PERFORMANCE ASSESSMENT OF PID CONTROLLERS

PERFORMANCE ASSESSMENT OF PID CONTROLLERS W. Tan, H. J. Marquez, and T. Chen Abtract: Criteria baed on diturbance rejection and ytem robutne are propoed to ae the performance of PID controller. The robutne i meaured by a two-block tructured ingular value, and the diturbance rejection i meaured by the minimum ingular value of the integral gain matrix. Example how that the criteria can be applied to a variety of procee, whether they are table, integrating or untable; ingle-loop or multi-loop. Key Word: PID Control; Tuning; Performance; Robutne; Structured Singular Value. Introduction PID controller are widely ued in the indutry due to their implicity and eae of re-tuning on-line []. In the pat four decade there are numerou paper dealing with the tuning of PID controller. Project Sponored by the Scientific Reearch Foundation for the Returned Overea Chinee Scholar, SEM, and by the Reearch Foundation for the Doctoral Program, NCEPU Department of Automation, North China Electric Power Univerity, Zhuxinzhuang, Dewai, Beijing 226, China. E-mail: wtan@ieee.org Department of Electrical and Computer Engineering, Univerity of Alberta, Edmonton, AB T6G 2V4, Canada. E-mail: marquez@ece.ualberta.ca Department of Electrical and Computer Engineering, Univerity of Alberta, Edmonton, AB T6G 2V4, Canada. E-mail: tchen@ece.ualberta.ca

See, for example, reference [2 9] for table procee; reference [ 7] for integrating and untable procee; and reference [8 23] for multivariable procee. A natural quetion arie: How can the PID etting obtained by different method be compared? Or more generally, how can the performance of a controller be aeed? In proce control, minimum variance ha been ued a a criterion for aeing cloed-loop performance for decade [24, 25]. Thi criterion i a valuable meaure of ytem performance but it pay little attention to the traditional performance uch a etpoint tracking and diturbance rejection. Beide, another important factor of ytem performance robutne i not addreed directly. Clearly a criterion that can be ued for table, integrating or untable; ingle- and/or multi-loop procee would be highly deirable. Thi criterion hould include time domain property a well a frequency domain robutne pecification. For ingle-loop procee, the integral error i a good meaure of ytem performance and the gain-phae margin i a good robutne meaure. Thu, a combination of thee two element can erve a a criterion for ytem performance aement. A comparion of the gain-phae margin of ome well-known PID tuning method ha been reported in [26]. But unfortunately gain and phae margin are not uitable for multiloop procee. In thi paper we will propoe criteria to ae ytem performance. The criteria reflect diturbance rejection performance and ytem robutne. The robutne i meaured by a two-block tructured ingular value, and the diturbance rejection i meaured by the minimum ingular value of the integral gain matrix. Example how that the criteria can be applied to a variety of procee, whether they are table, integrating or untable; ingle-loop or multi-loop. 2

2 Performance Aement of Cloed-loop Sytem It i well-known that a well-deigned control ytem hould meet the following requirement beide nominal tability: Diturbance attenuation Setpoint tracking Robut tability and/or robut performance The firt two requirement are traditionally referred to a performance and the third, robutne of a control ytem. 2. Performance The integral error i a good meaure for evaluating the etpoint and diturbance repone. The following are ome commonly ued criteria baed on the integral error for a tep etpoint or diturbance repone: IAE : ITAE : ISE : ITSE : ISTE : e(t) dt t e(t) dt e(t)2 dt te(t)2 dt t2 e(t) 2 dt Thee criteria, however, are not uitable for multivariable procee, ince each criterion i defined for a ingle-loop proce. 3

