Control of Time-Delay Systems by Means of Modified Smith Predictors: A Simple Matlab Toolbox

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1 Control of TimeDelay Sytem by Mean of Modified Smith Predictor: A Simple Matlab Toolbox Radek Matušů and Roman Prokop Abtract Thi paper i focued on decription of a imple Matlab toolbox for control of timedelay ytem by mean of modified Smith predictor. More pecifically, the program contain three method, namely modified Smith predictor for untable and integrating procee with time delay, modified PIPD Smith predictor for procee with long dead time, and modified Smith predictor deign by Coefficient Diagram Method (CDM). The oftware offer Graphical Uer Interface (GUI) for convenient controller deign with ubequent direct applicability in prepared Simulink cheme. It capabilitie are demontrated on everal imulation example. The decribed product i downloadable on the Internet and can be freely ued for reearch and educational purpoe. Keyword Timedelay ytem, modified Smith predictor, control deign, toolbox, Matlab, Simulink. T I. INTRODUCTION IMEDELAY ytem commonly appear in many practical control application and thu alo in a lot of related reearch work, e.g. [] [7]. They are alo called deadtime ytem, ytem with aftereffect, etc. [8]. Unfortunately, the preence of time delay can everely complicate control deign becaue it impair the tability and performance of the control loop. One of the claical tool for overcoming the time delay i known a the Smith predictor. Thi effective compenation tructure ha been publihed a early a in the late 95 [9], []. Since that time, a number of variou modification with improved feature have been introduced. Some of them can be found in [] [7]. Beide, the adaptive verion of Smith predictor i preented e.g. in [8], [9]. The principal aim of thi paper i to preent a imple Matlab program which i uitable for controller deign and imulation under aumption of three modification of Smith predictor. The package include the following method: Radek Matušů, and Roman Prokop are with the Centre for Security, Information and Advanced Technologie (CEBIA Tech), Faculty of Applied Informatic, Toma Bata Univerity in Zlín, nám. T. G. Maaryka 5555, 76 Zlín, Czech Republic. The contact i: rmatuu@fai.utb.cz. The work wa performed with financial upport of reearch project NPU I No. MSMT7778/4 by the Minitry of Education of the Czech Republic and alo by the European Regional Development Fund under the Project CEBIATech No. CZ..5/../3.89. modified Smith predictor for untable and integrating procee with time delay [4], modified PIPD Smith predictor for procee with long dead time [5], and modified Smith predictor deign by Coefficient Diagram Method (CDM) [6]. The oftware i freely downloadable from the web ite []. Actually, it i a tranlated and a bit improved verion of the program created under the cope of the Mater Thee []. The capabilitie are demontrated by mean of three control example. The previou work decribing thi program wa already preented at conference [], [3] and a a part of the monograph chapter [4]. However, thi paper deal with the topic in more detail. Furthermore, application of everal modified Smith predictor to control of ytem with timevarying delay and comparion with another control approach are provided e.g. in [5] [7]. The paper i organized a follow. In Section, baic theoretical background for three modification of Smith predictor i provided. The Section 3 then decribe the program itelf. Further, the three imulation example illutrating the capabilitie of the toolbox are preented in Section 4. And finally, Section 5 offer ome concluion remark. II. MODIFIED SMITH PREDICTORS The claical Smith predictor repreent wellknown tool for compenation of timedelay term [9], []. Neverthele, the Smith predictor fail for untable or integrating procee under preence of diturbance and it i very enitive to modelling error. Some of the original Smith predictor imperfection have been reduced by it modification which appeared in the cientific literature. Among many other, work [] [7] contributed to embellihment of the Smith predictor. Conequently, thee modification have brought the improvement in etpoint and diturbance repone for variou cenario. Within thi paper, three elected modified Smith predictor [4] [6] which have been implemented into the program are going to be briefly decribed. The common feature of all thee modification i that they utilize more ophiticated and complicated tructure with additional controller when compared with the original Smith predictor. However, the ue of mathematical model of the controlled plant including time ISSN:

