Novel robust approach–based dynamic global terminal synergetic tracking control for uncertain second-order systems: Wind turbine system and power converter applications

Abstract

This article provides a systematic approach for synthesizing a novel robust synergetic control mechanism. This controller employs a dynamic global terminal synergetic surface with additional sinusoidal function. The overall macrovariable gives the system a rapid approaching rate, maintains the finite time stability, and provides a chattering-free control law. The stability analysis is validated for any uncertain second-order system via indirect approach of Lyapunov theorems. Furthermore, a hyperbolic tangent nonlinear disturbance observer is designed in order to estimate any lumped disturbance. The feasibility and the effectiveness of the proposed composite control law is tested and evaluated through simulations for simple model, wind turbine system and DC-DC buck converter. Finally, the simulation results show that the presented dynamic global terminal synergetic control–based hyperbolic tangent nonlinear disturbance observer can effectively reject the disturbances, enhance the robustness, and guarantee the tracking accuracy and rapidity.

Introduction

It is widely acknowledged that the vast majority of dynamic systems exhibit nonlinear behavior. Nevertheless, the challenge of controlling such systems with uncertainties remains a crucial research area in the field of control engineering. In recent years, scientific communities have designed a number of unconventional control algorithms, including sliding mode control (SMC),^1–5 fuzzy logic type I and type II controllers,^6,7 model predictive control,^8–10 model independent control,^11,12 and back-stepping control. ¹³ All these approaches have found use in diverse applications; for instance, in Cuvillon et al., ⁸ an offset-free nonlinear model predictive control (NMPC) was designed to mitigate platform vibration and reduce the tracking error along complex trajectories of cable-driven parallel robots. The experimental results show that the proposed offset-free NMPC yields lower tracking errors than the nominal NMPC. The work presented in Tahamipour et al. ¹⁴ studies a three prismatic-series-prismatic parallel robot, in which the authors propose a generalized hyperbolic fuzzy type II control. The study established that the mentioned synergistic framework of fuzzy type II offers superior handling of unknowns and entails a simpler computational complexity compared to other fuzzy systems.

Structure variable control (SVC) has emerged as the most widely employed and successful control approach among the several methods presented for highly nonlinear systems, cascaded system, and hybrid ¹⁵ due to its robustness and simplicity. ¹⁶ However, SVCs are vulnerable to low convergence speed rate and serious chattering issues resulting from excessive switching control gain. ¹⁷ In literature, several sliding mode controllers with new structure of surfaces and reaching laws have been discussed for addressing the aforementioned drawbacks and ensuring the finite time stability and enhancing the performance rapidity. In Wang et al., ¹⁸ Mozayan et al. ¹⁹ suggested improved reaching law based on exponential variations for controlling permanent magnet synchronous motor (PMSM) drive and permanent magnet synchronous generator (PMSG) wind turbine system, respectively. In Pan et al., ²⁰ Halledj and Bouafassa ²¹ developed novel reaching laws with dynamic power terms in order to force the system states through the sliding surface as quick as possible, especially when the states are close to the attractor region. Although these mechanisms significantly enhance the convergence speed, the chattering effects still persist. On the other side, various types of SMCs with nonlinear surfaces have been studied^22,23 to guarantee the tracking errors of systems. The terminal sliding mode scheme is commonly used for this purpose, but it suffers from slow convergence speed and singularity issues. Many scholars have tried to address the aforementioned problems. In Guo et al., ²⁴ the singularity issue has been successfully avoided via nonsingular terminal SMC in hydraulic servo systems. Pan and Zhang ²⁵ designed a nonsingular fast terminal SMC (FTSMC) for controlling two link rigid robot manipulator. Moreover, Barambones et al. ²⁶ proposed a high-order SMC based on super-twisting algorithm for controlling the wind turbine speed and improving the maximum power extraction.

Despite the significant efforts made by scientific scholars to solve the issues in SVC approach, the chattering problem persists due to the discontinuous term in controller, leading to high oscillation in control law and consequently deteriorating the system performance. Numerous approaches have been proposed in literature to reduce the control law oscillation. Babes et al. ²⁷ developed adaptive switching gain based on fuzzy universal approximation intending to find the optimal switching gain for reducing the chattering. However, reducing chattering increases the system’s sensitivity to uncertainty. ²⁸

Based on the preceding analysis, it has been determined that synergetic theories offers the best solution for eliminating the chattering phenomenon.^29,30 Synergetic control (SC) is a dependent model approach originating from control science that depends on self-organization theory and the usage of nonlinear characters of system dynamics; it is similar to SMC and retains all the advantages of SMC such as robustness against the unknowns, simple in practical implementation, and simple stability analysis guarantees. ³¹ Since SC runs at fixed constant frequency, this controller becomes more efficient in suppressing the chattering than SVCs. This criterion gives a significant path to the researchers in order to apply it in various applications.^32–34 In classical models, synergetic controllers have suffered from infinite time convergence which may result in undesirable performances. For possible enhancement, a nonlinear macrovariable could be suggested in order to increase the speed convergence like terminal SC and nonsingular terminal SC. Hagh et al. ³⁵ developed a new robust proportional integral as attractor region, combined with nonsingular terminal synergetic as macrovariable for nth order nonlinear systems. This study solved the problem of infinite time convergence and enhanced the robustness against uncertainties. However, it should be noted that this controller requires a complex computation, and it would fail at some rapid time-variant disturbances. In a similar study, Asl et al. ³⁶ applied fuzzy type II to tune the parameters of nonsingular synergetic controller, providing a critical solution when system parameters vary over time. Moreover, Rajendran et al. ³⁷ proposed a terminal integral synergetic controller for wind turbine at region II using two mass model with speed estimator in order to improve the maximum power point taking (MPPT) of wind systems, reduce the control input, and drive train oscillations.

In disturbance point of view, a compensation of external disturbances is one of the major aspects in control theory. In literature, robust controllers have been proposed in order to overcome mismatched disturbance such as $H_2$ and $H_{\infty }$ ³⁸ and linear matrix inequality (LMI) approaches. ³⁹ Unfortunately, numerous mismatched disturbances with fast random values are hard to estimate using classical robust regulators. Recently, nonlinear anti-disturbance control laws have gained more attention because they can handle many disturbance types. For that reason, they used different powerful nonlinear observers such as extended state observer, nonlinear disturbance observer, fractional adaptive observers, and so on. These structures are added in feedforward path of the controllers in order to estimate those disturbances. In Wang et al. ¹⁸ and Halledj and Bouafassa, ⁴⁰ the authors used an extended hyperbolic tangent state observer for estimating both the measured speed and load disturbances of PMSM drive. Halledj and Bouafassa ²¹ and Wang et al. ⁴¹ designed an extended state observer for predicting the variation of mismatched disturbances in DC-DC buck converter. Almost all the work in literature, combine SMC with different types of disturbances, which will not completely handle with chattering issues.

Motivated by the above analysis, this article is devoted to suggest a novel dynamic robust controller based on SC for second-order uncertain systems to suppress the chattering, ensure finite time stability, and increase speed convergence rate. Moreover, it can be used to estimate very fast disturbances and solve the problems faced in previous controller described in literature. ³⁵

The main contributions of this article are summarized in threefold as:

• A novel nonlinear global terminal SC with dynamic state variant coefficients is developed, which can guarantee global finite time stability and enhance both transient and steady state response.

• A hyperbolic tangent nonlinear disturbance observer (HTNDO) is designed for predicting any time-variant lumped disturbance even though that has a fast time response.

• A novel composite control scheme is suggested by integrating a novel dynamic global terminal synergetic control (DGTSC) and HTNDO in order to form a new anti-disturbance control based on nonlinear dynamic synergetic controller rather than SMC.

The rest of the article is organized as follows: the composite control DGTSC-HTNDO scheme is detailed in the section “Controller design and stability analysis.” The “Results and discussion” section highlights the dynamic response of the suggested control algorithm when applied to second-order system, that is, a simple model, wind turbine system, and DC-DC buck converter. Finally, the conclusion of this article is given in the final section.

Controller design and stability analysis

As mentioned earlier in the “Introduction” section, one of the key drawback of the SMCs is the chattering phenomena, and in response of this issue, SCs have been proposed. In this section, a novel dynamic SC with nonlinear disturbance observer for uncertain second-order system is introduced. This structure ensures chattering-free, finite time convergence speed, fast response, and robustness against any bounded lumped disturbances.

Dynamic global SC

Considering uncertain nonlinear second-order system expressed as

$$\{\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{x}}_1=x_2}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2=f\left(x\right)+b\left(x\right)u+d(x,t)}\end{array}$$

(1)

where $x=[\begin{array}{cc}{x}_1\hfill & x_2\hfill \end{array}{]}^{\mathrm{T}}$ , $f(x)$ , $b(x)$ , u, and $d(x,t)$ present state vector, nonlinear system dynamics function, nonlinear control function, control signal, and lumped disturbance, respectively. The aim is to develop a control law based on synergetic, so that the system states may track the desired trajectory in the presence of lumped disturbance.

The error dynamic of equation (1) can be expressed as follows

$$\begin{array}{c}{e}_1=x_1-x_{1\mathrm{ref}}\hfill \\ {\stackrel{·}{e}}_1=e_2=x_2-{\stackrel{·}{x}}_{1\mathrm{ref}}\hfill \\ {\stackrel{··}{e}}_1={\stackrel{·}{e}}_2=f\left(x\right)+b\left(x\right)u+d(x,t)-{\stackrel{··}{x}}_{1\mathrm{ref}}\hfill \end{array}$$

(2)

Now, the proposed dynamic macrovariable ${\mathrm{\Psi }}_1(e_1,e_2)$ is presented as

$${\mathrm{\Psi }}_1(e_1,e_2)=e_2+\alpha \left(e\right).e_1+\beta \left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)+c\left(e\right).\sin \left(e_1\right)$$

(3)

where $\alpha (e)>0$ , $\beta (e)>0$ , and $c(e)>0$ present the dynamic coefficients of ${\mathrm{\Psi }}_1(e_1,e_2)$ and $0<\sigma <1$ .

In order to design an SC that drives the error dynamic to macrovariable, a manifold is required and it can written as

$$M_1\left({\mathrm{\Psi }}_1\right)=\lambda {\stackrel{·}{\mathrm{\Psi }}}_1(e_1,e_2)+{\mathrm{\Psi }}_1(e_1,e_2)=0$$

(4)

where $\lambda >0$ that enables the developer to select the convergence rate. The solution of equation (4) is just a decayed exponential function with a speed determined by $\lambda$ . For further enhancement of the controller and enabling a finite time convergence, the proposed manifold is expressed in equation (5) as follows

$$M_2\left({\mathrm{\Psi }}_1\right)=\lambda ({\stackrel{·}{\mathrm{\Psi }}}_1(e_1,e_2){)}^{\frac{p}{q}}+{\mathrm{\Psi }}_1(e_1,e_2)=0$$

(5)

where $p/q$ should be satisfied $1<(p/q)<2$ .

To solve $M_2({\mathrm{\Psi }}_1)$ , we need to find out the first derivative of ${\mathrm{\Psi }}_1(e_1,e_2)$

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\mathrm{\Psi }}}_1(e_1,e_2)={\stackrel{·}{e}}_2+\stackrel{·}{\alpha }\left(e\right).e_1+\alpha \left(e\right).e_2}\\ \multicolumn{1}{c}{+\stackrel{·}{\beta }\left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)+\sigma \beta \left(e\right)e_2|e_1{|}^{\sigma -1}}\\ \multicolumn{1}{c}{+\stackrel{·}{c}\left(e\right)\sin \left(e_1\right)+c\left(e\right).e_2.\cos \left(e_1\right)}\end{array}$$

(6)

Substituting equation (2) in equation (6), yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\mathrm{\Psi }}}_1(e_1,e_2)=f\left(x\right)+b\left(x\right)u+d(x,t)-{\stackrel{··}{x}}_{1\mathrm{ref}}+\stackrel{·}{\alpha }\left(e\right).e_1}\\ \multicolumn{1}{c}{+\alpha \left(e\right).e_2+\stackrel{·}{\beta }\left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)+\sigma \beta \left(e\right)e_2|e_1{|}^{\sigma -1}}\\ \multicolumn{1}{c}{+\stackrel{·}{c}\left(e\right)\sin \left(e_1\right)+c\left(e\right).e_2.\cos \left(e_1\right)}\end{array}$$

(7)

By substituting equation (7) in equation (5), the mentioned manifold expression can be rewritten as described in equation (8) as

$$\begin{array}{c}\lambda [f\left(x\right)+b\left(x\right)u+d-{\stackrel{··}{x}}_{1\mathrm{ref}}+\stackrel{·}{\alpha }\left(e\right).e_1+\alpha \left(e\right).e_2\hfill \\ \multicolumn{1}{c}{+\stackrel{·}{\beta }\left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)+\sigma \beta \left(e\right)e_2|e_1{|}^{\sigma -1}+\stackrel{·}{c}\left(e\right)\sin \left(e_1\right)}\\ \multicolumn{1}{c}{+c\left(e\right).e_2.\cos \left(e_1\right){]}^{\frac{p}{q}}+{\mathrm{\Psi }}_1(e_1,e_2)=0}\end{array}$$

(8)

The control law suggested can be computed through straightforward calculations as

$$u\left(t\right)=\frac{-1}{b\left(x\right)}\left[\begin{array}{c}f\left(x\right)+d-{\stackrel{··}{x}}_{1\mathrm{ref}}+\stackrel{·}{\alpha }\left(e\right).e_1+\alpha \left(e\right).e_2+\stackrel{·}{\beta }\left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)+\hfill \\ \multicolumn{1}{c}{\sigma \beta \left(e\right)e_2|e_1{|}^{\sigma -1}+\stackrel{·}{c}\left(e\right)\sin \left(e_1\right)+c\left(e\right).e_2.\cos \left(e_1\right)-{\left(\frac{-1}{\lambda }{\mathrm{\Psi }}_1(e_1,e_2)\right)}^{\frac{q}{p}}}\end{array}\right]$$

(9)

Assumption 1. The lumped disturbance $d(x,t)$ and its first derivative $\stackrel{·}{d}(x,t)$ are bounded. $\left|d\right|<l$ , $\left|\stackrel{·}{d}\right|<m$ .