Conider the unity feedback ytem (poibly multi-loop) hown in Fig.. Since diturbance rejection i more common in indutrial procee than etpoint tracking, the performance of the ytem may be evaluated by it ability to reject diturbance. d G d r _ e K G + y Figure : Typical unity feedback configuration The tranfer function from d to y i T yd =(I + GK) G d () Aume our controller K ha integral action, we can decompoe it a K()=K i / + K m () (2) where K i i the integral gain and K m i the part of the controller without integral action. Then at low frequency, we have σ((i + GK) G d )( jω)) jω σ((g( jω)k i ) G d ( jω)) σ(g d ( jω)) jω σ(g( jω))σ(k i ) (3) 4

where σ( ) and σ( ) denote the maximum and minimum ingular value of a matrix, repectively. In indutrial procee, the diturbance uually occur at low frequency, o to reject a diturbance, the mot important element of a controller i it integral gain, or pecifically, the minimum ingular value of the integral gain, thu it can erve a a meaure of ytem performance. A pointed out in [] for a ingle-loop proce, e(t)dt = K i o the integral gain i related to the integral of the error (IE). Moreover, if the repone i critically damped, IE would be equal to IAE. So the minimum ingular value of the integral gain i a natural extenion a a performance meaure to multi-loop procee. 2.2 Robutne For robut tability, a common choice of repreenting uncertainty for a multivariable ytem i the multiplicative perturbation, and the maximum ingular value of the complementary enitivity matrix i a meaure of robutne againt thi kind of uncertainty, which i uually frequencydependent, and uited for the unmodeled dynamic intead of parameter variation. The coprime factor uncertainty can repreent model uncertainty in a better way [27]. The uncertain model i repreented a: G =( M + M ) (Ñ + N ) (4) where G = M Ñ i a left normalized coprime factorization of the nominal plant model, and the uncertainty tructure i =[ M N ], < γ (5) 5

Then the ytem i robutly table if and only if ε := I K (I + GK) M γ (6) So ε can erve a a meaure of ytem robutne. However, we note that thi uncertainty clearly ignore the tructure of M and N. Suppoe M = W, N = W 2 2 (7) and define = 2 (8) then =[ W W 2 ] (9) For thi uncertainty tructure, we have (I + G K) = I + W [ ] W 2 2 I K (I + GK) M (I + GK) ( + M W ) () By the definition of tructured ingular value [28], the cloed-loop ytem i robutly table for 6

all < γ if and only if µ I K (I + GK) M [ W W 2 ] < γ () If we chooe pecial weighting a follow: W = M;W 2 = Ñ (2) and define ε m := µ I K (I + GK) [ I G ] (3) Then ε m i a better meaure of ytem robutne. We note that now the cla of uncertain plant can be repreented a G =( M + M ) (Ñ + Ñ 2 )=(I + ) G(I + 2 ) (4) o it can repreent imultaneou input multiplicative and invere output multiplicative uncertainty. If we treat a diturbance a a model uncertainty, then it can alo repreent imultaneou input and output diturbance. For a ingle-loop ytem, it can be hown that ε m = max( S( jω) + T( jω) ) (5) ω where S and T are the enitivity and complementary enitivity function of the cloed-loop ytem, repectively. The value approache at low and high frequencie and the maximum 7

occur at the mid-range frequencie. Compared with the uual indicator uch a M, the peak of the enitivity function, or M p, the peak of the complementary enitivity function, the meaure i more appropriate ince it bound both M and M p imultaneouly. In ummary we can ae the performance of a controller by evaluating the minimum ingular value of it integral gain matrix, and ae the robutne by the robutne meaure ε m defined in (3). We mainly concern with the diturbance repone. The etpoint repone can alway be improved by uing a etpoint filter or a etpoint weighting. The dicuion above ugget that we can deign an optimal PID controller by olving the following optimization problem: maxσ(k i ) (6) under the contraint µ I K (I + GK) [ I G ] < γ m (7) where γ m i a given robut tability requirement. The problem amount to maximizing the integral action under the contraint of a certain degree of robut tability, a generalization of the idea ued in [29, 3] for ingle-loop procee. The problem propoed i a nonconvex optimization problem thu it i not eay to olve directly. However, the loop-haping H approach provide a olution to a uboptimal problem. Detail can be found in [3]. 3 Illutrative Example In thi ection, we will apply the criteria propoed in the previou ection to analyze the PID controller etting for ome typical procee. 8

