Received June 10, 2021, accepted June 25, 2021, date of publication June 29, 2021, date of current version July 20, 2021.
Digital Object Identifier 10.1109/ACCESS.2021.3093427
Analytic Time Domain Specifications PID
Controller Design for a Class of 2nd Order Linear
Systems: A Genetic Algorithm Method
CHIA-LING LEE
AND CHAO-CHUNG PENG
Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan
Corresponding author: Chao-Chung Peng (ccpeng@mail.ncku.edu.tw)
This work was supported by the Ministry of Science and Technology under Grant MOST 107-2221-E-006-114-MY3 and Grant MOST
108-2923-E-006-005-MY3.
ABSTRACT In this paper, an analytical approach of a Proportional-Integral-Derivative (PID) controller
design for a servo motor position control with precise desired performance demands is presented. When
designing linear controllers, the resulting closed-loop system is often considered as an approximation of a
standard 2nd order system to simplify the gain design process. However, in reality, model discrepancies could
exist in such approximation, which inevitably lead to performance mismatches between the actual system’s
behavior and the desired one. In order to solve this issue, this paper presents an analytical PID controller
solution, which is able to achieve desired time domain specifications precisely. In addition, a disturbance
is added to the system and the associated PID controller will be designed to reject disturbance. Given
a 2nd order linear system along with a PID controller, the related analytical solutions are derived first.
Next, the associated fitness functions and constraints are constructed under given prescribed time domain
specifications. Lastly, the PID control gains are determined using Genetic Algorithm. To demonstrate the
effectiveness of the proposed solution, many numerical simulations are performed. A set of control gains
is also found using the standard 2nd order system formulas in order to prove the accuracy of the presented
method. Moreover, a comparison study with respect to the Matlab PID Tuner is considered to illustrate
the advantage of the proposed PID design scheme. The main contributions of this study include precisely
satisfying given control performance demands with exact analytical solutions, avoiding the existing model
discrepancies between the standard 2nd order system and the exact closed-loop model. Simulations firmly
prove that the proposed method can fulfill users’ different control demands as well as remove the frequently
used trial-and-error strategy.
INDEX TERMS PID analytical solution, genetic algorithm, PID tuning, servo motor control, time domain
specifications.
I. INTRODUCTION
The well-known proportional-integral-derivative (PID) controller is one of the most commonly used control algorithms
in linear/nonlinear systems [1] and in industrial fields [2].
It also plays a significant role in control systems related to
training and education. PID controller has served as one of
the major control methods for several engineering industries
such as mechanical, electrical, chemical, aerospace, and so
forth. The linear servo motor system is also a widely used
The associate editor coordinating the review of this manuscript and
approving it for publication was Nasim Ullah
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.
candidate that the PID was integrated for speed as well as
position control [3]–[5].
Several controllers consisting of extended concepts from
the traditional PID controller have been proposed. These
include fractional-order PID [6], [7], higher-order PID [8],
integer-order PID [9], etc. However, in general, the traditional PID controller is still broadly used because of its
low-cost implementation, feasibility, and robustness. Most
importantly, the traditional PID controller is relatively easy
for students and even engineers to comprehend as well
as apply in real control systems. Therefore, the traditional
PID configuration is considered and verified in this paper.
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
VOLUME 9, 2021
C.-L. Lee, C.-C. Peng: Analytic Time Domain Specifications PID Controller Design
The disturbance rejection, a very popular topic in control systems, is also discussed [10]. The purpose of disturbance rejection is to cope with expected and unexpected forces, which
cause the system to drift away from its desired steady-state
value.
Nevertheless, the most commonly seen issue when using
the PID controller is the selection of gain parameters. To deal
with this problem, many classical control theories such as
root-locus, phase lead/lag, Nyquist, and pole-placement [11]
techniques can be used to assist with the gain determinations. The Ziegler-Nichols rule is also a popular method for
PID tuning [12]. For practical implementations, most of the
methods widely used in industrial processes are based on
experience of field operators, actual experimental results,
or both. Unfortunately, approximated time domain specifications such as overshoot, peak time, and settling time can
roughly be achieved. As a result, fine tunings, which may be
time consuming, are usually unavoidable.
From the viewpoint of control performance evaluations,
time response specifications are generally accepted as the
most forthright way when observing behaviors of control
applications. In order to achieve the desired performance
specifications precisely when applying the traditional PID
controller, in this study, the Genetic Algorithm (GA) is
applied. GA is a search-based technique with the purpose of
solving optimization problems containing linear and nonlinear constraints, of which include PID controller parameter
tuning [13]–[15]. It has also been applied for disturbance
rejection [16]. GA consists of several advantages such as its
ability to avoid trapping in local optima and its wide solution
space search.
