Inverse dynamic models approach for chaotic system synchronization in the presence of uncertain parameters is considered. The problem is identifying and compensating unknown state-dependent parametric disturbance describing an unmodelled dynamics that generates chaotic motion. Based on the method of inverse model control, disturbance observers and compensators are synthesized. A control law is proposed that ensures the stabilization of chaotic system movement along master reference trajectory. The results of computational simulation of controlled Rösller attractor synchronization are also presented.
Controlled systems and processes with chaotic dynamics are a matter of unflagging interest in modern control theory and practice [1, 2]. The problem of synchronization of chaotic systems is intensively studied; in this case, control law is designed in such a way that the controlled variables of the slave system follow the reference output of the master system or nonlinear oscillating system stabilized along given reference trajectory in the presence of uncertainties and external disturbances [3, 4].
A typical model of a chaotic system is a linear system with additional nonlinear components dependent on the state, the presence of which determines the appearance of chaotic regimes [5]. Because the system nonlinearity may be treated as a parametric disturbance of nominal model, chaos synchronization problem may be reduced to the disturbance rejection problem, namely, unknown and unmeasurable disturbances eliminating from the systems output along with reference signal tracking.
Recently a number of model-based control methods have been developed for disturbance rejection taking into account the requirements of accuracy, dynamic performance, stability and robustness [6, 7]. In this paper the inverse model control approach [8] is applied for chaotic systems synchronization. Inverse models are used for both parametric disturbance identification and compensation, which made it possible to synthesize disturbance decoupling controller, ensure reference signal tracking.
The proposed approach was studied through computational modeling using the example of a controlled Rösller attractor with signal and parametric disturbances.
Consider a state-space model of controlled chaotic system
with a distinguished nonlinear component, which causes the
emergence of chaotic dynamics and interpreted as an
uncertain parametric disturbance
x(t) = Ax(t) + Bu(t) + Nf (x(t),δ),
y (t)
c
=Cx(t), y (t)
m
=Mx(t),
(
output controlled and measured variables respectively.
Disturbance f (x(t),δ) ∈ R q may be treated as unknown
input signal for system (
Matrices SCB (α1) = CAα1−1B, SMN (α2 ) = MAα2 −1N
are known as Markov parameters of system (
Without loss of generality, for simplicity reason, we will assume that rank SCB =rank m, SMN
=m,where SCB •
=(
Consider two main inverse model problems:
Chaotic system identification, namely, obtaining unknown parametric disturbance estimate ˆf (t) using available measurements ym (t) and known control •
Chaotic system
control, namely, control law u(y(t), y* (t), ˆf (t)) design, which ensure control signal u(t) ; goal achieving
lim || ec (t) ||2 ≤ ε* , t → ∞. where ec (t)
=y* (t) − yc (t) – control error, y* (t) –
setpoint signal given by the reference model
y∗ (t) =A∗ ⋅ y∗ (t) + yref (t) ,
(
+ RNS M+N ⋅ ym (t) + RΠN B ⋅ u(t), ˆx( t ) = P ⋅ ym (t) + Q ⋅ x (t), ˆf (t) =CN [ym (t) − MAQ ⋅ x (t) −
− MAP ⋅ ym (t) − SMBu(t)], where Π Ν = In − NS M+N M , ΩN = I p − SMN S M+N , CN =S M+N + N + PΩN .
From (
=(x(t),t) f − ˆf (t) are given ex (t)
=F ( R ) ⋅ ex (t), ex (t) = Q ⋅ ex (t) e f (t)
=−CN MAQ ⋅ ex (t).
III. INVERSE MODEL-BASED CONTROLLER DESIGN
The disturbance rejection control law will be constructed as a function of reference signal and disturbance estimate: u* (t) =SC−B1 ⋅[yref (t) + CA ˆx(t) − SCN ˆf (t)], CA
=A∗C − CA.
If system structure non-singularity condition takes place ε* – some sufficiently small constant.
In the chaos synchronization problem, reference model
(
Dynamic system with state vector x (t) ∈ R n−q
x (t) = AI x (t) + B I u(t) + B1I y(t) + B2I y (t),
fˆ (t) =C I xk + D I u(t) + D1I y(t) + D2I y (t),
(
Then fˆ (t) may be treated as unknown input signal f (t)
dynamic estimate, obtained by inverse model (
III. INVERSE DYNAMIC MODEL DESIGN fˆ (t) − f (t)
2 aggregate matrix.
→ 0 , if t → ∞ , where Rn−q×n – some
RT ) = n .
Take state vector estimate in the form
ˆx(t) = P ⋅ ym (t) + Q ⋅ x (t), where matrices P ∈ R n× p , Q ∈ R n×n− p are such that
MP =I p , RQ =In− p , PM + QR =In , MQ =p,n− 0 p , RP =0n−p,p.
be aggregated
auxiliary is some aggregate matrix, so minimal-order unknown-input observer (UIO) [9]: x (t) =Fx (t) + G1 ym (t) + Hym (t) + G0u(t).
The UIO (
=GM , RN − HMN =0, G − RB =0, G =G − FH .
