High order sliding-mode control for uncertain nonlinear systems with relative degree three

Commun Nonlinear Sci Numer Simulat 17 (2012) 3406–3416

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High order sliding-mode control for uncertain nonlinear systems with relative degree three Rui Ling ⇑, Meirong Wu, Yan Dong, Yi Chai College of Automation, Chongqing University, Chongqing 400030, China

a r t i c l e

i n f o

Article history: Received 17 September 2011 Received in revised form 20 December 2011 Accepted 21 December 2011 Available online 5 January 2012 Keywords: High order sliding-mode control Finite-time control Uncertain nonlinear systems Chattering free

a b s t r a c t A novel high order sliding-mode control is proposed based on finite state machine by combination of relay algorithm and an improved second order algorithm to solve uncertain nonlinear system stabilization with relative degree three in finite time. This approach drives finite state machines to switch according to sliding variable and first order derivative of it, forces sliding variable, first order derivative and second order derivative of it to zero, without the knowledge of second order derivative of sliding variable, and stabilizes the system in finite time. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Sliding-mode control (SMC) [1] is one of the main methods for nonlinear system under uncertainty conditions. But the high-frequency finite amplitude control signal generated by the traditional sliding-mode control results in the chattering phenomenon. In order to overcome the chattering problem in the traditional sliding-mode control, higher order slidingmode control was introduced [2]. Some second order sliding-mode algorithms, such as twisting [2,3], sub-optimal [4], super-twisting [5,6] and so on, were proposed. The theory research [7–12] and implementation [13–17] related to high order sliding-mode have already been carried out widely, while the high order sliding-mode control for uncertain nonlinear system with relative degree three was successfully implemented rarely. Two approaches [18,19] were proposed for the finitetime stabilization of the perturbed triple integrator with requiring the perfect knowledge of s; s_ and €s. Bartolini et al. [20] introduced an approach for uncertain nonlinear system with relative degree three only with requiring the knowledge of the sign of s; s_ and €s. The proper switching logic between ‘‘Anosov Unstable’’ (AU) and ‘‘Modified Twisting’’ was designed to driving the triple integrator to the zero dynamics manifold in finite time. However, the sign of €s is difficult to be measured because the sliding-mode variables s; s_ and €s are bounded by the addictive measurement noise which may affect the measurement of the sign of €s in real control system, specially, when €s  0. So the measurement noise may decrease the performance of real control system, deteriorate the accuracy, even destroy the stability of closed-loop system. An improved second order SMC (ISOSMC) approach [21] solves the stabilization problem for uncertain nonlinear systems with relative degree two by requiring just the knowledge of s. In this paper, we demonstrate that a proper switching logic between AU and the ISOSMC approach in [21] can solve the finite-time stabilization problem for uncertain nonlinear systems with relative degree three without the knowledge of €s.

⇑ Corresponding author. E-mail address: [email protected] (R. Ling). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.12.017

R. Ling et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3406–3416

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The paper organized as follows. In Section 2, the problem formulation and the derivation of the main result are presented. In Section 3, the convergence properties of the proposed algorithm are analyzed. Section 4 contains some simulation results and in Section 5, some concluding remarks are given. 2. A stabilization switching controller for the perturbed triple integrator Consider the following perturbed triple integrator:

s_ 1 ¼ s2 ; s_ 2 ¼ s3 ; s_ 3 ¼ uðtÞ þ cðtÞu;

ð1Þ

where u 2 R is the scalar control variable and u(t), c(t) are unknown bounded functions satisfying the inequalities for some known constants U, Cm and CM

juðtÞj 6 U;

0 < Cm 6 cðtÞ 6 CM :

ð2Þ

Sliding-mode variable s1 ands2 are measurable, s3 is unknown. Define:

8 u > 0; > < 1; signðuÞ ¼ 0 u ¼ 0; > : 1; u < 0:

ð3Þ

The triple integrator with uncertain perturbations and limited information is steered to the origin in finite time by the proper switching logic between following two controls shown in Fig. 1. It is a finite states machine, which consists of six states:  þ  AU; Sþ M ; SM ; Sm ; Sm and Input State. This finite states machine is divided into two parts on the top level. One is Anosov Unstable. Another is the Improved Second Order Sliding-Mode Control. The two parts are activated respectively depending on two switching conditions C1 and C2. Anosov Unstable (AU):

uI ¼ U 0 signðs1 Þ;

ð4Þ

Improved Second Order Sliding-Mode Control (ISOSMC): The control law of uII which is like a finite states machine containing five states is shown in Fig. 1.

U 1 ¼ gU;

ð5Þ

Fig. 1. Schematic representation of ISOSMC.

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R. Ling et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3406–3416

where

g P ð1=2Þð1=dCm  ð1  dÞCM Þ; 

d 2 ð0:75; 1Þ \



CM ;1 ; Cm þ CM

N P 4; U 0 2 R; U 1 2 R; N 2 R; U 0 > 0; U 1 > 0: The proposed controller in Fig. 1 can be formalized by means of a finite states machine in Fig. 1 combined with AU and ISOSMC. AU is a state which can be activated by switching condition C1. ISOSMC consists of five states which can be activated by switching condition C2. The control law of ISOSMC takes one of the opposite constant values ±U1 depending on the current state. The Input State is the entrance of ISOSMC when ISOSMC is activated and also the entrance of the proposed control algo þ  rithm. In the states Sþ M and SM , uII = U1. In the states Sm and Sm ; uII ¼ U 1 . Each state transition, represented by an arc, is fired when the associated switching condition, which consists of a proper inequality involving s1, s2, s1M, s2M and/or s2m and N, is satisfied. During some transitions, s2M or s2m is reset to the actual maximum or minimum value of s2, and s1M is reset to the actual maximum or minimum value of s1. Algorithm 1. (1) The ISOSMC is activated until s1(t) P 0 and s2(t) P 0 or s1(t) 6 0 and s2(t) 6 0 (C1). (2) The AU is activated until s1(t) 6 bjs1Mj(C2). (3) Go to (1). C1 and C2 are the switching conditions between two controls. The s1M is the maximum or minimum value of s1 over a proper receding-horizon time interval and defined as

s1M ðt 0 ; tÞ ¼ signðs1 ðsÞÞ max js1 ðsÞj:

ð6Þ

t 0 6s6t

s2M is maximum value of s2 in the control law of uII. s2m is minimum value of s2 in the control law of uII. They are defined as

s2M ðt 0 ; tÞ ¼ max s2 ðsÞ;

ð7Þ

s2m ðt 0 ; tÞ ¼ min s2 ðsÞ:

ð8Þ

t 0 6s6t

t 0 6s6t

  First, point’’ P 1 ¼ ðs1 t 1 ; 0; 0Þ is achieved in finite time t1 under the ISOSMC control [21], with, in gen alocal ‘‘equilibrium   eral, s1 t1 – 0. If s1 t1 > 0, the expected behavior of sliding variable s1 ðtÞ; t > t1 , is shown in Fig. 2. At the beginning of the Pi   at t ¼ ti , the AU control is switched on. The ISOSMC control is activated at the point sc1 with s1 ðtÞ ¼  bs  1 ti , in general,  s2(t) – 0, s3(t) – 0. The sliding-mode variable s2(t) and s3(t) are steered to be zero, which means P 2 ¼ ðs1 t 2 ; 0; 0Þ is achieved, in finite time by the ISOSMC. When a zero crossing of s1(t) occurs, the AU control will be activated again. With properly U0, U1, d and b, we can iterate the above reasoning to claim that the proposed control scheme enforces a sequence of local ‘‘equi    librium point’’ P i ¼ s1 ti ; 0; 0 fulfilling the contraction property.

       s1 t  6 ns1 t   n 2 ð0; 1Þ: i i1

ð9Þ

As a result the system trajectory converges toward the origin of the s1–s2 plane and the s2–s3 plane.

