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RETRACTED: Controlling chaotic and hyperchaotic systems via energy regulation
Chaos, Solitons and Fractals 14 (2002) 1015–1025 www.elsevier.com/locate/chaos
Controlling chaotic and hyperchaotic systems via energy regulation L. Laval *, N.K. M’Sirdi Laboratoire de Robotique de Paris, UPMC-UVSQ-CNRS URA 1778, 10-12, av. de l’Europe, 78140 Velizy, France Accepted 16 January 2002
Abstract This paper focuses on a new control approach to steer trajectories of chaotic or hyperchaotic systems towards stable periodic orbits or stationary points of interest. This approach mainly consists in a variable structure control (VSC) that we extend by explicitly considering the system energy as basis for both controller design and system stabilization. In this paper, we present some theoretical results for a class of nonlinear (possibly chaotic or hyperchaotic) systems. Then some capabilities of the proposed approach are illustrated through examples related to a four-dimensional hyperchaotic system. 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Many applications in communication (e.g. encryption of data for secure communications), soft-computing (e.g. pattern recognition via oscillators based neural networks), music (e.g. sound waves generation),. . . involve the use of dynamical systems as core engine of computational processes or others. In such contexts, the exploration of wide ranges of potential combinations or accession to extended computation capabilities involve to perform controlled or uncontrolled (but bounded) trajectories going around a large (closed) region of the system state space. This motivates the increasing interest for the use of chaotic systems (e.g. [3,5,9,21,23–25]) due to their intrinsic properties, the existence of low-order and low complexity mathematical descriptions of such systems, an ease of construction of electronic circuits to generate chaotic waveforms (see [13,19]), etc. However, one major problem is to control these chaotic systems to steer their trajectories towards specific orbits or stationary states of interest. Since the seminal work of H€ ubler and coauthors [1], who first demonstrated that an appropriate external input is efficient to force a chaotic system to perform desired motions, many ideas and methods for controlling chaos have then been proposed (see [5,12,15] and references therein): OGY method [4], time-delay feedback control [6], methods coming from classical control theory (adaptive control [7,8], H1 control [10], sliding mode control [11,14,23–25]), etc. An overview of these methods then leads to notice some drawbacks or constraints such as: • The use of a linearized model of the system as basis for control law design (in a such case, stability properties remain local ones). • The requirement of an analytical description of the targeted orbits. • The need of an accurate system modelization. • The lack of robustness against system parameters uncertainties or external disturbances, etc.
*
Corresponding author. Tel.: +33-1-39254828; fax: +33-1-39254967. E-mail address: [email protected] (L. Laval).
0960-0779/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 0 3 4 - 6
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Then, in this paper, we focus on a new control approach, referred to as energy based sliding mode control (ESMC for short) [16,17], which mainly consists in a variable structure control (VSC) design that we extend by an explicit consideration of system energy as basis for both controller design and system stabilization. The control objective is then to regulate the energy with respect to a shaped nominal representation, implicitly related to system trajectories. As a result, for any given (admissible) initial conditions, the controlled system trajectories converge towards and are maintained in a global invariant set included in a selected subspace of the state space. Then, according to controlled system asymptotic properties, dimension of the targeted subspace and Poincare–Bendixon theorem, 1 the (controlled) system trajectories may be driven to an equilibrium point (i.e. a stationary state), an asymptotically stable limit cycle or be themselves (stable) periodic or quasi-periodic orbits. It is then important to point out that the proposed control method does not require any mathematical expression of the targeted trajectory. Nevertheless, some geometrical properties of the stabilized orbits or localization of stationary points can be shaped by tuning some control law parameters and the magnitude of the system energy to target. Moreover, the intrinsic properties of the control scheme (that is VSC) lead to a good robustness against modeling uncertainties and external disturbances. This paper is organized as follows. Section 2 introduces some theoretical results related to the ESMC approach, for a class of nonlinear (possibly chaotic or hyperchaotic) systems. Section 3 illustrates some capabilities of the proposed approach through examples related to a four-dimensional hyperchaotic system. Finally, in Section 4, some concluding remarks are given. 2. Theoretical framework This paper considers the class of (four-dimensional) nonlinear autonomous systems of the form: X_ ¼ F ðX ; U Þ ¼ f ðX Þ þ U ; n
is the state vector partitioned as X ¼ ½X1T X2T T with X1 2 Rm vector of initial conditions, f ðX Þ ¼ ½f1T ðX1 ; X2 Þ f2T ðX1 ; X2 ÞT q
where X 2 R ð1 6 n 6 4Þ 4 mÞ. X ð0Þ 2 Rn is the f2 ðX1 ; X2 Þ 2 C 1 ðRp Þ. U 2 R ðq ¼ m þ pÞ is the vector of control inputs. Moreover we consider the following assumptions.
ð1Þ p
ð0 6 m 6 3Þ and X2 2 R ðp ¼ with f1 ðX1 ; X2 Þ 2 C 1 ðRm Þ and
Assumption A1. The system is, at least, locally observable and controllable. Assumption A2. The energy of the system can be represented by a Lyapunov function V which can be splitted into two added parts V1 and V2 (i.e. V ¼ V1 þ V2 ) related to scalar positive functions VT ðX1 Þ and VIS ðX2 Þ, respectively (i.e. V1 ¼ g1 ðVT ðX1 ÞÞ and V2 ¼ g2 ðVIS ðX2 ÞÞ). Assumption A3. Scalar positive functions VT ðX1 Þ and VIS ðX2 Þ have continuous first derivatives which can be expressed as V_T ¼ X_ 1T W1 X1 ; V_IS ¼ X_ T W2 X2 ;
ð2Þ ð3Þ
2
where W1 2 Rm m and W2 2 Rp p are diagonal matrices with strictly positive real values along the diagonal. We now state our main results. Lemma 1. Consider the autonomous system (1) and Assumptions A1–A3. Moreover, consider the following control structure: T ð4Þ U ¼ uTT uTIS with uT ¼ C1 signðVT ÞX1 f^1 ðX1 ; X2 Þ; uIS ¼ C2 signðVIS V ÞX2 f^2 ð0; X2 Þ; IS
1
In case of two-dimensional targeted subspaces.
