A homogeneous extended state estimator-based super-twisting sliding mode compensator for matched and unmatched uncertainties

Abstract

In this research work, an output tracking problem of a kind of nonlinear motion control systems influenced by exogenous uncertainties using second-order super-twisting sliding mode control is studied. It is shown that when second-order super-twisting sliding mode control is implemented with finite-time convergent homogeneous extended state observer, the second-order sliding mode is achieved on the selected sliding manifold with efficient disturbance attenuation from the output. The presented control structure is tested on the air-gap control of an electromagnetic levitation suspension system using MATLAB platform. The observations prove the efficacy of the proposed algorithm providing excellent robust control efficiency along with precise attenuation of various disturbances.

Introduction

Modern real-world systems are generally characterized by complicated nonlinear and multi-variable dynamics with various uncertainties such as parametric perturbations, unmodeled dynamics, measurement noises, linearization and approximation errors, and external disturbances. ¹ For example, see vessel dynamic positioning system, ² robot manipulator, ³ DC-DC power converter, ⁴ missile system, ⁵ spacecraft, ⁶ permanent magnet synchronous motor (PMSM), ⁷ hard-disk drive, ⁸ magnetic levitation (MAGLEV) suspension transportation vehicle,^9,10 and many more practical systems. These uncompensated uncertainties need to be treated properly otherwise they may become detrimental to the desired control performances which could further lead to system instability. ¹¹ Hence, to obtain intelligent and high-fidelity actuated devices, the need of interference attenuation techniques such as adaptive control, ¹² robust control, ¹¹ $H_2/H_{\infty }$ control ¹³ sliding mode control (SMC), ¹⁴ backstepping control, ¹⁵ and so on have received considerable attention from motion controller designers. Although these exquisite methods have been efficiently applied in various aforementioned uncertain systems, they mainly target to achieve robust stability on the price of nominal performance. ¹⁶ Among these control methods, SMC is a popular robust control technique because of its simple design, easier implementation, reduced order dynamics, robustness toward the matched disturbances and the plant uncertainties. ¹⁷ It should be pointed out that the SMC methods have mainly two concerns during their practical implementation such as the presence of high-frequency chattering phenomenon ¹⁸ and providing complete robustness only to the matched disturbances. ¹⁹

For proper and efficient treatment of the first concern of chattering phenomenon, recently, the high-order sliding mode control (HOSMC) ²⁰ methods have found popularity among chattering attenuation strategies in the literature. These HOSMC methods are widely tested on various applications such as hypersonic vehicles, ²¹ wind turbine, ²² cable-driven manipulators, ²³ vehicle suspension, ²⁴ quadcopters, ²⁵ and so on. However, among HOSMC family members, the super-twisting control (STC) ²⁶ has proven its effectiveness in which control input is applied in the second-order of the sliding variable. Its advantages include effective compensation of Lipschitz perturbations, requirement of only single output variable information, finite-time state focalization to the origin and continuous control signal generation for minimizing chattering phenomenon. ²⁷

The second concern of handling unmatched uncertainty is justified because it has been observed that the major uncertainties present in many MCS such as rail track input in MAGLEVs, ²⁸ load torque in PMSM, ²⁹ external wind disturbances in flight control system, ³⁰ and so on do not adhere to so-called matching condition and hence robust performance out of SMC may not be achieved. ³¹ Hence, due to the importance of nullifying mismatched uncertainties for getting precise and effective performance from MCS, many decent SMC strategies focusing the mismatched disturbance attenuation have been reported in the literature, for example, Riccati based approach, ³² LMI-based, ³³ $H_{\infty }$ control based, ³⁴ adaptive approach, ³⁵ and so on. However, these approaches are designed to handle mismatched uncertainties if they are vanishing type, that is, $H_2$ norm-bounded which may not be the case in real-time system, for example, in MAGLEV transportation system, the considered lumped uncertainties may not have zero steady-state and hence not $H_2$ norm-bounded. ³⁶ Consequently, these methods may not be able to provide nominal performance of the overall control systems.

Another widely used method to address this issue is integral sliding mode control (I-SMC) in which integral type of sliding surface is used for guiding the states to equilibrium from the first instant of time. Because of the simple design and better robustness feature, this method is widely applied in many practical systems.^37–39 However, it is commonly known that integral action brings adversity in control action such as disturbance amplification, large overshoot and extended settling time.

To address the issue of robustness of MCS in the presence of mismatched uncertainty, an alternative approach based on estimation/observer technique known as disturbance-observer-based control (DOBC) ⁴⁰ approach has been presented. Due to its extraordinary advantage of obtaining robustness without loosing nominal control performance, several control methods with different observers designs such as nonlinear disturbance observer (NLDO), ⁴¹ equivalent-input-disturbance control, ⁴² extended state observer (ESO), ⁴³ sliding mode based observers ⁴⁴ and so on have been proposed in the literature. However, it is noticed that many DOBC designs concentrate only the systems with matched disturbances; there are fewer results focusing systems with mismatched ones.^45,46 Moreover, aforementioned DOBC methods are heavily dependent on the plant information for estimation and control design. Among them the ESO, originated from Active Disturbance Rejection Control (ADRC), uses very less prior information that too only the system relative degree ⁴³ which makes it popular in theoretical ⁴⁷ as well as practical studies.^48,49 The conventional linear ESO (LESO) further modified as nonlinear ESO (NLESO) and Generalized Extended State observer (GESO) control ⁵⁰ in order to address the issue of mismatched uncertainties that are even not expressed in the standard integral chain form. Furthermore, the adaptive ESO ⁵¹ also exhibit its efficiency in solving the uncertainty estimation problem. While the ESO methodology has been often used widely, the mathematical analysis of stability, convergence time and so on are not performed rigorously. Guo and Zhao ⁵² have proposed the stability analysis of a nonlinear ESO with system modeling uncertainties, but it results in the asymptotic stability under various complex assumptions. A finite-time ESO ⁵³ has been designed for projecting mismatched uncertainties. However, it requires the high-order time derivatives of the uncertainties to be bounded with certain constant number.

Owing to the above discussions, the authors tried to address the issue of mismatched uncertainties present in the MCS using finite-time convergent ESO to obtain better convergence speed and stability performances.

The primary contributions of this paper are briefly emphasized as follows:

1. Motivated from the existing finite-time convergent observers, the conventional ESO is modified with the help of homogeneity principle and termed as homogeneous extended state observer (HESO).

2. The need of system information is kept as minimum as possible by selecting a dynamic sliding manifold which is designed with the help of system output and observer predicted nominal states.

3. A continuous and chattering free guidance law is used for the states to reach the proposed dynamic sliding manifold in finite time.

4. The overall control structure is kept simple and effective by combining the HESO with the conventional STC for counteracting the adversities of exogenous matched as well as unmatched disturbances.

The remaining article is cataloged as follows: Section “Problem description and control objective” formulates the general MCS problem and its control objective is specified. Section “The proposed controller design” proposes the HESO-based second-order STC (HESO-STC) for the systems with uncertainties. The section “Performance analysis of different ESOs” shows the comparative analysis of LESO, NLESO and HESO applied on a standard servo motor with the help of MATLAB simulation. In section “An electromagnetic suspension (EMS) vehicle example,” an application of electromagnetic suspension transportation system is presented for testing the proposed control algorithm. To showcase the efficacy of the proposed law, simulations results are studied in section “Simulation results.” Finally, the ending remarks are summarized in section “Conclusions.”

Problem description and control objective

Consider the following general MCS with input relative degree of $\rho$ under the presence of both matched and unmatched disturbances, represented as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\eta }}_i={\eta }_{i+1}+d_i,\forall i=1,\dots ,\rho -1}\\ \multicolumn{1}{c}{{\stackrel{·}{\eta }}_{\rho }=a(\eta )+b(\eta )u+d_{\rho }}\\ \multicolumn{1}{c}{y={\eta }_1}\end{array}$$

(1)

where $\eta =[{\eta }_1,\dots ,{\eta }_{\rho }{]}^T$ is the state vector, control input is represented by u, y denotes the system output. The term $d_i$ is $(\rho -i{)}^{\mathrm{th}}$ -order differentiable and bounded mismatched disturbance and $d_{\rho }$ is the matched disturbance entering in the control input channel. The functions $a(\eta )$ and $b(\eta )\ne 0$ are functions of $\eta$ which are differentiable everywhere.

Assumption 1

The disturbance $d_i$ is continuous and differentiable which comply the bounded condition as $\frac{d^j{d}_i}{d{t}^j}\le {\mathrm{\Delta }}_i$ for some positive constant ${\mathrm{\Delta }}_i$ , $i=1,2\dots \rho$ and $j=0,1\dots \rho +1-i$ .

Remark 1

The above asssumption on the disturbance $d_i$ is reasonably justified because major disturbances in MCS are due to discontinuous nature of friction which can be modeled with piece-wise continuous models such as LuGre model. ⁵⁴

The main intention is to construct a sturdy, finite-time HOSM-based feedback controller u which could compel the output y of the system (1) to stabilize at a pre-defined position ${\eta }_d$ in the finite-time $t_f$ that is, which ensures equation (2) under the influence of disturbances

$$\underset{t\to t_f}{lim}\Vert P_{\mathrm{err}}\Vert =\underset{t\to t_f}{lim}\Vert ({\eta }_1-{\eta }_d)\Vert =0$$

(2)

where $P_{\mathrm{err}}=({\eta }_1-{\eta }_d)$ is the position error.

The proposed controller design

To initiate the design of the rugged, finite-time convergent HESO-based super-twisting sliding mode controller (HESO-STC) for the system (1), a novel sliding manifold is designed as

$$\sigma =c_1{\eta }_1+\sum _{i=2}^{\rho }{c}_i{\hat{\eta }}_i$$

(3)

where the parameters $c_i>0,(i=1,\dots ,\rho )$ with $c_{\rho }=1$ are designed such that the polynomial $p_0(s)=s^{\rho }+c_{\rho -1}{s}^{\rho -1}+\dots +c_{2}s+c_1=0$ is Hurwitz.

The variables $[{\hat{\eta }}_2,{\hat{\eta }}_3,\dots ,{\hat{\eta }}_{\rho }{]}^T$ are the estimations of the states $[{\eta }_2,{\eta }_3,\dots ,{\eta }_{\rho }{]}^T$ from an observer which will be discussed later.

Remark 2

From author’s point of view, the considered sliding manifold (equation (3)) is different from the conventional sliding surface because the selected manifold depends only on the information of the MCS output and estimated states from the observer/estimator. Hence, it is justified to say that equation (3) needs minimum information from the system and depends only on the outputs.

Next, the design approach of HESO—a $(\rho +1{)}^{\mathrm{th}}$ —order finite-time observer for estimating the states $[{\hat{\eta }}_2,{\hat{\eta }}_3,\dots ,{\hat{\eta }}_{\rho }{]}^T$ is discussed.

HESO

The HESO is a special type of ESO which follows the design procedure of homogeneous state observer. ⁵⁵ The general design of $(\rho +1{)}^{\mathrm{th}}$ —order HESO for the system (1) with disturbances is given as

$$\begin{array}{l}{\stackrel{·}{\widehat{n}}}_1={\widehat{n}}_2+{\mathrm{\lambda }}^{\rho -1}{k}_1\left[\frac{n_1-{\widehat{n}}_1}{k_0}\right]a\\ {\stackrel{·}{\widehat{n}}}_2={\widehat{n}}_3+{\mathrm{\lambda }}^{\rho -2}{k}_2\left[\frac{n_1-{\widehat{n}}_1}{k_0}\right]2a-1\\ ⋮\\ {\stackrel{·}{\widehat{n}}}_{\rho -1}={\widehat{n}}_{\rho }+{\mathrm{\lambda }}^1{k}_{\rho -1}\left[\frac{n_1-{\widehat{n}}_1}{k_0}\right](\rho -1)a-(\rho -2)\\ {\stackrel{·}{\widehat{n}}}_{\rho }={\widehat{n}}_{\rho +1}+{\mathrm{\lambda }}^0{k}_{\rho }\left[\frac{n_1-{\widehat{n}}_1}{k_0}\right]\rho a-(\rho -1)+a(\widehat{n})+b(\widehat{n})u\\ {\stackrel{·}{\widehat{n}}}_{\rho +1}={\mathrm{\lambda }}^{-1}{k}_{\rho +1}\left[\frac{n_1-{\widehat{n}}_1}{k_0}\right](\rho +1)a-\rho \end{array}\}$$