Fig. 2. The expected behavior in s1–s2 plane.

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In order to enforce the trajectory of the triple integrator to converge like the expected behavior in Fig. 2, the following facts will be proved.   reached with the initializing ISOSMC.  A. The first local ‘‘equilibrium point’’ P1 ¼ s1 ðt 1 Þ; 0; 0 is globally     B. Starting from an local ‘‘equilibrium point’’ P i ¼ s1 ðt i Þ; 0; 0 , the next local ‘‘equilibrium point’’ Piþ1 ¼ s1 ðt iþ1 Þ; 0; 0 is reached infinitetime under the combined controller switching between the two controls. C. Sequence s1 ðti Þ fulfills the contraction property (9). D. The convergence process takes place in finite time. 3. Convergence properties of the proposed algorithm Theorem 1. Consider system pffiffiffiffiffiffiffiffiffiffiffi ffi (1) and (2). Set the contraction rate n 2 (0, 1), define the following quantities:

D1 ¼ 

U 1 1  dðCM þ Cm Þ

ðU  Cm U 1 Þ2 ðU þ CM U 1 Þ2

ð10Þ

;

3

1

D2 ¼ 6ð1  bÞðCM U 0 þ Cm U 1 Þ2 ðU þ CM U 0 Þ2 ; D3 ¼ 4U  dU þ 2CM U 1  2Cm U 1 þ dCM U 1 þ 2dCm U 1 ;  3 F  Cm U 1 þ dCM U 1 þ dCm U 1 2 D4 ¼ ; U þ CM U 1 ð CM U 0 þ Cm U 1 Þ D5 ¼ ð U  Cm U 1 Þ

ð11Þ ð12Þ ð13Þ ð14Þ

and the following inequalities are satisfied:

U0 >

U ; Cm

ð15Þ 1

ðD1 D2 D3 D4 ÞðD2 D4 ðd  1ÞÞ2 < 1; ðD5 ðb  1Þ þ 2D25 ðb  1Þ þ 1Þ   1 þ 2D25 ðb  1Þ þ D5 ðb  1Þ > 0: 0<

ð16Þ ð17Þ

Then, the application of Algorithm 1 steers s1(t), s2(t) and s3(t) to zero in finite time. Proof. It follows the steps A–D. Step A: Let t0 be the initial time instant, input state is activated. Assume that s1(t) P 0 and s2(t) P 0, AU is activated. The closed-loop system with the AU can be seen as a perturbed double integrator (sliding-mode variables s2(t) and s3(t)) controlled by the AU Algorithm and cascaded by an integrator giving s1(t) as its output. When the AU is activated, the trajectory of the uncertain system (1) and (2) are governed by the following differential inclusion.

s_ 1 ¼ s2 ; s_ 2 ¼ s3 ; s_ 3 2 ½U; U þ ½Cm ; CM uI :

ð18Þ

Considering (15), sliding-mode variables s1(t), s2(t) and s3(t) will be decreased by the AU. The condition C2 will be satisfied in finite time, which activates the ISOSMC. The behavior of sliding-mode variable s2(t) and s3(t) by the ISOSMC is analyzed in       [21], which will be steered     to zero  in finite time. The ‘‘equilibrium point’’ P 1 ¼ s1 t1 ; 0; 0 is reached in finite time t 1 < 1.  Step B: Let P i ¼ s1 ti ; 0; 0 ; i ¼ 1; 2; . . ., denote the subsequent intersections with axis (s2(t) = 0). Without loss of       generality, assume that s1 t1 > 0 and let P 1 ¼ s1 t1 ; 0; 0 be the starting point. According to Algorithm 1, the AU is active firstly, with a sufficiently large U0 such that CmU0 > U. The system trajectories refer to the following Fig. 3. The continuous line is a possible trajectory in the s1–s2 plane featuring the typical clockwise rotation. The dashed lines (a) is minorant curve

 and lines (b) is majorant curve in the time interval t1 ; t 01 . Curves (a) and (b) are the limit solutions of (18) in the fourth quadrant of the s1–s2 plane:

ðU  CM U 0 Þ 6 s_ 3 6 ðU  Cm U 0 Þ;

ð19Þ

Curve (a):

s_ 3 ¼ ðU  CM U 0 Þ;   U CM U 0 2 þ t ; s2 ¼  2 2     U CM U 0 3 þ t þ s1 t 1 ; s1 ¼  6 6

ð20Þ ð21Þ ð22Þ

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Fig. 3. Typical system trajectories in the s1–s2 plane with the AU controller.

Curve (b):

s_ 3 ¼ ðU  Cm U 0 Þ;   U Cm U 0 2  t ; s2 ¼ 2 2     U Cm U 0 3 t þ s1 t1 : s1 ¼  6 6

ð23Þ ð24Þ ð25Þ

The actual and limit trajectories initially go farther from the horizontal axis (see Fig. 3). As s1(t) is decreasing, s1M     remains constant. The ISOSMC is activated when the switching condition s1 t01 6 bs1 t1  is satisfied. s1(t) is reset to the actual maximum value of s1. Then s2(t) and s3(t) are converged to zero by the ISOSMC in finite time [21], which implies     the next ‘‘equilibrium point’’ P 2 ¼ s1 t 2 ; 0; 0 is reachable in finite time.

     Next, it can be derived that s2(t) 6 0 in the time interval t 01 ; t 2 such that s1 t 2 < s1 t1 , which implies       Piþ1 ¼ s1 ðt iþ1 Þ; 0; 0 is reached in finite time. So the next local ‘‘equilibrium point’’ lies in the left-hand of s1 t 1 . Step C: The system trajectory lies in the third quadrant of the s2–s3 plane since uI = U0 when the AU controller is     activated (see Fig. 4). After leaving the local ‘‘equilibrium point’’ P1 ¼ s1 t 1 ; 0; 0 , sliding-mode variable s2(t) and s3(t) will decrease and vary between a minimum and maximum value because of uncertainties in the third quadrant of the s2–s3

 plane (see Fig. 4). The system trajectory is confined between limit curves (a) and (b). In the time interval t1 ; t 01 , by considering (19)–(25), the following relationships hold:

t01  t 1 2

 1   1 !  6ðb  1Þs1 t 1 3 6ðb  1Þs1 t 1 3 ; ; ðU þ CM U 0 Þ ð U  Cm U 0 Þ

  s1 t01 ¼ bs1 ðt1 Þ;

13

13  2    2 1  1  6ðb  1Þs1 t 1 ðU þ CM U 0 Þ 6 s2 t01 6  6ðb  1Þs1 t 1 ðCm U 0  UÞ ;  2 2  1  1    0 6ðb  1Þs1 t1 3 6ðb  1Þs1 t 1 3  ðU þ CM U 0 Þ 6 s3 t 1 6 ðU  Cm U 0 Þ : ðU þ CM U 0 Þ ðU  Cm U 0 Þ

ð26Þ ð27Þ ð28Þ ð29Þ

     At t ¼ t01 , when the switching condition s1 t01 6 bs1 t 1  is satisfied, the ISOSM controller is activated. s2(t) will decrease firstly and then increase. A zero crossing of s3(t) will occur when t ¼ t 011 . The following relationships hold:

      s3 t 01 s3 t 01 t0  t 0 2 ; ; 11 1 ðCm U 1  UÞ ðU þ CM U 1 Þ 0  s3 t11 ¼ 0;

