Analytical design and analysis of mismatched smith predictor


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Analytical design and analysis of mismatched smith predictor

  1. 1. ISA Transactions 40 (2001) 133±138 design and analysis of mismatched Smith predictor Weidong Zhang *, Xiaoming Xu Department of Automation, Shanghai Jiaotong University, Shanghai 200030, PR ChinaAbstract In this paper, an analytical design method is developed for the mismatched Smith predictor on the basis of theinternal model control (IMC) method. Design formulas are given and design procedure is signi®cantly simpli®ed. Oneimportant merit of the proposed method is that the response of the closed loop system can be easily adjusted. Therelation between the controller parameter and the system response is monotonous. In addition, necessary and sucientcondition for the robust stability of mismatched Smith predictor is also given. It is shown that the proposed methodcan provide good performance for perfectly matched system and a better response for mismatched system. Severalnumerical examples are given to illustrate the proposed method. # 2001 Elsevier Science Ltd. All rights reserved.Keywords: Linear system; Time delay; Smith predictor; Internal model control; Robustness1. Introduction system. For example, [4] systematically studied the robust performance of the Smith predictor within Synthesis and tuning of control structures for the IMC structure using several design methods;single input±single output (SISO) systems com- [5] presented a simple criterion for the tuning ofprises the bulk of process control problems. In the Smith predictor when the plant time delay is notpast, hardware considerations dictated the use of precisely known; [6] discussed robust PID tuningPID controllers [1], but through the use of com- problem for the Smith predictor in the presence ofputers, controllers have now advanced to the stage model uncertainty; and [7] developed a two degree-where virtually any conceivable control policy can of-freedom robust Smith predictor. Recently, Wangbe implemented. Due to the progress, the design of et al. [8] proposed a new scheme. It employs aSmith predictor [2] is widely studied in the past deliberately mismatched model to enhance perfor-decades. Though the Smith predictor o€ers mance over a perfectly matched system whilepotential improvement in the closed loop perfor- using a simple primary controller.mance over conventional controllers, it su€ers Based on the IMC theory, this paper willfrom a sensitivity problem. In the face of inevi- develop an analytical design procedure for thetable mismatches between the model and actual mismatched Smith predictor proposed by [8]. Forprocess, the closed loop performance can be very controller design, simplicity, as well as optimality,poor [3]. A lot of work has been done in relation is important. Compared with the empiricalto the robustness issues of the Smith predictor method of [8] the analytical method can give design formulas and thus simpli®ed the design * Corresponding author. Tel.: +86-21-6293-3329; fax: +86- procedure. On the other hand, the result of IMC is21-6293-2138. suboptimal. This provides possibility of improving E-mail address: (W. Zhang). the closed loop performance.0019-0578/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.PII: S0019-0578(00)00045-8
