Two-loop Design for Dual-rate Cascade System

Transcription

1 FAT3.3 Two-loop Design fo Dual-ate Cascade System Sho Ito Takao Sato Nozomu Aaki Yasuo Konishi Depatment of Mechanical Engineeing, Gaduate School of Engineeing, Univesity of Hyogo, 67 Shosha, Himeji, Hyogo 67-8, Japan ( Abstact: In this study, we evaluated a new data-diven appoach fo designing a dual-ate cascade contol system. A dual-ate cascade contol system consists of inne and oute loops, whee the update inteval of the inne loop contolle is shote than that of the oute contolle. In the poposed method, fictitious efeence iteative tuning is used to optimize the contolle paametes in the inne and oute loops. Hence, the contolle paametes ae designed using only the contol data, and no ing pocedue is needed. The contolle paamete fo the inne loop is optimized fist, and then the oute loop paamete is optimized. Because the inne loop is updated faste than the oute loop in the dual-ate system, the contol pefomance of the poposed dual-ate design is supeio to that of conventional single-ate systems. The effectiveness of the poposed method was demonstated using numeical examples. Keywods: Cascade system, data-diven contol, dual-ate system, PID contol, iteative tuning methods, tansient esponses, sampled-data system. INTRODUCTION Data-diven contol has advantages elative to the based appoach because the contol system is designed using the contol data diectly and ing is not needed. The leading data-diven methods ae iteative feedback tuning (IFT) (Buyne, 3; Hjalmasson, ; Hjalmasson et al., 998), vitual efeence feedback tuning (VRFT) (Campi et al., ), noniteative coelation-based tuning (NCbT) (Kaimi et al., 7), and fictitious efeence iteative tuning (FRIT) (Souma et al., ; Kaneko et al., ). Because of its usefulness, the data-diven appoach has been widely used in vaious situations. One of the challenges in systems contol is to design the cascade contol system shown in Fig., which has inne and oute contol loops. Although the contol stuctue of the cascade contol system is moe complex than that of a singlemeasuement system, it has many industial applications, such as contol of pocesses (Hyl and Wagneova, 6; Hu et al., 7) and mechanical systems (Gama et al., 3). Conventional data-diven methods fo the cascade contol system wee designed as a single-ate system (Nguyen et al., 6; Kinoshita et al., 7; Nguyen and Kaneko, 7), in which all data signals ae updated synchonously. Theefoe, when conventional single-ate data-diven appoaches ae used, the update intevals of the inne and oute loops must be equivalent even if the update inteval of the inne loop can be shote than that of the oute loop Ling et al. (). Although the FRIT design method was poposed fo a dual-ate system (Ito et al., 8), the study did not addess the design of a cascade contol system with dual-ate sampling. Theefoe, this study evaluated a dualate data-diven appoach fo a cascade contol system u o u i yi y o Co C i P i P o Fig.. Cascade contol system Fig.. Conventional feedback contol system C with an inne loop update inteval that is shote than that of the oute loop. This pape is oganized as follows. An oveview of the standad FRIT method is given in Section, and Section 3 descibes a cascade contol system. The poposed datadiven appoach fo the dual-ate cascade contol system is given in Section. The effectiveness of the poposed method is demonstated in Section 5, and concluding emaks ae given in Section 6.. SINGLE-RATE FRIT The standad single-ate FRIT method (Souma et al., ; Kaneko et al., ) was designed using a fictitious efeence input (Safonov and Tsao, 997) fo the singleloop feedback contol system shown in Fig., whee the plant P is unknown and the contolle C is govened by the contolle paamete θ. In this section, the pocedue fo contol design with FRIT is intoduced biefly. u P y 8 Intenational Fedeation of Automatic Contol 58