(4)

where $\alpha \in (1-\frac{1}{\rho },1)$ , ${\kappa }_0={\mathrm{\lambda }}^{\rho }$ , $(\mathrm{\lambda },k_i)\in {\Re }^{+}$ are adjustable parameters selected as per Moreno ⁵⁶ and u is the system (1) control input. Here the notation $\lfloor a{\rceil }^b$ means it is equal to $|a{|}^b\mathrm{sign}(a)$ . The parameters $k_i,i=1,2,\dots (\rho +1)$ are properly tuned to make following matrix Hurwitz

$$E=\left[\begin{array}{ccccc}-k_1\hfill & 1\hfill & 0\hfill & \dots \hfill & 0\hfill \\ \multicolumn{1}{c}{-k_2}& 0\hfill & 1\hfill & \dots \hfill & 0\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{-k_{\rho }}& 0\hfill & 0\hfill & \dots \hfill & 1\hfill \\ \multicolumn{1}{c}{-k_{\rho +1}}& 0\hfill & 0\hfill & \dots \hfill & 0\hfill \end{array}\right]$$

(5)

The baseline idea of the HESO is same as that of ESO ⁵⁷ in which the observer states $[{\hat{\eta }}_1,{\hat{\eta }}_2,\dots {\hat{\eta }}_{\rho },]$ are considered as the estimations of the system states $[{\eta }_1,{\eta }_2,\dots ,{\eta }_{\rho }]$ whereas an extended state ${\hat{\eta }}_{\rho +1}$ estimates the total disturbances $d_i$ in the system.

Convergence analysis

In order to establish the convergence of the HESO equation (4), definitions of the homogeneity principle and finite-time stability along with relevant assumption and lemmas are presented herewith.

Definition 1

A function $\mathbb{V}$ : ${\Re }^n\to \Re$ is termed as homogeneous with degree d relative to weights $\{r_i>0{\}}_{i=1}^n$ , if $\forall i=1,2,\dots n$ ⁵⁸

$$\mathbb{V}({\varkappa }^{r_1}{x}_1,{\varkappa }^{r_2}{x}_2,\dots ,{\varkappa }^{r_n}{x}_n)={\varkappa }^d\mathbb{V}(x_1,x_2,\dots ,x_n)$$

(6)

for all $\varkappa >0$ and $(x_1,x_2,\dots ,x_n)\in {\Re }^n$ .

A vector field $\mathbb{G}$ : ${\Re }^n\to \Re$ is called homogeneous of degree d relative to weights $\{r_i>0{\}}_{i=1}^n$ , if $\forall i=1,2,\dots n$

$$\mathbb{G}({\varkappa }^{r_1}{x}_1,{\varkappa }^{r_2}{x}_2,\dots ,{\varkappa }^{r_n}{x}_n)={\varkappa }^{d+r_i}{\mathbb{G}}_i(x_1,x_2,\dots ,x_n)$$

(7)

for all $\varkappa >0$ and $(x_1,x_2,\dots ,x_n)\in {\Re }^n$ , where ${\mathbb{G}}_i$ is the $i^{\mathrm{th}}$ element of $\mathbb{G}$ .

If $\mathbb{V}$ satisfies equation (6) and is differentiable with respect to $x_n$ , then the partial derivative of $\mathbb{V}$ in $x_n$ follows

$$\begin{array}{l}{\varkappa }^{r_n}\frac{\partial }{\partial x_n}\mathbb{V}({\varkappa }^{r_1}{x}_1,{\varkappa }^{r_2}{x}_2,\dots ,{\varkappa }^{r_n}{x}_n)\\ ={\varkappa }^d\frac{\partial }{\partial x_n}\mathbb{V}(x_1,x_2,\dots ,x_n)\end{array}$$

(8)

Definition 2

The following system

$$\stackrel{·}{x}=f(x(t)),x(0)=x_0\in {\Re }^n$$

(9)

is globally finite-time stable, if it is Lyapunov stable, and $\forall x_0\in {\Re }^n\exists T(x_0)>0$ such that the solution of equation (9) satisfies $\underset{t\to T(x_0)}{lim}x(t)=0$ , and $x(t)=0\forall t\in [T(x_0),\infty )$ .

Let us denote the error variable as $e_1={\eta }_1-{\hat{\eta }}_1$ and error of the $j^{\mathrm{th}}$ state as $e_j={\eta }_j-{\hat{\eta }}_j$ for $j=2,\dots ,\rho$ , the observer dynamics equation (4) can be represented with change in coordinates as

$$\begin{array}{c}{\stackrel{·}{e}}_1=-{\mathrm{\lambda }}^{\rho -1}{k}_1\lfloor \frac{e_1}{k_0}{\rceil }^{\alpha }+e_2+d_1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{e}}_2=-{\mathrm{\lambda }}^{\rho -2}{k}_2\lfloor \frac{e_1}{k_0}{\rceil }^{2\alpha -1}+e_3+d_2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\stackrel{·}{e}}_{\rho -1}=-{\mathrm{\lambda }}^1{k}_{\rho -1}\lfloor \frac{e_1}{k_0}{\rceil }^{(\rho -1)\alpha -(\rho -2)}+e_{\rho }+d_{\rho -1}}\\ \multicolumn{1}{c}{{\stackrel{·}{e}}_{\rho }=-{\mathrm{\lambda }}^0{k}_{\rho }\lfloor \frac{e_1}{k_0}{\rceil }^{\rho \alpha -(\rho -1)}-{\hat{n}}_{\rho +1}+d_{\rho }+\tilde{a}}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{n}}}_{\rho +1}=-{\mathrm{\lambda }}^{-1}{k}_{\rho +1}\lfloor \frac{e_1}{k_0}{\rceil }^{(\rho +1)\alpha -\rho }}\end{array}\}$$

(10)

where $\tilde{a}=a(\eta )-a(\hat{\eta })$ is considered as Lipschitz bounded, that is, $|\stackrel{·}{\stackrel{~}{a}}|\le L$ , L is a constant.