2 delay term in the inner loop i preerved in all three modification. Furthermore, the ame model i uppoed a a nominal ytem during controller deign. The provided decription of the method i more or le general. Then, the pecific relation for deign of controller have been uually prederived for firt and econd order timedelay plant. They can be found either in the related literature [4] [6] or directly in the ource code of the program []. A. Method : Modified Smith Predictor for Untable and Integrating Procee with Time Delay Firt of the implemented modified Smith predictor wa preented in [4]. It i ueful for untable and integrating procee and it tructure i hown in fig.. w Gc ( ) n G () c Plant ( ) Gr G ( ) c Θ r ( ) Model G ( ) m Θ m ( ) G ( ) c Fig. modified Smith predictor tructure [4] The really controlled ytem i formally eparated into two block repreenting timedelayfree tranfer function Gr () and timedelay term Θ r (). Analogically, it mathematical model in the inner loop conit of Gm () and Θ m(). Signal w, n, y tand for reference value, diturbance in the input of the controlled plant, and output ignal, repectively. The ame notation will be adopted alo for the other two modification. The controller G () c i utilized to tabilize the untable pole. The other controller Gc () and G () c then enure reference tracking and diturbance rejection, repectively, by conidering the inner loop a an openloop table ytem. Moreover, the ignal outgoing from the controller G () c can be interpreted a an etimation of the diturbance n. The deign approach i baed on tandard form of the cloed loop ytem repone in the meaning of Integral Squared Time Error (ISTE) criterion and on the Nyquit tability analyi [4]. B. Method : Modified PIPD Smith Predictor for Procee with Long Dead Time The econd modification of the Smith predictor, uggeted in [5], ue the tructure with trio of controller a hown in Fig., where G () c i a PI controller, G () c i a PD (or only P where it i appropriate) controller and G () c3 i the diturbance controller introduced in [3]. y w G () c G () c n G () c3 Plant G () r Θ(,) t Model G () m Θ() Fig. modified Smith predictor tructure [5] Actually, the tructure diplayed in Fig. and are practically the ame from the block algebra point of view. However, the paper a well a the program itelf ha adopted the original viualization of the author [4], [5]. The tructure from Fig., which replace the conventional controller by a PIPD tructure, hould outperform a PID controller in ome common SISO ytem. Generally, the ynthei i baed on uage of tandard form for obtaining the optimal cloedloop tranfer function parameter in the meaning of ISTE criterion, i.e. an algebraically imple approach to control ytem deign i applied [5]. C. Method 3: Modified Smith Predictor Deign by Coefficient Diagram Method (CDM) Finally, the third modified Smith predictor utilize the deign by CDM. It wa publihed in [6] and it take advantage of the control tructure with trio of controller depicted in Fig. 3. w G ( ) c G () c G () c3 Plant () Gr Θ r () Model G ( ) m Θ m ( ) Fig. 3 modified Smith predictor tructure [6] The CDM deign itelf i baed on the following four tudie [6]: Coefficient diagram: It i a emilogarithmic diagram which allow invetigating the tability and repone of ytem in a ingle graph. The vertical axi logarithmically how the coefficient of characteritic polynomial, tability indice, tability limit and equivalent time contant while the horizontal axi repreent the order value correponding to each of coefficient. Modification of Keler tandard form: The form developed by Keler in 96 ha decreaed the ocillation and overhoot compared to the original Graham ITAE form. In thi approach, a new form called Standard Manabe Form i ued. Thi deign n y y ISSN:

hould reult in quite table and robut repone with mall ettling time. Lipatov tability analyi: The effect of coefficient variation can not be een clearly for higher order ytem.

However, the main difference i the Manabe form. The reader intereted in CDM can find all neceary background and detail of the procedure in [6], [8], [9] and related literature. A.

time, reference ignal, diturbance) will appear ee Fig. 5. III. DESCRIPTION OF THE PROGRAM The program can be freely downloaded from the web page [].

