Disturbance attenuation–based sliding mode control with disturbance observer for mismatched uncertain system

Abstract

A disturbance attenuation–based sliding mode control approach with an extended disturbance observer is proposed for systems with mismatched uncertainties. A novel adaptive sliding surface consisting of the disturbance estimation is presented to eliminate the effect of mismatched disturbance in the sliding mode. The proposed method exhibits the following two attractive features. First, the asymptotical stability of adaptive sliding mode can be guaranteed even if the disturbance estimation error of the disturbance observer exists. Second, the nominal performance of the proposed approach is close to that of the traditional sliding mode control method in the absence of uncertainties. Finally, simulation results of the numerical and application examples show that the proposed nonlinear sliding mode control approach has the better dynamic performance as well as robustness and chattering reduction compared with other nonlinear sliding mode control methods.

Keywords

Sliding mode control mismatched uncertainties chattering reduction disturbance observer robustness

Introduction

It is well known that sliding mode control (SMC) is robust to matched uncertainties since it has the property of invariance on the sliding mode in the presence of matched uncertainties and disturbances. SMC method as an effective and popular control strategy has been widely used in diverse fields in order to obtain satisfactory performance.^1,2 However, uncertainties existing in control systems may not necessarily satisfy the matching criteria, examples can be seen in a lot of practical systems such as permanent magnet synchronous motors,^3,4 MAGnetic LEVitation (MAGLEV) suspension system,⁵ flexible joint manipulator,⁶ and control system for missiles.^7,8 When the system is subjected to unmatched uncertainty, invariance and robustness with traditional SMC would fail due to the fact that SMC is not satisfied with matching condition.

In order to eliminate the effect, the mismatched uncertainties in practical applications, many authors are committed to researching activities to address the problem of mismatched uncertainty in SMC. In general, these methods can be divided into the following two categories.

The first category mainly uses different control strategies to reduce the impact of mismatched uncertainties on the stability of the system, such as linear matrix inequality (LMI)-based approach,⁹ the Riccati approach,¹⁰ fuzzy and neural network based control,^11–13 adaptive approach,^14–17 invariant ellipsoid method,¹⁸ and integral SMC approach.^19–22 The uncertainty of the mismatch uncertainties requires the bounds or vanity of the H2 norm in above methods. However, it is not a reasonable assumption of the actual system. For example, lumped uncertainties may have nonzero steady-state values, and there are no bounded H2 norms.^23,24

The second category to address the problem of mismatched uncertain systems is usually use the method composed of the backstepping and SMC method.^25,26 The key idea of these approaches is based on a virtual control method in frame of the backstepping approach and the state to reach the surface and slide to the desired equilibrium using the SMC method. SMC method plays an important role in these methods. It is an effective control method that the traditional sliding mode with the disturbance observer (DO) for mismatched nonlinear systems.^27,28 The approach can reduce the effect of the mismatched uncertainties by employing the disturbance estimation of DO. However, the dynamic performance in the sliding mode is seriously subject to the estimation error of the mismatch uncertainties.

In this article, a new approach is proposed based on the second category method, and the main contributions are listed as follows:

The adaptive sliding surface with DO is proposed in order that the sliding motion of SMC is insensitive to the mismatched disturbance.

The effect of estimation error associated with mismatched uncertainties can be eliminated so as to guarantee the asymptotical stability of system.

The sliding surface can be designed when the initial states are unknown.

The control with adaptive term is modified to alleviate the chattering problem further. The proposed approach can be implemented when the bound on the estimation of uncertainty is not known exactly.

The remaining part of the article is organized as follows: the problem and objective are stated in section “Problem formulation.” The control approach consisting of a novel adaptive sliding surface with DO is described. The stability is analyzed in section “Disturbance attenuation–based SMC method.” The performance is illustrated by simulation examples in section “Numerical example.” Finally, conclusions are reported in section “Conclusion.”

Problem formulation

Consider the following system

\begin{matrix} {\overset{\cdot}{x}}_{1} & = x_{2} + d (t) \\ {\overset{\cdot}{x}}_{2} & = a (x) + b (x) u \\ y & = x_{1} \end{matrix}

(1)

where $x_{1}$ and $x_{2}$ are the states, u is the control input, $d (t)$ is the unmeasurable disturbance, y is the output, and $b (x) \neq 0$ .