ð5Þ ð6Þ
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where uT 2 Rm 1 , uIS 2 Rp 1 are vectors of control inputs. f^1 ðX1 ; X2 Þ and f^2 ð0; X2 Þ are vectors of continuous functions which represent (local) equivalent system dynamics. 2 C1 2 Rm m and C2 2 Rp p are diagonal matrices with strictly positive real values, and VIS is a positive constant which characterizes a desired magnitude of energy. Then, (i) all solutions of the controlled system asymptotically converge towards and are maintained in a global invariant set XIS included in the same subspace as X2 and defined by 3 hVIS VIS i ¼ 0. (ii) the energy of the controlled system converges towards a neighborhood eIS of VIS .
Proof of Lemma 1. With respect to Assumption A2, let us consider a Lyapunov function candidate V ¼ V1 þ V2 such as V1 ¼ 12VT2 ðX1 Þ and V2 ¼ 12ðVIS ðX2 Þ VIS Þ2 . First, let us focus on V1 ¼ 12VT2 . Then, V_1 ¼ V_T VT :
ð7Þ
From Assumption A3 and system definition (1), the derivative V_T can be expressed as V_T ¼ X_ 1T W1 X1 () V_T ¼ ðf1 ðX1 ; X2 Þ þ uT ÞT W1 X1 :
ð8Þ
Then, substituting (5) into (8) leads to T V_T ¼ f1 ðX1 ; X2 Þ C1 signðVT ÞX1 f^1 ðX1 ; X2 Þ W1 X1 :
ð9Þ
As f^1 ðX1 ; X2 Þ represents (local) equivalent system dynamics according to f1 ðX1 ; X2 Þ, then the averaged value of f1 ðX1 ; X2 Þ f^1 ðX1 ; X2 Þ is zero (i.e. hf1 ðX1 ; X2 Þ f^1 ðX1 ; X2 Þi ¼ 0). Thus, in the mean, V_T ’ C1 signðVT ÞX1T W1 X1 :
ð10Þ
Substituting (10) into (7) then leads to V_1 ’ C1 signðVT ÞX1T W1 X1 VT :
ð11Þ
Therefore, V_1 is negative semi-definite and V1 is positive definite. Consequently, from sliding mode theory (cf. [20,22]), VT converges towards a vicinity of 0 in a finite time t1 > 0. Moreover, it comes from (10) that X1 is bounded and also goes to a vicinity of zero (as V_T goes to zero). Now, let us focus on V2 ¼ 12ðVIS VIS Þ2 . Then, V_2 ¼ V_IS ðVIS VIS Þ:
ð12Þ
From Assumption A3 and system definition (1), the derivative V_IS can be formulated as V_IS ¼ X_ 2T W2 X2 () V_IS ¼ ðf2 ðX1 ; X2 Þ þ uIS ÞT W2 X2 :
ð13Þ
Substituting (6) into (13) leads to T V_IS ¼ f2 ðX1 ; X2 Þ C2 sign VIS VIS X2 f^2 ð0; X2 Þ W2 X2 T T () V_IS ¼ f2 ðX1 ; X2 Þ f^2 ð0; X2 Þ W2 X2 C2 sign VIS VIS X2 W2 X2 : Now, let us recall that: • f^2 ð0; X2 Þ represents (local) equivalent system dynamics • VT goes to a vicinity of 0 in a finite time t1 . 2 3 4
4
when the averaged value of X1 is zero.
Deduced from analysis of the true system and its modelization. hi : averaged value of . According to f2 ðX1 ; X2 Þ.
ð14Þ
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Then, D
E f2 ðX1 ; X2 Þ f^2 ð0; X2 Þ ¼ 0 8t P t1 :
ð15Þ
Thus, for t P t1 , V_IS can be expressed (in the mean) as T V_IS ’ C2 sign VIS VIS X2 W2 X2 :
ð16Þ
Substituting this relation into (12) then leads to T V_2 ’ C2 sign VIS VIS X2 W2 X2 ðVIS VIS Þ:
ð17Þ
Therefore, according to positiveness of C2 and W2 elements, V_2 is negative semi-definite and V2 is positive definite (8t P t1 ). Consequently, from sliding mode theory (cf. [20,22]), the sliding surface defined by S2 ðX Þ ¼ VIS VIS ¼ 0 is attractive and VIS converges towards a vicinity of VIS in a finite time t2 P t1 > 0. Thus, we can conclude that, in a finite time t2 P t1 > 0, the controlled system trajectories are driven to and maintained in an invariant set XIS defined by hVIS VIS i ¼ 0 and included the same subspace as X2 . Moreover, as the system energy is bounded and its representation (related to VT ðX1 Þ and VIS ðX2 Þ) goes to a vicinity of VIS in a finite time, this energy also converges towards a neighborhood eIS of VIS (where eIS is related to the discrepancy between the true system energy and its nominal representation). Remark 1. Lemma 1 states that the solutions of the controlled system converge towards some global invariant sets. Thus, according to (controlled) system asymptotic properties, dimension of the targeted subspace (related to X2 ) and Poincare–Bendixon theorem, 5 the system trajectories under control may be driven to an equilibrium point (i.e. a stationary state), an asymptotically stable limit cycle or be themselves stable periodic or quasi-periodic orbits. Remark 2. By inspection of (6), it appears that convergence towards the targeted energy VIS is mainly supported by control law uIS . In order to improve the speed of convergence (i.e. the convergence rate), we propose the following result. Corollary 1. Lemma 1 holds when replacing VIS into (6) by a scalar positive function V of the form: V ¼ VT þ VIS . Proof of Corollary 1. Let us consider proof of Lemma 1 and replace the function V2 ¼ 12ðVIS VIS Þ2 by V2 ¼ 12ðV VIS Þ2
with V ¼ VT þ VIS ;
then V_2 ¼ V_ ðV VIS Þ () V_2 ¼ ðV_T þ V_IS ÞðVT þ VIS VIS Þ
ð18Þ
as uT , VT and V_T go to a vicinity of 0 in a finite time t1 > 0, thus, in the mean, V_2 ffi ðV_IS ÞðVIS VIS Þ
8t P t1 :
End of proof is then straightforward.