  2. 2. 134 W. Zhang, X. Xu / ISA Transactions 40 (2001) 133±138 The paper is organized as follows. In the more strict proof was given by [10]. De®ne thefollowing section the assessment of the achievable IMC controllerperformance of classical Smith predictor isformulated. In the third section an equivalent R…s† Q…s† ˆ …3†representation of the mismatched Smith predictor 1 ‡ Gmo …s†R…s†is given and the design formulas are derived ana-lytically. The robustness is also discussed. Several Then the two schemes are equivalent to eachexamples are provided in the fourth section, in other. The closed loop transfer function can bewhich the new controller is compared with that of written as[8]. Finally, conclusions are given in Section 5. T…s† ˆ Gm …s†Q…s† …4†2. Assessment of achievable performance The primary objective of any feedback control scheme is to make the di€erence between the con- The structure of classical Smith predictor is trolled outputs y(s) and desired setpoint r(s) asshown in Fig. 1, where R(s) is the controller, G(s) small as possible. What is meant by ``small canis the actual plant, Gm (s) is the model, and Gmo be de®ned in terms of various performance speci-(s) is the delay free part of Gm(s). The closed ®cations for the closed loop system. In IMCloop transfer function between the setpoint r(s) method, the performance objective is de®ned as „and the output y(s) is H2 optimal, i.e. min …r…t† À y…t††2 dt. Assume that the plant is stable. Factor the model R…s†G…s†T…s† ˆ …1† 1 ‡ R…s†…Gmo …s† À Gm …s† ‡ G…s†† Gmo …s† ˆ G‡ …s†GÀ …s† mo mo …5† In the case of perfect modeling, i.e. such that G‡ …s† contains all right half plane zeros moGm …s† ˆ G…s†, the closed loop transfer function is and is all pass. The suboptimal IMC controllergiven by can be expressed as R…s†Gm …s† Q…s† ˆ J…s†=GÀ …s† mo …6†T…s† ˆ …2† 1 ‡ R…s†Gmo …s† where J(s) is a user-speci®ed ®lter. Let m be theThis implies that the characteristic equation is free relative degree of GÀ …s†. The ®lter with asympto- moof the time delay so that the primary controller tic tracking ability is usually selected asR(s) can be designed with respect to Gm(s). Theachievable performance can thus be greatly J…s† ˆ 1=…ls ‡ 1†m …7†improved over a conventional system withoutdelay compensation. The Smith predictor can be related to IMC Here l is a positive real constant. The controllerstructure, which has been discussed by [9] and a of Smith predictor can then be obtained through Q…s† R…s† ˆ …8† 1 À Gmo Q…s† By tuning l, one can adjust the nominal perfor- mance and robust performance monotonously. In the case of perfect match, the nominal perfor- mance can arbitrarily approach optimality by Fig. 1. Structure of Smith predictor. decreasing l.
  3. 3. W. Zhang, X. Xu / ISA Transactions 40 (2001) 133±138 1353. Analytical design of modi®ed Smith predictor which is a PID controller for the default value of n Let the delay free part of the model be …o s ‡ 1†2 Ko R…s† ˆ …15†Gmo …s† ˆ ; n ˆ 1; 2; 3 …9† l2 s 2‡ …2l À o †s …o s ‡ 1†n Furthermore, apart from the introduced certainwhere n is speci®ed by the user. The default value uncertainty there always exists other uncertaintiesof n is suggested to be 2 by [8]. Factor it into in practice. Assume thatGmo …s† ˆ Gmo1 …s†Gmo2 …s†; where
  4. 4. G…s† À Gm …s†
  5. 5. K0
  6. 6. G …s†
  7. 7. ˆ l…j!† 4l…!†Gmo1 …s† ˆ …10† m o s ‡ 1 where l…!† is the bound on the multiplicative 1 uncertainty l…j!† (it consists of certain uncertaintyGmo2 …s† ˆ …11† …o s ‡ 1†nÀ1 and other uncertainties). The nominal transfer function of the closed loop system isAn equivalent of the mismatched Smith predictorproposed by [8] is show in Fig. 2. Gm …s†R…s† T…s† ˆ …16† Though a certain uncertainty is introduced by 1 ‡ Gmo1 …s†R…s†the simpli®ed model Gm (s) in the scheme, we canalso design it by IMC method. Regard Gm (s) as Then the closed loop system is robustly stable ifthe nominal plant. The IMC controller is and only if [9]
  8. 8. Gm …j!†R…j!†
  9. 9. 1 …o s ‡ 1†n
  10. 10. Q…s† ˆ …12†
  11. 11. 1 ‡ G …j!†R…j!†