3. A firt-order plu deadtime (FOPDT) proce Conider a proce with the following model: P()= + e.5 (8) Table how the PID etting tuned by the following well-known tuning rule: ) Ziegler-Nichol (Z-N) [2]. 2) Cohen-Coon (C-C) [3]. 3) Internal model control (IMC) [5]. The IMC method ha a tuning parameter. The maller it i, the better performance the cloed-loop ytem will have, and the le robut the cloedloop ytem i. Here the tuning parameter λ i choen a.25 of the delay, the mallet value a uggeted by [5]. 4) Gain-phae margin (GPM) [4, 8]. Since different pair of gain-phae margin will reult in different PID etting, here we chooe the tuning formula given in [7] where the gain-phae margin i optimized. 5) Optimum integral error for load diturbance (ISE-load, ISTE-load, ITAE-load) [7, 32]. 6) Optimum integral error for etpoint change (ISE-etpoint, ISTE-etpoint, ITAE-etpoint) [7, 32]. It can be oberved that the reulting controller can be divided into three group: i) Controller tuned by IMC, GPM, ISTE-etpoint and IAE-etpoint method have mall integral gain and mall robutne meaure. 9

Table : PID etting for example K p T i T d K i ε m IMC 2.25.2.6 3.93 GPM.938.64.28.665 3.92 ISE-etpoint.952.989.264.973 3.646 ISTE-etpoint.94.52.26.684 3.78 IAE-etpoint.674.24.92.39 2.544 ITAE-etpoint 2.393.2.75.992 4.54 Z-N 2.285.855.24 2.67 3.92 C-C 2.97.29.67 2.833 6.24 ISE-load 2.885.532.285 5.422 9.852 ISTE-load 2.876.642.23 4.477 6.698 IAE-load 2.673.852.2 3.39 5.88 ITAE-load 2.475.83.83 3.45 4.258 ii) Controller tuned by Z-N, ITAE-etpoint, ITAE-load and ISE-etpoint method have medium integral gain and medium robutne meaure. iii) Controller tuned by C-C, ISE-load, ISTE-load and IAE-load method have large integral gain and large robutne meaure. Fig. 2 how the the cloed-loop ytem repone of all the PID controller for a tep etpoint change of magnitude at t = following a tep load diturbance of magnitude at t = for the nominal model and for the perturbed cae that the deadtime increae by 2%. It i oberved that the integral gain and robutne meaure given by the firt group are too mall, thu the cloedloop ytem are very robut but the load rejection performance can be further improved. The integral gain and robutne meaure given by the third group are too large, thu the cloed-loop ytem how ocillatory repone and are not robut. The econd group give proper integral gain and robutne meaure.

.4.4.2.2.8.6.8.6.4.2 IMC G P ISTE etpoint IAE etpoint.4.2 IMC G P ISTE etpoint IAE etpoint 2 4 6 8 2 4 6 8 2.6 2 4 6 8 2 4 6 8 2 2.4.2.8.6.8.6.4.2.8.4.2 Z N ITAE etpoint ITAE load ISE etpoint.6.4.2 Z N ITAE etpoint ITAE load ISE etpoint 2 4 6 8 2 4 6 8 2 2 2 4 6 8 2 4 6 8 2 2.5.8.6 2.4.2.8.5.6.4 C C ISE load ISTE load IAE load.5 C C ISE load ISTE load IAE load.2 2 4 6 8 2 4 6 8 2 2 4 6 8 2 4 6 8 2 (a) Nominal model (b) Deadtime increae by 2% Figure 2: Repone of different PID etting for example

3.2 A proce with complex pole Second-order plu deadtime procee (SOPDT) are harder to tune than FOPDT procee due to the exitence of (poibly) underdamped complex pole. Nonethele, the criteria apply alo to uch procee. To illutrate, conider a proce with the following model: P()= 2 +. + e.2 (9) It ha a very mall damping ratio, thu repreent a heavily ocillatory proce. Table 2: PID etting for example 2 K p T i T d K i ε m γ m = 3.996.49.985.46 2.942 γ m = 4 3.825.55.7 2.468 3.92 γ m = 5 5.365.575.68 3.45 4.97 Table 2 how the PID etting tuned by olving the (ub)optimization problem propoed at the end of the previou ection with different value of γ m. We do not claim that the olution are optimal. The uboptimal olution are jut ued to illutrate the impact of the robutne meaure on ytem performance. Fig. 2 how the the cloed-loop ytem repone of all the PID controller for a tep etpoint change of magnitude at t = following a tep load diturbance of magnitude at t = 2 for the nominal model and for the perturbed cae that the deadtime increae by 3%. It i oberved that the integral gain and robutne meaure computed with γ m = 3 are too mall, thu the cloedloop ytem are very robut but the load rejection performance can be further improved. The integral gain and robutne meaure computed with γ m = 5 are too large, thu the repone of the cloed-loop ytem i ocillatory and not robut. γ m = 4 give a good trade-off. Extenive imulation how that for a table proce the robutne meaure ε m defined in (3) 2

