34 Inverse Dynamic Models in Chaotic Systems Identification and Control Problems Leonid Lyubchyk1, Galyna Grinberg2 1. Department of Computer Mathematics and Data Analisys, National Technical University “Kharkiv Polytechnic Institute”, UKRAINE, Kharkiv, 2 Kirpichova str, email: lyubchik.leonid@gmail.com 2. Department of Economic Cybernetics and Management, National Technical University “Kharkiv Polytechnic Institute”, UKRAINE, Kharkiv, 2 Kirpichova str, email: glngrinberg@gmail.com Abstract: Inverse dynamic models approach for chaotic Rösller attractor with signal and parametric disturbances. system synchronization in the presence of uncertain parameters is considered. The problem is identifying and II. PROBLEM STATEMENT compensating unknown state-dependent parametric Consider a state-space model of controlled chaotic system disturbance describing an unmodelled dynamics that with a distinguished nonlinear component, which causes the generates chaotic motion. Based on the method of inverse emergence of chaotic dynamics and interpreted as an model control, disturbance observers and compensators uncertain parametric disturbance are synthesized. A control law is proposed that ensures the stabilization of chaotic system movement along master x (t ) = Ax(t ) + Bu (t ) + Nf (x(t ),δ) , (1) reference trajectory. The results of computational = y (t ) Cx = (t ) , y (t ) Mx(t ) , simulation of controlled Rösller attractor synchronization c m where x(t ) ∈ R – chaotic system state vector, u (t ) ∈ R – are also presented. n m Keywords: chaotic system, synchronization, disturbance, identification, inverse model, unknown-input observer. control variables vector, f (x(t ),δ) ∈ R q – state-dependent parametric disturbance with uncertain parameters δ , I. INTRODUCTION yc (t ) ∈ R r , ym (t ) ∈ R p – output controlled and Controlled systems and processes with chaotic dynamics measured variables respectively. are a matter of unflagging interest in modern control theory Disturbance f (x(t ),δ) ∈ R may be treated as unknown q and practice [1, 2]. The problem of synchronization of chaotic systems is intensively studied; in this case, control input signal for system (1). law is designed in such a way that the controlled variables of α −1 α −1 the slave system follow the reference output of the master Matrices SCB (α1 ) = CA 1 B, S MN (α 2 ) = MA 2 N system or nonlinear oscillating system stabilized along given are known as Markov parameters of system (1). reference trajectory in the presence of uncertainties and Without loss of generality, for simplicity reason, we will external disturbances [3, 4]. = assume that rank SCB m, = rank S MN m, where A typical model of a chaotic system is a linear system with additional nonlinear components dependent on the = state, SCB S= CB (1) , S MN S MN (1). the presence of which determines the appearance of chaotic Consider two main inverse model problems: regimes [5]. Because the system nonlinearity may be treated • Chaotic system identification, namely, obtaining as a parametric disturbance of nominal model, chaos unknown parametric disturbance estimate f̂ (t ) using synchronization problem may be reduced to the disturbance rejection problem, namely, unknown and unmeasurable available measurements y (t ) and known control disturbances eliminating from the systems output along with m reference signal tracking. signal u (t ) ; Recently a number of model-based control methods have • Chaotic system control, namely, control law been developed for disturbance rejection taking into account u (y (t ) , y* (t ), ˆf (t )) design, which ensure control the requirements of accuracy, dynamic performance, stability and robustness [6, 7]. In this paper the inverse model control goal achieving approach [8] is applied for chaotic systems synchronization. lim || ec (t ) || 2 ≤ ε* , t → ∞. (2) Inverse models are used for both parametric disturbance identification and compensation, which made it possible to