mse 490 final project paper
TRANSCRIPT
A Review of Project Paper #24 A simple method of tuning PID controllers for stable and unstable FOPTD systems (R. Padma Sree, et al.)
MSE 490 – Advanced PID Control
Kjell Sadowski
12/04/2014
Abstract
This paper reviews the PI/PID controller design method proposed by R.Padma Sree, M. N. Srinivas and M.
Chidambaram in their paper titled: A simple method of tuning PID controllers for stable and unstable FOPTD systems.
A derivation of the theory for the proposed method is displayed and further analyzed. Results of the relevant
sections are replicated using simulations in MATLAB to check the correspondence of them to the results
they are reporting. Further analysis will show the discrepancies between results from the simulations and the
ones reported in the paper. A discussion of the results will be presented along with a brief literature review.
Page i
CONTENTS
List of Figures ...................................................................................................................................................................... ii
List of Tables ....................................................................................................................................................................... ii
Glossary ................................................................................................................................................................................ ii
1 Introduction ................................................................................................................................................................ 1
2 Problem Statement .................................................................................................................................................... 1
3 Theory.......................................................................................................................................................................... 2
3.1 Derivation of Proposed Method .................................................................................................................... 2
3.2 Derivation of Two-Parameter PID Tuning Method .................................................................................. 5
3.3 Analysis of Proposed Method ........................................................................................................................ 6
4 Simulation Studies...................................................................................................................................................... 7
4.1 PI Controller ..................................................................................................................................................... 7
4.2 PID Controller ................................................................................................................................................ 10
5 Analysis of Results ................................................................................................................................................... 14
6 Discussion ................................................................................................................................................................. 16
7 Literature survey ...................................................................................................................................................... 16
8 Conclusion ................................................................................................................................................................ 16
9 Bibliography .............................................................................................................................................................. 17
Appendix A: MATLAB Error Calculation Code ......................................................................................................... 18
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LIST OF FIGURES
Figure 1 - PI Controller Set Point Response .................................................................................................................. 8 Figure 2 - PI Controller Regulatory Response ............................................................................................................... 8 Figure 3 - PI Controller Uncertainty in Parameters Regulatory Response ................................................................ 9 Figure 4 - PI Controller Uncertainty in Parameters Set Point Response ................................................................. 10 Figure 5 - PID Controller Set Point Response ............................................................................................................ 11 Figure 6 - PID Controller Regulatory Response.......................................................................................................... 12 Figure 7 - PID Controller Parameter Uncertainty Regulatory Response ................................................................ 13 Figure 8 - PID Controller Parameter Uncertainty Set Point Response ................................................................... 14
LIST OF TABLES
Table 1 - Analysis of System Types ................................................................................................................................. 6 Table 2 - PI Controller Errors .......................................................................................................................................... 7 Table 3 - PI Controller Parameter Uncertainty Errors ................................................................................................. 9 Table 4 - PID Controller Errors ..................................................................................................................................... 11 Table 5 - PID Controller Parameter Uncertainty Errors ........................................................................................... 12 Table 6 - PI Controller Simulation Results Comparison ............................................................................................ 14 Table 7 - PI Controller Parameter Uncertainty Simulation Results Comparison ................................................... 15 Table 8 - PID Controller Simulation Results Comparison ........................................................................................ 15 Table 9 - PID Controller Parameter Uncertainty Simulation Results Comparison ............................................... 15
GLOSSARY
FOPTD – First Order Plus Time Delay
IAE – Integral Absolute Error
IMC – Internal Model Control
ISE – Integral Square Error
ITAE – Integral Time Absolute Error
PID – Proportional Integral Derivative
SOPTD – Second Order Plus Time Delay
Z-N – Ziegler-Nichols
Page 1
1 INTRODUCTION
The main idea behind the development of this method was to design a tuning method for a FOPTD model
which was simple and could be applied to both stable and unstable systems, which would be designed for set
point tracking and would provide an overall good response. This method would be able to be used to
calculate tuning parameters for both PI and PID type controllers.
In this report, the proposed tuning method of the analyzed paper [1] and the results for a stable FOPTD PI
and PID controller will be reproduced and analyzed along with the results from the related simulations.
Although this paper provides an analysis and results for unstable systems, this data will not be presented
because it is not within the scope of this analysis.
