Design of PID controllers satisfying gain margin and sensitivity constraints on a set of plants

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1 Available online at Automatica Brief paper Deign of PID controller atifying gain margin and enitivity contraint on a et of plant O. Yaniv a, M. Nagurka b; a Department of Electrical Engineering Sytem, Faculty of Engineering, Tel Aviv Univerity, Tel Aviv 69978, Irael b Department of Mechanical and Indutrial Engineering, Marquette Univerity, Milwaukee, WI , USA Received 12 October 2002; received in revied form 16 July 2003; accepted 28 Augut 2003 Abtract Thi paper preent a method for the deign of PID-type controller, including thoe augmented by a lter on the D element, atifying a required gain margin and an upper bound on the complementary enitivity for a nite et of plant. Important propertie of the method are: i it can be applied to plant of any order including non-minimum phae plant, plant with delay, plant characterized by quai-polynomial, untable plant and plant decribed by meaured data, ii the enor aociated with the PIterm and the D term can be dierent i.e., they can have dierent tranfer function model, iii the algorithm relie on explicit equation that can be olved eciently, iv the algorithm can be ued in near real-time to determine a controller for on-line modication of a plant accounting for it uncertainty and cloed-loop pecication, v a ingle plot can be generated that graphically highlight tradeo among the gain margin, complementary enitivity bound, low-frequency enitivity and high-frequency enor noie amplication, and vi the optimal controller for a practical denition of optimality can readily be identied.? 2003 Elevier Ltd. All right reerved. Keyword: PID control; Robutne; Gain and phae margin; Senitivity 1. Introduction Method for tuning PIand PID controller have been reported widely and active reearch continue due to the extenive ue of uch controller in indutry. The tuning method can be divided into two main categorie, thoe emphaizing gain and phae margin pecication and thoe emphaizing enitivity pecication. Deign technique baed on gain and phae margin pecication include thoe by Ho, Hang, and Cao 1995a and Ho, Hang, and Zhou 1995b. They developed imple analytical formulae to tune PIand PID controller for commonly ued rt-order and econd-order plu dead time plant model to meet gain and phae margin pecication. Ho, Hang, and Zhou 1997 and Ho, Lim, and Xu 1998 preented tuning formulae for the deign Thi paper wa not preented at any IFAC meeting. Thi paper wa recommended for publication in revied form by Aociate Editor Martin Guay under the direction of Editor Frank Allgower. Correponding author. Tel.: addree: yaniv@eng.tau.ac.il O. Yaniv, mark.nagurka@marquette.edu M. Nagurka. of PID controller that atify both robutne and performance requirement. Crowe and Johnon 2002 preented an automatic PIcontrol deign algorithm to atify gain and phae margin baed on a converging algorithm. Suchomki 2001 developed a tuning method for PIand PID controller that can hape the nominal tability, tranient performance, and control ignal to meet gain and phae margin. Although gain and phae margin pecication are claical meaure of robutne, they may fail to guarantee a reaonable bound on the enitivity. Thi point wa conidered by everal reearcher. Ogawa 1995 ued the QFT-framework to propoe a PIdeign technique that ati- e a bound on the enitivity for an uncertain plant. Poulin and Pomerleau 1999 developed a PIdeign methodology for integrating procee that bound the maximum peak reonance of the cloed-loop tranfer function. The peak reonance contraint i equivalent to bounding the complementary enitivity, which can be converted to bounding the enitivity. Cavicchi 2001 preented a deign method for bounding the enitivity while achieving deired teady-tate performance. Although the method can be applied to meaured data, plant uncertainty i not /$ - ee front matter? 2003 Elevier Ltd. All right reerved. doi: /j.automatica