The main purpose of this work is to achieve precise time
domain specifications for a class of 2nd order linear systems
controlled by a PID controller. A servo motor model is considered as a plant for demonstration. An external disturbance
is applied to the system. Related time domain analytical
solutions are then derived accordingly. These solutions in
addition to time domain specifications are further considered simultaneously to construct related fitness functions
as well as constraints. Without any solution approximation,
the GA is applied for the optimal PID gain determination.
The main contributions of this paper can be highlighted
as follows: (a) analytical solutions subject to time domain
specifications when applying the PID controller for a class
of 2nd order systems are derived; (b) constrained fitness
functions are then constructed to meet the users’ demands
without approximations; (c) a Genetic algorithm solver is
applied to find the specific control gains, which meet time
domain specifications precisely; (d) different from the existing classical control design methods, the proposed method
is capable of realizing time domain specifications without
the use of trial-and-error strategies. Finally, a comparison
study with respect to the Matlab PID Tuner is considered to illustrate the advantage of the proposed PID tuning
algorithm.
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II. SERVO MOTOR DESCRIPTION AND PARAMETER
IDENTIFICATION
A. SERVO MOTOR DESCRIPTION
The mechanical equation of a servo motor is described as
J θ̈ (t) + C θ̇ (t) = u (t)
(1)
where θ represents the angular position which is available via
the encoder, J denotes the motor inertia and C is the viscous
friction coefficient. Before the design of the PID controller,
all parameters should be available in advance. Therefore,
system parameter identification needs to be addressed.
Furthermore, in order to simulate the real situation for
servo motor control, only the angle position is available via
an encoder. Therefore, the rotation speed ω (k) is estimated
by applying the central difference derivative
θ (k + 1) − θ (k − 1)
(2)
ω (k) =
2Ts
where Ts is the sampling interval.
B. PARAMETER IDENTIFICATION
Consider the bilinear transform discretization
2 1 − z−1
s≈
(3)
Ts 1 + z−1
Substituting (3) into (1) yields the following discrete-time
differential equation
CTs
Ts
ω− (k) = −
ω+ (k) +
u+ (k)
(4)
2J
2J
where
ω− (k) = ω (k) − ω (k − 1)
ω+ (k) = ω (k) + ω (k − 1)
u+ (k) = u (k) + u (k − 1)
(5)
Afterwards, forming (4) as the following compact form
 

ω+ (k)
u+ (k)
ω− (k)
 ω− (k + 1)   ω+ (k + 1) u+ (k + 1)  opt 
 
 X1

=

..
..
..
opt



 X2
.
.
.
| {z }
ω− (k + n)
ω+ (k + n) u+ (k + n)
{z
} |
{z
} X2×1
|

Y(n+1)×1
A(n+1)×2
(6)
where
Ts
CTs
opt
, X2 =
(7)
2J
2J
To find the optimal estimates, apply the least squares
solution
−1
X opt = AT A
AT Y
(8)
opt
X1
=−
Eventually, the parameters of servo motor are obtained by
Ts
opt ,
opt
2Ĵ X1
Ts
(9)
2X2
The estimated parameters will be used in the PID control
gain determination.
Ĵ =
Ĉ = −
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C.-L. Lee, C.-C. Peng: Analytic Time Domain Specifications PID Controller Design
III. DESIGN OF PID CONTROLLER UNDER TIME DOMAIN
SPECIFICATIONS
A. CONTROLLER DESIGN
Consider a standard feedback system consisting of a 2nd
order plant Gp (s) and a controller Gc (s) with the reference
command Rθ (s) as shown in FIGURE 1, where D (s) denotes
the external disturbance.
In terms of the definition of M , N , P, Q, T , U , V , Z and
the detailed derivations, please refer to the Appendix.
B. TIME DOMAIN SPECIFICATIONS
As illustrated in FIGURE 2, time domain specifications for
control system designs often contain certain characteristics
related to the system time response. In this paper, the characteristics for a step response focus on overshoot Mp , peak
time tp , and settling time ts .
FIGURE 1. System structure.