ð19Þ
As a result, it comes from Corollary 1 that control law uT may be involved in convergence towards the targeted magnitude of energy (that is VIS ), before VT reaches a neighborhood of zero. 2.1. Discussion According to Lemma 1 and its proof, VT and its derivative converge towards a vicinity of zero (in a finite time) if we can define a local approximation f^1 ðX1 ; X2 Þ of f1 ðX1 ; X2 Þ such as f^1 ðX1 ; X2 Þ f1 ðX1 ; X2 Þ is zero in the mean. In a such case, X1 also goes to a vicinity of zero (cf. relation (10)). This means that controlled system trajectories go around a subspace (of the state space) whom dimension is restricted to the same one as X2 .
5
In case of two-dimensional targeted subspaces.
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It is then worthwhile to investigate the case for which f^1 ðX1 ; X2 Þ f1 ðX1 ; X2 Þ is different from zero (in the mean). In a such context, relation (10) can be expressed as V_T ¼ C1 signðVT ÞX1T W1 X1 þ dðX1 ; X2 Þ;
ð20Þ
where dðX1 ; X2 Þ denotes an error function of the form dðX1 ; X2 Þ ¼ f^1 ðX1 ; X2 Þ f1 ðX1 ; X2 Þ: The study of global convergence then leads to consider the two following potential cases: (1) Case 1: If kC1 signðVT ÞX1T W1 X1 k P kdðX1 ; X2 Þk, then V_T is negative semi-definite or negative definite, leading to a stable behavior of the controlled system. (2) Case 2: If kC1 signðVT ÞX1T W1 X1 k < kdðX1 ; X2 Þk, then V_T is positive definite, leading to an unstable behavior (i.e. divergence of the system trajectory). However, it is important to point out that divergence of the system trajectory mainly corresponds to an increase of kX1 k (as uIS tends to slow down the increase of X2 components). Therefore, if the growth rate of C1 signðVT ÞX1T W1 X1 is greater than those of dðX1 ; X2 Þ, then the system trajectory tends to reach a some states for which case (1) is valid. This can lead to an alternation between two local phases: a (local) divergent phase and a (local) phase of convergence towards a stable state. As a result, a such alternation can yield to a global, stable, quasi-periodic behavior, for which the system trajectory goes around a closed subspace (of the state space) whom dimension is greater than those of X2 . Remark 3. The previous analysis can be considered as basis for designing or tuning the gain matrix C1 components. Indeed, as dðX1 ; X2 Þ may characterize both some uncertainties on system modeling and external disturbances, tuning of C1 can be oriented such as the inequality of case (1) is always fulfilled. 3. Example: control of a hyperchaotic circuit This section aims at providing an example, related to the control of a hyperchaotic system, to illustrate some capabilities of the ESMC approach. To this end, let us consider the electronic circuit depicted through Figs. 1 and 2 [18].
Fig. 1. The hyperchaotic circuit.
Fig. 2. Current–voltage characteristic of the nonlinear diode.
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This circuit can be defined by the following set of ordinary differential equations [18]: C1 v_ 1 ¼ f ðv2 v1 Þ i1 ; C2 v_ 2 ¼ f ðv2 v1 Þ i2 ; L1 i_1 ¼ v1 þ Ri1 ;
ð21Þ
L2 i_2 ¼ v2 ; where v1 ; v2 ; i1 and i2 represent the voltage across C1 , the voltage across C2 , the current through L1 and the current through L2 , respectively. f is a piecewise continuous function related to the diode characteristic. This function can be expressed as f ðxÞ ¼ m0 x þ 12ðm1 m0 Þðj x þ Ej j x EjÞ;
ð22Þ
where E ¼ 1 represents the breakpoint voltage of the nonlinear diode. In this example, let us consider the systems parameters (see [2]): C1 ¼ 0:5, C2 ¼ 0:05, L1 ¼ 1 and L2 ¼ 2=3, R ¼ 1, m0 ¼ 3 and m1 ¼ 0:2. This circuit then exhibits a typical hyperchaotic behavior (see Fig. 3), emphasized by computing the Lyapunov exponents (two of them are positive [2]). In order to apply the ESMC approach, let us consider now the state vector X ¼ ½v1 v2 i1 i2 T and the following expression of the system energy, Esyst ¼ 12C1 v21 þ 12C2 v22 þ 12L1 i21 þ 12L2 i22 :
ð23Þ
The crucial point of the control design then consists in partitioning the state vector into two sub-vectors (namely X1 and X2 ). Indeed, definition of X2 is directly related to the subspace (of the state space) in which the system trajectories has to be stabilized. However, some subspaces may not admit some stable invariant sets with a finite magnitude of energy. Therefore, such a partitioning implies necessary to take into account the physical characteristics of the system. For instance, inspection of the hyperchaotic circuit of Fig. 1 can lead to consider two couples of inductor–capacitor to allow some energy transfers (such as storing the energy alternatively in the inductor and capacitor). According to that, the state partitioning is then of the form X1 ¼ ½v1 i1 T and X2 ¼ ½v2 i2 T . Moreover, with respect to relation (23) and Assumptions A2 and A3, let us define the following Lyapunov function candidate V ðX Þ and its splitted components (V ðX Þ ¼ V1 ðX1 Þ þ V2 ðX2 Þ) such as
1 2 1 2 L1 2 v þ i ; V1 ðX1 Þ ¼ VT with VT ¼ 2 2 1 C1 1
Fig. 3. Projection on the plane (v1 ; v2 ) of the hyperchaotic system trajectory.
