1.
Research Article
An optimal PID controller via LQR for standard second order plus time
delay systems
Saurabh Srivastava, Anuraag Misra, S.K. Thakur, V.S. Pandit n
Variable Energy Cyclotron Center, 1/AF, Bidhan Nagar, Kolkata 700064, India
a r t i c l e i n f o
Article history:
Received 17 July 2013
Received in revised form
21 September 2015
Accepted 19 November 2015
Available online 4 December 2015
This paper was recommended for publica-
tion by Dr. Ahmad B. Rad.
Keywords:
Linear system
PID controller
System matrix
Linear Quadratic Regulator (LQR)
Time delay
Closed-loop
a b s t r a c t
An improved tuning methodology of PID controller for standard second order plus time delay systems
(SOPTD) is developed using the approach of Linear Quadratic Regulator (LQR) and pole placement
technique to obtain the desired performance measures. The pole placement method together with LQR is
ingeniously used for SOPTD systems where the time delay part is handled in the controller output
equation instead of characteristic equation. The effectiveness of the proposed methodology has been
demonstrated via simulation of stable open loop oscillatory, over damped, critical damped and unstable
open loop systems. Results show improved closed loop time response over the existing LQR based PI/PID
tuning methods with less control effort. The effect of non-dominant pole on the stability and robustness
of the controller has also been discussed.
& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Proportional-Integral-Derivative (PID) controller, though very
old design, is still one of the favorite and most widely used con-
troller for many industrial process control applications. This is due
to its simple structure, satisfactory control effect and acceptable
robustness [1–3]. The PID controller is easier to understand due to
intuitive simplicity of the algorithm and simple meaning of its
tuning parameters proportional (Kp), integral (Ki) and derivative
(Kd). In order to provide good and robust performance these PID
parameters are required to be tuned individually to match the
process dynamics. An improper PID setting results in sluggish,
oscillatory time response and poor robustness. The PID controller
tuning first proposed by Ziegler–Nichols [4] has been improved by
several researchers [5]. As the high performance is always desired
from the controller and due to the availability of fast computa-
tional power, tuning methods based on optimization approach
have received more attention in the recent years [6,7]. Many
techniques have been developed and still research is going on for
better tuning of the PID controller using complex numerical
optimization procedures [8,9].
The design techniques based on linear Quadratic Regulator
(LQR) are well known in modern control theory and have been
widely used in many applications [10–12]. In a recent article Saha
et al. [11] have obtained the PID parameters for second order
systems via LQR using the dominant pole placement technique.
However, their approach is applicable only for systems having no
time delay. Most of the real industrial plants have time delay in
their transfer function. Since the presence of time delay in a
control loop is a source of instability and performance degrada-
tion, it is, therefore, necessary to design the PID controller opti-
mally to achieve good stability. Many researchers have worked on
the tuning of controller for the systems having time-delay [13–20]
with pole placement and mentioned the challenges due to the
presence of exponential term in the characteristics equation which
leads to the infinite roots. They have used different approaches to
design the controller with some limitations. He et al. [10] have
proposed an analytical method to tune the PI/PID parameters in an
optimal way using LQR techniques with user specified closed loop
damping ratio and natural frequency for the first order plus time
delay (FOPTD) model. His method is based on the decomposition
of state equation in two parts one for toL and another for tZL in
such a way that the state equation for tZL become independent of
L and then applied the usual LQR approach for obtaining the PID
parameters for FOPTD. They have compared simulation results of
their method with the gain-phase margin method [21] and pre-
sented much improved results.
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2015.11.020
0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
n
Corresponding author.
E-mail addresses: saurabh@vecc.gov.in (S. Srivastava),
pandit@vecc.gov.in (V.S. Pandit).
ISA Transactions 60 (2016) 244–253

2.
Most of the real plants can be more closely approximated using
second order plus time delay (SOPTD) model compared to FOPTD
model. The SOPTD processes are very rich in dynamics as they
include under damped, critically damped and over damped sys-
tems. Very few tuning rules are available for such processes. He
et al. [10] have also extended their approach for SOPTD systems by
equating the larger process pole with the derivative term of the
PID controller and then applied the PI tuning approach using LQR
to obtain other two parameters. This approach works satisfactory
for SOPTD model if the system poles have real roots, but does not
provide the optimum parameters of the PID controller as one of
the PID parameters is prefixed. This technique cannot be applied
for SOPTD systems with complex poles (such as highly oscillatory
processes) of the system as they are always in pairs and cannot be
eliminated with single complex zero of the controller.
In the present work we have combined the concept of LQR
based PI/PID tuning method together with the dominant pole
placement approach to derive the PID parameters analytically for
SOPTD systems. It is shown that the present technique gives a
good closed loop time response for various processes as compared
with the existing PI/PID tuning methods using LQR. In order to
illustrate the utility of the present technique, simulations per-
formed in MATLAB [22] have been presented for different types of
SOPTD models such as critically damped and over-damped pro-
cesses as well as processes having complex poles. The effect of
non-dominant pole on the control signal and on the stability of the
closed loop system has also been discussed.
2. LQR based PID controller design for SOPTD processes
In this section we briefly outline the LQR solution for time delay
systems formulated by He et al. [10] 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 con-
troller is 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.
