Non Linear Controllers for a Class of Chaotic Systems
E. Scorila1
1
2
D.P. Iracleous2
and
G.P. Syrcos1
Faculty of Applied Sciences, Technological Institute of Piraeus,
P. Ralli and Thivon 250, GR 12244 Egaleo, GREECE
Hellenic Naval Academy, Military Institutions of University Education
Terma Hatzikyriakou, GR 18539, Piraeus, GREECE
Abstract
Lorenz like systems is a common class of dynamical systems exhibiting chaotic behavior for some
selection of system parameters. The Permanent Magnet Synchronous Motor (PMSM) drive system is a
practical engineering system described by a set of equations of this class and therefore falls into the
category of systems with a probable chaotic behavior.
In this paper feedback linearization is studied for these systems, necessary conditions for feedback
stabilization are stated and a suitable control scheme is proposed. Proportional and integral (PI) control is
used to maintain the system output at desired setting value. A practical example is described, whereas the
effectiveness of the proposed method is demonstrated by extensive simulation results.
Keywords. Chaos control, PMSM (permanent magnet synchronous motor), feedback linearization.
1. Introduction
Chaos is an apparently strange behavior arising in
the response trajectory of a deterministic system
when two arbitrarily close starting points diverge
exponentially, so that its future behavior becomes
eventually unpredictable. In a more mathematical
terminology a system response is defined as chaotic
if it has sensitive dependence on initial conditions,
is topologically transitive, and its periodic points
are dense [1]. In other words, it is unpredictable,
indecomposable, and yet contains regularity. The
system trajectory is exponentially unstable and
neither periodic nor asymptotically periodic. That
is, it oscillates irregularly without settling down.
This behaviour is generally not acceptable to
engineering systems because of its unpredictable
nature. A typical chaotic system is Lorenz chaos
[9], a three state nonlinear system. This system
describes the process of heat convection in the
atmosphere and it is proved to exhibit chaotic
behaviour in a certain range of parameter selection.
Controlling chaos is a challenging engineering issue
[3-6] since it occurs in particular situations in
nonlinear systems, which is the case of most
practical systems. Unfortunately there is no general
control theory applicable to non linear systems
however various approaches have been developed
for certain system classes.
Feedback control has been applied to chaotic
systems [3]. In this paper Lorenz like chaotic
systems are studied and a suitable control strategy
is proposed. Permanent magnet synchronous motors
(PMSM) is an engineering system modeled by
Lorenz like dynamical equations, thus, exhibiting
chaotic behavior for certain parameter selection.
PMSM
have
noteworthy
performance
characteristics that should make this motor more
widely used in many applications today, however
the induction motors is still the choice in industry
where a large stock is available. PMSM are more
compact, lightweight and do not require space for
installation and maintenance of gear unit however,
more research on their control is needed before they
become widely used. Passive control theory was
tested in [2] whereas the feedback linearization
scheme is used here.
In this paper we review the Lorenz system, the
[A,B] model of PMSM drive and compare their
mathematical description. We present the feedback
linearization methodology and apply it to PMSM
drive. Simulations studies validate the effectiveness
of the proposed controller. Robustness issues are
tested and finally some key points are summarized