Assumption 1

The disturbance in system (1) is bounded and defined by $d^{*} = sup_{t > 0} | d (t) |$ .

Assumption 2

The disturbance in system (1) is continuous and satisfies

| \frac{d^{j} d (t)}{d t^{j}} | \leq μ j = 0, 1, 2, \dots, r

(2)

where $μ$ is a positive number.

Remark 1

The class of disturbance in Assumption 2 is much larger than the disturbance in Assumption 1. It may be noted that the disturbance in Assumption 2 is not required to know the bound $μ$ .

The objective is to design control u so that the system output is not affected by the mismatched uncertainty $d (t)$ .

SMC with DO

The sliding mode surface for system (1) affected by the mismatched disturbance is designed as follows⁵

σ = x_{2} + c x_{1} + \hat{d}

(3)

where $c > 0$ is a control parameter to be designed and $\hat{d}$ is the disturbance estimate of $d (t)$ . Then, DO can be written as follows⁵

\begin{matrix} \overset{\cdot}{p} = - l g_{2} p - l [g_{2} lx + f (x) + g_{1} (x) u] \\ \hat{d} = p + lx \end{matrix}

(4)

where $f (x) = [x_{2} a (x)]^{T}$ , $x = [x_{1} x_{2}]^{T}$ , $g_{1} (x) = [0 b (x)]^{T}$ , $g_{2} = [1 0]^{T}$ , and $\hat{d}$ , p, and l are the estimations of the disturbance $d (t)$ , the internal state of nonlinear observer (4), and the observer gain to be designed when $l g_{2} > 0$ holds, respectively.

Accordingly, the SMC law is designed as follows⁵

u = - b^{- 1} (x) [a (x) + c (x_{2} + \hat{d}) + ksgn (σ)]

(5)

Combining equations (1), (3), (4), and (5) leads to

\overset{\cdot}{σ} = - ksgn (σ) + (c + l g_{2}) (d - \hat{d})

(6)

According to equation (6), the states in system (1), which stay initially outside the sliding surface, will reach the sliding surface $σ = 0$ in finite time when the switch gain in the control law (equation (5)) satisfies $k > (c + l g_{2}) (d - \hat{d})$ .

When $σ = 0$ , the sliding motion in equation (3) is obtained as follows

{\overset{\cdot}{x}}_{1} = - c x_{1} + d - \hat{d}

(7)

Remark 2

The state in equation (7) cannot be driven to the desired equilibrium point asymptotically based on the sliding mode control with disturbance observer (DO-SMC) law (equation (5)) when the derivation of the disturbance $d (t)$ is bounded and satisfies $lim_{t \to \infty} \overset{\cdot}{d} (t) \neq 0$ . As a consequence, the DO-SMC (equation (4)) is sensitive to estimation error of the mismatched disturbance.

Extended DO-modified SMC

The modified sliding mode surface for system (1) affected by the mismatched disturbance is designed as follows⁶

\begin{matrix} σ^{*} = σ - σ (0) e^{- α t} \end{matrix}

(8)

where $σ$ is given in equation (3) and $α$ is a positive constant. The extended DO can be defined as follows⁶

\begin{matrix} \hat{d} & = p_{11} + l_{11} x_{1} \\ {\overset{\cdot}{p}}_{11} & = - l_{11} (x_{2} + \hat{d}) + \hat{\overset{\cdot}{d}} \\ \hat{\overset{\cdot}{d}} & = p_{12} + l_{12} x_{1} \\ {\overset{\cdot}{p}}_{12} & = - l_{12} (x_{2} + \hat{d}) \end{matrix}

(9)

where $\hat{d}$ and $\hat{\overset{\cdot}{d}}$ are the estimators of the disturbances $d (t)$ and $\overset{\cdot}{d} (t)$ , respectively, $p_{11}$ and $p_{12}$ are auxiliary variables, and $l_{11}$ and $l_{12}$ are positive constants. The estimation error of the extended DO is bound by $| | e_{d} | | \leq λ$ , where $e_{d} = [d - \hat{d}, \overset{\cdot}{d} - \hat{\overset{\cdot}{d}}]^{T}$ , $λ > 0$ is relative with $μ$ .⁶

The extended disturbance observer–modified sliding mode control (EDO-MSMC) law is designed as follows⁶

\begin{array}{l} u = - b^{- 1} (x) \\ [a (x) + c (x_{2} + \hat{d}) + α σ (0) e^{- α t} + k_{l} σ^{*} + k_{s} s g n (σ^{*})] \end{array}

(10)

Similarly, combining equations (1), (8), and (10) leads to

{\overset{\cdot}{σ}}^{*} = - k_{l} σ^{*} - k_{s} sgn (σ^{*}) + c (d - \hat{d}) + \overset{\cdot}{\hat{d}}

(11)

where $k_{l} > 0$ and $k_{s} > | c (d - \hat{d}) + \overset{\cdot}{\hat{d}} |$ .

