On simultaneous parameter identification and state estimation for cascade state affine systems

Transcription

1 Amercan Control Conference Westn Seattle Hotel, Seattle, Washngton, USA June 11-13, WeAI1.9 On smultaneous parameter dentfcaton and state estmaton for cascade state affne systems M. GHANES, G. ZHENG and J. DE LEON-MORALES Abstract In ths paper, an adaptve observer s proposed to solve the problem of smultaneous parameter dentfcaton and state estmaton for a class of cascade state affne systems. Suffcent condtons are gven n order to guarantee the exponental convergence of the proposed observer. Furthermore, smulaton results are gven llustratng the performance of the proposed observer when t s appled n the synchronzaton and dentfcaton problem of Rossler s chaotc system. Index Terms Adaptve observer, synchronzaton, dentfcaton, Rossler s chaotc system I. INTRODUCTION Nonlnear observer desgn s an mportant problem n the theory of systems. It s clear that no general procedure exsts to construct a nonlnear observer for a general nonlnear system. However, for a class of nonlnear systems, under some extra assumptons, such as Lpchtz, persstent exctng..., t s possble to construct the observer. For the purpose of state estmaton, many works have been devoted [1],[13],[9],[11]. However, when dealng wth the problem of parameter dentfcaton, t becomes more dffcult. The smultaneous parameter dentfcaton and state estmaton problem has been attracted the attenton of varous research groups, snce t s very useful to treat many practcal problems, such as fault detecton, sgnal transmsson or control, and recently for synchronzaton of chaotc systems. Motvated by ths nterest, several approaches have been proposed to smultaneously estmate the state and dentfy the parameters. In [], author has proposed a novel adaptve observer wth an approprate adaptaton law for the unknown parameters. And n [1], the unknown parameters were treated as the extended state of the system, n such a way the exsted classcal nonlnear observer desgn methods can be appled to the augmented system. All these methods can be used to desgn nonlnear observer for a large class of nonlnear system, such as lnear tme nvarant/varant systems, and a class of state affne systems. In ths paper, our goal s frstly to desgn an adaptve observer n order to estmate the unmeasurable state varables of the system and to dentfy the unknown parameters smultaneously for a class of cascade state affne systems. M. GHANES s wth ECS/ENSEA, Avenue du Ponceau, 951 Cergy- Pontose, France. G. Zheng s wth INRIA Rhône-Alpes, Inovallée, 55 avenue de l Europe, Montbonnot Sant Martn, 333 St. Ismer Cedex, France. J. DE LEON-MORALES s wth Unversdad Autónoma de Nuevo Leon, Apdo. Postal 1-F, San Ncolas de Los Garza; N. L 51, Mexco. Furthermore, some suffcent condtons n order to guarantee the exponental convergence