to keep the state on the sliding surface when there are no disturbances taken into account. Therefore, by setting $\stackrel{·}{s}=0$ , and according the second-order model, $u_{\mathrm{eq}}$ can be obtained as

$$u_{\mathrm{eq}}=(c_{1}A\left(e\right)+c_2)(x_2-{\stackrel{·}{x}}_d)-{\alpha }_N{x}_2-{\beta }_N-{\stackrel{··}{x}}_d,$$

(11)

In order to avoid chattering, a second-order super-twisting algorithm with a continuous function of the sliding variable and the integral of the discontinuous time derivative of the sliding function has been developed for the systems with relative degree one. Thus, the super-twisting control not only ensures all the main properties of the first-order sliding mode control but also that the control effort is continuous. The variable gain super-twist algorithm further improves the chatter phenomenon as the gain can vary with perturbation.

A novel fixed time stable variable gain super-twisting switch control law is proposed, that is, faster compensation for gradient boundary dependent state perturbations with variable exponential of sliding variable. The mathematical representation is addressed as

$$u_{\mathrm{sc}}=(c_{1}A\left(e\right)+c_2)h_1\left(x\right){\varphi }_1\left(s\right)+{\int }_0^t{h}_2\left(x\right){\varphi }_2\left(s\right)\mathrm{dt}$$

(12)

where,

$$\begin{array}{c}{\varphi }_1\left(s\right)={\left[s\right]}^{\frac{{\sigma }_2{s}^2}{1+{\mu }_2{s}^2}}+h_{3}s\hfill \\ \multicolumn{1}{c}{{\varphi }_2\left(s\right)={\varphi }_1\prime \left(s\right){\varphi }_1\left(s\right)=(\psi \left(\mathrm{s}\right)+h_3)({\left[s\right]}^{\frac{{\sigma }_2{s}^2}{1+{\mu }_2{s}^2}}+h_{3}s),}\end{array}$$

(13)

with

$$\begin{array}{c}\psi \left(s\right)=\left({\left[s\right]}^{\frac{{\sigma }_2{s}^2}{1+{\mu }_2{s}^2}}\right)\prime =\frac{{\sigma }_2\left|s\right|\stackrel{·}{s}(2\ln \left|s\right|+1+{\mu }_2{s}^2)}{{(1+{\mu }_2{s}^2)}^2}{\left|s\right|}^{\frac{{\sigma }_2{s}^2}{1+{\mu }_2{s}^2}}=A\left(s\right)\stackrel{·}{s}\hfill \end{array}$$

(14)

and $\frac{{\sigma }_2}{1+{\mu }_2}=\phi >1$ . When the exponent of function ${\varphi }_1$ , $\frac{{\sigma }_2{s}^2}{1+{\mu }_2{s}^2}=\frac{1}{2}$ the controller becomes a standard variable super twisting controller.

The switching control law $u_{\mathrm{sc}}$ is used to cancel disturbances and bring the sliding variable $s$ to 0. Meanwhile, the tracking error $e$ and the state motion to the sliding surface.

• Remark 1: For the plant with matched and mismatched disturbances, the designed sliding surface (9) with a variable exponential coefficient guarantees a fast response, which is used for precision and stability. The designed switching controller (12), (13) uses a novel structure of the variable gain super twisting algorithm to converge fast and steadily. The variable-gain super-twisting structure by power function also has an advantage over the traditional super-twisting for variable disturbances. Figure 1 and equivalent (14) show that $u_{\mathrm{eq}}$ contains both feedback and feedforward terms, which not only guarantee the completion of the sliding mode controller but also improve the performance of the plant. All the merits of our proposed method will be verified by simulations.

Stability and convergence analysis

This subsection will analyze the globally fixed-time stability and convergence time of the FXSTSMC in the system (8).

Substituting (10) into (9), The dynamics derivative of $s$ is addressed as

$$\begin{array}{c}\stackrel{·}{s}=-(c_{1}A\left(e\right)+c_2)h_1\left(x\right){\varphi }_1\left(s\right)-{\int }_0^t{h}_2\left(x\right){\varphi }_2\left(s\right)\mathrm{dt}\hfill \\ +\frac{\Delta g}{g\left(x\right)}({\alpha }_N{x}_2+{\beta }_N+{\stackrel{··}{x}}_d-(c_{1}A\left(e\right)+c_2)(x_2-{\stackrel{·}{x}}_d)\hfill \\ -(c_{1}A\left(e\right)+c_2)h_1\left(x\right){\varphi }_1\left(s\right)-{\int }_0^t{h}_2\left(x\right){\varphi }_2\left(s\right)\mathrm{dt})\hfill \\ +d_2\left(t\right)+d_3(x_2,t)+{\stackrel{·}{d}}_1\left(t\right)+(c_{1}A\left(e\right)+c_2)d_1(t).\hfill \end{array}$$

(15)

The equation can be rewritten as

$$\begin{array}{c}\stackrel{·}{s}=-(c_{1}A\left(e\right)+c_2)(h_1\left(t\right){\varphi }_1\left(s\right)+q_1\left(t\right))\hfill \\ -{\int }_0^t{h}_2\left(t\right){\varphi }_2\left(s\right)+q_2(x_2,t),\hfill \end{array}$$

(16)

with

$$\begin{array}{c}\begin{array}{c}{q}_1\left(t\right)=\frac{\Delta g}{g\left(x\right)}{h}_1\left(x\right){\varphi }_1\left(s\right)-\frac{\frac{\Delta g}{g\left(x\right)}({\beta }_N+{\stackrel{··}{x}}_d)+d_2+{\stackrel{·}{d}}_1}{c_{1}A\left(e\right)+c_2}-d_1\hfill \end{array}\hfill \\ \begin{array}{c}{q}_2(x_2,t)=-\frac{\Delta g}{g\left(x\right)}{\int }_0^t{h}_2\left(x\right){\varphi }_1\left(s\right)\mathrm{dt}+(\frac{\Delta g}{g\left(x\right)}{\alpha }_N-\Delta \alpha )x_2\hfill \end{array}\hfill \end{array}$$

and $d_1$ , $d_2$ for short $d_1(t)$ , $d_2(t)$ respectively.