.4.5.2.8.6.4.5.2 5 5 2 25 3 35 4 5 5 2 25 3 35 4 (a) Nominal (b) Deadtime increae by 3% Figure 3: Repone of different PID etting for example 2 (dahed: γ m = 3; olid: γ m = 4; dahdotted: γ m = 5) hould lie between 3 and 5 to have a good compromie between performance and robutne. 3.3 A firt-order delayed untable proce (FODUP) PID tuning for integrating and untable procee i much harder than that for table procee. There are few imple tuning formula a thoe in the cae of table procee available in the literature. Here we conider a firt-order delayed untable proce a an illutrating example: P()= e.4 (2) Table 3 how the PID etting tuned by typical method found in the literature, and the PID etting tuned by olving the optimization problem propoed at the end of the previou ection with γ m = 5. Alo hown are the correponding integral gain and the robutne meaure for each PID etting. From the table we oberve that only the controller tuned by R-L, H-C, IMC and the propoed method are robut enough. Now a good compromie between performance 3

Table 3: PID etting for example 3 K p T i T d K i ε m De Poar and O Malley (P-M) [].459 2.667.25.547.46 Rottein and Lewin (R-L) [] 2.25 5.76.2.39 4. Poulin and Pomerleau (P-P) [2] 2.25 4.738.427 9.4 Huang and Chen (H-C) [3] 2.636 5.673.8.465 5.7 Tan et.al. [5] 2.428 2.38.98.2 7.6 IMC [6] 2.634 2.52.54.45 5.29 ITSE-etpoint [7] 3.48.86.2.743 9.76 Propoed 2.467 4.8.5.64 4.64 and robutne require that the robutne meaure for the untable proce lie between 4 and 6 compared with between 3 and 5 for the table procee. The IMC method ha the bet load rejection, a hown by the cloed-loop ytem repone in Fig. 4(a) for a tep etpoint change of magnitude at t = following a etp load diturbance of magnitude at t = 2 for the nominal model. The PID controller by the H-C method wa hown to be wore than that by the IMC method in performance and robutne [6], o it repone i omitted here. The PID controller tuned by the R-L method ha the bet robutne, a hown by the cloed-loop ytem repone in Fig. 4(b) for a perturbed model with the deadtime increaed by 2%. The new etting by the propoed method ha the bet compromie between performance and robutne. 3.4 A multivariable proce The criteria can alo be ued to compare PID etting for multivariable procee. To illutrate, conider the ditillation column model reported by Wood and Berry [33]: G()= 2.8e 6.7+ 6.6e 7.9+ 8.9e 3 2+ 9.4e 3 4.4+ (2) 4

2.5 3 2 2.5.5 2.5.5.5 5 5 2 25 3 35 4 5 5 2 25 3 35 4 (a) Nominal (b) Deadtime increae by 2% Figure 4: Repone of different PID etting for example 3 (olid: propoed; dahed: R-L; dahdotted: IMC) The proce i highly coupled and attract much attention in the literature. Table 4 how the PID etting tuned by variou method found in the literature, and the PID etting deigned by olving the optimization problem propoed at the end of the previou ection with γ m = 4. It i clear that the PID controller given in [23, 33] have very large robutne meaure, and thoe given in [8, 22] have too mall integral action. For the ret etting, the propoed PID ha the bet diturbance ejection, which can be hown in Fig. 5. To tet the robut performance of the controller, uppoe the proce delay change, and the perturbed model become G p ()= 2.8e 2 +6.7 6.6e +.9 8.9e 4 +2 9.4e 4 +4.4 (22) The diturbance repone for all the controller are hown in Fig. 6. Again, the new etting by the propoed method ha the bet compromie between performance and robutne. 5