2 112 O. Yaniv, M. Nagurka / Automatica conidered, and the procedure t only imple compenation tructure. Crowe and Johnon 2001 reported a deign approach to nd a PI/PID controller that bound the enitivity while atifying a phae margin condition. Kritianon and Lennarton 2002 emphaized the need to bound the enitivity and complementary enitivity. They uggeted the ue of an optimization routine to deign PI/PID controller with low-pa lter on the derivative gain to optimize for control eort, while rejecting diturbance and bounding the enitivity. They alo gave tuning rule for non-ocillatory table plant or plant with a ingle integrator. Atrom, Panagopoulo, and Hagglund 1998 and Panagopoulo, Atrom, and Hagglund 2002 decribed a numerical method for deigning PIcontroller baed on optimization of load diturbance rejection with contraint on enitivity and weighting of the etpoint repone. Other tuning method have been propoed. Yeung, Wong, and Chen 1998 preented a non-trial and error graphical deign technique for controller deign of the lead-lag tructure that enable imultaneou fulllment of gain margin, phae margin and croover frequency pecication. Guillermo, Silva, and Bhattacharyya 2002 developed a theorem to calculate all tabilizing PID controller for rt-order delayed plant. However, uncertainty, enitivity and margin were not dicued. Thee paper and other apply gain and phae margin contraint in nding PIand PID controller deign. Some add limitation on the complementary enitivity. However, there are everal dierence between approache reported in the literature and the idea propoed here. Firt, the approach preented here bound the enitivity of the cloed-loop tranfer function for all frequencie, not jut at the croover frequencie where the gain and phae margin are atied. It i poible that the gain and phae margin condition are met with a given PI/PID deign, but the enitivity can be very high. Second, the approach account for plant uncertainty, with the controller deign atifying the pecication for a et of plant. Third, the algorithm can be applied to plant of any order including plant with pure delay, untable plant, and plant given by meaured data. Fourth, it allow for dierent enor model for the PIterm and the D term. Fifth, the approach relie on explicit equation, rather than optimization routine, to determine the et of all poible controller. Sixth, ince the algorithm ue explicit equation that can be olved eciently, it i very fat and uitable for near-real time implementation. Seventh, it i poible to extend the method to deign cacaded loop and other control tructure. 2. Problem tatement and motivation Conider the open-loop tranfer function, L, L=a[P 1 +bp 2 ]: 1 If P 1 =1+k i =P and P 2 =P, 1 correpond to the open-loop tranfer function of plant, P, with a PID controller C= ak i + a + ab: 2 It i poible to ue dierent enor for the D term and for the PIterm. In thi cae, if the tranfer function of the enor aociated with the D term i taken a H with H being a delay and/or a low-pa lter to decreae noie and the tranfer function of the enor for the PIterm i unity, then P 1 =1+k i =P, P 2 =HP, and the controller can be written a C= ak i + a + abh: 3 The gain and phae margin condition, the typical meaure of robutne, are replaced by a condition on the cloed-loop enitivity inequality, 1 1+kL 6 M for =j!;! 0; k [1;K]; 4 where the enitivity bound M 1and the gain uncertainty of the plant, k, i in the interval [1;K]. Yaniv 1999 how that 4 guarantee the following margin M GM = 20 log 10 K+20log 10 ; M 1 PM = 2 arcin[2m 1 ]: Inequality 4 i a more encompaing meaure of robutne than gain and phae margin. It place a bound on the enitivity at all frequencie, not jut at the two frequencie aociated with the gain and phae margin. Two deign problem are conidered here: 1 Determine all a; b pair that atify 4 where the pair P 1 ;P 2 i uncertain in the ene that it belong to a nite et of pair P 1m ;P 2m ;m=1;:::;n, and, in particular, extract an optimal pair a 0 ;b 0 for a given optimality criterion. 2 Replacing L in4 by L=a [ 1+ k i P 1 +bp 2 H ] ; = a[ P 1 +b P 2 ] 5 determine all a, b, k i and H of a given tructure that atify 4, and, in particular, extract an optimal olution a 0, b 0, k i0 and H 0. Thee two problem for the pecial cae of P 2 = P 1, without conidering plant and gain uncertainty, were olved by Atrom et al and Panagopoulo et al to determine a ingle controller for the cae of maximum k i a or a, where the olution were not obtained explicitly but numerically with MathWork Matlab 5 Optimization Toolbox.