Theorem 1. The proposed dynamic SC law described in equation (9) can force the synergetic manifold described in equation (5) to converge to zero in finite-time duration.

Proof. Select the Lyapunov candidate as $V_1({\mathrm{\Psi }}_1)=(1/2){{\mathrm{\Psi }}_1}^2$ , then the first time derivative of $V_1({\mathrm{\Psi }}_1)$ is ${\stackrel{·}{V}}_1({\mathrm{\Psi }}_1)={\mathrm{\Psi }}_1{\stackrel{·}{\mathrm{\Psi }}}_1$ . Substituting equation (7) into ${\stackrel{·}{V}}_1$ gives

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1\left({\mathrm{\Psi }}_1\right)=}\\ \multicolumn{1}{c}{{\mathrm{\Psi }}_1\left[\begin{array}{c}f\left(x\right)+b\left(x\right)u+d(x,t)-{\stackrel{··}{x}}_{1\mathrm{ref}}+\stackrel{·}{\alpha }\left(e\right).e_1+\alpha \left(e\right).e_2\hfill \\ \multicolumn{1}{c}{+\stackrel{·}{\beta }\left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)+\sigma \beta \left(e\right)e_2|e_1{|}^{\sigma -1}}\\ \multicolumn{1}{c}{+\stackrel{·}{c}\left(e\right)\sin \left(e_1\right)+c\left(e\right).e_2.\cos \left(e_1\right)}\end{array}\right]}\end{array}$$

(10)

Inserting the equation (9) into equation (10), yields to

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1\left({\mathrm{\Psi }}_1\right)={\mathrm{\Psi }}_1{(-\frac{1}{\lambda }{\mathrm{\Psi }}_1)}^{\frac{q}{p}}}\\ =-{\left(\frac{1}{\lambda }\right)}^{\frac{q}{p}}{\left({\mathrm{\Psi }}_1\right)}^{1+\frac{q}{p}}\hfill \end{array}$$

(11)

As it can be seen from equation (11), ${\stackrel{·}{V}}_1({\mathrm{\Psi }}_1)$ is less or equal to zero whenever $\lambda >0$ and $1<(p/q)<2$ .

To further elucidate the information provided by equation (11), a different representation can be obtained through the following rewriting

$$\begin{array}{c}{\stackrel{·}{V}}_1\left({\mathrm{\Psi }}_1\right)=-{\left(\frac{1}{\lambda }\right)}^{\frac{q}{p}}{\left[\sqrt{2}{V}_1^{0.5}\right]}^{1+\frac{q}{p}}\hfill \\ \le -{\left(\frac{1}{\lambda }\right)}^{\frac{q}{p}}{\left(\sqrt{2}\right)}^{1+\frac{q}{p}}{V}_1{\left({\mathrm{\Psi }}_1\right)}^{\frac{p+q}{2p}}\hfill \\ \le -L_a.V_1{\left({\mathrm{\Psi }}_1\right)}^r\hfill \end{array}$$

(12)

where $L_a=(1/\lambda {)}^{\frac{q}{p}}(\sqrt{2}{)}^{1+\frac{q}{p}}$ and $r=(\mathrm{p}+\mathrm{q})/(2\mathrm{p})$ .

Lemma 1 (Hagh et al. ³⁵ and Yu et al. ⁴² ): If there is a differentiable, continuous, and positive definite function g(x) that satisfies

The following inequality

$$\stackrel{·}{g}\left(x\right)\le L_g.g(x{)}^{r_g}$$

(13)

where $L_b>0$ and $0<r_g<1$ . Then for any starting point $x_s$ , $g(x)$ will satisfy the inequalities described in equation (14)

$$\begin{array}{c}{g}^{1-r_g}\left(x\right)\le g^{1-r_g}\left(x_s\right)-L_g(1-r_g)(x-x_s),x_s<x<x_f\hfill \\ g\left(x\right)\equiv 0\forall x\ge x_f\hfill \end{array}$$

(14)

In which, $x_f$ is the final point, and it can be expressed as $x_f=x_s+\frac{g^{1-r_g}\left(x_s\right)}{L_g(1-r_g)}$

According to lemma 1, we can conclude by analogy that $x\equiv t$ , $L_g\equiv L_a$ , and $r_g\equiv r$ . The proposed manifold will tend to zero in finite time duration $t_1$

$$t_1=t_f=\frac{V_1^{1-r}}{L_a(1-r)}\forall t_0=0$$

(15)

Proof is completed.

Furthermore, it can be also deduced that the proposed macrovariable ${\mathrm{\Psi }}_1(e_1,e_2)$ is equal to zero whenever $t\ge t_1$ . From the above analysis, we are ready now to study the behavior of the proposed dynamic macrovariable at the steady state ( ${\mathrm{\Psi }}_1(e_1,e_2)$  = 0). So, characteristic and stability analysis of ${\mathrm{\Psi }}_1(e_1,e_2)$  = 0 will be discussed in the next section.

Characteristic analysis

When ${\mathrm{\Psi }}_1(e_1,e_2)$  = 0, then the equation of macrovariable can be rewritten as

$$e_2\left(t\right)=-\alpha \left(e\right)e_1\left(t\right)-\beta \left(e\right)|e_1\left(t\right){|}^{\sigma }\mathrm{sign}\left(e_1\left(t\right)\right)-c\left(e\right)\sin \left(e_1\left(t\right)\right)$$

(16)

In order to guarantee the error dynamics can quickly converge to the origin regardless of the initial conditions, the dynamic coefficients $\alpha (e)$ , $\beta (e)$ , and $c(e)$ should be selected properly. Pan et al. ²⁰ and Pan and Zhang ²⁵ used dynamic coefficients $\alpha (e)$ and $\beta (e)$ . These functions are good enough for improving the speed convergence. However, the convergence speed is decaying whenever the error dynamics approach the origin. To address this issue, we added $c(e)\sin (e_1(t))$ function in order to speed up the convergence rate at $\left|e\right|<ϵ$ for all $0<ϵ<1$ . In this work, sinusoidal function has the ability to enhance the speed globally

$$\begin{array}{c}\alpha \left(e\right)=\frac{2\alpha }{1+\exp (-{\xi }_1(\left|e_1\right|-1))}\hfill \\ \beta \left(e\right)=\frac{2\beta }{1+\exp \left({\xi }_2(\left|e_1\right|-1)\right)}\hfill \\ c\left(e\right)=\frac{4c\exp (-{\xi }_1\left|e_1\right|)}{1+\exp \left({\xi }_2(\left|e_1\right|-1)\right)}\hfill \end{array}$$

(17)

In which, $\alpha >0,\beta >0,{\xi }_1>0,{\xi }_2>0$ and c>0.