of the proposed adaptve observer are gven. Besdes, accordng to [], the problem of synchronzaton of chaotc systems has been related to the concept of observer desgn from theoretcal pont of vew. Therefore, n order to hghlght the feasblty of the proposed observer, we apply t to treat the problem of chaotc synchronzaton. In fact, the topc of synchronzaton of chaotc system has attracted many researchers attenton snce the work of [3]. After that, many applcatons of chaotc synchronzaton have been developed, such as chaotc secure communcaton based on synchronzaton of chaotc system [], [], [7], [5] and [1]. Inspred by these works, n ths paper, secondly, we also try to deal wth the synchronzaton problem for Rossler s chaotc system based on the proposed observer n order to not only estmate the state of system, but also dentfy ts unknown parameters at the same tme. The paper s organzed as follows: In secton, some basc notatons about the consdered nonlnear systems are ntroduced. And secton 3 s devoted to adaptve observer desgn for such class of cascade state affne systems. In secton, an llustratve example dealng wth the synchronzaton and dentfcaton parameter problem of Rossler s chaotc systems s gven. Smulaton results are presented n order to emphasze the performance of the proposed adaptve observer. II. NOTATIONS In ths paper, we are nterested n desgnng an exponental observer for the followng cascade state affne system: ż = Ay,u,z,θ z + β y,u,z,θ + ϕy,u,z,θ θ y = Cz 1 where z R n, u R l,y R p, θ R q are respectvely the state, known nput, output of the system and the parameter, functon A, β, ϕ and C are the matrces of approprate dmensons, and the components of matrx A, and vectors β and ϕ are contnuous functons dependng on u, y and z 1,...,z 1,θ 1,...,θ 1, for 1 p and unformly bounded, wth z 1 y 1 θ 1 z =.. z p y =.. y p θ =.. θ p //$5. AACC. 5

2 = Ay,u,z,θ dag A 1 y,u,, A p y,u,z 1,...,z p 1,θ 1,...,θ p 1 C = dag C 1,, C p, Assume that ϕ s persstency exctng so that there exst γ, > γ 1, >, T, > and t and some postve defnte matrx Σ, such that the followng nequalty holds γ 1, I t+t, t for all t t and for 1 p. Λ T sc T Σ sc Λ sds γ, I β y,u = = ϕy,u,z,θ dag β 1 y,u. β p y,u,z1,...,z p 1,θ 1,...,θ p 1 ϕ 1 y,u,, ϕ p y,u,z 1,...,z p 1,θ 1,...,θ p 1 where z R n, θ R q, y R, A, β, ϕ, C are the matrces of approprate dmensons, for 1 p, u R l. And p =1 n = n, p =1 q = q. In the sequel, the dependence of u,y,z 1,...,z 1,θ 1,...,θ 1 and ẑ 1, ˆθ 1,..., ˆθ 1, wth A, ϕ, β and Â, ˆϕ, ˆβ wll be omtted n order to lghten notatons. III. ADAPTIVE OBSERVER FOR CASCADE SYSTEMS Before ntroducng our man result, we establsh the followng assumptons: Assumpton A1. If the nput s persstently exctng, n the sense that there exst α, > α 1, >, T 1, > and t such that for all ntal condton x, the followng condton for all t t s satsfed: t+t 1, t α 1, I Ψ T u,x, s,t