  12. 12. 4 ; V! …17† …ls ‡ 1†n mo1 l…!† This implies that there is a low bound larger thanOn the other hand, zero for l in a uncertainty system. R…s†Q…s† ˆ …13† 1 ‡ Gmo1 …s†R…s† 4. Examples Therefore, the controller is derived analytically Three typical plants from [8] will be used in thisas follows section to illustrate the proposed method. The perfectly matched system is studied in example 1. …o s ‡ 1†n For comparison, the mismatched cases are studiedR…s† ˆ …14† …ls ‡ 1†n À …o s ‡ 1†nÀ1 in example 2 and 3 for both the proposed method and Wangs method. Since l has a monotonous relation with the response of the closed loop sys- tem, one can easily get the required response. Example 1 Ð Consider the following process [8] eÀ4s G…s† ˆ …s ‡ 1†5 Take l=0.1, l=0.5 and l=1, respectively, for the Fig. 2. Structure of modi®ed Smith predictor. proposed Smith predictor. When the system is
  13. 13. 136 W. Zhang, X. Xu / ISA Transactions 40 (2001) 133±138perfectly matched, we have 1 Gm …s† ˆ eÀ5:39s …2:43s ‡ 0:995†2 …s ‡ 1†5Q…s† ˆ …ls ‡ 1†5 and Wangs controller [8] is 6:25s ‡ 2:56 …s ‡ 1†5 R…s† ˆR…s† ˆ s…s ‡ 2:29† …ls ‡ 1†5 À …s ‡ 1†4 Take l=0.6667 and l=1 for the proposed SmithA unit step setpoint change is introduced at predictor. We havet=0, and a 10% step disturbance is introduced atthe plant input at t=80. It is seen that the …2:43s ‡ 0:995†2 Q…s† ˆproposed method can provide arbitrarily good …ls ‡ 1†2performance by decreasing l (Fig. 3). As a matterof fact, for perfectly matched case, if l tends to be …2:43s ‡ 0:995†2zero, the closed loop transfer function has an in®- R…s† ˆ l2 s 2 ‡ …2l À 2:43†snity bandwidth and thus the performance tend tobe optimal [9]. When there exists uncertainty, thesucient and necessary condition (17) shows that A unit step setpoint change is introduced at t=0,the smaller l is the worse the robustness is. In this and a 10% step disturbance is introduced at thecase the designer should increase l from a very plant input at t=60. The closed loop responses ofsmall positive real monotonously until a satis®ed the two systems are shown in Fig. 4. It is seen thattrade-o€ between performance and robustness is for l=0.6667 the proposed method provides aobtained. better response than that of [8] and for l=1 a little Example 2 Ð Suppose that the process is better response is obtained.given by Example 3 Ð Consider a non-minimum phase process with time delay [8] 1G…s† ˆ …s ‡ 1†10 Às ‡ 1 À2s G…s† ˆ e …s ‡ 1†5The simpli®ed process model is computed as [8]Fig. 3. Performance of perfectly matched system (Solid line: Fig. 4. Multiple lag process (Solid line: l=1, dotted line:l=0.1, dotted line: l=0.5, dashed line l=1). l=0.6667, dashed line Wangs method).
  14. 14. W. Zhang, X. Xu / ISA Transactions 40 (2001) 133±138 137The mismatched model is given by [8] and robust performance by l and thus make a tra- deo€ between the two con¯icting objectives. 1Gm …s† ˆ eÀ5:07s …1:46s ‡ 0:999†2 5. Conclusionsand Wangs controller [8] is Several problems are discussed in this paper: 5:85s ‡ 4 1. It has been shown that for perfectly matchedR…s† ˆ model, the performance of the closed loop system s…s ‡ 2:83† can arbitrarily approach the optimal. Thus, it is not necessary to utilize the mismatch for improv-Take l=0.6667 and l=1 for the proposed Smith ing the system performance.predictor. Therefore, 2. Practical plants are of high order and models are usually low order. To improve the system per- …1:46s ‡ 0:999†2 formance, [8] presented a mismatched Smith pre-Q…s† ˆ …ls ‡ 1†2 dictor structure. It is found that the structure can be related to the IMC. A new design procedure is …1:46s ‡ 0:999†2 then developed in the framework of IMC theoryR…s† ˆ and analytical results are provided. Simulations l2 s 2 ‡ …2l À 1:46†s show that improved response is obtained. 