3 O. Yaniv, M. Nagurka / Automatica Deign methodology In order to determine the a; b value for which the cloed-loop ytem i table and 4 i atied, conider rt the pecial cae of no gain uncertainty, i.e., K = 1, and a ingle plant pair P 1 ;P 2. Subtituting 1 into 4 after plitting P 1 and P 2 for =j! into real and imaginary part, P 1 j!=a 1!+jB 1!; P 2 j!=a 2!+jB 2! give, 1 + aa 1 + aba 2 2 +ab 1 + abb 2 2 1=M 2 0: 6 For an a; b pair which i on the boundary region of the allowed a; b value, an! exit uch that 6 i an equality. Moreover, ince at that particular!, 6 i minimum, it derivative with repect to! at the ame! i zero. Thi obervation wa alo ued in Atrom et al to deign a controller for maximum a. Thu, 1 + aa 1 + aba 2 Ȧ 1 + bȧ 2 +ab 1 + abb 2 Ḃ 1 + bḃ 2 =0: 7 Solving 7 for a give a = 2Ȧ 1 + bȧ 2 Ḋ 1 + Ḋ 3 b + Ḋ 2 b ; 8 2 where D 1 = A B1; 2 D 2 = A B2; 2 D 3 =2A 1 A 2 + B 1 B 2 : Subtituting 8 into the equality of 6 give a fourth-order equation for each!. x 4 b 4 + x 3 b 3 + x 2 b 2 + x 1 b + x 0 =0; 9 where x 4 = 2E 2 Ḋ 2 + D 2 F 2 + QH 4 ; x 3 = D 2 F 1 2E 3 Ḋ 2 + QH 3 +4D 3 Ȧ 2 2 2E 2 Ḋ 3 ; x 2 = 2E 3 Ḋ 3 + QH 2 + D 2 F 0 +4D 1 Ȧ D 3 Ȧ 1 Ȧ 2 2E 1 Ḋ 2 2E 2 Ḋ 1 ; x 1 =4D 3 Ȧ 2 1 2E 3 Ḋ 1 + QH 1 2E 1 Ḋ 3 +8D 1 Ȧ 1 Ȧ 2 ; x 0 = 2E 1 Ḋ 1 + QH 0 +4D 1 Ȧ 2 1; Q =1 M 2 ; E 1 =2Ȧ 1 A 1 ; E 2 =2Ȧ 2 A 2 ; E 3 = Ȧ 1 A 2 + Ȧ 2 A 1 ; F 0 =4Ȧ 2 1; F 1 =8Ȧ 1 Ȧ 2 ; F 2 =4Ȧ 2 2; H 0 = Ḋ 2 1; H 1 =2Ḋ 1 Ḋ 3 ; H 2 =2Ḋ 1 Ḋ 2 + Ḋ 2 3; H 3 =2Ḋ 3 Ḋ 2 ; H 4 = Ḋ 2 2: The allowed a; b region for a given M value can be calculated a follow: For a given!, olve 9 for b. Noting Fig. 1. Region of a; b value for M =1:46, equivalent to 40 phae margin or greater and 10 db gain margin or greater. Lower haded region i with additional 6 db plant gain uncertainty K =2 for a total of 16 db or greater. that b ha four olution for a given!, pick the poitive real olution and ue 8 to nd their correponding a. Select the a; b pair for which the reulting cloed-loop ytem i table and 4 i atied. Searching over a range of frequencie,!, give two vector that are a function of!, a!;b! which lie on the boundary of the allowed a; b region. Note that for an a; b on the boundary, one of the following condition can occur: i increaing a i inide the region, ii decreaing a i inide the region, or iii neither increaing nor decreaing a i inide the region. Thu, for two point, a 1 ;b and a 2 ;b, on the boundary, any a [a 1 ;a 2 ] and b i a pair within the region only if i increaing a 1 i within the region, ii decreaing a 2 i within the region, and iii there exit no a; b point on the boundary for any a a 1 ;a 2. Since, a will be hown later, the optimal pair lie on the boundary of the a; b region, internal point are not of interet Example Conider an armature-controlled DC motor with the input being motor current and the output being poition. The motor tranfer function i P=e 0:001 = 2. It i required to nd the region of the a; b pair uch that the complementary enitivity M 6 1:46, which allow for gain uncertainty k [1;K] and the pair for which a i maximum. Thi i equivalent to at leat 40 phae margin and [ logk] db gain margin. The plant i P 1 = 1+ k i P; P 2 =P 10 Fig. 1 depict the boundary of the allowed a; b pair for k i = 80 and K = 1 the a; b value fall in both haded region. Fig. 1 can alo be ued to nd the a; b value which atify any gain uncertainty contraint k [1;K]. For