The effects of the proposed parameters on the behavior of ${\mathrm{\Psi }}_1(e_1,e_2)$ can be described graphically. Setting all the parameters as $\alpha =\beta =\mathrm{c}=1,{\xi }_1=1.3,{\xi }_2=1.5$ , it can be seen from Figure 1 that the coefficients have different influences on the proposed macrovariable.

Figure 1.

Parameter coefficients of ${\mathrm{\Psi }}_1(e_1,e_2)$ .

When $\left|e\right|>1$ , coefficient $\alpha (e)$ exerts a stronger influence compared to the other coefficients. In other words, the term $\alpha (e)e_1(t)$ plays the major role and brings the error dynamics into the attractor region. When $ϵ<\left|e\right|<1$ , $\beta (e)$ the second term will activate and it will play the leading role due to the decreasing of the first term. When $\left|e\right|<ϵ$ , the term $c(e)\sin (e_1)$ is added for compensating the degradation of $\alpha (e)$ . Hence, the convergence speed rate is kept high in the whole space. To demonstrate the effectiveness of ${\mathrm{\Psi }}_1(e_1,e_2)$ , a comparison with other macrovariables in equation (18) are introduced for near and far initial conditions as shown in Figure 2

$$\begin{array}{c}{\mathrm{\Psi }}_2(e_1,e_2)=e_2+\alpha .e_1\hfill \\ {\mathrm{\Psi }}_3(e_1,e_2)=e_2+\alpha .e_1+\beta |e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)\hfill \\ {\mathrm{\Psi }}_4(e_1,e_2)=e_2+\alpha \left(e\right).e_1+\beta \left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)\hfill \\ {\mathrm{\Psi }}_5(e_1,e_2)=e_2+\alpha \left(e\right).e_1+\beta \left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)+c.\sin \left(e_1\right)\hfill \end{array}$$

(18)

where ${\mathrm{\Psi }}_2,{\mathrm{\Psi }}_3,{\mathrm{\Psi }}_4,{\mathrm{\Psi }}_5$ refer to SMC surface, FTSMC surface, dynamic fast terminal sliding mode control (DFTSMC), and DFTSMC with additional sinusoidal function, respectively. Setting the parameters as follows: $\alpha =\beta =\mathrm{c}=1,{\xi }_1=1.3,{\xi }_2=1.5$ , $\sigma =0.6$ . For the initial conditions, we choose $e_1(0)=1$ and $e_1(0)=70$ .

Figure 2.

Error dynamics trajectory at $e_1(0)=1$ and $e_1(0)=70$ , respectively.

In control engineering, the analytic proof is necessary, and the stability analysis using Lyapunov method will be introduced.

Theorem 2. For disturbed system, the error dynamics $e_1$ and $e_2$ described in equation (16). The system will converge to neighborhood of the origin in finite-time duration.

Proof. The stability is proven by considering the Lyapunov function $V_2(e)=(1/2)e_1^2$ , its time derivative is ${\stackrel{·}{V}}_2(e)=e_1{e}_2$ . Substituting equation (16) gives

$${\stackrel{·}{V}}_2\left(e\right)=-[\alpha \left(e\right)e_1^2+\beta \left(e\right)|e_1\left(t\right){|}^{\sigma +1}+c\left(e\right)e_1\sin \left(e_1\right)]<0$$

(19)

It is clear to see from equation (19) and Figure 3, the system described in equation (16) is globally asymptotic stable (GAS).

Figure 3.

The time derivative of $V_2$ .

As we mentioned before, the dynamic coefficients have a major/minor playing role. Therefore, this mechanism works according to the value of error dynamics. Therefore, the macrovariable ${\mathrm{\Psi }}_1(e_1,e_2)$ is divided into three stages:

• Stage I: $e_1(0)\to \left|e_1\right|=\Omega$ , ( $\Omega$ is generally equal to 1)

The term $\alpha (e).e_1$ has the leading role

$$\begin{array}{c}{\int }_0^{t_a}\mathrm{dt}\le {\int }_{\Omega }^{e_1\left(0\right)}\frac{d\left(\left|e_1\right|\right)}{\alpha \left(e\right)\left|e_1\right|}<{\int }_{\Omega }^{e_1\left(0\right)}\frac{d\left(\left|e_1\right|\right)}{\alpha \left|e_1\right|}\hfill \\ t_a=\frac{\ln \left(\left|e_1\left(0\right)\right|\right)}{\alpha }\hfill \end{array}$$

(20)

• Stage II: $\left|e_1|=\Omega \to |e_1\right|=ϵ$

The term $\beta (e)|e_1{|}^{\sigma }\mathrm{sign}(e_1)$ play the leading role

$$\begin{array}{c}{\int }_0^{t_b}\mathrm{dt}\le {\int }_{ϵ}^{\Omega }\frac{d\left(\left|e_1\right|\right)}{\beta \left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)}<{\int }_{ϵ}^{\Omega }\frac{d\left(\left|e_1\right|\right)}{\beta |e_1{|}^{\sigma }}\hfill \\ t_b=\frac{1}{\beta (1-\sigma )}[|\Omega {|}^{1-\sigma }-|ϵ{|}^{1-\sigma }]=\frac{1}{\beta (1-\sigma )}[1-|ϵ{|}^{1-\sigma }]\hfill \end{array}$$

(21)

• Stage III: $\left|e_1|=ϵ\to |e_1\right|=0$

The third term of equation (3) is added to the second term and both of them play the major role

$$\begin{array}{c}{\int }_0^{t_c}\mathrm{dt}\le {\int }_0^{ϵ}\frac{d\left(\left|e_1\right|\right)}{\beta \left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)+c\left(e\right)\sin \left(e_1\right)}<{\int }_0^{ϵ}|e_1{|}^{3}d\left(\left|e_1\right|\right)\hfill \\ t_c=\frac{1}{4}|ϵ{|}^4\hfill \end{array}$$

(22)

Note: $t_c$ is very small value and it can be neglected.

Thus, the total time for ${\mathrm{\Psi }}_1(e_1,e_2)$ can be computed as

$$\begin{array}{c}{t}_T=t_a+t_b+t_c\approx t_a+t_b\hfill \\ =\frac{\ln \left(\left|e_1\left(0\right)\right|\right)}{\alpha }+\frac{1}{\beta (1-\sigma )}[1-|ϵ{|}^{1-\sigma }]\hfill \end{array}$$

(23)

It is clearly noticed that $t_T$ is lesser than the total time of sliding motion discussed in Pan et al. ²⁰

Proof is completed.

