Non-Fragile Sliding Mode Control of Uncertain Chaotic Systems

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 𝐻₂ 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 [6–9].

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 [6–11]. 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 𝐻₂ 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 [15–17]. 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 𝐻₂ 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. 𝑅^𝑛 and 𝑅^𝑛×𝑚 denote, respectively, the 𝑛-dimensional Euclidean space and the space of 𝑛×𝑚 real matrices. For a real symmetric matrix 𝑀, 𝑀>0 means 𝑀 is positive definite. 𝐿₂[0,∞) is the space of square-integrable vector functions over [0,∞), and 𝐿_∞ is the space of essential bounded functions. ∗ represents the symmetric form of matrix.

2. Problem Formulation

Consider the following chaotic system:𝑥̇(𝑡)=𝐴𝑥+𝐵(𝑓𝑧(𝑥(𝑡))+=Δ𝐶𝑓𝑥((𝑥𝑡)),+𝑢(𝑡))+𝐷𝑤(𝑡),(1) where 𝑥(𝑡)∈𝑅^𝑛 is the state vector, 𝑢(𝑡)∈𝑅^𝑚 is the control input. 𝑧(𝑡)∈𝑅^𝑞 is the system output. 𝑓(𝑥) is a nonlinear real-valued function. 𝐴,𝐵,𝐶,and𝐷 are matrices with compatible dimensions and (𝐴,𝐵) is completely controllable. 𝐵 is of full column rank. 𝑤(𝑡) is a bounded disturbance. Δ𝑓(𝑥) represents the uncertainty part, satisfying ‖Δ𝑓(𝑥)‖≤𝜌 with a positive constant 𝜌.

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

Figure 1 

The block diagram of the system (1).

The following lemmas are necessary for future discussion.

Lemma 1 (see [8]). For a given matrix 𝑆=^𝑆11𝑆12_{𝑆𝑇12𝑆22} with 𝑆₁₁=𝑆^𝑇11, 𝑆₂₂=𝑆^𝑇22, then the following conditions are equivalent: (1)𝑆<0, (2)𝑆₂₂<0, 𝑆₁₁−𝑆₁₂𝑆⁻¹²²𝑆^𝑇12<0,(3)𝑆₁₁<0, 𝑆₂₂−𝑆^𝑇12𝑆⁻¹¹¹𝑆₁₂<0.

Lemma 2 (see [17]). Let Ω,Γ and Ξ be matrices of appropriate dimensions, and let Ω be symmetrical. Then Ω+Γ𝐹(𝑡)Ξ+(Γ𝐹(𝑡)Ξ)^𝑇<0,(2) for an uncertain matrix 𝐹(𝑡) satisfying 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼 if and only if there exists a scalar 𝜀>0 such that Ω+𝜀ΓΓ^𝑇+1𝜀Ξ^𝑇Ξ<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 𝐻₂-norm bound 𝛾. 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:𝑠(𝑡)=𝐺𝑥(𝑡)−𝑥(0)−^𝑡0(𝐴+𝐵(𝐾+Δ𝐾(𝑠)))𝑥(𝑠)𝑑𝑠,(4) where 𝐺 is a matrix such that 𝐺𝐵 is invertible. 𝐾>0 is a gain matrix to be chosen later, and the matrix Δ𝐾(𝑡) represents the norm-bounded parameter uncertainty.

In this paper, the following two classes of parameter uncertainty will be considered:(a)Δ𝐾(𝑡) is with the norm-bounded additive form Δ𝐾(𝑡)=Δ₁=𝐻₁𝐹₁(𝑡)𝐸₁,(5) where 𝐻₁ and 𝐸₁ are known matrices and 𝐹₁(𝑡) is an unknown matrix satisfying 𝐹^𝑇1(𝑡)𝐹₁(𝑡)≤𝐼;(6)(b)Δ𝐾(𝑡) is with the norm-bounded multiplicative form Δ𝐾(𝑡)=Δ₂=𝐻₂𝐹₂(𝑡)𝐸₂𝐾,(7) where 𝐻₂ and 𝐸₂ are known matrices, and 𝐹₂(𝑡) is an unknown matrix satisfying 𝐹^𝑇2(𝑡)𝐹₂(𝑡)≤𝐼.(8)