Furthermore, let us follow the new coordinates as

$$\begin{array}{c}{\xi }_1=e_1\hfill \\ \multicolumn{1}{c}{{\xi }_i=e_i+\sum _{j=0}^{i=2}\frac{d^j{d}_{i-1-j}}{d{t}^j}}\\ ⋮\hfill \\ \multicolumn{1}{c}{{\xi }_{\rho +1}=-{\hat{\eta }}_{\rho +1}+\tilde{a}+\sum _{j=0}^{\rho -1}\frac{d^j{d}_{\rho -j}}{d{t}^j}}\end{array}\}$$

(11)

where, $i=2,\dots ,\rho$

Using equation (11), the coordinates of the system (10) are transformed as

$$(e_1,e_2,\dots ,e_{\rho },e_{\rho },{\hat{\eta }}_{\rho +1})\mapsto ({\xi }_1,{\xi }_2,\dots ,{\xi }_{\rho },{\xi }_{\rho +1}),$$

which immediately follows that

$$\begin{array}{c}{\stackrel{·}{\xi }}_1=-{\mathrm{\lambda }}^{\rho -1}{k}_1\lfloor \frac{{\xi }_1}{k_0}{\rceil }^{\alpha }+{\xi }_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\xi }}_2=-{\mathrm{\lambda }}^{\rho -2}{k}_2\lfloor \frac{{\xi }_1}{k_0}{\rceil }^{2\alpha -1}+{\xi }_3}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\stackrel{·}{\xi }}_{\rho -1}=-{\mathrm{\lambda }}^1{k}^{\rho -1}\lfloor \frac{{\xi }_1}{k_0}{\rceil }^{(\rho -1)\alpha -(\rho -2)}+{\xi }_{\rho }}\\ \multicolumn{1}{c}{{\stackrel{·}{\xi }}_{\rho }=-{\mathrm{\lambda }}^0{k}_{\rho }\lfloor \frac{{\xi }_1}{k_0}{\rceil }^{\rho \alpha -(\rho -1)}+{\xi }_{\rho +1}}\\ \multicolumn{1}{c}{{\stackrel{·}{\xi }}_{\rho +1}=-{\mathrm{\lambda }}^{-1}{k}_{\rho +1}\lfloor \frac{{\xi }_1}{k_0}{\rceil }^{(\rho +1)\alpha -\rho }+{\gamma }_1}\end{array}\}$$

(12)

where ${\gamma }_1=\stackrel{·}{\stackrel{~}{a}}+\sum _{j=1}^{\rho }\frac{d^j}{d{t}^j}{d}_{\rho +1-j}$ is the lumped disturbances.

Using the Lemma 4.2 of Bhat and Bernstein, ⁵⁸ Lemma 2, 3 and Theorem 2 of Guo and Zhao, ⁵² it can be shown that error equation (12) is a disturbed representation of global finite-time stable system $\stackrel{·}{y}=F_i(y),y\in {\Re }^{n_i+1}$ . Guo and Zhao ⁵² have shown that for the homogeneous global finite-time stable system, there exist a positive definite, radial unbounded, differentiable function $V_i:{\Re }^n\to \Re$ such that $V_i(x)$ is homogeneous of degree ${\gamma }_i$ w.r.t. to weights $\{r_{i,j}{\}}_{j=1}^{n_i}$ and the Lie derivative of $V_i$ along the vector field $F_i$ is negative definite. Moreover, $L_f{V}_i$ is homogeneous of degree ${\gamma }_i-d_i$ w.r.t. weights $\{r_{i,j}{\}}_{j=1}^{n_i}$ .

To test the above narration and check the homogeneity of $\partial V/\partial x_n$ , we consider $\rho =2,\mathrm{\lambda }=1,k_1=3,k_2=3,k_3=1$ in equation (12), then the vector field $F(\xi )$ is consider as

$$F(\xi )=\left[\begin{array}{c}{\xi }_2-{\varphi }_1(\xi )\hfill \\ \multicolumn{1}{c}{{\xi }_3-{\varphi }_2(\xi )}\\ \multicolumn{1}{c}{-{\varphi }_3(\xi )}\end{array}\right]$$

(13)

where, ${\varphi }_1(\xi )=3\lfloor \frac{{\xi }_1}{{\kappa }_0}{\rceil }^{\alpha }$ , ${\varphi }_2(\xi )=3\lfloor \frac{{\xi }_1}{{\kappa }_0}{\rceil }^{2\alpha -1}$ and ${\varphi }_3(\xi )=\lfloor \frac{{\xi }_1}{{\kappa }_0}{\rceil }^{3\alpha -2}$ . It is evident that the vector field $F(\xi )$ equation (13) is homogeneous of degree $(\alpha -1)$ with respect to the weights $\{1,\alpha ,2\alpha -1\}$ . Since the matrix E equation (5) is Hurwitz and for some $\alpha =(\frac{2}{3},1)$ , the system $\stackrel{·}{\xi }=F(\xi )$ is finite-time stable. Also, there exists a positive definite, radially unbounded function $V:{\Re }^3\to \Re$ such that V is homogeneous of degree $\gamma$ with respect to the weights $\{1,\alpha ,2\alpha -1\}$ , and $\frac{\partial V(y)}{\partial y_1}(y_2-g_1(y_1))+\frac{\partial V(y)}{\partial y_2}(y_3-g_2(y_1))-\frac{\partial V(y)}{\partial y_3}{g}_3(y_1)$ is negative definite and homogeneous of degree $(\gamma +\alpha -1)$ . Using Definition 2 and homogeneity of V, it can be shown that $|\frac{\partial V(y)}{\partial y_3}|$ is homogeneous of degree $(\gamma +1-2\alpha )$ . Moreover, the procedure to have proper selection of the gains $\mathrm{\lambda },\alpha$ for forcing the error dynamics to converge to zero in finite time has also been elaborated by Guo and Zhao. ⁵⁹ Consequently, it is clearly understood that the state ${\xi }_j$ equation (12) will go to zero in finite time. The condition for finite-time convergence of the observer state is given through following lemma:

Lemma 1

Considering the nonlinear system (1) with bounded and continuous disturbance $|d_i|$ , the HESO (4) is finite-time convergent if there exist ${\mathrm{\lambda }}_0>0$ s.t. for any $\mathrm{\lambda }\in (0,{\mathrm{\lambda }}_0)$ , there exists $T_{\mathrm{\lambda }}>0$ and $|{\eta }_j-{\hat{\eta }}_j|\le K_{j}v$ , $j=1,2,\dots ,\rho \forall t\ge T_{\mathrm{\lambda }}$ , where ${\eta }_j$ is the system states and ${\hat{\eta }}_j$ is the prediction of states. ⁵⁹ Constant $K_j$ is linked with the system (1) initial states, $v>0$ depends on $\mathrm{\lambda }$ and $T_{\mathrm{\lambda }}$ depends on $\mathrm{\lambda },{\eta }_j,{\hat{\eta }}_j$

Now, the overall design of the HESO-based second-order STC is elaborated.

Design of HESO-based second-order STC

In this section, a HESO (equation (4)) based continuous and finite-time control law relying on $2^{\mathrm{nd}}$ -order STC is designed which rejects both matched and unmatched system uncertainties and confirms that the control objective (equation (2)) is ensured. For this purpose, the following theorem is proposed:

Theorem 1

Considering the system (1) and the sliding surface defined as equation (3) and control law u as

$$\begin{array}{c}\multicolumn{1}{c}{u=-b^{-1}(\eta )(c_1{\hat{\eta }}_2+a(\hat{\eta })+\sum _{i=2}^{\rho }{c}_i({\hat{\eta }}_{i+1}\dots }\\ +{\mathrm{\lambda }}^{\rho -i}{k}_i\lfloor \frac{e_i}{{\kappa }_0}{\rceil }^{i\alpha -(i-1)}))+b^{-1}{u}_{\mathrm{sm}1}\hfill \end{array}$$

(14)

with

$$\begin{array}{c}\multicolumn{1}{c}{u_{\mathrm{sm}1}=-k_1|\sigma {|}^{\frac{1}{2}}\mathrm{sign}(\sigma )+u_{\mathrm{sm}2}}\\ \multicolumn{1}{c}{{\stackrel{·}{u}}_{\mathrm{sm}2}=-k_2\mathrm{sign}(\sigma )}\end{array}$$

(15)

where, $k_1>0$ and $k_2>\Xi$ are controller gains with $\Xi$ is some positive constant, then the HOSM is initiated on the sliding surface in finite time in the presence of matched and unmatched uncertainties.

Proof

The time-derivative of the sliding surface (equation (3)) is given as

$$\stackrel{·}{\sigma }=c_1{\stackrel{·}{\eta }}_1+\sum _{i=2}^{\rho }{c}_i{\stackrel{·}{\hat{\eta }}}_i\mathrm{with}{c}_{\rho }=1$$

(16)

Substituting equation (1) in equation (16), we get

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\sigma }=c_1{\eta }_2+c_1{d}_1+a(\hat{\eta })+b(\hat{\eta })u\dots }\\ +\sum _{i=2}^{\rho }{c}_i({\hat{\eta }}_{i+1}+{\mathrm{\lambda }}^{\rho -1}{k}_i\lfloor \frac{e_i}{{\kappa }_0}{\rceil }^{i\alpha -(i-1)})\hfill \end{array}$$

Since, with proper tuning, the HESO (equation (4)) will predict the actual data of the states hence substituting ${\eta }_2={\hat{\eta }}_2$ in the first equation of $\stackrel{·}{\sigma }$ will allow us to re-write the above equation as

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\sigma }=c_1{\hat{\eta }}_2+c_1{d}_1+a(\hat{\eta })+b(\hat{\eta })u\dots }\\ +\sum _{i=2}^{\rho }{c}_i({\hat{\eta }}_{\rho +1}+{\mathrm{\lambda }}^{\rho -1}{k}_i\lfloor \frac{e}{{\kappa }_0}{\rceil }^{i\alpha -(i-1)})\hfill \end{array}$$

(17)

Substituting the control input equation (14) in equation (17), we get

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\sigma }=c_1{d}_1-k_1|\sigma {|}^{\frac{1}{2}}\mathrm{sign}(\sigma )+u_{\mathrm{sm}2}}\\ \multicolumn{1}{c}{{\stackrel{·}{u}}_{\mathrm{sm}2}=-k_2\mathrm{sign}(\sigma )}\end{array}$$

(18)

Defining a new variable as $x_1=\sigma$ and $x_2=c_1{d}_1+u_{\mathrm{sm}2}$ and rewriting the above equation, we get

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{x}}_1=-k_1|x_1{|}^{\frac{1}{2}}\mathrm{sign}(x_1)+x_2}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2=-k_2\mathrm{sign}(x_1)+c_1{\stackrel{·}{d}}_1}\end{array}$$

(19)

Equation (19) is a second-order super-twisting algorithm.

Using Theorem 2, Page 1036 of Moreno and Osorio ⁶⁰ and its proof, it can be presented that the following Lyapunov function

$$V(x)={\zeta }^{T}P\zeta$$

(20)

is quadratic, strict and robust with symmetric and positive definite matrix P, will satisfy

$$\stackrel{·}{V}\le -|x_1{|}^{1/2}{\zeta }^{T}Q\zeta$$

(21)

almost everywhere, for symmetric and positive definite matrix Q. Moreover, the time taken by a trajectory originating at $x_0$ will meet to origin in a finite time smaller than ${\tilde{t}}_f(x_0)$ given by

$$\begin{array}{c}{\tilde{t}}_f(x_0)=\frac{2}{\mathrm{\Lambda }}{V}^{1/2}(x_0);\mathrm{\Lambda }=\frac{{\mathrm{\lambda }}_{\min }^{1/2}\{P\}{\mathrm{\lambda }}_{\min }\{Q\}}{{\mathrm{\lambda }}_{\max }\{P\}}\hfill \end{array}$$

(22)

The gains $k_1>0$ and $k_2>c_1\mathrm{\Delta }$ , $\mathrm{\Delta }>0$ along with the matrices P and Q of the Lyapunov function can be selected in accordance with the procedure given in Moreno and Osorio. ⁶⁰ Consequently, it is proved that the sliding variable $x_1$ and $x_2$ will go to zero in the finite time.