where e= c (t ) y* (t ) − yc (t ) – control error, y* (t ) – set- synthesize disturbance decoupling controller, ensure point signal given by the reference model reference signal tracking. The proposed approach was studied through y ∗ (t ) = A∗ ⋅ y∗ (t ) + yref (t ) , (3) computational modeling using the example of a controlled ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 35 ε* – some sufficiently small constant. we can obtained the minimal-order state and disturbance observer in the form of system (1) inverse model [10]: In the chaos synchronization problem, reference model (3) can be considered as a master system [3]. x = (t ) RΠ N AQ ⋅ x (t ) + RΠ N AP ⋅ ym (t ) + n−q Dynamic system with state vector x (t ) ∈ R + + RNS MN ⋅ y m (t ) + RΠ N B ⋅ u (t ), x (t ) = AI x (t ) + B I u (t ) + B1I y (t ) + B2I y (t ), x̂( t ) = P ⋅ ym (t ) + Q ⋅ x (t ) , (11) (4) fˆ (t ) =C I xk + D I u (t ) + D1I y (t ) + D2I y (t ), (t ) C N [y m (t ) − MAQ ⋅ x (t ) − f̂= − MAP ⋅ ym (t ) − S MB u (t )], will be referred to as inverse dynamic model of system (1), if + + where Π Ν = I n − NS MN M , Ω N = I p − S MN S MN 2 the following conditions take place: x (t ) − Rx(t ) → 0 , , + 2 fˆ (t ) − f (t ) → 0 , if t → ∞ , where Rn −q×n – some = C N S MN + N + PΩN . From (1), (11) it follows, that estimate errors vectors aggregate matrix. (t ) x(t ) − ˆx(t )= ex= , e f (t ) f (x(t ) ,t ) − ˆf (t ) are given Then fˆ (t ) may be treated as unknown input signal f (t ) by the equations: dynamic estimate, obtained by inverse model (2). III. INVERSE DYNAMIC MODEL DESIGN ex (t ) F ( R ) ⋅ ex (t ) , = ex (t )= Q ⋅ ex (t ) (12) Let =z (t ) Rx(t ) ∈ R n − p be aggregated auxiliary variables, where R is some aggregate matrix, so e f (t ) = −C N MAQ ⋅ ex (t ). that rank M ( T RT = n . ) III. INVERSE MODEL-BASED CONTROLLER DESIGN Take state vector estimate in the form The disturbance rejection control law will be constructed as a function of reference signal and disturbance estimate: x̂(t ) = P ⋅ ym (t ) + Q ⋅ x (t ), (5) −1 u* (t ) = SCB ⋅ [yref (t ) + C A ˆx(t ) − SCN ˆf (t )], n× p n×n − p where matrices P ∈ R ,Q∈R are such that (13) C A A∗C − CA. = = I p , RQ MP = I n − p , PM + QR = In , (6) If system structure non-singularity condition takes place =MQ 0= p ,n − p , RP 0n − p ,p .  Im −1 SCB SCN  We obtain the aggregated vector z (t ) estimate x (t ) by rank S = m + q, S =   (14) minimal-order unknown-input observer (UIO) [9]:  C N S MB I q  x (t ) =Fx (t ) + G1 ym (t ) + Hy m (t ) + G0u (t ). (7) or equivalently The UIO (7) parameters are determined from disturbance −1 det Φ ≠ 0 , Φ= I q − C N S MB SCB SCN , (15) estimate invariance conditions [9, 10] ( R − HM ) A − F ( R − HM ) = GM , disturbance estimate may be eliminated from the controller equations, which is therefore be regarded as disturbance (8) RN − HMN =0 , G − RB =0 , G =G − FH . decoupling controller. 0 1 In reality, situations often arise when conditions (14), (15) A solution of linear matrix equations (8) are obtained as are not met. In such a case the realizable control law may be obtained using the disturbance estimates, dynamically =F RΠ= AQ, G RB, transformed by the internal auxiliary "fast" filter with small N 0 parameters. AP, H RNS + , As a result, realizable controller are designed by including =G RΠ= (9) in its structure an additional internal low-pass filter with 1 N MN small time constant [11]: Π = I − BS + M, N n MN −1 u* (t ) =SCB ⋅ [y ∗ (t ) + C A ˆx(t ) − SCN f (t )], (16) Taking the unknown disturbance estimate as  εf (t ) =− f (t ) + (1 − μ) ⋅ ˆf (t ) , ( ) N + ˆx (t ) − Axˆ (t ) − Bu (t ) , ˆf (t= ) (10) ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 36 where 0 < ε << 1, 0 < µ << 1 - small filter parameters. f̂= x2 (t ) + π 2 y1 (t ) , 1 (t ) From (13), (16) follows, that disturbance compensator (22) equation with internal additional filter take the form: f̂ 2 (t ) = y 2 (t ) + cy2 (t ) − x2 (t ) − π 2 y1 (t ) − u2 (t ). εu (t ) =−μu (t ) + (1 − μ) ⋅ [φ1 (t ) + As a result disturbance decoupling controller with internal filter equation is obtained in the form: −1 + SCB SCN φ 2 (t )], εu (= t ) ν1 x1 (t ) − x2 (t ) + (ζ1 − π 2 )y1 (t ) − α1 y2 (t ) , (t ) u (t ) + φ1 (t ) , u=* (17) u2 (t ) =u (t ) + ν1 x1 (t ) − 2 x2 (t ) + −1 φ1 (t ) = SCB ⋅ [yref (t ) + C A ˆx(t )], (23) + (ζ1 − 2π 2 ) ⋅ y1 (t ) + (c − α1 − ε −1 ) ⋅ y2 (t ) , φ 2 (t ) =C N ⋅ [y m (t ) − MAQ ⋅ x (t ) − MAP ⋅ ym (t )]. ζ1 =α1 +ν1π1 − 1, ν1 =k − α1 − a III. CHAOTIC SYSTEM INVERSE MODEL CONTROL Proposed disturbance observer and decoupling controller are investigated by computational simulation. As an example of proposed approach consider inverse Simulation results for Rösller attractor model parameters a = 0.2 , c = −5.7 , observer and controller model control of the Rösller attractor under uncertainties: x1 (t ) =− x2 (t ) − x3 (t ) , parameters π1 = −2 , ε = 0.01= −1, π 2 = ,μ 0=, k 2.2 , x2 (t ) = x1 (t ) + ax2 (t ) + u1 (t ) + f1 (t ) , (18) , α1 6 are α 0 5= and reference model parameters = x3 (t ) = −cx3 (t ) + u2 (t ) + f1 (t ) + f 2 (x1 (t ) ,x3 (t )) , presented below. Disturbance f1 (t ) was modeled input signal disturbance where (t )) δc x3 (t ) + (1 + δ x )x1 (t )x3 (t ) , as a step wave function, reference model input signal yref (t ) (t ) δ f , f 2 (x1 (t ) ,x3 = f1= are input and parametric disturbances respectively adopted in the form of harmonic function. At Fig.1, 2 the state variables and phase plane of controlled with δ f , δc , δ x uncertain parameters. Rösller disturbed attractor are presented. Using the measurements y1 (t ) = x1 (t ) , y2 (t ) = x3 (t ) find the control so the controlled output yc (t ) = x1 (t ) will * track set-point signal y (t ) , generated by reference model y∗ (t ) + α1 y* (t ) + α 0 y* (t ) =  yref (t ) . (19) The control law, which ensures attractor synchronization with reference model, is the following: u2 (t )= (α 0 − 1) ⋅ ˆx1 (t ) + (k − a − α1 ) ⋅ ˆx2 (t ) + + (c − α ) ⋅ ˆx (t ) − 2 ˆf (t ) − f (t ) − y (t ) , 1 3 1 2 ref (20) u1 (t ) = −kxˆ 2 (t ) , Fig.1. Dynamics of the disturbed attractor. State variable y1 (t )  εf (t ) =− f (t ) + (1 − μ) ⋅ ˆf 2 (t ) , The state estimates for system (18), obtained by reduced- order UIO, are: x1 (t ) = ρ1 x1 (t ) + x2 (t ) + + (1 + π1ρ1 +π 2 ) ⋅ y1 (t ) + π1 y2 (t ) , x2 (t ) =π 2 x1 (t ) + π1π 2 y1 (t ) + π 2 y2 (t ) , (21) x̂1 (t ) = y1 (t ) , ˆx1 (t ) = x1 (t ) + π1 y1 (t ) , ˆx3 (t ) = y2 (t ) , where ρ1 = (π1 + a − k ) , π1 π 2 are tuning parameters. Fig.2. Dynamics of the disturbed attractor. Phase plane (y1 (t ) , y2 (t )) Corresponding disturbance estimates are ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 37 Disturbances estimations obtained by (21), (22) are Simulation results for chaotic system synchronization depicted in Fig. 3, 4 and control and output variables problem demonstrated high accuracy of disturbances obtained in accordance the control law (20), (23) are decoupling for broad range of parameters deviation. presented at Fig. 5, 6. IV. CONCLUSION In this paper we presented inverse model-based approach to chaotic systems synchronization problem. The proposed method allows us to decompose the problem into the stage of structural synthesis of inverse models and their parametric synthesis or optimization. This significant advantage of the method of inverse dynamic models is the possibility of real- time reconstruction of signals of complex shape in the absence of information on their structure. This, in turn, makes it possible to efficiently solve problems of compensation of nonlinear state-dependent, which makes it possible to suppress sources of chaotic dynamics and simplifies the Fig.3. Disturbances estimation f1 (t ) , ˆf1 (t ) in open-loop system solution of synchronization problems. Thus proposed approach seems to be quite universal and can be used to solve various problems of controlling chaotic systems. 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