Remark 3. From (4), we have 𝑠(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 Δ𝐾(𝑡) into account. Moreover, these existing results can be considered as a special form of the integral-type sliding surface designed in this paper, when Δ𝐾(𝑡) is zero. In fact, uncertainty Δ𝐾(𝑡) 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 𝑠(̇𝑡)=0. Then, the equivalent control law can be obtained as follows: 𝑢_eq(𝑡)=(𝐾+Δ𝐾(𝑡))𝑥(𝑡)−(𝐺𝐵)⁻¹𝐺𝐷𝑤(𝑡)−𝑓(𝑥)−Δ𝑓(𝑥).(9) Substituting (9) into the system (1) and denoting 𝐷=𝐷−𝐵(𝐺𝐵)⁻¹𝐺𝐷, sliding mode dynamics and the output equation can be formulated as 𝑥̇(𝑡)=(𝐴+𝐵𝐾𝑧(+𝑡)𝐵=Δ𝐶𝐾𝑥((𝑡𝑡))).𝑥(𝑡)+𝐷𝑤(𝑡),(10)

Definition 4. Sliding mode dynamics (10) is asymptotically stable with a generalized 𝐻₂-norm bound 𝛾 if the following conditions hold.(i)When 𝑤(𝑡)=0, sliding mode dynamics (10) is asymptotically stable.(ii)Sliding mode dynamics (10) has a given generalized 𝐻₂ (or 𝐿₂−𝐿_∞) disturbance rejection level; that is, the 𝐿₂−𝐿_∞ gain from disturbance input 𝑤(𝑡) to control output 𝑧(𝑡) of the system (10) is less than a given value. It is to make, under the zero-valued initial condition, the following inequality hold: sup_{𝑤∈𝐿2−0}‖𝑧‖^2∞‖𝑤‖²²<𝛾²,(11) where ‖𝑤‖²²=^∞0𝑤^𝑇(𝑡)𝑤(𝑡)𝑑𝑡,‖𝑧‖^2∞=sup_𝑡𝑧^𝑇(𝑡)𝑧(𝑡).(12)

Theorem 5. Consider the system (10) and a given constant 𝛾>0. If there exist matrices 𝑃>0 and 𝐾 such that the following matrix inequalities are true Θ=⎡⎢⎢⎣𝑃(𝐴+𝐵𝐾+𝐵Δ𝐾(𝑡))+(𝐴+𝐵𝐾+𝐵Δ𝐾(𝑡))^𝑇<0,⎡⎢⎢⎣𝑃𝐶_𝑇∗𝑃𝑃−𝐷𝐼⎤⎥⎥⎦(13)∗𝛾²𝐼⎤⎥⎥⎦>0,(14) then the sliding mode dynamics (10) is asymptotically stable with a generalized 𝐻₂-norm bound 𝛾.

Proof. Consider the following Lyapunov functional candidate: 𝑉₁(𝑡)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡).(15) The time derivative of 𝑉₁(𝑡) along the trajectories of system (10) with 𝑤(𝑡)=0 is given by 𝑉₁̇(𝑡)=𝑥^𝑇(𝑡+)(𝑃𝐴(+𝐴𝐵+𝐾𝐵+𝐾𝐵+Δ𝐵𝐾Δ(𝐾𝑡))(_𝑇𝑡))𝑃𝑥(𝑡).(16)
From (13), Θ<0 implies 𝑃(𝐴+𝐵𝐾+𝐵Δ𝐾(𝑡))+(𝐴+𝐵𝐾+𝐵Δ𝐾(𝑡))^𝑇𝑃<0, then we have 𝑉₁̇(𝑡)<0,(17) which implies sliding mode dynamics (10) is asymptotically stable, when 𝑤(𝑡)=0.
In order to construct the generalized 𝐻₂ (𝐿₂−𝐿_∞) performance index, we assume the zero-valued initial condition, then 𝑉₁(𝑡)|_𝑡=0=0. Now, we consider the following cost performance: 𝐽=𝑉₁(𝑡)−^𝑡0𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠.(18) For any nonzero 𝑤(𝑡)∈𝐿₂[0,∞), we have 𝐽==^𝑡0𝑉₁̇(𝑠)−𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠_𝑡0𝜉^𝑇Θ𝜉𝑑𝑠,(19) where 𝜉=[𝑥^𝑇(𝑠)𝑤^𝑇(𝑠)]^𝑇.
From (13), we obtain 𝐽=𝑉₁(𝑡)−^𝑡0𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠<0.(20) Equation (20) leads to 𝑥^𝑇(𝑡)𝑃𝑥(𝑡)=𝑉₁(𝑡)<^𝑡0𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠.(21) For (14), by Lemma 1, we have 𝐶^𝑇𝐶<𝛾²𝑃(22) which leads to 𝑧^𝑇(𝑡)𝑧(𝑡)=<𝑥𝛾^𝑇₂(𝑡)𝐶^𝑇𝐶𝑥(𝑡)<𝛾₂𝑥^𝑇(𝑡)𝑃𝑥(𝑡)^𝑡0<𝛾₂𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠^∞0𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠.(23) For any time 𝑡≥0, taking the maximum value of 𝑧^𝑇(𝑡)𝑧(𝑡), we have ‖𝑧‖^2∞<𝛾²‖𝑤(𝑡)‖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 𝐻₂-norm bound 𝛾.