Performance analysis of different ESOs

In this section, a comparative analysis on the performances of different ESOs viz. linear ESO (LESO), nonlinear ESO (NLESO) and homogeneous ESO (HESO) are demonstrated with the help of a classical servo system.

A classical servo system is represented as

$$J\stackrel{··}{\theta }(t)=u(t)-d(t)$$

(23)

where, $J>0$ is the moment of inertia, $u(t)$ is control input, $\theta (t)$ represents the actual angular position and $d(t)$ is the lumped disturbances. Expressing equation (23) in an alternate form as

$$\stackrel{··}{\theta }(t)=\mathrm{bu}(t)+f(t)$$

(24)

where, $b=1/J$ , $f(t)=-(1/J)d(t)$ is a function with bounded derivative. For representing equation (24) in a standard canonical form, let the angular position $x_1=\theta$ , the angular speed $x_2=\stackrel{·}{\theta }$ then equation (24) can be written as

$$\begin{array}{c}{\stackrel{·}{x}}_1=x_2\hfill \\ {\stackrel{·}{x}}_2=f(t)+\mathrm{bu}(t)\hfill \end{array}\}$$

(25)

Considering $x_3=f(t)$ as disturbance state and $\rho (t)=\stackrel{·}{f}(t)$ , then equation (25) can be re-written as

$$\begin{array}{c}{\stackrel{·}{x}}_1=,x_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2=x_3+\mathrm{bu}(t)}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_3=\rho (t)}\end{array}\}$$

(26)

The structure of different extended state observers considered for estimation analysis of states $x_1,x_2$ and $x_3$ are presented as follows:

The LESO is designed as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\hat{x}}}_1={\hat{x}}_2-\frac{{\alpha }_1}{\epsilon }({\hat{x}}_1-\theta )}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{x}}}_2={\hat{x}}_3-\frac{{\alpha }_2}{{\epsilon }^2}({\hat{x}}_1-\theta )+\mathrm{bu}(t)}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{x}}}_3=-\frac{{\alpha }_3}{{\epsilon }^3}({\hat{x}}_1-\theta )}\end{array}\}$$

(27)

The NLESO is designed as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\hat{x}}}_1={\hat{x}}_2-\frac{{\alpha }_1}{\epsilon }({\hat{x}}_1-\theta )+\epsilon \varphi (\frac{{\hat{x}}_1-\theta }{{\epsilon }^2})}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{x}}}_2={\hat{x}}_3-\frac{{\alpha }_2}{{\epsilon }^2}({\hat{x}}_1-\theta )+\mathrm{bu}(t)}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{x}}}_3=-\frac{{\alpha }_3}{{\epsilon }^3}({\hat{x}}_1-\theta )}\end{array}\}$$

(28)

The HESO is designed as

$$\begin{array}{c}{\stackrel{·}{\hat{x}}}_1={\hat{x}}_2+a_1{\epsilon }^1{\left[\frac{{\theta }_1-{\hat{x}}_1}{{\epsilon }^2}\right]}^k\hfill \\ {\stackrel{·}{\hat{x}}}_2={\hat{x}}_3+a_2{\epsilon }^0{\left[\frac{{\theta }_1-{\hat{x}}_1}{{\epsilon }^2}\right]}^{2k-1}+\mathrm{bu}(t)\hfill \\ {\stackrel{·}{\hat{x}}}_3=a_3{\epsilon }^{-1}{\left[\frac{{\theta }_1-{\hat{x}}_1}{{\epsilon }^2}\right]}^{3k-2}\hfill \end{array}\}$$

(29)

The simulation using MATLAB has been performed using ESO variants given in equations (27)–(29) on the system (26) and the comparative performances results are shown in Figures 1–3. The system model parameters are taken as J = 10, d(t) = 0.3sin(t), f(t) = 0.3sin(t). The initial states of the plant are assumed at (0.5,0) while that of the observers are considered at (0,0,0). The settings used during simulation tests are α₁ = 5.5, α₂ = 10.5, α₃ = 6.1, ϵ = 0.01 and k = 0.8. The function ϕ(·) is a nonlinear function as used by Guo and Zhao. ⁵² The settings are kept same for all three observers to have better demonstration of the comparative performances. The results show that with the same tuning parameters, the HESO observer equation (29) has outperformed others in estimating the states x₁, x₂ as well as disturbance x₃. Figure 1 depicts the faster convergence property of HESO. Figures 2 and 3 illustrate that the despite of little higher peak in HESO, still it is performing well in estimation of other states. The peak obtained during initial time of HESO estimation can be reduced by adjusting the parameter ϵ which governs the speed of convergence.

Figure 1.

Profile of states $x_1$ and ${\hat{x}}_1$ .

Figure 2.

Profile of states $x_2$ and ${\hat{x}}_2$ .

Figure 3.

Profile of states $x_3$ and ${\hat{x}}_3$ .

An electromagnetic suspension (EMS) vehicle example

Nonlinear EMS system model

The EMS vehicle system nonlinear model shown in Figure 4 ⁶¹ is given by

$$\begin{array}{c}\multicolumn{1}{c}{B=K_b\frac{I}{G};F=K_f{B}^2}\\ \multicolumn{1}{c}{\frac{\mathrm{dI}}{\mathrm{dt}}=\frac{V_c-I{R}_c+\frac{N_c{A}_p{K}_b}{G^2}(\frac{d{z}_t}{\mathrm{dt}}-\frac{\mathrm{dZ}}{\mathrm{dt}})}{\frac{N_c{A}_p{K}_b}{G}+L_c}}\\ \multicolumn{1}{c}{M_s\frac{d^{2}Z}{d^{2}t}=M_{s}g-F}\\ \multicolumn{1}{c}{\frac{\mathrm{dG}}{\mathrm{dt}}=\frac{d{z}_t}{\mathrm{dt}}-\frac{\mathrm{dZ}}{\mathrm{dt}}}\end{array}\}$$

(30)

where I represent the current, $Z,z_t$ are the positions of electromagnet and the rail, $(\mathrm{dZ}/\mathrm{dt}),(d{z}_t/\mathrm{dt})$ are the vertical velocities of electromagnet and the rail, G is the air-gap, F is the force, B is the flux density and the coil voltage is $V_c$ . The other system parameters used in equation (30) are shown in Table 1.