When $σ^{*} = 0$ in equation (8), the sliding motion can be obtained as follows

{\overset{\cdot}{x}}_{1} = - c x_{1} + d - \hat{d} + σ (0) e^{- α t}

(12)

Remark 3

EDO-MSMC can be obtained when the initial state of system is known. In addition, the state $x_{1}$ is not asymptotically stable and $| | x_{1} | | \leq λ$ , where $λ$ depends on the estimation error. It is noted that the estimation error of mismatched uncertainties will influence the stability of the state $x_{1}$ and the effect of chattering on the sliding mode cannot be totally eliminated in the above approaches; thus, the novel adaptive sliding surface with extended DO is proposed to improve the dynamical performance.

Disturbance attenuation–based SMC method

Adaptive SMC with extended DO

The novel adaptive sliding surface with extended DO (equation (9)) for system (1) is designed as follows

\bar{σ} = x_{2} + \frac{c x_{1}}{| x_{1} | + k^{2} e^{- β t}} + \hat{d}

(13)

where $\overset{\cdot}{k} = - (c γ | x_{1} | k e^{- β t} / (| x_{1} | + k^{2} e^{- β t})) - γ k$ , $k (0) > 0$ , $γ > 0$ , and $β > 0$ .

Stability of sliding mode dynamics

When the designed sliding surface $\bar{σ} = 0$ , it can be obtained as

{\overset{\cdot}{x}}_{1} = - \frac{c x_{1}}{| x_{1} | + k^{2} e^{- β t}} + d - \hat{d}

(14)

The other candidate Lyapunov function is as follows

V (x_{1}, k) = \frac{x_{1}^{2}}{2} + \frac{k^{2}}{2 γ}

(15)

Accordingly, the derivative of $V (x_{1}, k)$ in equation (15) is

\overset{\cdot}{V} (x_{1}, k) = - \frac{{cx}_{1}^{2}}{| x_{1} | + k^{2} e^{- β t}} + (d - \hat{d}) x_{1} + \frac{k \overset{\cdot}{k}}{γ}

(16)

Using adaptive parameter $\overset{\cdot}{k}$ into equation (16) yields

\begin{matrix} \overset{\cdot}{V} (x_{1}, k) & = - c | x_{1} | + (d - \hat{d}) | x_{1} | - k^{2} \\ \leq - c | x_{1} | + | d - \hat{d} | | x_{1} | - k^{2} \\ = - (c - | d - \hat{d} |) | x_{1} | - k^{2} \\ \leq - (c - λ) | x_{1} | - k^{2} \end{matrix}

(17)

When $c > λ$ , it can be verified that the sliding surface (17) is exponentially stable according to Lyapunov stability theory.

Stability of SMC

Theorem 1

Suppose that Assumption 2 holds for system (1) and the sliding mode is chosen as equation (13), the closed-loop system is asymptotically stable when the control law is designed as

\begin{matrix} u = & - b^{- 1} (x) [a (x) + \frac{ck e^{- β t} (x_{2} k + k \hat{d} - 2 x_{1} \overset{\cdot}{k} + x_{1} k β)}{{(| x_{1} | + k^{2} e^{- β t})}^{2}} \\ + k_{1} \bar{σ} + \frac{k_{2}}{| x_{1} | + k^{2} e^{- β t}} sgn (\bar{σ}) + k_{3} sgn (\bar{σ})] \end{matrix}

(18)

where $k_{1} > 0$ , $k_{2} > c λ$ , and $k_{3} > λ + μ$ .