Based on (16). Assumptions 1 and 2, we obtain that $\left|q_1\left(t\right)\right|$ and $\left|{\stackrel{·}{q}}_2(x_2,t)\right|$ are bounded and rewrite it as

$$\begin{array}{c}\left|q_1\left(t\right)\right|\le M_1\left|{\varphi }_1\left(s\right)\right|+M_2\hfill \\ \left|{\stackrel{·}{q}}_2(x_2,t)\right|\le M_3\left|{\varphi }_2\left(s\right)\right|+M_4\left|x_2\right|,\hfill \end{array}$$

(17)

where,

$$\begin{array}{c}{M}_1\ge \left|\frac{\Delta g}{g\left(x\right)}\right|\left|h_1\left(x\right)\right|,M_2\ge \left|d_1\right|-\frac{\left|\frac{\Delta g}{g\left(x\right)}\right||{\beta }_N+{\stackrel{··}{x}}_d|+\left|d_2\right|+\left|{\stackrel{·}{d}}_1\right|}{|c_{1}A\left(e\right)+c_2|},\hfill \\ M_3\ge \left|\frac{\Delta g}{g\left(x\right)}\right|\left|h_2\left(x\right)\right|,M_4\ge |\frac{\Delta g}{g\left(x\right)}{\alpha }_N-\Delta \alpha |\hfill \end{array}$$

• Theorem 1: Based on Assumptions 1–3, the second-order closed-loop system (8) with control input (10) is globally stable, and the sliding variable s will converge to the origin if the gains of the controller (13) satisfy as

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}{h}_1\left(x\right)=\frac{P_2}{(P_3^3-P_1{P}_2{P}_3)(c_1\mathrm{A}\left(e\right)+c_2)}\hfill \\ \multicolumn{1}{c}{\{\frac{1}{4}{({\lambda }_1{P}_3(c_1\mathrm{A}\left(e\right)+c_2)-{\lambda }_2{P}_2)}^2}\\ \multicolumn{1}{c}{-{\lambda }_1{P}_1{P}_3(c_1\mathrm{A}\left(e\right)+c_2)+{\lambda }_2{P}_3^2+\frac{P_1{P}_3^2}{P_2}\}}\\ \multicolumn{1}{c}{h_2\left(x\right)=-\frac{P_3{h}_1}{P_2}(c_1\mathrm{A}\left(e\right)+c_2)+\frac{P_1}{P_2}}\\ \multicolumn{1}{c}{h_3\ge 2\Omega {\sigma }_2(1+{\mu }_2)}\end{array}}\end{array}$$

(18)

with positive matrix

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{P}=\left[\begin{array}{cc}{P}_1\hfill & P_3\hfill \\ \multicolumn{1}{c}{P_3}& P_2\hfill \end{array}\right],P_1>0,P_3<0,P_1{P}_2<P_3^2,\Omega \in R^{+},}\\ c_2>2\Omega c_1(1+{\mu }_1)>0.\hfill \end{array}$$

Proof: We can rewrite the (16) as

$$\{\begin{array}{c}\stackrel{·}{s}=-(c_1\mathrm{A}\left(e\right)+c_2)*(h_1\left(x\right){\varphi }_1\left(s\right)+q_1\left(t\right))+z\hfill \\ \multicolumn{1}{c}{\stackrel{·}{z}=-h_2\left(x\right){\varphi }_2\left(s\right)+{\stackrel{·}{q}}_2(x_2,t).}\end{array}$$

(19)

Define a strict Lyapunov function with a quadratic form

$$V\left(z\right)=Z^T\mathrm{PZ}.$$

(20)

as $Z^T=\left[\begin{array}{cc}{\varphi }_1\left(s\right)\hfill & z\hfill \end{array}\right]$ , we can see:

$$\stackrel{·}{Z}=\left[\begin{array}{c}{\varphi }_1\prime \left(s\right)\stackrel{·}{s}\hfill \\ \multicolumn{1}{c}{\stackrel{·}{z}}\end{array}\right]={\varphi }_1\prime \left(s\right)\Gamma Z,$$

(21)

where $\Gamma =\left[\begin{array}{cc}-(c_{1}A\left(e\right)+c_2)h_1-\frac{q_1(c_{1}A\left(e\right)+c_2)}{{\varphi }_1\left(s\right)}\hfill & 1\hfill \\ \multicolumn{1}{c}{-h_2+\frac{{\stackrel{·}{q}}_2}{{\varphi }_2\left(s\right)}}& 0\hfill \end{array}\right]$ , $h_1\left(x\right)$ , $h_2\left(x\right)$ , $q_1\left(t\right)$ , $q_2(x_2,t)$ abbreviated to $h_1$ , $h_2$ , $q_1$ , $q_2$ respectively.

By substituting (21) into the derivative of the Lyapunov function, we have

$$\begin{array}{c}{\stackrel{·}{V}}_z={\varphi }_1\prime \left(s\right)Z^T({\Gamma }^{T}P+P\Gamma )Z=-{\varphi }_1\prime \left(s\right)Z^T\mathrm{QZ}\hfill \\ =-{\varphi }_1\prime \left(s\right)Z^T\left[\begin{array}{cc}{Q}_{11}\hfill & Q_{12}\hfill \\ \multicolumn{1}{c}{Q_{21}}& Q_{22}\hfill \end{array}\right]Z,\hfill \end{array}$$

(22)

where,

$$\begin{array}{c}Q=\hfill \\ \left[\begin{array}{cc}2P_1(c_{1}A\left(e\right)+c_2)(h_1+\frac{q_1}{{\varphi }_1\left(s\right)})+2P_3(h_2-\frac{{\stackrel{·}{q}}_2}{{\varphi }_2\left(s\right)})\hfill & *\hfill \\ \multicolumn{1}{c}{P_3(c_{1}A\left(e\right)+c_2)(h_1+\frac{q_1}{{\varphi }_1\left(s\right)})+P_2(h_2-\frac{{\stackrel{·}{q}}_2}{{\varphi }_2\left(s\right)})-P_1}& -2P_3\hfill \end{array}\right]\hfill \end{array}$$

and $Q$ is a symmetric matrix.