in the conclusion.
2. Preliminaries
I Lorenz chaos [4,9]
The celebrated autonomous Lorenz system is given
by the following set of eq.
x!1 = σ ( x2 − x1 )
x!2 = ρ x1 − x1 x3 − x2
x!3 = − β x3 + x1 x2
(1)
The system (1) has 3 states and 3 parameters
σ , ρ , β a selection of which, designates the chaotic
behavior of the system. The autonomous system has
two unstable and one stable equilibrium points [4].
II. PM synchronous motor model
Consider a PMSM model described as follows [7]
d
isd − ω Lsqisq
dt
d
usq = Rs isq + Lsq isq + ω ( Lsd isd + Ψ F ) (2)
dt
3
Te = p Ψ F isq + ( Ls d − Lsq ) isq isd
2
d
1
ω = ( − βω + Te − TL )
dt
J
usd = Rs isd + Lsd
(
)
where isd, isq are the d, q axes transformed currents,
usd, usq the transformed input voltages ω the motor
angular velocity, tW and tL are the electric and
external load torques.
Field flux in a PMSM is constant. In the case of
symmetric construction inductances Ls d , Lsq are
Rs
Stator resistance
20.785 Ω
ΨF
Field flux
0.4596 Wb
p
pole number
3
β
viscous friction constant
Ls d
Direct axis inductance
1e-3
Nm/rad/s
0.005 H
Lsq
Quadrature axis inductance
0.005 H
J
Rotor inertia
0.1 Kgr m2
TL
Nominal torque
10.1830 Nm
Isd,
Isq
Vsd,
Vsq
Nominal transformed stator
currents
Nominal transformed input
voltages
Nominal angular velocity
5A
ω
100 V, 180 V
157 rad/s
Table 1. List of PMSM drive parameters.
Considering the case where
ud = uq = TL = 0
(autonomous case) the system is clearly in Lorenz
form and exhibits chaotic behavior.
40
X1
20
0
0
50
time(sec) 100
Fig. 1 Typical Lorenz chaotic response.
equal, resulting in a further simplification of
equations (2).
Rewriting the system equations as state equations
we get
III. Feedback linearization to Lorenz systems
The system equations (3) are obviously nonlinear.
In the sequel, state transformation and feedback
linearization methods will be used to obtain an
equivalent linear system.
d
β
3
3
1 Feedback linearization is an approach to nonlinear
pΨ F isq +
Lsd − Lsq isqisd − TL
ω =− ω+
dt
J
2J
2J
J control design which algebraically transforms a
nonlinear dynamical system into a linear one [10].
Particularly, for a single input system which can be
L
R
1
1
d
isq = −
Ψ F ω − sd ωisd − s isq +
usq (3) represented as follows
dt
Lsq
Lsq
(
)
Lsq
Lsq
Lsq
1
d
R
isd = − s isd +
ωisq +
usd
dt
Lsd
Lsd
Lsd
A list of PMSM parameters used in this analysis is
given in Table 1.
y! = f1 ( y, z ) + w
z! = f 2 ( y , z ) + b( y, z )u
(4)
where u is the input, w is the external disturbance
 y
and   is the state vector where y may be vector
z
or scalar and z vector.
If b( y, z ) is an invertible mapping we can define
the control input as follows
u = b −1 ( y, z )[v − f 2 ( y , z )]
(5)
which modifies system (4) into the form
y! = f1 ( y, z ) + w
(6)
z! = v
where v is a new control input of the transformed
system (6).
We observe that feedback linearization can remove
the nonlinearities which appear in that equation
where the controlled input exists. To achieve a fully
linear system f1 ( y, z ) must be a linear function of
the state.
is calculated to maintain the rated steady state.
3. Controller design
I. Proportional - Integral controller (PI)
In order to achieve the tracking of the steady state
motor speed ω to be exactly at the value of a
desired reference speed r = ωr in the face of any
load disturbance, the following PI controller is
proposed [13].
v = K p x + k I ∫ [r − x]dt
(7)
However, defining the error as an extra state
p! = r + Cx
(8)
where C = [ −1 0
0] the PI controller has the
T
state feedback form
v = K [ x p]
(9)
where x and p are the states of the augmented
T
system
 x!   A 0   x   B 
D
0
 p!  = C 0   p  +  0  u +  0  w + 1  r
  