Proof

Taking the derivative of the sliding surface $\bar{σ}$ defined in equation (13) yields

\overset{\cdot}{\bar{σ}} = a (x) + bu + \frac{ck e^{- β t} (x_{2} k + kd - 2 x_{1} \overset{\cdot}{k})}{{(| x_{1} | + k^{2} e^{- β t})}^{2}} + \overset{\cdot}{\hat{d}}

(19)

Using the control law (18) into (19) leads to

\overset{\cdot}{\bar{σ}} = - k_{1} \bar{σ} - k_{2} sgn (\bar{σ}) + \frac{c k^{2} e^{- β t} (d - \hat{d})}{{(| x_{1} | + k^{2} e^{- β t})}^{2}} + \overset{\cdot}{\hat{d}}

(20)

The candidate Lyapunov function is as follows

V (\bar{σ}) = \frac{{\bar{σ}}^{2}}{2}

(21)

Accordingly, the derivative of $V (\bar{σ})$ in equation (21) is

\begin{matrix} \overset{\cdot}{V} (\bar{σ}) & = - k_{1} {\bar{σ}}^{2} - \frac{k_{2}}{| x_{1} | + k^{2} e^{- β t}} | \bar{σ} | - k_{3} | \bar{σ} | \\ + \frac{c k^{2} e^{- β t} (d - \hat{d}) \bar{σ}}{{(| x_{1} | + k^{2} e^{- β t})}^{2}} + \overset{\cdot}{\hat{d}} \bar{σ} \\ \leq - k_{1} {\bar{σ}}^{2} - \frac{k_{2} - c | d - \hat{d} |}{| x_{1} | + k^{2} e^{- β t}} - (k_{3} - | \overset{\cdot}{\hat{d}} |) | \bar{σ} | \end{matrix}

(22)

Considering that $| | e_{d} | | \leq λ$ and $| \overset{\cdot}{d} | \leq μ$ in the extended DO, then $| \overset{\cdot}{\hat{d}} | = | \overset{\cdot}{d} - \overset{\cdot}{\hat{d}} | + | \overset{\cdot}{d} | \leq λ + μ$ , equation (18) can be rewritten as follows

\overset{\cdot}{V} (\bar{σ}) \leq - k_{1} {\bar{σ}}^{2} - \frac{k_{2} - c λ}{| x_{1} | + k^{2} e^{- β t}} | \bar{σ} | - (k_{3} - λ - μ) | \bar{σ} |

(23)

Considering that $k_{1} > 0$ , $k_{2} > c λ$ , and $k_{3} > λ + μ$ , it can be derived from equation (23) that the system will reach the designed sliding surface $\bar{σ} = 0$ in finite time according to Lyapunov stability theory.

Remark 4

The sliding mode in the adaptive sliding mode control with extended disturbance observer (EDO-ASMC) method is insensitive to unmatched uncertainties even if the disturbance estimation error of the DO exists.

Numerical example

Example 1

Considering the following illustrative example^5,6

\begin{matrix} {\overset{\cdot}{x}}_{1} & = x_{2} + d (t) \\ {\overset{\cdot}{x}}_{2} & = - 2 x_{1} - x_{2} + e^{x_{1}} + u \\ y & = x_{1} \end{matrix}

(24)

Constant disturbance

Here, the proposed control method is evaluated using case 2.⁵ For this illustration, a constant disturbance $d = 0.5$ is applied at $t = 2 s$ and the initial state value is $x (0) = [0, 0]^{T}$ . For comparison studies, four methods including improved nonlinear sliding mode control (INSMC),²⁹ the DO-SMC,⁵ EDO-MSMC⁶, and the EDO-ASMC are employed in the control design for system (24). The control parameters of all the four control methods are listed in Table 1.

Table 1.

Control parameters for the numerical example case 1.

Controllers	Parameters
INSMC	$F = 5$ , $P = 0.1$ , $k = 1$ , $β = 1$ , $k_{1} = 0$ , $k_{2} = 3$
DO-SMC	$c = 5, k = 3, l = [6, 0]$
EDO-MSMC	$c = 5, k_{l} = 1, k_{s} = 1.5, l_{11} = 50, l_{12} = 10$
EDO-ASMC	$c = 1, β = 2, k_{1} = 1, k_{2} = 0.005, k_{3} = 1, l_{11} = 50, l_{12} = 10$

INSMC: improved nonlinear sliding mode control; DO-SMC: sliding mode control with disturbance observer; EDO-MSMC: extended disturbance observer–modified sliding mode control; EDO-ASMC: adaptive sliding mode control with extended disturbance observer.