2 PROBLEM STATEMENT
The tuning method outlined in this paper was design for closed loop control of a set point tracking problem
or formally known as the Servo problem. The authors report that the “simulation results show that the
method gives a similar response as that of Ziegler-Nichols (Z-N) method and better response than that of
IMC method” [1].
The assumptions that the authors make when deriving the method for determining the controller parameters
are:
The system can be forced to reach the set point: 𝑦(𝑞) 𝑦𝑟(𝑞)⁄ = 1, and
The transfer function is open-loop stable.
The approach which the authors of this paper have taken to calculate the PID parameters is very different
than what many other control systems researchers have derived in the past. Compared to a method like the
Ziegler Nichols Continuous Cycling Method [2] which calculates the parameters from the actual system, these
parameters rely solely on the FOPTD model being as accurate as possible. The controller parameters
calculated from this method will be very sensitive to small changes in the system, which would make this
method difficult to implement in systems that have very dynamic system states.
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3 THEORY
This section shows a complete derivation of the PID parameter equations presented in the analyzed paper [1],
a derivation of the two-parameter tuning equations and an analysis of the equations for determining the PID
parameters.
3.1 DERIVATION OF PROPOSED METHOD This section shows the derivation of the proposed method for the PID Controller parameter equations which
are presented in the analyzed paper [1]. Equations (1) to (14) in the analyzed paper are shown here and do not
correspond to the numbers attached to the equations in this section.
𝐹𝑂𝑃𝑇𝐷 𝑠𝑦𝑠𝑡𝑒𝑚:𝐹(𝑠) =𝑘𝑝𝑒−𝜏𝑑𝑠
𝜏𝑠+1
𝑃𝐼𝐷:𝑢(𝑠)
𝑒(𝑠)= 𝑘𝑐 [1 +
1
𝜏1𝑠+ 𝜏𝐷𝑠]
Closed Loop Transfer Function:
𝐹(𝑠) =𝑘𝑐𝑘𝑝𝑒−𝜏𝑑𝑠(1 +
1𝜏1𝑠
+ 𝜏𝐷𝑠)
[𝑘𝑐𝑘𝑝𝑒
−𝜏𝑑𝑠 (1 +1
𝜏1𝑠+ 𝜏𝐷𝑠)
(𝜏𝑠 + 1)+ 1] (𝜏𝑠 + 1)
(1)
= (𝑘𝑐𝑘𝑝𝑒−𝜏𝑑𝑠 (1 +
1𝜏1𝑠
+ 𝜏𝐷𝑠)
𝑘𝑐𝑘𝑝𝑒−𝜏𝑑𝑠 (1 +1
𝜏1𝑠+ 𝜏𝐷𝑠) + (𝜏𝑠 + 1)
) ∗ (𝑠
𝑠) (2)
= 𝑒−𝜏𝑑𝑠[𝑘𝑐𝑘𝑝𝑠 +
𝑘𝑐𝑘𝑝
𝜏𝐼+ 𝑘𝑐𝑘𝑝𝜏𝐷𝑠2
𝑒−𝜏𝑑𝑠 (𝑘𝑐𝑘𝑝𝑠 +𝑘𝑐𝑘𝑝
𝜏𝐼+ 𝑘𝑐𝑘𝑝𝜏𝐷𝑠2) + 𝜏𝑠2 + 𝑠
(3)
Setting a change of variables: 𝑞 = 𝑠𝜏 → 𝑠 =𝑞
𝜏
= 𝑒−𝜏𝑑𝑞𝜏 [
𝑘𝑐𝑘𝑝𝑞𝜏 +
𝑘𝑐𝑘𝑝
𝜏𝐼+ 𝑘𝑐𝑘𝑝𝜏𝐷
𝑞2
𝜏2
𝑒−𝜏𝑑𝑞𝜏 (
𝑘𝑐𝑘𝑝𝑞𝜏 +
𝑘𝑐𝑘𝑝
𝜏𝐼+ 𝑘𝑐𝑘𝑝𝜏𝐷
𝑞𝜏2
2) +
𝑞𝜏
2+
𝑞𝜏
] ∗ (𝜏𝑒0.5𝜖𝑞
𝜏𝑒0.5𝜖𝑞) (4)
= 𝑒−𝜏𝑑𝑞𝜏 [
𝑘𝑐𝑘𝑝𝑞 +𝑘𝑐𝑘𝑝𝜏
𝜏𝐼+
𝑘𝑐𝑘𝑝𝜏𝑑𝑞2
𝜏
𝑒−𝜏𝑑𝑞𝜏 (𝑘𝑐𝑘𝑝𝑞 +
𝑘𝑐𝑘𝑝𝜏𝜏𝐼
+𝑘𝑐𝑘𝑝𝜏𝑑𝑞2
𝜏 ) + 𝑞(𝑞 + 1)
(5)
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Making the following variable substitutions
𝑘1 = 𝑘𝑐𝑘𝑝
𝑘2 =𝑘1𝜏
𝜏𝐼
𝑘3 =𝑘1𝜏
𝜏𝐷
𝜖 =𝜏𝑑
𝜏
The 𝑒−𝜖𝑞 term is removed from the numerator of equation (5) and separated from the transfer function
because this function only provides a time shift. The closed loop transfer function of an open loop stable
transfer function is analyzed starting with equation (6).