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and 1 V2 ðX2 Þ ¼ VIS2 2
with VIS ¼
1 2
1021
C2 2 L2 2 v2 þ i2 : C1 C2
According to Lemma 1, the hyperchaotic system under control can then be defined by: C1 v_ 1 ¼ f ðv2 v1 Þ i1 þ uT1 ; C2 v_ 2 ¼ f ðv2 v1 Þ i2 þ uIS1 ; L1 _i1 ¼ v1 þ Ri1 þ uT2 ;
ð24Þ
L2 _i2 ¼ v2 þ uIS2 with uT1 ¼ C111 signðVT Þv1 f^11 ðv1 ; v2 ; i1 ; i2 Þ; uT ¼ C1 signðVT Þi1 f^12 ðv1 ; v2 ; i1 ; i2 Þ; 2
22
Fig. 4. Controlled system trajectory in the space (v1 ; v2 ; i1 Þ.
Fig. 5. Controlled system trajectory in the space (v1 ; i1 ; i2 ).
ð25Þ
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Fig. 6. Time waveform of function VT .
Fig. 7. Time waveform of function VIS .
uIS1 ¼ C211 signðVIS VIS Þv2 f^21 ð0; v2 ; 0; i2 Þ; uIS ¼ C2 signðVIS V Þi2 f^22 ð0; v2 ; 0; i2 Þ: 2
22
IS
By considering, for instance, an equivalent representation of system dynamics of the form: 1 1 f^11 ðv1 ; v2 ; i1 ; i2 Þ ¼ f ðv2 v1 Þ þ i1 ; ^ ^ C1 C1 ^ R 1 f^12 ðv1 ; v2 ; i1 ; i2 Þ ¼ v1 þ i1 ; L^1 L^1 1 f^21 ð0; v2 ; 0; i2 Þ ¼ i2 ; C^2 ^ f22 ð0; v2 ; 0; i2 Þ ¼ 0
ð26Þ
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Fig. 8. Controlled system trajectory in the space (v1 ; v2 ; i1 Þ.
Fig. 9. Controlled system trajectory in the space (v1 ; i1 ; i2 ).
and the following parameters • • • •
Initial conditions (arbitrary chosen): v1 ð0Þ ¼ 0:2, v2 ð0Þ ¼ 0:1, i1 ð0Þ ¼ 0:01 and i2 ð0Þ ¼ 0:1. Control law gains: C111 ¼ C122 ¼ 3, C211 ¼ C222 ¼ 2. Targeted magnitude of energy: VIS ¼ 5. Parameters estimated values: C^1 ¼ C1 , C^2 ¼ C2 , L^1 ¼ L1 and L^2 ¼ L2 .
We then obtain the simulation results of Figs. 4–7. By looking at Figs. 4 and 5, we can observe that the system trajectory converges towards an stationary point and stay at this point. This behavior is emphasized by the time waveform of system energy representations (namely, function VT and VIS ) as shown through Figs. 6 and 7. Indeed, Fig. 6 exhibits the convergence of VT towards a neighborhood of 0 in a finite time (t1 ’ 0:8 s) and Fig. 7 shows both the convergence and the stabilization of VIS in a neighborhood of the desired magnitude of energy (that is VIS ¼ 5).
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Fig. 10. Time waveform of function VT .
Fig. 11. Time waveform of function VIS .
In order to illustrate some ideas reported in Section 2.1, let us consider now the simulation results of Figs. 8–11. This result comes from the consideration of the following parameters: • An error rate of 10% on the estimated values of C1 and C2 . • The control law gains: C111 ¼ 5; C122 ¼ 1, C211 ¼ 5; C222 ¼ 1. • The same initial conditions and targeted magnitude of energy as in the previous example. Through Fig. 10, we can observe that function VT does not converge asymptotically towards zero to stay in, but becomes periodic (remaining, nevertheless, in a bounded neighborhood of zero). This leads the system to the stabilization of a periodic orbit as shown through Figs. 8 and 9, according to the desired magnitude of the targeted energy (that is: VIS ¼ 5). Moreover, we can note that the stabilized orbit goes around a three-dimensional subspace of the system state space (see Fig. 8), whose dimension is greater than those of vector X2 .
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4. Conclusion In this paper we have introduced some theoretical results related to a new approach to control chaotic or hyperchaotic systems. This approach, referred to as ESMC, is based on an explicit consideration of the system energy, and leads to the convergence, in a finite time, of the system trajectories towards a desired behavior of interest (that is stationary state, periodic orbit, etc.). It is worthwhile to note that this method does not require neither an analytical description of the targeted behavior nor an accurate modelization of system dynamics. Moreover, due to its intrinsic scheme (namely VSC), the control approach leads to a good robustness against some modeling uncertainties and external disturbances. Future research will be devoted to the control of coupled chaotic or hyperchaotic systems, by means of ESMC approach.
