DESIGN OF PID CONTROLLER FOR FOPDT AND IPDT SYSTEM

Transcription

1 DESIGN OF PID CONTROLLER FOR FOPDT AND IPDT SYSTEM A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF Master of Technology (Dual Degree) in Electrical Engineering By RUBEN ANDULNA Deartment of Electrical Engineering National Institute of Technology, Rourkela 205 P a g e

2 DESIGN OF PID CONTROLLER FOR FOPDT AND IPDT SYSTEM A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Technology (Dual Degree) in Electrical Engineering Under the Guidance of Prof. Sandi Gosh By RUBEN ANDULNA (70EE347) Deartment of Electrical Engineering National Institute of Technology, Rourkela 205 National Institute of Technology Rourkela P a g e 2

3 National Institute of Technology Rourkela Certificate This is to certify that the thesis entitled, Design of PID controller for FOPDT & IPDT system submitted by Mr. Ruben andulna in artial fulfilment of the requirements for the award of Master of Technology (Dual degree) degree in Electrical Engineering at the National Institute of Technology, Rourkela, is an authentic work carried out by me under my suervision and guidance. To the best of my knowledge the matter embodied in the thesis has not been submitted to any other University/Institute for the award of any degree or diloma. Date: Prof. SANDIP GOSH Deartment of Electrical Eng. National Institute of Technology, Rourkela Rourkela P a g e 3

4 ABSTRACT One of the ast control rocedures is the PID control which is used many industries. It can be comrehended on the grounds that it is tuneable effectively and the control structure is basic. In the meantime a few tasteful results have been demonstrated utilizing PID control as a art of control system, in mechanical control desite everything it has an has a widesread variety of resentations. As er a study it has been found that each control area requires PID tye for rocess control systems directed which was studied in 989. For a long time PID control has been an energetic study subject. Since numerous rocess lants have comarable dynamics which is PID controlled and it has been found from less lant data it is ossible to set accetable controller. In this few controller design techniques is been resented for PID-tye, and resulting details for the tuning algorithms is discussed. The PID control are all described fully, and some differences of the classic PID structure are resented. The erceived exerimental Ziegler Nichols tuning formula and for the PID controller design algorithms aroaches have been made for finding the corresonding FOPDT model. Some other simle PID setting formulae such as the Cohen Coon formula, Chien Hrones Reswick formula, Zhuang Atherton otimum PID controller, Wang Juang Chan formula and is resented. Some of the design techniques on PID control is resented, such as Smith redictor design and IMC control design. At long last, a few thoughts on the structure of the controller determinations for rocess control system are given. P a g e 4

5 ACNOWLEDGEMENT I have been very blessed to start my thesis work under the suervision and guidance of Prof. Sandi Ghosh. He introduced me to the field of Control systems, educated me with the methods and rinciles of research, and guided me through the details of PID controllers. Working with him, a erson of values has been a rewarding exerience. I am highly indebted and exress my dee sense of gratitude for his valuable guidance, constant insiration and motivation with enormous moral suort during difficult hase to comlete the work. I acknowledge his contributions and areciate the efforts ut by him for heling me comlete the thesis. I would like to take this oortunity to thank Prof. A..Panda, the Head of the Deartment for letting me use the laboratory facilities for my roject work. I am thankful to him for always extending every kind of suort to me. At this moment I would also like to exress my gratitude for my friends for heling me out in my difficulty during my thesis. They have always heled me in every-way they can during my exerimental hase of the work. RUBEN ANDULNA (70EE347) P a g e 5

6 CONTENT COVER PAGE CERTIFICATE..3 ABSTRACT..4 ACNOWLEDGEMENTS..5 LIST OF FIGURES...8 LIST OF TABLES 0. INTRODUCTION.. Introduction to Control...2 Closed loo SISO system..2.3 Proortional Control..3.4 Integral Control..5.5 Proortional lus Integral Control.6.6 Proortional lus Derivative Control.7.7 Proortional lus Integral lus Derivative Control 8.8 Motivation & Objective LITERATURE REVIEW PROCESS MODELLING Process modelling from resonse characteristics of lant Transfer function method FOPDT DESIGN & TUNING METHOD Different tuning rocedure Ziegler-Nichols tuning Chine-Hrones-Reswick PID tuning Cohen-Coon Tuning Wang-Juang-Chan tuning Otimal PID Controller Design Smith redictor design Internal Model Controller design Tuning of IPDT model SIMULATION OF FOPDT SIMULATION using P control SIMULATION using PI control P a g e 6

7 4.3 SIMULATION using PID control SIMULATION OF Otimal PID Controller Design CONCLUSION APPENDIX...47 A. Ste resonse of the rocess lant A.2 Smith redictor...48 A.3 Simulink for the comarison of set oint 49 A.4 Simulink for otimal control 50 BIBLIOGRAPHY.5 P a g e 7

8 LIST OF FIGURES Fig. No. Title Page No. Fig.. Fig..2 Fig..3 Fig..4 Fig..5 Fig..6 Fig..7 Fig..8 Fig..9 Fig..0 Fig.2. Fig.2.2 Fig.3. Fig.3.2 Fig.3.3 Fig.3.4 Inut and Outut of a lant to be controlled A feedback control system. A closed loo SISO system Controller with only P Resonse with a roortional controller Integral Control action Ste resonse with integral control action Proortional lus Integral Control action Transient resonse with P, I and P-I Control action with higher order rocess ste resonse of rocess lant Ste resonse of Process lant Vs FOPDT Parameters A and L obtained through ste resonse of lant Smith Predictor structure Ste resonse of FOPDT lant Basic PI controller Fig.3.5 Ste resonse of Y s and d 33 Fig.3.6 loss of stability when increases 33 Fig.3.7 Comarison of ste resonse for smith redictor and PI controller 35 P a g e 8

