Phương pháp điều chỉnh sử dụng trực tiếp dữ liệu thực nghiệm đối với bộ điều khiển imc trong hệ thống điều khiển tầng: Sự đạt được đồng thời các bộ điều khiển và mô hình đối tượng

Nội dung text: Phương pháp điều chỉnh sử dụng trực tiếp dữ liệu thực nghiệm đối với bộ điều khiển imc trong hệ thống điều khiển tầng: Sự đạt được đồng thời các bộ điều khiển và mô hình đối tượng

1. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) A DATA-DRIVEN APPROACH TO CASCADED INTERNAL CONTROLLERS: SIMULTANEOUS ATTAINMENT OF CONTROLLERS AND MODELS PHƯƠNG PHÁP ĐIỀU CHỈNH SỬ DỤNG TRỰC TIẾP DỮ LIỆU THỰC NGHIỆM ĐỐI VỚI BỘ ĐIỀU KHIỂN IMC TRONG HỆ THỐNG ĐIỀU KHIỂN TẦNG: SỰ ĐẠT ĐƯỢC ĐỒNG THỜI CÁC BỘ ĐIỀU KHIỂN VÀ MÔ HÌNH ĐỐI TƯỢNG Nguyen Thi Hien Vietnam National University of Agriculture Ngày nhận bài: 28/12/2020, Ngày chấp nhận đăng: 21/05/2020, Phản biện: TS. Nguyễn Ngọc Khoát Abstract: This paper proposes a data-driven parameter tuning of the internal model controllers (IMC) in cascade architecture with minimum phase processes. In order to perform the parameter tuning of the IMC, we utilize the fictitious reference iterative tuning (FRIT), which enables us to obtain the desired parameter of the controllers with only one-shot experiment data. The algorithm does not require mathematical process models but only a single set data collected from the closed loop system. Moreover, the proposed approach enables us to obtain both the optimal parameters of two controllers for the desired tracking property and mathematical models of the controlled process simultaneously. To show the validity of the proposal, we give illustrative examples. Keywords: Data-driven approach, FRIT, cascade control, IMC. Tóm tắt: Bài báo đề xuất sử dụng FRIT - một thuật toán dùng trực tiếp dữ liệu thực nghiệm để điều chỉnh thông số của bộ điều khiển IMC trong hệ thống điều khiển tầng với các đối tượng pha cực tiểu. Thuật toán đề xuất không đòi hỏi mô hình toán học của đối tượng điều khiển mà chỉ yêu cầu duy nhất một bộ dữ liệu vào/ra thu thập từ hệ thống. Kết quả nhận được là các bộ điều khiển với thông số tối ưu cho tín hiệu ra mong muốn của hệ thống, đồng thời nhận được mô hình toán học của đối tượng điều khiển. Từ khóa: Dữ liệu thực nghiệm, FRIT, điều khiển tầng, IMC. 1. INTRODUCTION and other advantages over single loop control systems [1]. Usually, the Cascade control has been implemented in controllers are tuned sequentially, the industry and different applications due to inner loop controller is tuned first to give their disturbance rejection, faster response a faster response than the outer loop, and Số 26 1
2. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) then, the primary controller is tuned a model-based procedure using IMC according to the resulting system. Thus, approach for synthesizing the controllers. tuning of cascade controllers is a time The suggested tuning procedure consuming task. determines the controller filter time On the other hand, internal model control constants to assure robust stability. (IMC, [2]) is one of the effective It is clear that most methods in mentioned approaches to the achievement of a studies require the process models, desired tracking property. The utilization thus the controller design asks of IMC for cascade control yields the an identification, which encounters robustness and flexibility in tuning difficulties in practice. In recent years, parameters. Thus, it provides a better design of a data-based control system system response than sequential tuning (without system identification) has been due to the adjustment of the inner loop proposed, such as iterative feedback has minimum effects on the outer loop. tuning (IFT, [6]), virtual reference In [3], Jeng et al. proposed an automatic feedback tuning (VRFT, [7]), and tuning method for cascade control fictitious reference iterative tuning (FRIT, systems based on a single closed loop step [8-9]) for the single loop control system. test. This method identifies the required In contrast to the iterative tuning method process information with the