Optimal H Filtering for Discrete-Time-Delayed Chaotic Systems via a Unified Model

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1 Optimal H Filtering for Discrete-ime-Delaye Chaotic Systems via a Unifie Moel Meiqin Liu Senlin Zhang Xiaofang ang Zhen Fan Dept. of Systems Science an Engineering College of Electrical Engineering Zhejiang University Hangzhou 37 P. R. China liumeiqin@zju.eu.cn slzhang@zju.eu.cn txf6@63.com fanzhen959@gmail.com Shiyou Zheng Raar an Avionics Institute of AVIC Wuxi 63 P. R. China seebank97@yahoo.com.cn Abstract his paper presents a unifie moel consisting of a linear ynamic system an a boune static nonlinear operator. Most iscrete-time chaotic systems such as chaotic neural networks Chua s circuits an Hénon map etc can be transforme into this unifie moel. Base on the H performance analysis of the estimation error system between the unifie moel an its improve Luenberger-like filter using the linear matrix inequality (LMI) approach the optimal H filter are esigne to estimate the states of iscrete-time chaotic systems with external isturbance. he H filter not only guarantees the asymptotic stability of the estimation error system but also reuces the influence of noise on the estimation error. wo numerical examples are exploite to illustrate the effectiveness of the propose filter esign schemes. Keywors-H filtering; estimation error; iscrete-time chaotic systems; time elay; neural networks. I. INRODUCION Chaotic system is a very complex ynamical nonlinear system an its response exhibits some specific features such as excessive sensitivity to initial conitions broa Fourier transform spectrums an irregular ientities of the motion in phase plane []. he chaotic systems an relevant properties have foun useful applications in many engineering areas such as secure communication biological systems power electronic evices an power quality igital communication chemical reaction analysis []. During past two ecaes research about control an synchronization of chaotic systems has gaine increasing attention [-]. However many existing approaches epen on the assumption that their states can be etecte by sensors which is not always the case in reality. herefore from a practical viewpoint the state estimate problem for chaotic systems has been vitally important. On the other han some noises or isturbances always exist in real systems that may cause instability an poor performance herefore the effect of the noises or isturbances must be also reuce in state estimation process for chaotic systems. In this regars recently many researchers aopte the H filter to estimate systems states in orer to guarantees the L gain less than a prescribe level. It shoul be point out that numerous research results regaring synchronization problems for chaotic systems base on the observer techniques have been evelope. For the observer-base synchronization a slave system is esigne such that its ynamics synchronizes that of the master system. From the viewpoint of control theory the slave system is an observer of the master system an the state of the master system can be estimate by the slave system. In this sense the synchronization problem can be viewe as a state estimation one [3]. here are many results regaring H synchronization problems for chaotic systems [-9]. However to our best knowlege the above aforementione methos an many other existing synchronization methos are only applie to the continuous-time chaotic systems. It seems not much (if any) stuy on the H synchronization or filtering for iscrete-time chaotic systems with external isturbance. It is well known that iscrete-time systems play a very important role in igital signal analysis an processing. Especially the iscrete-time chaotic systems are more suitable to realize timeseries preiction aaptive tracking an