ð30Þ ð31Þ

   2    2   ðCM U 0 þ CM U 1 þ 2UÞðU þ CM U 0 Þ 6s1 t 1 ð1  bÞ 3 ðCM U 0 þ Cm U 1 ÞðU þ CM U 0 Þ 6s1 t1 ð1  bÞ 3 6 s2 t 011  ;  2ðU þ Cm U 1 Þ 2ðU þ CM U 1 Þ U þ CM U 0 U þ CM U 0 ð32Þ

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Fig. 4. The transient in s2–s3 plane after leaving P 1 .

      s1 t 1 ðCM U 0 þ Cm U 1 Þðb  1Þ 2s1 t 1 ðCM U 0 þ Cm U 1 Þ2 ðb  1Þ þ 6 s1 t 011 ð U  CM U 1 Þ ð U  Cm U 1 Þ       7CM s1 t1 ðU 0  U 1 Þðb  1Þ 2C2M s1 t 1 ðU 0  U 1 Þ2 ðb  1Þ þ : 6 ð6b  5Þs1 t 1 þ U þ CM U 1 ðU þ CM U 1 Þ2

s1 ðt1 Þ þ

ð33Þ

  After leaving P t 011 , the system trajectory by the ISOSMC will converge to the origin of the s2–s3 plane. During the time

0 1  interval t 11 ; t11 , the following inequalities are derived:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 0  0  0  0   0  s2 t11 ðd þ 2Þ 2s2 t 11 ðF þ Gm U 1 Þðd  1Þ 1   0  s2 t11 ðd þ 2Þ 2s2 t11 ðF þ GM U 1 Þðd  1Þ s1 t11 þ 6 s1 t 11 6 s1 t11 þ ; 3ðU þ Gm U 1 Þ 3ðU þ GM U 1 Þ

ð34Þ

    s2 t111 ¼ ds2 t 011 ;

ð35Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi     qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2s2 t011 ðd  1ÞðU þ GM U 1 Þ 6 s3 t 111 6 2s2 t 011 ðd  1ÞðU þ Gm U 1 Þ:

ð36Þ S M

Since the system trajectories are very complex, assume that N ? 1, which means that the fire condition between and

1   2 S m is changed into s3 = 0. Under this hypothesis, in the time interval t 11 ; t 11 , the following inequalities are derived: 3 3                 s t 1 s t 1   3 ð 11 Þ þ Us2 t 111 s3 t 111  GM U 1 s2 t 111 s3 t 111 2   1  3 ð 311 Þ  Us2 t 111 s3 t 111  Gm U 1 s2 t 111 s3 t111   6 s þ s1 t 111 þ 3 6 s ; t t 1 11 1 11 ðU þ GM U 1 Þ2 ðU þ G m U 1 Þ2

ð37Þ

  ðU  Gm U 1 þ dGM U 1 þ dGm U 1 Þ     ð2dU  U þ GM U 1  dGM U 1  dGm U 1 Þ s2 t011 6 s2 t 211 6 s2 t011 ; U þ GM U 1 U  Gm U 1

ð38Þ

  s3 t211 ¼ 0:

ð39Þ

Considering (38), the sliding-mode variable s2(t) satisfies the contraction mapping

0<

ðU  Gm U 1 þ dGM U 1 þ dGm U 1 Þ ð2dU  U þ GM U 1  dGM U 1  dGm U 1 Þ 6e6 < 1: U þ GM U 1 U  Gm U 1

ð40Þ

The curve (a) is analyzed in order to obtain the expected converge behavior of s1(t) because it is the minimum value of the

 system trajectory. In the time interval t1 ; t 01 , for the curve (a), the following relationships hold:

13  2 1  6ðb  1Þs1 t1 ðU þ CM U 0 Þ ; 2  1  6ðb  1Þs1 t 1 3 : inf ks3 ðt0 Þk  ðU þ CM U 0 Þ ðU þ CM U 0 Þ

inf ks2 ðt0 Þk ¼ 

 Considering (41) and (42), in the time interval t01 ; t211 , the following relationships hold: 0 When t ¼ t11 ,

ð41Þ ð42Þ

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 2 2    s1 t 1 3 ðCM U 0 þ Cm U 1 ÞðU þ CM U 0 Þð6ð1  bÞÞ3 ; inf s2 t 011  ¼ 2 2ðU þ CM U 0 Þ3 ðU  Cm U 1 Þ

ð43Þ !

