Abstract

This paper is concerned with non-fragile sliding mode control of uncertain chaotic systems with external disturbance. Firstly, a new sliding surface is proposed, and sufficient conditions are derived to guarantee that sliding mode dynamics is asymptotically stable with a generalized š»2 disturbance rejection level. Secondly, non-fragile sliding mode controller is established to make the state of system reach the sliding surface in a finite time. Finally, an example is given to illustrate the effectiveness of the proposed method.

1. Introduction

Chaotic behavior is a seemingly random phenomenon of a deterministic system that is characterized by sensitive deaspendence on initial conditions. Many electronic, mechanical, and chemical systems exhibit chaotic dynamics. Therefore, chaos is a very interesting nonlinear phenomenon, and control of chaotic systems has been paid much attention by researchers since the pioneering work of Ott et al. [1]. The main aim of chaos control is to suppress chaotic behavior and to stabilize the chaotic system, and various effective techniques have been presented and successfully applied to obtain chaos control such as adaptive control [2], backstepping control [3], fuzzy control [4], optimal control [5], and sliding mode control [6ā€“9].

Among the above-mentioned methods, sliding mode control is a very effective approach to control chaos because of its attractive features such as fast response, good transient response, and insensitivity to variations in system parameters and external disturbances [6ā€“11]. Furthermore, in some engineering practices, for example, in order to avoid actuator saturation, one often guarantees that whenever the exogenous input has bounded energy, the controlled output should have a specified maximum peak value. In this case, generalized š»2 control is a good tool [12, 13].

Generally speaking, accurate controllers are required to stabilize control systems, so all of the controller coefficients are exact values in designing a desired controller. However, in practice, the uncertainty is not avoided, and it may be caused by many reasons, such as finite word length in digital systems, the imprecision inherent in analog systems, and the need for additional tuning of parameters in the final controller implementation [14]. The existence of uncertainty leads to the study of non-fragile control problem, that is, to design a controller such that the controller is insensitive to uncertainties. Many results about non-fragile control problems have been reported [15ā€“17]. However, so far, non-fragile sliding mode control problem for uncertain chaotic systems has been paid little attention.

In this paper, the problem of non-fragile sliding mode control of uncertain chaotic systems with external disturbance is considered. Firstly, a new integral-type sliding surface is proposed, and the reaching phase is eliminated, and the proposed sliding surface function includes a norm-bounded parameter uncertainty; this term is added not only to make the design of sliding surface more relaxed, but also to guarantee that the sliding mode dynamics has better robust performance. For both the cases with additive and multiplicative uncertainties, sufficient conditions are derived to make sliding mode dynamics stable with a generalized š»2 performance. Secondly, non-fragile sliding mode controller is established to make the state of system reach the sliding surface in a finite time. Finally, an example is given to illustrate the effectiveness of the proposed method.

Notations
The following notations will be used throughout this paper. š‘…š‘› and š‘…š‘›Ć—š‘š denote, respectively, the š‘›-dimensional Euclidean space and the space of š‘›Ć—š‘š real matrices. For a real symmetric matrix š‘€, š‘€>0 means š‘€ is positive definite. šæ2[0,āˆž) is the space of square-integrable vector functions over [0,āˆž), and šæāˆž is the space of essential bounded functions. āˆ— represents the symmetric form of matrix.

2. Problem Formulation

Consider the following chaotic system:š‘§Ģ‡š‘„(š‘”)=š“š‘„+šµ(š‘“(š‘„)+Ī”š‘“(š‘„)+š‘¢(š‘”))+š·š‘¤(š‘”),(š‘”)=š¶š‘„(š‘”),(1) where š‘„(š‘”)āˆˆš‘…š‘› is the state vector, š‘¢(š‘”)āˆˆš‘…š‘š is the control input. š‘§(š‘”)āˆˆš‘…š‘ž is the system output. š‘“(š‘„) is a nonlinear real-valued function. š“,šµ,š¶,andš· are matrices with compatible dimensions and (š“,šµ) is completely controllable. šµ is of full column rank. š‘¤(š‘”) is a bounded disturbance. Ī”š‘“(š‘„) represents the uncertainty part, satisfying ā€–Ī”š‘“(š‘„)ā€–ā‰¤šœŒ with a positive constant šœŒ.

