Second-order sliding mode controller design subject to mismatched term

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)

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Technical communique

Second-order sliding mode controller design subject to mismatched term✩ Shihong Ding a , Shihua Li b a

School of Electrical and Information Engineering, Jiangsu University, Zhenjiang 212013, PR China

b

School of Automation, Southeast University, Nanjing 210096, PR China

article

info

Article history: Received 23 June 2015 Received in revised form 15 March 2016 Accepted 15 July 2016 Available online xxxx Keywords: Finite time stability Second-order sliding mode Adding a power integrator

abstract This communique proposes a novel second-order sliding mode (SOSM) control method to handle sliding mode dynamics with mismatched term, so as to reduce the terms in the control channel. Meanwhile, it is shown that the proposed control approach can be used to design SOSM controllers under disturbances bounded by positive functions rather than conventional constant upper bounds. The finite-time stability of the sliding variables has been shown by using finite-time Lyapunov theory. The validity of the proposed approach is verified by controlling a Buck converter. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction The sliding mode control (SMC) has been paid much attention in recent years due to its robustness and simplicity (Li, Gao, Shi, & Zhao, 2014; Li, Shi, Yao, & Wu, 2016; Utkin, 2013; Yu & Long, 2015). However, the restriction on the relative degree of sliding variable and the chattering problem restrict the widespread applications of SMC (Shtessel, Edwards, Fridman, & Levant, 2013). To solve the above drawbacks, the SOSM control methodology has been developed in Emelyanov, Korovin, and Levantovsky (1986) and Levant (1993). On one hand, SOSM brings more flexibility on the choice of sliding variable. On the other hand, SOSM can be used to eliminate chattering problem (Shtessel et al., 2013). Due to the aforementioned two advantages, the SOSM control problems have been widely studied in recent years, such as (Basin & RodriguezRamirez, 2014; Levant, Li, & Yu, 2013; Pico, Pico-Marco, Vignoni, & De-Battista, 2013) and the reference therein. It can be clearly observed that the sliding mode dynamics considered in the existing SOSM control methods are obtained

✩ This work was supported by the NSF of China (61473080, 61573170), the PAPD of Jiangsu Higher Education Institutions, the Zhejiang Open Foundation of the Most Important Subjects, China Postdoctoral Science Foundation (2015M571687) and China Postdoctoral Science Special Foundation (2016T90427). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Tingshu Hu under the direction of Editor André Tits. E-mail address: [email protected] (S. Li).

http://dx.doi.org/10.1016/j.automatica.2016.07.038 0005-1098/© 2016 Elsevier Ltd. All rights reserved.

by directly taking two times derivative on the sliding variable. This also indicates that the existing SOSM control methods can only be used to deal with the matched terms. As a matter of fact, the two times derivative on the sliding variable may bring some disadvantages, because some useful information may be included in the first derivative of the sliding variable and the direct derivative may cause the terms in the control input channel much large. As a matter of fact, if the first derivative of sliding variable could be redesigned as a state variable plus a mismatched term, the above problem may be avoided, but the conventional SOSM control system will become a SOSM control system with mismatched term, which is not easy to handle. In this communique, the SOSM control design problem subject to a known mismatched term is considered by using a backstepping-like method. The controller design can be divided into two steps. In the first step, based on a growth condition for the mismatched term, a virtual controller is designed to stabilize the sliding variable. In the second step, the actual controller is constructed to assure that the virtual controller can be tracked in a finite time. The global finite-time stability of the sliding variables has been verified by Lyapunov theory. Afterwards, the SOSM controller is applied to the regulation problem of a Buck converter. The contributions of the paper are threefold. First of all, the derived SOSM method can reduce the control effort of the sliding mode controller. Secondly, it provides a SOSM algorithm handling the nonlinear systems with disturbances bounded by positive functions rather than frequently-used constant upper bound. Finally, it also presents the finite-time stability of the sliding variables.