3 hould reult in quite table and robut repone with mall ettling time. Lipatov tability analyi: The effect of coefficient variation can not be een clearly for higher order ytem. The condition for tability or intability of uch ytem, baed on Lipatov work, are included in CDM deign technique. Obtaining characteritic polynomial: A method imilar to pole placement i applied. However, the main difference i the Manabe form. The reader intereted in CDM can find all neceary background and detail of the procedure in [6], [8], [9] and related literature. A. Method After the election of the firt modification For Untable and Integrating TimeDelay Sytem [4], the window with type of controlled plant and fundamental propertie of the experiment (imulation time, reference ignal, diturbance) will appear ee Fig. 5. III. DESCRIPTION OF THE PROGRAM The program can be freely downloaded from the web page []. It i a one zipped file which contain 36 mfile, mdlfile and text file with baic intruction and reference. The oftware wa latly modified and verified under Matlab 7.9. (R9b) + Simulink. According to tet, it hould work properly in Matlab 7 (R4) and higher (there can arie problem with Simulink cheme in lower verion). After decompreion, it can be executed by go.m file. The main window of the Graphical Uer Interface (GUI) i depicted in Fig. 4. Here, a uer can elect the modification of Smith predictor which hould be ued for a whole control experiment. With regard to the firt and econd method, it hould be added that comparion of delay free part of etpoint repone with a cloedloop tandard form for a minimum ISTE criterion a uggeted in [3] (according to reference from [4]) lead to graphical relation() between ome parameter. Thee graph can be found e.g. in [5]. A thee relation are neceary to ue during controller computation, their graphical verion have been approximated by ixth order polynomial for the purpoe of the program (ee the mcode). Fig. 5 baic propertie of control experiment for the firt method [4] There are three ort of controlled ytem available for thi method. The popup menu with them i hown in Fig. 6. A can be een, the firt method allow chooing from untable firt order timedelay plant, untable econd order timedelay plant, and (firt order) integrating timedelay plant. Fig. 6 poible type of controlled ytem for the firt method [4] Fig. 4 initial GUI of the program In the next window, coefficient of the controlled ytem of pecific type and ome other additional parameter depending on the ued method can be et a depicted in Fig. 7. Here, the cae of untable econd order timedelay plant i illutrated. Neverthele, the program allow not only definition of nominal ytem, which i conidered a a model for control deign a well a a model in the inner control loop (according to Fig. 3), but alo adjutment of the perturbed ytem, which i ued a a really controlled plant (with potentially different coefficient). If thee ytem are the ame, the parameter of nominal ytem can be copied to the perturbed one uing a ingle button. ISSN:

Fig. 9 an example of warning for the firt method [4] Fig.

econd order timedelay plant After confirming the data from Fig.

Finally, a Simulink cheme (ee an example in Fig.

The control loop utilize preet and computed parameter o the imulation can be performed imply

Then, the reult can be een directly in the Scope block or exported for future proceing in a

4 Fig. 9 an example of warning for the firt method [4] Fig. 7 definition of parameter for nominal and perturbed ytem for the firt method [4] and untable econd order timedelay plant After confirming the data from Fig. 7, the parameter of controller are calculated and viualized (Fig. 8). Finally, a Simulink cheme (ee an example in Fig. ) i opened together with the controller parameter. The control loop utilize preet and computed parameter o the imulation can be performed imply by Start imulation button. Then, the reult can be een directly in the Scope block or exported for future proceing in a tandard manner. Fig. imulation cheme for the firt method [4] and untable econd order timedelay plant B. Method Fig. 8 controller parameter an example for the firt method [4] with parameter from fig. 7. Potentially, ome additional information can appear ee the example in Fig. 9. Fig. poible type of controlled ytem for the econd method [5] ISSN:

Regarding to the econd method PIPD Modification for Sytem with Long Dead Time [5], the window with type of controlled plant and fundamental propertie of the experiment i baically the ame a in the

Subequently, etting the parameter of controlled ytem i imilar to Fig. 7 and controller parameter are alo preented in the analogical form a in Fig. 8.