It can be seen from Figure 1 that the performances are similar for the DO-SMC, EDO-MSMC, and the EDO-ASMC, but DO-SMC and EDO-MSMC have the substantial chattering due to the fact that the switch control is a discontinuous process, while the chattering problem in the proposed EDO-ASMC can be almost eliminated. However, the INSMC method failed to drive the state $x_{1}$ to the desired equilibrium, which the INSMC method is sensitive to mismatched disturbance.

Figure 1.

Comparison among INSMC, DO-SMC, EDO-MSMC, and EDO-ASMC: (a) plot of x₁, (b) plot of x₂, and (c) plot of u.

Complex disturbance

In this case, the proposed method is evaluated using the complex mismatched disturbance.⁶

The unmatched disturbance can be defined as follows

d_{1} (t) = {\begin{matrix} \frac{t}{6} + \overset{2}{\sin} (2 t) - \cos (2 t) t < 2 \\ \frac{t}{6} + \overset{2}{\sin} (2 t) - \cos (2 t) + 1 t \geq 2 \end{matrix}

(25)

and a step change appears after 2 s. The initial state value is set as $x (0) = [1, 0]^{T}$ . The control parameters of all the three control methods are listed in Table 2.

Table 2.

Control parameters for the numerical example case 2.

Controllers	Parameters
DO-SMC	$c = 5, k = 5, l = [10, 0]$
EDO-MSMC	$c = 5, k_{l} = 3, k_{s} = 5, l_{11} = 50, l_{12} = 10$
EDO-ASMC	$c = 5, β = 2, k_{1} = 1, k_{2} = 0.005, k_{3} = 1, l_{11} = 50, l_{12} = 10$

DO-SMC: sliding mode control with disturbance observer; EDO-MSMC: extended disturbance observer–modified sliding mode control; EDO-ASMC: adaptive sliding mode control with extended disturbance observer.

It can be observed from Figure 2(a) that the state $x_{1}$ of the proposed control method can rapidly converge to desired equilibrium in the presence of complex mismatched disturbance; on the contrary, the adverse control effects, such as the undesired overshooting, oscillation, and unsatisfactory settling time, appear in the DO-SMC and the EDO-MSMC method. The performances of the state $x_{2}$ are similar for the three controls after 2.5 s in Figure 1(b). It is also noticed from Figure 2(c) that the DO-SMC and the EDO-MSMC result in the chattering in the control input response, while the chattering in the EDO-ASMC can be almost eliminated. Moreover, the proposed method is adapted best to the complex disturbance compared with the other two methods and is insensitive to the initial of the complex mismatched disturbance at t = 2 s.

Figure 2.

Comparison among DO-SMC, EDO-MSMC, and EDO-ASMC in the presence of complex disturbances: (a) plot of x₁, (b) plot of x₂, and (c) plot of u.

Example 2: MAGLEV suspension system

MAGLEV suspension dynamic model

The dynamic model of a MAGLEV suspension system, which is subjected to the mismatched disturbance, is as follows⁵

{\begin{matrix} \overset{\cdot}{x} = Ax + B_{u} u + B_{d} d \\ y = Cx \end{matrix}

(26)

where the states are the current, vertical electromagnet velocity, and air gap, that is, $x = [i, \overset{\cdot}{z}, (z_{t} - z)]$ , the input $u = u_{coil}$ is the voltage, and the track input $d = {\overset{\cdot}{z}}_{t}$ is the rail vertical velocity. The controlled variable is the variation of air gap, that is, $y = z_{t} - z$ . The detailed modeling produce can be found in the work by Michail.³⁰ Here, the state matrix A, the input matrix $B_{u}$ , the disturbance matrix $B_{d}$ , and the output matrix C are following

A = [\begin{matrix} \frac{- R_{c}}{L_{c} + K_{b} N_{c} \frac{A_{p}}{G_{0}}} & \frac{- K_{b} N_{c} A_{p} I_{0}}{G_{0}^{2} (L_{c} + K_{b} N_{c} \frac{A_{p}}{G_{0}})} & 0 \\ - 2 K_{f} \frac{I_{0}}{M_{s} G_{0}^{2}} & 0 & 2 K_{f} \frac{I_{0}^{2}}{M_{s} G_{0}^{3}} \\ 0 & - 1 & 0 \end{matrix}]

(27)