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Controlling chaotic and hyperchaotic systems via energy regulation L. Laval *, N.K. M’Sirdi Laboratoire de Robotique de Paris, UPMC-UVSQ-CNRS URA 1778, 10-12, av. de l’Europe, 78140 Velizy, France Accepted 16 January 2002
Abstract This paper focuses on a new control approach to steer trajectories of chaotic or hyperchaotic systems towards stable periodic orbits or stationary points of interest. This approach mainly consists in a variable structure control (VSC) that we extend by explicitly considering the system energy as basis for both controller design and system stabilization. In this paper, we present some theoretical results for a class of nonlinear (possibly chaotic or hyperchaotic) systems. Then some capabilities of the proposed approach are illustrated through examples related to a four-dimensional hyperchaotic system. 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Many applications in communication (e.g. encryption of data for secure communications), soft-computing (e.g. pattern recognition via oscillators based neural networks), music (e.g. sound waves generation),. . . involve the use of dynamical systems as core engine of computational processes or others. In such contexts, the exploration of wide ranges of potential combinations or accession to extended computation capabilities involve to perform controlled or uncontrolled (but bounded) trajectories going around a large (closed) region of the system state space. This motivates the increasing interest for the use of chaotic systems (e.g. [3,5,9,21,23–25]) due to their intrinsic properties, the existence of low-order and low complexity mathematical descriptions of such systems, an ease of construction of electronic circuits to generate chaotic waveforms (see [13,19]), etc. However, one major problem is to control these chaotic systems to steer their trajectories towards specific orbits or stationary states of interest. Since the seminal work of H€ ubler and coauthors [1], who first demonstrated that an appropriate external input is efficient to force a chaotic system to perform desired motions, many ideas and methods for controlling chaos have then been proposed (see [5,12,15] and references therein): OGY method [4], time-delay feedback control [6], methods coming from classical control theory (adaptive control [7,8], H1 control [10], sliding mode control [11,14,23–25]), etc. An overview of these methods then leads to notice some drawbacks or constraints such as: • The use of a linearized model of the system as basis for control law design (in a such case, stability properties remain local ones). • The requirement of an analytical description of the targeted orbits. • The need of an accurate system modelization. • The lack of robustness against system parameters uncertainties or external disturbances, etc.
*
Corresponding author. Tel.: +33-1-39254828; fax: +33-1-39254967. E-mail address: [email protected] (L. Laval).
0960-0779/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 0 3 4 - 6
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Then, in this paper, we focus on a new control approach, referred to as energy based sliding mode control (ESMC for short) [16,17], which mainly consists in a variable structure control (VSC) design that we extend by an explicit consideration of system energy as basis for both controller design and system stabilization. The control objective is then to regulate the energy with respect to a shaped nominal representation, implicitly related to system trajectories. As a result, for any given (admissible) initial conditions, the controlled system trajectories converge towards and are maintained in a global invariant set included in a selected subspace of the state space. Then, according to controlled system asymptotic properties, dimension of the targeted subspace and Poincare–Bendixon theorem, 1 the (controlled) system trajectories may be driven to an equilibrium point (i.e. a stationary state), an asymptotically stable limit cycle or be themselves (stable) periodic or quasi-periodic orbits. It is then important to point out that the proposed control method does not require any mathematical expression of the targeted trajectory. Nevertheless, some geometrical properties of the stabilized orbits or localization of stationary points can be shaped by tuning some control law parameters and the magnitude of the system energy to target. Moreover, the intrinsic properties of the control scheme (that is VSC) lead to a good robustness against modeling uncertainties and external disturbances. This paper is organized as follows. Section 2 introduces some theoretical results related to the ESMC approach, for a class of nonlinear (possibly chaotic or hyperchaotic) systems. Section 3 illustrates some capabilities of the proposed approach through examples related to a four-dimensional hyperchaotic system. Finally, in Section 4, some concluding remarks are given. 2. Theoretical framework This paper considers the class of (four-dimensional) nonlinear autonomous systems of the form: X_ ¼ F ðX ; U Þ ¼ f ðX Þ þ U ; n
is the state vector partitioned as X ¼ ½X1T X2T T with X1 2 Rm vector of initial conditions, f ðX Þ ¼ ½f1T ðX1 ; X2 Þ f2T ðX1 ; X2 ÞT q
where X 2 R ð1 6 n 6 4Þ 4 mÞ. X ð0Þ 2 Rn is the f2 ðX1 ; X2 Þ 2 C 1 ðRp Þ. U 2 R ðq ¼ m þ pÞ is the vector of control inputs. Moreover we consider the following assumptions.
ð1Þ p
ð0 6 m 6 3Þ and X2 2 R ðp ¼ with f1 ðX1 ; X2 Þ 2 C 1 ðRm Þ and
Assumption A1. The system is, at least, locally observable and controllable. Assumption A2. The energy of the system can be represented by a Lyapunov function V which can be splitted into two added parts V1 and V2 (i.e. V ¼ V1 þ V2 ) related to scalar positive functions VT ðX1 Þ and VIS ðX2 Þ, respectively (i.e. V1 ¼ g1 ðVT ðX1 ÞÞ and V2 ¼ g2 ðVIS ðX2 ÞÞ). Assumption A3. Scalar positive functions VT ðX1 Þ and VIS ðX2 Þ have continuous first derivatives which can be expressed as V_T ¼ X_ 1T W1 X1 ; V_IS ¼ X_ T W2 X2 ;
ð2Þ ð3Þ
2
where W1 2 Rm m and W2 2 Rp p are diagonal matrices with strictly positive real values along the diagonal. We now state our main results. Lemma 1. Consider the autonomous system (1) and Assumptions A1–A3. Moreover, consider the following control structure: T ð4Þ U ¼ uTT uTIS with uT ¼ C1 signðVT ÞX1 f^1 ðX1 ; X2 Þ; uIS ¼ C2 signðVIS V ÞX2 f^2 ð0; X2 Þ; IS
1
In case of two-dimensional targeted subspaces.