9 Fig. No. Title Page No. Fig.3.8 Fig.3.9 Fig.3.0 Comarison of bode lot for smith redictor and PI controller IMC configuration. Ste resonse of IMC Fig.3. Fig.3.2 Fig.4. Fig.4.2 Fig.4.3 Fig.4.4 Fig.4.5 Fig.4.6 PDF control structure Ste resonse of PDF controller Ste resonse using P controller Ste resonse using PI controller Ste resonse using PID controller Ste resonse using PI controller Ste resonse using PID controller Ste resonse using PID controller with D in feedback P a g e 9

10 LIST OF TABLES Fig. No. Title Page. No Table.. Table.2. Table.3. Table.4. Table.5. Table.6. Table.7. Table.8. set oint regulation for Chine-Hrones-Reswick disturbance rejection for Chine-Hrones-Reswick Cohen-Coon arameters for P, PI, PD, PID For set oint tracking PI Controller For set oint tracking PID Controller For set oint tracking with D in feedback ath using PID controller For disturbance rejection PI Controller For disturbance rejection PID Controller P a g e 0

11 . Introduction to Control:. INTRODUCTION Control designing manages Dynamic structures, for examle, cars, flying machine, shis and trains, for examle, refining sections and rincially in steel moving lants, electrical systems, for examle, ower system, generators, and motors and numerically controlled machines and robots. There are some variables which are deendent, called oututs, which is to be controlled, which must be made to act in a recommended manner. Case in oint it might be imortant to aoint the ressure and temerature in a rocess at different oints, or the ower system s voltage and frequency, to given desired unchangeable value. Some variables which are not deendent, called inuts, for examle, valve osition or voltage connected to the engine terminals, to direct and control the conduct of the system. There are disturbances influencing which are affecting the system are not known. These could be, for instance load variances in ower systems, disturbances influences, for examle, wind blows following u on a vehicle, on exosing and cooling lant outside climate conditions is acting, or the load torque fluctuating on a lift engine, as travellers enter and way out. The arameters contained in these comarisons and the mathematical statements deicting the lant elements, are not no doubt understood at all or, best case scenario known generally. System arameter changes as the set oint changes. The inut and outut of a lant to be controlled is given as. Unknown Disturbances Control inuts Plant oututs which is to be controlled Measurement Fig. Inut and outut of a lant which is to be controlled In Fig.. the oututs or inuts demonstrated can really be seaking to a signal of vectors. Control which is racticed by inut, which really imlies that the useful inut to the lant which is controlled is driven by available estimations which is roduced by a device. We can see the control system shown in Fig..2. P a g e

12 Disturbances Plant Controller oututs Controller Measurements Reference Inut Fig..2. Control system with feedback. The main urose of designing the control system so as to meet some criteria so that the outut can be. Set to a fixed value which is called as reference value; 2. Even though there is some unknown disturbances, reference value should be maintained; The first one is said to be tracking, the second one is said to be disturbance rejection,. If both the condition are met then the control system design can be a robust servomechanism..2 Closed loo SISO system: The single-inut single-outut (SISO) system is the essential control loo and can be simlified as in Fig..3 Here the disturbances resent in the system are ignored. Reference Inut + error outut Controller Process rt () - et () ut () ct () Fig..3 A closed loo SISO system Normally, a controller is essential to rocess the error signal such that the general system fulfils certain standards. Some of these criteria are:. Reduction in effect of disturbance signal. 2. Reduction in steady-state errors. 3. Sensitivity to arameter changes. The controllers have various structures so with a secific goal to accomlish favoured execution level various design techniques are there for lanning the controller. Anyway, P a g e 2

13 Proortional-Integral-derivative (PID) sort controller is the most famous among them. Actually in the modern control alication 95% controllers are of Proortional-Integral-Derivative [6]. As outut of the Proortional-Integral-Derivative controller u (t) can be stated in terms of e (t), as: t de( t) u( t) [ e( t) d e( ) d ] dt () i 0 Transfer function of the controller is: C( s) ( ds) (2) is The terms of the controller are defined as: = roortional gain, = Derivative time, and = Integral time. d In the subsequent segment we might try to learn the significance of the individual roortional, integral, derivative. For simlicity we consider first-order transfer function in the absence of time delay: Ps () (3) s.3 Proortional control: In the closed loo system only P control is considered: i Rs () + - Es () s Cs () Fig..4. Controller with only P Transfer function is: Cs ( ) s ' R( s) s s s ' Where A For a ste inut Rs () s A Rs () ' s( s) Or, A s ' ( ) ( e ) c t The system resonse is shown in Fig..5. (4) (5) P a g e 3

14 .4.2 A ste resonse with P controller closed loo oen loo amlitude A A/+ offset time It is aarent from eqn. (5) and Fig..5.. By a factor Fig..5.Resonse with a roortional controller the time resonse is enhanced (i.e. the time constant declines). 2. There is a steady state offset between reference and the outut = A A( ) 3. By increasing the roortional gain offset can be reduced; however oscillations can increment for systems with higher order. From error transfer function, the steady state error can be obtained and in terms of Lalace transform, the error function e(t) can be reresented as: A s A Es () (6) s s s s The steady state error can be evaluated by using final value theorem s A A ess lim e( t) lim se( s) lim (7) t s0 s0 s s Proortional band is defined as the band of error which causes a 00% variation in the controller outut exressed as a ercentage of range measurement. P a g e 4