2.1. LQR solution for SOPTD systems
A linear plant with time delay can be represented as
_XðtÞ ¼ AXðtÞþBuðtÀLÞ tZ0 ; ð1Þ
where A, B, X and L are the state transition matrix, control matrix,
state matrix and the time delay term respectively. For toL, no
control signal will be effective and thus we have Eq. (1) as control
free equation. Control signal will be effective only for tZL. So by
decomposing Eq. (1) into two components, one for toL and other
for tZL, we have
_XðtÞ ¼ AXðtÞ ; 0rtoL ; ð2Þ
_XðtÞ ¼ AXðtÞþBum
ðtÞ ; tZL ; ð3Þ
where um
ðtÞ ¼ uðtÀLÞ. Since Eqs. (2) and (3) are now delay free,
one can easily apply the standard LQR approach [12] for delay free
processes to find the optimum control vector um
ðtÞ subjected to
the minimization of the cost-function defined by
J ¼
Z 1
0
XT
ðtÞ Q XðtÞþumT
ðtÞ R um
ðtÞ
dt ; ð4Þ
where Q is the semi positive definite state weighting matrix and R
is the positive definite control weighting matrix. The LQR solution
gives the optimal control vector um
ðtÞ as
um
ðtÞ ¼ ÀRÀ 1
BT
PXðtÞ ; ð5Þ
where P is the symmetric positive definite Riccati coefficient
matrix which can be obtained by solving continuous algebraic
Riccati equation
AT
PþPAþQ ÀPBRÀ 1
BT
P ¼ 0 : ð6Þ
From (5) we can write
uðtÞ ¼ um
ðtþLÞ ¼ ÀRÀ1
BT
PXðtþLÞ : ð7Þ
Here we see that uðtÞ gives the control signal in the whole time
horizon of tZ0, however X(tþL) is not directly known at time t.
With the use of Eqs. (2), (3) and (5), X(tþL) can be expressed in
terms of X(t) [10]. The optimal control vector uðtÞ for the present
case, thus can be written as
uðtÞ ¼ ÀRÀ 1
BT
PeðAcÞt
eAðL ÀtÞ
XðtÞ ; 0rtoL ; ð8Þ
uðtÞ ¼ ÀRÀ 1
BT
PeðAcÞL
XðtÞ ; tZL ; ð9Þ
where
Ac ¼ AÀBRÀ 1
BT
P : ð10Þ
The beauty of above mathematical formulation lies in the fact
that the optimal control vector u(t) handles the delay part as given
by Eqs. (8) and (9). As the system matrix Ac given by Eq. (10) does
not contain any time delay for tZL, one can easily apply the
approach of direct pole placement to get `the desired closed loop
time performance measures. In order to obtain the optimal feed-
back gain uðtÞ we need to calculate eðAcÞt
and eAðLÀ tÞ
. By sub-
stituting um
ðtÞ from Eq.(5) into Eq. (3) we have for tZL,
_XðtÞ ¼ AcXðtÞ : ð11Þ
The matrix Ac can be determined by setting the characteristic
equation of the closed loop system ΔðsÞ ¼ sIÀAcj
equal to the
desired closed loop equation. For example, in the case of FOPTD
process, where the matrix Ac is a 2 Â 2 matrix, we have
ΔðsÞ ¼ sIÀAc ¼ ðsþp1Þðsþp2Þ ¼ ðs2
þ2ςclωclsþωcl
2
Þ ;
ð12Þ
where
p1 ¼ ζclωcl þiωcl
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1Àζ2
cl
q
; p2 ¼ ζclωcl Àiωcl
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1Àζ2
cl
q
;
with ζcl and ωcl as the desired closed loop damping ratio and
natural frequency.
For the SOPTD process the dimension of matrix Ac will be 3 Â 3.
Utilizing the help of dominant pole placement technique matrix Ac
can be evaluated in terms of known parameters ζcl and ωcl from
the equation
sIÀAc
¼ ðsþp1Þðsþp2Þðsþp3Þ
¼ ðsþmζclωclÞðs2
þ2sζclωcl þω2
clÞ ; ð13Þ
where the location of non-dominant pole p3 ¼ mζclωcl is placed m
times away from the real part of the dominant closed loop poles.
We call this m as the relative dominance and as per the literature
its value should be chosen around 3 or more [2].
2.2. Determination of state weighting and Riccati coefficient
matrices
In the case of second order process matrices Q, R and P are
generally taken as
Q ¼
q1 0 0
0 q2 0
0 0 q3
2
6
4
3
7
5; R ¼ r½ Š; P ¼
p11 p12 p13
p12 p22 p23
p13 p23 p33
2
6
4
3
7
5 : ð14Þ
In the optimal control it is a standard practice to design reg-
ulator by varying Q and keeping R fixed [10,11]. A schematic of
S. Srivastava et al. / ISA Transactions 60 (2016) 244–253 245

3.
closed loop system with PID controller for SOPTD process is shown
in Fig. 1.
The state variables for the present case are
XðtÞ ¼ ½x1ðtÞ x2ðtÞ x3ðtފT
; ð15Þ
where
x1ðtÞ ¼
Z
eðtÞ dt; x2ðtÞ ¼ eðtÞ; x3ðtÞ ¼
deðtÞ
dt
; ð16Þ
with error eðtÞ ¼ rðtÞÀyðtÞ. Here r(t) and y(t) are the reference and
output signals respectively.
From Fig. 1, the control signal can be expressed in terms of the
state variable as
uðtÞ ¼ Kpx2ðtÞþKix1ðtÞþKdx3ðtÞ : ð17Þ
The transfer function of the PID controller can be express in s
domain as
CðsÞ ¼ Kp þ
Ki
s
þKds : ð18Þ
In the case of unity output feedback system such as shown in
Fig. 1, if we put the reference signal rðtÞ ¼ 0 ; we have eðtÞ ¼ ÀyðtÞ :
With this condition, the second order transfer function with time
delay can be written as
GðsÞ ¼
yðsÞ
uðsÞ
¼
K eÀ sL
s2 þasþb
¼
ÀeðsÞ
uðsÞ
; ð19Þ
in which a ¼ 2ζolωol and b ¼ ω2
ol, where ζol and ωol are the
damping ratio and natural frequency of the open loop plant
respectively. Using Eq. (16) we can express Eq. (19) in terms of
state variables as
_x3ðtÞ ¼ Àax3ðtÞÀbx2ðtÞÀKuðtÀLÞ :
In terms of state-space formulation the derivative of the state
variables can be written as
_x1ðtÞ
_x2ðtÞ
_x3ðtÞ
2
6
4
3
7
5 ¼
0 1 0
0 0 1
0 Àb Àa
2
6
4
3
7
5
x1ðtÞ
x2ðtÞ
x3ðtÞ
2
6
4
3
7
5þ
0
0
ÀK
2
6
4
3
7
5 uðtÀLÞ : ð20Þ
Comparing Eq. (20) with Eq. (1), it is straightforward to
α ¼ rÀ1
K2
obtain matrices A and B as
A ¼
0 1 0
0 0 1
0 Àb Àa
2
6
4
3
7
5; B ¼
0
0
ÀK
2
6
4
3
7
5 : ð21Þ
Using Eqs. (10), (14) and (21) we have
sIÀAc
¼
s À1 0
0 s À1
η p13 bþη p23 sþaþη p33
: ð22Þ
where η ¼ rÀ 1
K2
: Now from Eqs. (22) and (13) we have
s3
þðaþη p33Þs2
þðbþη p23Þsþη p13 ¼
s3
þðð2þmÞζclωclÞs2
þðωcl
2
þ2mζcl
2
ωcl
2
Þsþmζclωcl
3
: ð23Þ
By comparing the coefficients of powers of s from both sides of
Eq. (23), the elements p13, p23 and p33 can be obtained as
p13 ¼
mζclω3
cl
η
;
p23 ¼
ω2
cl þ2mζ2
clω2
cl Àb
η
;
p33 ¼
ð2þmÞζclωcl Àa
η
: ð24Þ
The remaining three elements of the matrix P and three ele-
ments of the matrix Q can be obtain by solving Riccati equation Eq.
(6), which gives six equations for six variables in terms of known
parameters. With some algebraic manipulations we obtain,
p11 ¼
mζcl ω5
cl ð1þ2m ζ2
clÞ
η
;
p12 ¼
ð2þmÞ m ζ2
cl ω4
cl
η
;
p22 ¼
2 ω3
cl ðζcl þ2m ζ3
cl þm2
ζ3
cl ÞÀab
η
;
q1 ¼
m2
ζ2
cl ω6
cl
η
;
q2 ¼
ω4
clð1þ4m2
ζ4
cl À2m2
ζ2
cl ÞÀb
2
η
;
q3 ¼
ω2
clð4ζ2
cl þm2
ζ2
cl À2Þþ2bÀa2
η
: ð25Þ
2.3. Evaluation of eAðLÀ tÞ
In order to obtain the value of eAðL ÀtÞ
we proceed as follows.
eAðL ÀtÞ
¼ ℓÀ1
ðsIÀAÞÀ 1
h i
t ¼ LÀ t
¼ ℓÀ 1
1
s
ðsþ aÞ
sðsþ p01Þðsþ p02Þ
1
sðsþ p01Þðsþ p02Þ
0 sþ a
ðs þp01Þðsþp02Þ
1
ðs þp01Þðs þp02Þ
0 Àb
ðs þp01Þðsþp02Þ
s
ðs þp01Þðs þp02Þ
2
6
6
6
4
3
7
7
7
5
0
B
B
B
@
1
C
C
C
A
¼
f
0
11ðtÞ f
0
12ðtÞ f
0
13ðtÞ
f
0
21ðtÞ f
0
22ðtÞ f
0
23ðtÞ
f
0
31ðtÞ f
0
32ðtÞ f
0
33ðtÞ
2
6
4
3
7
5
t ¼ LÀ t
: ð26Þ
Here p01 and p02 are the poles of the open loop system (see Eq.
(19)) given by
p01 ¼
aÀ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2 À4b
p
2
; p02 ¼
aþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2 À4b
p
2
; ð27Þ
Using partial fraction approach f
0
11,f
0
12,f
0
13, f
0
21,f
0
22,f
0
23,f
0
31,f
0
32 and
f
0
33 can be evaluated as
f
0
11ðLÀtÞ ¼ 1 ;
f
0
12ðLÀtÞ ¼
p02 eÀp01ðLÀ tÞ
p01ðp01 Àp02Þ
À
p01 eÀp02ðL ÀtÞ
p02ðp01 Àp02Þ
þ
a
b
;
f
0
13ðLÀtÞ ¼
eÀ p01ðLÀ tÞ
p01ðp01 Àp02Þ
À
eÀp02ðLÀ tÞ
p02ðp01 Àp02Þ
þ
1
b
;
f
0
21ðLÀtÞ ¼ 0 ;
f
0
22ðLÀtÞ ¼
Àp02 eÀ p01ðL ÀtÞ
ðp01 Àp02Þ
þ
p01 eÀ p02ðLÀ tÞ
ðp01 Àp02Þ
;
f
0
23ðLÀtÞ ¼ À
eÀ p01ðL ÀtÞ
ðp01 Àp02Þ
þ
eÀ p02ðLÀ tÞ
ðp01 Àp02Þ
;
f
0
31ðLÀtÞ ¼ 0 ;
Kp
Ki
Kd
x3(t)
+
(t)
PIDControllerC(s)
sL
e
bass
K −
++2
x2(t)
x1(t)
s
1
s
+
+
+
-
(t) y(t)(t)
Plant G(s)
Fig. 1. Schematic of closed loop system with PID controller.
S. Srivastava et al. / ISA Transactions 60 (2016) 244–253246

4.
f
0
32ðLÀtÞ ¼
beÀ p01ðLÀ tÞ
ðp01 Àp02Þ
À
beÀp02ðL À tÞ
ðp01 Àp02Þ
;
f
0
33ðLÀtÞ ¼
p01eÀp01ðL À tÞ
ðp01 Àp02Þ
À
p02eÀp02ðLÀ tÞ
ðp01 Àp02Þ
: ð28Þ
2.4. Evaluation of eðAcÞt
Using Eqs. (10), (14) and (21) we have
Ac ¼ ðAÀBRÀ 1
BT
PÞ ¼
0 1 0
0 0 1
Àγ Àβ Àα
2
6
4
3
7
5:
where
γ ¼ η p13, α ¼ aþη p33 and β ¼ bþη p23.
Now
eðAcÞt
¼ ℓÀ 1
ðsIÀAcÞÀ1
h i
¼ ℓÀ 1 1
sIÀAc
s2
þαsþβ sþα 1
Àγ s2
þαs s
Àsγ ÀsβÀγ s2
2
6
4
3
7
5
0
B
@
1
C
A
¼
f 11ðtÞ f 12ðtÞ f 13ðtÞ
f 21ðtÞ f 22ðtÞ f 23ðtÞ
f 31ðtÞ f 32ðtÞ f 33ðtÞ
2
6
4
3
7
5 : ð29Þ
Using Eq. (13) in Eq. (29) and with some algebraic manipula-
tions, it is straightforward to get f11, f12, f13, f21, f22, f23, f31, f32 and f33
as
f 11ðtÞ ¼
X3
i ¼ 1
p2
i Àα pi þβ
Di
eÀpi t
;
f 12ðtÞ ¼
X3
i ¼ 1
Àpi þα
Di
eÀpi t
;
f 13ðtÞ ¼
X3
i ¼ 1
1
Di
eÀpi t
;
f 21ðtÞ ¼
X3
i ¼ 1
Àγ
Di
eÀpi t
;
f 22ðtÞ ¼
X3
i ¼ 1
pi
2
Àα pi
Di
eÀ pi t
;
f 23ðtÞ ¼
X3
i ¼ 1
Àpi
Di
eÀ pi t
;
f 31ðtÞ ¼
X3
i ¼ 1
γ pi
Di
eÀ pi t
;
f 32ðtÞ ¼
X3
i ¼ 1
β pi Àγ
Di
eÀ pi t
;
f 33ðtÞ ¼
X3
i ¼ 1
pi
2
Di
eÀ pi t
; ð30Þ
where
D1 ¼ Àðp1 Àp2Þðp3 Àp1Þ ;
D2 ¼ Àðp1 Àp2Þðp2 Àp3Þ ;
D3 ¼ Àðp3 Àp1Þðp2 Àp3Þ :
2.5. Evaluation of PID parameters for 0rtoL ;
Using Eqs. (28) and (30) in Eq. (8) the optimal value of control u
(t) for 0rtoL ; can be expressed as
uðtÞ ¼ ÀRÀ 1
BT
PeAc t
eAðLÀ tÞ
XðtÞ ;
¼ rÀ1
K
p13
p23
p33
2
6
4
3
7
5
T
f 11ðtÞ f 12ðtÞ f 13ðtÞ
f 21ðtÞ f 22ðtÞ f 23ðtÞ
f 31ðtÞ f 32ðtÞ f 33ðtÞ
2
6
4
3
7
5
Â
f
0
11ðLÀtÞ f
0
12ðLÀtÞ f
0
13ðLÀtÞ
f
0
21ðLÀtÞ f
0
22ðLÀtÞ f
0
23ðLÀtÞ
f
0
31ðLÀtÞ f
0
32ðLÀtÞ f
0
33ðLÀtÞ
2
6
4
3
7
5
x1ðtÞ
x2ðtÞ
x3ðtÞ
2
6
4
3
7
5 : ð31Þ
By comparing the coefficients of x1ðtÞ, x2ðtÞ and x3ðtÞ in Eqs. (17)
and (31), one can easily obtain the PID parameters for 0rtoL ;as
KiðtÞ ¼ rÀ1
K p13
X3
i ¼ 1
f 1iðtÞf
0
i1ðLÀtÞþp23
X3
i ¼ 1
f 2iðtÞf
0
i1ðLÀtÞ
þp33
X3
i ¼ 1
f 3iðtÞf
0
i1ðLÀtÞ
!
;
KpðtÞ ¼ rÀ 1
K p13
X3
i ¼ 1
f 1iðtÞf
0
i2ðLÀtÞþp23
X3
i ¼ 1
f 2iðtÞf
0
i2ðLÀtÞ
þp33
X3
i ¼ 1
f 3iðtÞf
0
i2ðLÀtÞ
!
;
KdðtÞ ¼ rÀ 1
K p13
X3
i ¼ 1
f 1iðtÞf
0
i3ðLÀtÞþp23
X3
i ¼ 1
f 2iðtÞf
0
i3ðLÀtÞ
þp33
X3
i ¼ 1
f 3iðtÞf
0
i3ðLÀtÞ
!
; ð32Þ
2.6. Evaluation of PID parameters for tZL
Similarly using Eqs. (9) and (30) the optimal control u(t) for
tZL can be evaluated as
uðtÞ ¼ ÀRÀ 1
BT
PeAcL
XðtÞ ;
¼ rÀ1
K
p13
p23
p33
2
6
4
3
7
5
T
f 11ðLÞ f 12ðLÞ f 13ðLÞ
f 21ðLÞ f 22ðLÞ f 23ðLÞ
f 31ðLÞ f 32ðLÞ f 33ðLÞ
2
6
4
3
7
5
x1ðtÞ
x2ðtÞ
x3ðtÞ
2
6
4
3
7
5 : ð33Þ
A comparison of coefficients of x1ðtÞ, x2ðtÞ and x3ðtÞ in Eqs. (33)
and (17) gives the PID parameters for tZL as
Ki ¼ rÀ1
K p13f 11ðLÞþp23f 21ðLÞþp33f 31ðLÞ
À Á
;
Kp ¼ rÀ 1
K p13f 12ðLÞþp23f 22ðLÞþp33f 32ðLÞ
À Á
;
Kd ¼ rÀ 1
K p13f 13ðLÞþp23f 23ðLÞþp33f 33ðLÞ
À Á
: ð34Þ
Note that for L¼0, the matrix elements fij ¼1 when i¼j and
fij ¼0 for iaj and Eq. (34) leads to the optimal PID parameters for
systems having no time delay.
3. Results and discussion
In order to demonstrate the application of the PID tuning
methodology proposed in this paper, we now present simulation
results for different processes performed using MATLAB. We have
considered the examples of under damped, critically damped and
over damped SOPTD processes. The present day control challenge
is to design a controller to tune unstable and highly oscillatory
processes [3]. Therefore, in Examples 4 and 5 we have discussed
two plants; one with unstable open loop time response and other
with highly oscillatory behavior.
3.1. Example 1: non-minimum phase process
In this example we will consider an over damped SOPTD model
of a non-minimum phase process and evaluate the PID parameters
for 0rtoL and tZL. The closed loop time response is compared
S. Srivastava et al. / ISA Transactions 60 (2016) 244–253 247

5.
with the previously developed LQR-based PI/PID tuning method
[10], where the derivative term of the PID controller for SOPTD
process was set equal to one of the process pole and thus is not
obtained in an optimum way. For fair comparison, the desired
closed loop damping ratio ζcl ¼0.8 and natural frequency
ωcl ¼0.793 rad/s are taken same in the simulation. The non-
dominant pole is placed 6 times away from the desired domi-
nant real poles i.e. m¼6.
The transfer function of the non-minimum phase process is
P1 ¼
1Às
ð1þsÞ2
ð2þsÞ
ð35Þ
and its corresponding over damped SOPTD model is
P1 ¼
1
s2 þ3sþ2
eÀ 1:64s
ð36Þ
Matrices A and B can be obtained from Eqs. (36) and (21) as
A ¼
0 1 0
0 0 1
0 À2 À3
2
6
4
3
7
5 and B ¼
0
0
À1
2
6
4
3
7
5 : ð37Þ
Using Eqs. (24) and (25), matrices P and Q with R¼[1] can be
evaluated as
Q ¼
5:7158 0 0
0 1:4889 0
0 0 9:8290
2
6
4
3
7
5;
P ¼
13:0394 12:1288 2:3908
12:1288 19:2785 3:4540
2:3908 3:4540 2:0732
2
6
4
3
7
5 : ð38Þ
The eigen values of matrices P and Q are
eig P½ Š ¼
1:4050
3:6579
29:3282
2
6
4
3
7
5 and eig Q½ Š ¼
5:7158
1:4889
9:8290
2
6
4
3
7
5 : ð39Þ
The positive eigen values of matrices P and Q indicate that the
positive definite condition of LQR is satisfied. Finally, the PID
parameters for 0rto1:64 s can be obtain using Eq. (32). The
time varying PID parameters are plotted in Fig. 2. The PID para-
meters for tZ1:64 s can be calculated using Eq. (34) as
½Kp Ki KdŠ ¼ 0:6984 0:4602 0:1543½ Š : ð40Þ
Fig. 2 shows the variation of PID parameters used in the
simulation at m ¼ 6. For comparison, PID parameters evaluated at
m ¼ 3 and m ¼ 10 are also shown. It is clear that values of all the
PID parameters are very high at the beginning (t ¼ 0s), followed by
a decrease with t up to t ¼ 1:64 s and then remain constant
thereafter. Note that design of PID controller at higher value of m
leads to lower values for all the PID parameters for tZ1:64 s
whereas the situation is completely reverse in the case of
to1:64 s.
The time response of the step input for process P1 with 20%
disturbance at t ¼ 40 s is shown in Fig. 3 by solid black line. The
observed behavior of the closed loop time response during the
initial period is due to the high values of initial PID parameters,
which are responsible for the decrease in the system rise time and
hence enhancement in the overshoot. Note that PID parameters
between 0rtoL are time varying and large initially. This leads to
a comparatively larger control efforts and may cause the actuator
saturation in some cases. It is also difficult to implement them
practically, particularly in analog domain. It is obvious that a
choice of constant PID parameters throughout eases the practical
implementation, needs low control effort and maintains the state
optimality for all values of tZL. The plots of time response using
only constant PID parameters throughout (i.e. for tZ0) obtained
for tZ1:64 s using Eq. (34) for various values of relative dom-
inance m are also shown in Fig. 3 for comparison. In the simulation
all other parameters are kept constant.
It can be seen from Fig. 3 that as the value of m is decreased
from 6 to 3, the PID controller based on constant parameters tries
to cope with the actual time varying PID controller and produces
overshoot with improved rise time. An increase in the value of m
reduces the overshoot but at the same time increases the rise time.
At higher values of m, say around m¼50, this effect saturates and
further increase in m has no significant effect on the time
response. Thus the choice of m depends upon a particular
requirement whether one needs fast rise time or less overshoot. In
our experience a good choice for m is between 3 and 10.
It is interesting to point out here that increase in the rise time
with m in the delayed processes is just opposite to the LQR based
PID tuning with no delay [11], where an increase in m decreases
the rise time of the closed loop time response. For a given process,
our simulation results indicate that an increase in the value of m
results in the lower values of PID parameters as shown in Fig. 2
and thus a reduction in the control effort. This fact can also be
explained using Eqs. (9), (24) and (25) where an increase in m
increases the value of matrix elements of P. This finally causes
reduction in the control effort u(t) due to the presence of the term
PeðAcÞL
which decreases with increase in the value of elements of
matrix P. Note that in the case of delay free process eðAcÞL
¼ 1 and u
(t) is proportional to matrix P.
0.511.522.533.544.5
0.0
3.0
6.0
Kp
0.51.01.52.02.53.03.54.5
0.0
1.0
2.0
3.0
Ki
0.0 1.0 2.0 3.0 4.0 5.0
0.0
0.5
1.0
1.5
Time (s)
Kd
m = 10
m = 6
m = 3
t = 1.64 s
Fig. 2. PID parameters Kp, Ki and Kd as a function of time.
0 10 20 30 40 50 60 70
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Time(s)
y(t)
Time varying PID with m=6
Constant PID with m=6
Constant PID with m=5
Constant PID with m=4
Constant PID with m=3
Constant PID with m=10
Constant PID with m=50
Constant PID with m=200
Fig. 3. Comparison of time responses of time varying PID with constant PID
parameters for process P1 at ζcl ¼ 0:8 and ωcl ¼ 0:793 rad=s at different m.
S. Srivastava et al. / ISA Transactions 60 (2016) 244–253248

6.
Observing the simulation results shown in Fig. 3, it appears that
the time varying part of PID parameters though, improves the rise
time of the closed loop time response, but at the same time it
produces substantial overshoot as compared to the cases where
only constant PID parameters are used. As it will be easy to
implement practically, we therefore, in subsequent examples
consider only constant value of PID parameters evaluated for tZL
using Eq. (34).