Observer design

In this section, a HTNDO is designed for estimating lumped disturbances and improving the anti-disturbance ability of control system. This type of observer has simple structure, good estimation effects when it compared to conventional nonlinear disturbance observers,^35,43,44 and is able to suppress measurement noise. ⁴⁵

Lemma 2 (Karimi et al. ³⁹ ): Consider a nonlinear system, given by

$$\begin{array}{c}{\stackrel{·}{\mathrm{\Phi }}}_1={\mathrm{\Phi }}_2\hfill \\ {\stackrel{·}{\mathrm{\Phi }}}_2=-a_1\tanh \left(b_1{\mathrm{\Phi }}_1\right)-a_2\tanh \left(b_2{\mathrm{\Phi }}_2\right)\hfill \end{array}$$

(24)

If $a_1,a_2,b_1,b_2>0$ , the corresponding system in equation (24) is asymptotic stable at (0,0).

The presented HTNDO in Chen et al. ⁴⁵ is helpful in the proposed control design. The mathematical presentation is described in equation (25) as

$$\begin{array}{c}{\stackrel{·}{z}}_2=f\left(x\right)+b\left(x\right)u+z_3\hfill \\ {\stackrel{·}{z}}_3=-R^2[a_1\tanh \left(b_1(z_2-x_2)\right)+a_2\tanh (b_2{z}_3/R)]\hfill \end{array}$$

(25)

In which, $a_1,a_2,b_1,b_2,R>0$ , $z_2$ estimates $x_2$ and $z_3$ estimates $d(x,t)$ .

Using some basic calculus, when $R\to \infty$ , the magnitude of estimated disturbance change also approaches to infinity $(\left|{\stackrel{·}{z}}_3\right|\to \infty )$ , which implies that $z_3$ changes more rapidly than ${\stackrel{·}{z}}_2$ . Hence, we can conclude that the proposed observer satisfies lemma 2 $(\underset{R\to \infty }{lim}{\stackrel{··}{z}}_2={\stackrel{·}{z}}_3)$ and HTNDO is asymptotic stable.

Based on the output of HTNDO presented in equation (25), the composite anti-disturbance control (DGTSC-HTNDO) is designed as

$$\begin{array}{c}\multicolumn{1}{c}{u\left(t\right)=}\\ \multicolumn{1}{c}{\frac{-1}{b\left(x\right)}\left[\begin{array}{c}f\left(x\right)+L{z}_3-{\stackrel{··}{x}}_{1\mathrm{ref}}+\stackrel{·}{\alpha }\left(e\right).e_1+\alpha \left(e\right).e_2+\stackrel{·}{\beta }\left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)\hfill \\ \multicolumn{1}{c}{+\sigma \beta \left(e\right)e_2|e_1{|}^{\sigma -1}+\stackrel{·}{c}\left(e\right)\sin \left(e_1\right)+c\left(e\right).e_2.\cos \left(e_1\right)}\\ \multicolumn{1}{c}{-{\left(\frac{-1}{\lambda }{\mathrm{\Psi }}_1(e_1,e_2)\right)}^{\frac{q}{p}}}\end{array}\right]}\end{array}$$

(26)

Where: L is normal number, $\stackrel{·}{\alpha }(e),\stackrel{·}{\beta }(e)$ , and $\stackrel{·}{c}(e)$ are expressed in Appendix 1.

Theorem 3. The anti-disturbance control law described in equation (26) can force the system described in equation (1) to converge into the attractor point in finite time.

Proof. Substituting equation (26) into the time derivative of $V_1$ , yields

$$\begin{array}{c}{\stackrel{·}{V}}_1={\mathrm{\Psi }}_1[[d(x,t)-L{z}_3\left(t\right)+{\left(\frac{-1}{\lambda }{\mathrm{\Psi }}_1\right)}^{\frac{q}{p}}]\hfill \\ =-{\left(\frac{1}{\lambda }\right)}^{\frac{q}{p}}{\mathrm{\Psi }}_1^{\frac{q}{p}+1}+{\mathrm{\Psi }}_1[d(x,t)-L{z}_3\left(t\right)]\hfill \end{array}$$

(27)

Inserting equation (25) into equation (27) gives

$${\stackrel{·}{V}}_1=-{\left(\frac{1}{\lambda }\right)}^{\frac{q}{p}}{\mathrm{\Psi }}_1^{\frac{q}{p}+1}+{\mathrm{\Psi }}_1\left[d\left(x,t\right)-L\left(\underset{R\to \infty }{\lim }\underset{0}{\overset{T}{\int }}\left\{-R^2\left(\begin{array}{l}{a}_1\tanh \left(b_1\left(z_2-x_2\right)\right)\\ +a_2\tanh \left(\frac{b_2{z}_3}{R}\right)\end{array}\right)\right\}\right)\right]$$

(28)

Whenever R and T are large enough, $L{z}_3\to d(x,t)$ . Hence

$${\stackrel{·}{V}}_1\le -{\left(\frac{1}{\lambda }\right)}^{\frac{q}{p}}{\mathrm{\Psi }}_1^{\frac{q}{p}+1}$$

(29)

Based on lemma 1 and lemma 2, we can conclude that ${\mathrm{\Psi }}_1\to 0$ in finite time $t_1$ .

Proof is completed.

Results and discussion

In this section, the proposed anti-disturbance composite controller is applied to simple model, wind turbine system, and DC-DC buck converter. The simulation results validate the effectiveness and feasibility of the suggested scheme.

Simple model

Consider a simple model with the following state space presentation

$$\begin{array}{c}{\stackrel{·}{x}}_1=x_2\hfill \\ {\stackrel{·}{x}}_2=u+d\left(t\right)\hfill \end{array}$$

(30)

The lumped disturbance is supposed to be $d\left(t\right)=\{\begin{array}{c}\multicolumn{1}{c}{0,t<2.5}\\ \multicolumn{1}{c}{\sin (1.5t),2.5\le t<6}\\ \multicolumn{1}{c}{1,t\ge 6}\end{array}$

A DGTSC-HTNDO is developed for stabilizing the system described in equation (30)

$$u\left(t\right)=-\left[\begin{array}{c}L{z}_3-{\stackrel{··}{x}}_{1\mathrm{ref}}+\stackrel{·}{\alpha }\left(e\right).e_1+\alpha \left(e\right).e_2+\stackrel{·}{\beta }\left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)+\sigma \beta \left(e\right)e_2|e_1{|}^{\sigma -1}\hfill \\ \multicolumn{1}{c}{+\stackrel{·}{c}\left(e\right)\sin \left(e_1\right)+c\left(e\right).e_2.\cos \left(e_1\right)-{\left(\frac{-1}{\lambda }{\mathrm{\Psi }}_1(e_1,e_2)\right)}^{r_1}}\end{array}\right]$$

(31)

The parameters of both controller and observer are set as $\alpha =\beta =1$ , $\mathrm{c}=1.5,{\xi }_1=0.2,{\xi }_2=0.4$ , $r_1=1/1.15$ , $\lambda =0.1$ , $\sigma =0.6,L=1,a_1=a_2=2,b_1=b_2=1.5,R=50$ .