Substituting (18) into $Q$ to obtain

$$\begin{array}{c}{Q}_{11}=\frac{(P_3{\lambda }_1(c_1\mathrm{A}\left(e\right)+c_2)-P_2{\lambda }_2{)}^2}{-2P_3}+2P_1(c_1\mathrm{A}\left(e\right)+c_2)\hfill \\ ({\lambda }_1+\frac{q_1}{{\varphi }_1})-2P_3({\lambda }_2+\frac{{\stackrel{·}{q}}_2}{{\varphi }_2}),\hfill \end{array}$$

(23)

From Assumption 1 and Assumption 2, we rewrite equation (10) as follows,

$$\begin{array}{c}\frac{\left|q_1\right|}{\left|{\varphi }_1\left(s\right)\right|}\le M_1+\frac{M_2}{\left|{\varphi }_1\left(s\right)\right|}={\lambda }_1\hfill \\ \frac{\left|{\stackrel{·}{q}}_2\right|}{\left|{\varphi }_2\left(s\right)\right|}\le M_3+\frac{M_4}{\left|{\varphi }_2\left(s\right)\right|}\left|x_2\right|={\lambda }_2,\hfill \end{array}$$

(24)

From Lemma 1 if $c_2>2{\sigma }_1{c}_1(1+{\mu }_1)>0$ , we attain $Q_{11}>0$ .

And the symmetric matrix $Q$ gives

$$\begin{array}{c}Q=(P_3{\lambda }_1(c_1\mathrm{A}\left(e\right)+c_2)-P_2{\lambda }_2{)}^2\hfill \\ -{(\frac{q_1}{{\varphi }_1\left(s\right)}{P}_3(c_{1}A\left(e\right)+c_2)-P_2\frac{{\stackrel{·}{q}}_2}{{\varphi }_2\left(s\right)})}^2\hfill \\ -4P_1{P}_3(c_{1}A\left(e\right)+c_2)({\lambda }_1+\frac{q_1}{{\varphi }_1\left(s\right)})\hfill \\ +4P_3^2({\lambda }_2\left(x\right)+\frac{{\stackrel{·}{q}}_2}{{\varphi }_2\left(s\right)}).\hfill \end{array}$$

(25)

so $\mathrm{Q}>0$ , $Q$ is a positive definite.

To further analyze $\stackrel{·}{V}\left(z\right)$ . Next, confirm the characteristics of ${\varphi }_1\prime \left(s\right)$ is a positive definite matrix. Variable exponential function ${\left|s\right|}^{\frac{{\sigma }_2{s}^2}{1+{\mu }_2{s}^2}}$ , with exponent above 1, the parabola’s central symmetry over the first and third quadrant. If define the lowest point as $S_A$ at the first quadrant of the curve. The function monotonically decreases in $0\le S\le S_A$ and monotonically increases in $S\ge S_A$ , so as the third quadrant. Therefore, we separate $s$ into two periods

• (1)  $\left|\mathrm{s}\right|\ge {\mathrm{S}}_{\mathrm{A}}$

According to the power function curve, it is easy to see $\psi \left(\mathrm{s}\right)\ge 0$ due to monotonically increasing of the original function, so that ${\varphi }_1\prime \left(s\right)=\psi \left(s\right)+h_3\ge h_3>0$ .

• (2)  $\left|\mathrm{s}\right|<{\mathrm{S}}_{\mathrm{A}}<1$

In this range $0\le {\left|s\right|}^{\frac{{\sigma }_2{s}^2}{1+{\mu }_2{s}^2}}\le 1,0\le \frac{{\sigma }_2}{{(1+{\mu }_2{s}^2)}^2}{\left|s\right|}^{\frac{{\sigma }_2{s}^2}{1+{\mu }_2{s}^2}}\le {\sigma }_2$ ;, and based on Lemma 1, $|A(s)|\le 2(1+{\mu }_2)$ ; since $s$ is continuously bounded, assume $\left|\stackrel{·}{s}\right|\le \Omega ,\Omega \in R^{+}$ . According to (18), ${\varphi }_1\prime \left(s\right)>0$ .

In summary, if all the coefficients satisfy Theorem 1, the $\stackrel{·}{V}<0$ at any time, so $Z^T=\left[\begin{array}{cc}{\varphi }_1\left(s\right)\hfill & z\hfill \end{array}\right]$ convergence to 0, as a consequence of the track error $e$ and differential $\stackrel{·}{e}$ converge to 0, the state vector tracks the desired vector, and the closed-loop is global stability.

Theorem 2: Based on Theorem 1, the second-order closed-loop system (8) is globally fixed-time stability and the settling time satisfies the following inequality:

$$\begin{array}{c}\mathrm{T}\left(x_0\right)\le \frac{1}{c_2(\eta -1)}\ln (1+\frac{c_2}{c_1-D_1^{+}})\hfill \\ \multicolumn{1}{c}{+\frac{1}{c_2}\ln (1+\frac{c_2}{c_1{E}^{-\frac{{\sigma }_1}{2E}}-D_1^{+}})-\frac{2}{{\varpi }_2}\ln |1-\frac{{\varpi }_2{P}_1}{{\varpi }_1}|,}\end{array}$$

(26)

with ${\varpi }_1=2\Omega {\sigma }_2(1+{\mu }_2)\frac{{\lambda }_{\max }\left\{Q\right\}}{{\lambda }_{\min }^{\frac{3}{2}}\left\{P\right\}},{\varpi }_2=\frac{{\lambda }_{\min }\left\{Q\right\}}{{\lambda }_{\max }\left\{P\right\}}{h}_3$ .

Proof: The SMC uses an appropriate controller to ensure the dynamical system constraint on the sliding surface, and then the system trajectories slide toward the equilibrium point. Therefore, the settling time for two cases is considered separately.

First, consider the system state of sliding motion on the sliding surface, that is,. $s=0$ , move from the initial state $x_0$ to the zero in a fixed-time $T\left(e\right)$ .

