EFFECT OF THIRD POLE PLACEMENT ON THE CHARACTERISTICS
OF CLOSED LOOP CONTROL SYSTEM
S. Srivastava*, A. Misra, S.K. Thakur and V.S. Pandit
Variable Energy Cyclotron Centre, Kolkata-700064, India.
Abstract
In this paper tuning of PID controller is optimized using
the optimal approach of Linear Quadratic Regulator
(LQR) and the dominant pole placement approach for the
second order systems with time delay. Simulation has
been performed in MATLAB for the buck converter
based power supply. The effect of third pole on the
stability of closed loop system has been studied in detail
in the case of mismatch between the process delay time
and the delay time at which the controller is designed. It
is found that if the PID controller is designed by choosing
a higher location of the third pole with one of the real part
of the dominant poles, the stability of the closed loop
system improves with slight increase in rise time.
designed according to the classical control theory. We
then extend this approach utilizing the dominant pole
placement technique to find the optimal PID parameters
for SOPTD systems. A linear plant with time delay can be
characterized by the state-space representation as
 (t )  AX(t )  Bu(t  L)
X
where X(t )  x1 (t ) x2 (t ) x3 (t )T with state variables
defined as
de(t )
.
(2)
x1 (t )  e(t ) dt , x2 (t )  e(t ) , x3 (t ) 
dt
Let the TF of the SOPTD system is given as

INTRODUCTION
G(s) 
Various types of power supplies used in accelerators
require stable operation under the different load
conditions and other types of fluctuations. Therefore, a
robust controller is mandatory. Proportional-IntegralDerivative (PID) controller is one of the widely used
controllers for the control of processes with various
dynamics. For efficient functioning one needs an optimal
PID controller as it incorporates extra robustness in the
power supply with required output at minimum cost [1, 2,
3]. In this paper we, present the methodology of PID
tuning based on the optimal approach of LQR and
dominant pole placement technique applicable to second
order plus time delay (SOPTD) model with user specified
closed loop damping ratio ‘ζcl’ and natural frequency
‘ωcl’. We have used the PI/PID tuning method proposed
by He et al. [5] for the first order plus time delay
(FOPTD) model and developed the tuning procedure for
SOPTD model using dominant pole placement approach
[6]. In this paper we present the simulation results of
optimal PID controller for a prototype buck converter
SOPTD model. We have chosen SOPTD model since the
transfer function (TF) of the power supply can be
estimated more accurately as SOPTD model [4] compared
with the FOPTD model. The effect of third pole
placement on the robustness of the PID controller has
been discussed with simulation results.
y (s)
K e  sL
 2
.
u ( s ) s  as  b
___________________________________________
*[email protected]
(3)
where K, a, b and L are the parameters of the plant G(s).
From Fig. 1, if the Ref (t) is made equal to zero then
output y(s) will be –e(s). Using y(s) = –e(s) and Eqs. (2)
and (3), the state space representation of the derivative of
state variable can be written as
0   x1 (t )   0 
 x1 (t )  0 1
 x (t )  0 0
  x (t )   0  u (t  L) .
1
 2  
 2  

 x3 (t )  0  b  a   x3 (t )   K 
(4)
Comparing Eq. (4) with Eq. (1), the state transition matrix
A and control matrix B are
0
0 1
 0 


A  0 0
1  , B   0  .
0  b  a 
 K 
PID TUNING USING LQR
Fig. 1 shows the standard closed loop block diagram of
SOPTD systems. We have followed the method of He et
al. [5] for the FOPDT models where the motion equation
is reformulated into a first order differential equation that
contains no time delay and then the optimal controller is
(1)
Figure 1: Closed loop time delay system.
(5)
Using the procedure outlined in Ref [5], the optimal value
of control u(t) for Eq. (1), subject to the minimization of
cost function

J
 X
T
q1 

(t ) Q X(t )  u (t ) R u(t ) dt
T
p22 
(6)
0
q2 
(2cl3  cl  4m cl3 cl3  2m 2 cl3 cl3  ab)
r 1K 2
m 2  cl2 cl6
r 1K 2
cl4  4m 2  cl4 cl4  b 2  2m 2 cl2 cl4
r 1K 2
is given by
1
u(t )  R B Pe
T
( Ac )t A ( L t )
e
X(t ) 0 ≤ t < L,
u(t )  R 1 B T Pe ( Ac ) L X(t )
t ≥L
(7)
(8)
where R is the control weighting matrix in form [r] and Q
is the state weighting matrix in diagonal form. P is the
symmetric matrix which can be obtained from the
solution of continuous algebraic Riccati equation [1],
1
A P  PA  Q  PBR B P  0
T
T
(9)
In Eqs. (7) and (8) we need the values of A and Ac which
can be obtain utilizing dominant pole placement approach
[6] and by equating the characteristic equation to the
desired dominant poles as
sI  A c  sI  ( A  BR 1B T P)  (s  P1 )( s  P2 )( s  P3 )
 (s  m cl cl )(s 2  2s cl cl  cl2 )
(10)
where m is known as relative dominance. It affects the
location of third pole on the left half of complex plane.
Equating the coefficients of the powers of s in Eq. (10)
gives three elements p11 , p12 , p13 of matrix P. Using
these three values and utilizing Eq. (9) the other elements
of matrices P and Q can be obtain as
q1 0
Q   0 q2
 0 0
0
 p11