Define the plant as a servo motor, of which the transfer
function (TF) is
1
(10)
s (Js + C)
The reference command as a step input
r
(11)
Rθ (s) =
s
is considered in this work where r is the input size. Thus, the
system error can be expressed as E(s) = Rθ (s) − θ (s).
A disturbance as a step function
d
D (s) = e−τ s
(12)
s
is applied where d is the input size. Note that τ denotes the
time-delay coefficient.
The TF of the PID controller Gc (s) is
kI
Gc (s) = kP +
+ kD s
(13)
s
where kP is the proportional gain, kI represents the integral
gain, and kD denotes the derivative gain.
Hence, the closed-loop TF can be written as
Gc Gp
Gp
θ (s) =
Rθ (s) +
D (s)
(14)
1 + Gc Gp
1 + Gc Gp
The output solution for reference command and disturbance will be derived separately.
The output solution for the reference command is
θ1 (t) = r − Peλ1 t − e−Mt [Q cos (Nt) + T sin (Nt)] (15)
The output solution introduced by the disturbance is
Ueλ1 t 0 + Ve−Mt 0 cos Nt 0
0
θ2 (t) = H t
(16)
0
+Ze−Mt sin Nt 0
where t 0 = t − τ . H represents the Heaviside step function
(
0, t < τ
H (t − τ ) =
(17)
1, t ≥ τ
Combining both solutions derived from reference command and disturbance yields the analytical solution
θ (t) = θ1 (t) + θ2 (t)
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(18)
FIGURE 2. Design of overshoot, peak time and settling time.
The definition of overshoot Mp is the maximum value
the response overshoots the steady-state value. Peak time tp
is the time required for the response to reach the point of
maximum overshoot. Due to the effects from disturbances,
two different values of each specification
will be considered.
In other words, Mp1 , tp1 , ts1 and Mp2 , tp2 , ts2 represent
what happens to the system before and after the disturbance,
respectively.
To express overshoot analytically, recall that
Mp = θ tp − θss /θss × 100%
(19)
where θss is the steady-state value that can be calculated by
θss = lim sθ (s) = κ
(20)
s→0
According to the definition, it is evident that overshoot
occurs when θ̇ (t) is zero. Therefore, taking the derivative
of (18) separately gives
θ̇1 (t)
= −λ1 Pe
λ1 t
−Mt
+e
(QM − NT ) cos (Nt)
+ (QN + TM ) sin (Nt)
(21)
θ̇2 (t)
=H t
0
λ1 Ue
λ1 t 0
+e
−Mt 0
(NZ − MV ) cos Nt 0 − (NV + MZ ) sin Nt 0
(22)
In (21) and (22), it has been assumed that the derivative
of H is zero since the disturbance in (12) is time-invariant.
Peak time tp1,2 is then obtained by satisfying
θ̇1 tp1 = 0 & θ̇2 tp2 − τ = 0
(23)
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C.-L. Lee, C.-C. Peng: Analytic Time Domain Specifications PID Controller Design
Substituting tp1,2 into (19) gives
λ 1 tp
−Mtp Q cos Ntp × 100% (24)
Mp1 = −Pe
−e
+T sin Ntp
λt
Ue 1 p2 + Ve−Mtp2 cos
Ntp2 × 100%
Mp2 =
(25)
+Ze−Mtp2 sin Ntp2
Eq. (24)-(25) is the analytical solution for overshoot.
Settling time ts is the time it takes for the time response
curve to reach and stay within a particular range of the
steady-state value of the system.
θ (ts ) = θss × (100 ± α) %
(26)
where α represents different values of percentages that can be
applied in the following derivations.
IV. GENETIC ALGORITHM FORMULATION FOR PID
PARAMETERS DESIGN
Based on the derivations shown in Section III, it can be found
that design of PID control gains to precisely fulfill the time
domain specifications is not a simple task.
To satisfy the nonlinear equations given the time domain
specifications, the GA is applied for PID gain searching.
The fitness function and constraints are formed to ensure the
system fulfill users’ desired performance demands including peak time tp,d , overshoot Mp,d , and settling time ts,d .
In the following segments, three different cases will be discussed separately with their corresponding fitness function
and constraints.
A. REFERENCE COMMAND WITHOUT DISTURBANCE
In this case, the reference command in (11) is considered.
The input size of disturbance is set to be 0.
D (s) = 0
(27)
The desired specifications are tp,d1 , Mp,d1 , ts,d1 .
The fitness function can be expressed as
θ1 (t) − θss
(28)
where t = ts,d1 + ρ. ρ is the tolerance of the time required for
the system to reach its steady-state after the desired settling
time (5% of ts,d1 is recommended).