Table 4: PID controller etting for example 4 [2] [22] [8] [2] [33] [23] [3] [ PID controller σ(k i ) ε m.84 +.469.2.229 ] +.82.674 +.59.537.66.55. 2.647 [.2833 +.285 (.45 +.285 ] ).954 +.97 (.2 +.48.56 3.927 ) [.375( + 8.29 ) ].75( + 23.6 ).32 4.3 [.83( +.7 ) ].72( +.7 ). 4.52 [.2( + 4.44 ) ].4( + 2.67 ).5 7.55 [.637( + 3.84 ) ].96( + 7.4 ).3 8.756 [.2796 +.334 (.85 ] +.98) (.38 +..36.3 3.988 ) (.89 + +.493 5+ ) 3 2.5 2.5.5 2 3 4.5 5 2 3 4 5 6 7 8 (a) Input diturbance at t channel 6 2 3 4 5 6 7 8 (b) Input diturbance at 2nd channel Figure 5: Proce diturbance repone for example 4: nominal cae (olid: propoed; dahdotted: [2]; dahed: [2]) 6

3.5 3 3 2 2.5 2.5.5 2 3 4.5 5 6.5 2 3 4 5 6 7 8 (a) Input diturbance at t channel 7 2 3 4 5 6 7 8 (b) Input diturbance at 2nd channel 4 Concluion Figure 6: Proce diturbance repone for example 4: perturbed cae (olid: propoed; dahdotted: [2]; dahed: [2]) Criteria baed on diturbance rejection and ytem robutne were propoed to ae the performance of PID controller. The robutne i meaured by a two-block tructured ingular value, and the diturbance rejection i meaured by the minimum ingular value of the integral gain matrix. Example howed that the criteria can be applied to a variety of procee, whether they are table, integrating or untable; ingle-loop or multi-loop. It wa alo oberved that robutne meaure hould lie between 3 and 5 to have a better compromie on performance and robutne for table procee, and between 4 and 6 for untable and integrating procee. 7

Reference [] K. J. Atrom & H. Hagglund, PID controller: theory, deign, and tuning. (Reearch Triangle Park, NC: Intrument Society of America, 2nd ed., 995). [2] J. G. Ziegler & N. B. Nichol, Optimum etting for automatic controller, Tran. A.S.M.E., 62(3), 942, 759 768. [3] G. H. Cohen & G. A. Coon, Theoretical conideration of retarded control, Tran. A.S.M.E., 75(4), 953, 827 834. [4] K. J. Atrom & T. Hagglund, Automatic tuning of imple regulator with pecification on phae and amplitude margin, Automatica, 2(5), 984, 645 65. [5] M. Morari & E. Zafiriou, Robut proce control. (Englewood Cliff NJ.: Prentice-Hall, 989). [6] C. C. Hang, K. J. Atrom, & W. K. Ho, Refinement of the Ziegler-Nichol tuning formula, IEE Proc. Contol Theory and Application, 38(2), 99, 8. [7] M. Zhuang & D. P. Atherton, Automatic tuning of optimum PID controller, IEE Proc. Control Theory and Application, 4(3) 993, 26 224. [8] W. K. Ho, C. C. Hang, & L. S. Cao, Tuning of PID controller baed on gain and phae margin pecification, Automatica, 3(3), 995, 497 52. [9] W. Tan, J. Z. Liu, & P. K. S. Tam, PID tuning baed on loop-haping H control, IEE Proc. Control Theory and Application, 45(4), 998, 485 49. [] A. M. D. Poar & M. O Malley, Controller of Ziegler-Nichol type for untable procee, Int. J. Control, 49(6), 989, 273 284. 8