To show the effectiveness of the present method, we now
compare our results with the previously developed LQR based
PI/PID tuning method at same values of closed loop damping ratio
and natural frequency. The optimal PID controller for process P1
with ζcl ¼ 0:8 ; ωcl ¼ 0:793 rad=s and m ¼ 6 obtained for tZ1:64 s
is
C1½presentŠ ¼ 0:6984þ
0:4602
s
þ0:1543 s ;
and the PID controller used in Ref.[10] is
C1½He et al:Š ¼ 0:6138þ
0:5561
s
þ1:0 s :
Fig. 4(a) compares the step responses with 20% disturbance at
t ¼ 40 s. It is easy to observe that the present method gives very
less overshoot, only 4% as compared to the 14% of the earlier
method. Note that the value of derivative gain in controller C1½He
et al:Š is larger than the controller C1½presentŠ. From the simula-
tion it is clear that a choice of larger derivative gain not necessarily
reduced the overshoot. The main reason for the reduction in
overshoot is the optimal tuning of derivative parameter Kd in the
present case, which was taken as one of the real pole of the open
loop system in earlier case and thus, was not optimum one. Due to
the optimal design of all the three parameters in the present
method, there is almost 46% reduction in the settling time toge-
ther with a substantial reduction in the overshoot.
Fig. 4(b) compares the control energy required to achieve good
closed loop time response. Since the cost-function is optimized
properly in the present method, the required control energy is also
less. PID parameters and closed loop performance measures such
as percentage overshoot (%OS), settling time (Ts) and rise time (Tr)
are presented in Table 1 for comparison.
3.2. Example 2: higher order process
Now we consider a higher order process [1] given by
P2 ¼
1
ð1þsÞ8
: ð41Þ
The corresponding over damped SOPTD model of this process is
P2 ¼
0:3360
s2 þ1:3878sþ0:3360
eÀ 4:3s
: ð42Þ
The controller parameters for this model calculated using the
method presented by [10] where one of the pole is taken equal to
the Kd are given in Table 1. The optimal PID controller for the
above process using present method for m ¼ 4 and tZ4:3 s is
C2 ¼ 0:3919þ
0:0912
s
þ0:2834s : ð43Þ
In both the cases same values for desired closed loop para-
meters ζcl ¼ 0:9 ; ωcl ¼ 0:3 rad=s are used. The eigen values of
matrices P and Q for this model are also positive and therefore
satisfying the condition of LQR. Fig. 5(a) compares of the step
response with 20% disturbance at t ¼ 70 s. Due to optimal design
of all the three parameters, overshoot is almost negligible with an
improvement in rise time and disturbance rejection. The control
effort required for desired time response, plotted in Fig. 5(b), is
also slightly less in the present optimization method.
In order to test the present method with large time delay, we
have varied the time delay of process P2 from 4.3 s to 44.3 s in
steps of 10 s and performed simulations. In all the cases fixed
value of ωclL ¼ 1:3 is used. Simulation results indicate that the
satisfactory closed loop time response. As usual, we found the
response time to become slow as the time delay L increases.
0.0
0.5
1.0
y(t)
0 10 20 30 40 50 60 70 80
0.5
1.0
1.5
2.0
2.5
Time (s)
u(t)
Present
He et. al.
a
b
Fig. 4. Time response and controller response for process P1 with 20% disturbance
at t ¼ 40 s.
Table 1
Closed loop performance measures.
Processes Kp Ki Kd ζcl ωclL m %OS Tr(s) Ts(s)
P1 (He et al.) P1
(Present)
0.6138 0.5561 1 0.8 1.3 14 4.5 15
0.6984 0.4602 0.1543 0.8 1.3 6 4 4.5 8
P2 (He et al.) P2
(Present)
0.2873 0.0851 1.0753 0.9 1.3 4 16 23
0.3919 0.0912 0.2834 0.9 1.3 4 0 12 20
P3 (He et al.) P3
(Present)
1.7342 2.1759 1 0.98 0.4 35 1.2 8
3.7238 1.9858 1.6867 0.98 0.4 4 15 1.1 5
0.0
0.5
1.0
y(t)
0 20 40 60 80 100 120 140
0.0
0.5
1.0
Time (s)
u(t)
Present
He et. al.
Fig. 5. Time response and controller response for higher order process P2 with 20%
disturbance at t ¼ 70 s.
S. Srivastava et al. / ISA Transactions 60 (2016) 244–253 249

7.
3.3. Example 3: critically damped SOPTD process
Consider a critically damped SOPTD process [5] given by
P3 ¼
eÀ0:2s
ð1þsÞ2
: ð44Þ
The optimal PID controller designed for ζcl ¼ 0:98 ; ωcl ¼ 2 rad=s
and m ¼ 4 is
C3 ¼ 3:7238þ
1:9858
s
þ1:6867s : ð45Þ
Fig. 6(a) shows the comparison of step responses of the criti-
cally damped SOPTD process with 20% disturbance at t ¼ 20 s.
Clearly, the present method gives an improved performance. Both
the overshoot and settling time are improved by considerable
amount (see Table 1) with slight improvement in the rise time and
disturbance rejection time. Although the present tuning method
takes slightly more control signal initially (Fig. 6(b)), but one can
easily verify that the total control cost is almost identical in both
the cases.
3.4. Example 4: unstable SOPTD process
Now we consider an unstable plant [23] given by
P4 ¼
1:5
ð0:5sþ1ÞðsÀ1Þ
eÀ 0:3s
: ð46Þ
With some algebraic manipulation we can easily write Eq. (46)
in standard second order TF as given in Fig. 1 and get the value of
a¼1, b¼ À2 and K¼3. The optimal LQR based PID controller
obtained with ζcl ¼ 0:9 ; ωcl ¼ 0:8rad=s and m ¼ 4 is
C4½presentŠ ¼ 1:2153þ
0:1688
s
þ0:5682 s : ð47Þ
The controller designed by adopting the method of He et al.
[10] taking Kd ¼ 2, the larger real system pole with same ζcl and
ωcl is given by
C4½He et al:Š ¼ 0:5619þ
0:0824
s
þ2:0 s : ð48Þ
The time response plotted in Fig. 7 clearly shows the advantage
of the proposed method for control of unstable plant dynamics.
Expect for slightly higher percentage overshoot all other closed
loop performance measures are quite reasonable (Tr ¼ 1:8 s, %OS
¼ 150 and Ts ¼ 5 s). Simulation results indicate that the range of
ωclL is limited. In the case of stable system the appropriate range is
ωclL A ð1:0; 1:5Þ and for the case of unstable systems it is
ωclL A ð0:1; 0:4Þ.