For further verifying the feasibility, others controllers as SMC $(u_1(t))$ , FTSMC $(u_2(t))$ , and DFTSMC ²⁰ $(u_3(t))$ are also designed for comparison purpose

$$\begin{array}{c}{u}_1\left(t\right)=-\left[\begin{array}{c}-{\stackrel{··}{x}}_{1\mathrm{ref}}+\alpha e_2+k_1(b^{\left|{\mathrm{\Psi }}_2\right|}-1)\mathrm{sign}\left({\mathrm{\Psi }}_2\right)+k_2|{\mathrm{\Psi }}_2{|}^{m\left({\mathrm{\Psi }}_2\right)}\mathrm{sign}\left({\mathrm{\Psi }}_2\right)\hfill \\ \multicolumn{1}{c}{+({\eta }_{\max }+\eta )\mathrm{sign}\left({\mathrm{\Psi }}_2\right)}\end{array}\right]\hfill \\ u_2\left(t\right)=-\left[\begin{array}{c}-{\stackrel{··}{x}}_{1\mathrm{ref}}+\alpha e_2+\sigma \beta e_2|e_1{|}^{\sigma -1}+k_1(b^{\left|{\mathrm{\Psi }}_3\right|}-1)\mathrm{sign}\left({\mathrm{\Psi }}_3\right)+k_2|{\mathrm{\Psi }}_3{|}^{m\left({\mathrm{\Psi }}_3\right)}\mathrm{sign}\left({\mathrm{\Psi }}_3\right)\hfill \\ \multicolumn{1}{c}{+({\eta }_{\max }+\eta )\mathrm{sign}\left({\mathrm{\Psi }}_3\right)}\end{array}\right]\hfill \\ u_3\left(t\right)=-\left[\begin{array}{c}-{\stackrel{··}{x}}_{1\mathrm{ref}}+\stackrel{·}{\alpha }\left(e\right).e_1+\alpha \left(e\right).e_2+\stackrel{·}{\beta }\left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)+\sigma \beta \left(e\right)e_2|e_1{|}^{\sigma -1}\hfill \\ \multicolumn{1}{c}{+k_1(b^{\left|{\mathrm{\Psi }}_4\right|}-1)\mathrm{sign}\left({\mathrm{\Psi }}_4\right)+k_2|{\mathrm{\Psi }}_4{|}^{m\left({\mathrm{\Psi }}_4\right)}\mathrm{sign}\left({\mathrm{\Psi }}_4\right)+({\eta }_{\max }+\eta )\mathrm{sign}\left({\mathrm{\Psi }}_4\right)}\end{array}\right]\hfill \end{array}$$

(32)

where $b=1.01,k_1=1.1,k_2=0.7,\eta =1,{\eta }_{\max }=0.7,m\left(\mathrm{\Psi }\right)=1.5+\frac{0.6}{1+e^{800{\mathrm{\Psi }}^2}}-\frac{2.6}{1+e^{2{\mathrm{\Psi }}^{400}}}$ .

From Figure 4(a) and (b), it can be seen that the system states converge rapidly into (0,0) when DGTSC-HTNDO is applied. Figure 4(c) and (d) present both phase trajectory and marco variable of the closed loop system. Figure 4(c) shows that the state $x_2$ oscillation are well suppressed compared with other SMCs. Moreover, Figure 4(d) presents the macrovariable change with respect to time, and it can be clearly observed that ${\mathrm{\Psi }}_1$ is smoothly converged to zero without any oscillations as shown in the zoomed part. For the chattering phenomenon, the proposed composite controller has successfully eliminated the chattering in control law as depicted in Figure 4(e). In disturbance point of view, we can see that hyperbolic tangent disturbance observer is better in disturbance estimations than the conventional nonlinear disturbance observer ³⁵ as depicted in Figure 4(f). For more clarification, the performance indices are listed in Table 1.

Figure 4.

(a) State $x_1$ , (b) state $x_2$ , (c) phase trajectory, (d) macro-variable, (e) control law, and (f) lumped disturbance.

Table 1.

Performance indices.

	$t_s\left(s\right)$	$x_{\mathrm{ss}1}$	$x_{\mathrm{ss}2}$	$p_{\mathrm{\Psi }}$
SMC	3.4	0.01	0.1	$0.08$
FTSMC	1.29	0.0023	0.049	$0.08$
DFTSMC	1.14	0.0019	0.05	$0.07$
DGTSC	0.8	−1.2. 10-6	−1.0. 10-4	$0.00$

$t_s\left(s\right):$ settling time; $x_{\mathrm{ss}1},x_{\mathrm{ss}2}$ : steady state error of $x_1,x_2$ ; $p_{\mathrm{\Psi }}$ : peak ripple of $\mathrm{\Psi }$ ; SMC: sliding mode control surface; FTSMC: fast terminal sliding mode control surface; DFTSMC: dynamic fast terminal sliding mode control; DGTSC: dynamic global terminal synergetic control.

Wind turbine system

Wind turbine system mainly consisted of three subsystems: aerodynamic model, generator subsystem, and driven-train subsystem.

• i. Aerodynamic subsystem

Usually, the aerodynamics power $P_a$ is approximated as

$$P_a=\frac{1}{2}\rho \pi R^2{v}^3{C}_p({\lambda }_m,\beta )$$

(33)

in which, $\rho ,R,v,$ and $\beta$ refer to air density, rotor radius, wind speed, and pitch angle, respectively. ${\lambda }_m$ refers to tip speed ratio and $C({\lambda }_m,\beta )$ stands for power coefficient. The mathematical expressions for ${\lambda }_m$ and $C({\lambda }_m,\beta )$ can be written as follows

$$\begin{array}{c}{\lambda }_m=\frac{w_{r}R}{v}\hfill \\ C_p({\lambda }_m,\beta )=c_1(\frac{c_2}{{\lambda }_i}-c_3-c_4)e^{\frac{c_5}{{\lambda }_i}}+c_6{\lambda }_m\hfill \end{array}$$

(34)

where $w_r$ is the rotor speed, ${\lambda }_i^{-1}=\frac{1}{{\lambda }_m+0.08\beta }-\frac{0.035}{{\beta }^3+1}$ , $c_1=0.5176,c_2=116,c_3=0.4,c_4=5,c_5=21\mathrm{and}{c}_6=0.0068$ .

• ii. Generator subsystem

The generator model can be expressed by the following differential equation ⁴⁶

$$\tau {\stackrel{·}{T}}_{\mathrm{em}}+T_{\mathrm{em}}=T_{\mathrm{em}-\mathrm{ref}}$$

(35)

where $T_{\mathrm{em}-\mathrm{ref}}$ presents referenced electromagnetic torque. $\tau$ denotes the torque coefficient.