0 1
A solution of linear matrix equations (
F =RΠAQ, G
N 0 G 1 =RΠAP, H
N Π =I − BS + M ,
N n MN
=RB, =RNS+ ,
MN Taking the unknown disturbance estimate as
ˆf (t) = N + ( ˆx(t) − Aˆx(t) − Bu(t)) ,
We obtain the aggregated vector z(t) estimate x (t) by
rank S
=m + q, S
(
=
Im CN SMB
SC−B1 SCN
Iq or equivalently
det Φ ≠ 0, Φ = Iq − CN SMB SC−B1 SCN , (15) disturbance estimate may be eliminated from the controller equations, which is therefore be regarded as disturbance decoupling controller.
In reality, situations often arise when conditions (14), (15) are not met. In such a case the realizable control law may be obtained using the disturbance estimates, dynamically transformed by the internal auxiliary "fast" filter with small parameters.
As a result, realizable controller are designed by including in its structure an additional internal low-pass filter with small time constant [11]: u* (t) =SC−B1 ⋅[y∗ (t) + CA ˆx(t) − SCN f (t)],
εf (t) =− f (t) + (1 − μ) ⋅ ˆf (t), (16) where 0 < ε << 1,
0 < µ << 1 - small filter parameters.
From (13), (16) follows, that disturbance compensator equation with internal additional filter take the form: ˆf1(t )
=x2 (t ) + π2 y1(t ), ˆf2 (t ) = y2 (t ) + cy2 (t ) − x2 (t ) − π2 y1(t ) − u2 (t ). f1(t ) =δ f , f2 (x1(t ), x3 (t )) =δc x3 (t ) + (1 + δ x )x1(t )x3 (t ), as a step wave function, reference model input signal yref (t ) (17)
As a result disturbance decoupling controller with internal filter equation is obtained in the form: εu (t ) =ν1x1(t ) − x2 (t ) + (ζ1 − π2 )y1(t ) − α1 y2 (t ), u2 (t ) =u (t ) + ν1x1(t ) − 2 x2 (t ) +
+ (ζ1 − 2π2 ) ⋅ y1(t ) + (c − α1 − ε−1) ⋅ y2 (t ), ζ1 =α1 +ν1π1 −1,
ν1 =k − α1 − a
Proposed disturbance observer and decoupling controller are investigated by computational simulation.
Simulation results for Rösller attractor model parameters a = 0.2, c = −5.7 , observer and controller parameters π1 =−1, π2 =−2, ε = 0.01, μ =0, k =2.2 , (18) and reference model parameters α0 =5,α1 =6 are presented below.
Disturbance f1(t ) was modeled input signal disturbance (22) (23) εu(t ) =−μu(t ) + (1 − μ) ⋅[φ1(t ) +
+ SC−B1 SCN φ2 (t )], u* (t ) =u(t ) + φ1(t ), φ1(t )
=SC−B1 ⋅[yref (t ) + CA ˆx(t )], φ2 (t ) =CN ⋅[ym (t ) − MAQ ⋅ x (t ) − MAP ⋅ ym (t )].
III. CHAOTIC SYSTEM INVERSE MODEL CONTROL
As an example of proposed approach consider inverse model control of the Rösller attractor under uncertainties: x1(t )
=−x2 (t ) − x3 (t ), x2 (t ) = x1(t ) + ax2 (t ) + u1(t ) + f1(t ), x3 (t )
=−cx3 (t ) + u2 (t ) + f1(t ) + f2 (x1(t ), x3 (t )), where are input and parametric disturbances respectively with δ f ,δc ,δ x uncertain parameters.
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 ) + α0 y* (t ) =yref (t ) .
(19)
The control law, which ensures attractor synchronization with reference model, is the following: u2 (t ) = (α0 −1) ⋅ ˆx1(t ) + (k − a − α1) ⋅ ˆx2 (t ) + + (c − α1) ⋅ ˆx3 (t ) − 2 ˆf1(t ) − f2 (t ) − yref (t ), (20) u1(t ) = −kˆx2 (t ),
εf (t ) =− f (t ) + (1 − μ) ⋅ ˆf2 (t ),
The state estimates for system (18), obtained by reducedorder UIO, are: x1(t ) = ρ1x1(t ) + x2 (t ) +
+ (1 + π1ρ1 +π2 ) ⋅ y1(t ) + π1 y2 (t ), x2 (t ) =π2 x1(t ) + π1π2 y1(t ) + π2 y2 (t ), (21) ˆx1(t ) = y1(t ), ˆx1(t ) =x1(t ) + π1 y1(t ), ˆx3 (t )
=(t y2 ), where ρ1 = (π1 + a − k ), π1 π2 are tuning parameters.
Corresponding disturbance estimates are adopted in the form of harmonic function.
At Fig.1, 2 the state variables and phase plane of controlled Rösller disturbed attractor are presented.
Disturbances estimations obtained by (21), (22) are depicted in Fig. 3, 4 and control and output variables obtained in accordance the control law (20), (23) are presented at Fig. 5, 6.
Fig.3. Disturbances estimation f1(t), ˆf1(t) in open-loop system Fig.4. Disturbances estimation f2(t), ˆf2 (t) in open-loop system Fig.6. Set-point signal y* (t) and output variables yc (t)
Simulation results for chaotic system synchronization problem demonstrated high accuracy of disturbances decoupling for broad range of parameters deviation.
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 realtime 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 solution of synchronization problems. Thus proposed approach seems to be quite universal and can be used to solve various problems of controlling chaotic systems.
The implementation of the proposed control requires differentiating the measured output signals in real time, for which differentiators based on sliding modes can be used. Further development of the proposed approach is associated with the development of robust methods for inverse models design under conditions of uncertain deviations of the parameters of the chaotic object model.