   
 
 
(10)
Now, applying standard linear control design
techniques, such as pole placement or optimal
control, state feedback gain can be determined
resulting to the following close loop system
 x!   A + BK p
 p!  =  C
  
KI   x   D 
0
+   w+  r



0   p  0 
1
(11)
II . Feedback linearization to PMSM
Clearly, comparing equations (3) and (4) feedback
linearization method can be applied to PMSM drive
system. The control input vector is defined in eq (5)
[ y ] ≡ [ω ] ,
where
z 
I 
 2

[ z ] =  z1  ≡  Isd 
sq

and
u 
[u ] ≡ usd  .

sq

The mappings f1, f2 and b are
ω 
 
0  isq 
(12)

isd 
Lm
Rs 
 1
 − L Ψ F ω − L ωisd − L isq 
sq
sq
sq
 (13)
f 2 ( y, z ) = 


Rs
−
isd + ωisq


Lsd


 1

0 
L
sq

(14)
b( y , z ) = 

1 
 0

Lsd 

 β
f1 ( y, z ) =  −
 J
3
pΨ F
2J
and K is the feedback control gain matrix of the
system [A,B]
 β
− J

 0
 0

 −1
3
pΨ F
2J
0
0
0
0
0
0
0
 0
 
0  1
 L
0  '  sq
0  0
 
0  
 0
0 

0 


1 
Lsd 

0 
(15)
This is called canonical form.
The pair is a controllable 4x2 linear system.
The new input v is defined as linear control of the
deviations of the states nominal values, i.e.
T
v = K ω isq − isq 0 isd − isd 0 ω − ωr 
Any standard control design method, such as pole
placement technique, optimal control or a
combination of methods can be used to determined
the control feedback gain matrix K.
4. Simulation studies
Case I. Load torque change
The PMSM drive system with the PI controller is
operating at the steady state with command input
r=ωr=1500 rpm and the external load torque is
increased by 20%. The response of the rotor speed
is shown in fig. 2-a with a solid line where the
reference is shown with a dotted line. As it is
expected by the action of the integral term of the
controller, the system reaches exactly the reference
input ω r =1500 rpm after a short transient. In Fig 2b current is shown which increases to a new steady
state. Fig. 2-c shows the response of the integral
error and Fig. 2-d and 2-e show the input voltages
ud and uq for this case.
Case II. Command input change
The PMSM drive system with the PI controller is
operating at the steady state where the command
input is initially selected at the point ω r =1500 rpm.
Again, at the time t=1 sec a decrease step of 20% of
the initial set point ω r occurs, as shown in Fig. 3-a
with a dotted line.
The rotor speed tracks the new set point after a
transient period (Fig. 3-a solid line), while the
motor currents (Fig. 3-b) and the input voltages
(Fig. 3-c and 3-d) reach the new steady state.
Obviously, the external load torque is assumed to
be constant.
40
Int(error)
0
-40
0.00
time (sec)
10.00
Fig2-c Integral error state response.
360
Vd (V)
320
280
0.00
160
5.00
5.00
time (sec)
10.00
Fig. 2-d The necessary ud input to perform
feedback linearization.
w (rad/sec)
101
150
Vq (V)
100
140
0.00
5.00
time (sec)
10.00
Fig. 2-a Dotted line is the reference input and solid
line is the system response.
99
0.00
5.00
time (sec)
10.00
Fig. 2-e The necessary uq input to perform
feedback linearization.
8
I (A)
160
w (rad/sec)
6
120
4
0.00
5.00
time (sec)
10.00
Fig. 2-b Dotted line is the reference input and solid
line is the system response.
80
0.00
5.00
time (sec)
10.00
Fig. 3-a Dotted line is the reference input and solid
line is the system response.
8
Extensive simulation results confirm our design in
both the case of torque and angular velocity change.
I (A)
4
0
0.00
5.00
time (sec)
10.00
Fig. 3-b Motor current is depicted.
400
Vd (V)
200
0
0.00
5.00
time (sec)
10.00
Fig. 3-c The necessary ud input to perform feedback
linearization.
.
102
Vq (V)
100
98
0.00
5.00
time (sec)
10.00
Fig. 3-d The necessary uq input to perform feedback
linearization.
5. Conclusions
Lorenz like chaotic systems can be controlled by
suitable designed feedback controllers. These
nonlinear controllers can not only successfully
remove chaos but also design the closed loop
response.
This control procedure is applied and tested on a
PMSM drive described by Lorenz like dynamical
equations. Feedback linearization method reduces
the PMSM system into an equivalent linear model
where a suitably designed proportional and integral
controller provides the quick and smooth follow up
of the reference input. The control is transformed
back for the original nonlinear system resulting in a
nonlinear control that preserves the stabilizing
properties of the linear performance.
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