Multi-loop PI/PID Controller Design Based On Direct .
Proceedings of the World Congress on Engineering and Computer Science 2008WCECS 2008, October 22 - 24, 2008, San Francisco, USAMulti-loop PI/PID Controller Design Based onDirect Synthesis for Multivariable SystemsTruong Nguyen Luan Vu and Moonyong LeeAbstract—In this paper, a new analytical method based onthe direct synthesis approach is proposed for the design of amulti-loop proportional-integral-derivative (PID) controller.The proposed design method is aimed to achieve a desiredclosed-loop response for the multiple-input, multiple-output(MIMO) processes with multiple time delays. The idealmulti-loop controller is firstly designed in terms of relative gainand desired closed-loop transfer function. Then the standardmulti-loop PID controller is obtained by approximating theideal multi-loop controller by the Macraulin series expansion.Simulation study demonstrates the effectiveness of the proposedmethod the in multi-loop PID controller design. The multi-loopPID controller designed by the proposed method shows a fast,well-balanced, and robust response with the minimum integralabsolute error (IAE).Index Terms—Multi-loop PI/PID controller, Direct synthesis,Multivariable system. IMC-PID tuning, Robust controllerdesign.I. INTRODUCTIONThe multi-loop PI/PID controllers, sometimes called asdecentralized PI/PID controllers, have been widely utilizedfor processes with modest interactions for many decadesbecause of many practical advantages such as a simplecontrol structure, fewer tuning parameters, robustness againstsensor/actuator failure, and easy understanding. Hence, manymulti-loop design methods have been reported in the processcontrol literature. However, most of the existing designmethods are based on the extension of single-input,single-output (SISO) PI/PID controller design methods.The modification of the Ziegler-Nichols (Z-N) method [1]with a detuning factor to meet the stability and performanceof the multi-loop control system is a typical one of this class.In the family of the modified Ziegler-Nichols method [2]-[4],the desired critical point has to be determined by identifyingthe critical gain and frequency and then the multi-loopcontrollers are tuned by the Z-N tuning method with aweighting factor. However, a common disadvantage in thesemethods is that they try to cope with the interaction effect bydetuning while neither dynamic nor static interactions isincorporated in the design stage.Another widely used approach is the extension ofsingle-loop relay tuning to MIMO case [5],[6]. This approachis straightforward because it directly combines a single-looprelay auto-tuning and a sequential tuning, wherein themulti-loop control system is tuned sequentially loop by loop,closing the ith loop while it is tuned and the jth loop has toopen [5]. However, the poor output responses can beobtained when the MIMO system has large multiple timedelays which is one of main causes for the strong dynamicinteractions.It is well known that the integral model control (IMC)method [7] is very effective to design the IMC-PID controllerfor taking into account time delays and closed-loopinteractions. Recently, several methods [8], [9] which extendthe IMC-PID method of the SISO case to the MIMO case, arereported.In this paper, a simple but efficient design method formulti-loop PI/PID controller is presented which exploitsprocess interactions for the improvement of loopperformance. The proposed method is based on the directsynthesis approach [10],[11] in which the multi-loop PI/PIDcontroller is designed based on the desired closed-looptransfer function [8],[9],[12]. The resulting analytical designrule includes a frequency-dependent relative gain array [13],[14] that provides information of dynamic interactions usefulfor estimating the controller parameters.R1- R2 Y1Gc1(s)Y2-Gc2(s)G(s)YnRnGcn(s) -Fig.1 Multi-loop control system.ndManuscript received July 22 , 2008. This work was supported by BrainKorea (BK) 21.Truong Nguyen Luan Vu and Moonyong Lee are with the School ofDisplay & Chemical Engineering, Yeungnam University, 214-1, Dae-dong,Gyeongsan , Gyeongbuk 712-749, Korea (corresponding author to -mails:tnluanvu@yahoo.com, mynlee@yu.ac.kr ).ISBN: 978-988-98671-0-2WCECS 2008