• iii. Train driven subsystem

The dynamic properties of driven-train system can be written as Zhang et al. ⁴⁷

$$\begin{array}{c}{J}_r{\stackrel{·}{w}}_r=T_a-K_r{w}_r-T_{\mathrm{ls}}\hfill \\ J_g{\stackrel{·}{w}}_g=T_{\mathrm{hs}}-K_g{w}_g-T_g\hfill \end{array}$$

(36)

$K_r,K_g,J_r$ , and $J_g$ refer to turbine rotor damping factor, generator rotor damping factor, rotational inertia of wind turbine, and rotational inertia of rotor generator, respectively. $T_a$ and $T_g$ stand for mechanical torque and electromagnetic torque of generator. $w_g,T_{\mathrm{ls}}$ , and $T_{\mathrm{hs}}$ present angular velocity of rotor generator, low speed torque, and high speed torque, respectively. The gearbox ratio is given by

$$n_g=\frac{w_g}{w_r}=\frac{T_{\mathrm{ls}}}{T_{\mathrm{hs}}}$$

(37)

It come then

$$J_t{\stackrel{·}{w}}_r=T_a-K_t{w}_r-T_g$$

(38)

where

$$\begin{array}{c}{J}_t=J_r+n_g^2{J}_g\hfill \\ K_t=K_r+n_g^2{K}_g\hfill \end{array}$$

(39)

In order to get the dynamical model of wind turbine system, we need to combine equations (33), (35), and (38)

$$\begin{array}{c}{\stackrel{·}{x}}_1=x_2\hfill \\ {\stackrel{·}{x}}_2=\frac{\rho \pi R^5}{2J_t}\frac{C_p\left({\lambda }_m\right)}{{\lambda }_m^3}{x}_2^2-\frac{K_t}{J_t}{x}_2-\frac{n_g}{J_t}u\left(t\right)+d\hfill \end{array}$$

(40)

where $x_1=\theta$ , $x_2=w_r$ , $d(t)=\sin (1.5t)$ for all $t\ge 4s$ , $f\left(x\right)=\frac{\rho \pi R^5}{2J_t}\frac{C_p\left({\lambda }_m\right)}{{\lambda }_m^3}{x}_2^2-\frac{K_t}{J_t}{x}_2$ , and $b\left(x\right)=-\frac{n_g}{J_t}$ .

The experimental parameters of wind turbine system are discussed and listed in Ma et al. ⁴⁶

In this case, we are ready to force the angular speed $w_r$ to track the desired trajectory. The shape of the desired trajectory depends on wind speed signal. Therefore, we will apply two different signals of wind speed $v(t)$ . The first case, a $12\mathrm{m}/\mathrm{s}$ is excited to the system which gives reference angular speed around $4.492\mathrm{m}/\mathrm{s}$ . The second try yields to an almost realistic signal as $v(t)=5\cos (t)+5$ which gives $w_{r-\mathrm{ref}}=x_{2\mathrm{ref}}=1.87\cos (t)+1.87\mathrm{m}/\mathrm{s}$ .

For comparison purpose, the controller described in equation (26) is applied for second-order wind turbine system as

$$u\left(t\right)=\frac{J_t}{n_g}\left[\begin{array}{c}\frac{\rho \pi R^5}{2J_t}\frac{C_p\left({\lambda }_m\right)}{{\lambda }_m^3}{x}_2^2-\frac{K_t}{J_t}{x}_2+L{z}_3-{\stackrel{··}{x}}_{1\mathrm{ref}}+\stackrel{·}{\alpha }\left(e\right).e_1+\alpha \left(e\right).e_2+\hfill \\ \multicolumn{1}{c}{+\stackrel{·}{\beta }\left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)+\sigma \beta \left(e\right)e_2|e_1{|}^{\sigma -1}}\\ \multicolumn{1}{c}{+\stackrel{·}{c}\left(e\right)\sin \left(e_1\right)+c\left(e\right).e_2.\cos \left(e_1\right)-{\left(\frac{-1}{\lambda }{\mathrm{\Psi }}_1(e_1,e_2)\right)}^{r_1}}\end{array}\right]$$

(41)

In which, ${\stackrel{··}{x}}_{1\mathrm{ref}}=0$ for all $x_{1\mathrm{ref}}=4.49t$ , and it is equal to ${\stackrel{··}{x}}_{1\mathrm{ref}}=-1.87\sin (t)$ for $x_{1\mathrm{ref}}=1.87\sin (t)+1.87t$ . The parameters of observer and control law are set as $\alpha =\beta =1$ , $c=4,{\xi }_1=0.3,{\xi }_2=0.4$ , $r_1=1/1.01$ , $\lambda =0.1$ , $\sigma =0.6,L=1,a_1=a_2=2,b_1=b_2=1.5,R=50$ .

Figure 5 presents the performance tracking of wind turbine system via different advanced controllers. As we have seen, the proposed DGTSC-HTNDO can force all of rotor speed and power coefficient to quickly reach the desired values $w_{\mathrm{ref}}=4.492\mathrm{m}/\mathrm{s}$ and ${\lambda }_{\max }=8.1$ , respectively, within $0.5\mathrm{s}$ without any oscillations or overshoots as shown in Figure 5(b) and (c). Moreover, the position state $x_1$ takes around $0.2\mathrm{s}$ to reach the desired linear function as depicted in Figure 5(a). All these indices have been dramatically improved via the proposed control compared with the simulation results in Ma et al. ⁴⁶ or with other SMCs. Figure 5(d) exhibits the free chattering control law of DGTSC-HTNDO. In practice, various internal and external fluctuations like frictions in shafts and lack of hydraulics are often characterized as time-variant disturbances. When $t\ge 4\mathrm{s}$ , it is noteworthy that these fluctuations begin as shown in Figures 5(e) and 6(d). However, the proposed observer estimates this disturbance, which can aid the proposed controller to increase its robustness and reject these fluctuations. For almost realistic situation (case II), Figure 6 presents the performance tracking of wind turbine system under wind speed $v(t)=5\cos (t)+5(\mathrm{m}/\mathrm{s})$ . According to Figure 6(c), the free chattering composite controller can guarantee all of position state and speed state to track quickly the reference functions within $0.38\mathrm{s}$ without any fluctuation or overshoots as shown in Figure 6(a) and (b).

Figure 5.

Performances of case 1: (a) Position state, (b) angular velocity $w_r$ , (c) tip speed ratio, (d) control law, and (e) observer estimation.

Figure 6.

Performances of case 2: (a) Position state, (b) angular velocity $w_r$ , (c) control law, and (d) observer estimation.

DC-DC buck converter

In this section, another application is used to further illustrate the feasibility of the composite control scheme. The average model of the circuit depicted in Figure 7 can be expressed as

$$\begin{array}{c}{\stackrel{·}{x}}_1=\frac{-1}{L}{x}_2+\frac{1}{L}{v}_{i}u\left(t\right)\hfill \\ {\stackrel{·}{x}}_2=\frac{-1}{R_{B}C}{x}_2+\frac{1}{C}{x}_1\hfill \end{array}$$

(42)

where $x_1$ , $x_2$ , $u(t)$ , and $v_i$ stand for inductor current $i_L$ , output voltage $v_0$ , duty cycle, and input voltage, respectively.