ð5Þ ð6Þ
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where uT 2 Rm 1 , uIS 2 Rp 1 are vectors of control inputs. f^1 ðX1 ; X2 Þ and f^2 ð0; X2 Þ are vectors of continuous functions which represent (local) equivalent system dynamics. 2 C1 2 Rm m and C2 2 Rp p are diagonal matrices with strictly positive real values, and VIS is a positive constant which characterizes a desired magnitude of energy. Then, (i) all solutions of the controlled system asymptotically converge towards and are maintained in a global invariant set XIS included in the same subspace as X2 and defined by 3 hVIS VIS i ¼ 0. (ii) the energy of the controlled system converges towards a neighborhood eIS of VIS .
Proof of Lemma 1. With respect to Assumption A2, let us consider a Lyapunov function candidate V ¼ V1 þ V2 such as V1 ¼ 12VT2 ðX1 Þ and V2 ¼ 12ðVIS ðX2 Þ VIS Þ2 . First, let us focus on V1 ¼ 12VT2 . Then, V_1 ¼ V_T VT :
ð7Þ
From Assumption A3 and system definition (1), the derivative V_T can be expressed as V_T ¼ X_ 1T W1 X1 () V_T ¼ ðf1 ðX1 ; X2 Þ þ uT ÞT W1 X1 :
ð8Þ
Then, substituting (5) into (8) leads to T V_T ¼ f1 ðX1 ; X2 Þ C1 signðVT ÞX1 f^1 ðX1 ; X2 Þ W1 X1 :
ð9Þ
As f^1 ðX1 ; X2 Þ represents (local) equivalent system dynamics according to f1 ðX1 ; X2 Þ, then the averaged value of f1 ðX1 ; X2 Þ f^1 ðX1 ; X2 Þ is zero (i.e. hf1 ðX1 ; X2 Þ f^1 ðX1 ; X2 Þi ¼ 0). Thus, in the mean, V_T ’ C1 signðVT ÞX1T W1 X1 :
ð10Þ
Substituting (10) into (7) then leads to V_1 ’ C1 signðVT ÞX1T W1 X1 VT :
ð11Þ
Therefore, V_1 is negative semi-definite and V1 is positive definite. Consequently, from sliding mode theory (cf. [20,22]), VT converges towards a vicinity of 0 in a finite time t1 > 0. Moreover, it comes from (10) that X1 is bounded and also goes to a vicinity of zero (as V_T goes to zero). Now, let us focus on V2 ¼ 12ðVIS VIS Þ2 . Then, V_2 ¼ V_IS ðVIS VIS Þ:
ð12Þ
From Assumption A3 and system definition (1), the derivative V_IS can be formulated as V_IS ¼ X_ 2T W2 X2 () V_IS ¼ ðf2 ðX1 ; X2 Þ þ uIS ÞT W2 X2 :
ð13Þ
Substituting (6) into (13) leads to T V_IS ¼ f2 ðX1 ; X2 Þ C2 sign VIS VIS X2 f^2 ð0; X2 Þ W2 X2 T T () V_IS ¼ f2 ðX1 ; X2 Þ f^2 ð0; X2 Þ W2 X2 C2 sign VIS VIS X2 W2 X2 : Now, let us recall that: • f^2 ð0; X2 Þ represents (local) equivalent system dynamics • VT goes to a vicinity of 0 in a finite time t1 . 2 3 4
4
when the averaged value of X1 is zero.
Deduced from analysis of the true system and its modelization. hi : averaged value of . According to f2 ðX1 ; X2 Þ.
ð14Þ
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Then, D
E f2 ðX1 ; X2 Þ f^2 ð0; X2 Þ ¼ 0 8t P t1 :
ð15Þ
Thus, for t P t1 , V_IS can be expressed (in the mean) as T V_IS ’ C2 sign VIS VIS X2 W2 X2 :
ð16Þ
Substituting this relation into (12) then leads to T V_2 ’ C2 sign VIS VIS X2 W2 X2 ðVIS VIS Þ:
ð17Þ
Therefore, according to positiveness of C2 and W2 elements, V_2 is negative semi-definite and V2 is positive definite (8t P t1 ). Consequently, from sliding mode theory (cf. [20,22]), the sliding surface defined by S2 ðX Þ ¼ VIS VIS ¼ 0 is attractive and VIS converges towards a vicinity of VIS in a finite time t2 P t1 > 0. Thus, we can conclude that, in a finite time t2 P t1 > 0, the controlled system trajectories are driven to and maintained in an invariant set XIS defined by hVIS VIS i ¼ 0 and included the same subspace as X2 . Moreover, as the system energy is bounded and its representation (related to VT ðX1 Þ and VIS ðX2 Þ) goes to a vicinity of VIS in a finite time, this energy also converges towards a neighborhood eIS of VIS (where eIS is related to the discrepancy between the true system energy and its nominal representation). Remark 1. Lemma 1 states that the solutions of the controlled system converge towards some global invariant sets. Thus, according to (controlled) system asymptotic properties, dimension of the targeted subspace (related to X2 ) and Poincare–Bendixon theorem, 5 the system trajectories under control may be driven to an equilibrium point (i.e. a stationary state), an asymptotically stable limit cycle or be themselves stable periodic or quasi-periodic orbits. Remark 2. By inspection of (6), it appears that convergence towards the targeted energy VIS is mainly supported by control law uIS . In order to improve the speed of convergence (i.e. the convergence rate), we propose the following result. Corollary 1. Lemma 1 holds when replacing VIS into (6) by a scalar positive function V of the form: V ¼ VT þ VIS . Proof of Corollary 1. Let us consider proof of Lemma 1 and replace the function V2 ¼ 12ðVIS VIS Þ2 by V2 ¼ 12ðV VIS Þ2
with V ¼ VT þ VIS ;
then V_2 ¼ V_ ðV VIS Þ () V_2 ¼ ðV_T þ V_IS ÞðVT þ VIS VIS Þ
ð18Þ
as uT , VT and V_T go to a vicinity of 0 in a finite time t1 > 0, thus, in the mean, V_2 ffi ðV_IS ÞðVIS VIS Þ
8t P t1 :
End of proof is then straightforward.