15 .4 Integral Control: For closed loo system, the integral control is demonstrated in Fig..6. Rs () + - Es () i s s Cs () Fig..6. Integral Control action Continuing the same as in eqn. (4), C() s is( s) (8) 2 R() s is is i ( s) We can see from above that closed loo systems order is increased by so, it may cause instability as the rocess dynamic becomes higher order. For inut ste A Rs () s A is( s) A Es () s is( s) s ( s) e ss lim se( s) 0 s0 i Due to inut ste the steady state error decreases to zero, it is the significant advantage of this integral control. Anyhow, all together, the resonse of the system is slow, oscillatory and unstable. The ste resonse due to integral control is demonstrated in Fig..7. (9).4.2 ste resonse with integral control I Control A amlitude time Fig..7. Ste resonse with integral control action P a g e 5

16 .5 Proortional Plus Integral (P-I) Control: With Proortional lus integral controller the closed loo system is demonstrated in Fig..8. Rs () + - Es () ( ) s i s Cs () Fig..8. Proortional lus Integral Control action As here we have two control actions P and I, P hels in quick resonse and I hels in reducing steady state error to zero. The transfer function of the error of the system can be stated as: Es ( ) is( s) (0) 2 R( s) ( is) s i ( ) is is( s) Additionally, the closed control loo characteristic equation for Proortional-Integral control is 2 s ( ) s = 0, () i i Daming constant is obtained as: i ( ). (2) 2 Daming constant for simle integral control is i ( ) 2 At the oint when these two are looked at, one can undoubtedly observe that the daming constant can be increased by changing the term. So we confirm that the steady state error can be zero by utilizing Proortional-Integral control and all together, we see imrovement in the transient resonse. The system outut resonse due to Proortional, Integral and Proortional-Integral control for same lant is thought about from the reresentation indicated in Fig Transient resonse with P, I and P-I I PI P A amlitude time Fig..9. Transient resonse with P, I and P-I P a g e 6

17 .6 Proortional lus Derivative (P-D) Control: Transfer function of P-D controller is given by: C( s) ( s) (3) d P-D control transfer function Ps () really is not excetionally helful, since it can t s decrease the steady state error to zero. But the closed loo system stability can be imroved for higher order system using P-D controller. Let Ps () at Fig.8, closed loo transfer function 2 Js with roortional control is Cs () 2 Js (4) 2 R() s Js 2 Js 2 Characteristics equation is given as Js = 0; resonse is oscillatory, closed loo transfer function with P-D is: ( ds) Cs () 2 Js ( ds) (5) 2 R( s) ( ds) Js ( ds ) 2 Js 2 Characteristics equation is Js ( s) = 0; that will give a closed loo stable resonse. Rs () + - Es () d Js 2 Cs () Fig..0. Control action with higher order rocess.7 Proortional-Integral-Derivative (PID) control: It is now clear that the required erformance can be obtained by a roer combination of P,I and D action. PID control transfer function is: C( s) ( ds) (6) is It is a low order control system, however its alicability is widesread, and it can be utilized as a art of any kind of Single Inut Single Outut system. A large number of Multile Inut Multile Outut systems are initially subdivided into a few Single Inut Single Outut loos and for each loo PID controllers are intended. Proortional Integral Derivative controllers have additionally be discovered that it should be robust, and this is the reason why it finds wide suitability for modern rocedures. The method of tuning PID arameters would be taken in later chater. P a g e 7

18 It is not that necessary that we ought to utilize all the control art. Truth be told, in a large ortion of the cases, a basic Proortional-Integral control will be adequate. A broad guidance for the choice of mode of controller to be used, is rescribed []. Choice of controller mode:. Proortional Controller: It is basic for regulation, easy tuning. Anyhow, steady state error is introduced. It is suggested that if the transfer function which is having single dominant ole or having a ole at origin. 2. Integral Control: It is relatively slow and no steady state error is observed. It will be oerative for quick rocess, having noise level high. 3. Proortional-Integral (P-I) Control: Integral action alone results in faster resonse. It is widely used for rocess industries because they do not have large time constants for controlling the variables for examle level control, flow control etc. 4. Proortional-Derivative (P-D) Control: For larger time constants this P-D controller is used. It has more raid resonse and less offset comared to roortional controller. If measurement is noisy one should be careful in using derivative control. 5. Proortional-Integral-Derivative (P-I-D) Control: It alication is widesread however it s tuning is a touch troublesome. It is mostly helful for controlling moderate variables, as H, temerature, and so forth in rocess industries..8 Motivation & Objective: The motivation behind this roject is to observe different kinds of lant in the real world. As in the modern day alication we come across several control machines and we think of new methods of controlling so, I made a study on different control methods for FOPDT and IPDT lant model. For FOPDT through Zeigler-Nichols tuning method, the objective was to find the controller arameters to decay the first overshoot to 0.25 times the original overshoot after oscillation. Chine-Hrones-Reswick tuning method focuses on the main roblem consisting of how to regulate set-oint and how to reject the disturbances. Cohen-Coon main aroach was to find three dominant oles it should be a air of comlex oles and one real ole such that for load disturbance rejection, the amlitude decay ratio becomes /4 th and the integral error is also minimized. The objective behind this Otimal PID Controller Design methods is to select the Proortional-Integral-Derivative controller arameters which hels in minimizing an integral cost functional. IMC design objective is to minimize the tracking error. The objective of PDF controller is to result in smooth resonse to every set-oint change and gives maximum robustness whenever there is uncertain arameter..9 LITERATURE REVIEW The mathematical model of any real time rocesses can be classified as stable systems, unstable systems and systems with dead time. The PID controller is very imortant in control engineering alication and is widely used in many industries. An excellent account of many ractical asects of PID control is given in PID Controllers: Theory, Design and Tuning by Astrom and Hagglund [0]. After the study of PID controller, Xu, H., Datta, A., and Bhattcharyya, S. P. [22] exlained the study of PID stabilization of linear P a g e 8