3.5. Example 5: highly oscillatory SOPTD process
Here we consider a SOPTD process with highly oscillatory open
loop response [24] with transfer function given by
P5 ¼
1
s2 þsþ5
eÀ0:1s
: ð49Þ
Our aim is to design a controller with very small percentage
overshoot and settling time. Since the roots of the process are
complex, the method used by He et al. for SOPTD process cannot
be applied here. The controller with ζcl ¼ 0:9 ; ωcl ¼ 1:5 rad=s and
m ¼ 4 is
C5 ¼ 3:9434þ
5:8325
s
þ3:6339s : ð50Þ
Simulation result presented in Fig. 8 shows a remarkable time
response (Tr ¼ 1:5 s, %OS ¼ 2 and Ts ¼ 3:3 s) for process P5.
0.0
0.5
1.0
1.5
y(t)
0 5 10 15 20 25 30 35 40
1.0
2.0
3.0
4.0
Time (s)
u(t)
Present
He et. al.
Fig. 6. Time response and controller response for process P3 with 20% disturbance
at t ¼ 20 s.
-4.0
0.0
4.0
y(t)
0 10 20 30 40 50
-4.0
0.0
4.0
Time (s)
u(t)
Present
He et. al.
Fig. 7. Time response and controller response for process P4 with 20% disturbance
at t ¼ 30 s.
0.0
0.4
0.8
1.2
y(t)
0 10 20 30 40 50
2.0
3.0
4.0
5.0
Time (s)
u(t)
Fig. 8. Time response and controller response for process P5 with 20% disturbance
at t¼30 s.
S. Srivastava et al. / ISA Transactions 60 (2016) 244–253250

8.
3.6. Comparison with other time domain tuning methods
In order to check the relative merits and demerits of the pre-
sent method, simulations have been performed for processes P1,
P2, P3 and P5 using other time domain tuning methods [1,3] such
as Integral of Square Error (ISE), Integral of Time Square
Error (ITSE), Integral of Absolute Error (IAE), Integral of Time
Absolute Error (ITAE). We have used the fmincon() function
of the MATLAB [22] optimization toolbox for finding the sets
of optimized PID controller parameters subject to a given
time domain performance index based cost-function. In all the
cases the optimization started with same initial value of PID
parameters equal to 0.3. i.e. Kp ¼0.3, Ki ¼0.3, Kd ¼0.3. Results
are compared in Fig. 9. It is easy to observe that present
method gives overall satisfactory closed loop time response.
In other cases the response time is fast but with substantial
overshoot and oscillations. We have also calculated the
control energy using the square of the MATLAB function norm
(u(t),2). Except for process P3, where ITAE and ITSE require
slightly less control energy, controllers designed with present
method need comparatively less control energy for all
other cases.
X Data
y(t)
0.0
0.4
0.8
1.2
IAE
ISE
ITAE
ITSE
Present
Time (s)
0 10 20 30 40 50 60 70 80
u(t)
0.0
0.8
1.6
2.4
3.2
a
b X Data
y(t)
0.0
0.4
0.8
1.2
IAE
ISE
ITAE
ITSE
Present
Time (s)
0 20 40 60 80 100 120 140
u(t)
0.0
0.4
0.8
1.2
1.6
a
b
X Data
y(t)
0.0
0.4
0.8
1.2
IAE
ISE
ITAE
ITSE
Present
Time (s)
0 10 20 30 40
u(t)
0.0
1.8
3.6
5.4
7.2
a
b X Data
y(t)
0.0
0.4
0.8
1.2
IAE
ISE
ITAE
ITSE
Present
Time (s)
0 10 20 30 40 50
u(t)
0
2
4
6
8
10
a
b
P1 P2
P3 P5
Fig. 9. Time response and controller response for processes P1, P2, P3 and P5 obtained using methods based on different time domain performance measures.
S. Srivastava et al. / ISA Transactions 60 (2016) 244–253 251

9.
3.7. Example 6: robustness test
Most of the real plants operate in a wide range of operating
conditions and it is required that the controller must be able to
stabilize the system with slight change in the operating conditions.
In such situation, the robustness of the closed loop system is an
important feature. The purpose of studying this process is to check
the robustness property of the optimal LQR-PID controller when
there is a mismatch between the delay time of the process and the
delay time for which the PID controller is designed. We consider
an under damped SOPTD process given by
P6 ¼
9
s2 þ1:2sþ9
eÀ 2s
: ð51Þ
It is easy to observe that the open loop system poles are
complex. We have designed the PID controller with closed loop
parametric demand of ζcl ¼ 0:98 ; ωcl ¼ 2 rad=s. The optimal LQR
based PID controller obtained with m ¼ 3 is
C6 ¼ 0:0979þ
0:1913
s
þ0:0111s : ð52Þ
Fig. 10(a) shows the closed loop step response of the under
damped SOPTD process P4 with 20% disturbance at t ¼ 30 s and
the corresponding control effort is plotted in Fig. 10(b). It can be
readily seen that the stabilization of load disturbance by present
controller is quite satisfactory.
We have also studied the robustness of the present controller
by varying the mismatched delay time Lm from 0.5 s to 4 s covering
both sides of the actual delay L ¼ 2 s for which the controller is
designed. Results of simulation are presented in Fig. 10(c) for
comparison. The time response with designed parameters is
shown by black solid line. It can be readily seen that an increase in
the value of time Lm from the designed value of L ¼ 2 s, causes an
overshoot in the time response and finally leads to the oscillation
if the value of Lm becomes larger. In contrast, the mismatched
value of Lm less than L is responsible for the increase in the rise
time as well as in the settling time and thus making the response
sluggish.