B_{u} = {[\begin{matrix} \frac{1}{L_{c} + K_{b} N_{c} \frac{A_{p}}{G_{0}}} & 0 & 0 \end{matrix}]}^{T}

(28)

B_{d} = {[\begin{matrix} \frac{K_{b} N_{c} A_{p} I_{0}}{G_{0}^{2} (L_{c} + K_{b} N_{c} \frac{A_{p}}{G_{0}})} & 0 & 1 \end{matrix}]}^{T}

(29)

C = [\begin{matrix} 0 & 0 & 1 \end{matrix}]

(30)

The physical meanings of the parameters in equations (27)–(30) are listed in Table 3. The major external disturbance in the MAGLEV suspension system comes from deterministic inputs to the suspension in the vertical direction. Such deterministic inputs are the transitions onto the track gradients. Deterministic input components considered here are referred Michail³⁰ and shown in Figure 3. They represent a gradient of $5 %$ at a vehicle speed of 15 m/s, while the jerk level is $1 m / s^{3}$ . The control specifications of the MAGLEV system under consideration of the deterministic track input are given in Table 4.

Table 3.

Parameters of MAGLEV suspension system.

Parameters	Meaning	Value
$M_{s}$	Carriage mass	1000 kg
$F_{0}$	Nominal force	9810 N
$G_{0}$	Nominal air gap	0.015 m
$R_{c}$	Coil’s resistance	$10 Ω$
$B_{0}$	Nominal flux density	1 T
$L_{c}$	Coil’s inductance	0.1 H
$I_{0}$	Nominal current	10 A
$N_{c}$	Number of turns	2000
$V_{0}$	Nominal voltage	1000 V
$A_{p}$	Pole face area	$0.01 m^{2}$

MAGLEV: MAGnetic LEVitation.

Figure 3.

Track input to the suspension with a vehicle speed of 15 m/s and 5% gradient.

Table 4.

Constraints for MAGLEV suspension system.

Constraints	Value
Maximum air gap deviation $((z_{t} - z)_{p})$	$\leq 0.0075 m$
Maximum input coil voltage $((u_{coil})_{p})$	$\leq 300 V (3 I_{0} R_{c})$
Settling time $(t_{s})$	$\leq 3 s$
Air gap steady-state error $((z_{t} - z)_{ess})$	0

MAGLEV: MAGnetic LEVitation.

Control design

In order to implement the proposed method for the MAGLEV system, the following coordinate transformation is introduced

ξ = Px

(31)

where $P = [\begin{matrix} C & CA & C A^{2} \end{matrix}]^{T}$ .

Using this coordinate transformation, the MAGLEV system can be rewritten as follows

\overset{\cdot}{ξ} = \bar{A} ξ + {\bar{B}}_{u} u + {\bar{B}}_{d} d

(32)

where $\bar{A} = PA P^{- 1}$ , ${\bar{B}}_{u} = P B_{u}$ , and ${\bar{B}}_{d} = P B_{d}$ . Using the parameters in Table 3 into equation (32) leads to

{\begin{matrix} {\overset{\cdot}{ξ}}_{1} = ξ_{2} + d \\ {\overset{\cdot}{ξ}}_{2} = ξ_{3} \\ {\overset{\cdot}{ξ}}_{3} = C A^{3} P^{- 1} ξ + C A^{2} B_{u} u + C A^{2} B_{d} d \end{matrix}

(33)

The following extended DO (9) is employed to estimate the disturbance in the MAGLEV system.

In order to use the proposed method for high-order system (33), the backstepping method is also applied to design the control law.

First, the proposed method is used for the states $ξ_{1}$ and $ξ_{2}$ in system (33), and the sliding mode $σ_{ξ} = ξ_{2} + (c ξ_{1} / (| ξ_{1} | + k^{2} e^{- β t})) + \hat{d}$ , where $c > λ$ , $k_{1} > 0$ , $k_{2} > c λ$ , and $k_{3} > λ + μ$ , can be obtained.