19 time invariant lants with time delay with various tuning methods for different tyes of lants like FOPDT, IPDT and FOIPDT. There is a vast mathematical literature on the analysis of stability of time-delay systems which we have not included. We refer the reader to the excellent and comrehensive recent work Stability of Time-Delay Systems by Gu, haritonov, and Chen [] for these results. The control of time delay systems is still being a challenge to imrove its time domain conditions. The survey exoses that the tuning techniques are different for different kind of systems, systems like first order lus dead time delay and others. The set of tuning rules alicable for the first order lus dead time delay systems are not alicable for IPDT and FOIPDT systems. This means we have to follow different tuning rules for different kind of systems. If there is a arameter variation for any nominal lant, conventional controller are unable to maintain the stability of the system. For this kind of situation we need to design a robust controller where a single controller in able to control the whole lant family. While designing a robust controller we need to kee in mind of its robust stability and erformance. Since both the robust stability and erformance are inversely roortional to each other, the otimization between these two becomes an interesting one. There has been several tuning methods emirically roosed, every tuning aroach has its own significance, Zeigler-Nichols [20] aroach was that after one oscillation, decay the first overshoot to 0.25 times of its original value. Chine-Hrones-Reswick [9] tuning method focused on how to regulate set-oint and how to reject the disturbances. Cohen-Coon [8] tuning method aroach was to decay the amlitude ratio for load disturbance so, the load disturbance is rejected also to minimize the integrator error. Zhuang, M., and Atherton, D. P. [4] also roosed otimal PID controller design method because there aroach was to minimize the integral cost function by choosing the PID controller. The controller arameters are determined by minimizing the integral erformance 2 criteria such as ISE, ISTE, IST E. Both the set-oint and the load disturbance rejection design secifications are given in this thesis. The obtained results are take on both for tuning uroses and for the evaluation of the erformances of an earlier PID controller. D.E.Rivera, M.Morari and S.Skogestad [7] suggested the IMC design where an internal model is referred which is basically the original lant whose time delay is been aroximated by ade first-order aroximation to minimize the tracking error. Smith redictor control design invented by O.J.M.Smith in 957, this is a tye of redictive controller for ure time delay. Then other tye of lant resulted i.e. integral lus dead time lant who s tuning can t be done by the above rocedures so.g.arvanitis, G.Syrkos, I.Z.Stellas and N.A.Sigrimis [8] have done some tuning rocedures using Pseudo Derivative Feedback controller where integral control is in forward ath and the roortional and derivative is in feedback, equations are formed and the arameters for PI and PID are extracted.the objective of PDF controller is to result in smooth resonse to every set-oint change and gives maximum robustness whenever there is uncertain arameter. IPDT [3]-[6] model has many advantages in the field of tuning, this kind of model has the ability to reresent various systems to be controlled by PID controllers. As IPDT contains only P a g e 9

20 two arameters one is gain and the other is time delay therefore it is easy to identify, L.Wang and W.R.Cluett roosed some tuning rocedure for IPDT model [2] For higher order controller its real time imlementation becomes difficult in many alications such as aerosace, chemical industries, sace vehicles etc. For satisfying some of the robust rinciles, lower order controller with minimum tuning arameters are resented. As the structure of the PID controller is fixed our work is to find stable values of roortional gain ( ) and the integral gain ( ) for the first order lus dead time ), derivative gain ( d delay lant for set oint resonse and load disturbance rejection and for the integral lus dead time delay for smooth resonse for every set-oint change. Simulation results obtained for different tuning rocedures and analysed and also a smith redictor aroach for the system is roosed. i P a g e 20

21 2. PROCESS MODELLING 2. PROCESS MODELLING FROM RESPONSE CHARACTERISTICS OF PLANT In control alications used in industries the lant is modelled as a first-order or second-order system with time delay and the controller is either of the P, PI or the PID tye. From the model it can be seen that this model (23) is helful for the design of a Proortional- Integral-Derivative control due to the accessibility of a straightforward equation. The technique used in Sec for the conclusion to find L & T of a lant it is easy to use the lot of the ste resonse of the lant. Though in current scenario we need not cut the model u to this form to find at Proortional-Integral-Derivative arameters of controllers. In this section, successful and regularly utilized calculation is resented. 2.. Transfer function method: Let us take the first-order lus dead time lant model Ls ke Gs ( ). Ts First-order and second-order derivatives with resect to s, ' G () s T L, G( s) Ts " ' 2 2 G ( s) G ( s) T G( s) G( s) ( Ts ) 2. (7) Evaluating the values at s=0 yields ' G (0) Tar T L, G(0) T G (0) G(0) " 2 2 Tar Where T ar, = average residence time. From revious equation, L Tar T. and from G (0) DC gain value can be evaluated. The key to the FOPDT model is in this way acquired by utilizing the Gs () derivatives in the above formula. A large selection of lant can be roughly modelled by FOPDT in real time rocess control system. Equation of the first-order lus dead time model: (8) Ls G() s e Ts Where =gain; L= time delay; T= time constant; (9) P a g e 2

22 We need to find the controller arameters using some of the tuning formulae. Matlab is used to trace the resonse of lant versus time. Some basic calculation have to be done for finding lant model arameter FOPDT (first order lus dead time): For examle, to find the arameters, L and T by alying a ste resonse to the lant model through an exeriment ( L a T ). Finding arameter of FOPDT: Process transfer function of a lant is [9] 0 Gs () ( s 4)( s 3)( s 2)( s ) (20) For ste resonse of system matlab code is used and as the steady-state value of (see in Aendix A.). Ste resonse: 0.45 Ste Resonse System: G Time (seconds): 2.22 Amlitude: Amlitude System: G Time (seconds):.3 Amlitude: Time (seconds) Fig.2.. ste resonse of rocess lant t = the time at gain(c) =0.3 *steady state gain () t 2 = the time at gain(c) = 0.6 *steady state gain () Find T and L T 3( t2 t) 2 P a g e 22