Theorem 6. Consider the system (10), uncertainty Δ𝐾(𝑡) in (5), and a given constant 𝛾>0. If there exist matrices 𝑋>0, 𝑌 and a scalar 𝜀₁>0 such that the following linear matrix inequalities (LMIs) are true ⎡⎢⎢⎢⎢⎣𝐴𝑋+𝑋𝐴^𝑇+𝐵𝑌+𝑌^𝑇𝐵^𝑇+𝜀₁𝐵𝐻₁𝐻^𝑇1𝐵^𝑇𝐷𝑋𝐸^𝑇1∗∗−∗𝐼−𝜀0₁⎡⎢⎢⎣𝑋𝑋𝐶_𝑇𝐼⎤⎥⎥⎥⎥⎦<0,(25)∗𝛾²𝐼⎤⎥⎥⎦>0,(26) then the sliding mode dynamics (10) is asymptotically stable with a generalized 𝐻₂-norm bound 𝛾. Furthermore, the gain matrix 𝐾 is chosen as 𝐾=𝑌𝑋⁻¹.

Proof. Pre- and postmultiplying (13) and (14) by diag{𝑋,𝐼} with 𝑋=𝑃⁻¹, respectively, we have ⎡⎢⎢⎣(𝐴+𝐵𝐾+𝐵Δ𝐾(𝑡))𝑋+∗𝑋(𝐴+𝐵𝐾+𝐵Δ𝐾(𝑡))^𝑇−𝐷𝐼⎤⎥⎥⎦<0,(27) and (26).
From (27), we can obtain ⎡⎢⎢⎣(𝐴+𝐵𝐾)𝑋+∗𝑋(𝐴+𝐵𝐾)^𝑇−𝐷𝐼⎤⎥⎥⎦+⎡⎢⎢⎣𝐵𝐻₁0⎤⎥⎥⎦𝐹¹+⎛⎜⎜⎝⎡⎢⎢⎣𝐵𝐻₁(𝑡)𝐸₁𝑋00⎤⎥⎥⎦𝐹¹(𝑡)𝐸₁𝑋0⎞⎟⎟⎠^𝑇<0.(28) From Lemma 2, (28) holds if and only if there exists a scalar 𝜀₁>0 such that the following inequality holds: ⎡⎢⎢⎣(𝐴+𝐵𝐾)𝑋+∗𝑋(𝐴+𝐵𝐾)^𝑇−𝐷𝐼⎤⎥⎥⎦+𝜀₁⎡⎢⎢⎣𝐵𝐻₁0⎤⎥⎥⎦⎡⎢⎢⎣𝐵𝐻0¹⎤⎥⎥⎦^𝑇+1𝜀₁⎡⎢⎢⎣𝑋𝐸^𝑇10⎤⎥⎥⎦𝐸¹𝑋0<0,(29) which, according to Lemma 1, is equivalent to (25).
The proof is completed.

Theorem 7. Consider the system (10), uncertainty Δ𝐾(𝑡) in (7), and a given constant 𝛾>0. If there exist matrices 𝑋>0, 𝑌 and the scalar 𝜀₂>0 such that the following linear matrix inequalities (LMIs) are true ⎡⎢⎢⎢⎢⎣𝐴𝑋+𝑋𝐴^𝑇+𝐵𝑌+𝑌^𝑇𝐵^𝑇+𝜀₂𝐵𝐻₂𝐻^𝑇2𝐵^𝑇𝐷𝑌𝐸^𝑇2∗∗−∗𝐼−𝜀0₂⎡⎢⎢⎣𝑋𝑋𝐶_𝑇𝐼⎤⎥⎥⎥⎥⎦<0,(30)∗𝛾²𝐼⎤⎥⎥⎦>0,(31) then the sliding mode dynamics (10) is asymptotically stable with a generalized 𝐻₂-norm bound 𝛾. Furthermore, the gain matrix 𝐾 is chosen as 𝐾=𝑌𝑋⁻¹.