ð19Þ
As a result, it comes from Corollary 1 that control law uT may be involved in convergence towards the targeted magnitude of energy (that is VIS ), before VT reaches a neighborhood of zero. 2.1. Discussion According to Lemma 1 and its proof, VT and its derivative converge towards a vicinity of zero (in a finite time) if we can define a local approximation f^1 ðX1 ; X2 Þ of f1 ðX1 ; X2 Þ such as f^1 ðX1 ; X2 Þ f1 ðX1 ; X2 Þ is zero in the mean. In a such case, X1 also goes to a vicinity of zero (cf. relation (10)). This means that controlled system trajectories go around a subspace (of the state space) whom dimension is restricted to the same one as X2 .
5
In case of two-dimensional targeted subspaces.
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It is then worthwhile to investigate the case for which f^1 ðX1 ; X2 Þ f1 ðX1 ; X2 Þ is different from zero (in the mean). In a such context, relation (10) can be expressed as V_T ¼ C1 signðVT ÞX1T W1 X1 þ dðX1 ; X2 Þ;
ð20Þ
where dðX1 ; X2 Þ denotes an error function of the form dðX1 ; X2 Þ ¼ f^1 ðX1 ; X2 Þ f1 ðX1 ; X2 Þ: The study of global convergence then leads to consider the two following potential cases: (1) Case 1: If kC1 signðVT ÞX1T W1 X1 k P kdðX1 ; X2 Þk, then V_T is negative semi-definite or negative definite, leading to a stable behavior of the controlled system. (2) Case 2: If kC1 signðVT ÞX1T W1 X1 k < kdðX1 ; X2 Þk, then V_T is positive definite, leading to an unstable behavior (i.e. divergence of the system trajectory). However, it is important to point out that divergence of the system trajectory mainly corresponds to an increase of kX1 k (as uIS tends to slow down the increase of X2 components). Therefore, if the growth rate of C1 signðVT ÞX1T W1 X1 is greater than those of dðX1 ; X2 Þ, then the system trajectory tends to reach a some states for which case (1) is valid. This can lead to an alternation between two local phases: a (local) divergent phase and a (local) phase of convergence towards a stable state. As a result, a such alternation can yield to a global, stable, quasi-periodic behavior, for which the system trajectory goes around a closed subspace (of the state space) whom dimension is greater than those of X2 . Remark 3. The previous analysis can be considered as basis for designing or tuning the gain matrix C1 components. Indeed, as dðX1 ; X2 Þ may characterize both some uncertainties on system modeling and external disturbances, tuning of C1 can be oriented such as the inequality of case (1) is always fulfilled. 3. Example: control of a hyperchaotic circuit This section aims at providing an example, related to the control of a hyperchaotic system, to illustrate some capabilities of the ESMC approach. To this end, let us consider the electronic circuit depicted through Figs. 1 and 2 [18].
Fig. 1. The hyperchaotic circuit.
Fig. 2. Current–voltage characteristic of the nonlinear diode.
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This circuit can be defined by the following set of ordinary differential equations [18]: C1 v_ 1 ¼ f ðv2 v1 Þ i1 ; C2 v_ 2 ¼ f ðv2 v1 Þ i2 ; L1 i_1 ¼ v1 þ Ri1 ;
ð21Þ
L2 i_2 ¼ v2 ; where v1 ; v2 ; i1 and i2 represent the voltage across C1 , the voltage across C2 , the current through L1 and the current through L2 , respectively. f is a piecewise continuous function related to the diode characteristic. This function can be expressed as f ðxÞ ¼ m0 x þ 12ðm1 m0 Þðj x þ Ej j x EjÞ;
ð22Þ
where E ¼ 1 represents the breakpoint voltage of the nonlinear diode. In this example, let us consider the systems parameters (see [2]): C1 ¼ 0:5, C2 ¼ 0:05, L1 ¼ 1 and L2 ¼ 2=3, R ¼ 1, m0 ¼ 3 and m1 ¼ 0:2. This circuit then exhibits a typical hyperchaotic behavior (see Fig. 3), emphasized by computing the Lyapunov exponents (two of them are positive [2]). In order to apply the ESMC approach, let us consider now the state vector X ¼ ½v1 v2 i1 i2 T and the following expression of the system energy, Esyst ¼ 12C1 v21 þ 12C2 v22 þ 12L1 i21 þ 12L2 i22 :
ð23Þ
The crucial point of the control design then consists in partitioning the state vector into two sub-vectors (namely X1 and X2 ). Indeed, definition of X2 is directly related to the subspace (of the state space) in which the system trajectories has to be stabilized. However, some subspaces may not admit some stable invariant sets with a finite magnitude of energy. Therefore, such a partitioning implies necessary to take into account the physical characteristics of the system. For instance, inspection of the hyperchaotic circuit of Fig. 1 can lead to consider two couples of inductor–capacitor to allow some energy transfers (such as storing the energy alternatively in the inductor and capacitor). According to that, the state partitioning is then of the form X1 ¼ ½v1 i1 T and X2 ¼ ½v2 i2 T . Moreover, with respect to relation (23) and Assumptions A2 and A3, let us define the following Lyapunov function candidate V ðX Þ and its splitted components (V ðX Þ ¼ V1 ðX1 Þ þ V2 ðX2 Þ) such as
1 2 1 2 L1 2 v þ i ; V1 ðX1 Þ ¼ VT with VT ¼ 2 2 1 C1 1
Fig. 3. Projection on the plane (v1 ; v2 ) of the hyperchaotic system trajectory.