[] G. E. Rottein & D. R. Lewin, Simple PI and PID tuning for open-loop untable ytem, Ind. Eng. Chem. Re., 3(8), 99, 864 869. [2] E. Poulin & A. Pomerleau, PID tuning for integrating and untable procee, IEE Proc. Control Theory and Application, 43(5), 996, 429 434. [3] H. P. Hung & C. C. Chen, Control-ytem ynthei for open-loop untable proce with time-delay, IEE Proc. Control Theory and Application, 44(4), 997, 334 346. [4] L. Wang & W. R. Cluett, Tuning PID controller for integrating procee, IEE Proc. Control Theory and Application, 44(5), 997, 385 392. [5] W. Tan, Y. Q. Yuan, & Y. G. Niu, Tuning of PID controller for untable proce, Proc. of the IEEE International Conf. on Control Application (CCA), Hawaii, USA, 2 24, 999. [6] Y. Lee, J. Lee, & S. Park, PID controller tuning for integrating and untable procee with time delay, Chem. Eng. Sci., 55(7), 2, 348 3493. [7] A. Viioli, Optimal tuning of PID controller for integral and untable procee, IEE Proc. Control Theory and Application, 48 (2), 2, 8 84. [8] W. L. Luyben, Simple method for tuning SISO controller in multivariable ytem, Ind. Eng. Chem. Re., 25(), 986, 654 658. [9] M. Zhuang & D. P. Atherton, PID controller deign for a TITO ytem, IEE Proc. Control Theory and Application, 4(2), 994, 2. [2] Z. J. Palmor, Y. Halevi, & N. Kraney, Automatic tuning of decentralized PID controller for TITO procee, Automatica, 3(7), 995, 4. 9

[2] Q. G. Wang, Q. Zou, T. H. Lee, & Q. Bi, Autotuning of multivariable PID controller from decentralized relay feedback, Automatica, 33(3), 997, 39 33. [22] J. W. Dong & G. B. Broilow, Deign of robut multivariable PID controller via IMC, Proc. American Control Conference, Albuquerque, New Mexico, 338 3383, 997. [23] I. L. Chien, H. P. Huang, & J. C. Yang, A imple multiloop tuning method for PID controller with no proportional kick, Ind. Eng. Chem. Re., 38(4), 999, 456 468. [24] T. Harri, Aement of cloed loop performance, Can. J. Chem. Eng., 67(9), 989, 856 86. [25] B. Huang & S. L. Shah, Performance aement of control loop: theory and application. (London: Springer-Verlag, 999). [26] W. K. Ho, O. P. Gan, E. B. Tay, & E. L. Ang, Performance and gain and phae margin of well-known PID tuning formula, IEEE Tran. Control Sytem Technology, 4(4), 996, 473 477. [27] D. C. McFarlane & K. Glover, Robut controller deign uing normalized coprime factorization decription. (Springer-Verlag, 99). [28] J. C. Doyle, Analyi of feedback ytem with tructured uncertaintie, IEE Proc. Control Theory and Application, 29(3), 982, 242 25. [29] K. J. Atrom, H. Panagopoulo, & T. Hagglund, Deign of PI controller baed on nonconvex optimization, Automatica, 34(5), 998, 585 6. [3] H. Panagopoulo & K. J. Atrom, PID control deign and H loop haping, in Proc. IEEE Conf. on Control Apllication, Hawaii, USA, 3 8, 999. 2

[3] W. Tan, T. Chen, & H. J. Marquez, Robut controller deign and PID tuning for multivariable ytem, Aian J. Control, 4(4), 22, 439 45. [32] C. A. Smith & A. B. Corripio, Principle and practice of automatic proce control. (New York: Wiley, 985). [33] R. K. Wood & M. W. Berry, Terminal compoition control of a binary ditillation column, Chem. Eng. Sci., 28(6), 973, 77 7. Biographie Wen Tan received hi B.Sc. degree in applied mathematic and M.Sc. degree in ytem cience from the Xiamen Univerity, China, and Ph.D. degree in automation from the South China Univerity of Technology, China, in 99, 993, and 996, repectively. From October 994 to February 996, he wa a Reearch Aitant with the Department of Mechanical Engineering and Electronic Engineering, Hong Kong Polytechnic Univerity, Hong Kong. After June 996, he joined the faculty of the Power Engineering Department at the North China Electric Power Univerity, China, where he wa a Lecturer until December 998 and an Aociate Profeor from January 999. From January 2 to December 2, he wa a Potdoctoral Fellow in the Department of Electrical and Computer Engineering at the Univerity of Alberta, Canada. He i currently a Profeor with the Automation Department of the North China Electric Power Univerity (Beijing), China. Hi reearch interet include robut and H control with application in indutrial procee. Horacio J. Marquez Tongwen Chen 2