secure communication etc. herefore here we will combine the H control concept an Lyapunov (or Lyapunov-Krasovskii) stability theory to investigate the optimal H filtering problem for a class of iscrete-time chaotic systems with external isturbances. We are inspire by the stanar neural network moel (SNNM) in [9-] an put forwar a unifie moel which is the interconnection of a linear ynamic system an a boune static nonlinear operator. Most iscrete-time chaotic systems with time elays such as chaotic neural networks Chua s circuits an Hénon map etc can be transforme into this unifie moel to be H filter esigne in a unifie way. he contributions of this paper inclue: (i) A unifie moel is presente to escribe ifferent kins of chaotic systems. (ii) he improve Luenberger-like filter is less conservative than general filters. (iii) he H filter will reuce the effect of noise or isturbance with boune energy on the estimation accuracy. Notation: he superscript stans for matrix transposition. l [ ) is the space of square integrable vectors. R n enotes n imensional Eucliean space an R n m is the set of all n m real matrices. I enotes ientity matrix of 553

2 appropriate orer. enotes the symmetric parts. iag{ } stans for a block-iagonal matrix. he notations X>Y an X Y where X an Y are matrices of same imensions mean that the matrix XY is positive efinite an positive semiefinite respectively. If X R p an Y R q C(X; Y) enotes the space of all continuous functions mapping R p R q. where K R n l K R n l K 3 R L l an K R L l are the filter gains to be etermine to meet certain performance criteria xk ˆ( ) an zˆ( k ) enote the estimates of x(k) an z(k) respectively. By efining the error vector e(k)x(k) ˆx (k) we get the following ynamic equations that e(k) satisfies. II. PROBLEM FORMULAION he unifie moel we suggeste consists of a linear ynamic system an a boune static nonlinear operator: xk ( + ) Axk ( ) + Axk ( τ) + Bpφ( ξ) + Bww ξ Cqx + Cqx( k τ) + Dpφ( ξ) + Dqww () ek ( + ) Aek ( ) + Aek ( τ ) + Bp f( ξ ) + Bww KDv ξ ξ ξˆ Cqe + Cqe( k τ ) + Dp f( ξ ) + Dqww K3Dv zk ( ) zk ( ) zk ˆ( ) Cek z ( ) (5) with the initial conition function x(k)ϖ(k) k [τ ] where x(k) R n is the system state A R n n A R n n B p R n L B w R n m C q R L n C q R L n D p R L L an D qw R L m are the corresponing state-space matrices ξ R L is the input of nonlinear function φ φ C(R L ; R L ) is nonlinear function satisfying φ() w(k) R m is the isturbance input which belongs to l [ ) L N is the number of nonlinear functions τ R is the time elay ϖ(k) is the given function on [τ ]. Remark : While w(k) the nonlinear moel () unifies linear systems several well-known iscrete-time intelligent systems incluing ynamic neural networks or fuzzy moels with or without time elays iscrete-time Chua s circuits an Hénon map. Ref. [9]-[3] illustrate that some intelligent systems are special examples of (). We will provie some other examples in Section IV. Suppose that the measurement of the system () by the sensor is of the form: yk ( ) Cxk ( ) + Dυ () where y(k) R l is the measurement output υ(k) R s is the measurement noise that belongs to l [ ). C R l n an D R l s are known constant matrices. he signal to be estimate is the combination of the system state escribe as follows: z C x (3) where z(k) R r is the non-measurable signal to be estimate C z R r n is a constant matrix. We construct the following improve Luenberger-like filter for z(k): xk ˆ( + ) Axk ˆ( ) + Axk ˆ( ) ( ˆ τ + Bpφ ξ) + K ˆ ˆ [ yk ( ) Cxk ( )] + K[ yk ( τ) Cxk ( τ)] ξ ˆ C ˆ( ) ˆ( ) ( ˆ qx k + Cqx k τ + Dpφ ξ) () + K ˆ ˆ 3[ yk ( ) Cxk ( )] + K[ yk ( τ) Cxk ( τ)] zk ˆ( ) Cxk ˆ z ( ) z where A A KC A A KC Cq Cq K3C Cq Cq KC 3 3 D iag{ D D} f ( ξ ) φ( ξ) φ( ξˆ ) v(k)[υ (k) υ (kτ)] l [ ) an z is the estimation error. K [ K K ] K [ K K ] If there exists a positive scalar