Figure 7.

DC-DC buck converter circuit.

The main aim is to force the system to follow a lower output voltage than its input value. Therefore, the error dynamic presentation is required

$$\begin{array}{c}{\stackrel{·}{e}}_1=e_2\hfill \\ {\stackrel{·}{e}}_2=-\frac{e_1-v_{\mathrm{ref}}}{\mathrm{LC}}-\frac{e_2}{R_{B}C}-\frac{v_i}{\mathrm{LC}}u\left(t\right)\hfill \end{array}$$

(43)

Let $e_1=v_{\mathrm{ref}}-x_1$ , where $v_{\mathrm{ref}}$ refers to the output reference voltage value. The parameters of buck converter are set as $R_B=8\Omega ,C=1100\mathrm{uF},L=2\mathrm{mH},v_i=15\mathrm{V}$ .

Next, according to DGTSC-HTNDO technique, the controller is designed as

$$\begin{array}{c}\multicolumn{1}{c}{u\left(t\right)=}\\ \multicolumn{1}{c}{\frac{\mathrm{LC}}{v_i}\left(\begin{array}{c}-\frac{e_1-v_{\mathrm{ref}}}{\mathrm{LC}}-\frac{1}{R_{B}C}{e}_2+L{z}_3-{\stackrel{··}{v}}_{\mathrm{ref}}+\stackrel{·}{\alpha }\left(e\right).e_1+\alpha \left(e\right).e_2+\hfill \\ \multicolumn{1}{c}{+\stackrel{·}{\beta }\left(e\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)+\sigma \beta \left(e\right)e_2|e_1{|}^{\sigma -1}+\stackrel{·}{c}\left(e\right)\sin \left(e_1\right)}\\ \multicolumn{1}{c}{+c\left(e\right).e_2.\cos \left(e_1\right)-{\left(\frac{-1}{\lambda }{\mathrm{\Psi }}_1(e_1,e_2)\right)}^{r_1}}\end{array}\right)}\end{array}$$

(44)

The composite control coefficients are chosen as $\alpha =100,\beta =100$ , $c=6,{\xi }_1=1.2,{\xi }_2=1.7$ , $\lambda =0.000001$ , $\sigma =0.6,L=1,a_1=a_2=2,b_1=b_2=1.5,R=50$ .

To validate again the effectiveness of the proposed control algorithm in power electronic devices, two tests scenarios are illustrated to prove its benefits in terms of robustness and shorten time response. In addition, an anti-disturbance FTSMC based on improved quick reaching law ²¹ is designed for comparison purpose as

$$s_a={\mathrm{\Psi }}_3(e_1,e_2)+N(e_1,e_2)$$

(45)

With: $N\left(x\right)=\alpha \tanh \left({\Omega }_1(\left|e_1\right|-1)\right)e_1-\beta \tanh \left({\Omega }_2(\left|e_1\right|-1)\right)|e_1{|}^{\sigma }\mathrm{sign}\left(e_1\right)$

$$u_a\left(t\right)=-\frac{\mathrm{LC}}{V_{\mathrm{in}}}\left[\begin{array}{c}\frac{e_1}{\mathrm{LC}}+\frac{e_2}{\mathrm{RC}}-\frac{V_{\mathrm{ref}}}{\mathrm{LC}}-(K_1(b^{\left|s_a\right|}-1)+K_2|s_3{|}^{m\left(s_a\right)})\mathrm{sign}\left(s\right)-\hfill \\ \multicolumn{1}{c}{\frac{2\alpha e_2(1+e^{-2{\Omega }_1(\left|e_1\right|-1)}(1+2{\Omega }_1\left|e_1\right|))}{{(1+e^{-2{\Omega }_1(\left|e_1\right|-1)})}^2}-\frac{2\beta \sigma |e_1{|}^{\sigma -1}{e}_2(1+e^{2{\Omega }_2(\left|e_1\right|-1)}(1-\frac{2{\Omega }_2}{\sigma }\left|e_1\right|))}{{(1+e^{-2{\Omega }_1(\left|e_1\right|-1)})}^2}}\end{array}\right]$$

(46)

All the controller coefficients are listed in Halledj and Bouafassa. ²¹

• i. First scenario: in this test, we set the following conditions: $V_{\mathrm{ref}}=5\mathrm{V}$ ,

$$R_B=\{\begin{array}{c}\multicolumn{1}{c}{8\Omega t<1.0\mathrm{s}}\\ \multicolumn{1}{c}{4\Omega 1.5\le t<1.5\mathrm{s}}\\ \multicolumn{1}{c}{8\Omega t\ge 1.5\mathrm{s}}\end{array}$$

Figure 8 presents the output voltage of DC-DC buck converter under resistance variation. It can be seen that the proposed controller improves the robustness and convergence speed rate. In addition, the settling time of buck system becomes lesser than $0.02\mathrm{s}$ and the composite controller takes $0.005\mathrm{s}$ to recover the desired voltage when $R_B$ is increased or decreased by $R_B/2$ . The output voltage shoots are also ameliorated. So that, the system can raise up only by $0.02\mathrm{V}$ or can fall down only by $0.08\mathrm{V}$ .

Figure 8.

Output voltage under resistance change.

• ii. Second scenario

In this experiment, we try to select again $5\mathrm{V}$ as desired output voltage and a variable capacitor is equal to $C=\{\begin{array}{c}\multicolumn{1}{c}{1100\mathrm{uF}t<1}\\ \multicolumn{1}{c}{1100\pm 110\mathrm{uF}t\ge 1}\end{array}$ . From Figure 9(a) and (b), we can see that the DGTSC-HTNDO ameliorates again both the transient response (settling time is less than $0.02\mathrm{s}$ ) and robustness (recovery time is lesser than $0.017\mathrm{s}$ ) under capacitance change. In conclusion, DGTSC-HTNDO is preferable to the other DFTSMC ²⁰ and NFTSMC-NQRL ²¹ for tracking control of DC-DC buck converter.

Figure 9.

(a), (b) Output voltage under increasing and decreasing of capacitance value, respectively.

Conclusion

In this article, a DGTSC-HTNDO is investigated for uncertain second-order systems. The presented approach uses a synergetic macrovariable, augmented with additional sinusoidal function, to drive system states toward a desired point and improve the convergence rate, especially when the system trajectory is in vicinity of equilibrium point. To mitigate the effects of lumped perturbation, a HTNDO is designed to accurately predict disturbance. This estimation is used in the proposed closed loop control law. Results confirm that the suggested algorithm outperforms variable structure controllers in terms of robustness, accuracy, and convergence rapidity. Furthermore, the proposed synergetic regulator addresses the issue of chattering phenomenon. Finally, it is pertinent to note that the SC approach could be also extended to other complicated nth order systems, which would be the subject of future study.