4 114 O. Yaniv, M. Nagurka / Automatica Amplitude db pectrum i typically concentrated at low frequencie, while the meaurement noie pectrum i typically more ignicant at high-frequencie. It follow that an optimal controller can be found by weighting the performance at low frequencie and of noie at high frequencie. Since the high-frequency noie i proportional to ab and low-frequency performance to 1=a, a practical optimal criterion can be J =1=a+ab whoe olution mut lie on the boundary of the a; b curve. When i mall enough or zero meaning the enor noie i neglected, the optimal olution i the maximum poible a. Thi i the criterion correponding to the Nichol plot of the example Phae deg Fig. 2. Nichol plot for M =1:46 and K = 2, correponding to 40 phae margin or greater and 16 db gain margin or greater. The open-loop mut not enter the haded region in order to atify the M; K contraint. example, if 6 db gain uncertainty i deired K =2, then for any b, the allowed a value hould be 6 db le in order to cope with the increae of a between 0 and 6 db. The a; b region will therefore be the lower haded region depicted in Fig. 1 where the upper curve i hifted down by 6 db. The maximum a for K = 1 occur at a; b = 94:2 db, and for K = 2 occur at a; b = 84:9 db, 0.011, giving the controller deign correponding to lowet enitivity at low frequencie. Qualitatively, thi mean that the price of protecting the ytem from a poible gain uncertainty of 6 db i increaing the low-frequency enitivity by 9:5 db while the high-frequency noie decreae by 48%. The open-loop Nichol plot for maximum a and K = 2 i hown in Fig. 2 for verication Extenion to complementary enitivity pec It i poible to replace the enitivity margin contraint 4 by the complementary enitivity, klj! 1+kLj! 6 M;! 0; k [1;K]: 11 The following lemma how that L = L 0 atie 4 ifand only if L = M 2 =M 2 1L 0 atie 11. Lemma 1. The pair a; b olve problem 1 2 tated in Section 2 if and only if the pair [M 2 1=M 2 ]a; b olve problem 1 2 where 4 i replaced by Optimization The anwer to the quetion Which i the bet a; b pair? of coure depend on the optimization criterion. Seron and Goodwin 1995 note that In general, the proce noie 5. Deign methodology for PID controller with low pa on D term In the previou ection, a deign methodology of a PID controller whoe three parameter are a; b; k i wa given auming the k i term i known ak i i the Iterm of the PID, ee 5. The extenion to include a lter, H, onthe D parameter of the PID i again by earching over both the k i and H ee Eq. 5. The idea i to chooe the tructure of the lter H, for example, H=p=+por H=p 2 = 2 + p + p 2, and earch over the parameter p. The quetion then i how bet to chooe the p value for the earch. Since the reaon for introducing the lter H 0 i to limit the enor noie amplication of the D term and/or reduce high-frequency reonance, it i recommended to perform an iterative earch on p a follow: tarting with very large p, meaure the noie and if it i too large decreae p. When reaching an acceptable noie level a rened earch can be conducted around the atifactory p. A reaonable range for thi earch on p for rt- or econd-order lter can be calculated a follow: If! 0 i the larget frequency where the open-loop phae i 180 for a given k i where H 0 =1, the earch hould not exceed about p =10! 0 becaue above that value the low-pa lter phae at frequencie larger than! 0 i neglected le than 5. The anwer to the quetion of how bet to chooe the k i value for the earch i baed on the following equation: the PID controller i a 0 1+ k i + b = a b 1+ k i= 1+b 12 if P 2 P 2 thi i an approximation. Let a 0 ;b 0 denote the optimal olution for k i = 0 and! 0 it lowet croover frequency. Find the range of k i value whoe phae contribution to 12 at! 0 i between 45 and about 1, that i, the k i value for which arg 1+ k i=j! 0 1+bj! 0 = 45 ; 1 : 13 Ue thee two k i value a the larget and lowet value for the earch on k i.