2 The design objective of FRIT is to have the plant output follow the efeence T d duing a finite inteval, and hence the objective function is given as follows: J(θ) = T d y(θ) N () whee is the efeence input and x N := N N k= x(k) fo a discete time signal x. Then the following poblem is solved to obtain the optimal paamete θ : ag min J(θ) () θ The contolle C(θ ) with the initial paamete θ is assumed to be designed so that the closed-loop system is stable. Using the initial contolle paamete, the contol loop is executed fo the efeence input, and the contolled input data u(θ ) and output data y(θ ) ae collected. Based on the initial contolled data value, a fictitious efeence input is calculated as follows: (θ) =C(θ) u(θ )y(θ ), (3) whee the closed-loop plant output is always y(θ ) fo any θ when (θ) is used as the efeence input. Using (θ), a new objective function is defined as follows: J F (θ) = T d (θ) y(θ ) N () Eq. () is minimized offline since it consists of only the initial input and output data. The input and output elationship y(θ )=Pu(θ )and (θ) ae substituted into Eq. (), and J F (θ) is eaanged as follows: ( ) J F (θ) = Td y(θ ) G cl (5) N G cl = PC PC, whee G cl denotes the closed-loop tansfe function fom the efeence input to the plant output y. Theefoe, Eq. (5) is intepeted as minimizing the eo between the closed-loop system and the desied efeence based on the initial output data. The use of nonlinea optimization gives the optimal contolle paamete. 3. DUAL-RATE CASCADE CONTROL SYSTEM This study addesses a contol system design fo a dualate cascade contol system whee the update intevals of the inne and oute loops ae diffeent. The cascade contol system is implemented using a digital compute. The poposed method is designed using the following assumption: Assumption. The update inteval of the oute loop is twice the update inteval of the inne loop. Fom Assumption, The input and output intevals ae as illustated in Fig. 3. The block diagam of the dualate cascade contol system is shown in Fig., whee P i and P o ae unknown plants in the inne and oute loops, espectively, and C i and C o ae the inne and oute loop contolles, espectively. In the cascade contol system, is the efeence input; u i and y i ae the input and output of the inne loop, espectively; and u o and y o ae the input and output of the oute loop, espectively. Moeove, because of Assumption, the down-sample S d and upsample S u ae used to intepolate the diffeent signal intevals. Fig. 3. Dual-ate input and output intevals uo ui yi Co S u Ci Pi S d Po Fig.. Dual-ate cascade contol system The inne and oute contolles ae assumed to be govened by contolle paametes θ i and θ o, espectively, and the contol inputs ae detemined by the following contol laws: u i = C i (θ i )(u o y i ) (6) u o = C o (θ o )( y o ) (7) Thus, the contol input u o (k) in the oute loop is the efeence input fo the inne contolle C i (θ i ). This study poposes a method fo detemining the contolle paametes θ i and θ o using only input and output data in the dual-ate cascade contol system.. TWO-LOOP DESIGN FOR DUAL-RATE CASCADE CONTROL SYSTEM The objective function of the dual-ate cascade contol system is defined as follows: J(θ i,θ o )= T d y o (θ i,θ o ) N (8) Because the update intevals of the inne and oute loops ae diffeent, the contolle paametes ae optimized in seies. In the poposed method, fistly the contolle paamete in the inne loop is optimized based on the fist contol esult, and next the contolle paamete in the oute loop is optimized based on the second contol esult using the optimized inne loop contolle paamete instead of the initial value. This study poposes a new design method in which the contolle paamete in the inne loop is optimized independently of the oute loop paamete, even though the inne and oute loops influence each othe. This section descibes why the contolle paamete in the inne loop is designed independently of the oute loop contolle paamete. The objective functions of the inne and oute loops ae defined as follows: J i (θ i )= T di u o y i (θ i ) N i (9) J o (θ o )= T do y o (θ o ) N o, () whee N o =N i because of Assumption. The initial contolle paametes θi ini and θo ini ae assumed to be designed to stabilize both closed loops, and the input data u st i (θi ini ) and output data yi st (θi ini ) in the inne loop ae obtained fom the fist execution of the cascade contol system. The oute loop input data u st o (θini o )and yi yo uo ui yo 58

3 output data yo st (θo ini ) ae also obtained but not used fo tuning the inne loop. Using the data obtained fom the fist execution, the fictitious efeence input fo the inne loop is defined as follows: ũ o (θ i )=C i (θ i ) u st i (θi ini )yi st (θi ini ) () In Eq. (), ũ o (θ i ) may have to be expessed as ũ o (θ o ) because it is detemined as the contolle paamete in the oute loop. Howeve, since it is used as the fictitious signal, it is expessed as a function of θ i, which is the only paamete tuned in the fist optimization. Using Eq. (), a new objective function fo the inne loop to eplace Eq. (9)isdefinedasfollows: J if (θ i )= T di ũ o (θ i ) yi st (θi ini ) N i. () By optimizing Eq. () fo θ i, we obtain the optimal contolle paamete fo the inne loop, θi. Because Eq. () is optimized using a fictitious efeence input, the optimization in the inne loop is independent of the contolle paamete in the oute loop. Next, using both θi and θo ini, the second cascade contol instuction is executed fo the oute loop, and we obtain u nd o (θo ini)andynd o (θo ini ). Using the obtained data, the fictitious efeence input and a new objective function fo the oute loop ae defined as follows: (θ o )=C o (θ o ) u nd o (θo ini (θ ini )y nd o (θ ini o ) (3) J of (θ o )= T do (θ o ) yo nd o ) N o. () Based on the optimization of Eq. (), we obtain the optimal contolle paamete in the oute loop, θo. Because the contol esult of the oute loop is influenced by the contolle in the inne loop, the contolle paamete of the oute loop is detemined using the optimized contolle paamete of the inne loop. Howeve, the contolle paamete of the inne loop is obtained independently of that in the oute loop because the contol input fom the oute loop is the efeence input fo the inne loop. The poposed contol design algoithm is summaized as follows: () Select initial contolle paametes θi ini and θo ini so that both closed loops ae stabilized. () Execute the fist contol instuction fo the inne loop using the initial contolle paametes to obtain contol data u st i and yi st in the inne loop. (3) Calculate θi in the inne loop based on the optimization of Eq. (). () Execute the second contol instuction fo the oute loop afte eplacing θi with θini i to obtain contol data u nd o and yo nd in the oute loop. (5) Calculate θo in the oute loop based on the optimization of Eq. (). 5. NUMERICAL EXAMPLE The contolled plants in the inne and oute loops ae given as follows: P i = s (5).5s P o = s (6) s To contol the inne and oute loops, the following discetetime popotional integal deivative (PID) contolles ae used: Initial oute output.5 Table. Initial PID paametes C o... C i.. Initial oute output single ate dual ate Fig. 5. Initial oute loop output u i = C i (θ i )(u o y i ) (7) [ ] C i (θ i )= z z θ i (8) θ i =[K Pi K Ii K Di ] T (9) u o = C o (θ o )( y o ) () [ ] C o (θ o )= z z θ o () θ o =[K Po K Io K Do ] T, () whee z denotes the backwad shift opeato. The update intevals of the inne and oute loops ae s and s, espectively. Fo compaison with the poposed method, a single-ate system was also designed in which the update intevals of both the inne and oute loops ae s. The efeence input was. fom s to s and fom s to s. To tune the contolle paametes based on the input and output data, the initial paamete values must be selected so that the closed-loop system is stable. The initial PID paametes selected by tial and eo ae shown in Table. These paametes wee used in both the singleate and dual-ate systems. The output data of the oute loop plant using the initial PID paametes ae shown in Fig. 5, whee the solid and dashed lines ae the oute loop output tajectoies of the single-ate and dual-ate systems, espectively. Based on the initial contolled data, the PID paametes wee tuned to have the output of the oute loop follow two efeence outputs. The efeence s wee established with slow and fast tansfe functions, T dslow and T dfast, espectively, and thei contol esults wee compaed. T dslow = (s ) (3) T dfast = single ate dual ate (.8s ). () 583

4 Table. Single-ate PID paametes fo slow C o C i Oute input: u o single ate dual ate Table 3. Dual-ate PID paametes fo slow. C o C i Inne input: u i Inne output: y i single ate dual ate single ate dual ate Oute output: y o single ate dual ate desied output Fig. 6. Oute loop output fo slow Table. Single-ate PID paametes fo fast C o C i The same efeence was used in both the inne and oute loops. Because the contol system was implemented using a digital compute, dead-time fo the sampling was appended to the efeence s. The tuned single-ate and dual-ate PID paametes fo the slow efeence ae shown in Table and Table 3, espectively. The contol esults obtained using the tuned PID paametes ae shown in Fig. 6 and Fig. 7, whee the solid and the dashed lines ae the tajectoies of the single-ate and dual-ate systems, espectively. Fig. 6 also shows the desied output tajectoy of the slow efeence (dashed-dotted line). The figue shows that both the single-ate and dual-ate oute outputs followed the efeence output without eo. Next, the PID paametes wee tuned fo the fast efeence, and