Proceedings of the World Congress on Engineering and Computer Science 2008WCECS 2008, October 22 - 24, 2008, San Francisco, USAII.MULTI-LOOP PI/PID CONTROLLER DESIGNA. The multi-loop feedback controller design for desiredset-point responsesConsider a general transfer function matrix for stable, square,and multi-delays MIMO processes represented as followingmatrix: g11 ( s ) g12 ( s ) g (s) g (s)22G (s) 21 g n1 ( s ) g n 2 ( s )g1n ( s ) g 2 n ( s ) g nn ( s ) (1)From a standard block diagram of multi-loop feedbackcontrol shown in Fig. 1, the closed-loop transfer functionmatrix can be written as(H( s) G( s)G c ( s) I G(s)G c ( s)) 1(2) ( s ) of a diagonal structureConsider a transfer function Hfor a desired closed-loop response. Then the feedbackcontroller to give the desired closed-loop response can bestraightforwardly found by rearranging (2). However, theresulting controller is generally not a diagonal (ordecentralized) form. By taking off all off-diagonal elementsfrom the resulting centralized controller, one can obtain amulti-loop (or decentralized) feedback controller as follows:{ (s) I H ( s ) 1G C ( s ) diag G -1 ( s ) H }adj G (s)G (s)(4)of adjG ( s ) hii ( s ) G C ( s ) diag G ( s ) 1 hii ( s ) (6)Therefore, each element of the multi-loop controller can bederived asG ii ( s ) hii ( s ) g ci ( s ) G (s) 1 hii ( s ) (7)From Bristol [14], the diagonal element of thefrequency-dependent relative gain array for G(s) iscalculated byΛ ii ( s) gii ( s)G ii ( s)G ( s)(8)Hence, by substituting (8) into (7), each element of themulti-loop controller can be obtained as h ( s) g ci ( s ) Λ ii ( s ) gii 1 ( s ) ii 1 hii ( s ) (9)According to the IMC theory [7], the desired closed-looptransfer function hii ( s ) is chosen ashii ( s ) ewhere z k , θii sqifi ( s ) k 1 s zk,s zk*i 1, 2, , n(10)zk* and θ ii denote the right-haft-plane (RHP)zeros of the (i, i)th diagonal element of the process transferfunction matrix, the corresponding complex conjugate ofRHP zeros, and the time delay term, respectively; qi denotesthe number of z k ; f i ( s ) is the IMC filter of the ith loop andchosen simply asadjG G and G is the cofactorcorresponding to gij in G ; G ( s ) denotes the determinantwherecorresponds to the desired servo closed-loop transferfunction for each loop.Substituting (4) and (5) into (3) gives(3)It is clear that the controller by (3) gives a closed-loopresponse closer to the desired one as process interactions areinsignificant. Since the multi-loop controllers are usuallyapplied to processes with modest interactions, this approachcan have validity.Note that the multi-loop controller by (3) is not a standardPID form. The controller above consists of two parts. i.e., 1G 1 ( s ) and H ( s ) I H ( s ) .G 1 ( s ) can be written asG 1 ( s ) ( s ) andwhere hii is each diagonal element of HijjiG ( s) . ( s) I H (s) Furthermore, H 1can be expressed interms of diagonal element as 1 hii (s) (s) I H (s) 1 H -1H (s) I diag 1 h (s) ii ISBN: 978-988-98671-0-2(5)fi ( s) 1( λi s 1)ri(11)The IMC filter time constant λi , which is also equivalent tothe closed-loop time constant, is an adjustable parameter toachieve the adequate tradeoff between performance androbustness.Substituting (10) and (11) into (9), the multi-loop controllerof the ith loop can be rewritten byWCECS 2008