From (9), $s=0$ yields

$$\stackrel{·}{e}=-c_1{\left[e\right]}^{\frac{{\sigma }_1{e}^2}{1+{\mu }_1{e}^2}}-c_{2}e+d_1.$$

(27)

Choose the following Lyapunov function candidate:

$$V(e)=e^2$$

(28)

In according with (7), the derivative of (28) is

$$\stackrel{·}{V}\left(e\right)=2e\stackrel{·}{e}=-2c_1{\left[e\right]}^{\frac{{\sigma }_1{e}^2}{1+{\mu }_1{e}^2}+1}-2c_2{e}^2+2d_{1}e.$$

(29)

• 1.  $V\left(e\right)\le 1$

Consider the case thus $1+{\mu }_1{e}^2\ge 1,\frac{{\sigma }_1{e}^2}{1+{\mu }_1{e}^2}\le {\sigma }_1{e}^2,\min \left({\left|e\right|}^{\frac{{\sigma }_1{e}^2}{1+{\mu }_1{e}^2}}\right)\ge \min \left({\left|e\right|}^{{\sigma }_1{e}^2}\right)=E^{-\frac{{\sigma }_1}{2E}}$ . ³² We rewrite (29)

$$\begin{array}{c}\stackrel{·}{V}\left(e\right)=-2c_1{\left|e\right|}^{\frac{{\sigma }_1{e}^2}{1+{\mu }_1{e}^2}}\left|e\right|-2c_2{e}^2+2d_{1}e\hfill \\ \le -2(c_1{E}^{-\frac{{\sigma }_1}{2E}}-D_1^{+})\left|e\right|-2c_2{e}^2\hfill \\ \stackrel{·}{V}\left(e\right)\le -2(c_1{E}^{-\frac{{\sigma }_1}{2E}}-D_1^{+})V^{\frac{1}{2}}\left(e\right)-2c_{2}V\left(e\right),\hfill \end{array}$$

(30)

It follows that

$$\frac{\mathrm{dt}}{\mathrm{dV}\left(e\right)}\ge \frac{1}{-2(c_1{E}^{-\frac{{\sigma }_1}{2E}}-D_1^{+})V^{\frac{1}{2}}\left(e\right)-2c_{2}V\left(e\right)}.$$

(31)

For $0\le e\le 1$ , then we have:

$$\begin{array}{c}{T}_1\left(e\right)\le \hfill \\ {\int }_{V\left(e_0\right)}^0\frac{1}{-2(c_1{E}^{-\frac{{\sigma }_1}{2E}}-D_1^{+})V^{\frac{1}{2}}\left(e\right)-2c_{2}V\left(e\right)}\mathrm{dV}\left(e\right)\hfill \\ =\frac{1}{2}{\int }_0^{V\left(e_0\right)}\frac{1}{(c_1{E}^{-\frac{{\sigma }_1}{2E}}-D_1^{+})V^{\frac{1}{2}}\left(e\right)+c_{2}V(e)}\mathrm{dV}\left(e\right)\hfill \end{array}$$

(32)

Let $\omega =V^{\frac{1}{2}}\left(e\right)$ , so $d\omega =\frac{1}{2}{V}^{-\frac{1}{2}}\left(e\right)\mathrm{dV}\left(e\right),\mathrm{dV}\left(e\right)=2V^{\frac{1}{2}}\left(e\right)d\omega$ , the (32) yield

$$\begin{array}{c}{\int }_0^1\frac{V^{\frac{1}{2}}\left(e\right)}{(c_1{E}^{-\frac{{\sigma }_1}{2E}}-D_1^{+})V^{\frac{1}{2}}\left(e\right)+c_{2}V\left(e\right)}d\omega \hfill \\ ={\int }_0^1\frac{1}{(c_1{E}^{-\frac{{\sigma }_1}{2E}}-D_1^{+})+c_2\omega }d\omega \hfill \\ =\frac{1}{c_2}\ln (1+\frac{c_2}{c_1{E}^{-\frac{{\sigma }_1}{2E}}-D_1^{+}}).\hfill \end{array}$$

(33)

• 2.  $V\left(e\right)\ge 1$

Consider the case, denote $\frac{{\sigma }_1{e}^2}{1+{\mu }_1{e}^2}=\gamma$ , as $\left|e\right|>1$ and $\frac{{\sigma }_1{e}^2}{1+{\mu }_1{e}^2}+1>\frac{{\sigma }_1}{1+{\mu }_1}+1>2$ that is, $\frac{\gamma +1}{2}>\frac{\eta +1}{2}>1,e<e^{\eta +1}$ (30) leads to:

$$\begin{array}{c}\stackrel{·}{V}\left(e\right)\le -2(c_1-D_1^{+}){\left|e\right|}^{\eta +1}-2c_2{e}^2\hfill \\ \stackrel{·}{V}\left(e\right)\le -2(c_1-D_1^{+})V^{\frac{\eta +1}{2}}\left(e\right)-2c_{2}V\left(e\right),\hfill \end{array}$$

(34)

It follows that

$$\frac{\mathrm{dt}}{\mathrm{dV}\left(e\right)}\ge \frac{1}{-2(c_1-D_1^{+})V^{\frac{\eta +1}{2}}\left(e\right)-2c_{2}V\left(e\right)}.$$

(35)

For $\left|e\right|\ge 1$ , then we have:

$$\begin{array}{c}{T}_2\left(e\right)\le \hfill \\ {\int }_{V\left(e_0\right)}^1\frac{1}{-2(c_1-D_1^{+})V^{\frac{\eta +1}{2}}\left(e\right)-2c_{2}V\left(e\right)}\mathrm{dV}\left(e\right)\hfill \\ =\frac{1}{2}{\int }_1^{+\infty }\frac{1}{(c_1-D_1^{+})V^{\frac{\eta +1}{2}}\left(e\right)+c_{2}V\left(e\right)}.\hfill \end{array}$$

(36)