𝑦(𝑞)
𝑦𝑟(𝑞)= 𝑒−𝜖𝑞 [
(𝑘1𝑞 + 𝑘2 + 𝑘3𝑞2)𝑒0.5𝜖𝑞
(𝑘1𝑞 + 𝑘2 + 𝑘3𝑞2)𝑒−0.5𝜖𝑞 + (𝑞 + 1)𝑞𝑒0.5𝜖𝑞
] (6)
Taylor series expansions
𝑒0.5𝜖𝑞 = 1 + 0.5𝜖𝑞 + 0.125𝜖2𝑞2 + 0.0208333𝑞3𝜖^3 (7)
𝑒−0.5𝜖𝑞 = 1 − 0.5𝜖𝑞 + 0.125𝜖2𝑞2 − 0.0208333𝑞3𝜖^3 (8)
Since the objective of the transfer function is to reach the point where the system reaches the set point and
settles there, 𝑦(𝑞) 𝑦𝑟(𝑞)⁄ = 1 will be sets and the analysis will be carried using this result and numerator and
denominator of equation (9) will be equated.
𝑦(𝑞)
𝑦𝑟(𝑞)= 1 = 𝑒−𝜖𝑞 [
(𝑘1𝑞 + 𝑘2 + 𝑘3𝑞2)𝑒0.5𝜖𝑞
(𝑘1𝑞 + 𝑘2 + 𝑘3𝑞2)𝑒−0.5𝜖𝑞 + (𝑞 + 1)𝑞𝑒0.5𝜖𝑞
] (9)
(𝑞2 + 𝑞)𝑒0.5𝜖𝑞 + (𝑘1𝑞 + 𝑘2 + 𝑘3𝑞2)𝑒−0.5𝜖𝑞 = (𝑘1𝑞 + 𝑘2 + 𝑘3𝑞
2)𝑒0.5𝜖𝑞 (10)
𝑞2(𝑒0.5𝜖𝑞 + 𝑘3𝑒−0.5𝜖𝑞) + 𝑞(𝑒0.5𝜖𝑞 + 𝑘1𝑒
−0.5𝜖𝑞) + 𝑘2𝑒−0.5𝜖𝑞
= 𝑞2(𝑘3𝑒−0.5𝜖𝑞) + 𝑞(𝑘1𝑒
−0.5𝜖𝑞) + +𝑘2𝑒−0.5𝜖𝑞
(11)
(0.5𝜖 + 0.125𝜖2 − 0.5𝜖𝑘3 + 0.125𝜖2𝑘1 − 0.0208333𝜖3𝑘2)𝑞3
+ (0.5𝜖 + 𝑘3 − 0.5𝜖𝑘1 + 0.125𝜖2𝑘2 + 1)𝑞2 + (𝑘1 − 0.5𝜖𝑘2 + 1)𝑞 + 𝑘2
= (0.5𝜖𝑘3 + 0.125𝜖2𝑘1 + 0.0208333𝜖3𝑘2)𝑞3
+ (𝑘3 + 0.5𝜖𝑘1 + 0.125𝜖2𝑘2)𝑞2 + (𝑘1 + 0.5𝜖𝑘2)𝑞 + 𝑘2
(12)
𝑞1: 𝑘1 − 0.5𝜖𝑘2 + 1 = 𝑘1 + 0.5𝜖𝑘2
1 = 𝜖𝑘2
𝑘2 =1
𝜖 (13)
Page 4
𝑞2: 0.5𝜖 + 𝑘3 − 0.5𝜖𝑘1 + 0.125𝜖2𝑘2 + 1 = 𝑘3 + 0.5𝜖𝑘1 + 0.125𝜖2𝑘2
0.5𝜖 − 0.5𝜖𝑘1 + 1 = 0.5𝑘1
0.5𝜖 + 1 = 𝜖𝑘1
𝑘1 =1
𝜖+ 0.5 (14)
𝑞3: 0.5𝜖 + 0.125𝜖2 − 0.5𝜖𝑘3 + 0.125𝜖2𝑘1 − 0.0208333𝜖3𝑘2
= 0.5𝜖𝑘3 + 0.125𝜖2𝑘1 + 0.0208333𝜖3𝑘2
0.5𝜖 + 0.125𝜖2 − 0.0208333𝜖3 (1
𝜖) = 0.5𝜖𝑘3 + 0.0208333𝜖3 (
1
𝜖)
0.5𝜖 + 0.08333𝜖2 = 𝜖𝑘3
𝑘3 = 0.5 + 0.08333𝜖 (15)
𝑘1 = 𝑘𝑐𝑘𝑝 =1
𝜖+ 0.5
𝑘𝑐𝑘𝑝 =𝜏
𝜏𝑑+ 0.5 (16)
𝑘2 =𝑘1𝜏
𝜏𝑑=
1
𝜖=
𝜏
𝜏𝑑
𝜏𝐼 = 𝜏𝑑 (1
𝜖+ 0.5)
𝜏𝐼 = 𝜏 + 0.5𝜏𝑑 (17)
𝑘3 =𝑘1𝜏𝐷
𝜏= 0.5 + 0.08333𝜖
𝜏𝐷 =𝜏0.5 + 0.08333𝜏 (
𝜏𝑑𝜏 )
𝑘1=
𝜏0.5 + 0.08333𝜏𝑑𝜏𝜏𝑑
+ 0.5(𝜏𝑑
𝜏𝑑) =
𝜏𝜏𝑑0.5 + 0.08333𝜏𝑑2
𝜏 + 0.5𝜏𝑑
𝜏𝐷 =𝜏𝑑0.5(𝜏 + 0.1667𝜏𝑑)
𝜏 + 0.5𝜏𝑑 (18)