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S. Ding, S. Li / Automatica (

The rest of the paper is organized as follows. Section 2 briefly presents the problem formulation. Main contribution to design the sliding mode controller is given in Section 3. Section 4 shows an example to control Buck converter. Finally, the concluding remarks are given in Section 5. 2. Problem formulation Consider the nonlinear system of the following form x˙ = f (t , x) + g (t , x)u, s = s(t , x) where x ∈ Rn and u ∈ R are the state and control input, respectively; f (t , x) and g (t , x) are smooth functions; the smooth function s is the measured output, which has a relative degree ∂s ∂s r = 2 with respect to controller u. Let s1 = s and s2 = ∂ t1 + ∂ x1 x˙ , one has s˙1 = s2 , s˙2 = a(t , x) + b(t , x)u

(1) ∂˙s

with smooth functions a(t , x) = s˙2 |u=0 and b(t , x) = ∂ u2 being not exactly known, but satisfy: Assumption 2.1. There exist three positive constants a¯ , b and b¯ such that |a(t , x)| ≤ a¯ , b ≤ b(t , x) ≤ b¯ . According to Levant (1993), the conventional SOSM control is to design u such that the nonempty set s1 = s˙1 = 0 can be kept. The SOSM control of nonlinear systems has been widely studied recently, resulting in several algorithms, including twisting algorithm (Emelyanov et al., 1986), super-twisting algorithm (Basin & Rodriguez-Ramirez, 2014; Levant, 1993), sub-optimal algorithm (Bartolini, Pisano, Punta, & Usai, 2003), etc. It can be observed from conventional SOSM control methods that a direct derivative has usually been imposed on the term ∂s ∂s s2 = ∂ t1 + ∂ x1 x˙ no matter what it contains. This may bring some problems. One problem is that the term existing in s2 will be removed to s¨1 , which implies that a larger switching gain in controller u will be required to suppress the disturbances. In addition, the noise included in s2 may be enlarged by taking derivative. The above problems may be solved if we can divide the ∂s ∂s term ∂ t1 + ∂ x1 x˙ into two parts: the well-chosen term depending on s1 as the first part and the rest as the second part. Then taking the second part as the new variable and the first part as the mismatched term, system (1) can be written as s˙1 = s2 + c (t , s1 ), s˙2 = a(t , x) + b(t , x)u ∂ s1 ∂t

(2)

∂ s1 x˙ ∂x

)

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Remark 2.2. The criterion of choosing c (t , s1 ) is whether it will reduce the terms in the control channel and satisfy Assumption 2.2. Consequently, we usually should know some information about the mismatched term c (t , s1 ), or even the full information. When c (t , s1 ) is determined, we can take s2 = s˙1 − c (t , s1 ). If there are disturbances included in s2 , an observer can be used to estimate s2 . The observer design is feasible, because the term c (t , s1 ) in many cases is only dependent on the measurable variable s1 , and it can be cancelled directly in observation error system. Remark 2.3. Note that a hypothesis for choosing s2 is that the control input should exactly appear in s˙2 . This also implies that the relative degree of the sliding variable s1 is required to be two. To this end, the choice of the sliding variable is similar to the conventional SOSM algorithms. The purpose of this communique is to design a suitable SOSM controller for system (2) under mismatched term such that s1 = s˙1 = 0 can be kept in a finite time. Finally, we list three lemmas. To simplify the expression, we denote ⌊x⌉α = sign(x)|x|α . Lemma 2.1 (Ding, Li, & Zheng, 2012). If p1 > 0 and 0 < p2 ≤ 1, then for ∀x, y ∈ R, we have |⌊x⌉p1 p2 − ⌊y⌉p1 p2 | ≤ 21−p2 |⌊x⌉p1 − ⌊y⌉p1 |p2 . Lemma 2.2 (Qian & Lin, 2001). Let c and d be positive constants. Given any function γ > 0, the following inequality holds |x|c |y|d ≤ c c γ |x|c +d + c +d d γ − d |y|c +d , ∀x, y ∈ R. c +d Lemma 2.3 (Hardy, Littlewood, & Polya, 1952). Let p be a real number with 0 < p < 1. Then for ∀xi ∈ R, i = 1, . . . , n, we have (|x1 | + · · · |xn |)p ≤ |x1 |p + · · · + |xn |p . 3. Main result In this section, by using adding a power integrator method (Qian & Lin, 2001), we will propose a novel SOSM controller for system (2) using a step-by-step way. Theorem 3.1. Considering the SOSM dynamics (2) with Assumption 2.2, there exist positive functions β2 (x, s1 ) ≥ β1 (s1 ) and positive constants a ≥ r1 = 2r2 > 0 such that u = −β2 (x, s1 ) sign(⌊s2 ⌉a/r2 + β1 (s1 )⌊s1 ⌉a/r1 )

(3)

provides for the finite-time establishment of SOSM s1 = s˙1 = 0.

where + = s2 + c (t , s1 ) with s2 and a continuous differentiable function c (t , s1 ) being properly chosen catering to the control design. Here the functions a(t , x), b(t , x) and c (t , s1 ) satisfy

Proof. Step 1. We choose a C 1 Lyapunov function as follows

Assumption 2.2. There exist a positive constant b and continuous functions a¯ (x) ≥ 0, ρ(s1 ) ≥ 0 such that |a(t , x)| ≤ a¯ (x), b(t , x) ≥ b, |c (t , s1 )| ≤ ρ(s1 )|s1 |1/2 .