The associated constraints are shown as follows.
θ̇1 tp,d1 = 0
(29)
θ1 tp,d1 − Mp,d1 = 0
(30)
θ1 ts,d1 − θss − α 100 = 0
(31)
|θ1 (t) − θss | − α 100 < 0
(32)
where (29) ensures that the desired peak time satisfies its
definition expressed in (23), (30) is applied based on (24),
(31) is set to fulfill the settling time requirement. Note that
(32) must be guaranteed when t > ts,d1 to ensure the time
response curve will indeed stay within the α% range.
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B. PURE DISTURBANCE REJECTION
In this case, the problem turns into a regulation problem,
which means the input size of the reference command is 0.
Rθ (s) = 0
(33)
The disturbance in (12) is applied.
Note that the steady-state value θss is recalculated by
θss = lim sθ (s) = 0
s→0
(34)
The desired specifications are tp,d2 , Mp,d2 , ts,d2 . The
time-delay coefficient τ is also considered.
The fitness function can be expressed as
θ2 (t) − θss
where t = ts,d2 + ρ.
The associated constraints are shown as follows.
θ̇2 tp,d2 − τ = 0
θ2 tp,d2 − τ − Mp,d2 = 0
θ2 ts,d2 − τ − θss = α 100
|θ2 (t) − θss | − α 100 < 0
(35)
(36)
(37)
(38)
(39)
where (36)-(38) share the same feature as (29)-(31). In (39),
t > ts,d2 is applied.
C. POSITIONING IN THE PRESENCE OF EXTERNAL
DISTURBANCE
This case is a combination of two previous cases. The reference command in (11) and the disturbance in (12) are
considered.
The specifications Mp,d1 , tp,d1 , ts,d1 and Mp,d2 , tp,d2 , ts,d2
are applied before and after the disturbance happens, respectively. The time-delay coefficient is τ .
The fitness function can be expressed as
θ1 (t) + θ2 (t − τ ) − θss
(40)
where t = ts,d1 + ρ.
The associated constraints are shown as follows.
θ̇1 tp,d1 = 0 & θ̇2 tp,d2 − τ = 0
(41)
θ1 tp,d1 − Mp,d1 < 0 & θ2 tp,d2 − τ − Mp,d2 < 0 (42)
(
|θ1 (t) − θss | − α 100 < 0,
t = ts,d1 + ρ1
θ1 ts,d1 + θ2 (t) − θss − α 100 < 0, t = ts,d2 + ρ2
(43)
In (43), the tolerance ρ1,2 guarantees that the time response
curve will indeed stay within the α% range when t > ts,d2 .
Note that the value of ρ1 has a constraint of
ρ1 < τ − ts,d1
(44)
to ensure it doesn’t conflict with the physical response after
the appearance of disturbance.
Since there are only three variables to meet six specifications, the constraints in this case are set differently from
previous cases. The constraints in (42) serve to let the system
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C.-L. Lee, C.-C. Peng: Analytic Time Domain Specifications PID Controller Design
have overshoots that are less than the desired overshoots,
instead of being the exact same as previous cases. In addition,
the constraints for settling time are not as strict. The equation
that ensures the settling time of the system will be the exact
desired settling time, such as (31) and (38), has been removed.
In terms of upper bound and lower bound in GA, it is
important to choose the correct range. Incorrect selection will
decrease the efficiency when searching for feasible solutions
or even lead to infeasible solutions. Therefore, kP , kI , kD ∈
[0, ∞] is applied for all three cases in this paper to guarantee
that the controller parameters are physically feasible.
V. NUMERICAL SIMULATIONS
In this section, numerical simulations for both with and without disturbances are performed to verify the work mentioned
in previous sections.
A flowchart is made and shown in FIGURE 3 for the
purpose of demonstrating how the controller design process
works.
Step 3 (Problem Formulation Applying GA): Based on the
demands in (48), the fitness function is
θ1 (t) − θss
(49)
where t = ts,d + 0.05 × ts,d = 1.05. A 5% of ts,d is chosen
for the tolerance ρ here.
According to (48), the resulting constraints are expressed as
θ̇1 (0.5) = 0
θ1 (0.5) − 12% = 0
|θ1 (1) − θss | − 0.02 = 0
|θ1 (t) − θss | − 0.02 < 0
(50)
(51)
(52)
(53)
where (53) is the associated constraint when t > 1.