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and 1 V2 ðX2 Þ ¼ VIS2 2
with VIS ¼
1 2
1021
C2 2 L2 2 v2 þ i2 : C1 C2
According to Lemma 1, the hyperchaotic system under control can then be defined by: C1 v_ 1 ¼ f ðv2 v1 Þ i1 þ uT1 ; C2 v_ 2 ¼ f ðv2 v1 Þ i2 þ uIS1 ; L1 _i1 ¼ v1 þ Ri1 þ uT2 ;
ð24Þ
L2 _i2 ¼ v2 þ uIS2 with uT1 ¼ C111 signðVT Þv1 f^11 ðv1 ; v2 ; i1 ; i2 Þ; uT ¼ C1 signðVT Þi1 f^12 ðv1 ; v2 ; i1 ; i2 Þ; 2
22
Fig. 4. Controlled system trajectory in the space (v1 ; v2 ; i1 Þ.
Fig. 5. Controlled system trajectory in the space (v1 ; i1 ; i2 ).
ð25Þ
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Fig. 6. Time waveform of function VT .
Fig. 7. Time waveform of function VIS .
uIS1 ¼ C211 signðVIS VIS Þv2 f^21 ð0; v2 ; 0; i2 Þ; uIS ¼ C2 signðVIS V Þi2 f^22 ð0; v2 ; 0; i2 Þ: 2
22
IS
By considering, for instance, an equivalent representation of system dynamics of the form: 1 1 f^11 ðv1 ; v2 ; i1 ; i2 Þ ¼ f ðv2 v1 Þ þ i1 ; ^ ^ C1 C1 ^ R 1 f^12 ðv1 ; v2 ; i1 ; i2 Þ ¼ v1 þ i1 ; L^1 L^1 1 f^21 ð0; v2 ; 0; i2 Þ ¼ i2 ; C^2 ^ f22 ð0; v2 ; 0; i2 Þ ¼ 0
ð26Þ
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Fig. 8. Controlled system trajectory in the space (v1 ; v2 ; i1 Þ.
Fig. 9. Controlled system trajectory in the space (v1 ; i1 ; i2 ).
and the following parameters • • • •
Initial conditions (arbitrary chosen): v1 ð0Þ ¼ 0:2, v2 ð0Þ ¼ 0:1, i1 ð0Þ ¼ 0:01 and i2 ð0Þ ¼ 0:1. Control law gains: C111 ¼ C122 ¼ 3, C211 ¼ C222 ¼ 2. Targeted magnitude of energy: VIS ¼ 5. Parameters estimated values: C^1 ¼ C1 , C^2 ¼ C2 , L^1 ¼ L1 and L^2 ¼ L2 .
We then obtain the simulation results of Figs. 4–7. By looking at Figs. 4 and 5, we can observe that the system trajectory converges towards an stationary point and stay at this point. This behavior is emphasized by the time waveform of system energy representations (namely, function VT and VIS ) as shown through Figs. 6 and 7. Indeed, Fig. 6 exhibits the convergence of VT towards a neighborhood of 0 in a finite time (t1 ’ 0:8 s) and Fig. 7 shows both the convergence and the stabilization of VIS in a neighborhood of the desired magnitude of energy (that is VIS ¼ 5).
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Fig. 10. Time waveform of function VT .
Fig. 11. Time waveform of function VIS .
In order to illustrate some ideas reported in Section 2.1, let us consider now the simulation results of Figs. 8–11. This result comes from the consideration of the following parameters: • An error rate of 10% on the estimated values of C1 and C2 . • The control law gains: C111 ¼ 5; C122 ¼ 1, C211 ¼ 5; C222 ¼ 1. • The same initial conditions and targeted magnitude of energy as in the previous example. Through Fig. 10, we can observe that function VT does not converge asymptotically towards zero to stay in, but becomes periodic (remaining, nevertheless, in a bounded neighborhood of zero). This leads the system to the stabilization of a periodic orbit as shown through Figs. 8 and 9, according to the desired magnitude of the targeted energy (that is: VIS ¼ 5). Moreover, we can note that the stabilized orbit goes around a three-dimensional subspace of the system state space (see Fig. 8), whose dimension is greater than those of vector X2 .
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4. Conclusion In this paper we have introduced some theoretical results related to a new approach to control chaotic or hyperchaotic systems. This approach, referred to as ESMC, is based on an explicit consideration of the system energy, and leads to the convergence, in a finite time, of the system trajectories towards a desired behavior of interest (that is stationary state, periodic orbit, etc.). It is worthwhile to note that this method does not require neither an analytical description of the targeted behavior nor an accurate modelization of system dynamics. Moreover, due to its intrinsic scheme (namely VSC), the control approach leads to a good robustness against some modeling uncertainties and external disturbances. Future research will be devoted to the control of coupled chaotic or hyperchaotic systems, by means of ESMC approach.
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