Equations (16), (17) and (18) describe the parameters for the system gain, reset time and derivative time. If
the time delay (𝜏𝑑) and time constant (𝜏) are known, all of these parameters can be calculated.
Page 5
3.2 DERIVATION OF TWO-PARAMETER PID TUNING METHOD This section shows the derivation of the two-parameter controller optimization method presented in the
paper of analysis [1]. This section describes the development of equation (17) in the studied paper.
Starting with equation (12), the left-hand side is multiplied by a constant, α#. q1 is multiplied by α1 and q2 and
q3 are multiplied by α2, which has a linear relation to α1 defined by the relation:
𝛼2 = 𝛽𝛼1 (19)
The expansion of equation (12) follows
(0.5𝜖 + 0.125𝜖2 − 0.5𝜖𝑘3 + 0.125𝜖2𝑘1 − 0.0208333𝜖3𝑘2)𝛼2𝑞3
+ (0.5𝜖 + 𝑘3 − 0.5𝜖𝑘1 + 0.125𝜖2𝑘2 + 1)𝛼2𝑞2 + (𝑘1 − 0.5𝜖𝑘2 + 1)𝛼1𝑞
+ 𝑘2 = (0.5𝜖𝑘3 + 0.125𝜖2𝑘1 + 0.0208333𝜖3𝑘2)𝑞
3
+ (𝑘3 + 0.5𝜖𝑘1 + 0.125𝜖2𝑘2)𝑞2 + (𝑘1 + 0.5𝜖𝑘2)𝑞 + 𝑘2
(20)
𝛼1(𝑘1 − 0.5𝜖𝑘2 + 1) = 𝑘1 + 0.5𝜖𝑘2
𝛼1𝑘1 − 0.5𝜖𝑘2𝛼1 + 𝛼1 = 𝑘1 + 0.5𝜖𝑘2
−𝛼 = 𝑘1 − 𝛼𝑘1 + 0.5𝜖𝑘2 + 0.5𝜖𝑘2𝛼
−𝛼 = (1 − 𝛼1)𝑘1 + (1 + 𝛼1)0.5𝜖𝑘2 (21)
𝛼2(0.5𝜖 + 𝑘3 − 0.5𝜖𝑘1 + 0.125𝜖2𝑘2 + 1) = (𝑘3 + 0.5𝜖𝑘1 + 0.125𝜖2𝑘2)
𝛼2(1 − 0.5𝜖) = 𝑘3 − 𝛼2𝑘3 + 0.5𝜖𝑘1 + 0.5𝜖𝑘1𝛼2 + 0.125𝜖2𝑘2 − 0.125𝜖2𝑘2𝛼2
𝛼2(1 − 0.5𝜖) = (1 − 𝛼2)𝑘3 + 0.5𝜖(1 + 𝛼2)𝑘1 + 0.125𝜖2(1 − 𝛼2)𝑘2 (22)
(1
0.5𝜖) (0.5𝜖 + 0.125𝜖2 − 0.5𝜖𝑘3 + 0.125𝜖2𝑘1 −
0.25
12𝜖3𝑘2)𝛼2
= (0.5𝜖𝑘3 + 0.125𝜖2𝑘1 +0.25
12𝜖3𝑘2)(
1
0.5𝜖)
𝛼2 + 0.25𝜖𝛼 − 𝑘3𝛼 + 0.25𝜖𝑘1𝛼2 −0.25
6𝜖2𝑘2𝛼2 = 𝑘3 + 0.25𝜖𝑘1
𝛼(1 + 0.25𝜖) = 𝑘3(1 + 𝛼2) + 0.25𝜖𝑘1(1 − 𝛼2) +0.25
6𝜖2𝑘2𝛼2 (23)
Page 6
Using equations (21), (22) and (23) to form a matrix of the linear equations:
[
1 − 𝛼1 0.5(1 + 𝛼1) 0
0.5𝜖(1 + 𝛼2) 0.125𝜖2(1 − 𝛼2) 1 − 𝛼2
0.25𝜖(1 − 𝛼2)0.25
6𝜖(1 + 𝛼2) 1 + 𝛼2]
[
𝑘1
𝑘2
𝑘3
] = [