0  , P   p12
 p13
q3 
p12
p22
p23
p13 
p23 
p33 
where,
p11 
p13 
p 33 
(11)
q3 
,
4 cl2 cl2  m 2 cl2 cl2  2b  a 2  2cl2
r 1K 2
(12)
Finally, the PID parameters can be obtain using Eq. (7) by
equating the coefficient of state variables with the PID
parameters Kp, Ki and Kd as shown in Fig. 1 i.e
u(t )  K p x2 (t )  Ki x1 (t )  K d x3 (t )
(13)
Eq. (7) is not used for evaluation of PID parameters as it
is only applicable for 0 ≤ t < L, and it gives time varying
PID parameters which is difficult to implement in analog
circuit. Utilizing, Eqs. (5), (8), (9) and (12) the PID
parameters are
K p  r 1 K  p13 H 12 ( L)  p 23 H 22 ( L)  p 33 H 32 ( L) 
Ki  r 1K  p13H11( L)  p23H21( L)  p33H31( L)
K d  r 1 K  p13 H13 ( L)  p 23 H 23 ( L)  p33 H 33 ( L) (14)
where H ij (L) are the matrix elements of e (A c )t at t = L.
with i,j = 1,2,3.
SIMULATION RESULTS
In order to illustrate the above PID tuning algorithm for
SOPTD systems, we have considered the example of a
typical buck converter based power supply that is widely
used in particle accelerator technology [7] as shown in
Fig. 2. RL and RC are the series equivalent resistance of
inductor L and capacitor C respectively. Switch sw is the
switching element and Rload is the load.
m cl cl5 (1  2m cl2 )
(2  m) m  cl2 cl4
,
p

12
r 1K 2
r 1K 2
m cl cl3
1
R K
2
, p23 
cl2  2m cl2 cl2  b
(2  m) cl  cl  a
R 1 K 2
R 1 K 2
,
Figure 2: A standard buck converter based power supply.
The transfer function of the buck converter having rating
(10V/10A) is estimated using the system identification
toolbox of the MATLAB [8]. The estimated SOPTD
model of the buck converter is
G( s) 
1.667  10 6 e 0.0004s
s 2  629 .8s  1.755  10 6
.
(15)
Utilizing Eqs (4), (12) and (14) the PID parameters
obtained with choosing closed loop natural frequency cl
=1324.8 rad/s (≥
K p
Ki
b [9,10]),  cl = 0.98, with m = 10 are
 
K d  3.55 2.55  10
3
0.0023

e (A c ) L decreases with increase in m which reduces the
optimal control u(t) and hence increases closed loop
system rise time. Fig. 4 shows the stability and robustness
of the proposed PID controller with the mismatch in the
delay time L. L is the delay at which the controller is
designed. Lp shows the percentage perturbation
introduced to the delay time L. It can be easily concluded
from the Fig. 4 that if the PID controller is designed at
higher value of third pole location i.e. more toward left
half complex plane (higher value of m), the controller is
more robust in the case of mismatch in the time delay.
CONCLUSION
(16)
The parameters of PID controller have been derived
analytically using the optimal approach of LQR and the
dominant pole placement technique for the SOPTD
systems. A buck converter is estimated to a standard
SOPTD model. The effect of third pole on the stability of
closed loop system is studied in detail in the case of
mismatch between the process delay time and the delay
time at which the controller is designed. It is observed
from the simulation results that if the controller is
designed with higher value of m, the closed loop time
response will be more stable as compared with the
controller designed at lower value of m. The only
compromise is the increase in the system rise time.
REFERENCES
Figure 3: Step response of buck converter with different
relative dominance m.
Figure 4: Stability and robustness with mismatched in the
delay L.
Fig. 3 shows the step response of buck converter with
different values of relative dominance m. It can be seen
that as the location of third pole is shifted towards left
half plane the rise time is increased and overshoot is
decreased. The reason for this is the occurrence of
exponential term e (A c ) L in Eq. (7). The matrix element of
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