CT Σ sc Ψ u,x, s,t ds α, I where Ψ u,x, denotes the transton matrx for the system z = A y,u,z 1,...,z 1,θ,...,θ 1 z, y = C z and Σ s some postve defnte bounded matrx, for 1 p. Assumpton A. Consderng matrx Λ = dag Λ 1,...,Λ p, where Λ s a matrx defned by Λ = A S 1 C T Σ C Λ + ϕ Assumpton A3. We assume that the components of z and θ are bounded.e. there exst postve constants such that z δ z, and the followng nequaltes hold θ δ θ Â A δj A e j + Aj e θj ˆβ β δ β j e j + β ˆϕ ϕ δ ϕ j e j + ϕ where e = z ẑ and e θ = θ ˆθ, for 1 p. Theorem 1: Consder system 1, f assumptons A1, A and A3 are satsfed, then the followng system ẑ = A y,u,ẑ, ˆθ ẑ + β y,u,ẑ, ˆθ ˆθ + ϕ y,u,ẑ, ˆθ where + S 1 C T + ΛΓ 1 Λ T C T Σy Cẑ ẑ = Âẑ + ˆβ + ˆϕ ˆθ + S 1 C T + Λ Γ 1 Λ T CT Σ y C ẑ Ṡ = ρ S ÂT S S Â + C T Σ C Λ = Â S 1 C T Σ C Λ + ˆϕ Γ = λ Γ + Λ T CT Σ C Λ ˆθ = Γ 1 Λ T CT Σ y C ẑ s an exponental observer for system 1, where ρ and λ are suffcently large postve constants and Σ are some postve defnte matrces for 1 p. Proof: Set e = e 1,...,e p T and e θ = e θ1,...,e θp T, the state estmaton error and the parameter estmaton error respectvely, where e = z ẑ and e θ = θ ˆθ, and θ s a constant, for 1 p. For a general case of, for 1 p, the estmaton error dynamcs for e = z ẑ and e θ = θ ˆθ can be obtaned as follows ė = A S 1 C TΣ C Λ Γ 1 Λ T CT Σ C e +ϕ e θ + Â A z + ˆβ β +ˆϕ ϕ θ

3 and ė θ = Γ 1 Λ T C T Σ C e Introducng the followng change of varable we get ǫ = e Λ e θ 3 ǫ = A S 1 C T Σ C ǫ + Â A z + ˆβ β + ˆϕ ϕ θ Because S and Γ are some postve defnte matrces accordng to Assumptons A1 and A, a canddate Lyapunov functon can be chosen as follows v = ǫ T S ǫ + e T θ Γ e θ Hence ts tme dervatve s gven by v = ρ ǫ T S ǫ λ e T θ Γ e θ ǫ + Λ e θ T C T Σ C ǫ + Λ e θ Â +ǫ T A S z + ˆβ β +ˆϕ ϕ θ Accordng to Assumptons A1, A and A3, we obtan v ρ ǫ T S ǫ λ e T θ Γ e θ + ǫ S δz δj A e j + Aj e θj + ǫ S δ β j e j + β 1 + ǫ S δθ δ ϕ j e j + ϕ whch can be rearranged as follows v ρ ǫ T S ǫ λ e T θ Γ e θ + ǫ S δ z δ A j + δ β j + δθ δ ϕ j e j + ǫ S δ z Aj + β j + δθ ϕ j e θj Hence, usng the followng nequalty x y 1 x + y we get where v ρ ǫ T S ǫ λ e T θ Γ e θ + L j,1 + L j, ǫ + L j,1 e j L j, e θj 1 + L j,1 = S δ z δj A + δ β j + δθ δ ϕ j L j, = S δ z Aj + β j + δθ ϕ j for 1 p, 1 j 1, wth L 1 j,1 = L1 j,1 =. By applyng x + y x + y and after straghtforward computatons, we obtan 1 L j,1 + L j, v ρ ǫ T S ǫ whch follows η λ e T θ Γ e θ L j,1 + ǫ T j S j ǫ j η j L j,1 Λ j + L j, + e T θ j Γ j e θj where µ = mn κ,j = max χ j v µ v + κ,j v j 1 L j,1 +L j, η,λ ρ L j,1 η j, L j,1 Λj +L j, χ j, wth κ 1,j =. and As a result, the whole Lyapunov functon can be chosen as v = p =1 v, whose tme dervatve s v = p =1 v, and we have p v µ v + κ,j v j =1 p p = µ + v =1 j=+1 κ j, Consequently, f µ = mn µ µ > p j=+1 κ j,, 1 p, then we obtan v µv < and ths ends the proof. 7

4 IV. APPLICATION INTO CHAOTIC SYNCHRONIZATION AND PARAMETER IDENTIFICATION As an llustratve example of adaptve state affne observaton, let us consder the synchronzaton and parameter dentfcaton problem of a chaotc system. It s well-known that a chaotc system s a nonlnear determnstc system havng a complex and unpredctable behavor. The senstve dependence on ntal condtons and the parameter varatons s a promnent feature of chaotc behavor. In the sequel, we apply the proposed observer to smultaneously estmate states and dentfy parameters for a gven chaotc system. A. Chaotc system Consderng the followng 3-dmensonal autonomous Rossler s chaotc system descrbed by [15] ẋ 1 = x x 3 ẋ = x 1 + ax ẋ 3 = b + x 3 x 1 c where x 1 3 are the state varables, and a,b,c are all postve real constant parameters. The actual system parameters of are set to a =., b =., and c = 5.7 for exhbtng the chaos phenomenon. B. Change of coordnate and transformaton nto cascade system By consderng the followng change of coordnate z 1,1 x 1 z1 = z z 1, = x z,1 x + x 3 where z 1 R, z R, the chaotc system can be rewrtten nto cascade system 1 as follows: ż 1 = A 1 y,uz 1 + β 1 y,u + ϕ 1 y,uθ 1 ż = A y,u,z 1,θ 1 z + β y,u,z 1,θ 1 +ϕ y,u,z 1,θ 1 θ y 1 = z 1, y = z where y 1 R and y R, wth A 1 y,u =, 1 y β 1 y,u =, ϕ 1 y,u =, y 1 θ 1 = a, and A y,u,z 1,θ 1 = z 1,1, β y,u,z 1,θ 1 = z 1,1 + y 1 a z 1,1, ϕ y,u,z 1,θ 1 = 1 y y 1, θ = b c. 5 From fgure, the chaotc system 5 wth the parameters a =., b =., c = 5.7 and ntal condtons z 1,1 =, z 1, = 3, z =, exhbts the chaotc dynamcs. Now, for ths cascade system 5, the proposed adaptve observer s appled and desgned as follows: and ẑ 1 = Â1 y,uẑ 1 + ˆβ 1 y,u + ˆϕ 1 y,u ˆθ 1 + S1 1 CT 1 + Λ 1 Γ 1 Λ T 1 C1 T Σ1 y 1 C 1 ẑ 1 Ṡ 1 = ρ 1 S 1 ÂT 1 S 1 S 1  1 + C1 T Σ 1 C 1 Λ 1 = Â1 S1 1 CT 1 Σ 1 C 1 Λ 1 + ˆϕ 1 Γ 1 = λ 1 Γ 1 + Λ T 1 C1 T Σ 1 C 1 Λ 1 ˆθ 1 = Γ 1 1 ΛT 1 C1 T Σ 1 y 1 C 1 ẑ 1 ẑ = A y,u,ẑ 1, ˆθ 1 ẑ + ˆβ y,u,ẑ 1, ˆθ 1 +ˆϕ y,u,ẑ 1, ˆθ 1 ˆθ + S 1 CT + Λ Γ 1 Λ T C T Σ y C ẑ Ṡ = ρ S ÂT S S  + C T Σ C Λ =  S 1 CT Σ C Λ + ˆϕ Γ = λ Γ + Λ T C T Σ C Λ ˆθ = Γ 1 ΛT C T Σ y C ẑ where ẑ1 = [ẑ 1,1,ẑ 1, ] T,Â1 = A 1 y,u, ˆβ 1 = β 1 y,u, ˆϕ 1 = ϕ 1 y,u, ˆθ 1 = â,c 1 = 1 ẑ R, = ẑ 1,1, ˆβ 1 = ẑ 1,1 + y 1 a ẑ 1,1, ˆϕ 1 = 1 y y 1, ˆθ = ˆb ĉ,c = 1. C. Smulaton results The am here s to llustrate the smulaton results obtaned by the proposed cascade observer when t s appled n the synchronzaton and parameter dentfcaton problem of Rossler s chaotc system. The chosen numercal values ntal condtons and parameters for Rossler s chaotc cascade system 5 are gven n secton IV-B whle the values of ts observer are as follows. The ntal condtons are ẑ 1,1 = ẑ 1, = ẑ = ; ˆθ 1 = â =, ˆθ = ˆb ĉ = 1. S 1 = I, Λ 1 =, Γ 1 = 1I ; S = 1, Λ = 1, Γ = 1I. The gan were chosen as ρ 1 =, λ 1 = 15; ρ =, λ = 15. The smulaton results obtaned wth the proposed observer are llustrated n Fgs. 1 to 7.