help of B- (IFT), which requires many control spline series expansions of the step executions, the VRFT and FRIT require responses. Then, two PID controllers are only one-shot experiment. While VRFT tuned using an IMC method. Lee et al. [4] considers error between the virtual input proposed IMC - based PID tuning rules and actual one, FRIT focuses on error that enable simultaneous tuning of between the fictitious output and the primary and secondary controllers. Their actual one. method is based on process models for Compared to a model-based approach, in cascade control systems. The main point the data-based methods, the controller is of simultaneously tuning cascade directly designed based on the controllers is to approximate the inner experimental data, thus the modeling step loop dynamics with the inner loop design is omitted and problems of under target. Such an approximation allows modeling encountered in practice are obtaining a process model for the tuning avoided. Moreover, due to the special of of primary controller. However, this IMC structure, it is expected that a data- approximation may be inaccurate because driven approach to the IMC yields not the implemented secondary PID controller only a controller but also a mathematical cannot guarantee meeting the inner loop model of the plant. In [9], Kaneko et al. design target. In [5], Cesca et al. proposed have succeeded in applying a data-driven 2 Số 26
3. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) FRIT for the single loop IMC, which the following notation N 2 yields simultaneous attainment of optimal 2 1 w:. w k controller and a plant model. In [10-11], N  N k 1 Nguyen et al. developed FRIT for cascade control systems, two PI controllers are 2. PRELIMINARIES simultaneously tuned to get the desired 2.1. Internal model control for cascade performance. As an application of the systems data-driven FRIT for cascade systems, the An IMC for a cascade system is shown in speed of DC motor is controlled in [11]. Fig.1 [3], [5]. In this figure, C and C are However, the results in [10-11] are only 1 2 the IMC controllers, P and P are the controllers, no any process model is 1 2 achieved. process for the loops. P2 is a process From these backgrounds, we propose a model of the inner loop and PB is the data-driven approach of FRIT for IMC equivalent process model of the outer parameter tuning in cascade systems. The loop. ru, and yy21, are the reference processes we treat here are linear, time- signal, the input, and the outputs, invariant, stable and minimum phase. The respectively. algorithm does not require mathematical The closed loop transfer function for the process models but only a set of inner loop is determined as: experimental data collected from the closed-loop system. Particularly, it is CP22 G2 (1) expected that the application of FRIT for 1 CPP2 2 2 cascaded IMC leads to both optimal controllers for achievement of a desired performance and mathematical models that reflect dynamics of the actual process. [Notations] Let  and n denote the set Figure 1. Internal model control of real numbers and that of real vectors of for cascade structure size n, respectively. For a time series w, we use wt() to describe the value of w at The transfer function PB is a model of time t . For a transfer function G , the equivalent process PB composed of the output y of G with respect to u is inner loop and the primary plant P1 denoted with y Gu for the enhancement connected in series, namely: CP of the readability. For a time series PP 22 (2) B11 CPP w w , w 2 , , w N  , we use 2 2 2 Số 26 3
4. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) The transfer function G from r to y 1 1 ry 1 CPFP ,  1 2 P 2 2 2 P n2 can be expressed as: 2s 1 CCPP1 2 2 1 Gry (6) 1 CPPCCPPCPCCPPP22 2 1221 1B 12B2 2 where n must be selected to ensure that (3) 2 the IMC controller is proper.  adjusts the Using the transfer function relations for 2 the inner and outer loop, the respective speed of the closed response in the inner IMC controllers are derived to satisfy the loop and it should be tuned to meet the set point and disturbance rejection desired performance. requirements. The IMC controller design for the outer loop is based on the process of the outer 2.2. Assumptions loop PB , which composes of the inner Consider the case P1 and P2 are linear, loop and the primary process P1 time-invariant, stable and minimum connected in series, then a model P also phase, they are unknown except degrees B of the numerator and the denominator. depends on P and 2 . The IMC Assume that the process models P1 and controller C1 is designed such that the P2 are parameterized with a tunable closed loop transfer function of the outer T TT loop G follows the reference model T . vector P: P1 P2 as: ry d  From the result in Kaneko et al. [9], we a s a10 s a P1(), P1    (4) construct the controller C1 as: b s b1 s 1 1 CPT ,,   (7) and: 1 P 2 B P 2 d k a'' s a s a The reference model Td should have the P(), k 10 k l (5) 2 P2 l form: bl ' s b1 ' s 1 1 T  1 Td n (8) where P1 a a 0 b b 1  1 1s 1 T kl 1 and P2 akl''''. a 0 b b 1   where n1 must be selected to guarantee For the inner loop, from the result the controller C1 proper. In a cascaded by Azar et al. [1] and Lee et al. [4], the IMC structure, if the reference model Td IMC controller is obtained and augmented is given, the controllers C1 and C2 1 by a filter F  as shown depend on both P and 2 . For 2 n2 2s 1 convenience, we use the following following: notation: 4 Số 26
5. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) expressed. The main idea of the FRIT P (9) scheme is to construct the model- 2 reference criterion in the fictitious domain The closed loop system in the cascaded [8]. IMC structure with a tunable parameter vector is illustrated in Fig. 2. The Consider a conventional closed loop system as Fig. 3, where ru, and y are input u and the outputs yy, also 21 the reference signal, the input, and the depend on the parameter vector , so we output, respectively. The controller C is denote them as u() and yy( ), ( ) , 21 parameterized by a vector since the respectively. controlled plant model is unknown. Figure 3. A conventional closed loop system Figure 2. A cascaded IMC system with a tunable vector with a tunable vector First, set an initial parameter vector 0 of 2.3. Problem setting the controller and perform a one-shot The objective of this paper is to find a experiment on the closed loop system to parameter vector to attain the design obtain the data u() 0 and y( 0 ). The output, which is represented by a controller C() 0 is assumed to stabilize reference model Td , with the direct use of 0 experimental data. The model-reference the closed loop system such that u() criterion is described as: and y() 0 are bounded. By using the data 0 0 J()() y T r 2 u() and y() , the fictitious refence 1dN (10) signal r() is computed as: Since controllers include the process r()()()() C 1 u 0 y 0 (11) models internally, it is expected that we can also simultaneously obtain For a given reference model Td , the cost appropriate models of the actual process. function is described by: For this purpose, FRIT, which is briefly 2 J():()() y 0 T r (12) explained in the next section, is utilized. FdN 3. FICTITIOUS REFERENCE Then we minimize JF () to achieve the ITERATIVE TUNING - FRIT [8] optimal parameter vector * , which In this section, the brief review of FRIT is yields a desired controller. Note that the Số 26 5
6. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) cost function (12) with the fictitious be expressed as: reference signal r() in Eq. 11 requires 11 Td P B P 2 FP 2 P 1 0 0 G only the initial data u() and y() . ry 1 1 1 1 1 PFP2 22 P TPPFPPT dB2 21dd2 TPFP 22 P This means that the minimization of Eq. 12 can be performed off-line by using T P 11 P FP P only one-shot experimental data. As for d B 2 2 1 1 1 1 the relationship between the minimization 1 Td 1 F 1 P 2 P 2 T d P B P 2 FP 2 P 1 of J() and that of JF () , it was shown (13) in Theorem 3.1 by Souma et al. [8] that Since the left hand side is equal to Td , * * J( ) 0 is equivalent to JF ( ) 0 Eq. 13 yields: (see Theorem 3.1 in [8] for the detailed 1 1 1 proof and discussions). 