23 L= ( t t ) 2 L a T From ste resonse =0.467 t =.3 sec t 2 =2.2 sec And L=0.855 sec T=.365 sec We have FOPDT equation as: G() s e.365s 0.855s 0.45 Ste Resonse Amlitude Process FOPDT Time (seconds) Fig.2.2. ste resonse of Process lant Vs FOPDT After the modification of rocess lant transfer function to a FOPDT transfer function it is clear from resonse that in the FOPDT it shows a clear delay at time of starting. As most of the lant are of accumulated with dead time so this is the reason behind the conversion of the rocess lant to FOPDT model. It is exciting to note that desite the fact that a large ortion of these systems give suitable results, the set of all Proortional-Integral-Derivative controllers for these first-order models with time delay has been exlained in the next chater. P a g e 23

24 3. DESIGN AND TUNING METHODS 3. DIFFERENT TUNING PROCEDURE: As discussed in the earlier chater how to model a lant, after modelling we have to control the lant by using PID controller and as PID controller has three arameters we have to find those arameters with the hel of some tuning rocedures. For finding controller arameters same tuning rocedure can t be used for all tyes of lant model. For each lant model different tuning formula is used. 3.. Ziegler- Nichols method: The Proortional-Integral-Derivative controller is realised as follows: i C() s ds s Where = roortional gain, i = integral gain, and d = derivative gain. In this Ziegler-Nichols it is only valid to oen loo lants which are stable [20] as it is an oenloo tuning done by exerimentation. In this our rior thing is to find the arameters A and L which we can get it through the lants ste resonse as shown in Fig..8. Firstly we should determine the oint where it shows the maximum sloe and draw a tangent, this tangent intersects with the vertical axis roduces A and intersection with the horizontal axis roduces L. By now after we find A and L we can find the Proortional-Integral-Derivative arameters to control. Maximum sloe A L Fig.3.. lant ste resonse to get A and L. Emirically obtained formulas are there to roduce Proortional-Integral-Derivative control arameters from which we observe that after one oscillation there is a decay in its first overshoot of 0.25 times the original value. P a g e 24

25 Tuning formula Controller tye For ste resonse For frequency resonse P PI PID /a 0.9/a.2/a i d 3L 2L L/2 0.5c 0.4c 0.6c i 0.8Tc 0.5Tc d 0.2Tc Here only using ste resonse; controller arameters are found out. Then using Simulink outut ste resonse the model lant is taken. We have FOPDT equation as: G() s e.365s a = 0.92 P = s PI = 4.706( ) 2.56s PID = N s N / s 3..2 Chine-Hrones-Reswick tuning: This method focus on the main roblem consisting of how to regulate set-oint and how to reject the disturbances. Also regarding seed of resonse and overshoot an additional comment comarable with the Ziegler Nichols tuning formula, the time constant T is been used clearly in this CHR method [9]. Closed-loo resonse which is more heavily damed, guarantees for an ideal lant and the one which is having high resonse seed without overshoot is considered as overshoot of 0% and other with good resonse seed with 20% overshoot is considered as overshoot of 20%. Set oint regulation Controller tye Overshoot of 0% Overshoot of 20% P PI PID 0.3/a 0.35/a 0.6/a i.2t T 0.5L d 0.7/a 0.6/a 0.95/a i d T.4T 0.47L P a g e 25

26 Disturbance rejection Controller tye Overshoot of 0% Overshoot of 20% P PI PID 0.3/a 0.6/a 0.95/a i d 4L 2.4L 0.42L 0.7/a 0.7/a.2/a i d 2.3L 2L 0.42L Table. Set oint regulation for Chine-Hrones-Reswick: Controller tye Overshoot of 0% Overshoot of 20% P PI PID i d i d Table 2. Disturbance rejection for Chine-Hrones-Reswick: Controller tye Overshoot of 0% Overshoot of 20% P PI PID i d i d Cohen-Coon Tuning algorithm: Cohen-Coon method [8] is a dominant ole design method and tries to locate some oles to attain definite erformance control. The. It is based on first order lus dead time model: G() s e Ts Ls This tuning method aroach was to decay the amlitude ratio for load disturbance so, the load disturbance is rejected also to minimize the integrator error. This gives good load disturbance rejection, Proortional-Integral-Derivative arameters in relation to, T, and L: L L a, T L T P a g e 26

27 Controller tye i d P a ( τ τ ) PI 0.9 ( τ ) a τ PD.24 ( + 0.3τ a PID.35 ( + 0.8τ ) a τ 3.3 3τ +.2τ L ) τ L τ 0.87τ 2.5 2τ 0.39τ L τ L 0.8τ Table 3. Cohen-Coon arameters for P, PI, PD, PID Controller tye i d P PI PD 7.09 PID Wang-Juang-Chan method of tuning: Name itself says that this tuning method is suggested by Wang, Juang, and Chan [9]. For choosing the Proortional-Integral-Derivative control arameters it is an easy & effective method which is built on the otimum Integral-Time-Absolute-Error criterion. The controller arameters can give by, if the arameters, L & T of the lant are known ( T / L)( T L / 2) ( T L) (2) L i T 2 d L T 2 L T 2 P a g e 27