−𝛼1
𝛼2(1 − 0.5𝜖)
𝛼2(1 − 0.25𝜖)] (24)
As stated in the analyzed paper [1], 𝛽 = 0.6 for the PI controller equations and 0.8 for the PID controller
equations. Insufficient parameters are presented in the paper to calculate the equations for the PI controller
so only the equations will be presented.
𝑘𝑐𝑘𝑝 = 0.9719𝜖−0.8915 (25)
𝜏𝐼
𝜏= 0.7719𝜖4 − 3.6608𝜖3 + 6.5791𝜖2 + 5.1652𝜖 + 2.8059 (26)
For the PID controller design, 𝛽 = 0.8 as stated above, 𝛼1 = 1.2 and 𝛼2 is calculated using equation (19).
The 𝜖 value corresponds to the model and is approximately but not exactly 0.5. The equations for the PID
controller developed using this method are:
𝑘𝑐𝑘𝑝 = 1.377𝜖−0.8422 (27)
𝜏𝐼
𝜏= 1.085𝜖0.4777 (28)
𝜏𝐷
𝜏= 0.3899𝜖 + 0.0195 (29)
3.3 ANALYSIS OF PROPOSED METHOD
In order to understand the effect of varying time delay and time constant on equations (16), (17) and (18), the
limits of the gains have been taken and the results are summarized in Table 1.
Table 1 - Analysis of System Types
System Type Limit System Gain Integral Time Derivative Time
Very slow lim𝜏𝑑→∞
𝑓(𝜏, 𝜏𝑑) 𝑘𝑐𝑘𝑝 = 0.5 𝜏𝐼 = 0.5𝜏𝑑 𝜏𝑑 = 1
Very fast lim𝜏𝑑→0
𝑓(𝜏, 𝜏𝑑) 𝑘𝑐𝑘𝑝 = ∞ 𝜏𝐼 = ∞ 𝜏𝑑 = 𝜏𝑑/6
Large time delay lim𝜏→∞
𝑓(𝜏, 𝜏𝑑) 𝑘𝑐𝑘𝑝 = 0.5 𝜏𝐼 = ∞ 𝜏𝑑 = ∞
Short time delay lim𝜏→0
𝑓(𝜏, 𝜏𝑑) 𝑘𝑐𝑘𝑝 = ∞ 𝜏𝐼 = 𝜏 𝜏𝑑 = 1
In the case where a system is very slow, the controller uses small gains to control the system. Derivative
action is the same as the system gain and the integral gain is the inverse of the time delay. So a slow system
with a small time delay will be mainly controlled by the integral action and a slow system with a large time
delay will use a very sluggish controller, where the derivative action will be the driving force.