Let $z=V^{\frac{1-\eta }{2}}\left(e\right)$ , so $\mathrm{dz}=\frac{1-\eta }{2}{V}^{-\frac{1+\eta }{2}}\left(e\right)\mathrm{dV}\left(e\right),\mathrm{dV}\left(e\right)=\frac{2}{1-\eta }{V}^{\frac{\eta +1}{2}}\left(e\right)\mathrm{dz}$ , (36) gives

$$\begin{array}{c}\frac{1}{1-\eta }{\int }_1^0\frac{V^{\frac{\eta +1}{2}}\left(e\right)}{(c_1-D_1^{+})V^{\frac{\eta +1}{2}}\left(e\right)+c_{2}V\left(e\right)}\mathrm{dz}\hfill \\ =\frac{1}{\eta -1}{\int }_0^1\frac{1}{(c_1-D_1^{+})+c_{2}z}\mathrm{dz}\hfill \\ =\frac{1}{c_2(\eta -1)}\ln (1+\frac{c_2}{c_1-D_1^{+}}).\hfill \end{array}$$

(37)

Combine the above two periods (33) and (37), start at $x_0$ reaches the origin in a fixed-time satisfying $T\left(x\right)$ :

$$\begin{array}{c}T\left(e\right)=T_1\left(e\right)+T_2\left(e\right)\hfill \\ \le \frac{1}{c_2(\eta -1)}\ln (1+\frac{c_2}{c_1-D_1^{+}})+\frac{1}{c_2}\ln (1+\frac{c_2}{c_1{E}^{-\frac{{\sigma }_1}{2E}}-D_1^{+}}).\hfill \end{array}$$

(38)

Secondly, we consider the sliding variable state starting at the initial state $s_0$ approach to sliding surface $s=0$ in the fixed-time $T(s)$ .

The $\Vert Z{\Vert }_2^2$ is the Euclidean norm of $Z$ , we obtain

$$\begin{array}{c}{\lambda }_{\min }\left\{P\right\}\Vert Z{\Vert }_2^2\le Z^T\mathrm{PZ}\le {\lambda }_{\max }\left\{P\right\}\Vert Z{\Vert }_2^2\hfill \\ -{\lambda }_{\min }\left\{Q\right\}\Vert Z{\Vert }_2^2\ge -Z^T\mathrm{QZ}\ge -{\lambda }_{\max }\left\{Q\right\}\Vert Z{\Vert }_2^2,\hfill \end{array}$$

(39)

while

$$\left|{\varphi }_1\left(s\right)\right|\le Z_2\le \frac{V^{\frac{1}{2}}\left(e\right)}{{\lambda }_{\min }^{\frac{1}{2}}\left\{P\right\}}$$

(40)

In according with (39), (40), the derivative of (22) is

$$\stackrel{·}{V}\left(z\right)=-{\varphi }_1\prime \left(s\right)Z^T\mathrm{QZ}\le -{\varphi }_1\prime \left(s\right)\frac{{\lambda }_{\min }\left\{Q\right\}}{{\lambda }_{\max }\left\{P\right\}}V\left(z\right)$$

(41)

Using (19), then (35) yields,

$$\begin{array}{c}\stackrel{·}{V}\left(e\right)\le 2\Omega {\sigma }_2(1+{\mu }_2)\frac{{\lambda }_{\max }\left\{Q\right\}}{{\lambda }_{\min }^{\frac{3}{2}}\left\{P\right\}}{V}^{\frac{3}{2}}\left(z\right)-\frac{{\lambda }_{\min }\left\{Q\right\}}{{\lambda }_{\max }\left\{P\right\}}{h}_{3}V\left(z\right)\hfill \\ ={\varpi }_1{V}^{\frac{3}{2}}\left(z\right)-{\varpi }_{2}V\left(z\right),\hfill \end{array}$$

(42)

$$\frac{\mathrm{dt}}{\mathrm{dV}\left(z\right)}\ge \frac{1}{{\varpi }_1{V}^{\frac{3}{2}}\left(z\right)-{\varpi }_{2}V\left(z\right)},$$

(43)

$$T\left(s\right)\le {\int }_{V\left(s_0\right)}^{V\left(0\right)}\frac{1}{{\varpi }_1{V}^{\frac{3}{2}}\left(z\right)-{\varpi }_{2}V\left(z\right)}\mathrm{dV}\left(z\right).$$

(44)

The Lyapunov function is Euclidean norm $V\left(z\right)=P_1{\varphi }_1^2\left(s\right)+2P_3{\varphi }_1\left(s\right)z+P_2{z}^2$ . when the sliding variable $s=0$ , then $V_z\left(0\right)=P_1$ .

Let $\rho =V^{-\frac{1}{2}}\left(z\right),d\rho =-\frac{1}{2}{V}^{\frac{3}{2}}\left(z\right)\mathrm{dV}\left(z\right)$ , so $\mathrm{dV}\left(z\right)=-2V^{\frac{3}{2}}\left(z\right)d\rho$ , the time was calculated (44) leads to,

$$\begin{array}{c}T\left(s\right)\le -2{\int }_0^{P_1}\frac{V^{\frac{3}{2}}\left(z\right)}{{\varpi }_1{V}^{\frac{3}{2}}\left(z\right)-{\varpi }_{2}V\left(z\right)}d\rho =-2{\int }_0^{P_1}\frac{1}{{\varpi }_1-{\varpi }_2\rho }d\rho \hfill \\ =-\frac{2}{{\varpi }_2}\ln |1-\frac{{\varpi }_2{P}_1}{{\varpi }_1}|.\hfill \end{array}$$

(45)

In the closed-loop system (8) with the controller from (10) to (14), from (45) the system starts at the sliding variable initial state $s_0$ reaches the sliding surface in a fixed-time satisfying $T\left(s\right)=-\frac{2}{{\varpi }_2}\ln |1-\frac{{\varpi }_2{P}_1}{{\varpi }_1}|$ . In addition, it is deduced from (38) that the state variable started at $x_0$ move toward the origin in a fixed-time $T\left(x\right)\le \frac{1}{c_2(\eta -1)}\ln (1+\frac{c_2}{c_1-D_1^{+}})+\frac{1}{c_2}\ln (1+\frac{c_2}{c_1{E}^{-\frac{{\sigma }_1}{2E}}-D_1^{+}})$ . Finally, the total time for the closed-loop system reaches the equilibrium trajectory $(0,-d_1)$ in a fixed-time $T=T\left(x\right)+T\left(s\right)$ , which is bounded by (26).