28 = , i =.7925, d = Otimal PID Controller Design [4]: This method tries to find the PID arameters which minimizes the integral cost function. 2 ( ) [ n n (, )] 0 J t e t dt Where = vector having the arameters of the controller and e (, t) signifies the signal error. Another influence is due to Pessen [3], who utilized IAE rincile: J( ) e(, t) dt 0 To reresent time function in Lalace transform we make use of Parseval's Theorem to minimize the cost function [2]. To minimize the integral cost function as soon as the integration gets started, the Proortional-Integral-Derivative controller arameters are adjusted. (22) (23) Set-Point otimum PID tuning: For PI controller: b a ( L ), i T k T a b ( L / T) 2 2 (24) For PID controller: a L T L b b3 ( ), i, d a3t ( ) k T a2 b2( L / T) T (25) The values for a, b, a2, b2 for set oint regulation for all the controller tyes their aramters deending uon the range of L/T is given in[9]. Disturbance rejection PID controller: PI controller: a L T L b b2 ( ), i ( ) (26) k T a2 T Proortional-Integral-Derivative controller: a L T L L 2 3 ( ) b, ( ) b, 3 ( ) b i d a T (27) k T a2 T T The values for a, b, a2, b2 for disturbance rejection for PI and PID controller aramters deending uon the range of L/T is given in[9] Controller arameter on the basis of this tuning: P a g e 28

29 Table 4. For set oint tracking PI Controller: Criterion ISE ISTE IST 2 E i d Table 5. For set oint tracking PID Controller Criterion ISE ISTE IST 2 E i d Table 6. For set oint tracking with D in feedback ath using PID controller: Criterion ISE ISTE IST 2 E i d Table 7. For disturbance rejection using PI Controller: Criterion ISE ISTE IST 2 E i d Table 8. For disturbance rejection using PID Controller: Criterion ISE ISTE IST 2 E i d P a g e 29

30 3..6 Smith Predictor design Using the above arameters for and i ; The first-order lus dead time with a.365 second time constant and second time delay. The Smith Predictor control structure is u y0 y P d + ys - e C G e s y y d Fig.3.2. Smith Predictor structure F dy By using the matlab code (see in Aendix A.2): The ste resonse of the first-order lus dead time lant is 0.45 Ste Resonse From: u To: y Amlitude Time (seconds) Fig Ste resonse of FOPDT lant P a g e 30

31 PI Controller: In rocess control Proortional-Integral (PI) control is a commonly used technique. The PI control standard diagram is shown in fig.6.3. ys - + u e C P y Fig.3.4. Basic PI control structure C is a comensator with two tuning arameters roortional gain and an integral time i. Here we have taken 0% overshoot. and i values from Chine-Hrones-Reswick PID tuning algorithm for With =.7, i =.8 The feedback loo is closed and it is been simulated to observe the resonses to the ste change in the reference signal ys and outut disturbance signal d by which we can evaluate PI controller erformance..4 From: ys Ste Resonse From: d Amlitude To: y Time (seconds) Fig.3.5. Ste resonse of ys and d P a g e 3

32 The closed-loo resonse has tolerable overshoot but is somewhat slow (it settles in about 2 seconds). To increase the seed of the resonse we should start increasing the gain because of this it can lead to instability. but.8 Loss of stability w hen increasing From: ys To: y Amlitude Time (seconds) Fig.3.6. loss of stability when increases Because of the dead time, PI controller erformance is not u to the mark because the actual outut y is not getting matched with the reference set oint ys. The Smith Predictor rocedures an internal model G to guess the resonse which is delayfree y of the rocess. Before it matches this rediction y with the reference set oint ys to decide what tunings are needed (control u). By taking in consideration of rejecting their disturbances which are external, the Smith redictor also relates the actual outut of the rocess with a rediction y which takes the dead time into justification. The ga dy=y-y is fed back via a filter F and contributes the error signal e. Smith Predictor requirements: A model G which is the rocess and an estimate tau of the rocess dead time satisfactory settings for the comensator and filter dynamics (C and F) Based on the rocess model, we use: G() s e.365s 0.855s P a g e 32

33 For F, to cature low frequency disturbances we use a first-order filter with a 20 second time constant. F 20s For C, we re-design the PI controller with the overall lant seen by the PI controller, which includes dynamics from P, G, F and dead time. With the hel of the Smith Predictor control structure we are able to increase the oen-loo bandwidth to achieve faster resonse and increase the hase margin to reduce the overshoot. Process P e.365s Model redicted G.365s 0.855s ; D 0.855s e Design PI controller with 0.08 rad/s bandwidth and 90 degrees hase margin Comarison of PI Controller vs. Smith Predictor: To equate two designs, first derive the transfer function of the closed-loo from ys,d to y for the Smith Predictor architecture. To facilitate the task of connecting all the blocks involved, name all their inut and outut channels and let connect do the wiring:.4.2 From: ys Ste Resonse From: d Smith Predictor PI Controller 0.8 Amlitude To: y Time (seconds) Fig.3.7.Comarison of ste resonse for smith redictor and PI controller P a g e 33

34 0 Bode Diagram From: ys To: y 0 Magnitude (db) Smith Predictor PI Controller Phase (deg) Frequency (rad/s) Fig.3.8.Comarison of bode lot for smith redictor and PI controller. P a g e 34

35 3..7 IMC Design: In rocess control alication IMC design has become famous [7]. In this G(s) is FOPDT, in IMC it is suitable for oen-loo stable control systems. The Internal model control consists of ^ () Gs a stable internal model controller arameter Q(s) and is the model of the lant. F(s) is internal model controller filter selected to make Q(s) F(s) roer by imroving the robustness. F( s) Q( s) Cs ( ). ^ F( s) Q( s) G( s) (28) IMC design main objective was to select Q(s) which hels in minimizing the tracking error r- y. r + y - e Fs () Qs () u Gs () ^ () Gs ^ y + - The following lant is to be controlled: Fig.3.8. IMC configuration G() s e.365s 0.855s By Pade aroximation, 0.855s s e s (29) (30) ^ Gs () Which is the internal model whose transfer function is ^ Gs () 0.78s s.7925s.365s Qs () (3) (32) P a g e 35