Regarding the PID gain searching, the Matlab Genetic
Algorithm Toolbox is applied. Note that several optimization options are adjusted to ensure the accuracy of the optimal solution. These include setting constraint tolerance, max
stall generations, max generations, and function tolerance.
Their parameters are set to be 10−2 , 150, 500 and 10−3 ,
respectively.
The parameters returned by the GA are shown as follows.
(kP , kI , kD ) = (8.2588, 0.5760, 1.8459)
(54)
Therefore, the PID controller that meets the desired performance demands in (54) is
FIGURE 3. Procedure for designing PID controllers using GA.
Dc (s) = 8.2588 +
Step 1 (Parameters Identification): Consider the system
structure in FIGURE 1. The reference parameters J = 0.3
and C = 0.5 are chosen. Simulink in Matlab is used to
generate measurement data for parameter identification.
By following the procedure shown in Section II, a set of
estimated parameters Ĵ = 0.3018 and Ĉ = 0.5045 are
obtained. The plant for PID controller design is shown as
follows.
1
G (s) =
(45)
s (0.3018s + 0.5045)
0.576
+ 1.8459s
s
(55)
Step 4 (Validate the Result With Numerical Simulations):
The simulation displayed in FIGURE 4 illustrates the time
response and its time domain specifications with the plant
described in (54) and the controller form in (55).
A. REFERENCE COMMAND WITHOUT DISTURBANCE
Let the input size of the reference command be 1.
1
Rθ (s) =
(46)
s
Note that the disturbance is not included in this case.
That is,
D(s) = 0
(47)
Step 2 (Consider Desired Demands): The following
desired performance demands are applied.
tp,d = 0.5, Mp,d = 12%, ts,d = 1, α = 2%
(48)
The goal is to find a PID controller which makes the
parameters kP , kI , kD meet the specifications in (48) precisely
without any deviation.
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FIGURE 4. Numerical simulation applying the proposed GA based PID
control parameter design scheme.
It is obvious that the time response is able to reach the
desired demands in (48) precisely.
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C.-L. Lee, C.-C. Peng: Analytic Time Domain Specifications PID Controller Design
For comparison, the classical standard 2nd order formulas
are also applied to show the influence caused by model
discrepancies.
Consider the general form of a standard 2nd system
G2nd (s) =
ωn2
s2 + 2ζ ωn + ωn2
(56)
where ζ and ωn stand for damping ratio and natural frequency,
respectively.
The well-known standard 2nd order system formulas for
time domain specifications including overshoot and settling
time are provided in (57).
!
4
πζ
, ts =
Mp = exp − p
(57)
2
ζ
ω
n
1−ζ
To approximate a 3rd order system expressed in (14) as a
2nd order system, rewrite (14) as
DPID (s)
θ (s)
=
· G2nd (s)
Rθ (s)
s − λ01
FIGURE 5. Numerical simulation applying the standard 2nd order system
formulas.
(58)
where DPID (s) consists of PID control gains, λ01 represents
the root to be determined, and G2nd (s) denotes the remaining
2nd order transfer function as shown in (56).
Applying the desired performance demands Mp,d and ts,d
to (57) gives the corresponding ζ and ωn . Therefore, a standard 2nd order system which satisfies the desired demands is
obtained.
Afterwards, with the purpose of approximating the system
behavior of (59) as a 2nd order system, the dominant pole
approximation is applied.
Rewrite (58) as a form with two roots
θ(s)
DPID (s)
ωn2
=
·
0
0
Rθ (s)
s − λ1
s − λ2 s − λ03
(59)
To allow λ02 and λ03 to become the dominant poles in this
system, let λ01 satisfy the following requirement.
λ01 = k · max Re λ02 , Re λ03
(60)
where k ∈ R. The value of k decides how much alike this
3rd order system is compared to a 2nd order system. If k is
too small, the system behavior won’t be close enough to a
2nd order system; on the other hand, if k is too big, it would
lead to high gain control. For generality, k ∈ [10, 20] is
recommended.
In this simulation,
k = 15 is chosen. Therefore, a set of
kP0 , kI0 , kD0 = (112.0061, 617.2041, 13.9819) is obtained.
The simulation result is shown in FIGURE 5. It is evident
that even though the desired overshoot can be achieved, there
exist unacceptable design mismatches for both peak time and
settling time. This is because when a 3rd order system is
approximated as a 2nd order system, only the denominator
part is considered. However, when coping with real systems,
the numerator also contains control gains that can affect system behaviors, leading to unexpected time response. By this
comparison study, it firmly demonstrates that the proposed
VOLUME 9, 2021
FIGURE 6. MATLAB PID Tuner Toolbox for comparison study.