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

3.2. Sliding Mode Controller Design

The input 𝑢(𝑡) in system (1) should be appropriately designed such that the state 𝑥(𝑡) in system (1) reaches the sliding surface 𝑠(𝑡)=0 within a finite time. The non-fragile sliding mode controller 𝑢(𝑡) is as follows:𝑢(𝑡)=(𝐾+Δ𝐾(𝑡))𝑥(𝑡)−𝑓(𝑥)−(𝐺𝐵)⁻¹×𝛼𝑠(𝑡)+(‖G𝐷‖⋅‖𝑤(𝑡)‖+‖𝐺𝐵‖⋅𝜌+𝛽)sgn(𝑠(𝑡)),(32) where sgn(𝑠(𝑡))=[sign(𝑠₁),sign(𝑠₂),…,sign(𝑠_𝑚)]^𝑇, 𝛼>0 and 𝛽>0.

Theorem 8. Consider the system (1). If the input 𝑢(𝑡) is taken as (32), then the trajectory of the system (1) converges to the sliding surface 𝑠(𝑡)=0 in a finite time.

Proof. Consider a Lyapunov function candidate as follows: 𝑉₂(𝑡)=12𝑠^𝑇(𝑡)𝑠(𝑡).(33) Calculating the time derivative of 𝑉₂(𝑡) along the trajectory of system (1), we have 𝑉₂̇(𝑡)=𝑠^𝑇(𝑡)+[𝐺−𝐺𝐵𝐵(𝑢((𝐾𝑡)++Δ𝑓𝐾(𝑥()𝑡)+)𝑥Δ(𝑓𝑡)(+𝑥)𝐺)]𝐷.𝑤(𝑡)(34)
Substituting (32) into (34) yields 𝑉₂̇(𝑡)=≤𝑠‖^𝑇𝑠((𝑡𝑡)+−−)‖[−(𝐺𝐺‖⋅𝐺𝐵𝐵𝐺(‖𝐵𝑓𝑓𝐷𝐺(((‖𝐾𝐷𝑥𝑥))⋅‖++−‖⋅𝑤Δ𝐺𝛼‖𝐾𝑠(𝑤𝐵𝑡()𝑡((‖()𝑡𝑡𝐾)))+‖+𝑥‖+(Δ𝐺𝑡)‖𝐾𝐵+𝐺‖(𝐵𝐺𝑡)⋅‖𝐷)𝜌𝑥⋅𝑤+(𝜌𝑡())𝛽𝑡−))+s𝛼g‖𝐺n𝑠𝐵((𝑠𝑡Δ)(‖𝑡𝑓)₂)(𝑥)−(‖𝐺𝐷‖⋅‖𝑤(𝑡)‖+‖𝐺𝐵‖⋅𝜌+𝛽)‖𝑠(𝑡)‖.(35) Then, we have 𝑉₂̇(𝑡)≤−𝛼‖𝑠(𝑡)‖²−𝛽‖𝑠(𝑡)‖.(36) According to sliding mode theory, we conclude that the trajectory of the system (1) converges to the sliding surface 𝑠(𝑡)=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:𝑥̇(𝑡)=𝐴𝑥+𝐵(𝑓𝑧(𝑥(𝑡))+=Δ𝐶𝑓𝑥((𝑥𝑡)),+𝑢(𝑡))+𝐷𝑤(𝑡),(37) where𝐶=0𝐴0=−⎡⎢⎢⎢⎢⎣−0006.1−,210.9Δ2𝑓−(𝑥101.)2=⎤⎥⎥⎥⎥⎦,sin𝐵𝑥=₃⎡⎢⎢⎢⎢⎣001⎤⎥⎥⎥⎥⎦,𝐷=⎡⎢⎢⎢⎢⎣111⎤⎥⎥⎥⎥⎦,(𝑡),𝑤(𝑡)=𝑒^−𝑡10+𝑡.(38)

Without loss of generality, we consider Δ𝐾(𝑡) is with the norm-bounded additive form, where 𝐻₁=[010], 𝐸₁=−0.1 and 𝐹₁(𝑡)=0.1sin𝑡, 𝐹₁(𝑡) satisfies 𝐹^𝑇1(𝑡)𝐹₁(𝑡)=0.01sin²𝑡<1. Furthermore, we choose 𝐺=[001] and 𝛾=0.3162.

Then, solving LMIs (25) and (26) yields 𝐾=4.6545−0.8134−0.8913.(39)

The initial value is 𝑥(0)=[30−1]^𝑇. The simulation results are shown in Figures 2, 3, 4, and 5.

Figure 2 

The phase curves of uncontrolled Genesio's system.

Figure 3 

State 𝑥(𝑡) of the controlled Genesio's system.

Figure 4 

The control input 𝑢(𝑡).

Figure 5 

Sliding mode 𝑠(𝑡).

Figure 2 shows the phase curves of uncontrolled Genesio's system. Figure 3 shows the time response of the state 𝑥(𝑡) 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 𝐻₂ 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.