36 Since Q(s) is imroer and to get the suitable we have to negotiate between robustness and erformance. Zafiriou & Morari [5] have suggested an at choice to select, >0.2T and >0.25L. Fs () 0.274s The equivalent feedback controller becomes L ( Ts)( s) Cs () 2 s( L ) (33) (34) From the above equation we get the arameters for a standard PID controller: =3.80 i = d = IMC ste resonse PID amlitude time Fig.3.9.ste resonse of IMC P a g e 36

37 3..8 Integrator lus dead time (IPDT) Model: ds A generally faced lant which is modelled mathematically G() s e is denoted as the s IPDT model. IPDT lant cannot be tuned by the earlier tuning rocedures. As there is already an integrator so no need of another integrator to remove a steady state error for a ste inut. The following IPDT model is exerimentally obtained transfer function of a temerature rocess rig, by controlling the temerature at a articular junction using PID setting in the controller we obtained an inut and outut data in excel file. Using the inut-outut data with the hel of matlab system identification we obtained this transfer function s G() s e (35) s To control integrating lus dead-time model we should use Pseudo-Derivative Feedback (PDF) structure. The methods used for tuning this PDF structure is simle and results in smooth resonse to every set-oint change and gives maximum robustness whenever there is uncertain arameter. IPDT [3]-[6] model has many advantages in the field of tuning, this kind of model has the ability to reresent various systems to be controlled by PID controllers. As IPDT contains only two arameters one is gain and the other is time delay therefore it is easy to identify. If the systems having large time constants over critical range of frequency that is near ultimate frequency, IPDT model can be aroximated. As we are going on saying that IPDT model is simle but there is less number of tuning aroaches comared to FOPDT model. The Ziegler- Nichols methods leads to oscillation and becomes unstable even there is a small erturbations in the arameters of the model. IPDT model tuning based on the coefficients matching of the owers of s in numerator and denominator is discussed in [7], to avoid overshoot. + + L(s) I s G () s Rs () - Es () + U(s) Ys () - s... s n D, n D, D,0 PDF control structure Fig.3.0. PDF control structure In this our aims should be focussed on two forms of PDF structure, in the first only the roortional control is in the feedback and it denoted as PD-0F and the second forms consists of roortional and derivative control in feedback and it is denoted as PD-F. P a g e 37

38 One by one each tuning method is discussed and arameters are found. As shown in the figure the controller is PD-0F when Di, =0, for i=,,n- and D,0 0 and the controller is PD-F when D,0 0, D, = d 0 and, i=2 n-. For the above we should analyse for both the controller for the above shown IPDT model. PD-0F Controller Settings for IPDT models [8]: Di =0, for The PD-0F controller arameters can be chosen as I (8 ) (8 ) d d (36) Where α is an adjustable arameter, in order to obtain referred daming ratio (see [8], for details). Alternative PD-0F Controller for IPDT models: The PD-0F controller arameters I 4 8 d 8 2 d (37) = , I = Where 2 2 PD-F Controller Settings for IPDT models: The PD-F controller arameters I d d d (38) Where γ is an adjustable arameter, in order to obtain referred daming ratio (see [8], for details). = 69.95, I = , d = P a g e 38

39 .4.2 ste resonse PI PID amlitude time Fig.3.. ste resonse of PDF controller Using Pseudo Derivative Feedback controller for IPDT lant model we got to know that PD- F which is equivalent to PID has a good rise time comared to PI and minimum overshoot and settles faster than PI controller. P a g e 39

40 4. Simulation of FOPDT Simulation is done using Simulink. Using above mentioned tuning formula and we have comared the P controllers of all the above tuning rules and made analysis, after that similarly for PI and PID.The solution to the roortional control case is develoed first because it serves as a steing stone for tackling the more comlicated cases of stabilization using a PI or a PID controller. The roortional control stabilization roblem for first-order systems with time delay can be solved using other techniques such as the Nyquist criterion and its variations. The aroach resented here, however, allows a clear understanding of the relationshi between the time delay exhibition by a system and its stabilization using a constant gain controller. The objective of finding arameters through different tuning methods and analysing which method control erformance in good and stable. SIMULATION using P control [similar to Aendix (A.3)]:.4.2 ste resonse using P controller Z-N CHR 0% CHR 20% Cohen coon amlitude time Fig.4..ste resonse using P controller As we observe from Fig.4. using P controller Cohen coon is faster comared to other tuning rules but it also tends to large overshoot and in Chine-Hrones-Reswick 0% overshoot, comaratively has minimum overshoot but it is slow in reonse. As it is a P controller it introduces steady state error it is difficult for all the above tuning rules the Ziegler-Nichols, the P a g e 40

41 Cohen-coon,the Chine-Hrones-Reswick 0% overshoot and 20% overshoot to get good control erformance. SIMULATION using PI controller [similar to Aendix (A.3)]:.4.2 ste resonse using PI controller Z-N CHR 0% CHR 20% COHEN amlitude time Fig.4.2.ste resonse using PI controller While using a PI controller to control the first-order lus dead time lant we observe that there is a quick resonse due to P control and the steady state error is zero due to I control, Ziegler- Nichols takes around 5 sec, Chine-Hrones-Reswick 20% overshoot and Cohen-coon has almost equal settling time of sec, Chine-Hrones-Reswick 0% overshoot takes 3 sec to settle. Ziegler-Nichols resonds faster than other tuning methods mentioned above. While doing the simulation it is imortant to select the controllers deending uon the tye of tuning of the PI controller to achieve desired controller erformance while maintaining closed loo stability. P a g e 4