Figure 4.

MAGLEV suspension vehicle. ⁶²

Table 1.

Parameters of the MAGLEV suspension vehicle.

Parameters	Meaning	Value
$M_s$	Boggie mass	1000 $\mathrm{Kg}$
$K_b$	Flux constant	0.0015 $\mathrm{T}.\mathrm{m}/\mathrm{A}$
$K_f$	Force constant	9810 $\mathrm{N}/{\mathrm{T}}^2$
$R_c$	Resistance of coil	10 $\Omega$
g	Gravitational const.	9.81 $\mathrm{m}/{\mathrm{s}}^2$
$L_c$	Inductance of coil	0.1 $\mathrm{H}$
$N_c$	coil turns	2000
$A_p$	Area of pole	0.01 ${\mathrm{m}}^2$
$B_0$	Nominal flux density	1.0 T
$F_0$	Minimal force	9810 N
$I_0$	Minimal current	10 A
$G_0$	Minimal air-gap	0.015 m
$V_0$	Minimal voltage	100 V

Linearized model

To transform the nonlinear MAGLEV model (equation (30)) into the selected design formation (equation (1)), the model linearization technique described by Michail ⁶¹ is used to apply the designed control technique. The linearized dynamic model of MAGLEV suspension vehicle operating with nominal values tabulated in Table 1 is written as

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{x}=\mathrm{Ax}+B_{u}u+B_{d}d(t)+\mathrm{\Delta }\mathrm{Ax}+O(x,u,d)}\\ \multicolumn{1}{c}{y=\mathrm{Cx}}\end{array}$$

(31)

where the states $x=[i,\stackrel{·}{z},(z_t-z){]}^T$ represent the current variations, electromagnet vertical velocity, and air-gap, the input $u=u_c$ denotes the voltage, $d(t)={\stackrel{·}{z}}_t$ is the disturbances generated due to vertical velocity of rail, the air gap variation $y=z_t-z$ is the controlled variable, $\mathrm{\Delta }A$ is the uncertainty matrix and $O(x,u,d)$ is the nonlinear function of high-order non-linearities. The matrices of system (31) are selected as

$$\begin{array}{c}A=\left[\begin{array}{ccc}\frac{-R_c}{L_c+K_b{N}_c\frac{A_p}{G_0}}\hfill & \frac{-K_b{N}_c{A}_p{I}_o}{G_0^2(L_c+K_b{N}_c\frac{A_p}{G_0})}\hfill & 0\hfill \\ \multicolumn{1}{c}{-2K_f\frac{I_0}{M_s{G}_0^2}}& 0\hfill & 2K_f\frac{I_o^2}{M_s{G}_0^3}\hfill \\ \multicolumn{1}{c}{0}& -1\hfill & 0\hfill \end{array}\right]\hfill \\ B=\left[\begin{array}{c}\frac{1}{L_c+K_b{N}_c\frac{A_p}{G_0}}\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right];B_d=\left[\begin{array}{c}\frac{K_b{N}_c{A}_p{I}_0}{G_0^2(L_c+K_b{N}_c\frac{A_p}{G_0})}\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{1}\end{array}\right]\hfill \\ C=\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\hfill \end{array}$$

(32)

The control requirement is to maintain the gap position in presence of the major disturbances which is the track input to the suspension originated vertically. The control specifications for the EMS under the influence of the deterministic track input are tabulated in Table 2. ⁶¹

Table 2.

Control specifications for EMS vehicle.

Constraints	Value
Max. air-gap change $(z_t-z{)}_p$	$\le 0.0075\mathrm{m}$
Max. coil voltage given $(u_{\mathrm{coil}}{)}_p$	$\le 300\mathrm{V}(3I_0{R}_c)$
Settling time $(t_s)$	$\le 3\mathrm{s}$
Steady air-gap error $(z_t-z{)}_{e_{\mathrm{ss}}}$	$=0$

Controller design

The high-order nonlinear expression $O(x,u,d)$ in equation (31) is considered as a lumped disturbances consisting exogenous disturbances and parametric variation, denoted as

$$d_l=B_{d}d+\mathrm{\Delta }\mathrm{Ax}+O(x,u,d)$$

(33)

It is worth to mention that the lumped disturbances (equation (33)) are generally very feeble as compared with the strong system dynamics and hence it is reasonable to assume that such disturbances can be compensated by the presented control method. Using equations (31) and (33), the complete dynamic model of the EMS vehicle is represented as

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{x}=\mathrm{Ax}+B_{u}u+B_l{d}_l}\\ \multicolumn{1}{c}{y=\mathrm{Cx}}\end{array}$$

(34)

where $B_l=I$ is a $3\times 3$ identity matrix. Next, by coordinate transformation method using $\eta =\mathrm{Tx}$ with $T=[C;\mathrm{CA};C{A}^2{]}^T$ , the original system is transformed into Byrnes-Isidori normal form ⁶³ with both matched and unmatched disturbances. The system (34) is then represented as

$$\stackrel{·}{\eta }=\overline{A}\eta +{\overline{B}}_{u}u+{\overline{B}}_l{d}_l$$

(35)

where $\overline{A}=\mathrm{TA}{T}^{-1}$ , ${\overline{B}}_u=T{B}_u$ and ${\overline{B}}_l=T{B}_l$ .

Substituting equation (32) in equation (35) gives

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\eta }}_1={\eta }_2+d_{l3}}\\ \multicolumn{1}{c}{{\stackrel{·}{\eta }}_2={\eta }_3-d_{l2}}\\ \multicolumn{1}{c}{{\stackrel{·}{\eta }}_3=C{A}^3{T}^{-1}\eta +C{A}^2{B}_{u}u+C{A}^2{B}_l{d}_l}\end{array}\}$$

(36)

where vector $d_l=[d_{l1},d_{l2},d_{l3}{]}^T$ represents lumped disturbances in the respective channels. It can be clearly observed from equation (36) that the EMS vehicle is affected by both matched $(C{A}^2{B}_l{d}_l)$ and mismatched $(d_{l3},d_{l2})$ disturbances which are not possible to attenuate by traditional SMC or even by well-known I-SMC method.