Simulations results

Sliding surface comparison

The first to investigate the proposed sliding surface with simulations. For comparison, the simulations about the tracking error on the four sliding surfaces. According to the literature,^25,30 the formulated as follow: Proposed fixed-time terminal sliding surface (FTSS) (9), the conventional linear sliding surface (LSS) $s=\stackrel{·}{e}+c_{2}e$ , the integral sliding surface (ISS) $\mathrm{s}=\stackrel{·}{e}+c_1{\int }_0^{t}e+c_{2}e$ , the terminal sliding surface (TSS) $s=\stackrel{·}{e}+c_1{\left[e\right]}^{\frac{1}{2}}+c_{2}e$ . Without loss of generality, parameters of the sliding surface are set as $c_1=2,c_2=4.5,{\sigma }_1=2,{\mu }_1=0.1$ , and initial condition $s\left(0\right)=3$ . The control law for four cases uses the same switching control law to reduce the disturbance $u_{\mathrm{sc}}=k_1\mathrm{sign}\left(s\right)+k_2{\left|s\right|}^{\alpha }\mathrm{sign}\left(s\right)+k_3{\left|s\right|}^{\beta }\mathrm{sign}\left(s\right)+k_{4}s$ , and the parameters of these are $k_1=2,k_2=2,k_3=2,k_4=2$ . The system of the mismatched and matched disturbances are $d_1=\sin \left(5t\right),d_2=\sin \left(t\right)$ respectively at $2.5-5s$ .

From Figure 2(a), it is apparent that the tracking error on the LSS convergences to the equilibrium point at a prolonged rate. The tracking error on the ISS converges fast, but the trajectory is with a rather significant overshoot, and it takes a long time to converge. The TSS, its trajectory converges faster than ISS, and it has a much smaller overshoot. However, the above three schemes can not restrain the disturbance well. The trajectory of the sliding variable can be clearly seen in the enlarged Figure 2(b). As for our proposed sliding surface, tracking error converges the fastest, has the smallest overshoot, and is the least sensitive to disturbances.

Figure 2.

(a) State variable $x(t)$ versus times with different sliding surface and (b) vertical enlargement of the top figure.

The second is to investigate the proposed sliding surface that converges in the fixed time with simulations. In order to verify, the simulations about the tracking error on the proposed fixed-time terminal sliding surface (FTSS) (9) with different initial conditions. Consider $s(0)=0.5,1,3,5,10$ five cases. Choose the same parameters as the above FTSS. According to (38), the state variable $s$ reaches the origin in a fixed time $T\le 0.8961s$ . From Figure 3 it can be seen that when the initial state is far away from $x(0)=3$ , the three convergence curves $\mathrm{x}(0)=3,5,10$ basically coincide, and the state can quickly converge to near the sliding surface at the first few moments of control input joined. When the state is far from the origin, the variable exponent coefficient function of FTSS plays a major role and can quickly converge. The linear function plays a main role in the motion process of the state variable approaching the origin. It can be seen that FTSS can converge in a fixed time, which is consistent with the maximum convergence time deduced above.

Figure 3.

(a) State variable x(t) versus times with different initial condition and (b) enlargement of the top figure.

Control law comparison

This part to investigate the proposed control law with simulations. For comparison, select a sliding surface (9), with $c_1=1,c_2=3,{\sigma }_1=2,{\mu }_1=0.1$ , and three different switching controllers were used for the (10) and tested in simulation. In 1–2 s of the system, the mismatched and matched disturbances are $d_1=\sin \left(5t\right),d_2=\sin \left(t\right)$ . A proposed FXSTSMC given in (11)–(13), a linear correction term, with constants $h_3=8,{\sigma }_2=4,{\mu }_1=1$ , and variable gains $h_1,h_2,h_3$ defined by (18), with the parameters ${\sigma }_2=4,{\mu }_2=1,a=4,P_1=5,P_2=-1$ ; A stand super-twisting sliding mode control (STSMC) $u_{\mathrm{sc}}=k_1({\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)+k_{3}s)+{\int }_0^t{k}_2(\frac{1}{2}\mathrm{sign}\left(s\right)+\frac{3}{2}{k}_3{\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)+k_3^{2}s)\mathrm{dt}$ , the control with constant gains $k_1=10,k_2=10,k_3=8$ ; A first-order sliding mode control (FOSMC) $u_{\mathrm{sc}}=k_1\mathrm{sign}\left(s\right)+k_2{\left|s\right|}^{\alpha }\mathrm{sign}\left(s\right)+k_3{\left|s\right|}^{\beta }\mathrm{sign}\left(s\right)+k_{4}s$ , The control with fixed gains $k_1=20,k_2=20,k_3=20,k_4=20$ .

The quantification of the control input is chattering suppression and system performance. As can be seen from Figure 4 since the sliding mode variables are the same, the moving time from the initial state to the origin of the sliding mode variables is similar, and they are not sensitive to the input of the control law. Figure 5 shows three different algorithms of control input. From Figure 5(a) it can be seen as a whole that although the second-order sliding mode input can not completely eliminate chattering, but the amplitude and frequency are improved. The Figure 5(b) reflects the process of the state variable on the sliding mode surface without disturbance. At 0.2–0.3 s, the first-order sliding mode control input has obvious chattering, and the amplitude is large. The Figure 5(c) reflects the effect of adding the disturbance on the control input. It can be seen that the method proposed in this paper is better than the other two methods in amplitude and frequency. Therefore, it can be proved that the second-order sliding mode is more effective to improve chattering than the first-order sliding mode control input and constant gain super-twisting.

Figure 4.

Position tracking error versus times with different switching control law.

Figure 5.

(a) Control input u(t) versus times with different switching control law, (b) between 0.2s and 0.3s the control input u(t) with different switching control law and (c) between 1.4s and 1.5s the control input u(t) with different switching control law.