Proceedings of the World Congress on Engineering and Computer Science 2008WCECS 2008, October 22 - 24, 2008, San Francisco, USAqi z s θii se k* 1k 1 zk s (12) gci (s) Λii (s)gii (s)qi zs ri θii sk ( λi s 1) e z* s k 1 k Note that in (12), the non-minimum portion of gii ( s ) isC. Example of two-input, two-output (TITO) caseThe TITO multi-delay processes are very popular in theprocess industry. In this section, the TITO multi-delayprocesses with the first-order plus delay time (FOPDT)dynamics are considered. The multi-loop feedback controllercan be obtained from (12) ascancelled out with the time delay and RHP zero zk in thenumerator so that the controller has neither causality norstability problem.B. Reduction to the multi-loop PI/PID controllerΛ ii ( s) (13)qi z s θii se k* k 1 zk s (14)pii (s) sΛii (s)gii 1(s) qi zsri θii sk ( λi s 1) e z* s k 1 k where (20)the1(21)K12 K 21 (T11s 1)(T22 s 1) θei se1 K11 K 22 (T12 s 1)(T21s 1)effectivedelayθ eiisdefinedbyθ ei θ12 θ 21 θ11 θ 22 .Substituting (21) into (20), an analytical tuning rule of themulti-loop PI controller can be obtained by using (17) and(18) asFurthermore, (13) can be expanded by using the Maclaurinseries asK Ci {θp′′ (0)1 g ci ( s ) pii (0) spii′ (0) s 2 ii (15)2s K Ii Since the standard form of multi-loop PID controller isgiven byΛ ii (0) 2 K ii (λi θ ii ) 22ii(22)} 2 Λ ii ( 0 )( λi θ ii ) K ei (Tei θ ei ) Tii Λ ii (0)Kii (λi θii )(23)K ei denotes the interaction quotient [15] andK KK ei 12 21 . The effective time constant Tei is definedK11 K 22where(16)The proposed PID controller is found by the comparisonbetween (15) and (16).asTci T jj Tij T ji , j i .It is noted thatK Ci diag { pii′ (0) }(17)K Ii diag { pii (0)}(18)K Di diag { pii′′ (0) / 2}(19)From (17), (18), and (19), it is straightforward to design themulti-loop PI/PID controller for various multivariableprocesses with delays.ISBN: 978-988-98671-0-2Kii1 θii s ( λi s 1) erespectively. The order of the IMC filter is selected as 1 forthe controller to be realizable.The (i, i)th element of the frequency-dependent relativegain array is calculated byThus, (s) 1 K sK s 2K GCiIiCiDi s (Tii s 1) where K ii and Tii denote the gain and time constant of gii ,For n x n processes with multi-delays, the proposedmulti-loop controller can be found by the followingprocedure:The multi-loop feedback controller can be rewritten asgci ( s) s 1 pii ( s )g ci ( s) Λ ii ( s)Λ ii (0) corresponds to the diagonalelement of the steady-state relative gain array (RGA) byBristol [14].III. ROBUST STABILITY ANALYSISThe robustness of control system is one of the mostimportant issues in any controller design because thedynamics of real plants usually have many sources ofWCECS 2008
Proceedings of the World Congress on Engineering and Computer Science 2008WCECS 2008, October 22 - 24, 2008, San Francisco, USAuncertainty, which cause a poor performance or eveninstability in control systems. In this study, the well knownrobustness analysis [16], [17] is introduced for faircomparison with other existing controller design methods.The robust stability can be examined under the outputmultiplication uncertainty. For the multi-delay process withthe output multiplicative uncertainty of 0 , the upper boundof robust stability can be written by{(In this example,γ(25)is chosen as 0.53 both for the proposedand J. Lee [16] design methods. From (24),)} 1γ σ ( 0 ) 1/σ I G( jω)Gc ( jω) G( jω)Gc( jω) (1.3 e -0.3 s 7s 1 4.3e -0.35 s 9.2s 1 -2.2 e - s 7s 1G(s) -2.8e -1.8 s 9.5s 1) 1 σ I G( jω)Gc( jω) , ω 0 (24)where G ( jω )G c ( jω ) is invertible.For fair comparison, the degree of robust stability will behold at the same level for all design methods compared. In thesimulation study, the proposed multi-loop PI/PID controlleris tuned by adjusting the closed-loop time constant λi so thatthe γ value of the proposed control system should be kept assame as or at least larger than those by the other comparativemethods.