5 O. Yaniv, M. Nagurka / Automatica ki a db Amplitude db a b p x 10 6 Fig. 3. Region of ak i ; abp value for M =1:46 and K =1. Remark 1. There may exit control application where the plant ha a high-frequency reonance that mut be attenuated becaue i if it i not attenuated the achievable cloed-loop performance i retricted and ii the reonance might generate non-tolerable aliaing phenomena in dicrete ytem. Thi attenuation can be done uing a notch lter or a low-pa lter on the D term or on both the D and P term. The algorithm propoed here upport thi kind of lter where H can be applied either on D or on the D and P term Continuation of example for deigning H It i next of interet to evaluate the tradeo between high-frequency noie, that i a quantity proportional to the multiplication ab by the enor noie of the D term ltered by H. For implicity, it i aumed that thi noie i abp and the tradeo i conidered for k i = 100. The frequency! 0 i 1500 rad=, and thu the earch i in the range p [15; 000; 1500]. Fig. 3 depict the boundary of the allowed ak i ; abp pair for dierent p. Uing a low-pa lter with p = 5000 intead of p =15; 000 decreae the noie by a factor of three, while decreaing the performance by 2:1 db. If the optimal criterion i that the high-frequency noie mut be le than 1: , then p = 5000 give the bet performance. It controller parameter are k i a = 30:9 db; abp=1: and k i =100. It open-loop Nichol plot i hown in Fig Extenion to plant given by meaured data If a plant identied at a lit of frequencie i given and it i not poible to nd a tate-pace model or the accuracy of a choen model i not good enough, it i till poible to deign a controller. One option i to interpolate and/or pline t the plant pair correponding to the known frequencie Phae deg Fig. 4. Nichol plot for M =1:46 and K = 1. Frequencie are marked in rad/. The open-loop tranfer function mut not enter the haded region in order to atify the M and K contraint. and replace the derivative appearing in 7 by a numerical derivative. Another option i baed on the fact that any a; b pair on the boundary of the a; b region mut atify 1 + Lj! = 1=M, from which the following obervation can be made: i uing the notation Lj! = x + jy, Lj!=a[A 1!+bA 2!+jB 1!+bB 2!]; 14 then 1+Lj! =1=M if and only if x y 2 = M 2 or equivalently x = 1 + co =M and y = in =M, where [0; 2], ii any x +jy on the circle x y 2 = M 2 mut atify y=x 6 1=M 2 1 and x 0 and iii olving 14 for b give b = B 1x A 1 y A 2 y B 2 x = B 1 A 1 y=x : 15 A 2 y=x B 2 Baed on the above three obervation, the propoed deign method i: 1 pickadenelitofy=xintheinterval y=x 6 1=M Solve for b uing 15. From all poible b value, pick only the poitive b for which the ign of x, that i, of aa 1!+bA 2! i negative. 3 Subtitute b in 6 to get the following quadratic equality on a Q =1 1=M 2 a 2 [A 1 + ba 2 2 +B 1 + bb 2 2 ]+a[a 1 + ba 2 ]Q =0 and olve for a. 4 All the a; b pair for which the cloed-loop ytem i table and 6 i atied at all meaured frequencie lie on the boundary of the a; b region. 5 Repeat the above tep for all meaured data point to get a lit of controller, that i a lit of a; b pair. Apply the optimal criterion on thi lit to determine the optimal olution.