Controller comparison

In this section, the numerical simulation results verify the effectiveness of the proposed scheme for tracking position and speed in a second-order system (8) with matched and mismatched perturbations. In addition, the convergence time and the chattering effect of the closed-loop system (8) are analyzed. To make the results more convincing, we compared three other schemes recently proposed by scholars to show the advantage of our method. The four controllers are listed as follows:

• Case 1: fixed-time super twisting sliding mode control (FXSTSMC)

  sliding surface: $s=\stackrel{·}{e}+c_1{\left[e\right]}^{\frac{{\sigma }_1{e}^2}{1+{\mu }_1{e}^2}}+c_{2}e$ ,

  controller: $u\left(x\right)=-\frac{1}{g\left(x\right)}(u_{\mathrm{eq}}+u_{\mathrm{rl}})$ ,

$$\begin{array}{c}{u}_{\mathrm{eq}}=f\left(x\right)+(c_1\mathrm{A}\left(e\right)+c_2)*(x_2-{\stackrel{·}{x}}_d)-{\stackrel{··}{x}}_d,\\ u_{\mathrm{sc}}=(c_1\mathrm{A}\left(e\right)+c_2)h_1(t,x){\varphi }_1\left(s\right)+{\int }_{0h_2}(t,x){\varphi }_2\left(s\right)\mathrm{dt},\\ {\varphi }_1\left(s\right)={\left[s\right]}^{\frac{{\sigma }_2{s}^2}{1+{\mu }_2{s}^2}}+h_{3}s,{\varphi }_2\left(s\right)={\varphi }_1\prime \left(s\right){\varphi }_1\left(s\right),\end{array}$$

 the parameters choose: ${\sigma }_1=1.2,{\mu }_1=0.1,{\sigma }_2=2,{\mu }_2=0.5,c_1=3,c_2=8,h_3=6$ , consider the positive define matrix $P=\left[\begin{array}{cc}{P}_1\hfill & P_3\hfill \\ \multicolumn{1}{c}{P_3}& P_2\hfill \end{array}\right]=\left[\begin{array}{cc}a+4b^2\hfill & -2b\hfill \\ \multicolumn{1}{c}{-2b}& 1\hfill \end{array}\right],a=6,b=0.5$ ;

• Case 2 ¹⁴ : Theorem 2, fixed-time sliding mode control (FXSMC)

  sliding surface: $s=\stackrel{·}{e}+c_1{\left[e\right]}^{\frac{{\lambda }_1{e}^2}{1+{\mu }_1{e}^2}}$ .

  controller: $u\left(x\right)=-\frac{1}{g\left(x\right)}(u_{\mathrm{eq}}+u_{\mathrm{rl}})$ ,

  $\begin{array}{c}\multicolumn{1}{c}{u_{\mathrm{eq}}=f\left(x\right)+c_{1}A\left(e\right)\stackrel{·}{e}-{\stackrel{··}{x}}_d,u_{\mathrm{sc}}=k_1{\left[s\right]}^{\frac{{\lambda }_2{s}^2}{1+{\mu }_2{s}^2}}}\\ +c_{1}A\left(e\right)k_2\mathrm{sign}\left(s\right),\hfill \end{array}$

  The parameters: $c_1=5.5,k_1=30,k_2=2,{\mu }_1=1,{\lambda }_1=3,{\mu }_2=1,{\lambda }_2=4$ ;

• Case 3 ³³ : Theorem 1, linear sliding mode control (LSMC)

 sliding surface: $s=\stackrel{·}{e}+c_{1}e$ .

 controller: $u\left(x\right)=-\frac{1}{g\left(x\right)}(u_{\mathrm{eq}}+u_{\mathrm{rl}})$ ,

  $u_{\mathrm{eq}}=f\left(x\right)+c_1\stackrel{·}{e}-{\stackrel{··}{x}}_d,u_{\mathrm{sc}}=k_1{\left[s\right]}^{\beta }+k_2{\left[s\right]}^{\alpha }+k_{3}s+k_4\mathrm{sign}\left(s\right),$

The parameters: $k_1=3000,k_2=3000,k_3=3000,k_4=3000,\alpha =1,\beta =1.5$ ;

• Case 4 ³⁴ : backstepping sliding mode control (BSMC)

 sliding surface: $s=\stackrel{·}{{\hat{z}}_2}+k_1{\hat{z}}_1$ .

 controller: $u\left(x\right)=-\frac{1}{g\left(x\right)}(k_1\left(z_2-c_1{z}_1\right)+\overline{F}sign\left(s\right)-{\stackrel{..}{x}}_d+c_1{\stackrel{.}{z}}_1$ $+h(s+k_2\mathrm{sign}\left(s\right)))$

The parameters: $c_1=100,k_1=150,k_2=50,h=10,\overline{F}=50$ .

In this simulation, taking $f\left(x\right)=x_1,g\left(x\right)=1$ in the system (2), set the desired tracking signal as $x_d=\sin \left(t\right)$ . The disturbances are imposed according to the following order: after 5 s, the mismatched disturbance $d_1\left(t\right)=\sin \left(5t\right)$ , and the matched disturbance $d_2\left(t\right)=\sin \left(10t\right)$ are added into the system. The initial states are $x\left(0\right)={[1,0]}^T$ .

The simulation results for FXSTSMC and the other three different schemes are shown in Figures 6–8. Figure 6 illustrates the response of state variable position tracking and velocity tracking. Except for LSMC, other methods tracked the reference position signal in a short time. The speed tracking FXSTSMC method performed much better than the other three. According to Theorem 2, the settling time (26) is computed as $T\le 2.6472s$ . Seen from Figure 7, the settling time is less than 0.2 s in the simulation. The convergence error of the four schemes is shown in Figure 7; the tracking errors of all scheme systems are goes to 0 without steady-state errors. However, a subgraph (a) from a vertical zoom at ${10}^4$ times shows that the position tracking error convergence of LSMC and BSMC is unsatisfactory. The LSMC converges asymptotically, while the BSMC cannot suppress disturbances when the position tracking errors change if the disturbance changes suddenly. Zooming in again on the time region from time 8.45 to 8.5 s, the subgraph (b) can see that the FXSTSMC and FXSMC schemes maintain position tracking accuracy, the position tracking error of the LSMC method did not converge to the origin, and the BSMC strategy vibrated around the equilibrium region. Subgraph (c) shows the speed tracking error within the time 8.45 to 8.5 s, higher precision is achieved for the FXSTSMC and LSMC speed tracking errors compared with the other two. To summarize, the current method outperforms the comparison methods in terms of convergence time, control precision, and anti-disturbance, as evidenced by the aforementioned results.