Page 7
A very fast system will not require a PID controller according to these tuning rules. The integral action
becomes zero and the derivative action becomes very small compared to the system gain so control using the
difference in error, or proportional action, will be enough to control the system.
A system with a very large time delay will be controlled by a PD controller where the derivative action is the
main driving component of the controller.
A system with a very short time delay relies mainly on the system gain and derivative action. Integral action
may also play a major part in the system action if the system has a very small time constant.
It is also interesting to note that the analyzed paper [1] makes no mention of using a filter on the derivative of
the controller; thus, all simulation studies will be formed without using one for the proposed method.
4 SIMULATION STUDIES
This section shows all of the replicated results from the paper of study [1]. All results were replicated using
MATLAB to be as similar as possible to the original figures given that enough information was provided in
the section descriptions. Comments regarding major deviations in the results will be made in Section 5.
4.1 PI CONTROLLER The simulation results of the PI controller development using equations (25) and (26), which were partially
developed in Section 3.2, are shown in this section. Graphs for both the set point response and the regulatory
response are shown. Graphs for parameter uncertainty are also provided where each FOPTD model
parameter is increased by 20% individually.
Table 2 provides the information for the errors and Figure 1 and Figure 2 show the results of the proposed
method compared with the Ziegler-Nichols [3] and Abbas [4] PI controller development methods. Table 3
provides ISE error data for the responses to parameter variations. Figure 3 and Figure 4 are the graphs which
describe these responses.
Table 2 - PI Controller Errors
Method Servo Problem Regulatory Problem
ISE IAE ITAE ISE IAE ITAE
Present 1.4227 2.0192 3.1423 0.3576 0.8125 1.3198 Z-N 1.7699 3.6769 17.7768 0.3070 0.9335 3.4799 Abbas 2.1632 3.5847 10.9859 0.7102 1.9993 7.4831
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Figure 1 - PI Controller Set Point Response
Figure 2 - PI Controller Regulatory Response
Page 9
Table 3 - PI Controller Parameter Uncertainty Errors
Method Servo Problem Regulatory Problem
ISE values for uncertainty in ISE values for uncertainty in
120% kp 120% τ 120% τd 120% kp 120% τ 120% τd
Present 1.4458 1.4840 1.7610 0.4719 0.3176 0.3918 Z-N 1.5626 1.8189 1.8964 0.4156 0.2621 0.3288 Abbas 1.8911 2.2488 2.2849 0.8911 0.6653 0.7342
Figure 3 - PI Controller Uncertainty in Parameters Regulatory Response
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Figure 4 - PI Controller Uncertainty in Parameters Set Point Response
4.2 PID CONTROLLER This section contains the simulation results for the PID controller using no tuning parameters and two tuning
parameters, and comparing them against the results for the Ziegler-Nichols [3] and IMC [5] [6] methods. The
graphs are based off of equations (16), (17) and (18) for the model with no tuning parameters and equations
(27), (28) and (29) for the model with two tuning parameters. Graphs for parameter uncertainty are also
provided where each FOPTD model parameter is increased by 20% individually.
Table 4 provides the errors for the set point and regulatory responses of all four systems and Figure 5 and
Figure 6 show the graphs of the responses. Table 5 provides the ISE error information for the set point
tracking and regulatory responses with parameter variation; Figure 7 and Figure 8 show the graphical
responses.
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Table 4 - PID Controller Errors
Method Servo Problem Regulatory Problem
ISE IAE ITAE ISE IAE ITAE
Presenta 0.6781 0.9691 0.7913 0.0833 0.4247 0.7433 Presentb 0.7520 1.1165 0.9582 0.0966 0.4076 0.5650 Z-N 0.6939 1.0176 0.8876 0.1173 0.5112 0.8511 IMC 0.7146 0.9968 0.7675 0.1202 0.4754 0.6868
a No tuning parameters b Two tuning parameters
Figure 5 - PID Controller Set Point Response
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Figure 6 - PID Controller Regulatory Response
Table 5 - PID Controller Parameter Uncertainty Errors
Method Servo Problem Regulatory Problem
ISE values for uncertainty in ISE values for uncertainty in
120% kp 120% τ 120% τd 120% kp 120% τ 120% τd
Presenta 0.7686 0.6860 0.8961 0.1217 0.0729 0.0897 Presentb 0.8252 0.7800 1.0587 0.1141 0.0946 0.1179 Z-N 0.7750 0.7126 0.9070 0.1504 0.1112 0.1217 IMC 0.7051 0.7591 0.8622 0.1453 0.1132 0.1381
a No tuning parameters b Two tuning parameters
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Figure 7 - PID Controller Parameter Uncertainty Regulatory Response
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Figure 8 - PID Controller Parameter Uncertainty Set Point Response
5 ANALYSIS OF RESULTS
This section presents a comparison of the results of the simulations against those which were provided in the
analyzed paper [1]. The discrepancies which exist are hypothesized to be due to the fact that the analyzed
paper was written 10 years ago. This time discrepancy may allow for the development of better numerical
methods used to evaluate the mapping of transfer functions.
It is also worth noting that the authors of the paper did not provide any information about which
mathematical function was used to test the disturbance rejection of the controller. A square impulse, ramp,
impulse and sinc function were tried but only the ramp impulse provided a response that was stable as well as
visually similar to those produced by the authors.
Table 6, Table 7, Table 8 and Table 9 provide the results comparisons in relative percentages to the numbers
presented in the analyzed paper.
Table 6 - PI Controller Simulation Results Comparison
Method Servo Problem Regulatory Problem
ISE IAE ITAE ISE IAE ITAE
Present -2% -1% -4% -51% -47% -71% Z-N 0% 0% -1% -75% -75% -86% Abbas 0% 0% -1% -56% -44% -59%
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Table 7 - PI Controller Parameter Uncertainty Simulation Results Comparison
Method Servo Problem Regulatory Problem
ISE values for uncertainty in ISE values for uncertainty in
120% kp 120% τ 120% τd 120% kp 120% τ 120% τd
Present -4% -1% -1% -51% -55% -57% Z-N -2% 0% -2% -71% -78% -75% Abbas 0% 0% -1% -54% -59% -58%
Table 8 - PID Controller Simulation Results Comparison
Method Servo Problem Regulatory Problem
ISE IAE ITAE ISE IAE ITAE
Presenta 0% 1% 6% -24% -15% -15% Presentb -1% 0% 0% 0% 11% 12% Z-N 0% 0% 3% 16% 20% 32% IMC -4% -4% 3% -31% -20% -28%
a No tuning parameters b Two tuning parameters
Table 9 - PID Controller Parameter Uncertainty Simulation Results Comparison
Method Servo Problem Regulatory Problem
ISE values for uncertainty in ISE values for uncertainty in
120% kp 120% τ 120% τd 120% kp 120% τ 120% τd
Presenta 2% -1% 0% -8% -31% -35% Presentb 0% -1% 0% 20% -18% 30% Z-N 2% -1% 0% 23% 15% -5% IMC -4% -4% -4% -27% -33% -33%
a No tuning parameters b Two tuning parameters
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6 DISCUSSION
Aside from the issue regarding the disturbance rejection modelling described in Section 5, the results of the
system appeared to look very good for PID controllers when compared to the Ziegler-Nichols and IMC
models and for PI controllers when compared to Ziegler-Nichols and Abbas models. The results of the
proposed method do exhibit some high overshoot but fast settling time also.
It should be strongly noted that there are issues with the stability of PID (no tuning and two tuning
parameter models) and PI controllers developed using this method. As seen in Figure 4, Figure 7, and Figure
8, these controllers do not display strong stability characteristics for parameter fluctuation; however, the
disturbance rejection capability of PI controllers does not seem to be affected, as seen in Figure 3.
In order to implement this algorithm, the following conditions should be met:
1. Adaptive control must be used, and
2. The system must not change faster than the adaptive control can recalculated the FOPTD model
Adaptive control will ensure that the values which the controller are using are most appropriate for the
current state of the system and thus ensure the best possible response.
This type of method is suggested only for systems that can tolerate an underdamped response, require a quick
settling time and have the capacity to use adaptive control. This method would not be appropriate for a
system that has highly fluctuating parameters and does not need such a fast settling time. A more
conservative and stable method such as IMC may be a better choice for such a system.
7 LITERATURE SURVEY
A survey of the papers provided for this project produced a paper which described the development of a
simple and analytic controller tuning method. The paper titled Simple analytic rules for model reduction and PID
controller tuning by S. Skogestad [7] describes a method which uses the IMC controller framework to create a
simple set of tuning rules for FOPTD and SOPTD systems. These rules are simple equations where the input
parameters are governed by relations which describe the types of output behaviors; for example, 𝜏𝐼 = 8𝜃 will
produce a system that provides “a good trade-off between disturbance response and robustness” [7].
The advantage of this type of method is that the results for the model can be accurately predicted based on
the type of system input. The disadvantage is that there are probably few real-life models which can satisfy
such relations on a consistent basis. In comparison with the method in the analyzed paper [1], the method
proposed by Skogestad provides more predictable results for more rigidly defined models, whereas the
proposed method is more flexible and provides good results but the controller is not as robust.
8 CONCLUSION
This paper has reviewed the theory and simulation results depicted in the paper: A simple method of tuning PID
controllers for stable and unstable FOPTD systems by R. Padma Sree, et al. A detailed derivation of the theory has
been presented to show the steps required to reach the proposed equations along with an analysis of those
equations. The models which were tested in the paper were reviewed and replicated using MATLAB
simulation. The results from these simulations are presented with the discrepancies in the data. The set point
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simulations were easily replicated with slightly better performance but the regulatory response showed much
better results than what was shown in the paper. This is probably due to the fact that the input disturbance
signal was not disclosed in the paper and it therefore had to be estimated. A discussion section presents an
analysis of the results and a suggestion that this method of PID parameter tuning is best for systems which
can handle overshoot, require a fast settling time and have built-in adaptive control. Finally a literature review
is conducted and the current method is compared against a method provided by S. Skogestad [7] which gives
tuning rules that provide predictable outputs for stable systems. When compared against the proposed
method, the proposed method is more flexible but less robust and the S. Skogestad method is more robust
but less flexible.
9 BIBLIOGRAPHY
[1] R. Padma Sree, M. N. Srinivas and M. Chidambaram, "A simple method of tuning PID controllers for
stable and unstable FOPTD systems," Department of Chemical Engineering, Indian Institute of
Technology, Madras, 204.
[2] J. Bennett, A. Bhasin, J. Grant and W. Chung Lim, "PID Tuning Classical - ControlsWiki," 16 10 2007.
[Online]. Available: https://controls.engin.umich.edu/wiki/index.php/PIDTuningClassical#Ziegler-
Nichols_Method.
[3] J. G. Ziegler and N. B. Nichols, "Optimum settings for automatic controllers," ASME Transactions, vol.
64, no. 759, 1942.
[4] A. Abbas, "A new set of controller tuning relations," ISA Transactions, vol. 36, no. 183, 1997.
[5] D. E. Rivera, M. Morari and S. Skogestad, "IMC-PID controller design," Industrial Engineering and
Chemical Process, Design and Development, vol. 25, no. 252, 1986.
[6] Q.-C. Wang, C. C. Hang and X.-P. Yang, "Single loop controller design via IMC principles," Automatica,
vol. 37, no. 2041, 2001.
[7] S. Skogestad, "Simple analytic rules for model reduction and PID controller tuning," Modeling,
Identification and Control, vol. 24, no. 2, pp. 85-120, 2004.
[8] A. Rad, MSE 490: Advanced PID Controllers - Lecture 4, Simon Fraser University: Surrey, 2014.
[9] A. Rad, MSE 490: Advanced PID Controllers - Lecture 3, Surrey: Simon Fraser University, 2014.
[10] G. H. Cohen and G. A. Coon, "Theoretical investigation of retarded control," Transactions of ASME, vol.
75, no. 827, 1953.
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APPENDIX A: MATLAB ERROR CALCULATION CODE
% Calculates the ISE, IAE and ITAE for a curve which
function [ISE, IAE, ITAE] = CurveResults(Curve_t,Curve_y)
if(abs(Curve_y(end)) < 0.1) set_point = 0; %Set point for a disturbance rejection scenario else set_point = 1; %Set point for a servo response scenario end
% Calculate error error = Curve_y - set_point;
% IAE Calculation temp = cumtrapz(Curve_t,abs(error)); IAE = temp(end); clear temp
% ISE Calculation temp = cumtrapz(Curve_t,error.^2); ISE = temp(end); clear temp;
% ITAE Calculation temp = cumtrapz(Curve_t,Curve_t.*abs(error)); ITAE = temp(end); clear temp;
end