6 116 O. Yaniv, M. Nagurka / Automatica The drawback of thi technique, compared to the one uing a model, i that the computation time can be much longer. 7. Concluion In thi paper, explicit equation are provided for determining controller of the claical form, i.e., PI, PID, and PID with D ltered, that tabilize a given et of plant and atify both gain margin contraint and a bound on the complementary enitivity. The algorithm t any plant dimenion including pure delay, untable plant, continuou plant, dicrete plant, and plant given by meaured data. The outcome of applying the algorithm i a lit of controller from which an optimal controller can be extracted for many practical optimization criteria. Moreover, tradeo among high-frequency enor noie, low-frequency enitivity, the parameter of the PID and the lter on D are directly preented by deign graph a ingle plot uce for a ingle lter. Since the algorithm ue explicit equation, it can be executed very fat, and a uch the controller deign can be updated in near real-time to reect change in plant uncertainty and/or cloed-loop pecication. Ho, W. K., Hang, C. C., & Zhou, J. H Self-tuning PID control of a plant with underdamped repone with pecication on gain and phae margin. IEEE Tranaction on Control Sytem Technology, 54, Ho, W. K., Lim, K. W., & Xu, W Optimal gain and phae margin tuning for PID controller. Automatica, 348, Kritianon, B., & Lennarton, B Robut and optimal tuning of PI and PID controller. IEE Proceeding Control Theory Application, 1491, Ogawa, S PIcontroller tuning for robut performance. IEEE Conference on control application, September 28 29, Albany, NY pp Panagopoulo, H., Atrom, K. J., & Hagglund, T Deign of PIcontroller baed on contrained optimization. IEE Proceeding Control Theory and Application, 1491, Poulin, E., & Pomerleau, A PIetting for integrating procee baed on ultimate cycle information. IEEE Tranaction on Control Sytem Technology, 74, Seron, M., & Goodwin, G Deign limitation in linear ltering. Proceeding of the 34th CDC, New Orlean, LA, December. Suchomki, P Robut PIand PID controller deign in delta domain. IEE Proceeding Control Theory Application, 1485, Yaniv, O Quantitative feedback deign of linear and nonlinear control ytem. Dordrecht: Kluwer Academic Publiher. Yeung, K. S., Wong, K. W., & Chen, K. L A non-trial-and-error method for lead-lag compenator deign. IEEE Tranaction on Education, 411, Acknowledgement M. Nagurka i grateful for a Fulbright Scholarhip for the academic year allowing him to purue thi reearch at The Weizmann Intitute of Science Rehovot, Irael. Reference Atrom, K. J., Panagopoulo, H., & Hagglund, T Deign of PIcontroller baed on non-convex optimization. Automatica, 345, Cavicchi, T. J Minimum return dierence a a compenator deign tool. IEEE Tranaction on Education, 442, Crowe, J., & Johnon, M. A Automated PIcontrol tuning to meet claical performance pecication uing a phae locked loop identier. Proceeding of the American Control Conference, Arlington, VA, June 25 27, 2001 pp Crowe, J., & Johnon, M. A Toward autonomou PIcontrol atifying claical robutne pecication. IEE Proceeding Control Theory Application, 1491, Guillermo, J., Silva, A. D., & Bhattacharyya New reult on the ynthei of PID controller. IEEE Tranaction on Automatic Control, 472, Ho, W. K., Hang, C. C., & Cao, L. S. 1995a. Tuning of PID controller baed on gain and phae margin pecication. Automatica, 313, Ho, W. K., Hang, C. C., & Zhou, J. H. 1995b. Performance and gain and phae margin of well-known PItuning formula. IEEE Tranaction on Control Sytem Technology, 32, Oded Yaniv received hi B.S. in Mathematic and Phyic in 1974 from the Jerualem Univerity, Irael and hi M.Sc. in Applied Phyic in 1976 and Ph.D. in Applied Mathematic in 1984 from The Weizmann Intitute of Science, Rehovot, Irael. In 1988 he joined the Department of Electrical Engineering-Sytem at Tel Aviv Univerity at Tel Aviv Irael, where he i a profeor of electrical engineering. He ha extenive experience in feedback control ytem having conulted and worked in indutry. Mark Nagurka received hi B.S. and M.S. in Mechanical Engineering and Applied Mechanic from the Univerity of Pennylvania Philadelphia, PA in 1978 and 1979, repectively, and a Ph.D. in Mechanical Engineering from M.I.T. Cambridge, MA in He taught at Carnegie Mellon Univerity Pittburgh, PA from , and wa a Senior Reearch Engineer at the Carnegie Mellon Reearch Intitute Pittburgh, PA from In 1996, Dr. Nagurka joined Marquette Univerity Milwaukee, WI where he i an Aociate Profeor of Mechanical and Biomedical Engineering. Dr. Nagurka i a regitered Profeional Engineer in Wiconin and Pennylvania, a Fellow of the American Society of Mechanical Engineer ASME, and a former Fulbright Scholar at The Weizmann Intitute of Science Irael. Hi reearch interet include mechatronic, control ytem deign, human/machine interaction, and vehicle dynamic.