Since in the present LQR based PID method, we have an extra
tuning factor that is the value of relative dominance m which one
can utilized to improve the robustness of the time response in the
case of a mismatch in the delay time. To explain the effect of m on
robustness of the controller we have designed another optimal
LQR-PID controller keeping all the parameter same except the
value of m. The optimal PID controller for m ¼ 10 is
Cm10 ¼ 0:0658þ
0:1586
s
þ0:0029s : ð53Þ
A comparison of time response curves for cases Lm ¼ 4 s, m ¼ 3
and Lm ¼ 4 s, m ¼ 10 clearly indicates that a controller designed at
higher m shows less overshoot and thus will be more robust in the
case of mismatch between the process delay time and the delay
time at which the controller is designed. However, the penalty one
has to pay is the increase in the rise time.
4. Conclusion
In this paper an improved design methodology of PID con-
troller for standard SOPTD system has been developed by com-
bining the optimal approach of LQR and the dominant pole pla-
cement technique. The proposed tuning method allows more
flexible pole placement, which results in better time response. The
PID parameters have been calculated analytically using user
defined closed loop damping ratio and natural frequency. It is
demonstrated by simulation that present tuning methodology
gives improved closed loop time response with less control effort
as compared to the earlier developed LQR based PI/PID tuning
method. Simulation results indicate that present method works
well for most of the SOPTD models such as under-damped, criti-
cally-damped, over-damped, unstable and highly oscillatory pro-
cesses. It is observed from the simulation that most appropriate
range of ωclL for stable SOPTD processes is ωclL A ð1:0; 1:5Þ and for
the unstable SOPTD process is ωclL A ð0:1; 0:4Þ. A comparison of
simulations results with other time domain performance indices
indicates that the present methods gives an overall better closed
loop time response with comparatively less control effort.
It is observed that the location of non-dominant pole (value of
m) affects the closed loop time response provided all others
parameters are kept constant. An increase in the value of m,
increases the rise time with a substantial control on the overshoot.
This observed behavior of the closed loop time response with m in
the case of processes with time delay is completely opposite to the
cases of delay free processes. A slightly higher value of m adds an
extra robustness to the closed loop time response in the case of
mismatch between the process delay time and the delay time at
which the controller is designed. The proposed analytical tuning
method to obtain optimum PID parameters for SOPTD process will
be helpful for the on-line applications. We like to point out here
that the present approach cannot be applied to integrating pro-
cesses because they cannot be represented in the form of standard
second order transfer function.
Fig. 10. The plots of (a) time response and (b) controller response for under
damped SOPTD process P6 with 20% disturbance at t ¼ 30 s. (c) Time response of
process P6 at various mismatched delay time.
S. Srivastava et al. / ISA Transactions 60 (2016) 244–253252

10.
References
[1] Astrom KJ, Hagglund T. PID controller, theory design and tuning. NC: Instru-
ment Society of America Research Triangle Park; 1995.
[2] Norman SN. Control systems engineering. USA: John, Wiley Sons, Inc.; 2011.
[3] Vilanova R, Visioli A. PID control in the third millennium. London: Springer;
2012.
[4] Ziegler JG, Nichols NB. Optimum settings for automatic controllers. Trans
ASME 1942;64:759–68.
[5] Dey C, Mudi KR. An improved auto-tuning scheme for PID controllers. ISA
Trans 2009;48(4):396–409.
[6] Pan I, Das S, Gupta A. Tuning of an optimal fuzzy PID controller with stochastic
algorithms for networked control systems with random time delay. ISA Trans
2011;50:28–36.
[7] Shabani H, Vahidi B, Ebrahimpour M. A robust PID controller based on
imperialist competitive algorithm for load-frequency control of power sys-
tems. ISA Trans 2013;52:88–95.
[8] Das S, Pan I, Halder K, Das S, Gupta A. LQR based improved discrete PID
controller design via optimum selection of weighting matrices using fractional
order integral performance index. Appl Math Model 2013;37:4253–68.
[9] Zhao YM, Xie WF, Tu XW. Performance-based parameter tuning method of
model-driven PID control systems. ISA Trans 2012;51:393–9.
[10] He JB, Wang QG, Lee TH. PI/PID controller tuning via LQR approach. Chem Eng
Sci 2000;55(13):2429–39.
[11] Saha S, Das S, Das S, Gupta A. A conformal mapping based fractional order
approach for sub-optimal tuning of PID controllers with guaranteed dominant
pole placement. Commun Nonlinear Sci Numer Simul 2012;17:3628–42.
[12] Naidu DS. Optimal control system. CRC Press, USA; 2003.
[13] Cai. GP, Huang JZ, Yang SX. An optimal control method for linear systems with
time delay. Comput Struct 2003;81:1539–46.
[14] Panda RC, Yu CC, Huang HP. PID tuning rules for SOPDT systems: review and
some new results. ISA Trans 2004;43(2):283–95.
[15] Michiels W, Engelborghs K, Vansevenant P, Roose D. Continuous pole place-
ment for delay equation. Automatica 2002;38:747–61.
[16] Yeung KS, Wong WT. Root-locus plot of systems with time delay. Electron Lett
1982;18(11):480–1.
[17] Engelborghs K, Dambrine M, Roose D. Limitations of a class of stabilization
methods for delay systems. IEEE Trans Autom Control 2001;46(2):336–9.
[18] Suh Il H, Bien Z. A root-locus technique for linear systems with delay. IEEE
Trans Autom Control 1982;27(1):205–8.
[19] Michiels W, Vyhlídal T, Zítek P. Control design for time-delay systems based
on quasi-direct pole placement. J Process Control 2010;20(3):337–43.
[20] Sen M, De La. On pole-placement controllers for linear time-delay systems
with commensurate point delays. Math Probl Eng 2005;1:123–40.
[21] Ho WK, Hang CC, Cao LS. Tuning of PID controllers based on gain and phase
margin specications. Automatica 1995;31(3):497–502.
[22] MATLAB toolbox: user's guide. Mathworks, Inc.; 2009.
[23] Xiang C, Wang QG, Lu X, Nguyen LA, Lee TH. Stabilization of second-order
unstable delay processes by simple controllers. J Process Control 2007;17:675–
82.
[24] Wang QG, Zhang Z, Astrom KJ, Chek LS. Guaranteed dominant pole placement
with PID controllers. J Process Control 2009;19(2):349–52.
S. Srivastava et al. / ISA Transactions 60 (2016) 244–253 253