     ðCM U 0 þ Cm U 1 Þ2 ðb  1Þ ðCM U 0 þ Cm U 1 Þðb  1Þ inf s1 t 011  ¼ s1 t 1 1 þ 2 þ ; ðU  Cm U 1 Þ ð U  Cm U 1 Þ 2

ð44Þ

when t ¼ t211 ,      inf s1 t211  ¼ s1 t 011 

  U 1 s2 t011 ðCM þ Cm Þ 2



1   2s2 t011 ðU  Cm U 1 Þð1  dÞ 2 ð4U  dU þ 2CM U 1  2Cm U 1 þ dCM U 1 þ 2dCm U 1 Þ;

3ðU þ CM U 1 Þ ðU  Cm U 1 Þ      ðU  Cm U 1 þ dCM U 1 þ dCm U 1 Þ : inf s2 t211  ¼ s2 t 011 U þ CM U 1

ð45Þ ð46Þ

When the system trajectory reaches the next local ‘‘equilibrium point’’ P2 by the ISOSMC, the following equation is derived.

P 2     0  1   U1 1 n¼1 s2 ðt 1n ÞðCM þ Cm Þ inf s1 t2  ¼ s1 t 011  2s2 t 11 ðU  Cm U 1 Þð1  dÞ 2 ð4U  dU þ 2CM U 1  2Cm U 1 2 3ðU þ CM U 1 Þ ðU  Cm U 1 Þ þ dCM U 1 þ 2dCm U 1 Þ;

ð47Þ

where

  1 X  2  s2 t 011  s2 t 1n ¼ : 1e n¼1

ð48Þ

Considering (10)–(14), (40) and (47), (48), the following equations are derived.

      inf s1 t 011  ¼ s1 t 1 1 þ 2D25 ðb  1Þ þ D5 ðb  1Þ ;        D1 D2 D3 : inf s1 t 2  ¼ s1 t 011 1  3D4  3

ð49Þ ð50Þ

If conditions (15)–(17) are satisfied, the following relationship is derived.

    s1 t2 ¼ n0 s1 t1 ;

ð51Þ

    D1 D2 D3 0 < n0 ¼ 1 þ 2D25 ðb  1Þ þ D5 ðb  1Þ 1  < 1: 3D4  3

ð52Þ

where

  It implies s1 t 2 > 0 and P2 lies in the right-hand of the s1–s2 plane.         Step D: By the Algorithm 1, the crossing points s1 t 1 and s1 t 2 satisfy the condition 0 < s1 t1 < s1 t 2 . During the   transient between Pi and Piþ1 , the sliding-mode variable.

s2 6 0:

ð53Þ

    By considering (52), the proposed control scheme can enforce a sequence of singular points P 1 ¼ s1 t i ; 0; 0 ; i ¼ 1; 2; . . . for some 0 < n0 < 1, satisfying:

0 < n0 ¼

  s1 t iþ1   < 1; s1 t i

i ¼ 1; . . . ; n:

ð54Þ

As a result, the convergence of system trajectory takes place in finite time. The contraction rate holds:

n 2 ðn0 ; 1Þ:   If s1 t 1 < 0, the same qualitative converging behavior is obtained. The proof is completed. 4. Simulations Considering the following fifth-order nonlinear system:

ð55Þ h

R. Ling et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3406–3416

Fig. 5. s1 Variable time evolution.

Fig. 6. s2 Variable time evolution.

Fig. 7. s3 Variable time evolution.