5 z 1,1 estmate 3 1 z z1, Tme s Fg. 1. z1,1 and ts estmate z1,1.9 a estmate z 1,.1 estmate 1 Tme s Fg Fg.. Three-dmensonal phase portrat of z1,1, z1, and z wth ntal condtons z1,1 =, z1, = 3, and z =. 1 1 Tme s z1, and ts estmate Fg. 5. Parameter a and ts estmate b estmate.1.5 z =z,1 estmate 1 1 Tme s Fg. 3. Tme s z and ts estmate Fg.. 9 Parameter b and ts estmate 1

6 c estmate 1 1 Tme s [9] H. Hammour, J. De Leon-Morales, Observers synthess for state affne systems, n IEEE CDC 199, pp [1] G. Besancon, J. De Leon-Morales, et al, On adaptve observers for state affne systems, Internatonal journal of control, vol 79, no.,, pp [11] G. Besancon, Remarks on nonlnear adaptve observer desgn, System control letters, 1, pp. 71-,. [1] U. Parltz, L.O. Chua, Lj. Kocarev, K. S. Halle and A. Sang, Transmsson of dgtal sgnals by chaotc synchronzaton, Int. Journal of Bfurcatons and Chaos Vol., pp , 199. [13] G. Bastn and M. Gevers, Stable adaptve observers for nonlnear tme varyng systems, IEEE TAC, vol 33, no 7. 19, pp:5-5. [1] G. Kresselmeer, Adaptve observers wth exponental rate of convergence, IEEE TAC, pp. -, [15] We Der Chang, Parameter dentfcaton of Rossler s chaotc system by an evolutonary algorthm, Chaos, Soltons and Fractals, 9, pp ,. Fg. 7. Parameter c and ts estmate Fgs 1, and 3 llustrate the smulaton results of the estmaton of state varables,.e. the problem of synchronzaton. The parameter dentfcaton results are depcted from Fgs. 5, and 7. Smulaton results show that the proposed cascade state affne observer performs well. In all cases, state varables z 1,1, z 1,, z and parameters a, b, c ndeed appear to be well estmated. V. CONCLUSION In ths paper, the problem of adaptve observer for a class of cascade state affne systems has been dscussed. Suffcent condtons have been gven n order to guarantee the exponental convergence of the adaptve observer. Furthermore, t has been shown that the adaptve observer has an arbtrarly tunable rate. One practcal nterest of such observer s to study the synchronzaton and parameter dentfcaton problem of chaotc systems. An example of synchronzaton and parameter dentfcaton of Rossler s chaotc system has been studed n order to llustrate the feasblty of the proposed observer. REFERENCES [1] Q G.Y., Du S.Z., Chen G.R. et al. On a four-dmensonal chaotc system. Chaos, Soltons and Fractals, 35. [] Njmejer H. and Mareels I. M. Y. An observer looks at synchronzaton. IEEE Trans. on Crcuts and Systems-1: Fundamental theory and Applcatons, Vol, No 1, pp -91, 1997 [3] Pecora L. M. and Carroll T. L. Synchronzaton n chaotc systems. Physcal Revew Letters, 1-, 199. [] L. Kovarev, K. S. Eckert, L. O. Chua and U. Parltz, Expermental demonstraton of secure communcatons va chaotc synchronzaton, Int. J. Bfurcaton and Chaos, , 199. [5] Feldmanne U., Hasler M. and Schwarz W. Communcaton by chaotc sgnals: The nverse system approach. Int. J. Crcut Theory and Applcatons ,199. [] Parltz U., Chua L. O., Kocarev L. et al. Transmsson of dgtal sgnals by chaotc synchronzaton. Int. J. Bfurcaton and Chaos, , 199. [7] Wu C. W. and Chua L. O. A smple way to synchronze chaotc systems wth applcatons to secure communcatons systems. Int. J. Bfurcaton and Chaos 3, , [] Q. Zhang, Adaptve observers for MIMO lnear tme-varyng systems, IEEE TAC, 7, pp ,. 5