V˙ 1 (s1 ) = ⌊s1 ⌉

Note that c (t , s1 ) is continuous differentiable. Let s¯2 = s2 + c (t , s1 ). Then system (2) can be rewritten as s¨1 = a(t , x)+b(t , x)u+˙c (t , s1 ), which is also understood in Filippov sense. By the definition of SOSM in Levant (1993), the controller is called the SOSM controller if u can be designed such that s1 = s˙1 = 0. Remark 2.1. By Assumption 2.2, it is clear that the disturbances in conventional SOSM satisfying Assumption 2.1 have been extended to be bounded by positive functions rather than positive constants. This implies the method proposed in the paper can be used to design SOSM controllers for nonlinear systems subject to disturbances bounded by positive functions.

2ρ+r2

V1 (s1 ) = 2ρ+1 r |s1 | r1 with ρ ≥ a ≥ r1 = 2r2 > 0. Taking 2 derivative of V1 (s1 ) along system (2) produces r

≤ ⌊s1 ⌉

2ρ−r2 r1 2ρ−r2 r1

s2 + ⌊s1 ⌉

2ρ−r2 r1

c ( t , s1 )

(s2 − s∗2 ) + ⌊s1 ⌉

2ρ−r2 r1

2ρ r

s∗2 + ρ(s1 )s1 1

where s2 is a virtual control law. Design s2 = −β1 (s1 )⌊ξ1 ⌉ β1 (s1 ) being a smooth function and satisfying ∗

∗

(4) r2 /a

with

β1 (s1 ) ≥ ρ(s1 ) + β0 , β0 > 0

(5) 2ρ

and ξ1 = ⌊s1 ⌉a/r1 . It follows from (4) that V˙ 1 (s1 ) ≤ −β0 ξ1 a + 2ρ−r2 a

(s2 − s∗2 ). Step 2. Choose V2 (s1 , s2 ) = V1 (s1 ) + W2 (s1 , s2 ) with W2 (s1 , s2 ) =  s2  a/r  a/r  2aρ ⌊κ⌉ 2 − s∗2 2 dκ. Similar to the proof of Propositions s∗ ⌊ξ1 ⌉

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B.1 and B.2 in Qian and Lin (2001), it is easy to verify that the function V2 (s1 , s2 ) is C 1 , positive definite and proper. Taking derivative of V2 (s1 , s2 ) along system (2) yields

∂ W2 (s1 , s2 ) ∂ W2 (s1 , s2 ) s˙1 + s˙2 ∂ s1 ∂ s2 2ρ 2ρ−r2 2ρ ∂ W2 (·) ≤ −β0 ξ1 a + ⌊ξ1 ⌉ a (s2 − s∗2 ) + s˙1 + ⌊ξ2 ⌉ a s˙2 (6) ∂ s1  a/r2 . Next we estimate each term on the with ξ2 = ⌊s2 ⌉a/r2 − s∗2 right hand side of (6).   r  a   a × r2  × 2 Note that |s2 − s∗2 | = ⌊s2 ⌉ r2 a − s∗2 r2 a . It follows from V˙ 2 (s1 , s2 ) = V˙ 1 (s1 ) +

2ρ−r2

Lemma 2.1 that ⌊ξ1 ⌉ a (s2 − s∗2 ) ≤ 21−r2 /a |ξ1 | using Lemma 2.2, it can be concluded that

⌊ξ1 ⌉

2ρ−r2 a

1

(s2 − s∗2 ) ≤

with c1 (β0 ) = 21−

4

r2 r a 1

2ρ/a

β0 ξ1 

4ρ

2ρ−r2 a

2ρ/a

+ c1 (β0 )ξ2

(4ρ−r1 )21−r2 /a ρβ0

r1  4ρ− r

|ξ2 |r2 /a . By (7)

.