Fig. 8. The u variable time evolution.

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R. Ling et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3406–3416

Fig. 9. The system trajectory in s1–s2 plane.

x_ 1 ¼ x2 ; x_ 2 ¼ x3 ; x_ 3 ¼ x2 þ x3 þ g1 þ g2 þ

x3 þ ð3 þ 2  cosðx3 þ g1 ÞÞu; 1 þ x23

ð56Þ

g_ 1 ¼ g2 ; g2 ¼ g1  g31  g32 þ x2 þ x3 : The initial conditions are

½x1 ð0Þ; x2 ð0Þ; x3 ð0Þ ¼ ½5; 3; 1; ½g1 ð0Þ; g2 ð0Þ ¼ ½1; 0:5:

ð57Þ

The lower and upper bound of control gain are CM = 5, Cm = 1. The tuning procedure outlined in Theorem 1 was performed using U = 6. Define the sliding-mode variables are as follows

s ¼ ½s1 ; s2 ; s3  ¼ ½x1 ; x2 ; x3 :

ð58Þ

The sliding variable dynamics can be reduced:

s_ 1 ¼ s2 ; s_ 2 ¼ s3 ; s_ 3 ¼ uðtÞ þ cðtÞu;

ð59Þ

where

uðtÞ ¼ x2 þ x3 þ g1 þ g2 þ

x3 ; 1 þ x23

cðtÞ ¼ ð3 þ 2  cosðx3 þ g1 ÞÞ; g_ 1 ¼ g2 ; g2 ¼ g1  g31  g32 þ x2 þ x3 :

ð60Þ

The control task is to drive the s = [s1, s2, s3] to the origin in finite time despite that s3 is unknown.

Fig. 10. The system trajectory in s2–s3 plane.

R. Ling et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3406–3416

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Fig. 11. Steady behavior of s1(f = 1 kHz).

Fig. 12. Steady behavior of s1(f = 10 kHz).

It is possible to implement the convergence trajectory with U0 = 10, U1 = 150, N = 1000, d = 0.75, b = 0.8 and the switch frequency f = 1 kHz. The contraction rate 0.3970 < n < 1 can be calculated by (51). Figs. 5–7 show the time evolution of sliding-mode variable s1, s2, s3. They are converged to zero in finite time. Fig. 5 shows the transient when the AU or ISOSMC is activated. Fig. 6 shows the local ‘‘equilibrium points’’. Fig. 8 shows the time evolution of the control variable u. Fig. 9 shows the system trajectory in s1–s2 plane, which is similar to the expected behavior in Fig. 2. Fig. 10 shows the system trajectory in s2–s3 plane. Fig. 11 shows the steady behavior of s1 when f = 1 kHz. Fig. 12 shows the steady behavior of s1 and the accuracy improvement when f = 10 kHz. 5. Conclusion A novel high order SMC scheme has been proposed to cope with some classes of relative degree three uncertain nonlinear systems just requiring the knowledge of sliding-mode and its derivative. The sliding-mode variables can be driven to zero in finite time by a combined discontinuous controller. Simulations confirm the convergence and accuracy of sliding variable. Acknowledgements The authors are very grateful to the anonymous referee for his (or her) careful reading, detailed comments and helpful suggestions which helped to improve our manuscript. The authors are grateful for the support of Fundamental Research Funds for the Central Universities of China (CDJZR10170005), the Fundamental Research Funds for the Central of China (CDJRC10170007) and Chongqing Science and Technology Key Projects of China (CSTC2011AC6069). References [1] Utkin V. Sliding modes in control and optimization. Berlin, Germany: Springer-Verlag; 1992. [2] Levant A. Sliding order and sliding accuracy in sliding mode control. Int J Control 1993;58(6):1247–63. [3] Khan MK, Goh KB, Spurgeon SK. Second order sliding mode control of a diesel engine. Asian J Control 2003;5(4):614–9.

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