3. Necessary and sucient condition for robustA unit step setpoint change is introduced at t=0, stability is provided for the mismatched Smith pre-and a 10% step disturbance is introduced at the dictor system. If the exact uncertainty pro®le isplant input at t=40. The closed loop responses for known, the exact controller parameter can be cal-both cases are presented in Fig. 5. Again a per- culated. In the context of process control, one canformance improvement is achieved for l=0.6667 adjust the performance and robustness of the closedwith the proposed method and a similar response loop system in such a way: increase the controlleris obtained for l=1. parameter from a small positive real monotonously The usefulness of the method is not only in per- until a satisfactory response is obtained.formance enhancement, but also lie in that thedesigner can easily adjust the nominal performance Acknowledgements Project supported by National Natural Science Foundation of China (69804007) and Science and Technology Phosphor Program of Shanghai (99QD14012). References [1] K.J. Astrom, T. Hagglund, PID controllers: theory, design and tuning, 2nd Edition, ISA, NC, 1995. [2] O.J.M. Smith, Closer control of loops with dead times, Chem. Eng. Prog. Trans. 53 (5) (1957) 216±219. [3] J.E. Marshall, H. Gorecki, K. Walton, A. Korytowski, Time delay systems, Ellis Horwood Limited, New York, 1992.Fig. 5. Non-minimum phase process with time delay (Solid [4] D.L. Laughlin, D.E. Rivera, M. Morari, Smith predictorline: l=1, dotted line: l=0.6667, dashed line: Wangs design for robust performance, Int. J. Control 46 (2)method). (1987) 477±504.
  15. 15. 138 W. Zhang, X. Xu / ISA Transactions 40 (2001) 133±138 [5] C. Santacesaria, R. Scattolini, Easy tuning of Smith pre- [8] Q.G. Wang, Q. Bi, Y. Zhang, Re-design of Smith pre- dictor in presence of delay uncertainty, Automatica 29 dictor systems for performance enhancement, ISA Trans. (1993) 1595±1597. 39 (1) (2000) 79±92. [6] D.K. Lee, M.Y. Lee, S.W. Sung, I.B. Lee, Robust PID [9] M. Morari, E. Za®riou, Robust Process Control, Prentice tuning for Smith predictor in the presence of uncertain, J. Hall, Englewood Cli€s, NY, 1989. Process Control 9 (1) (1999) 79±85. [10] W.D. Zhang, Analytical Design for Process Control, Post- [7] W.D. Zhang, Y.X. Sun, X.M. Xu, Two degree-of-freedom Doctoral Research Report, Shanghai Jiaotong University, Smith predictor for processes with time delay, Automatica 1998. 34 (1998) 1279±1282. Weidong Zhang was born in Xiaoming Xu was born in Daqing, Peoples Republic of Shanghai, Peoples Republic of China in 1967. He received the China in 1957. He received the BS, MS and PhD degree from BS degree from Huazhong Zhejiang University in 1990, University of Science and 1993 and 1996, respectively. He Technology in 1982, and MS worked in National Key and Ph.D degree from Shang- Laboratory of Industrial Con- hai Jiaotong University in 1984 trol Technology as a post-doc- and 1987, respectively. From toral research fellow before 1988 to 1990 he worked in joining Shanghai Jiaotong Uni- Germany as an Alexander von versity in 1998 as an associate Homboldt research fellow. He professor in Automatic Con- joined Shanghai Jiaotong Uni- trol. Since 1999 he has been the versity in 1990 as a professor.youngest professor at the Department of Automation, Shang- He was the director of Department of Automation in 1993 andhai Jiaotong University. His current research interests include the deputy president of Electronics Information College,process control, robust control, ®eld bus, multi-agent, and Shanghai Jiaotong University in 1996. Since 1997 he has beendigital ®ltering. the deputy principal of Shanghai Jiaotong University. His cur- rent research interests include control theory, arti®cial intelli- gence, computer network, and digital signal processing.