1

Meanwhile, by Lemma 2.1 we know

  a     ∂ s∗ r2   ∂ W2 (·)  2ρ 1− r2 r2 2ρ   2   a |ξ | a + a −1  s˙1  . 2  ∂ s s˙1  ≤ a 2   ∂ s 1 1

(8)

Noting that s∗2 = −β1 (s1 )⌊ξ1 ⌉r2 /a , one has

 a a    a    ∂( s∗ r2 )   ∂β r2 (s )ξ  aβ r2 (s ) a 1    1 1 1 −1 2 1 |s1 | r1 . +  ≤   ∂ s1   ∂ s1 r1  

From Lemma 2.3, we also have |s2 | ≤ |ξ2 |

(9)

together with (9) and Assumption 2.2 implies

     a/r   ∂β a/r2 (s )  aβ a/r2 (s )  ∂( s∗ 2 )  1 1  1   r1 /a  2 1 |ξ1 |1−r1 /a | s˙1  ≤  ξ1  +   ∂ s1    ∂ s1 r1   r2 r2 r2 × |ξ2 | a + β1 (s1 )|ξ1 | a + ρ(s1 )|ξ1 | a . (10) Applying Lemma 2.2 to (10), it can be concluded that there exist two positive continuous functions c2 (s1 ) and c3 (s1 ) such that

 ∂(⌊s∗ ⌉a/r2 )  r r 1− 2 1− 2   2  ∂ s1 s˙1  ≤ c2 (s1 )|ξ1 | a + c3 (s1 )|ξ2 | a . This, together with    r2 2ρ  ∂W  2ρ (8), yields  ∂ s 2 s˙1  ≤ a 21−r2 /a|ξ2 | a + a −1 × c2 (s1 )|ξ1 |1−r2 /a + 1  c3 (s1 )|ξ2 |1−r2 /a . Applying Lemma 2.2 again leads to    ∂ W2  1 2ρ/a 2ρ/a   (11)  ∂ s s˙1  ≤ 4 β0 ξ1 + c4 (s1 )ξ2 1 with c4 (s1 ) is a positive continuous function. Substituting (7) and (11) into (6) gives V˙ 2 (s1 , s2 ) ≤ −

β0 2

ξ

2ρ/a 1

+ [c1 (β0 ) + c4 (s1 )]ξ

+ ⌊ξ2 ⌉ (a(t , x) + b(t , x)u).

(12)

We design u = −β2 (x, s1 ) sign(ξ2 )

(13)

with

β2 (x, s1 ) ≥

c1 (β0 ) + c4 (s1 ) + b

β0 2

+ a¯ (x)

.

3

Substituting controller (13) into (12) yields V˙ 2 (s1 , s2 ) ≤ −

+ξ

2ρ/a 2

).

By the fact that 1−r2 /a

2ρ+r2 a

 s2  s∗ 2

β0 2

2ρ/a

(ξ1

 a/r  2aρ 2ρ ⌊κ⌉a/r2 − s∗2 2 dκ ≤ |s2 − s∗2 ||ξ2 | a 2ρ+r2

=2 |ξ2 | , it can be verified that V2 (s1 , s2 ) ≤ 2(|ξ1 | a + 2ρ+r2 β1 |ξ2 | a ). Letting c = , we can obtain V˙ 2 (s1 , s2 ) + 2ρ 2ρ 2ρ+r2

2×2 2ρ+r2

2ρ

cV2 (s1 , s2 ) ≤ 0. Note that 2ρ+r2 ∈ (0, 1). It follows from the finite-time Lyapunov theory given in Bhat and Bernstein (2000) and Zhang, Han, and Zhu (2015) that system (2) can be globally finite-time stabilized by controller (13). Note that controller (13) can be rewritten as (3). It also implies that system (2) can be globally finite-time stabilized by controller (3). This completes the proof.  Remark 3.1. It should be noted that the conventional SMC and terminal SMC under mismatched disturbances have already been reported in Yang, Li, and Yu (2013) and Yang, Li, Su, and Yu (2013), respectively. We can observe that the disturbances considered in Yang, Li, and Yu (2013) and Yang, Li, Su et al. (2013) should have some global Lipschtiz constants. In this paper, we cannot find the proper global Lipschitz constant for the mismatched term. Thus the methods proposed in Yang, Li, and Yu (2013) and Yang, Li, Su et al. (2013) cannot be applied to system (2).