Second, the virtual control $ξ_{3 v}$ using the proposed method is chosen as follows

\begin{matrix} ξ_{3 v} = - \frac{ck e^{- β t} (k ξ_{2} - k \hat{d} - 2 ξ_{1} \overset{\cdot}{k} + ξ_{1} k β)}{{(| ξ_{1} | + k^{2} e^{- β t})}^{2}} - k_{1} σ_{ξ} - k_{2} sgn (σ_{ξ}) \end{matrix}

(34)

Let $e_{3} = ξ_{3} - ξ_{3 v}$ , the EDO-ASMC law for the MAGLEV suspension system is as follows

\begin{array}{l} u = - {(C A^{2} B_{u})}^{- 1} \\ [C A^{3} P^{- 1} ξ + C A^{2} B_{d} \hat{d} - {\dot{ξ}}_{3 v} + k_{1 e} e_{3} + k_{2 e} s g n (e_{3})] \end{array}

(35)

where $k_{1 e} > 0$ and $k_{2 e} > C A^{2} B_{d} | \hat{d} - d |$ .

Simulation results

In order to evaluate the performance of the proposed method, the proposed method is compared with the DO-SMC and the EDO-MSMC. The sliding surface of the DO-SMC method for the MAGLEV suspension system is designed as follows

σ_{1} = c_{1} ξ_{1} + c_{2} (ξ_{2} + \hat{d}) + c_{3} ξ_{3}

(36)

where the parameter $c_{i} (i = 1, 2, 3)$ satisfies the Hurwitz polynomial $c_{3} s^{2} + c_{2} s + c_{1} = 0$ . Then, DO can be obtained as follows

{\begin{matrix} \overset{\cdot}{p} = - L {\bar{B}}_{d} (p + L ξ) - L (\bar{A} ξ + {\bar{B}}_{u} u) \\ \hat{d} = p + L ξ \end{matrix}

(37)

where $\hat{d}$ is the disturbance estimate, p is an auxiliary vector, and L is the observer gain matrix to be designed. Accordingly, the corresponding DO-SMC law is as follows

\begin{array}{l} u = - {(C A^{2} B_{u})}^{- 1} \\ {C A^{3} P^{- 1} ξ + C A^{2} B_{d} \hat{d} + c_{3}^{- 1} [k_{σ} s g n (σ_{1}) + c_{1} (ξ_{2} + \hat{d}) + c_{2} ξ_{3}]} \end{array}

(38)

The sliding surface of the EDO-MSMC method for the MAGLEV suspension system is designed as follows

σ_{2} = σ_{21} + σ_{21} (0) e^{- α t}

(39)

where $σ_{21} = {\bar{c}}_{1} ξ_{1} + {\bar{c}}_{2} (ξ_{2} + \hat{d}) + {\bar{c}}_{3} ξ_{3} + {\hat{\overset{\cdot}{d}}}_{1}$ , $d_{1}$ and ${\hat{d}}_{1}$ are estimates of d and $\overset{\cdot}{d}$ , respectively, and the parameter $c_{i} (i = 1, 2, 3)$ also satisfies the Hurwitz polynomial ${\bar{c}}_{3} s^{2} + {\bar{c}}_{2} s + {\bar{c}}_{1} = 0$ .

The EDO can be obtained as follows

{\begin{matrix} {\hat{d}}_{1}^{j - 1} = p_{1 j} + l_{1 j} ξ_{1} \\ {\overset{\cdot}{p}}_{1 j} = - l_{1 j} (ξ_{2} + {\hat{d}}_{1}) + {\hat{d}}_{1}^{j} \\ {\overset{\cdot}{p}}_{13} = - l_{13} (ξ_{2} + {\hat{d}}_{1}) \end{matrix}

(40)

where $p_{1 j}$ is the auxiliary variable, $j = 1, 2$ , and $l_{1 r}$ is the observer gain matrix to be designed, $r = 1, 2, 3$ .

Accordingly, the corresponding EDO-MSMC law is as follows

\begin{array}{l} u = - {(C A^{2} B_{u})}^{- 1} \\ {C A^{3} P^{- 1} ξ + C A^{2} B_{d} {\hat{d}}_{1} + {\bar{c}}_{3}^{- 1} [α σ_{21} (0) e^{- α t} + k_{l} σ_{2} + k_{s} s g n (σ_{2}) + {\bar{c}}_{1} (ξ_{2} + {\hat{d}}_{1}) + {\bar{c}}_{2} (ξ_{3} + {\hat{d}}_{1})] + {\dot{\hat{\dot{d}}}}_{1}} \end{array}

(41)