42 SIMULATION using PID controller [see Aendix (A.3)]:.4 ste resonse using PID controller.2 amlitude time Z-N CHR 0% CHR 20% COHEN WANG Fig.4.3.ste resonse using PID controller By using PID controller we are able to minimize the overshoot, quick settling time and rise time. In the above lot we analyse that Wang-Juang-Chan has a good resonse comared to others as there is no overshoot and also settles by 9 sec. Chine-Hrones-Reswick 0% overshoot tuning has a quick resonse comared to above mentioned tunings. P a g e 42

43 5.5 Otimal PID Controller Design: Tuning methods based on the minimization of ISE guarantee small error and very fast resonse. However, the closed-loo ste resonse is very oscillatory, and the tuning can lead to excessive controller outut swings that cause rocess disturbances in other control loos. For Set oint tracking: PI controller ste resonse using PI controller ISE ISTE IST2E.2 amlitude time Fig.4.4.ste resonse using PI controller In the above lot it s been analysed that IST2E has settling time of 7 sec but it is slow in resonse and in ISE rise time is.5 sec almost equal to ISTE but lesser overshoot so its settling time is about sec and on the other ISTE has a large overshoot so it take more time for the quarter amlitude decay we can do this in matlab simulation [similar to Aendix (A.4)].. P a g e 43

44 .4.2 ste resonse using PID ISE ISTE IST2E amlitude time Fig.4.5.ste resonse using PID controller Furthermore Comaratively PID is showing better control erformance than PI. Here also IST2E settles down quickly and also has a minimum overshoot comared to ISE and ISTE but its resonse is slow. IST2E settles in about 7 sec. and ISE settles in 2 sec. and ISTE settles in 0 sec. ISE has a faster resonse comared to others as one can see from the above lot by simulation [see Aendix (A.4)]. P a g e 44

45 .4.2 ste resonse using PID with D in feeback ISE ISTE IST2E amlitude time Fig.4.6.ste resonse using PID controller As we can say that in feedback ath if we have derivative in the PID controller it may be easy & fast related to the tyical Proortional-Integral-Derivative controller but we don t get a good result in its erformance. Therefore if you are thinking of designing it do use a dedicated algorithm for good control erformance. Here IST2E shows a better resonse comared to ISE and ISTE as it has minimum overshoot so it settles quickly. P a g e 45

46 5. CONCLUSION: Project study on PID controller design for various lant model rovide a brief idea of lant modelling, tye of lant model and controllers (P, PI, PD and PID) tuning method used for the of the model lant. Discussed Plant modelling which will hel in modelling of many industrial lant. And tuning method used for that lant will hel to find out of controllers arameters. Resonse will suggest which tuning method is better for the lant. And also it will lay great roll in selecting of controller. For tuning of controllers of FOPDT Ziegler-Nichols tuning formula, Chine-Hrones-Reswick PID tuning algorithm, Cohen-Coon Tuning algorithm, Wang-Juang-Chan tuning formula and otimal PID controller design are used and what we observed is that for P controller tuning Cohen coon erforms better comared to other tuning and for PI controller tuning Zeigler Nichols tuning is best suited for controlling than others and for PID controller tuning CHR 0% overshoot resulting in quick resonse and better settling time for the exerimentally obtained IPDT model Pseudo Derivative Feedback controller is used and for this PID controller is having a good control behaviour comared to PI.. P a g e 46

47 Aendix Matlab Source code A. for finding ste resonse of the rocess lant. clc; close all; clear all; s=tf('s'); G=0/(s+4)/(s+3)/(s+2)/(s+); ste(g); k=dcgain(g); A.2 Smith Predictor deisgn. s = tf('s'); P = ex(-0.855*s) * 0.467/(.365*s+); P.InutName = 'u'; P.OututName = 'y'; P P = From inut "u" to outut "y": ex(-0.855*s) * s + Continuous-time transfer function. ste(p), grid on Ci = id(.749,.85); Ci Ci = + i * --- s with =.7, i =.8 Continuous-time PI controller in arallel form. P a g e 47

48 Ti = feedback([p*ci,],,,); % closed-loo model [ys;d]>y Ti.InutName = {'ys' 'd'}; ste(ti), grid on 3 = [.76;.80;.85]; of Ti3 = remat(ci.ti,3,); C3 = idstd(3,ti3); controllers T3 = feedback(p*c3,); T3.InutName = 'ys'; % try three increasing values % Ti remains the same % corresonding three PI ste(t3) title('loss of stability when increasing ') F = /(20*s+); F.InutName = 'dy'; F.OututName = 'd'; % Process P = ex(-0.855*s) * 0.467/(.365*s+); P.InutName = 'u'; P.OututName = 'y0'; % Prediction model G = 0.467/(.365*s+); G.InutName = 'u'; G.OututName = 'y'; D = ex(-0.855*s); D.InutName = 'y'; D.OututName = 'y'; % Overall lant S = sumblk('ym = y + d'); S2 = sumblk('dy = y0 - y'); Plant = connect(p,g,d,f,s,s2,'u','ym'); % Design PI controller with 0.08 rad/s bandwidth and 90 degrees hase margin Otions = idtuneotions('phasemargin',90); C = id(.749,.85); C.InutName = 'e'; C.OututName = 'u'; C C = + i * --- s P a g e 48

49 with =.7, i =.8 Continuous-time PI controller in arallel form. % Assemble closed-loo model from [y_s,d] to y Sum = sumblk('e = ys - y - d'); Sum2 = sumblk('y = y0 + d'); Sum3 = sumblk('dy = y - y'); T = connect(p,g,d,c,f,sum,sum2,sum3,{'ys','d'},'y'); Use STEP to comare the Smith Predictor (blue) with the PI controller (red): ste(t,'b',ti,'r--') grid on legend('smith Predictor','PI Controller') bode(t(,),'b',ti(,),'r--',{e-3,}) grid on legend('smith Predictor','PI Controller') A.3 P a g e 49