1. By
Dr.Warsame H.Ali
Roy G. Perry College of Engineering, Prairie View A&M University, Prairie View, TX
A CASCADED PLUS STATE FEEDBACK OPTIMAL CONTROLLER
FOR A THREE PHASE INDUCTION MOTOR BASED ON VECTOR
CONTROL ALGORITHM
07/22/15ICGST 2012 Presented July 17
2. Outline
Problem Statement
Overview of Literature
- induction motor
- Linearization
- Optimal Control
Proposed solution
Method used
Simulation results
Analysis
Conclusion & Recommendation
Q/A
07/22/15ICGST 2012 Presented July 17
3. Problem Statement
• To design an optimal 3-phase Induction motor controller
based on vector control algorithm that rejects disturbances,
eliminates steady state errors and improves dynamic
performance.
Design consists of proposed
- state feedback gain which provides optimal control
- plant cascaded with integrator to mitigate errors.
07/22/15ICGST 2012 Presented July 17
4. Overview of Literature
Induction motors
• Construction
- Stator
- Rotor
Advantages of induction motors
Simple design
- Series of three windings in the exterior stator
Low cost
- due to simple design
Reliability
- due to simple design
- no brushes to replace like a dc motor 07/22/15ICGST 2012 Presented July 17
5. Induction motor characteristics
1. The induced torque is zero at synchronous speed
2. The curve is nearly linear between no-load and full load
3. The rotor resistance is much greater than the reactance, so
the rotor current, torque increase linearly with the slip.
07/22/15ICGST 2012 Presented July 17
6. Motor Drives
Induction motors can only rotate at a synchronous speed (Ns)
speed that is related to the frequency by;
Ns= 120 x f/p
where p = # of poles in the stator
f = supply frequency in hertz
• To operate induction motors at speeds rather than the main
supply speed an electric drive is needed
Electric drives are an interconnected combination of
equipment that provides a means of adjusting the operating
speed of a mechanical load.
07/22/15ICGST 2012 Presented July 17
7. Electric drives consists of two main components
- controller
- inverter
Controllers are devices that use algorithms to control the
performance of electric motors in a predetermined way
Inverters supply the motor with power as prescribed by the
controller
Control algorithms are divided into two major categories;
- Scalar Control
- Vector Control
07/22/15ICGST 2012 Presented July 17
8. Optimal Control
In optimal control, one tries to find a controller that
provides the best possible performs with regards to a certain
criteria with respect to some given measure of performance.
An example is a controller that uses the least amount of
control-signal energy to take the output to zero.
In this case the measure of performance (optimality
criterion) would be the control-signal energy.
Linear Quadratic Regulator
The optimal control method chosen for the controller is a
Linear Quadratic Regulator.
07/22/15ICGST 2012 Presented July 17
9. In a state feedback version of the LQR, all the states can be
measured and are available for control. The cost is described
by a quadratic function that is a function of state and control
variables
LQR guarantees the stability of the closed loop system as
long as the system is controllable and observable.
The matrices A, B, C and D of a linear quadratic regulator
are constants based on the state space model, matrices Q and
R are weighting matrices which are also constants
ẋ = Ax +Bu
y = Cx +Du
.
07/22/15ICGST 2012 Presented July 17
10. Linearization
Jacobian Linearization of nonlinear systems is mostly used for
nonlinear system control because of the well established and
design analysis tools for linear systems.
The linearization method approximates the solution of the
non-linear induction motor by decomposing the non linear
model into a linear component and a non-linear component
The Jacobian linearization matrix does not fully represent the
dynamic system accurately because after linearization there is
an affine non linear term that is disregarded in controller
computations.
The non linear component is considered as a disturbance to
the linear component and as a result disregarded in
controller computations. 07/22/15ICGST 2012 Presented July 17
11. Proposed solution
A current fed induction motor mathematical model is
developed and its performance compared to an existing
voltage model
07/22/15ICGST 2012 Presented July 17
12. The induction motor nonlinear differential equations are
linearized using Jacobian linearization and transformed into
state space model.