GA based PID control gain design is able to precisely achieve
time domain specifications.
Then, we further consider the comparison study with the
Matlab PID Tuner from the view point of end-users. The
interface of the PID Tuner provided by Matlab R2018b is
shown in FIGURE 6, where FIGURE 6(a) displays the user
interface and FIGURE 6(b) illustrates the performance tuning
variables provided by the PID Tuner.
The comparison of the simulation results from the Matlab
PID Tuner and the proposed method is shown in FIGURE 7.
FIGURE 7 shows that the simulation result obtained from
the Matlab PID Tuner fails to meet the desired specifications
simultaneously. This is because only two adjustable tuning
tools are provided in the PID Tuner Toolbox. Moreover,
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C.-L. Lee, C.-C. Peng: Analytic Time Domain Specifications PID Controller Design
FIGURE 7. Numerical simulation applying the MATLAB PID tuner.
FIGURE 8. Numerical simulation applying the proposed GA based PID
control parameter design scheme.
while it is possible for users to achieve two of the desired
specifications, the process requires a certain amount of time
with trial-and-error. The features in the toolbox don’t allow
users to acquire the time domain specifications instantly,
which means the users have to adjust the parameters based on
visual observation. Thus, not only is this method more timeconsuming, the result is also less precise compared to the proposed method. On the contrary, as illustrated in FIGURE 4,
it shows that the desired time domain specifications can be
simultaneously achieved without any trial-and-error.
In order to show that the proposed method can be applied
to meet various desired specifications, a set of parameters to
fulfill aggressive response demands is chosen.
tp,d = 0.5, Mp,d = 50%, ts,d = 3, α = 2%
(61)
Based on (61), the fitness function is
θ1 (t) − θss
(62)
where t = ts,d + 0.05 × ts,d = 3.15.
According to (61), the resulting constraints are expressed as
θ̇1 (0.5) = 0
θ1 (0.5) − 50% = 0
|θ1 (3) − θss | − 0.02 = 0
|θ1 (t) − θss | − 0.02 < 0
(63)
(64)
(65)
(66)
where (66) is the associated constraint when t > 3.
The parameters returned by the GA are shown as follows.
(kP , kI , kD ) = (10.0891, 18.4405, 0.9522)
(67)
The simulation displayed in FIGURE 8 illustrates the time
response and its time domain specifications with the plant
described in (45) and the controller form in (67).
All the simulation verifications demonstrate that the proposed method is able to successfully achieve users’ time
domain demands precisely. The main advantage is that the
time domain specifications are able to be satisfied simultaneously via the derived analytical solution, where the associated
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PID parameters are solved by the proposed fitness functions.
Therefore, it saves quite a lot of time for trial-and-error
tuning.
B. PURE DISTURBANCE REJECTION
The input size of the reference command is 0.
Rθ (s) = 0
(68)
Consider a step disturbance with the size of 1 and let the
time-delay coefficient be 2.
1
(69)
s
Step 1 (Consider Desired Demands): The following
desired performance demands are applied.
D(s) = e−2s
tp,d2 = 0.5 + 2, Mp,d2 = 5%, ts,d2 = 1 + 2
(70)
The goal is to find a PID controller which makes the
parameters kP , kI , kD meet the specification in (70) precisely
without any approximation.
Step 2 (Problem Formulation Applying GA): Based on the
demands in (70), the fitness function is
θ2 (t) − θss
(71)
where t = ts,d + 0.05 × ts,d − 2 = 3.05. A 5% of ts,d is
chosen for the tolerance ρ here.
According to (70), the resulting constraints are expressed as
θ̇2 (0.5) = 0
θ2 (0.5) − 5% = 0
|θ2 (1)| = 0.02
|θ2 (t)| − 0.02 < 0
(72)
(73)
(74)
(75)
where (75) is the associated constraint when t > ts,d1 = 3.
Similarly, the Matlab Genetic Algorithm Toolbox is
applied here and the settings are the same as before.
VOLUME 9, 2021
C.-L. Lee, C.-C. Peng: Analytic Time Domain Specifications PID Controller Design
FIGURE 9. Numerical simulation applying GA.
FIGURE 10. Numerical simulation applying GA.