Figure 6.

position tracking and speed tracking response.

Figure 7.

Position tracking error and speed tracking error: (a) position tracking error is magnified ${10}^4$ times in the vertical direction, (b) position tracking error image enlarged from 8.45 to 8.5 s, and (c) speed tracking error image enlarged from 8.45 to 8.5 s.

Figure 8.

(a) Control input $u(t)$ versus time(s) and (b) control input image enlarged from 8 to 8.01 s.

Remark 2: The influence of four schemes on convergence performance is mainly caused by the different sliding mode surfaces. FXSTSMC and FXSMC both used sliding surfaces with variable exponent coefficients that not only tracked the reference signal in fixed time but also insensitive to matched and mismatched disturbances. The LSMC has a linear sliding surface, which makes the system converge asymptotically, and the speed is slow and cannot change rapidly with disturbance. The BSMC can speed up the convergence time by adjusting parameters, but it also causes the problem of low control precision and susceptibility to disturbance.

Compared with the four schemes, the control input and zoom-in figures of the four schemes can be seen in Figure 8, the control input magnitude of the proposed FXSTMC method is half that of the FXSMC method and one-fifth that of the LSMC. Although all four schemes inherit the chattering characteristics of the sliding mode control, the FXSTSMC scheme has less frequency of control input. The LSMC maintains a very high switching frequency from 8 to 8.01 s, and the frequency of the FXSMC is approximately more than two times that of the FXSTSMC. The result shows that FXSTSMC has better inhibition of the chattering effect.

Remark 3: The proposed method, the variable-gain super-twisting switching control law with a variable exponential function of the control gain is used, so that the gain of the control law could change rapidly with interference. In addition, the control rate inherits the characteristics of second-order function filtering, and the new control input is smoother.

Based on the above four control schemes of simulation, we find that the proposed scheme in this paper has the absolute advantages of a shorter settling-time, higher control precision, a better convergence effect, and smaller chattering when matched and mismatched disturbances occur. The system state converges to the origin in a fixed-time, which is impossible for linear sliding mode and traditional terminal sliding mode.

The FXSTSMC in simulations with practical systems

To verify the high performance of the proposed controller scheme, the FXSTSMC algorithm is applied to the simulation of two practical systems.

Example 1, the sliding surface, controller and results of Theorems 1 and 2 have been tested in simulation on a double integrator, that is, $f=0,g=1$ in the system. The following disturbances have been chosen $d_1=\sin \left(5t\right),d_2=2\sin \left(10t\right)$ so $D_1^{+}=1,D_2^{+}=2$ , The ode23e solver with a variable step of 10e−3 has been used. The parameters of the controller are as follows: $c_1=1,c_2=6,{\sigma }_1=3,{\mu }_1=1,{\sigma }_2=4,{\mu }_2=1$ . This set of parameters verifies the condition. The results of the simulations with the initial condition $x\left(0\right)={[1,3]}^T$ are played in Figure 9. it shows that after a reaching phase, the state $x_1$ converges to 0, and the dynamics of $x_2$ track the unknown mismatched disturbances $-d_1$ . For comparison purpose, one has implemented the fixed-time controller given in Moulay. ¹⁴ One can see that this settling-time is few smaller in the simulation. both the controllers are able to manage the mismatched disturbances d₁, However, regardless of $x_1$ or $x_2$ from the point of view of following accuracy, FXSTMC is much worse than FXSTSMC, and the jitter is strong, especially $x_2$ .

Figure 9.

Double integrator with FXSTSMC and FXSMC.

Example 2. the sliding surface, controller and results of Theorems 1 and 2 have been tested in simulation on the single inverted pendulum (SIP) used by Zuo et al. ³² as a benchmark. Consider measuring error add mismatched disturbances, a bounded perturbation of the form $d_1=\sin \left(5t\right),d_2=\sin \left(10x_1\right)+\cos \left(x_2\right)$ has been supposed to affect the system in the presented simulations. The control input (10)–(14), guaranteeing fixed time stability, has been tested by simulation, with initial conditions ${\mathrm{x}}_1\left(0\right)=1,{\mathrm{x}}_2\left(0\right)=\frac{\pi }{2}$ . With the design choices $c_1=1,c_2=6,{\sigma }_1=3,{\mu }_1=1,{\sigma }_2=4,{\mu }_1=1$ . The parameters of the matrix P are the same as in the previous example. The variable gains are chosen with (18). From the (26) bound the settling time is $T\le 6.002$ . Figures 10 and 11 displays the position, speed tracking and their error results obtained by the controller. The simulations show that the real convergence times are by far lower than the estimated bound because of the initial state is not far from the origin.

Figure 10.

Position tracking and speed tracking of the SIP.

Figure 11.

Position tracking error and speed tracking error of the SIP.

Conclusion

In this study, a fixed-time stability super-twisting sliding mode controller is proposed for a second-order system presence under unknown matched and mismatched perturbations. To design controller with robustness property and converge in a fixed-time, robust fixed-time stability sliding surfaces are obtained by using state variable variable exponential coefficients, and the variable gain super-twisting switching law is obtained by using a sliding variable variable exponential function. The stability analysis of the controller and the calculation of the upper bound of convergence time are all derived by Lyapunov method. The simulation results verify the superiority and feasibility of the FXSTSMC method, which achieved better control performance than the existing FXSMC, LSMC, and BSMC schemes in terms of convergence time, control precision, amplitude, and frequency of the control input. The proposed method is used for a fixed-time differentiator and a high precision fixed-time stability control. In future work, it is necessary to practice the controller in application.