λi are obtainedas 1.55 and 0.25 for loop 1 and loop 2, respectively. Allcontrol parameters are listed in Table I. Fig. 2 shows that theproposed method provides a stable and robust response. Asshown in Table I, the proposed design method gives the bestclosed-loop performance in terms of IAE under the same ormore robust stability.Table I: Controller parameters and performance indices bythe various methods: VL columnProposedJ. LeeSATKc-1.9, 5.45-1.31, 3.97-1.35, 3.36τI6.54, 8.652.26, 2.423.00, 1.33IAEt5.687.197.28γ0.530.530.40IAEt : total sum of IAE of each loop.Example 2.A multi-product distillation column forseparation of binary ethanol-water mixture was modeledexperimentally [18]. The transfer function matrix of the ORcolumn is given by(a) 0.66e 2.6s 6.7s 1 6.5s 1.11eG(s) 3.25s 1 9.2s 34.68e 8.15s 1 0.6e 3.5s 0.0049e s 8.64s 19.06s 1 2.36e 3s 0.01e 12s (26) 5s 17.09s 1 46.2e 9.4s 0.89(11.61s 1)e s 10.9s 1 (3.89s 1)(18.8s 1) For a fair comparison, the upper bound γ of robuststability of the proposed method is selected by 0.035 same asthose by the BLT [3] and Y. Halevi [6] methods.Accordingly, the closed-loop time constant λi is found as(b)Fig. 2 Closed-loop responses for VL column by sequentialset-point changes in loop 1 and loop 2.IV. SIMULATION STUDYExample 1: Consider the following Vinante and Luyben(VL) column studied by W. Luyben [3].ISBN: 978-988-98671-0-28.85, 8.85, and 1.65 for loop 1, 2, and 3, respectively.Fig. 3 compares the closed-loop time responses by severaldesign methods. The magnitude of step set-point wassequentially made on loop 1, 2 and 3 by 1, 1, and 20,respectively. The order of the IMC filter is chosen as 1 for allloops. As seen from Fig. 3, the proposed method yields asuperior closed-loop response over the other methods whilethose by the BLT and Y. Halevi methods are sluggish andunbalanced.WCECS 2008
Proceedings of the World Congress on Engineering and Computer Science 2008WCECS 2008, October 22 - 24, 2008, San Francisco, USAV. CONCLUSIONSIn this paper, a novel analytical design method is proposedfor the multi-delay processes. The proposed method isstraightforward and easy to implement on the multi-loopcontrol systems. The robust stability and performance can beefficiently fixed by adjusting a single parameter, i.e., theclosed-loop time constant.The time-domain simulation illustrates that the proposedcontrol system provides a fast and well-balanced closed-looptime [9][10][11](c)[12]Fig. 3 Closed-loop responses for OR column by sequentialset-point changes in loop 1, 2 and 3.Table II: Controller parameters and performance indices bythe various methods: OR 64.8119.596BLT1.510-0.2952.63016.4018.006.610Y. 3.503979.124γ0.0350.0350.035KcτI[17][18]J. G. Ziegler, N. B. Nichols, “Optimum settings for automaticcontrollers,” Trans. ASME , vol. 64, 1942, pp.759-768.A. Niederlinski, “A heuristic approach to the design of linearmultivariable interacting control systems,” Automatica, vol. 7, 1971,pp. 691-701.W. L. Luyben, “Simple method for tuning SISO controllers inmultivariable systems,” Ind. Eng. Chem. Process Des. Dev., vol. 25,1986, pp. 654-660.J. Lee, T. F. Edgar, “Continuation method for the modified Ziegler Nichols tuning of multi-loop control systems,” Ind. Eng. Chem.Res,vol. 44, 2005, pp. 7428-7434.A. P. Loh, C. C. Hang, C. K. Quek, and V. U. Vasnani, “Autotuning ofmulti-loopproportional - integralcontrollersusingrelayfeedback,” In. Eng. Chem. Res., , vol. 32, 1993, pp. 1102-1107.Y. Halevi, Z. J. Palmor, and T. Efrati, “Automatictuning ofdecentralized PID controllers for MIMO processes,” Journal ofProcess Control, vol. 7, 1997, pp. 119-128.M. Morari., E. Zafiriou, Robust process control, New Jersey: PrenticeHall-Englewood Cliffs, 1989, ch. 4.M. Lee, K. Lee, C. Kim, and J. Lee, “Analytical design of multi-loopPID controllers for desired closed-loop responses,” AIChE , vol.50,2004, pp. 1631-165.L. V. Truong-Nguyen, J. Lee, M. Lee, “Design of multi-loop PIDcontrollers based on the generalized IMC-PID method with Mpcriterion,” IJCAS, vol.5, 2007, pp. 212-218.J. G. Truxal, Automatic Feedback Control System Synthesis, NewYork: McGraw-Hill, 1955.D. E. Seborg, T. F. Edgar, D. A. Mellichamp, Process Dynamics andControl, New York: John Wiley & Sons. 1989.Q.G. Wang, T. H. Lee, C. Lin., Relay feedback: analysis, identificationand control, London : Springer, 2003.S. Skogestad, I. Postlethwaithe, Multivariable feedback controlanalysis and design, New York: John Wiley & Sons, 2005.E. H. Bristol, “Recent results on interactions in multivariable processcontrol,” Proceedings of the 71st annual AIChE meeting, Houston,USA, 1979.J.E Rijnsdorp, “Interaction in two-variable control system fordistillation columns-I and II,” Automatica, vol. 1, 1965, pp. 15-28.J. Lee, W. Cho, T. F. Edgar, “Multi-loop PI controller tuning forinteracting multivariable processes,” Comp. Chem. Eng., vol.22, 1998,pp.1711-1723.S. L. William, Control system fundamentals, Florida: CRC Press LLC,1999.B. A. Ogunnaike, J. P. Lemaire, M. Morari, and W. H. Ray, “Advancedmultivariable control of a pilot plant distillation column,” AIChE, vol.29,1983, pp.225-230.The IAE values listed in Table II also show the superiorityof the proposed method over the other existing methods.ISBN: 978-988-98671-0-2WCECS 2008
Multi-loop PI/PID Controller Design Based on Direct Synthesis for Multivariable Systems Abstract—In this paper, a new analytical method based on the direct synthesis approach is proposed for the design of a multi-loop proportional-integral-derivative (PID) controller.
PID-controller Today most of the PID controllers are microprocessor based DAMATROL MC100: digital single-loop unit controller which is used, for example, as PID controller, ratio controller or manual control station. Often PID controllers are integrated directly into actuators (e.g valves, servos)File Size: 1MBPage Count: 79Explore furtherWhen not to use PID-controllers - Control Systems .www.eng-tips.comPID Controller-Working and Tuning Methodswww.electronicshub.org(PDF) DC MOTOR SPEED CONTROL USING PID CONTROLLERwww.researchgate.netTuning for PID Controllers - Mercer Universityfaculty.mercer.eduLecture 9 – Implementing PID Controllerscourses.cs.washington.eduRecommended to you b
4.1 Simulink Block of PID Controller 58 4.2 Detailed Simulink Block of the System 60 4.3 Output of DC Motor without PID Controller 60 4.4 Detail Simulink Block of the System with PID Controller 61 4.5 Output of DC Motor without PID Controller 62 4.6 Sim
SPEED CONTROL WITH PID CONTROLLER A proportional-integral-derivative controller (PID controller) is widely used in industrial control systems. It is a generic control loop feedback mechanism and used as feedback controller. PID working principle is that it calculates an error
Plot System Responses . (time-domain response) or Bode plots (frequency-domain response). For 1-DOF PID controller types such as PI, PIDF, and PDF, PID Tuner computes system responses based upon the following single-loop control architecture: For 2-DOF PID controller types such as PI2, PIDF2, and I-PD, PID Tuner computes responses based upon .
fuzzy controller are 69.9 % and 67.9 % less than PID controller. 6. CONCLUSION Theses. Paper This paper presents the control of the level in a single tank using different two type controllers PID and fuzzy. From program simulation that built it was indicates that the fuzzy controller has more Advantages to the system than the PID controller.
Fig.4.1 Step response using P controller Fig.4.2 Fig.4.3 Fig.4.4 Fig.4.5 Step response using PID controller Fig.4.6 Comparison of bode plot for smith predictor and PI . Step response of IMC PDF control structure Step response of PDF controller Step response using PI controller Step response using PID controller Step response using PI controller
4. Design of PID Controller . Industrial PID controllers are usually available as a packaged, and it's performing well with the industrial process problems. The PID controller requires optimal tuning. Figure 3shows the diagram of a simple closed loop - cont
The American Board of Radiology . i The Diagnostic Radiology Milestone Project The Milestones are designed only for use in evaluation of resident physicians in the context of their participation in ACGME accredited residency or fellowship programs. The Milestones provide a framework for the assessment of the development of the resident physician in key dimensions of the elements of physician .