The parameters returned by the GA are shown below.
(kP , kI , kD ) = (13.1783, 31.3647, 3.9490)
(76)
Therefore, the PID controller that meets the desired performance demands in (70) is
31.3647
+ 3.9490s
(77)
s
Step 3 (Validate the Result With Numerical Simulations):
The simulation displayed in FIGURE 9 illustrates the time
response and its time domain specifications with the plant
described in (45) and the controller form in (77).
It is obvious that the time response is able to reach the
desired demands in (70) precisely.
Dc (s) = 13.1783 +
1
(78)
s
Consider a step disturbance with the size of 1 and let the
time-delay coefficient be 2.
Rθ (s) =
1
(79)
s
Step 1 (Consider Desired Demands): The following
desired performance demands are applied.
D(s) = e−2s
Mp,d2 = 5%, ts,d2 = 1.2 + 2 (80)
The goal is to find a PID controller which makes the
parameters kP , kI , kD meet the specification in (80) precisely
without any approximation.
Step 2 (Problem Formulation Applying GA): Based on the
demands in (80), the fitness function is
VOLUME 9, 2021
θ1 (0.3) − 12% < 0 & θ2 (0.5) − 5% < 0
(83)
(
|θ1 (0.8) − θss | − 0.02 = 0
(84)
|θ1 (0.8) + θ2 (3.3) − θss | − 0.02 = 0
(
|θ1 (t) − θss | − 0.02 < 0,
t = 0.8 + δ1
(85)
|θ1 (0.8) + θ2 (t) − θss | − 0.02 < 0, t = 3.3 + δ2
where (85) is the associated constraint when t > ts,d1 = 0.8.
The parameters returned by the GA are shown as follows.
(86)
21.2365
+ 4.2656s
(87)
s
Step 3 (Validate the Result With Numerical Simulations):
The simulation displayed in FIGURE 10 illustrates the time
response and its time domain specifications with the plant
described in (45) and the controller form in (87).
It is obvious that the time response is able to meet the
desired demands in (80) precisely.
Dc (s) = 19.6220 +
VI. CONCLUSION
Mp,d1 = 12%, ts,d1 = 1
θ1 (t) + θ2 (t − 2) − θss
(82)
Therefore, the PID controller that meets the desired performance demands in (80) is
Let the input size of the reference command be 1.
tp,d2 = 0.5 + 2,
θ̇1 (0.3) = 0 & θ̇2 (0.5) = 0
(kP , kI , kD ) = (19.6220, 21.2365, 4.2656)
C. POSITIONING IN THE PRESENCE OF EXTERNAL
DISTURBANCES
tp,d1 = 0.3,
where t = ts,d + 0.05 × ts,d = 0.84 (5% of ts,d is chosen for
the tolerance ρ here).
According to (80), the resulting constraints are expressed as
(81)
In this paper, we present analytical solutions of PID control
gain design for a class of 2nd order linear systems. The main
contribution of this work is that the designed PID control
gains are able to meet users’ demands precisely without
time-exhausted trial-and-error. From the application point of
view, the analytical solutions of PID control with the 2nd
order servo motor plant and a step disturbance are derived.
Moreover, the time domain specifications including peak
time, overshoot, and settling time are given as performance
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C.-L. Lee, C.-C. Peng: Analytic Time Domain Specifications PID Controller Design
demands. Three different scenarios including reference command without disturbance, pure disturbance rejection, reference command with disturbance are discussed individually.
GA is applied by using the proposed analytical solutions
as fitness function as well as constraints to determine the
controller parameters. Finally, numerical simulations are performed to verify the effectiveness of the proposed method.
The result shows that the designed PID controller can successfully reach the prescribed performance demands. Thus,
the position behavior of the motor, with and without disturbance, including both transient and steady-state are assured.
A comparison study between the proposed method and standard 2nd order system formulas as well as Matlab PID Tuner is
also provided to highlight the advantages of the proposed PID
design scheme. The result shows that the developed method
is able to achieve users’ demands precisely without the need
of trial-and-error. The proposed design procedures can be
applied to a wide range of related control topics and is suitable
for educational purposes of PID control system design. In
future works, the artificial neural network may be considered
for the design of PID parameters for nonlinear systems as well
as suppression of varying external loads.