s
p
e
e
d
time
07/22/15ICGST 2012 Presented July 17
13. The plant in state space model
a) Generic cascaded controller scheme
b) Controller design
a b
07/22/15ICGST 2012 Presented July 17
14. Pre-defined Controller
from ẋ = Ax +Bu
y = Cx
1 = u, A1 = 0, B1 = 1, C1 = [1;1;1], Eẋ C = 1
Controller and plant cascaded
07/22/15ICGST 2012 Presented July 17
15. Augmented System
State space model for the augmented system is
ẋe (t) = Ae + Beu1(t) + Eer(t)
ye(t) = y1(t) = Cexe(t)
Where
07/22/15ICGST 2012 Presented July 17
16. With the initial time set to zero and the terminal time is set
to infinity the problem is a infinite horizon problem and the
feedback law in this case is given by the performance
subject to the dynamics
ẋe = Aex +Beu
It has been shown in classical control theory that LQR
optimal control has a linear state feedback
u1(t) = – Kexe(t)
where K(t) is a properly dimensioned matrix given as
Ke(t)= R-1
* Be
T
* S
07/22/15ICGST 2012 Presented July 17
17. In which the matrix S > 0 is the solution for the Riccati
equation
0= -SAe - Ae
T
S + SBeR-1
Be
T
S – Q
While there are two solutions to the algebraic Ricatti
equation, the positive definite or positive semi- definite is
used to compute the feedback gain.
The resulting closed- loop system becomes
ẋe (t) = (Ae +BeKe) xe(t) + Eer(t)
which is asymptotically stable due to the property of LQR
design.
07/22/15ICGST 2012 Presented July 17
27. Regular controller torque after disturbance
t
o
r
q
u
e
Time in seconds
07/22/15ICGST 2012 Presented July 17
28. Cascaded controller response to a disturbance at time 2 sec
t
o
r
q
u
e
Time in seconds
07/22/15ICGST 2012 Presented July 17
29. Regular controller after disturbance at time 2sec
s
p
e
e
d
Time in seconds
07/22/15ICGST 2012 Presented July 17
30. Cascaded controller after disturbance at time 2 seconds
s
p
e
e
d
Time in seconds
07/22/15ICGST 2012 Presented July 17
31. Analysis
Dynamic performance
- regular controller takes time to settle down
- cascaded controller has no settling time
• Steady state performance
- regular controller has steady state error.
- cascaded controller eliminates steady state error
• Response to disturbance
- regular controller has an increased steady state error
after the disturbance.
- cascaded controller rejects the effects of disturbance.
07/22/15ICGST 2012 Presented July 17
32. conclusion
The cascaded plus state feedback controller gives better
dynamic performance, steady state performance and
responds better to a disturbance introduced to the plant
by rejecting the effects of the disturbance.
Future Work
Implementation of the proposed control scheme.
07/22/15ICGST 2012 Presented July 17
33. References
1. Jasem M. Tamimi and Hussein M. Jaddu., “Nonlinear optimal controller for a
three phase induction motor using quasilinearization,” proceedings of
International Symposium on Communications, Control and Signal Processing,
March 2006
2. Padmaraja Yedamale., “Speed Control of 3-Phase Induction Motor Using
PIC18 Microcontrollers,” Microchip Technology Inc, 2002
3. Dal Y Ohm., “Dynamic Model of Induction Motors for Vector Control,”
Drivetech, Inc,. Blacksburg, Virginia.
4. Jaroslav Lepka and Petr Steki., “3- Phase AC Induction Motor Vector Control
Using a 56F80x, 56F8100 or 56F8300 Device. Freescale Semiconductor, Inc.,
2004
5. P. C. Krause and C.H Thomas., “Simulation of Symmetrical Induction
Machinery”. November 1965
6. T. Benmiloud and A Omari., “New Robust Approach of Direct Field Oriented
Control of Induction Motor,” World Academy of Science and Engineering
Technology 2010
07/22/15ICGST 2012 Presented July 17
34. 7. N. Mohan, Electric Drives: An Integrative Approach, MNPERE, Minneapolis,
Minnesota, 2001.
8. Kevin Warwick, An introduction to control systems, River Edge, New Jersey:
World Scientific Publishing Co., 1996
9. Donald E Kirk, Optimal Control theory; An introduction. Mineola, New York:
Dover Publications Inc., 1998
10. B.K., Bose, Power Electronics and AC Drives, Prentice-Hall, Englewood Cliffs,
New Jersey, 1986.
11. F.L. Lewis, V.L. Syrmos, Optimal Control, Wiley-Interscience, New York,
1995.
12.Dr. Ing and O.I. Okoro., “Matlab Simulation of Induction Machines with Saturable
Leakage and Magnetizing Inductances” The Pacific Journal of Science and
Technology April 2003.
13.Burak Ozpineci and Leone M. Tolbert., “ Simulink Implementation of Induction
machine Model- A modular Approach”.IEEE
14.N. Mohan, Advanced Electric Drives: Analysis, Control and Modeling using Simulink,
MNPERE, Minneapolis, Minnesota, 2001. 07/22/15ICGST 2012 Presented July 17