The block diagram of the system (1) is given in Figure 1.

The following lemmas are necessary for future discussion.

Lemma 1 (see [8]). For a given matrix ī‚ƒš‘†=š‘†11š‘†12š‘†š‘‡12š‘†22ī‚„ with š‘†11=š‘†š‘‡11, š‘†22=š‘†š‘‡22, then the following conditions are equivalent: (1)š‘†<0, (2)š‘†22<0, š‘†11āˆ’š‘†12š‘†āˆ’122š‘†š‘‡12<0,(3)š‘†11<0, š‘†22āˆ’š‘†š‘‡12š‘†āˆ’111š‘†12<0.

Lemma 2 (see [17]). Let Ī©,Ī“ and Īž be matrices of appropriate dimensions, and let Ī© be symmetrical. Then Ī©+Ī“š¹(š‘”)Īž+(Ī“š¹(š‘”)Īž)š‘‡<0,(2) for an uncertain matrix š¹(š‘”) satisfying š¹š‘‡(š‘”)š¹(š‘”)ā‰¤š¼ if and only if there exists a scalar šœ€>0 such that Ī©+šœ€Ī“Ī“š‘‡+1šœ€ī€·Īžš‘‡Īžī€ø<0.(3)

3. Main Results

For system (1), firstly, a new sliding surface is designed, and sufficient conditions are given to guarantee that the sliding mode dynamics is asymptotically stable with a generalized š»2-norm bound š›¾. Secondly, non-fragile sliding mode controller is designed to drive the state arriving at the sliding surface in a finite time.

3.1. Sliding Surface Design

For system (1), we design the following integral-type sliding surface function:ī‚øī€œš‘ (š‘”)=šŗš‘„(š‘”)āˆ’š‘„(0)āˆ’š‘”0(ī‚¹,š“+šµ(š¾+Ī”š¾(š‘ )))š‘„(š‘ )š‘‘š‘ (4) where šŗ is a matrix such that šŗšµ is invertible. š¾>0 is a gain matrix to be chosen later, and the matrix Ī”š¾(š‘”) represents the norm-bounded parameter uncertainty.

In this paper, the following two classes of parameter uncertainty will be considered:(a)Ī”š¾(š‘”) is with the norm-bounded additive form Ī”š¾(š‘”)=Ī”1=š»1š¹1(š‘”)šø1,(5) where š»1 and šø1 are known matrices and š¹1(š‘”) is an unknown matrix satisfying š¹š‘‡1(š‘”)š¹1(š‘”)ā‰¤š¼;(6)(b)Ī”š¾(š‘”) is with the norm-bounded multiplicative form Ī”š¾(š‘”)=Ī”2=š»2š¹2(š‘”)šø2š¾,(7) where š»2 and šø2 are known matrices, and š¹2(š‘”) is an unknown matrix satisfying š¹š‘‡2(š‘”)š¹2(š‘”)ā‰¤š¼.(8)

Remark 3. From (4), we have š‘ (0)=0. It is worth to note that the integral-type sliding surface function (4) is different from the existing methods, for example, those presented in [10, 11], since it takes the term Ī”š¾(š‘”) into account. Moreover, these existing results can be considered as a special form of the integral-type sliding surface designed in this paper, when Ī”š¾(š‘”) is zero. In fact, uncertainty Ī”š¾(š‘”) is added not only to make the design of sliding surface more relaxed, but also to guarantee that the sliding mode dynamics has better robust performance.
Let Ģ‡š‘ (š‘”)=0. Then, the equivalent control law can be obtained as follows: š‘¢eq(š‘”)=(š¾+Ī”š¾(š‘”))š‘„(š‘”)āˆ’(šŗšµ)āˆ’1šŗš·š‘¤(š‘”)āˆ’š‘“(š‘„)āˆ’Ī”š‘“(š‘„).(9) Substituting (9) into the system (1) and denoting š·=š·āˆ’šµ(šŗšµ)āˆ’1šŗš·, sliding mode dynamics and the output equation can be formulated as Ģ‡š‘„(š‘”)=(š“+šµš¾+šµĪ”š¾(š‘”))š‘„(š‘”)+š·š‘¤(š‘”),š‘§(š‘”)=š¶š‘„(š‘”).(10)

Definition 4. Sliding mode dynamics (10) is asymptotically stable with a generalized š»2-norm bound š›¾ if the following conditions hold.(i)When š‘¤(š‘”)=0, sliding mode dynamics (10) is asymptotically stable.(ii)Sliding mode dynamics (10) has a given generalized š»2 (or šæ2āˆ’šæāˆž) disturbance rejection level; that is, the šæ2āˆ’šæāˆž gain from disturbance input š‘¤(š‘”) to control output š‘§(š‘”) of the system (10) is less than a given value. It is to make, under the zero-valued initial condition, the following inequality hold: supš‘¤āˆˆšæ2āˆ’0ā€–š‘§ā€–2āˆžā€–š‘¤ā€–22<š›¾2,(11) where ā€–š‘¤ā€–22=ī€œāˆž0š‘¤š‘‡(š‘”)š‘¤(š‘”)š‘‘š‘”,ā€–š‘§ā€–2āˆž=supš‘”ī€½š‘§š‘‡ī€¾.(š‘”)š‘§(š‘”)(12)

Theorem 5. Consider the system (10) and a given constant š›¾>0. If there exist matrices š‘ƒ>0 and š¾ such that the following matrix inequalities are true āŽ”āŽ¢āŽ¢āŽ£Ī˜=š‘ƒ(š“+šµš¾+šµĪ”š¾(š‘”))+(š“+šµš¾+šµĪ”š¾(š‘”))š‘‡š‘ƒš‘ƒš·āŽ¤āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ£āˆ—āˆ’š¼<0,(13)š‘ƒš¶š‘‡āˆ—š›¾2š¼āŽ¤āŽ„āŽ„āŽ¦>0,(14) then the sliding mode dynamics (10) is asymptotically stable with a generalized š»2-norm bound š›¾.

Proof. Consider the following Lyapunov functional candidate: š‘‰1(š‘”)=š‘„š‘‡(š‘”)š‘ƒš‘„(š‘”).(15) The time derivative of š‘‰1(š‘”) along the trajectories of system (10) with š‘¤(š‘”)=0 is given by Ģ‡š‘‰1(š‘”)=š‘„š‘‡(ī€ŗš‘”)š‘ƒ(š“+šµš¾+šµĪ”š¾(š‘”))+(š“+šµš¾+šµĪ”š¾(š‘”))š‘‡š‘ƒī€»š‘„(š‘”).(16)
From (13), Ī˜<0 implies š‘ƒ(š“+šµš¾+šµĪ”š¾(š‘”))+(š“+šµš¾+šµĪ”š¾(š‘”))š‘‡š‘ƒ<0, then we have Ģ‡š‘‰1(š‘”)<0,(17) which implies sliding mode dynamics (10) is asymptotically stable, when š‘¤(š‘”)=0.
In order to construct the generalized š»2 (šæ2āˆ’šæāˆž) performance index, we assume the zero-valued initial condition, then š‘‰1(š‘”)|š‘”=0=0. Now, we consider the following cost performance: š½=š‘‰1(ī€œš‘”)āˆ’š‘”0š‘¤š‘‡(š‘ )š‘¤(š‘ )š‘‘š‘ .(18) For any nonzero š‘¤(š‘”)āˆˆšæ2[0,āˆž), we have ī€œš½=š‘”0ī€ŗĢ‡š‘‰1(š‘ )āˆ’š‘¤š‘‡(ī€»=ī€œš‘ )š‘¤(š‘ )š‘‘š‘ š‘”0šœ‰š‘‡Ī˜šœ‰š‘‘š‘ ,(19) where šœ‰=[š‘„š‘‡(š‘ )š‘¤š‘‡(š‘ )]š‘‡.
From (13), we obtain š½=š‘‰1(ī€œš‘”)āˆ’š‘”0š‘¤š‘‡(š‘ )š‘¤(š‘ )š‘‘š‘ <0.(20) Equation (20) leads to š‘„š‘‡(š‘”)š‘ƒš‘„(š‘”)=š‘‰1(ī€œš‘”)<š‘”0š‘¤š‘‡(š‘ )š‘¤(š‘ )š‘‘š‘ .(21) For (14), by Lemma 1, we have š¶š‘‡š¶<š›¾2š‘ƒ(22) which leads to š‘§š‘‡(š‘”)š‘§(š‘”)=š‘„š‘‡(š‘”)š¶š‘‡š¶š‘„(š‘”)<š›¾2š‘„š‘‡(š‘”)š‘ƒš‘„(š‘”)<š›¾2ī€œš‘”0š‘¤š‘‡(š‘ )š‘¤(š‘ )š‘‘š‘ <š›¾2ī€œāˆž0š‘¤š‘‡(š‘ )š‘¤(š‘ )š‘‘š‘ .(23) For any time š‘”ā‰„0, taking the maximum value of š‘§š‘‡(š‘”)š‘§(š‘”), we have ā€–š‘§ā€–2āˆž<š›¾2ā€–ā€–š‘¤(š‘”)22.(24)
The proof is completed.

For sliding mode dynamics (10) with additive and multiplicative uncertainties, the following theorems give some sufficient conditions which guarantee the sliding mode dynamics is asymptotically stable with a generalized š»2-norm bound š›¾.

Theorem 6. Consider the system (10), uncertainty Ī”š¾(š‘”) in (5), and a given constant š›¾>0. If there exist matrices š‘‹>0, š‘Œ and a scalar šœ€1>0 such that the following linear matrix inequalities (LMIs) are true āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ£š“š‘‹+š‘‹š“š‘‡+šµš‘Œ+š‘Œš‘‡šµš‘‡+šœ€1šµš»1š»š‘‡1šµš‘‡š·š‘‹šøš‘‡1āˆ—āˆ’š¼0āˆ—āˆ—āˆ’šœ€1š¼āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ£<0,(25)š‘‹š‘‹š¶š‘‡āˆ—š›¾2š¼āŽ¤āŽ„āŽ„āŽ¦>0,(26) then the sliding mode dynamics (10) is asymptotically stable with a generalized š»2-norm bound š›¾. Furthermore, the gain matrix š¾ is chosen as š¾=š‘Œš‘‹āˆ’1.