γ such that J( w v) { z z γ [ w w v v] } (6) + < k for any nonzero w(k) l [ ) v(k) l [ ) with the initial state x(k)ϖ(k) k [τ ] an system (5) is asymptotically stable when w(k) v(k) then the L gain of system (5) oes not excee γ that is system () is an H filter for z(k). If we fin a minimal positive γ to satisfy the above conitions system () is an optimal H filter for z(k). III. MAIN RESULS In this paper we assume that the nonlinear functions in () are monotonically non-ecreasing an globally Lipschitz. hat is there exist a positive scalar u i such that φi( α) φi( β) α β u i α β R i L. (7) Accoring to (7) the nonlinear functions fi( ξ i) (i L) satisfy the following sector-boune conitions: fi( ξ i) ui i.e. f ξ i( ξ i) [ fi( ξ i) uξ i i] i u i >q i i L. (8) heorem : If there exist symmetric positive efinite matrices P an Γ iagonal positive semi-efinite matrix Σ matrices K K K 3 an K an a positive scalar γ that satisfy 55

3 APAP A PA A PBp + Cq ΣU Γ CC + + z z * A PA Γ A PBp + CqΣU M BpPBp + DpΣU * * UΣDp Σ + * * * * * * APBw APKD APB w APK D B ppbw + UΣDqw BpPKDUΣK3D < (9) BwPBw γ I BwPKD * D K PKD γ I where Uiag{u u u L } then system (5) with w(k) an v(k) is globally asymptotically stable an the upper boun on the L gain of the system (5) is finite an can be obtaine by minimizing γ with respect to γ P Γ Σ K K K 3 an K i.e. minimize γ subject to (9) P > Γ > Σ. () Proof: First consier system (5) with w(k) an v(k); that is ek ( + ) Aek ( ) + Aek ( τ ) + Bp f( ξ ) ξ Cqe + Cqe( k τ ) + Dp f( ξ ). Since e(k) an ξ () are solutions to () there exists at least one equilibrium locate at the origin i.e. e eq ξ eq. For system () we aopt the following Lyapunov-Krasovskii functional: (() ()) () () + ( + ) ( + ) iτ Vek ξ k e k Pe k e i k Γe i k () where P> an Γ>. hus e(k) ξ V(e(k) ξ )> an V(e(k) ξ ( k ) ) iff e(k) an ξ. From the sector conition (8) we have f ( ξ ) ε [ f ( ξ ) uξ ] (3) i i i i i i i where ε i (i L). From the equality (3) we have () εi fi ( ξi) εiui fi( ξi) ξi where he ifference of V(e(k) ξ ) along the solution to () is Δ Vek ( ( ) ξ ) V( e( k+ ) ξ ( k+ )) V( e ξ ) Ae + A ( ) ( ( )) e k τ + Bp f ξ k P Ae + Ae( k τ ) + Bp f ( ξ ) e Pe + e Γe e ( kτ) Γe( kτ) L L εi fi ( ξ i) εiui fi( ξ i) ξ i i i + e e ( k τ) f ( ξ) G e e ( kτ) f ( ξ) (5) APA P+ Γ A PA A PBp + CqΣU G * A PA Γ A PBp + CqΣU BpPBp + DpΣU * * + UΣDp Σ Σiag{ε ε ε L }. Next for system (5) uner zero initial conition J in Eq. (6) is equivalent to k { γ } J ( wk ( ) vk ( )) z ( kzk ) ( ) [ w( kwk ) ( ) + v ( kvk ) ( )] k { z ( kzk ) ( ) γ [ w ( kwk ) ( ) v ( kvk ) ( )] Vek ( ( ) ξ ) } { Ve ( ( ) ξ ( )) Ve ( () ξ ()) } + +Δ { z ()() k z k γ [ w () k w() k v ()()] k v k V(() e k ξ ()) k } + +Δ + k { z z γ k e C C e [ w w + v v] + + e e ( k τ) f ( ξ) G e e ( kτ) f ( ξ) Bwwk ( ) KDvk ( ) P Bwk w ( ) KDvk ( ) Bwk w ( ) KDvk ( ) P Aek ( ) + Aek ( τ ) + Bp f( ξ) + Ae + Ae( k τ ) + Bp f ( ξ ) P Bww KDv + w D Σ Uf( ξ ) + f ( ξ ) ΣUD w qw ( ) Σ ( ( )) ( ( )) Σ ( ) v k D K3 Uf ξ k f ξ k UK3Dv k qw } 555

4 e e ( k τ) f ( ξ) w v M k. (6) e e ( kτ) f ( ξ) w v Since M< in inequality (6) J(w(k) v(k))< for any [e (k) e (kτ) f (ξ (k)) w (k) v (k)] w(k) l [ ) an v(k) l [ ). With the well-known Schur complement [] M< is equivalent to APA P+ Γ A PA A PBp + CqΣU * A PA Γ A PBp + CqΣU BpPBp + DpΣU * * + UΣDp Σ * * * * * * * * * APBw APKD C z APB w APK D BpPBw + UΣDqw BpPKDUΣK3D <. (7) BwPBw γ I BwPKD * D KPKD γ I * * I Since G in Eqn. (5) is the principal minor of the left-han sie of Inequality (7) we have G<. So system (5) with w(k) an v(k) i.e. system () is globally asymptotically stable. We expect that γ reaches its minimum so that system (5) can reject the external isturbance as much as possible. It requires the solution of the eigenvalue problem (EVP) in the inequality () which is a convex optimization problem that can be solve using the MALAB LMI Control oolbox [5]. his completes the proof. Base on heorem we can obtain the following theorem to esign an optimal H filter for the nonlinear system () with the observation system (). heorem : here exists an optimal H filter () such that system (5) is globally asymptotically stable when w(k) v(k) an the upper boun of the L gain of system (5) is minimal provie that there exist symmetric positive efinite matrices P an Γ a iagonal positive semi-efinite matrix Σ matrices S S S 3 an S an a positive scalar γ that satisfy the following EVP: minimize γ (8) subject to P PASC PA SC * P + Γ + CC z z * * Γ * * * * * * * * * PBp PBw SD Cq ΣU C S3 CqΣU C S < (9) DpΣU + UΣDp Σ UΣDqw S3D * γ I * * γ I where S [ S S ] S [ S S ]. Furthermore the 3 3 parameters of the esire H filter () can be etermine by: K P S K P S K3 ( UΣ) S3 K ( UΣ) S. () Proof: With the well-known Schur complement [] the equality (9) in heorem is equivalent to P PA PA * P + Γ + CC z z * * Γ * * * * * * * * * PBp PBw PKD Cq ΣU CqΣU <. () DpΣU + UΣDp Σ UΣDqw UΣK3D * γ I * * γ I Defining S PK S PK S 3 UΣK 3 an S UΣK in () we can obtain heorem. We complete the proof. 556

5 x(k) 5 5 in which x(k) [x (k) x (k)] with the initial conition [x (k) x (k)] [..6] for /Δt k an Δt A. A B.. Figure shows the chaotic behavior of the system (). aking process noise w(k) into account which belongs to l [ ) the system () can be reformulate as follows: x(k) Figure. he phase curve of the chaotic neural network (3) with the initial conition [x (k) x (k)] [..6] for k (5 iterations have been plotte). x(k) an its estimate ime(k) Figure. State x (k) an its estimate xˆ ( k ). State x(k) Estimate of x(k) Remark : In most literature on H filter esign [6 7] it is assume that the estimate system () is asymptomatically stable in the absence of process noise w(k) an measurement noise v(k). Consequently it is impossible to assess the systems stability without a priori knowlege about the process an measurement noise. In our work this assumption can be remove. On the other han while constructing the filter structure we take full avantage of the information provie by system () an measurements from sensor as escribe in system () such as A A B p C q C q D p C an C z therefore only four parameters K K K 3 an K in the H filter () nee to be etermine. IV. ILLUSRAIVE EXAMPLES Example : We consier the following iscrete-time chaotic elaye Hopfiel neural network [8]. xk ( + ) ( +Δ ta) xk ( ) +ΔtAtanh( xk ( )) +ΔtB tanh( x( k / Δ t)) () xk ( + ) ( +Δ ta) xk ( ) +ΔtAtanh( xk ( )) +ΔtB tanh( x( k / Δ t)) + w. (3) We convert the elaye Hopfiel neural network (3) into expression () where A+ΔtA A B p [ΔtA ΔtB ] B w C q C q D p UI φ i (ξ i (k))tanh(x i (k)) i φ 3 (ξ 3 (k)) D qw tanh(x (k/δt)) φ (ξ (k))tanh(x (k/δt)). If x (k) is estimate we have [ ] z x. () We employ the following sensor to observe the system s information: yk ( ) xk ( ) + vk ( ) 3. (5) he process noise w(k) l [ ) an the measurement noise v(k) l [ ) are efine as follows. wk ( ) rsin( ke ) k (6) vk ( ) r k (7) where r an r are ranom numbers taken from a uniform istribution over [ ]. Base on the sensor (5) we obtain the following gains of optimal H filter () by virtue of heorem..5. K K K..9 3 K

6 .7 x3(k) - Estimation error x(k) x(k) Figure 3. rajectory of chaotic system (8) with.9 an initial conitions (9) (5 iterations have been plotte) ime(k) Figure 5. he estimation error zk ( ) x ˆ 3 x3. x3(k) an its estimate State x3(k) Estimate of x3(k) x()[ ] x()[ ] x()[ ] (9) the trajectory of the system (8) is shown in Figure 3. If we introuce the process noise w(k) l [ ) an w(k)sin(k) exp(k) into the system (8) it becomes: x( k + ).5 x + tanh( x3( k )) + w x( k + ).5 x + sin( x( k )) + w x3( k + ).5 x3 + tanh( x( k )) + w. (3) he optimal upper boun on the L gain of the system (3) with