Assumption 3.1. There exist three positive constants b, a¯ , ρ¯ such that |a(t , x)| ≤ a¯ , b(t , x) ≥ b, |c (t , s1 )| ≤ ρ| ¯ s1 |1/2 . Remark 3.2. Note that the widely used assumption for SOSM is Assumption 2.1, which is usually restricted to be in a local region, where the state variables are bounded. Note that s1 is usually a function of state variables. Similar to Assumption 2.1, we can find such a positive constant ρ¯ such that |c (t , s1 )| < ρ| ¯ s1 |1/2 holds at least locally. Then by the proof of Theorem 3.1, we can obtain the following corollary directly. Corollary 3.1. Considering the SOSM dynamics (2) with Assumption 3.1, there exist positive constants β2 > β1 > 0 and a ≥ r1 = 2r2 > 0 such that the following controller u = −β2 sign(⌊s2 ⌉a/r2 + β1 ⌊s1 ⌉a/r1 )

(14)

(15)

provides for the finite-time establishment of SOSM s1 = s˙1 = 0. Remark 3.3. By Assumption 2.1, it can be clearly observed that controller (15) can be regarded as a SOSM controller for system (1). Note that the nested SOSM or the controller based on a prescribed convergence law has a similar structure as (Shtessel et al., 2013) u = −β2 sign(s2 + β1 ⌊s1 ⌉1/2 ).

2ρ/a 2

2ρ a

–

Now let us consider the case that Assumption 2.2 reduces to the following assumption:

 r2 + β1 (s1 )|ξ1 | a . This,

r2 a

)

(16)

Apparently, controllers (15) and (16) are two kinds of different controllers. As a matter of fact, according to Levant (2007), controller (15) can be considered as non-singular terminal sliding mode (TSM), while controller (16) can be regarded as conventional TSM. Remark 3.4. In the literature, there are several well known SOSM algorithms, such as the algorithm with prescribed convergence law (Shtessel et al., 2013), the twisting algorithm (Emelyanov et al., 1986), the super-twisting algorithm (Levant, 1993; Moreno & Osorio, 2012), the suboptimal algorithms (Bartolini et al., 2003), etc.

4

S. Ding, S. Li / Automatica (

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The task here is to design a controller u such that the output voltage v0 will well track a desired reference voltage vref . To design a SOSM controller for Buck converter control system (17), the first thing is to choose a sliding variable. We now define the sliding variable s (i.e., the voltage error) as s = v0 − vref .

According to the conventional SOSM control method, by letting v s1 = s, s2 = v˙ 0 = C1 (iL − R0 ) we can obtain the dynamics of the sliding variable s directly as

Fig. 1. Circuit diagram of Buck converter.

s¨ = a(t , iL , v0 ) + b(t , iL , v0 )u,

One thing we want to stress is the SOSM algorithm proposed in this paper is different from the existing ones. First of all, the hypotheses on the disturbances are different. The algorithm proposed here needs the disturbances of the sliding mode dynamics to be bounded by some positive functions, while the existing algorithms require the constant upper bound assumptions. It implies that the method proposed in this paper has a wider application scope. Secondly, the proposed method can be used to deal with mismatched term, while similar property cannot be found in the existing algorithms. Lastly, Theorem 3.1 gives the finite-time Lyapunov stability analysis for the sliding variables, while only finite-time convergence is proposed for the most existing SOSM algorithms. Remark 3.5. If the mismatched term c (t , s1 ) equals zero, then Assumption 3.1 will reduce to the following assumption: Assumption 3.2. There exist a positive constant b and positive function a¯ (x) such that |a(t , x)| ≤ a¯ (x), b(t , x) ≥ b. It can be observed that Assumption 3.2 is a special case of Assumption 3.1. According to Corollary 3.1, controller (15) will remain working for system (1) with Assumption 3.2. Remark 3.6. It is seen that system (2) can also be regarded as a double integrator with a mismatched perturbation. It should be pointed out that the control design problem for nonlinear system with mismatched perturbations has already been reported, such as Cruz-Zavala and Moreno (2014), Estrada and Fridman (2010a) and Estrada and Fridman (2010b). As a matter of fact, the logic of method to handle mismatched perturbations in Cruz-Zavala and Moreno (2014) and Estrada and Fridman (2010a,b) and this paper is to some extent similar, because the key technique is a stepby-step virtual control approach. The main difference is that the mismatched perturbations of the nonlinear system in Cruz-Zavala and Moreno (2014) and Estrada and Fridman (2010a,b) should be suppressed, while the mismatched term of the sliding mode dynamics in this paper is in purpose introduced in order to reduce the uncertainties in the control channel. 4. SOSM control of a buck converter The Buck converters are one of the most important switched mode DC–DC converters, which have been widely used in applications, such as mobile power supply equipment, photovoltaic system, DC supply system, etc. Generally, Buck converter consists of a DC voltage source (Vin ), a controllable switch (S w ), a diode (D), an inductor (L), a capacitor (C ) and a load resistor (R), properly connected as shown in Fig. 1. The mathematical model of DC–DC Buck converter can usually be described as (Erickson & Maksimovic, 1999) diL dt