Moreover, the etting time (directly in econd) i alo adjutable parameter here ee Fig.

imulation cheme for the econd method [5] and econd order timedelay plant Finally, the imulation cheme for thi modification under aumption of econd order timedelay plant and controller with diturbance

5 Regarding to the econd method PIPD Modification for Sytem with Long Dead Time [5], the window with type of controlled plant and fundamental propertie of the experiment i baically the ame a in the previou cae (ee Fig. 5). The only alteration conit in the election of controlled ytem type. Now, the three choice according to Fig. are available. Subequently, etting the parameter of controlled ytem i imilar to Fig. 7 and controller parameter are alo preented in the analogical form a in Fig. 8. The example of Simulink control loop cheme for the econd method i provided in Fig.. rejection feature, can be choen by uer. Moreover, the etting time (directly in econd) i alo adjutable parameter here ee Fig. 4. Fig. 4 definition of parameter for nominal and perturbed ytem for the third method [6] and econd order timedelay plant Fig. imulation cheme for the econd method [5] and econd order timedelay plant Finally, the imulation cheme for thi modification under aumption of econd order timedelay plant and controller with diturbance rejection i hown in Fig. 5. C. Method 3 Finally, the third method Control Deign Uing Coefficient Diagram Method ha alo the ame window for adjutment of baic propertie of control experiment a in Fig. 5. However, a uer can elect only from the firt and econd order timedelay plant a depicted in Fig. 3. Fig. 5 imulation cheme for the third method [6] and econd order timedelay plant Fig. 3 poible type of controlled ytem for the third method [6] In the following tep, the window with pecific parameter of the nominal and perturbed plant i a bit different than for other two implemented method, becaue the preferred type of controller, in the meaning of without or with diturbance IV. SIMULATION EXAMPLES The capabilitie of the toolbox and the effectivene of the method are going to be illutrated on the three imulation example. In all cae, the reference ignal i conidered with the tep change from to in a third of a imulation time and then ome diturbance i injected to the input of the controlled plant during the lat third of the imulation time. All time contant are uppoed to be in econd. ISSN:

6 A. Method In the firt example, the modification [4] i employed. The controlled ytem i uppoed to be decribed by untable firt order timedelay tranfer function: 3 G( ) 5 The imulation time i 3 econd and the load diturbance n =.. The calculation lead to the trio of controller: + Gc ( ) = Gc( ) = 7.9 G ( ) =.474 c () () The control reult are viualized in Fig. 7 (for perturbed gain), Fig. 8 (for perturbed time contant) and Fig. 9 (for perturbed timedelay term). with preumption (ee the program): K i p = T =. (3) Time [] Fig. 7 method control reult for the perturbed ytem (4) The control reult obtained from the program are provided in Fig Time [] Fig. 6 method control reult for the nominal ytem () Subequently, the ame controller ha been applied to the perturbed plant. Firt to the plant with % higher gain: 3.3 GP( ) 5 Then to the plant with % higher time contant: 3 GP ( ) 5 And finally to the plant with % higher timedelay term: (4) (5) Time [] Fig. 8 method control reult for the perturbed ytem (5) Time [] 3 GP 3 ( ) 5.5 (6) Fig. 9 method control reult for the perturbed ytem (6) ISSN:

7 B. Method The econd example ue the modification [5]. The controlled object i now given in the form of ideal firt order timedelay integrator: G( ) (7) The imulation time i now econd and the load diturbance remain the ame a in the previou example n =.. The computed controller are a follow: + Gc( ) = G ( ) = G c c 3 ( ) = (8) Time [] Fig. method control reult for the perturbed ytem () with preumption, again: K i c = T =. (9) C. Method 3 Finally, the third example i focued on the application of modification [6]. Here, it ha been uppoed the econd order timedelay plant: The Fig. how the correponding output controlled ignal. Analogically to the firt example, the controller (8) have been applied alo to a perturbed plant. Now, it ha been the integrator with % higher both coefficient in numerator and timedelay term.. GP( ) e. = () 5 5 G( ) = e ( )( ) () The imulation time wa preet to 3 econd and the load diturbance till remained n =.. A the method allow precribing the etting time directly, it wa choen T = [ ]. Furthermore, the controller with diturbance rejection ability were preferred. The reulting regulator are then: The reult of control are preented in Fig.. G ( ) = c Gc ( ) = G c 3 ( ) = () Time [] Fig. method control reult for the nominal ytem (7) The control imulation output i depicted in Fig Time [] Fig. method 3 control reult for the nominal ytem () ISSN:

8 In the final imulation, all the perturbed ytem parameter K, a, a, TD (ee Fig. 4) are aumed % higher. So, the perturbed plant i: 5.5 GP( ) = e The reult of control are preented in Fig Time [] Fig. 3 method 3 control reult for the perturbed ytem (3) V. CONCLUSION (3) The paper ha decribed the imple Matlab toolbox for control of timedelay ytem uing modified Smith predictor. Three method (modified Smith predictor for untable and integrating procee with time delay, modified PIPD Smith predictor for procee with long dead time, and modified Smith predictor deign by CDM) have been implemented into GUI for convenient control deign with direct utilizability in Simulink environment. The baic capabilitie are hown by mean of imulation example. The toolbox i free to ue for reearch and educational purpoe. REFERENCES [] L. Pekař, R. Matušů, P. Dotálek, and J. Dolinay, The Nyquit Criterion for LTI TimeDelay Sytem in Proceeding of the 3th WSEAS International Conference on Automatic Control, Modelling and Simulation, Lanzarote, Spain,. [] L. Pekař, and E. Kurečková, Doe the Higher Order Mean the Better Internal Delay Rational Approximation? International Journal of Mathematic and Computer in Simulation, vol. 6, no. 3, pp. 536,. [3] B. Makki, and B. Makki, Control Deign for Uncertain Singularly Perturbed Sytem with Dicrete Time Delay WSEAS Tranaction on Sytem and Control, vol. 6, no., pp ,. [4] T. Hahimoto, and T. Amemiya, Stabilization of Linear Timevarying Uncertain Delay Sytem with Double Triangular Configuration WSEAS Tranaction on Sytem and Control, vol. 4, no. 9, pp , 9. [5] Y.Ch. Chang, and S. S. Chen, Static OutputFeedback Simultaneou Stabilization of Interval TimeDelay Sytem WSEAS Tranaction on Sytem, vol. 7, no. 4, pp. 8594, 8. [6] É. Gyurkovic, and T. Takác, Output Feedback Guaranteeing Cot Control by Matrix Inequalitie for DicreteTime Delay Sytem WSEAS Tranaction on Sytem, vol. 7, no. 7, pp , 8. [7] R. Bárcena, A. Etxebarria, Indutrial PCbaed realtime controller applied to econdorder and firtorder plu time delay procee WSEAS Tranaction on Sytem, vol. 7, no. 9, pp , 8. [8] J.P. Richard, Timedelay ytem: an overview of ome recent advance and open problem Automatica, vol. 39, no., pp , 3. [9] O. J. M. Smith, Cloed control of loop with dead time Chemical Engineering Progre, vol. 53, pp. 79, 957. [] O. J. M. Smith, A controller to overcome dead time ISA Journal, vol. 6, no., pp. 833, 959. [] K. Watanabe, and M. Ito, A procemodel control for linear ytem with delay IEEE Tranaction on Automatic Control, vol. 6, no. 6, pp. 669, 98. [] K. J. Åtröm, C. C. Hang, and B. C. Lim, A new Smith predictor for controlling a proce with an integrator and long deadtime IEEE Tranaction on Automatic Control, vol. 39, no., pp , 994. [3] M. R. Mataušek, and A. D. Micić, A modified Smith predictor for controlling a proce with an integrator and long deadtime IEEE Tranaction on Automatic Control, vol. 4, no. 8, pp. 993, 996. [4] S. Majhi, and D. P. Atherton, A new Smith predictor and controller for untable and integrating procee with time delay in Proceeding of the 37th IEEE Conference on Deciion & Control, Tampa, Florida, USA, 998. [5] I. Kaya, and D. P. Atherton, A new PIPD Smith predictor for control of procee with long dead time in Proceeding of the 4th IFAC World Congre, Beijing, China, 999. [6] S. E. Hamamci, I. Kaya, and D. P. Atherton, Smith predictor deign by CDM in Proceeding of the European Control Conference, Porto, Portugal,. [7] T. Liu, Y. Z. Cai, D. Y. Gu, and W. D. Zhang, New modified Smith predictor cheme for integrating and untable procee with time delay IEE Proceeding Control Theory and Application, vol. 5, no., pp. 3846, 5. [8] V. Bobál, P. Chalupa, P. Dotál, and M. Kubalčík, Adaptive Digital Smith Predictor in Proceeding of the 3th WSEAS International Conference on Automatic Control, Modelling and Simulation, Lanzarote, Spain,. [9] V. Bobál, P. Chalupa, P. Dotál, and M. Kubalčík, Deign and Simulation Verification of Selftuning Smith Predictor International Journal of Mathematic and Computer in Simulation, vol. 5, no. 4, pp. 3435,. [] R. Matušů, and R. Prokop, Control of TimeDelay Sytem Uing Modified Smith Predictor Software, Available from: ~matuu/delay.zip. [] R. Matušů, Control of timedelay ytem (Řízení ytémů dopravním zpožděním), Mater Thee, Faculty of Technology, Toma Bata Univerity in Zlín, Czech Republic,. (In Czech). [] R. Matušů, and R. Prokop, A Matlab Program for Control of Time Delay Sytem Uing Modified Smith Predictor in Annal of DAAAM for & Proceeding of the t International DAAAM Sympoium, Zadar, Croatia,. [3] R. Matušů, and R. Prokop, Implementation of Modified Smith Predictor into a Matlab Program in Proceeding of the 5th European Conference on Modelling and Simulation, Krakow, Poland,. [4] R. Matušů, and R. Prokop, Implementation of Control Deign Method into Matlab Environment in Cutting Edge Reearch in New Technologie, C. Voloencu, Ed., Rijeka, Croatia: InTech, pp. 7388,, ISBN [5] R. Matušů, and R. Prokop, Control of Sytem with TimeVarying Delay: A Comparion Study in Proceeding of the th WSEAS International Conference on Automatic Control, Modelling and Simulation, Catania, Italy,. [6] R. Matušů, and R. Prokop, Control of Periodically TimeVarying Sytem with Delay: An Algebraic Approach v. Modified Smith Predictor WSEAS Tranaction on Sytem, vol. 9, no. 6, pp. 6897,. ISSN:

9 [7] R. Matušů, and R. Prokop, Variou approache to control of ytem with timevarying delay International Journal of Modelling, Identification and Control, vol. 4, no. /, pp. 664,. [8] S. E. Hamamci, and A. Ucar, A robut modelbaed control for uncertain ytem Tranaction of the Intitute of Meaurement and Control, vol. 4, no. 5, pp ,. [9] S. Manabe, Coefficient diagram method in Proceeding of 4th IFAC Sympoium on Automatic Control in Aeropace, Seoul, Korea, 998. [3] D. P. Atherton, and S. Majhi, Tuning of optimum PIPD controller in Proceeding of the 3rd Portugee Conference on Automatic Control CONTROLO 98, Coimbra, Portugal, 998. Radek Matušů wa born in Zlín, Czech Republic in 978. He i a Reearcher at Faculty of Applied Informatic of Toma Bata Univerity in Zlín, Czech Republic. He graduated from Faculty of Technology of the ame univerity with an MSc in Automation and Control Engineering in and he received a PhD in Technical Cybernetic from Faculty of Applied Informatic in 7. He i the author or coauthor of more than 4 cientific journal paper, about 9 conference contribution, everal monograph chapter, etc. The main field of hi profeional interet include robut control ytem and application of algebraic method to control deign. Roman Prokop, born in 95, i a Full Profeor at Faculty of Applied Informatic of Toma Bata Univerity in Zlín, Czech Republic. He graduated from Czech Technical Univerity in Prague in 976 and received a PhD from Slovak Univerity of Technology in Bratilava in 983. He wa an Aociate Profeor from 996 and a Full Profeor from 4. He aim hi pedagogical and reearch work to automatic control theory, algebraic method in control deign and optimization. The main interet of the latet period are uncertain and robut ytem, autotuning of controller and timedelay ytem. ISSN:

Improving the Regulatory Response of PID Controller Using Internal Model Control Principles

Improving the Regulatory Response of PID Controller Using Internal Model Control Principles International Journal of Control Science and Engineering 9, 9(): 9-4 DOI:.59/j.control.99. Improving the Regulatory Repone of PID Controller Uing Internal Model Control Principle Arun R. Pathiran Dept.