Proof. Pre- and postmultiplying (13) and (14) by diag{š‘‹,š¼} with š‘‹=š‘ƒāˆ’1, respectively, we have āŽ”āŽ¢āŽ¢āŽ£(š“+šµš¾+šµĪ”š¾(š‘”))š‘‹+š‘‹(š“+šµš¾+šµĪ”š¾(š‘”))š‘‡š·āŽ¤āŽ„āŽ„āŽ¦āˆ—āˆ’š¼<0,(27) and (26).
From (27), we can obtain āŽ”āŽ¢āŽ¢āŽ£(š“+šµš¾)š‘‹+š‘‹(š“+šµš¾)š‘‡š·āŽ¤āŽ„āŽ„āŽ¦+āŽ”āŽ¢āŽ¢āŽ£āˆ—āˆ’š¼šµš»10āŽ¤āŽ„āŽ„āŽ¦š¹1ī‚ƒšø(š‘”)1ī‚„+āŽ›āŽœāŽœāŽāŽ”āŽ¢āŽ¢āŽ£š‘‹0šµš»10āŽ¤āŽ„āŽ„āŽ¦š¹1ī‚ƒšø(š‘”)1ī‚„āŽžāŽŸāŽŸāŽ š‘‹0š‘‡<0.(28) From Lemma 2, (28) holds if and only if there exists a scalar šœ€1>0 such that the following inequality holds: āŽ”āŽ¢āŽ¢āŽ£(š“+šµš¾)š‘‹+š‘‹(š“+šµš¾)š‘‡š·āŽ¤āŽ„āŽ„āŽ¦āˆ—āˆ’š¼+šœ€1āŽ”āŽ¢āŽ¢āŽ£šµš»10āŽ¤āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ£šµš»10āŽ¤āŽ„āŽ„āŽ¦š‘‡+1šœ€1āŽ”āŽ¢āŽ¢āŽ£š‘‹šøš‘‡10āŽ¤āŽ„āŽ„āŽ¦ī‚ƒšø1ī‚„š‘‹0<0,(29) which, according to Lemma 1, is equivalent to (25).
The proof is completed.

Theorem 7. Consider the system (10), uncertainty Ī”š¾(š‘”) in (7), and a given constant š›¾>0. If there exist matrices š‘‹>0, š‘Œ and the scalar šœ€2>0 such that the following linear matrix inequalities (LMIs) are true āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ£š“š‘‹+š‘‹š“š‘‡+šµš‘Œ+š‘Œš‘‡šµš‘‡+šœ€2šµš»2š»š‘‡2šµš‘‡š·š‘Œšøš‘‡2āˆ—āˆ’š¼0āˆ—āˆ—āˆ’šœ€2š¼āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ£<0,(30)š‘‹š‘‹š¶š‘‡āˆ—š›¾2š¼āŽ¤āŽ„āŽ„āŽ¦>0,(31) then the sliding mode dynamics (10) is asymptotically stable with a generalized š»2-norm bound š›¾. Furthermore, the gain matrix š¾ is chosen as š¾=š‘Œš‘‹āˆ’1.

Proof. The proof is similar to Theorem 6, and omitted.

3.2. Sliding Mode Controller Design

The input š‘¢(š‘”) in system (1) should be appropriately designed such that the state š‘„(š‘”) in system (1) reaches the sliding surface š‘ (š‘”)=0 within a finite time. The non-fragile sliding mode controller š‘¢(š‘”) is as follows:š‘¢(š‘”)=(š¾+Ī”š¾(š‘”))š‘„(š‘”)āˆ’š‘“(š‘„)āˆ’(šŗšµ)āˆ’1Ɨī€ŗī€»,š›¼š‘ (š‘”)+(ā€–Gš·ā€–ā‹…ā€–š‘¤(š‘”)ā€–+ā€–šŗšµā€–ā‹…šœŒ+š›½)sgn(š‘ (š‘”))(32) where sgn(š‘ (š‘”))=[sign(š‘ 1),sign(š‘ 2),ā€¦,sign(š‘ š‘š)]š‘‡, š›¼>0 and š›½>0.

Theorem 8. Consider the system (1). If the input š‘¢(š‘”) is taken as (32), then the trajectory of the system (1) converges to the sliding surface š‘ (š‘”)=0 in a finite time.

