Parametric optimization-based design

of PI controllers: application to a
DC servomechanism
Mohammad Tabatabaei
Department of Electrical Engineering, Khomeinishahr Branch, Islamic Azad University, Isfahan, Iran

Abstract
Purpose – This paper aims to propose an analytical method for designing parametric optimization-based proportional integral (PI) controllers.
Design/methodology/approach – In this method, a performance index containing the weighted summation of the integral square of the error and
its derivative is minimized. This performance index is analytically calculated in terms of the controller parameters by solving a Lyapunov equation.
Then partial derivatives of the performance index with respect to controller parameters are calculated. Equating these partial derivatives to zero
gives explicit relations for the PI controller parameters.
Findings – The experimental tests on a DC servomotor system are given to demonstrate the efficiency of the proposed method.
Originality/value – This paper proposes an analytical parametric optimization approach for designing PI controllers for the first time. The
application of the proposed method in a laboratory experiment is examined.
Keywords DC servomechanism, Integral square error, Lyapunov equation, Parametric optimization, PI controllers
Paper type Research paper

1. Introduction search-based PID controller has been constructed to reach the

desired transient response (Bagis, 2011). Binary-coded
Proportional integral (PI) and proportional integral derivative
extremal optimization algorithm has been used to calculate the
(PID) controllers have been widely used in industrial
optimum values for a PID controller (Zeng et al., 2014). Exact
applications (Astrom and Hägglund, 1995). Their simple
gradients have been used by Grimholt and Skogestad (2015) to
structure and provided tuning rules make them initial choices
obtain an optimal PID controller. Moharam et al. (2016)
to control real plants. A variety of methods have been proposed
proposed a hybrid differential evolution and particle swarm
to tune PI and PID controllers in the literature. According to a
common approach, these controllers can be tuned to attain optimization (PSO) with an aging leader and challenges to
some frequency specifications, such as gain and phase margin determine the PID controller parameters. To determine the
and gain or phase crossover frequencies (Mikhalevich et al., PID controller parameters, an integral time-absolute error
2015). Moreover, internal model control method has been performance index, subject to robustness constraints, has been
used to obtain some explicit relations for PI and PID controller minimized using the grey wolf optimization algorithm (Oliveira
parameters for some special case of plants (Najafizadegan et al., et al., 2016). A many-objective PSO algorithm has been
2017; Saxena and Hote, 2016). PID controllers and lead–lag proposed to design PI and PID controllers (Freire et al., 2017).
compensators have been designed to meet some time domain The H1 theory and constrained artificial bee colony algorithm
specifications such as maximum overshoot and settling time have been used for tuning the PID controller (Bijani and
(Tchamna and Lee, 2017). Moradi and Tabatabaei (2016) Khosravi, 2018). Sensitivity analysis and genetic algorithm
designed PI, proportional derivative and PID controllers for have been used to design PI controllers for some high-order
different plants using the Legendre orthogonal functions. physical plants (Perng et al., 2018).
Optimization methods have been used to design the family of Designing PI and PID controllers using the optimal control
PID controllers. Hast et al. (2013) obtained the PI and PID methods has been considered in the literature as well. PI and
controller parameters by convex–concave optimization of an PID controller parameters for low-order plus time-delay plants
integral error performance index, subject to robustness have been calculated by He et al. (2000) using a linear
constraints. Chang and Yan (2004) proposed an evolutionary quadratic regulator (LQR) approach. An LQR-based PID
programing algorithm for minimizing an integral absolute error controller has been proposed by Srivastava et al. (2016) for
index to find the PID controller parameters. A Tabu second-order plus time-delay systems. Srivastava and Pandit
(2017) proposed a two degree of freedom LQR-based PID
The current issue and full text archive of this journal is available on controller for integrating plants with time-delay considering
Emerald Insight at: www.emeraldinsight.com/1708-5284.htm performance/robustness.

World Journal of Engineering

Received 29 May 2018
16/3 (2019) 351–356 Revised 12 August 2018
© Emerald Publishing Limited [ISSN 1708-5284] 7 October 2018
[DOI 10.1108/WJE-05-2018-0178] Accepted 13 October 2018

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 Parametric optimization-based design World Journal of Engineering
Mohammad Tabatabaei Volume 16 · Number 3 · 2019 · 351–356
2 3
The optimization-based PI and PID controllers could not 0 1 0 ... 0 2 3 2 3
6 7 0 c0
give straightforward relations for the controller parameters in 6 0 0 1 ... 0 7 607 6 c1 7
6 7 6 7 6 7
6 7 .. 7
terms of the plant parameters. In this paper, a performance
A¼6
.. .. .. .. .. 7; b ¼ 6
6 . 7; c ¼
6 .. 7
6 7
index consisting of the weighted summation of the integral 6
6
. . . . . 7
7 6 7 6 . 7
6 ... 7 405 4 cn2 5
square error (ISE) and the integral square derivative error, 4 0 0 0 1 5
1 cn1
which could be called ISE-DE, is defined. Consider that the a0 a1 a2 . . . an1
error derivative in this performance index can improve the (3)
transient response of the closed-loop system. Now, a unit
negative feedback control structure with a PI controller is Now, consider that the controller C(s) is designed to minimize
considered. The mentioned performance index is analytically the following ISE-DE performance index:
calculated in terms of the plant and PI controller parameters by ð1 
solving a special Lyapunov equation. Setting the partial J¼ e2 ðtÞ 1 m e_ 2 ðtÞ dt (4)
0
derivatives of the performance index with respect to the PI
controller parameters to zero yields explicit relations for these where J is the performance index and m is an arbitrary positive
parameters in terms of the plant parameters. This algorithm is number. Increasing m can decrease error fluctuations.
performed for first- and second-order plants and its According to (2), relation (4) could be written as:
effectiveness for speed control of a laboratory DC motor is ð1  ð1
verified through experimental tests. J¼ eT ðtÞeðtÞ 1 m e_ T ðtÞ_e ðtÞ dt ¼ x T ðtÞQx ðtÞdt
The arrangement of this paper is as follows. The main 0 0
procedure for designing a controller using the parametric (5)
optimization approach is presented in Section 2. In Section 3,
the formulation of the proposed method for designing PI where
controllers for the first- and second-order plants is given. The Q ¼ cc T 1 m AT cc T A (6)
experimental tests on a DC servomechanism system are
presented in Section 4. Finally, the conclusion and future The solution of the state space equation (2) is:
perspectives are presented in Section 5.
x ðtÞ ¼ eAt b (7)

2. Parametric optimization-based design of Replacing (7) in (5) gives:

controllers
J ¼ b T Rb (8)
Consider the block diagram of the negative unit feedback control
structure shown in Figure 1. The transfer functions of the plant where
and controller are denoted by G(s) and C(s). The reference ð1
T
signal, the process output and the error signal are denoted by r(t), R¼ eA t QeAt dt (9)
y(t) and e(t), respectively. Now, consider that the reference signal 0

is a unit step signal. If the transfer functions of the plant and Consider that A is Hurwitz. Thus, the integral on the right side
controller are considered as rational functions, then the Laplace of (9) exists. It can be easily verified that the positive definite
transform of the error signal [E(s)] could be described as: symmetric matrix R = [rij], ij = 1, . . . n can be obtained by
cn1 sn1 1 . . . c1 s 1 c0 solving the following Lyapunov equation:
E ðsÞ ¼ (1)
sn 1 an1 sn1 1 . . . a1 s 1 a0
AT R 1 RA ¼ Q (10)
where ai, ci, i = 0,. . ., n – 1 are arbitrary real numbers, and n is
the degree of the closed-loop system. Moreover, consider that Now, according to (3) and (8), we have:
the closed-loop system is stable. J ¼ rnn (11)
According to the linear system theory, relation (1) can be
described with the following state space equations: Consider that the performance index depends on the controller
( parameters. For minimization of the performance index, its
x_ðtÞ ¼ Ax ðtÞ partial derivatives with respect to the controller parameters
(2)
eðtÞ ¼ cT x ðtÞ; x ð0Þ ¼ b should be zero. In other words, if the controller parameters are
denoted by zi, i = 1,. . .m, then we have:
where x(t) = [x1(t) . . . xn(t)]T is the state vector, and matrices
@J
A, b and c are defined as: ¼ 0; i ¼ 1; . . . ; m (12)
@zi

Figure 1 The unit negative feedback control structure The controller parameters (zi, i = 1,. . .m) can be obtained by
solving the algebraic relations given in (12). In the following
section, a PI controller is designed for the first-order plants
using this approach.

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Mohammad Tabatabaei Volume 16 · Number 3 · 2019 · 351–356

3. Proportional integral controller design using k

GðsÞ ¼ (21)
the parametric optimization method ðs 1 aÞðs 1 bÞ

In this section, the parametric optimization method is used to where k is a positive real number. Moreover, a and b can be
design PI controllers for some special case of plants. positive real numbers or complex conjugate numbers with a
positive real part. Considering the PI controller transfer
3.1 Proportional integral controller design for first- function given in (14), E(s) is calculated as:
order plants
Consider a first-order plant with the following transfer function: s2 1 ða 1 bÞs 1 ab
E ðsÞ ¼ (22)
s3 1 ða 1 bÞs2 1 ðab 1 kkp Þs 1 kki
k
GðsÞ ¼ (13)
s1a
Relations (3) and (22) lead to the following state space
where k and a are arbitrary positive real numbers. The transfer matrices:
function of the PI controller is given by: 2 3 2 3 2 3
0 1 0 0 ab
6 7 6 7 6 7
CðsÞ ¼ kp 1
ki
(14) A¼6 4 0 0 1 7; b 6 0 7; c ¼ 6 a 1 b 7
5 4 5 4 5
s
kki ðab 1 kkp Þ ða 1 bÞ 1 1
where kp and ki are the proportional gain and integrator (23)
coefficient. According to (13) and (14), E(s) in (1) can be
obtained as: Relations (6) and (23) give:
2 3
s1a a2 b2 1 m k2 k2i abða 1 bÞ 1 m k2 ki kp ab
E ðsÞ ¼ (15) 6 7
s 1 ða 1 kkp Þs 1 kki
2
Q¼6 4 abða 1 bÞ 1 m k ki kp
2 ða 1 bÞ2 1 m k2 k2p a1b7
5
According to (3) and (15), we have: ab a1b 1
" # " # " # (24)
0 1 0 a
A¼ ;b ¼ ;c ¼ (16)
kki ða 1 kkp Þ 1 1 The performance index is calculated in accordance with (10),
(11), (23) and (24) as:
According to (6) and (16), we have:  
m ða 1 bÞk2 k2i 1 kki a2 1 b2 1 m k2 k2p 1 ab 1 kkp 1 a2 b2 ða 1 bÞ
" 2 # J¼
a 1 m k2 k2i a 1 m k2 ki kp 2kki fða 1 bÞðab 1 kkp Þ  kki g
Q¼ (17)
a 1 m k2 ki kp 1 1 m k2 k2p (25)

According to (10), (11), (16) and (17), the performance index Relation (19) can be used for minimizing the performance
is obtained as: index J. Relations (19) and (25) lead to the following equations
in terms of kp and ki:
a2 1 m k2 k2i 1 1 m k2 k2p n o
J¼ 1 (18) k2 k2i ðab 1 kkp Þð1 1 m ða 1 bÞ2 Þ 1 a2 1 b2 1 m k2 k2p
2kki ða 1 kkp Þ 2ða 1 kkp Þ

To minimize J, the partial derivatives of the performance index 1 2kki a2 b2 ða 1 bÞ  a2 b2 ða 1 bÞ2 ðab 1 kkp Þ ¼ 0 (26)
with respect to controller parameters should be zero. Or:

@J @J k2 k2i f2 m kkp 1 1 1 m ða 1 bÞ2 g 1 kki ða 1 bÞ

¼ ¼0 (19)
@kp @ki  ða2 1 b2  m k2 k2p  2 m abkkp Þ 1 a2 b2 ða 1 bÞ2 ¼ 0 (27)

Solving relations in (19) gives the following PI controller

To find the PI controller parameters, relations (26) and (27)
parameters:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi should be solved numerically.
 pffiffiffiffi
a m 1 a2 m 2 1 m 1 1 2a m a
kp ¼ ; ki ¼ pffiffiffiffi 4. Simulation and experimental results
km k m
(20) In this section, some examples are presented to investigate the
effectiveness of the proposed PI controller.
In Example 1, the ability of the proposed controller to
3.2 Proportional integral controller design for second- control the angular velocity of a DC servomotor is verified.
order plants Figure 2 shows the components of the DC servomechanism
Now, consider a second-order plant describing the following system. It consists of a permanent magnet DC motor coupled
transfer function: with a tachometer to measure its angular velocity. To

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Mohammad Tabatabaei Volume 16 · Number 3 · 2019 · 351–356

Figure 2 The DC servomechanism system Figure 3 The motor angular velocities for different values of m

implement the designed PI controllers in MATLAB real-time

environment, Advantech analog to digital and digital to analog
interfaces are used. A magnetic brake causing angular velocity
decrement is used to investigate the robust performance of the
proposed controller in the presence of the load torque.
Figure 4 The control signals for different values of m in Example 1
A parametric identification approach is used to calculate the
following transfer function of the plant:
v ðsÞ 34:5
GðsÞ ¼ ¼ (28)
V ðsÞ s 1 10

where v (s) and V(s) are the Laplace transforms of the motor
angular velocity and the voltage applied to the servo amplifier
block, respectively. The command angular velocity is considered
as 1,000 RPM.
The PI controller parameters obtained with different values
of m ( m = 0.001,0.01,0.1,1,3,5) are given in Table I. The
motor angular velocities obtained with these PI controllers are
compared in Figure 3. The control signals are shown in
Figure 4. Decreasing the value of m increases the transient
response speed. But the oscillations in the transient response
and the initial value of the control signal could be decreased by
increasing the value of m . Figure 5 shows the effect of the load
torque (TL = 0.16 N.m) applied to the motor shaft for different
values of m . As can be seen in Figure 5, the effect of the load
torque can be eliminated for all values of m . However, this Figure 5 The angular velocities in the presence of load torque for
elimination is faster for smaller values of m . Figure 6 shows the different values of m
optimal value of performance index (J ) in terms of m . It is
obvious that decreasing m will decrease the optimal value of the
performance index. However, it may cause some oscillations in
the transient response and increase in the control effort.
Example 2 considers the second-order plant (21) with k = 3,
a = 1, b = 2. Table II shows the calculated PI controller
parameters for various values of m ( m = 0.1,0.5,1,5). Figure 7
compares the units step responses for the closed-loop system
obtained with the mentioned values of m . The corresponding

Table I PI Controller parameters for different values of m in Example 1

Values
PI coefficients m = 0.001 m = 0.01 m = 0.1 m = 1 m = 3 m = 5
kp 0.9166 0.2899 0.0917 0.029 0.0167 0.013
ki 9.166 2.8986 0.9166 0.2899 0.1673 0.1296

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Mohammad Tabatabaei Volume 16 · Number 3 · 2019 · 351–356


Figure 6 The (J – m ) plot for Example 1 Figure 8 The control signals for different values of m in Example 2

It is obvious that the plant poles are complex (a = 1 1 3j,

Table II PI Controller parameters for different values of m in Example 2 b = 1 – 3j). The PI controller parameters for different values
Values of m ( m = 0.5,1,5,20) are presented in Table III. The unit step
PI coefficients m = 0.1 m = 0.5 m =1 m =5 responses for the closed-loop system and control signals for
these values of m are shown in Figures 9 and 10, respectively. It
kp 3.1623 1.4142 1 0.4472 is obvious that increasing the value of m decreases the
ki 1.0263 0.6408 0.5 0.2595 oscillations in the transient response and the initial value of the
control signal, whereas it increases the settling time. Finally,
the effect of m on the transient response and the control signal
can be summarized in Table IV.
Figure 7 The unit step responses for different values of m in
Example 2
5. Conclusion
Optimization-based techniques for designing PI and PID
controllers provide good transient response for the closed-loop