these ae shown in Table and Table 5. The contol esults ae shown in Fig. 8 and Fig. 9. Fig. 8 shows that the output of the dual-ate system followed the efeence, but the output of the single-ate system deviated fom the desied output when the efeence input was changed. These esults show that the oute loop outputs followed the slow efeence in both the single-ate and dualate systems. Howeve, the tacking pefomance of the Fig. 7. Oute loop input and inne loop input/output fo slow Table 5. Dual-ate PID paametes fo fast Oute output: y o C o C i single ate dual ate desied output Fig. 8. Oute loop output fo fast dual-ate system was supeio to that of the single-ate system fo the fast efeence. 6. CONCLUSION This study poposes a two-loop data-diven design method fo a dual-ate cascade contol system, whee the update intevals of the inne and oute loops ae diffeent. Conventional cascade contol systems ae designed as singleate systems, whee all the update intevals in the inne and oute loops must be equal. In contast, the poposed methodcanbeusedtodesignadual-atesystemaswell as a single-ate system, and the inne and oute contolle paametes ae tuned in seies. In the dual-ate system, the inne contol loop is updated faste than the oute loop, so the contol pefomance of the poposed dual-ate method is impoved with espect to conventional singleate methods. The effectiveness of the poposed method was demonstated with numeical examples. 58

5 Oute input: u o Inne input: u i Inne output: y i single ate dual ate single ate dual ate single ate dual ate Fig. 9. Oute loop input and inne loop input/output fo fast ACKNOWLEDGEMENTS This study was suppoted by the Japan Society fo the Pomotion of Science, Kakenhi Gant No. JP6K65. REFERENCES Buyne, F. (3). Iteative feedback tuning fo intenal contolles. Contol Engineeing Pactice,, 3 8. Campi, M., Lecchini, A., and Savaesi, S. (). Vitual efeence feedback tuning (VRFT): a diect method fo the design of feedback contolles. Automatica, 38, Gama, F., Matins, J.C., Mianda, T., Tome, W.F., and Fenandes, M. (3). Speed and cuent contol of a pemanent-magnet dc sevo moto using a eal-time micocontolle. In th Intenational Confeence on Electical Engineeing, Computing Science and Automatic Contol (CCE). Hjalmasson, H. (). Iteative feedback tuning -an eview. Intenational Jounal of Adaptive Contol and Signal Pocessing, 6, Hjalmasson, H., Geves, M., Gunnasson, S., and Lequin, O. (998). Iteative feedback tuning: Theoy and applications. IEEE Contol Systems Magazine, 6. Hu, H., Zhang, J., Yang, Q., and Cai, Y. (7). Feedfowad dmc-pid cascade stategy fo main steam tempeatue contol system in fossil-fied powe plant. In 9th Chinese Contol And Decision Confeence (CCDC). Hyl, R. and Wagneova, R. (6). Design and implementation of cascade contol stuctue fo supeheated steam tempeatue contol. In 7th Intenational Capathian Contol Confeence (ICCC). Ito, S., Sato, T., Aaki, N., and Konishi, Y. (8). Data-diven contol system design fo a multi-input single-output dual-ate system based on input-multiple closed-loop data. IEEJ Tansactions on Electical and Electonic Engineeing, 3(), Kaneko, O., Wadagaki, Y., and Yamamoto, S. (). FRIT based PID paamete tuning fo linea time delay systems -simultaneous attainment of s and contolles-. In IFAC Confeence on Advances in PID Contol, Kaimi, A., Heusden, K.V., and Bonvin, D. (7). Noniteative data-diven contolle tuning using the coelation appoach. In Poc. of Euopean Contol Confeence, Kinoshita, T., Yamamoto, T., and Samavedham, L. (7). Design of a data-oiented cascade contol system. In 6th Intenational Symposium on Advanced Contol of Industial Pocesses, Taipei,Taiwan. Ling, K., Bingfang, W., Minghua, H., and Yu, Z. (). A pedictive contolle fo multiate cascade systems. In Poc. of the Ameican Contol Confeence, Boston. Nguyen, H. and Kaneko, O. (7). Fictitious efeence iteative tuning of cascade contolles fo nonminimum phase systems. In 6th Intenational Symposium on Advanced Contol of Industial Pocesses, Taipei,Taiwan. Nguyen, H., Kaneko, O., and Kitazaki, Y. (6). Vitual efeence feedback tuning fo cascade contol systems. Jounal of Robotics and Mechatonics, 8(5), Safonov, M. and Tsao, T. (997). The unfalsified contol concept and leaning. IEEE Tans. on Automatic Contol,, Souma, S., Kaneko, O., and Fujii, T. (). A new method of contolle paamete tuning based on inputoutput data, -fictitious efeence iteative tuning-. In Poceedings of IFAC Wokshop on Adaptation and Leaning in Contol and Signal Pocessing,