Proof. Consider a Lyapunov function candidate as follows: š‘‰21(š‘”)=2š‘ š‘‡(š‘”)š‘ (š‘”).(33) Calculating the time derivative of š‘‰2(š‘”) along the trajectory of system (1), we have Ģ‡š‘‰2(š‘”)=š‘ š‘‡[].(š‘”)āˆ’šŗšµ(š¾+Ī”š¾(š‘”))š‘„(š‘”)+šŗš·š‘¤(š‘”)+šŗšµ(š‘¢(š‘”)+š‘“(š‘„)+Ī”š‘“(š‘„))(34)
Substituting (32) into (34) yields Ģ‡š‘‰2(š‘”)=š‘ š‘‡[āˆ’(ī€»(š‘”)āˆ’šŗšµ(š¾+Ī”š¾(š‘”))š‘„(š‘”)+šŗš·š‘¤(š‘”)+šŗšµĪ”š‘“(š‘„)+šŗšµš‘“(š‘„)+šŗšµ(š¾+Ī”š¾(š‘”))š‘„(š‘”)āˆ’šŗšµš‘“(š‘„)āˆ’š›¼š‘ (š‘”)ā€–šŗš·ā€–ā‹…ā€–š‘¤(š‘”)ā€–+ā€–šŗšµā€–ā‹…šœŒ+š›½)sgn(š‘ (š‘”))ā‰¤ā€–š‘ (š‘”)ā€–ā‹…(ā€–šŗš·ā€–ā‹…ā€–š‘¤(š‘”)ā€–+ā€–šŗšµā€–ā‹…šœŒ)āˆ’š›¼ā€–š‘ (š‘”)ā€–2)āˆ’(ā€–šŗš·ā€–ā‹…ā€–š‘¤(š‘”)ā€–+ā€–šŗšµā€–ā‹…šœŒ+š›½ā€–š‘ (š‘”)ā€–.(35) Then, we have Ģ‡š‘‰2ā€–(š‘”)ā‰¤āˆ’š›¼ā€–š‘ (š‘”)2āˆ’š›½ā€–š‘ (š‘”)ā€–.(36) According to sliding mode theory, we conclude that the trajectory of the system (1) converges to the sliding surface š‘ (š‘”)=0 in a finite time and remains in it thereafter. Thus, the proof is completed.

4. Simulation

In this section, we use Genesio's chaotic system to verify the effectiveness of the method. Genesio's system with additional input is as follows:š‘§Ģ‡š‘„(š‘”)=š“š‘„+šµ(š‘“(š‘„)+Ī”š‘“(š‘„)+š‘¢(š‘”))+š·š‘¤(š‘”),(š‘”)=š¶š‘„(š‘”),(37) whereāŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ£001āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ£111āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ¦,ī‚ƒī‚„ī€·š‘„š“=010001āˆ’6āˆ’2.92āˆ’1.2,šµ=,š·=š¶=00āˆ’0.1,Ī”š‘“(š‘„)=sin3(ī€øš‘’š‘”),š‘¤(š‘”)=āˆ’š‘”.10+š‘”(38)

Without loss of generality, we consider Ī”š¾(š‘”) is with the norm-bounded additive form, where š»1=[010], šø1=āˆ’0.1 and š¹1(š‘”)=0.1sinš‘”, š¹1(š‘”) satisfies š¹š‘‡1(š‘”)š¹1(š‘”)=0.01sin2š‘”<1. Furthermore, we choose šŗ=[001] and š›¾=0.3162.

Then, solving LMIs (25) and (26) yields ī‚ƒī‚„š¾=4.6545āˆ’0.8134āˆ’0.8913.(39)

The initial value is š‘„(0)=[30āˆ’1]š‘‡. The simulation results are shown in Figures 2, 3, 4, and 5.

Figure 2 shows the phase curves of uncontrolled Genesio's system. Figure 3 shows the time response of the state š‘„(š‘”) of the controlled Genesio's system. Figure 4 shows the control input response curve. Sliding mode response curve is shown in Figure 5. From the simulation results, it is concluded that the proposed method is effective.

5. Conclusion

In this paper, the problem of non-fragile sliding mode control of uncertain chaotic systems with external disturbance is investigated. A new sliding surface is proposed, and sufficient conditions are derived for asymptotic stability with a generalized š»2 disturbance rejection level of sliding mode dynamics. Non-fragile sliding mode controller is established to make the state of system reach the sliding surface in a finite time. Finally, the simulation shows the effectiveness of the proposed method.

Acknowledgment

The authors are grateful for the support of the National Natural Science Foundation of China under Grants nos. 61074003 and 60904023.