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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 ex (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)
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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:
x1 (t ) =− x2 (t ) − x3 (t ) , parameters π1 = −2 , ε = 0.01=
−1, π 2 = ,μ
0=, k 2.2 ,
x2 (t ) = x1 (t ) + ax2 (t ) + u1 (t ) + f1 (t ) , (18) , α1 6 are
α 0 5=
and reference model parameters =
x3 (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:
x1 (t ) = ρ1 x1 (t ) + x2 (t ) +
+ (1 + π1ρ1 +π 2 ) ⋅ y1 (t ) + π1 y2 (t ) ,
x2 (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
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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.
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.
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ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic