This document proposes a new one-step method for tuning PI/PID controllers based on closed-loop experiments. It derives simple correlations between data from a proportional-only closed-loop step response experiment and PI/PID settings that provide good performance and robustness. Specifically:
1) A proportional-only controller is used to generate a step response with 10-60% overshoot. The gain, overshoot, peak time, and steady-state change are recorded.
2) Simulations show the proposed controller gain is proportional to the proportional gain used in the experiment, with the ratio dependent only on overshoot. Simple equations are derived relating overshoot and peak time to the PI/PID settings.
3
Design of a new PID controller using predictive functional control optimizati...ISA Interchange
An improved proportional integral derivative (PID) controller based on predictive functional control (PFC) is proposed and tested on the chamber pressure in an industrial coke furnace. The proposed design is motivated by the fact that PID controllers for industrial processes with time delay may not achieve the desired control performance because of the unavoidable model/plant mismatches, while model predictive control (MPC) is suitable for such situations. In this paper, PID control and PFC algorithm are combined to form a new PID controller that has the basic characteristic of PFC algorithm and at the same time, the simple structure of traditional PID controller. The proposed controller was tested in terms of set-point tracking and disturbance rejection, where the obtained results showed that the proposed controller had the better ensemble performance compared with traditional PID controllers.
Disturbance Rejection with a Highly Oscillating Second-Order Process, Part I...Scientific Review SR
This research paper aims at investigating disturbance rejection associated with a highly oscillating
second-order process. The PD-PI controller having three parameters are tuned to provide efficient rejection of a
step input disturbance input. Controller tuning based on using MATLAB control and optimization toolboxes.
Using the suggested tuning technique, it is possible to reduce the maximum time response of the closed loop
control system to as low as 0.0095 and obtain time response to the disturbance input having zero settling time.
The effect of the proportional gain of the PD-PI controller on the control system dynamics is investigated for a
gain ≤ 100. The performance of the control system during disturbance rejection using the PD -PI controller is
compared with that using a second-order compensator. The PD-PI controller is superior in dealing with the
disturbance rejection associated with the highly oscillating second-order process
Controller Tuning for Integrator Plus Delay Processes.theijes
A design method for PID controllers based on internal model control (IMC) principles, direct synthesis method (DS), stability analysis (SA) method for pure integrating process with time delay is proposed. Analytical expressions for PID controllers are derived for several common types of process models, including first order and second-order plus time delay models and an integrator plus time delay model. Here in this paper, a simple controller design rule and tuning procedure for unstable processes with delay time is discussed. Simulation examples are included to show the effectiveness of the proposed method
Closed-loop step response for tuning PID fractional-order filter controllersISA Interchange
Analytical methods are usually applied for tuning fractional controllers. The present paper proposes an empirical method for tuning a new type of fractional controller known as PID-Fractional-Order-Filter (FOF-PID). Indeed, the setpoint overshoot method, initially introduced by Shamsuzzoha and Skogestad, has been adapted for tuning FOF-PID controller. Based on simulations for a range of first order with time delay processes, correlations have been derived to obtain PID-FOF controller parameters similar to those obtained by the Internal Model Control (IMC) tuning rule. The setpoint overshoot method requires only one closed-loop step response experiment using a proportional controller (P-controller). To highlight the potential of this method, simulation results have been compared with those obtained with the IMC method as well as other pertinent techniques. Various case studies have also been considered. The comparison has revealed that the proposed tuning method performs as good as the IMC. Moreover, it might offer a number of advantages over the IMC tuning rule. For instance, the parameters of the frac- tional controller are directly obtained from the setpoint closed-loop response data without the need of any model of the plant to be controlled.
The objective of the paper is to investigate the possibility of using a 2DOF controller in disturbance rejection associated with delayed double integrating processes. The effect of time delay of the process in a range between 0.1 and 0.9 seconds is considered. The controller is tuned using MATLAB optimization toolbox with three forms of the objective function in terms of the error between the step time response of the closed-loop control system and the desired zero value. Using the proposed controller with the fractional delayed double integrating process indicates the robustness of the controller in the time delay range used. The 2DOF controller is able to complete with the PID plus first-order lag controller , but it can not compete with other types of controllers such as the I-PD and PD-PI controllers..
Keywords — Delayed double integrating process, 2DOF controller, controller tuning, MATLAB optimization toolbox, Control system performance.
Improving Structural Limitations of Pid Controller For Unstable ProcessesIJERA Editor
PID controllers have structural limitations which make it impossible for a good closed-loop performance to be achieved. A step response with high overshoot and oscillations always results. In controlling processes with resonances, integrators and unstable transfer functions, the PI-PD controller provides a satisfactory closed-loop performance. In this paper, a simple approach to extracting parameters of a PI-PD controller from parameters of the conventional PID controller is presented so that a good closed-loop system performance is achieved. Simulated results from this formation are carried out to show the efficacy of the technique proposed.
Design of a new PID controller using predictive functional control optimizati...ISA Interchange
An improved proportional integral derivative (PID) controller based on predictive functional control (PFC) is proposed and tested on the chamber pressure in an industrial coke furnace. The proposed design is motivated by the fact that PID controllers for industrial processes with time delay may not achieve the desired control performance because of the unavoidable model/plant mismatches, while model predictive control (MPC) is suitable for such situations. In this paper, PID control and PFC algorithm are combined to form a new PID controller that has the basic characteristic of PFC algorithm and at the same time, the simple structure of traditional PID controller. The proposed controller was tested in terms of set-point tracking and disturbance rejection, where the obtained results showed that the proposed controller had the better ensemble performance compared with traditional PID controllers.
Disturbance Rejection with a Highly Oscillating Second-Order Process, Part I...Scientific Review SR
This research paper aims at investigating disturbance rejection associated with a highly oscillating
second-order process. The PD-PI controller having three parameters are tuned to provide efficient rejection of a
step input disturbance input. Controller tuning based on using MATLAB control and optimization toolboxes.
Using the suggested tuning technique, it is possible to reduce the maximum time response of the closed loop
control system to as low as 0.0095 and obtain time response to the disturbance input having zero settling time.
The effect of the proportional gain of the PD-PI controller on the control system dynamics is investigated for a
gain ≤ 100. The performance of the control system during disturbance rejection using the PD -PI controller is
compared with that using a second-order compensator. The PD-PI controller is superior in dealing with the
disturbance rejection associated with the highly oscillating second-order process
Controller Tuning for Integrator Plus Delay Processes.theijes
A design method for PID controllers based on internal model control (IMC) principles, direct synthesis method (DS), stability analysis (SA) method for pure integrating process with time delay is proposed. Analytical expressions for PID controllers are derived for several common types of process models, including first order and second-order plus time delay models and an integrator plus time delay model. Here in this paper, a simple controller design rule and tuning procedure for unstable processes with delay time is discussed. Simulation examples are included to show the effectiveness of the proposed method
Closed-loop step response for tuning PID fractional-order filter controllersISA Interchange
Analytical methods are usually applied for tuning fractional controllers. The present paper proposes an empirical method for tuning a new type of fractional controller known as PID-Fractional-Order-Filter (FOF-PID). Indeed, the setpoint overshoot method, initially introduced by Shamsuzzoha and Skogestad, has been adapted for tuning FOF-PID controller. Based on simulations for a range of first order with time delay processes, correlations have been derived to obtain PID-FOF controller parameters similar to those obtained by the Internal Model Control (IMC) tuning rule. The setpoint overshoot method requires only one closed-loop step response experiment using a proportional controller (P-controller). To highlight the potential of this method, simulation results have been compared with those obtained with the IMC method as well as other pertinent techniques. Various case studies have also been considered. The comparison has revealed that the proposed tuning method performs as good as the IMC. Moreover, it might offer a number of advantages over the IMC tuning rule. For instance, the parameters of the frac- tional controller are directly obtained from the setpoint closed-loop response data without the need of any model of the plant to be controlled.
The objective of the paper is to investigate the possibility of using a 2DOF controller in disturbance rejection associated with delayed double integrating processes. The effect of time delay of the process in a range between 0.1 and 0.9 seconds is considered. The controller is tuned using MATLAB optimization toolbox with three forms of the objective function in terms of the error between the step time response of the closed-loop control system and the desired zero value. Using the proposed controller with the fractional delayed double integrating process indicates the robustness of the controller in the time delay range used. The 2DOF controller is able to complete with the PID plus first-order lag controller , but it can not compete with other types of controllers such as the I-PD and PD-PI controllers..
Keywords — Delayed double integrating process, 2DOF controller, controller tuning, MATLAB optimization toolbox, Control system performance.
Improving Structural Limitations of Pid Controller For Unstable ProcessesIJERA Editor
PID controllers have structural limitations which make it impossible for a good closed-loop performance to be achieved. A step response with high overshoot and oscillations always results. In controlling processes with resonances, integrators and unstable transfer functions, the PI-PD controller provides a satisfactory closed-loop performance. In this paper, a simple approach to extracting parameters of a PI-PD controller from parameters of the conventional PID controller is presented so that a good closed-loop system performance is achieved. Simulated results from this formation are carried out to show the efficacy of the technique proposed.
Design and Implementation of Discrete Augmented Ziegler-Nichols PID ControllerIDES Editor
Although designing and tuning a proportionalintegral-
derivative (PID) controller appears to be conceptually
intuitive, but it can be hard in practice, if multiple (and often
conflicting) objectives such as short transient and high
stability are to be achieved. Traditionally Ziegler Nichols is
widely accepted PID tuning method but it’s performance is
not accepted for systems where precise control is required. To
overcome this problem, the online gain updating method
Augmented Ziegler-Nichols PID (AZNPID) was proposed, with
the amelioration of Ziegler-Nichols PID’s (ZNPID’s) tuning
rule. This study is further extension of [1] for making the
scheme more generalized. With the help of fourth order
Runge-Kutta method, differential equations involved in PID
are solved which significantly improves transient performance
of AZNPID compared to ZNPID. The proposed augmented
ZNPID (AZNPID) is tested on various types of linear processes
and shows improved performance over ZNPID. The results of
the proposed scheme is validated by simulation and also
verified experimentally by implementing on Quanser’s real
time servo-based position control system SRV-02.
OPTIMAL PID CONTROLLER DESIGN FOR SPEED CONTROL OF A SEPARATELY EXCITED DC MO...ijscmcjournal
This paper presents a new approach to determine the optimal proportional-integral-derivative controller
parameters for the speed control of a separately excited DC motor using firefly optimization technique.
Firefly algorithm is one of the recent evolutionary methods which are inspired by the Firefly’s behavior in
nature. The firefly optimization technique is successfully implemented using MATLAB software. A
comparison is drawn from the results obtained between the linear quadratic regulator and firefly
optimization techniques. Simulation results are presented to illustrate the performance and validity of the
design method.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Level Control of Tank System Using PID Controller-A ReviewIJSRD
This paper discusses the review of level control of tank system using PID controller. PID controller use for one or more tank system. PID has fast response. Paper present different methods of level control. Eliminate the steady state error. It is most common way of solving problems of practical control systems.
Design and Implementation of Discrete Augmented Ziegler-Nichols PID ControllerIDES Editor
Although designing and tuning a proportionalintegral-
derivative (PID) controller appears to be conceptually
intuitive, but it can be hard in practice, if multiple (and often
conflicting) objectives such as short transient and high
stability are to be achieved. Traditionally Ziegler Nichols is
widely accepted PID tuning method but it’s performance is
not accepted for systems where precise control is required. To
overcome this problem, the online gain updating method
Augmented Ziegler-Nichols PID (AZNPID) was proposed, with
the amelioration of Ziegler-Nichols PID’s (ZNPID’s) tuning
rule. This study is further extension of [1] for making the
scheme more generalized. With the help of fourth order
Runge-Kutta method, differential equations involved in PID
are solved which significantly improves transient performance
of AZNPID compared to ZNPID. The proposed augmented
ZNPID (AZNPID) is tested on various types of linear processes
and shows improved performance over ZNPID. The results of
the proposed scheme is validated by simulation and also
verified experimentally by implementing on Quanser’s real
time servo-based position control system SRV-02.
OPTIMAL PID CONTROLLER DESIGN FOR SPEED CONTROL OF A SEPARATELY EXCITED DC MO...ijscmcjournal
This paper presents a new approach to determine the optimal proportional-integral-derivative controller
parameters for the speed control of a separately excited DC motor using firefly optimization technique.
Firefly algorithm is one of the recent evolutionary methods which are inspired by the Firefly’s behavior in
nature. The firefly optimization technique is successfully implemented using MATLAB software. A
comparison is drawn from the results obtained between the linear quadratic regulator and firefly
optimization techniques. Simulation results are presented to illustrate the performance and validity of the
design method.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Level Control of Tank System Using PID Controller-A ReviewIJSRD
This paper discusses the review of level control of tank system using PID controller. PID controller use for one or more tank system. PID has fast response. Paper present different methods of level control. Eliminate the steady state error. It is most common way of solving problems of practical control systems.
MODEL BASED ANALYSIS OF TEMPERATURE PROCESS UNDER VARIOUS CONTROL STRATEGIES ...Journal For Research
This paper analyze the temperature process in an empirical model. From the empirical model the system behavior is determined by transfer function and the basic controller strategies Ziegler-Nichols & Cohen-Coon method are implemented in it. With these tuning methods the best control strategies are obtained at the final stage by interfacing the system with NI-myRIO kit.
Optimised control using Proportional-Integral-Derivative controller tuned usi...IJECEIAES
Time delays are generally unavoidable in the designing frameworks for mechanical and electrical systems and so on. In both continuous and discrete schemes, the existence of delay creates undesirable impacts on the underthought which forces exacting constraints on attainable execution. The presence of delay confounds the design structure procedure also. It makes continuous systems boundless dimensional and also extends the readings in discrete systems fundamentally. As the Proportional-IntegralDerivative (PID) controller based on internal model control is essential and strong to address the vulnerabilities and aggravations of the model. But for an real industry process, they are less susceptible to noise than the PID controller.It results in just one tuning parameter which is the time constant of the closed-loop system λ, the internal model control filter factor. It additionally gives a decent answer for the procedure with huge time delays. The design of the PID controller based on the internal model control, with approximation of time delay using Pade’ and Taylor’s series is depicted in this paper. The first order filter used in the design provides good set-point tracking along with disturbance rejection.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
A New Approach for Design of Model Matching Controllers for Time Delay System...IJERA Editor
Modeling of physical systems usually results in complex high order dynamic representation. The simulation and design of controller for higher order system is a difficult problem. Normally the cost and complexity of the controller increases with the system order. Hence it is desirable to approximate these models to reduced order model such that these lower order models preserves all salient features of higher order model. Lower order models simplify the understanding of the original higher order system. Modern controller design methods such as Model Matching Technique, LQG produce controllers of order at least equal to that of the plant, usually higher order. These control laws are may be too complex with regards to practical implementation and simpler designs are then sought. For this purpose, one can either reduce the order the plant model prior to controller design, or reduce the controller in the final stage, or both. In the present work, a controller is designed such that the closed loop system which includes a delay response(s) matches with those of the chosen model with same time delay as close as possible. Based on desired model, a controller(of higher order) is designed using model matching method and is approximated to a lower order one using Approximate Generalized Time Moments (AGTM) / Approximate Generalized Markov Moments (AGMM) matching technique and Optimal Pade Approximation technique. Genetic Algorithm (GA) optimization technique is used to obtain the expansion points one which yields similar response as that of model, minimizing the error between the response of the model and that of designed closed loop system.
PID Control of Runaway Processes - Greg McMillan DeminarJim Cahill
On-line demo / seminar presented by ModelingAndControl.com's Greg McMillan on August 25, 2010.
Recorded version of presentation will be available post live session at: http://www.screencast.com/users/JimCahill/folders/Deminars
Distributed Control System Applied in Temperatur Control by Coordinating Mult...TELKOMNIKA JOURNAL
In Distributed Control System (DCS), multitasking management has been important issues
continuously researched and developed. In this paper, DCS was applied in global temperature control
system by coordinating three Local Control Units (LCUs). To design LCU’s controller parameters, both
analytical and experimental method were employed. In analytical method, the plants were firstly identified
to get their transfer functions which were then used to derive control parameters based on desired
response qualities. The experimental method (Ziegler-Nichols) was also applied due to practicable reason
in real industrial plant (less mathematical analysis). To manage set-points distributed to all LCUs, master
controller was subsequently designed based on zone of both error and set-point of global temperature
controller. Confirmation experiments showed that when using control parameters from analytical method,
the global temperature response could successfully follow the distributed set-points with 0% overshoot,
193.92 second rise time, and 266.88 second settling time. While using control parameters from
experimental method, it could also follow the distributed set-points with presence of overshoot (16.9%), but
has less rise time and settling time (111.36 and 138.72 second). In this research, the overshoot could be
successfully decreased from 16.9 to 9.39 % by changing master control rule. This proposed method can
be potentially applied in real industrial plant due to its simplicity in master control algorithm and presence
of PID controller which has been generally included in today industrial equipments.
Fuzzy gain scheduling control apply to an RC Hovercraft IJECEIAES
The Fuzzy Gain Scheduling (FGS) methodology for tuning the ProportionalIntegral-Derivative (PID) traditional controller parameters by scheduling controlled gains in different phases, is a simple and effective application both in industries and real-time complex models while assuring the high achievements over pass decades, is proposed in this article. The Fuzzy logic rules of the triangular membership functions are exploited on-line to verify the Gain Scheduling of the Proportional-Integral-Derivative controller gains in different stages because it can minimize the tracking control error and utilize the Integral of Time Absolute Error (ITAE) minima criterion of the controller design process. For that reason, the controller design could tune the system model in the whole operation time to display the efficiency in tracking error. It is then implemented in a novel Remote Controlled (RC) Hovercraft motion models to demonstrate better control performance in comparison with the PID conventional controller.
DESIGN OF PID CONTROLLERS INTEGRATOR SYSTEM WITH TIME DELAY AND DOUBLE INTEGR...ijics
In this paper first we investigate optimal PID control of a double integrating plus delay process and compare with the SIMC rules. What makes the double integrating process special is that derivative action is actually necessary for stabilization. In control, there is generally a trade-off between performance and
robustness, so there does not exist a single optimal controller. However, for a given robustness level (here defined in terms of the Ms-value) we can find the optimal controller which minimizes the performance J (here defined as the integrated absolute error (IAE)-value for disturbances). Interestingly, the SIMC PID controller is almost identical to the optimal pid controller. This can be seen by comparing the paretooptimal
curve for J as a function of Ms, with the curve found by varying the SIMC tuning parameter Tc.
Second, design of Proportional Integral and Derivative (PID) controllers based on internal model control (IMC) principles, direct synthesis method (DS), stability analysis (SA) method for pure integrating process with time delay is proposed. The performances of the proposed controllers are compared with the
controllers designed by recently reported methods. The robustness of the proposed controllers for the uncertainty in model parameters is evaluated considering one parameter at a time using Kharitonov’s theorem. The proposed controllers are applied to various transfer function models and to non linear model of isothermal continuous copolymerization of styrene-acrylonitrile in CSTR. An experimental set up of tank
with the outlet connected to a pump is considered for implementation of the PID controllers designed by
the three proposed methods to show the effectiveness of the methods.
DESIGN OF PID CONTROLLERS INTEGRATOR SYSTEM WITH TIME DELAY AND DOUBLE INTEGR...ijcisjournal
In this paper first we investigate optimal PID control of a double integrating plus delay process and compare with the SIMC rules. What makes the double integrating process special is that derivative action is actually necessary for stabilization. In control, there is generally a trade-off between performance and robustness, so there does not exist a single optimal controller. However, for a given robustness level (here defined in terms of the Ms-value) we can find the optimal controller which minimizes the performance J (here defined as the integrated absolute error (IAE)-value for disturbances). Interestingly, the SIMC PID controller is almost identical to the optimal pid controller. This can be seen by comparing the paretooptimal curve for J as a function of Ms, with the curve found by varying the SIMC tuning parameter Tc. Second, design of Proportional Integral and Derivative (PID) controllers based on internal model control (IMC) principles, direct synthesis method (DS), stability analysis (SA) method for pure integrating process with time delay is proposed. The performances of the proposed controllers are compared with the
controllers designed by recently reported methods. The robustness of the proposed controllers for the uncertainty in model parameters is evaluated considering one parameter at a time using Kharitonov’s theorem. The proposed controllers are applied to various transfer function models and to non linear model of isothermal continuous copolymerization of styrene-acrylonitrile in CSTR. An experimental set up of tank with the outlet connected to a pump is considered for implementation of the PID controllers designed by the three proposed methods to show the effectiveness of the methods.
Performance Based Comparison Between Various Z-N Tuninng PID And Fuzzy Logic ...ijsc
The objective of this paper is to compare the time specification performance between conventional
controller and Fuzzy Logic controller in position control system of a DC motor. The scope of this research
is to apply direct control technique in position control system. Two types of controller namely PID and
fuzzy logic PID controller will be used to control the output response. This paper was written to reflect on
the work done on the implementation of a fuzzy logic PID controller. The fuzzy controller was used to
control the position of a motor which can be considered for a general basis in any project design
containing logic control. Motor parameters were taken from a datasheet with respect to a real motor and a
simulated model was developed using Matlab Simulink Toolbox. The fuzzy control was also designed using
the Fuzzy Control Toolbox provided within Matlab, with each rule consisting of fuzzy sets conditioned to
provide appropriate response times with regards to the limitations of our chosen motor. The Fuzzy
Inference Engine chosen for our control was the Mamdani Minimum Inference engine. The results of the
control provided response times suitable for our application.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
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Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
2. Therefore, it is important to have other tuning method based
on the closed-loop experiment which gives better
performance and robustness. In this method it is simple to
obtain the PID tuning parameters in one step for improved
performance while satisfying the other concern during the
closed-loop experiment like reduces the number of trails, and
works for a wide range of processes. IMC-PID Controller
Tuning Rule
The motivation of this section is to develop IMC-PID
controller tuning for first order process with delay. In
process control, a first-order process with time delay is a
common representation of the process dynamics:
-
( )
1
s
ke
g s
s
θ
τ
=
+
(1)
Here k is the process gain, τ lag time constant and θ the time
delay. Most processes in the chemical industries can be
satisfactorily controlled using a PID controller:
( )
1
1c D
I
c s K s
s
τ
τ
= + +
(2)
The other structure of the PID controller like series form of
PID can easily be transform from Eq. (2). The conventional
feedback controller which is equivalent to the IMC controller
can be expressed by following relation.
( )
1
q
c s
gq
=
− %
(3)
where g% denotes the process transfer function, c and q are the
conventional and IMC controller, respectively. The IMC
controller is designed in two steps:
Step 1: The process model g% is decomposed into two parts:
M Ag p p=% (4)
where pm and pA are the portions of the model inverted and
not inverted, respectively, by the controller (pA is usually a
non-minimum phase and contains dead times and/or right
half plane zeros); pA(0)=1.
Step 2: The IMC controller is designed by
-1
Mq p f= (5)
The IMC filter f is usually given as 1( 1)r
cf sτ= + and cτ is an
adjustable parameter which controls the tradeoff between the
performance and robustness; r is selected to be large enough
to make the IMC controller semi-proper. Consider
approximation of the dead time term in Eq. (1) by first order
Pade approximation:
( )
1-
2
( )
1 1
2
k s
g s
s s
θ
θ
τ
=
+ +
(6)
The resulting IMC-PID tuning formula after simplification is
obtain in Eq.(7) for the given process in Eq. (1).
( )
2
2
c
c
K
k
τ θ
τ θ
+
=
+
(7a)
2
I c
θ
τ τ= + (7b)
2
D
τθ
τ
τ θ
=
+
(7c)
The PID controller designed on the basis of the IMC
principle provides excellent set-point tracking, but has a
sluggish disturbance response, especially for processes with
a small θ/τ ratio [4,6,9]. To improve the load disturbance
response Skogestad [6] recommended modifying the integral
time as
I cτ =4(τ +θ) (8)
Therefore, the integral time in Eq.(7b) is modified for the
improved disturbance
I cτ =min , 4(τ +θ)
2
c
θ
τ
+
(9)
τc= θ has been recommend which gives maximum sensitivity
(Ms)=1.70 approximately. The revised tuning method for the
PID controller tuning is given as:
2
3
cK
k
τ θ
θ
+
= (10a)
min , 8
2
I
θ
τ τ θ
= +
(10b)
2
D
τθ
τ
τ θ
=
+
(10c)
I. CLOSED-LOOP EXPERIMENT
This section is devoted for the development of the PI/PID
controller based on the closed-loop data which resembles the
proposed tuning method in Eq.(10). The simplest closed-
loop experiment is probably a setpoint step response (Fig. 1)
where one maintains full control of the process, including the
change in the output variable. The simplest to observe is the
time tp to reach the (first) overshoot and its magnitude, and
this information is therefore the basis for the proposed
method.
The proposed procedure is as follows (Shamsuzzoha and
Skogestad, [1]):
1. Switch the controller to P-only mode (for example,
increase the integral time τI to its maximum value or set the
integral gain KI to zero). In an industrial system, with
bumpless transfer, the switch should not upset the process.
2. Make a setpoint change that gives an overshoot between
0.10 (10%) and 0.60 (60%); about 0.30 (30%) is a good
value. Record the controller gain Kc0 used in the experiment.
Most likely, unless the original controller was quite tightly
tuned, one will need to increase the controller gain to get a
sufficiently large overshoot.
Note that small overshoots (less than 0.10) are not
considered because it is difficult in practice to obtain from
experimental data accurate values of the overshoot and peak
time if the overshoot is too small. Also, large overshoots
(larger than about 0.6) give a long settling time and require
2369
3. more excessive input changes. For these reasons we
recommend using an “intermediate” overshoot of about 0.3
(30%) for the closed-loop setpoint experiment.
3. From the closed-loop setpoint response experiment, obtain
the following values (see Fig. 1):
• Fractional overshoot, (∆yp - ∆y∞) /∆y∞
• Time from setpoint change to reach peak output
(overshoot), tp
• Relative steady state output change, b = ∆y∞/∆ys.
Here the output variable changes are:
∆ys: Setpoint change
∆yp: Peak output change (at time tp)
∆y∞: Steady-state output change after setpoint step test
To find ∆y∞ one needs to wait for the response to settle,
which may take some time if the overshoot is relatively large
(typically, 0.3 or larger). In such cases, one may stop the
experiment when the setpoint response reaches its first
minimum and record the corresponding output, ∆yu.
∆y∞ = 0.45(∆yp + ∆yu) (11)
The details about how to obtaining Eq.(11) is given in
Shamsuzzoha and Skogestad (2010).
II. CORRELATION BETWEEN SETPOINT RESPONSE AND THE
IMC-PID-SETTINGS
The objective of this paper is to provide a one step
procedure in closed-loop for controller tuning similar to the
Shamsuzzoha and Skogestad (2010) and Ziegler-Nichols
(1942) method. Thus, the goal is to derive a correlation,
preferably as simple as possible, between the setpoint
response data (Fig. 1) and the Proposed PID settings in Eq.
(10), initially with the choice τc=θ. For this purpose, we
considered 15 first-order with delay models g(s)=ke-θs
/(τs+1)
that cover a wide range of processes; from delay-dominant to
lag-dominant (integrating):
τ/θ=0.1,0.2,0.4,0.8,1.0,1.5,2.0,2.5,3.0,7.5,10.0,20.0,50.0,100
Since we can always scale time with respect to the time delay
(θ) and since the closed-loop response depends on the
product of the process and controller gains (kKc) we have
without loss of generality used in all simulations k=1 and
θ=1.
For each of the 15 process models (values of τ/θ), we
obtained the PID-settings using Eq. (10) with the choice
τc=θ. Furthermore, for each of the 15 processes we generated
6 closed-loop step setpoint responses using P-controllers that
give different fractional overshoots.
Overshoot= 0.10, 0.20, 0.30, 0.40, 0.50 and 0.60
In total, we then have 90 setpoint responses, and for each of
these we record four data: the P-controller gain Kc0 used in
the experiment, the fractional overshoot, the time to reach
the overshoot (tp), and the relative steady-state change, b =
∆y∞/∆ys.
Controller gain (Kc). We first seek a relationship between
the above four data and the corresponding proposed
controller gain Kc. Indeed, as illustrated in Fig. 2, where we
plot kKc as a function of kKc0 for the 90 setpoint
experiments, the ratio Kc/Kc0 is approximately constant for a
fixed value of the overshoot, independent of the value of τ/θ.
Thus, we can write
c
c0
K
=A
K
(12)
py∆
pt
y∞∆ sy∆
t
0t =
θ
uy∆
sy∆
Fig. 1. Closed-loop step setpoint response with P-only control.
0 20 40 60 80 100 120
0
10
20
30
40
50
60
70
kKc0
kKc
0.10 overshoot
kK
c
=1.1621kK
c0
0.20 overshoot
kK
c
=0.9701kK
c0
0.30 overshoot
kK
c
=0.841kK
c0
0.40 overshoot
kK
c
=0.7453kK
c0
0.50 overshoot
kK
c
=0.6701kK
c0
0.60 overshoot
kK
c
=0.6083kK
c0
Fig. 2. Relationship between P-controller gain kKc0 used in setpoint
experiment and corresponding proposed controller gain (Eq. 10a) kKc.
where the ratio A is a function of the overshoot only. In Fig.
3 we plot the value of A, which is obtained as the best fit of
the slopes of the lines in Fig. 2, as a function of the
overshoot. The following equation (solid line in Fig. 3) fits
the data in Fig. 2 well and given as:
A=[1.55(overshoot)2
-2.159 (overshoot)+1.35] (13)
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4. 0.1 0.2 0.3 0.4 0.5 0.6
0.7
0.8
0.9
1
1.1
overshoot (fractional)
A
y = 1.55*(overshoot)2
- 2.159*(overshoot) + 1.35
Fig. 3. Variation of A with overshoot using data (slopes) from Fig. 2.
0.1 0.3 0.5 0.6
0
0.1
0.2
0.3
0.4
0.5
Overshoot
θ/tp
0.43 (τI1
)
0.305 (τI2
)
τ/θ=0.1
τ/θ=8
τ/θ=100
τ/θ=1
Fig. 4. Ratio between delay and setpoint overshoot peak time (θ/tp) for P-
only control of first-order with delay processes (solid lines); Dotted lines:
values used in final correlations, Shamsuzzoha and Skogestad (2010).
Integral time (τI). It is interesting to find a simple
correlation for the integral time. The proposed method in Eq.
(10b) uses the minimum of two values, it seems reasonable
to look for a similar relationship, that is, to find one value
(τI1 =τ) for processes with a relatively large delay, and
another value (τI2 =8θ) for processes with a relatively small
delay including integrating processes.
(1) Process with relatively large delay: This case arise
when processes have a relatively large delay i.e., τ/θ<8, the
integral action in the proposed tuning rule is to use τI = (τ+
θ/2). Rearrangement of Eq.(10a) is given as
3
2
ckK θ θ
τ
−
= (14)
Adding both the side θ/2 in Eq.(14) and substitute (τ+
θ/2)=τI, we get
I 1.5 ckKτ θ= (15)
In Eq. (15), we also need the value of the process gain k, and
to this effect write
kKc= kKc0.Kc/ Kc0 (16)
Here, the value of the loop gain kKc0 for the P-control
setpoint experiment is given from the value of b:
c0
b
kK =
(1-b)
(17)
Substituting kKc from Eq. (17) and Kc/ Kc0=A into Eq. (15)
and given as
I
b
1.5A
(1-b)
τ θ= (18)
To prove this, the closed-loop setpoint response is ∆y/∆ys =
gc/(1+gc) and with a P-controller with gain Kc0, the steady-
state value is ∆y∞/∆ys = kKc0/(1+kKc0)=b and we derive
Eq.(17). The absolute value is included to avoid problems if
b>1, as may occur for an unstable process or because of
inaccurate data.
It is possible to obtain the value of time delay θ directly from
the closed-loop setpoint response, but usually this is not
always easy task. The reasonable correlation has been
developed by Shamsuzzoha and Skogestad [1] for the θ and
the setpoint peak time tp which is easier to observe.
For processes with a relatively large time delay (τ/θ<8), the
ratio θ/tp varies between 0.27 (for τ/θ= 8 with overshoot=0.1)
and 0.5 (for τ/θ=0.1 with all overshoots). For the
intermediate overshoot of 0.3, the ratio θ/tp varies between
0.32 and 0.50. A conservative choice would be to use
θ=0.5tp because a large value increases the integral time.
However, to improve performance for processes with smaller
time delays, we propose to use θ=0.43tp which is only 14%
lower than 0.50 (the worst case).
In summary, we have for process with a relatively large time
delay:
0.645
(1- )
I p
b
A t
b
τ = (19)
(2) Process with relatively small delay. Shamsuzzoha and
Skogestad [1] method and in this proposed tuning rule have
same integral action for the lag-dominant process. The
integral time for a lag-dominant (including integrating)
process with τ/θ>8 the proposed tuning rule for integral time
gives
τI2=8θ (20)
For τ/θ>8 we see from Fig. (4) that the ratio θ/tp varies
between 0.25 (for τ/θ=100 with overshoot=0.1) and 0.36 (for
τ/θ=8 with overshoot 0.6). We select to use the average value
θ= 0.305tp which is only 15% lower than 0.36 (the worst
case). Also note that for the intermediate overshoot of 0.3,
the ratio θ/tp varies between 0.30 and 0.32. In summary, we
have for a lag-dominant process
I2 pτ =2.44t (21)
Conclusion. Therefore, the integral time τI is obtained as the
minimum of the above two values:
I pτ =min 0.645 , 2.44t
(1- )
p
b
A t
b
(22)
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5. Derivative action (τD): Although a significant number of the
PID controllers switched off their derivative part but proper
use of derivative action can increase stability and improve
the closed-loop performance. The derivative action is very
important for slow moving loops where overshoot is
undesirable e.g., temperature loop. The motivation of this
section is to develop the approach for inclusion of the
derivative action from closed-loop data. In the proposed
study the derivative action is recommended for the process
having 1τ θ ≥ which can give performance improvement.
Substitute the value of 0.5Iτ τ θ= − into 1τ θ ≥ and after
rearrangement the resulting equation is
( )0.5
1Iτ θ
θ
−
≥ (23)
After simplification it is 1.5Iτ θ ≥ and resulting constrain is
1.0ckK ≥ . The corresponding closed-loop condition for the
derivative action is given as:
( )
b
A 1
1-b
≥ (24)
Case I: For approximately integrating process (τ>> θ),
where integral time is τI =8θ and in the closed-loop the time
delay θ= 0.305tp. The derivative time τD1 in Eq. (10c) can be
approximated as
1
0.305
0.15
2 2 2
p
D p
t
t
τθ θ
τ
τ
≈ = = = (25)
Case II: The processes with a relatively large delay, for this
case integral time τI=(τ+0.5θ) and time delay in closed-loop
is θ=0.43tp. For such cases the derivative action is
recommended only if τ/θ ≥ θ. Assuming the case when τ=θ
the τD2 is given from Eq. (10c) as
2 2
2
0.43
0.1433
2 3 3 3
p
D p
t
t
θ θ θ
τ
θ θ θ
≈ = = = =
+
(26)
The derivative action is only recommended for the process
having 1τ θ ≥ and in the closed-loop this criteria is
( )
b
A 1
1-b
≥
.
Summary: The derivative action for both the cases i.e., τD1
and τD2 are approximately same and the conservative choice
for the selection of τD is given as
( )
0.14 1
1-
D p
b
t if A
b
τ = ≥
(27)
Selection of Proportional Controller Gain (Kc0): An
overshoot of around 0.3 is recommended for the proposed
study. Sometimes achieving the P-controller gain (Kc0) via
trial and error which gives the overshoot around 0.3 can be
time consuming.
Therefore, an effective approach to get the value of Kc0
which gives the overshoot around 0.3 is very significant for
the proposed method. It is important to note that this
procedure requires initial information of the first closed-loop
experiment. Let’s assume for the first closed-loop test P-
controller gain of Kc01 is applied and resulting overshoot OS1
is achieved that is between 0.1 to 0.60 but not around 0.30.
Let the target overshoot be OS and the target P-controller
gain be Kc0. In the proposed closed-loop tuning method the
goal is to match the performance with IMC-PID tuning rule
and for this only maintains a constant P gain Kc, regardless
of the overshoot that resulted from the closed-loop setpoint
test. Ideally, Kc should be the same as that determined with
different overshoots from various closed-loop setpoint test
and the resulting correlation is given as:
( ) ( ) ( ) ( )2 2
1 1 01 01.55 OS 2.159 OS 1.35 1.55 OS 2.159 OS 1.35c cK K − + = − +
(28)
The above Eq.(28) gives a general guideline for choosing the
P-controller gain for the next closed-loop setpoint test. As it
is mentioned earlier the proposed method is good agreement
with the IMC-PID for the overshoot around 0.3. Therefore
the overshoot in Eq.(28) is set as 0.3 and after simplification
the gain for the next closed-loop test is recommended as:
( ) ( )( )2
0 1 1 011.19 1.45 OS 2.02 OS 1.27c cK K= − + (29)
It is important to note that we are not keen to achieve the
precise fractional overshoot of 0.3, so few trial is sufficient
to achieve the desire overshoot around 0.3 from above
equation.
III. SIMULATION
The proposed closed-loop tuning method has been tested on
broad class of the process model. It provides the acceptable
controller setting for all cases with respect to both the
performance and robustness. To show the effectiveness of
the proposed method two cases have been shown as a
representative example i.e., integrating with time delay and
higher order process with time delay. The simulation has
been conducted for three different overshoot (around 0.1, 0.3
and 0.6) and are compared with the recently reported method
of Shamsuzzoha and Skogestad[1].
Example 1: ( )
( )( )
2
1
6 1 2 1
s
s e
s s
−
− +
+ +
Example 2:
s
e
s
−
Figure 5 and 6 presents a comparison of the proposed
method with Shamsuzzoha and Skogestad [1] by introducing
a unit step change in the set-point at t = 0 and an unit step
change of load disturbance (at t = 100 for Example 1 and t =
50 for Example 2) at plant input. It is clear from Figure 5
and 6 that the proposed method gives better closed-loop
response for both the high order and integrating processes.
There are significant performance improvements in both the
case for the disturbance rejection while maintaining setpoint
performance.
The overshoot around 0.1 typically gives slower and more
robust PID-settings, whereas a large overshoot around 0.6
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6. gives more aggressive PID-settings. It is good because a
more careful step response results in more careful tunings
settings.
0 50 100 150 200
0.2
0.6
1
1.4
time
OUTPUTy
Shamsuzzoha and Skogestad method with F=1(overshoot=0.119)
Shamsuzzoha and Skogestad method with F=1 (overshoot=0.344)
Shamsuzzoha and Skogestad method with F=1(overshoot=0.608)
Proposed method with F=1 (overshoot=0.119)
Proposed method with F=1 (overshoot=0.344)
Proposed method with F=1 (overshoot=0.608)
Fig. 5. Responses for PID-control of high order process ( )
( )( )2
1
6 1 2 1
s
s e
s s
−
− +
+ +
,
Setpoint change at t=0; load disturbance of magnitude 1 at t=100.
IV. CONCLUSION
A simple approach has been developed for PI/PID controller
tuning by the closed-loop setpoint step using a P-controller
with gain Kc0. The PID-controller settings are then obtained
directly from following three data from the setpoint
experiment:
• Overshoot, (∆yp - ∆y∞) /∆y∞
• Time to reach overshoot (first peak), tp
• Relative steady state output change, b = ∆y∞/∆ys.
If one does not want to wait for the system to reach steady
state, one can use the estimate ∆y∞ = 0.45(∆yp + ∆yu).
The proposed tuning PID tuning method is:
c c0K =K A
I pτ =min 0.645 , 2.44t
(1- )
p
b
A t
b
( )
0.14 1
1-
D p
b
t if A
b
τ = ≥
where, A=[1.55(overshoot)2
-2.159 (overshoot)+1.35]
The proposed method works well for a wide variety of the
processes typical for process control, including the standard
first-order plus delay processes as well as integrating, high-
order, inverse response, unstable and oscillating process.
0 10 20 30 40 50 60 70 80 90 100
0
0.5
1
1.5
2
2.5
3
time
OUTPUTy
Shamsuzzoha and Skogestad method with F=1 (overshoot=0.108)
Shamsuzzoha and Skogestad method with F=1 (overshoot=0.302)
Shamsuzzoha and Skogestad method with F=1 (overshoot=0.60)
Proposed method with F=1 (overshoot=0.108)
Proposed method with F=1 (overshoot=0.302)
Proposed method with F=1 (overshoot=0.60)
Fig. 6. Responses for PID-control of integrating processes
s
e s−
, Setpoint
change at t=0; load disturbance of magnitude 1 at t=50.
V. REFERENCES
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simple and fast closed-loop approach for PID tuning”, Journal of
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