=

1 L

(uVin − v0 ),

d v0 dt

=

1 v0  iL − C R

where v0 is the output voltage and u is the controller.

(18)

(17)

with a(t , iL , v0 ) =



(19)



− LC1 v0 − RCiL2 and b(t , iL , v0 ) = VLCin . By Corollary 3.1 and letting a = r1 = 1, the controller can be designed 1

(RC )2

as u = −β2 sign(⌊˙s⌉2 + β1 s)

(20)

with proper constants β1 and β2 . On the other hand, by letting s1 = s, s2 = obtain

iL C

−

vref CR

, we can

s˙1 = s2 + c (t , s1 ), s˙2 = a(t , iL , v0 ) + b(t , iL , v0 )u v0

(21)

with a(t , iL , v0 ) = − CL , b(t , iL , v0 ) = , c ( t , s1 ) = − Note that iL and v0 are bounded. Thus, we can find proper constants V 1 a¯ = LCin , ρ¯ = max{ CR |s1 |1/2 } and b = VLCin such that Assumption 3.1 holds. Then, by letting a = r1 = 1, the SOSM controller for (17) can also be designed as Vin LC

1 s . CR 1

u = −β2 sign(⌊s2 ⌉2 + β1 s1 ).

(22)

By (5) and (14), we have β1 ≥ ρ¯ + β0 , β0 > 0 and β2 (·) ≥ β c1 (β0 )+c4 (β0 )+ 20 +¯a

(c1 (β0 ) + c4 (β0 ) + β0 /2) + 1. Noting that s1 will rapidly converge to zero, ρ¯ could be chosen as a smaller one. This implies that β1 can be also small. However, to keep the convergence of the sliding variable s1 , it will be better to choose β1 ≥ 1. On the other hand, due to the fact that LC is very small, β2 > 1 will be acceptable. b

=

LC Vin

Remark 4.1. For the closed loop system (21)–(22), the steady state error of s1 is determined by the value of R and the length of the sampling time. Choose a Lyapunov function as V (s1 ) = 21 s21 . It is not s2

 − | s | . It 2 CR implies that the state s1 will converge to the region |s1 | ≤ CR|s2 |. By difficult to verify V˙ (s1 )|(21) = − CR1 + s1 s2 ≤ −|s1 |



|s1 |

the definition of s2 , it can be concluded that there exists a positive constant γ¯ such that |s2 | ≤ γ¯ . Then, we obtain |s1 | ≤ CRγ¯ , which implies that the smaller R means the smaller steady state error. On the other hand, the steady state error is also affected by the length of the sampling time. The smaller sampling time will yield the smaller steady state error. To obtain a fair comparison, the magnitude of the controllers is restricted to be less than 5, because the input of the actuator is usually no more than 5 V. Then we choose L = 1000 µH, C = 1000 µF, Vin = 24 V, R = 30 , vref = 12 V. The control parameters for controllers (20) and (22) are chosen as β1 = 3, β2 = 10. Take the initial state as (iL (0), v0 (0)) = (0, 0). The simulation is carried out by using Euler method and the sampling time is 10−7 s. The simulation results under controllers (20) and (22) are shown in Figs. 2–3. It can be clearly seen from Figs. 2–3 that under controller (22), the output voltage can be rapidly tracked, while the tracking result under controller (20) is not satisfactory because the convergence is very slow. This is because the disturbances in system (19) are very large and controller (20) cannot yield enough control effort due to the control saturation.

S. Ding, S. Li / Automatica (

)

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5

variables more flexibly such that the terms that appeared in the control channel can be significantly reduced. The future work will be focused on extending the result to the higher-order sliding mode control systems. References

Fig. 2. Simulation result under controller (22).

Fig. 3. Simulation result under controller (20).

5. Conclusion A new second-order sliding mode method has been developed to deal with the sliding mode control systems subject to mismatched term. This method allows us to choose the sliding

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