The sliding surface for system (36) as per equation (3) is selected as

$$\sigma =c_1{\eta }_1+c_2{\hat{\eta }}_2+c_3{\hat{\eta }}_3$$

(37)

where ${\hat{\eta }}_2,{\hat{\eta }}_3$ represent the estimate of ${\eta }_2,{\eta }_3$ , respectively. The fourth order HESO (equation (4)) used to obtain the estimates ${\hat{\eta }}_2$ and ${\hat{\eta }}_3$ in presence of disturbances is given as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\hat{\eta }}}_1={\hat{\eta }}_2+{\mathrm{\lambda }}^2{k}_1\lfloor \frac{{\eta }_1-{\hat{\eta }}_1}{{\kappa }_0}{\rceil }^{\alpha }}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{\eta }}}_2={\hat{\eta }}_3+{\mathrm{\lambda }}^1{k}_2\lfloor \frac{{\eta }_1-{\hat{\eta }}_1}{{\kappa }_0}{\rceil }^{2\alpha -1}}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{\eta }}}_3={\hat{\eta }}_4+{\mathrm{\lambda }}^0{k}_3\lfloor \frac{{\eta }_1-{\hat{\eta }}_1}{{\kappa }_0}{\rceil }^{3\alpha -2}+a(\hat{\eta })+b(\hat{\eta })u}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{\eta }}}_4={\mathrm{\lambda }}^{-1}{k}_4\lfloor \frac{{\eta }_1-{\hat{\eta }}_1}{{\kappa }_0}{\rceil }^{4\alpha -3}}\end{array}$$

(38)

where $k_i$ , $i=1,2,3$ are the gains selected according to the procedure mentioned by Guo and Zhao. ⁵⁹

Based on the above analysis, proposed HESO-based $2^{\mathrm{nd}}$ -order STC (equation (14)) can be tested on a EMS vehicle system without any restriction.

Simulation results

Simulation results obtained from MATLAB are presented here to corroborate the efficacy of the overall system performance. To better showcase the claim, the proposed control method is compared with the Integral sliding mode control (I-SMC), ⁶⁴ a popular and effective strategy to suppress the mismatched uncertainties.

The I-SMC is generally described with the sliding surface ${\sigma }_{\mathrm{ism}}=c_0{\eta }_2+c_1{\eta }_1+c_2\int {\eta }_1$ , has the control law formulated as

$$u=-b^{-1}(\eta )[a(\eta )+c_0{\eta }_3+c_1{\eta }_2+c_2{\eta }_1+\mathrm{ksign}({\sigma }_{\mathrm{ism}})]$$

The simulations tests are carried out on a full nonlinear EMS system under the influence of a measurement noise environment. The tuned control parameters of both the strategies are enlisted in Table 3.

Table 3.

Controller parameters of the EMS vehicle.

Controller	Parameters
Proposed	$c_1=100$ , $c_2=10$ , $c_3=1$ , $\mathrm{\lambda }=0.001$ , $\kappa ={\mathrm{\lambda }}^3$
	$k_1=10.25$ , $k_2=35.3$ , $k_3=50$ , $k_4=24.1$
	$\alpha =0.8$
I-SMC	$c_0=200$ , $c_1=100$ , $c_2=20$ , $c_3=1$ , $k=80$

I-SMC: integral sliding mode control.

External disturbance rejection

During simulation, the track input considered is shown in Figure 5. ⁶¹ It shows 5% gradient at a vehicle speed of $15\mathrm{m}/\mathrm{s}$ , a vertical acceleration of $0.5\mathrm{m}/{\mathrm{s}}^2$ and a jerk level of $1\mathrm{m}/{\mathrm{s}}^3$ . Practically, the track input disturbance would change continuously due to the rail variations. Furthermore, an extra time-dependent track input disturbance $z_t=0.1\sin (\pi t)\mathrm{m}/\mathrm{s}$ is enforced on the vehicle at $t=4$ second to resemble the real scenario. The initial states are assumed as $[i(0),\stackrel{·}{z}(0),z_t(0)-z(0){]}^T=[0,0,0.003{]}^T$ . The simulation outputs from I-SMC and proposed control method are revealed in Figures 6 –10.

Figure 5.

Profile of vehicle track input with 5% slope at 15 m/s vehicle speed.

Figure 6.

Air-gap $(z_t-z)$ v/s time response of EMS vehicle with external disturbances.

Figure 7.

Current (i) v/s time response of EMS vehicle with external disturbances.

Figure 8.

Vertical velocity $(\stackrel{·}{z})$ v/s time response of EMS vehicle with external disturbances.

Figure 9.

Control input $(u_{\mathrm{coil}})$ of EMS vehicle with external disturbances.

Figure 10.

Actual and estimated disturbances by proposed observer.

Figures 6–8 represent the states of the system and clearly depict that the proposed control method has demonstrated the nominal performance recovery property. From Figure 6, it is evident that in presence of high-order time varying disturbances (after $t>4\mathrm{s}$ ), I-SMC fails to stabilize the air gap distance and also unable to compensate the disturbances effectively, whereas the proposed control shows the finite-time regulation of the gap. The control input from the proposed method is smooth and chattering free as illustrated in Figure 9. The I-SMC is effective in canceling the offset caused by track input disturbances. However, the proposed scheme demonstrates the better disturbance rejection capabilities than I-SMC. Moreover, Figure 10 illustrates the capability of the HESO in estimating the disturbances perfectly in finite time and eliminating their effects.

Conclusions

This paper investigated both matched and mismatched disturbance attenuation problem for MCSs. A new HESO has been combined with second-order STC to handle the high-order disturbances consequences on the output in the finite time. The main contribution is to design an effective strategy which incorporates the estimations of non-vanishing disturbances whose bounds are not known a priori and compel the states onto the sliding surface and the output reaches to equilibrium in the finite time even under the influence of high-order mismatched disturbances. Simulation results of a EMS vehicle system have exhibited that the presented control method is better than I-SMC in terms of improved dynamics and better normal performance in the presence of exogenous disturbances.