The initial state values of MAGLEV suspension system (26) are set as $[\begin{matrix} i (0) & \overset{\cdot}{z} (0) & (z_{t} - z) (0) \end{matrix}]^{T} = [\begin{matrix} 0 & 0 & 0.03 \end{matrix}]^{T}$ . The external disturbance, as displayed in Figure 3, is imposed on the system. For comparison studies, the DO-SMC and EDO-MSMC are also employed in the MAGLEV suspension system. The parameters of three control methods are listed in Table 5. Response curves of the states and inputs of the MAGLEV system are shown in Figure 4. When the system suffers from the mismatched disturbance, the proposed EDO-ASMC obtains better disturbance rejection performance than DO-SMC and EDO-MSMC. The chattering in the sliding mode has been almost eliminated in the EDO-ASMC, while the chattering still exists in the sliding mode of the DO-SMC and the EDO-MSMC.

Table 5.

Control parameters for the MAGLEV suspension in case 1.

Controllers	Parameters
DO-SMC	$c_{1} = 100, c_{2} = 20, c_{3} = 1, k = 60, l = [100, 0, 0]$
EDO-MSMC	${\bar{c}}_{1} = 100, {\bar{c}}_{2} = 20, {\bar{c}}_{3} = 1, k_{l} = 10, k_{s} = 30, l_{11} = 50, l_{12} = 10, l_{13} = 10$
EDO-ASMC	$c = 2, β = 2, k_{1} = 1, k_{2} = 0.01, k_{3} = 2, k_{1 e} = 1, k_{2 e} = 2, l_{11} = 50, l_{12} = 10, l_{13} = 10$

MAGLEV: MAGnetic LEVitation; DO-SMC: sliding mode control with disturbance observer; EDO-MSMC: extended disturbance observer–modified sliding mode control; EDO-ASMC: adaptive sliding mode control with extended disturbance observer.

Figure 4.

Comparison among DO-SMC, EDO-MSMC, and EDO-ASMC in the MAGLEV suspension system: (a) plot of the current i, (b) plot of vertical velocity $\overset{\cdot}{z}$ , (c) plot of the air gap $\overset{\cdot}{z} - z$ , and (d) plot of the voltage of the coil u_coil.

The other deterministic input components, as shown in Figure 5, are considered. They represent the jerk level which is sine wave with magnitude of $0.4 m / s^{3}$ . The control parameters of three control methods are listed in Table 6.

Figure 5.

Track input to the suspension with a jerk level with sine wave.

Table 6.

Control parameters for the MAGLEV suspension in case 2.

Controllers	Parameters
DO-SMC	$c_{1} = 100, c_{2} = 20, c_{3} = 1, k = 60, l = [100, 0, 0]$
EDO-MSMC	${\bar{c}}_{1} = 100, {\bar{c}}_{2} = 20, {\bar{c}}_{3} = 1, k_{l} = 10, k_{s} = 20, l_{11} = 50, l_{12} = 10, l_{13} = 10$
EDO-ASMC	$c = 2, β = 2, k_{1} = 0, k_{2} = 0.01, k_{3} = 2, k_{1 e} = 1, k_{2 e} = 2, l_{11} = 50, l_{12} = 10$

The response curves of the states and inputs in this case are shown in Figure 6. By comparing with the first simulation scenario, it can be observed from Figure 6 that the states of the MAGLEV system for the proposed EDO-ASMC have hardly affected by the mismatched disturbance with time variation, while the states of the system for the DO-SMC and the EDO-MSMC are severely affected by the mismatched disturbance. Moreover, with respect to the DO-SMC and the EDO-MSMC, the chattering in the sliding mode of the proposed EDO-ASMC is significantly eliminated.

Figure 6.

Conclusion

In this article, a new disturbance attenuation–based SMC approach has been proposed to reduce the effect of mismatched uncertainties on the control system. A new adaptive sliding mode surface, in which the disturbance estimation is designed according to the sliding motion along the sliding surface, is proposed for driving the states to the desired equilibrium point in the presence of mismatched uncertainties. Both numerical and application examples have been simulated to demonstrate the effectiveness as well as the superiorities of the proposed method. The results have shown that the proposed method exhibits the excellent properties of dynamical performance and chattering reduction compared with other nonlinear SMC methods. Further research will focus on extending the proposed approach to high-order systems.

Footnotes

Academic Editor: Nima Mahmoodi

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the National Natural Science Foundation of China (No. 61473226).

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