Table III PI Controller parameters for different values of m in Example 3

Values
PI coefficients m = 0.5 m =1 m =5 m = 20
kp 0.5657 0.4 0.1789 0.0894
ki 1.6569 1.3333 0.731 0.4022

Figure 9 The unit step responses for different values of m in Example 3

control signals are compared in Figure 8. As can be seen in

Figures 7 and 8, decreasing the value of m leads to increase in the
transient response speed. However, the oscillations in the
transient response and the initial value of the control signal are
decreased by increasing the value of m . This means that selecting
an appropriate value for m depends on the design requirements.
In Example 3, a plant with the following transfer function is
considered:
5
GðsÞ ¼ (29)
s2 1 2s 1 10

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Mohammad Tabatabaei Volume 16 · Number 3 · 2019 · 351–356

Figure 10 The control signals for different values of m in Example 3 swarm optimization”, International Journal of Control,
Automation and Systems, Vol. 15 No. 2, pp. 918-932.
Grimholt, C. and Skogestad, S. (2015), “Improved optimization-
based design of PID controllers using exact gradients”,
Computer Aided Chemical Engineering, Vol. 37, pp. 1751-1756.
Hast, M., Astrom, K.J., Bernhardsson, B. and Boyd, S. (2013),
“PID design by convex-concave optimization”, Proceedings
of the European Control Conference, Zurich, pp. 4460-4465.
He, J.B., Wang, Q.G. and Lee, T.H. (2000), “PI/PID
controller tuning via LQR approach”, Chemical Engineering
Science, Vol. 55 No. 13, pp. 2429-2439.
Mikhalevich, S.S., Baydali, S.A. and Manenti, F. (2015),
“Development of a tunable method for PID controllers to
achieve the desired phase margin”, Journal of Process Control,
Vol. 25, pp. 28-34.
Moharam, A., Hosseini, M.A. and Ali, H.A. (2016), “Design
of optimal PID controller using hybrid differential evolution
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Table IV The effect of m on the transient response and the control signal
challengers”, Applied Soft Computing, Vol. 38, pp. 727-737.
Moradi, R. and Tabatabaei, M. (2016), “Proportional integral
PI controller Transient response Transient response Control derivative controller design using Legendre orthogonal
performance speed oscillations effort functions”, Journal of Central South University, Vol. 23
Large values of l Low Low Low No. 10, pp. 2616-2629.
Small values of l High High High Najafizadegan, H., Merrikh-Bayat, F. and Jalilvand, A. (2017),
“IMC-PID controller design based on loop shaping via LMI
approach”, Chemical Engineering Research and Design,
system. The main drawback of these methods is that these Vol. 124, pp. 170-180.
cannot give straightforward relations for the controller Oliveira, P., Freire, H. and Pires, E.J.S. (2016), “Grey wolf
parameters in terms of the plant parameters. To overcome this optimization for PID controller design with prescribed robustness
drawback, this paper presents straightforward relations for PI margins”, Soft Computing, Vol. 20 No. 11, pp. 4243-4255.
controller parameters by minimizing an appropriate performance Perng, J.W., Hsieh, S.C., Ma, L.S. and Chen, G.Y. (2018),
index. Although this analytical method is performed only for “Design of robust PI control systems based on sensitivity
designing PI controllers for the first- and second-order plants, the analysis and genetic algorithms”, Neural Computing and
same approach can be proposed for higher-order plants or other Applications, Vol. 29 No. 4, pp. 913-923.
types of controllers as well. The experimental tests on a DC Saxena, S. and Hote, Y.V. (2016), “Simple approach to design
servomechanism system in a laboratory demonstrate the PID controller via internal model control”, Arabian Journal
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Srivastava, S. and Pandit, V.S. (2017), “A 2-Dof LQR based
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Freire, H., Oliveira, P. and Pires, E.J.S. (2017), “From single Mohammad Tabatabaei can be contacted at: tabatabaei@
to many-objective PID controller design using particle iaukhsh.ac.ir

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