1 Td P B P 2 FP 2 P 1 1 T d 1 F 1 P 2 P 2 (14) 4. FRIT FOR CASCADED INTERNAL MODEL CONTROL If we can achieve PP22 then PP11 4.1. Simultaneous attainment of simultaneously holds. (Q.E.D). controllers and process models 4.2. Utilization of FRIT for the Consider a cascade control system with simultaneous attainment IMC as Fig. 2. Under the assumption that Let consider a cascaded IMC system the processes are unknown and they are described in Fig. 2 with minimum phase parameterized by as Eq. 4 and Eq. 5, processes. Assume that we can collect the we give the following result. 0 0 0 input/output data u( ), y21 ( ), y ( ) Theorem 1: For a given reference from the closed loop system with an model T , assume that the controllers d initial setting 0 . By using a set of the are described as Eq. 6 and Eq. 7, initial data, we introduce the fictitious then GT() holds if and only if ry d reference signal r() described by: both PP11 () and PP22 () 1 1 0 1 0 simultaneously holds. r()()()()()() C1 C 2 u C 1 P 2 u C()()() 1 P u 0 P P u 0 Proof. It follows from Eq. 3 that the ‘if’ 2 B 2 B C()()()() 1 y 0 P y 0 y 0 part clearly holds, with a notice that 1 2 B 2 1 (15) together with Eq. 2, we see PPFB1 And we minimize the cost function: when PP22 . Thus, we focus on the ‘only if’ part. By implementing the 0 2 JF():()() y 1 T d r controllers described in Eq. 6 and Eq. 7, N (16) the transfer function Gry from r to y1 can Consider the meaning of the minimization 6 Số 26
7. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) of Eq. 16. We can see that the output of can be summarized as following: the system with respect to the signal 1. Parameterize the process with the expressed in Eq. 15 always equals to the unknown parameter vector as Eq. 4 initial one y () 0 for any parameter 1 and Eq. 5. vector 2. The controllers are also parameterized 0 Gry()()() r  y 1 (17) with respect to as Eq. 6 and Eq. 7. Indeed, we can validate (17) by using 3. Set an initial parameter vector 0 1 and perform the closed loop experiment the trivial relations: yy21()() and P1 to obtain a set of data 1 0 0 0 u( ), y21 ( ), y ( ). uy( ) 1 ( ). PP12 4. Compute the fictitious reference signal Substituting Eq. 17 to Eq. 16 enables us to r() by using Eq. 15. see that the cost function (16) can be also rewritten as: 5. Construct the cost function JF () as 2 Eq. 16 and minimize it by an off-line non- T Jy( ) 1d (0 ) (18) linear optimization. F1 Gry () N * 6. Obtain argminJF ( ) which This implies that the minimization of yields both desired controllers JF () in Eq. 16 equals to that of the CC12( ), ( ) and mathematical models relative error of the desired transfer PP12( ), ( ) of the actual process. function Td and the closed loop transfer function Gry with under the influence 5. SIMULATION RESULTS 0 of y1( ). In this section, we give examples to show the validity of the proposed approach. Note that the cost function (16) with The first-order and second-order, the fictitious reference signal r() in minimum phase plants are considered in Eq. 15 requires only a set of data Example 1 and Example 2, respectively. 0 0 0 u( ), y21 ( ), y ( ), which means the minimization of Eq. 16 can be performed 5.1. Example 1 off-line by using one-shot experimental Consider a cascade system with the data. unknown first-order, minimum phase 3 1 plants as: P and: P . 4.3. Algorithm 1 51s 2 21s The algorithm of the proposed approach Then they are parameterized as: Số 26 7
8. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) * Ki r , the optimal output y1() and the Pii (with 1, 2). For the inner i s 1 desired output Trd are drawn by the dot- 1 and-dash line, the solid line and the dotted loop we use the filter: F 2 . 2s 1 line, respectively. From Fig. 6, we see Assume that we can achieve the desired * that the actual output y1() and the transfer function of the inner loop, thus desired output Trd are almost the same, the outer internal model P has the B which implies that the desired controllers * 1 K1 are achieved by using . parameterized form: PB . 21ss 11 The unknown parameter vector here 0.8 u 0.4 T : KK    , and we use the  1 1 2 2 2  0 0 30 60 100 Time [s] 1 0.8 reference model: Td 2 for the 0.4 21s y2 0 system. 0 30 60 100 Time [s] With the initial parameter vector Figure 4. The input signal u( 0) and the output 0 T signal y ( ) in Example 1 0 22222 , we perform a one-shot 2 experiment on the cascade control system 1.5 00 to obtain the initial data uy( ),2 ( ) 1 0 and y1() , which are described in Fig. 4 Outputs 0.5 and Fig. 5 (the solid line). Note that the 0 0 controllers with the initial setting are 0 30 60 100 Time [s] assumed to be able to stabilize the Figure 5. The reference signal r (the dot-and- closed loop system such as to yield 0 dash line), the actual output y1( ) (the solid line) bounded input/output [8]. In Fig. 5, and the desired output Tdr (the dotted line) in Example 1 we also plot the reference signal r (the dot-and-dash line) and the 1.4 desired output Trd (the dotted line). By 1 applying the proposed algorithm, the optimal parameters are obtained as Outputs 0.5 * T 3.000 4.947 0.984 1.988 1.880 . We 0 0 30 60 100 implement these parameters to the system Time [s] in Fig. 2 and perform the final Figure 6. The reference signal r (the dot-and- dash line), the optimal output y ( *) (the solid experiment. The results are illustrated in 1 line) and the desired output Tdr (the dotted line) Fig. 6. In this figure, the reference signal in Example 1 8 Số 26
9. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) On the other hand, by using * , the plant * to the system in Fig. 2, we obtain the 3.000 optimal output described in Fig. 9. From models are obtained as: P 1 4.947s 1 this figure, we can see that the achieved * 0.984 output y1() (the solid line) can meet and: P2 . Compared with the 1.988s 1 the reference one Trd (the dotted line), poles and gains of the actual plant, we see that means the vector * yields optimal that they are also well-identified. controllers. From these results, we can see that the optimal parameter vector * yields both 1 u 0.5 controllers for a desired output and 0 0 30 Time [s] 60 100 mathematical models of the actual plants. 1 y2 0.5 5.2. Example 2 0 0 30 Time [s] 60 100 In this case, the proposed approach Figure 7. The input signal u(p0) and the output 0 is applied for the unknown second - signal y2(p ) in Example 2 s 1 order plants as: P1 2 and 3ss 5 1 1.4 0.5s 1 P . Thus, the parameterized 1 2 2ss2 3 1 models of the plants are: Outputs 0.5 12s 56s P1 2 and: P2 2 . 0 34ss 1 78ss 1 0 30 60 100 Time [s] We use the same form of the filter and reference model as in Figure 8. The reference signal r (the dot-and- dash line), the optimal output y (p*) (the solid example 1, then the unknown 1 line) and the desired output Tdr (the dotted line) T in Example 2 vector :  1 2 3 4 5 6 7 8  2  . With the initial setting: Moreover, by using * we obtain the 0 T 2 2 2 2 2 2 2 2 2 , we collect a 0.27s 0.997 plant models as: P1 2 0 0 0 1.849ss 4.144 1 set of data u( ), y21 ( ), y ( ) that 1.197s 0.979 are described in Fig. 7 and Fig. 8. The and P . It seems 2 2.236ss2 2.543 1 proposed algorithm is applied and we that, the poles and zeros of the actual obtain the optimal parameter vector as plants are not identified. Fig. 10 and Fig. * 0.270 0.997 1.849 4.144 1.197 0.979 11 show the frequency characteristics of T 2.236 2.543 1.185 . After implementing the actual plants and the obtained models. In these two figures, characteristics of Số 26 9
10. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) PP,() * and P() 0 are illustrated by 10 ii i 0 the dotted line, the solid line and the dot-and-dash line, respectively. It is seen (dB) Gain -20 0 that the frequency characteristics of Pi * and those of Pi () are almost the same -90 in frequency range of reference model (deg) Phase -2 -1 0 1 * 10 10 10 10 Td. That means the models Pi () Frequency (rad/sec) appropriately reflect the dynamics of the Figure 11. Frequency characteristics: P2 (the actual plants. * 0 dotted lines), P(2 ρ) (the solid lines), and P(2 ρ) (the dot-and-dash lines) in Example 2 1.4 6. CONCLUSIONS 1 In this paper, we have proposed a data- driven approach to the cascaded IMC Outputs 0.5 with fictitious reference iterative tuning (FRIT). The processes we consider here 0 0 30 60 100 are linear, time-invariant, stable and Time [s] minimum phase. The algorithm directly Figure 9. The reference signal r (the dot-and- * designs controllers based on the one-shot dash line), the optimal output y1(p ) (the solid line) and the desired output Tdr (the dotted line) input/output data collected from the in Example 2 closed-loop system, and it does not require an identification. The approach 10 enables us to obtain not only desired 0 controllers but also mathematical models -20 that reflect the dynamics of the actual Gain (dB) Gain process. -40 0 Future direction of this study is to extend the proposed method to various processes -90 (e.g. with unstable zeros and/or time- Phase (deg) Phase -2 -1 0 1 delay) to show its useful and effective. 10 10 10 10 Frequency (rad/sec) The comparison with other data-driven approaches will also be considered in the Figure 10. Frequency characteristics: P1 (the * future researches. dotted lines), P(1 ρ) (the solid lines), and 0 P(1 ρ) (the dot-and-dash lines) in Example 2 REFERENCES 10 Số 26
11. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) [1] A.T. Azar and F.E. Serrano, Robust IMC-PID tuning for cascade control systems with gain and phase margin specifications, Neural Computing and Applications, Springer, Vol. 25, 2014, pp. 983–995. [2] M. Morari and E. Rafiriou, Robust Process Control, PTR Prentice Hall, Englewood Cliffs, New Jersey, 1989. [3] J. Jeng and M. Lee, Simultaneous automatic tuning of cascade control systems from closed-loop step response data, Journal of Process Control, Vol. 22, 2012, pp. 1020–1033. [4] Y. Lee and S. Park, PID controller tuning to obtain desired closed loop responses for cascade control systems”, Ind. Eng. Chem. Res., Vol. 37, 1998, pp. 1859–1865. [5] M.R. Cesca and J.L. Marchetti, IMC design of cascade control, Proceedings of the European Symposium on Computer Aided Process Engineering – 15, Vol. 20, 2005, pp. 1243–1248. [6] H. Hjalmarsson, M. Gevers, S. Gunnarsson and O. Lequin, Iterative Feedback Tuning: Theory and Applications, IEEE Control Systems Magazine, Vol. 18, No. 4, 1998, pp. 26 – 41. [7] M.C. Campi, A. Lecchini, and S. M. Savaresi, Virtual reference feedback tuning: a direct method for design of feedback controllers, Automatica, Vol. 38, No. 8, 2002, pp. 1337–1346. [8] S. Souma, O. Kaneko and T. Fujii, A new method of controller parameter tuning based on input- output data- fictitious reference iterative tuning (FRIT), Proceedings of the 8th IFAC Workshop on Adaption and Learning Control and Signal Processing, 2004, pp. 788–794. [9] O. Kaneko, H.T. Nguyen, Y. Wadagaki, and S. Yamamoto, Fictitious Reference Iterative Tuning for Non-Minimum Phase Systems in the IMC Architecture: Simultaneous Attainment of Controllers and Models, SICE Journal of Control, Measurement, and System Integration, Vol. 5, No. 2, 2012, pp. 101–108. [10] H.T. Nguyen and O. Kaneko, Fictitious reference iterative tuning for cascade control systems, Proceedings of the SICE Annual Conference, 2015, pp. 774–777. [11] H.T. Nguyen and O. Kaneko, Fictitious reference iterative tuning for cascade PI controllers of DC motor speed control systems, IEEJ Transactions on Electronics, Information and Systems, Vol. 136, No. 5, 2016, pp. 710–714. Biography: Nguyen Thi Hien, received B.E. and M.E. degrees in Electrical Engineering from Vietnam National University of Agriculture, Vietnam in 2000 and 2002, respectively, and Ph.D. in Control Engineering from Kanazawa University, Japan in 2013. She is a lecturer at Faculty of Engineering, Vietnam National University of Agriculture, Hanoi, Vietnam. Her research interests include control systems and its applications. Số 26 11
12. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) 12 Số 26