the sensor (5) is.86. Simulation result is presente in Figure which shows the state x (k) an its estimate xˆ ( k )(i.e. zˆ( k )) ime(k) Figure. State x 3(k) an its estimate xˆ 3( k ). Su et.al [8] have provie the synchronization methos of the chaotic system (3) where Δt /7 however the synchronization error iverge when Δt>/7. Here our approaches can eal with the state estimation problem of the chaotic system () in which Δt that is heorem is less conservative than theorems in [8]. Example : Consier the following elaye chaotic system [9 3]: x( k + ).5 x + tanh( x3( k )) x( k + ).5 x + sin( x( k )) x3( k + ).5 x3 + tanh( x( k )). (8) while.5 system (8) will exhibit chaotic behavior. For example for.9 an initial conitions: We transform the chaotic system (3) into expression () where Aiag{.5.5.5} A B p iag{ } B w C q 3 3 C q D p 3 3 D qw 3 Uiag{} φ (ξ (k))tanh(x 3 (k)) φ (ξ (k)) x (k) +sin(x (k)) an φ 3 (ξ 3 (k))tanh(x (k)). If x 3 (k) is estimate we have [ ] z x. (3) We employ the following sensor to observe the system s information: yk ( ) xk ( ) + vk ( ). (3) he measurement noise vk ( ) l[ ) is efine in Eqn. (7). Accoring to heorem solving the EVP (8)-(9) we obtain the gains of esire H filter as follows: 558

7 .5.79 K..998 K K K When the filter () with the above K K K 3 an K is use to estimate the state x 3 (k) of chaotic sytem (3) the state x 3 (k) an its estimate xˆ 3 ( k ) are shown in Figure an estimation error z is shown in Figure 5. It can be seen that the effect of the noise w(k) an v(k) on the estimation error z can be restricte in the lowest level. Although Ref. [3] provie synchronization methos of chaotic system (8) the influence of the noise or isturbance on synchronization controller hasn t been consiere. Besies the gains of synchronization impulses were obtaine by trial an error in Ref. [3]. Our methos not only reuce the effect of the noise or isturbance but also solve the gains of the filters by the MALAB LMI Control oolbox [5]. V. CONCLUSION In this paper we have propose H filter esign metho for a class of elaye chaotic systems. We have presente a unifie moel to escribe these chaotic systems. By employing the Lyapunov functional metho combine with the H control concept an optimal H filter has been esigne to estimate the states of this unifie moel such that the upper boun on the L gain is minimize. As most iscrete-time chaotic systems with time elays can be transforme into this unifie moel H filter esign for these systems can be one in a unifie way. he resulting esign equations are a set of LMIs which can be solve by the MALAB LMI Control oolbox [5]. Here it is worth pointing out that there are no unifie ways about how to convert the chaotic systems into the unifie moel () but generally state-transformation is applie. ACKNOWLEDGMEN his work was supporte in part by the National Natural Science Founation of China uner Grants an 6875 the Zhejiang Provincial Natural Science Founation of China uner Grants R3 an Z93 the Program for New Century Excellent alents (NCE) in University uner Grant NCE--69 the Research Project of Zhejiang Provincial Eucation Department uner Grant Z9933 the Funamental Research Funs for the Central Universities uner Grants QNA36 an 9QNA the ASFC uner Grant 76 an the SRFDP uner Grant 55. his work was also supporte by the 5 alent Project of Zhejiang Province. REFERENCES [] A. Boulkroune an M. M Saa A fuzzy aaptive variable-structure control scheme for uncertain chaotic MIMO systems with sector nonlinearities an ea-zones Expert Systems with Applications vol. 38 no. pp November December. [] M. Yahyazaeh A. R. Noei an R. Ghaeri Synchronization of chaotic systems with known an unknown parameters using a moifie active sliing moe control ISA ransactions vol. 5 no. pp April. [3] L. M. 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