The output solution regarding reference command is
s2 +
(C+kD ) 2
s
J
C
Js
+
kP
kI
J s+ J
(88)
Examining the characteristic equation,
kI
(C + kD ) 2 kP
s + s+
=0
(89)
C (s) = s3 +
J
J
J
the roots are
1
10
λ1 = −
b+G+
(90)
3a
G
"
√
#
1
1
3
10
10
λ2,3 = −
b+ −
G+
±j
G−
3a
2
G
2
G
(91)
where the parameters used in (90) and (91) are listed as
follows
C + kD
kP
kI
a = 1, b =
, c= , d = ,
J
J
J
v
q
u
u
3 11 ±
121 − 4130
t
G=
,
2
10 = b2 − 3ac, 11 = 2b3 − 9abc + 27a2 d
(92)
Applying partial fraction expansion gives
θ1 (s) =
r
P
Qs + R
−
−
s
s − λ1
(s − λ2 ) (s − λ3 )
where
P=
99274
C
+ λ1
J
λ2 λ3 −1
λ1 − λ2 − λ3 +
,
λ1
R=
λ2 λ3
P
λ1
(94)
Rewrite (93) as
θ1 (s) =
P
Q (s + M ) s + (R − QM )
r
(95)
−
−
s
s − λ1
(s + M )2 + N 2
of which
!1/2
λ2 + λ3
(λ2 + λ3 )2
M = s−
, N = λ2 λ3 −
(96)
2
4
Taking the inverse Laplace of (95) gives
θ1 (t) = r − Peλ1 t − e−Mt [Q cos (Nt) + T sin (Nt)]
(93)
(97)
where
R − QM
(98)
N
Eq. (97) is the analytical solution for reference command.
Similarly, the output solution regarding disturbance is
T =
θ2 (s) =
Js3
d
e−τ s
+ (C + kD ) s2 + kP s + kI
Applying partial fraction expansion gives
Vs + W
U
+ 2
θ2 (s) =
e−τ s
s − λ1
s − (λ2 + λ3 ) s + λ2 λ3
APPENDIX
r
θ1 (s) = −
s
s3 +
Q = 1 − P,
(99)
(100)
of which
λ2 λ3 −1
d
λ1 − λ2 − λ3 +
,
U =
J λ1
λ1
λ2 λ3
d
W =
U−
λ1
J λ1
V = −U ,
(101)
Rewrite (100) as
U
V (s + M ) s + (W − VM ) −τ s
θ2 (s) =
+
e
(102)
s − λ1
(s + M )2 + N 2
Taking inverse Laplace yields
Ueλ1 t 0 + Ve−Mt 0 cos Nt 0
0
0
θ2 t = H t
0
+Ze−Mt sin Nt 0
(103)
where
W − VM
(104)
N
Eq. (103) is the analytical solution for disturbance.
t 0 = t − τ,
Z=
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VOLUME 9, 2021
CHIA-LING LEE was born in Taoyuan, Taiwan,
in 1998. She is currently pursuing the B.S. degree
with the Department of Aeronautics and Astronautics (DAA), National Cheng Kung University
(NCKU), Tainan, Taiwan. She is a member of the
Intelligent Embedded Control Laboratory, DAA,
NCKU. Her research interests include system
modeling and identification, analysis of dynamics and controls, autonomous systems, and flight
mechanics.
CHAO-CHUNG PENG was born in Kaohsiung,
Taiwan, in 1980. He received the B.S. degree
from the Department of Aeronautics and Astronautics (DAA), National Cheng Kung University
(NCKU), Tainan, Taiwan, in 2003, and the Ph.D.
degree, in 2009. From 2008 to 2009, he was a
Research Assistant with the Department of Engineering, Leicester University, U.K. From 2010 to
2012, he was a Postdoctoral Fellow with the
Department of Mechanical Engineering, NCKU.
He was a Senior Engineer with the Embedded System Development Section,
Measurement and Automation Department, ADLINK Technology, in 2012.
From 2014 to 2016, he was with the Automation and Instrumentation
System Development Section, Iron and Steel Research and Development
Department, China Steel Corporation (CSC). Since 2016, he has been an
Assistant Professor with the Department of Aeronautics and Astronautics,
NCKU. He was promoted as an Associate Professor, in August 2020. His
research interests include high-performance motion control and applications,
unmanned vehicle design, advanced flight control system development,
autonomous robotics, intelligence SLAM technology, system modeling, and
diagnosis. He was awarded a membership in the Phi Tau Phi Scholastic
Honor Society, in 2009, and the Excellent Young Engineering Professor
Award by Chinese Society of Mechanical Engineers (CSME), in 2019.
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