- Email: [email protected]
Fictitious Reference Iterative Tuning for Internal Model Controller
Adaptation and Learning in Control and Signal Processing — ALCOSP 2010 Antalya, Turkey, August 26-28, 2010
Fictitious Reference Iterative Tuning for Internal Model Controller Osamu Kaneko ⁄ Yusuke Wadagaki ⁄⁄ Shigeru Yamamoto ⁄ ⁄ Institute of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa, 920-1192, Japan (e-mail: {o-kaneko, shigeru}@t.kanazawa-u.ac.jp). ⁄⁄ Graduate School of Natural Science and Technology, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa, 920-1192 Japan (e-mail: [email protected])
Abstract: Internal model controller (which is abbreviated as IMC in the following ) is widely well-known as one of the easily-understandable controllers from the practical points of view. The reason for this is that IMC has a simple structure which works to decrease the error between the output of the actual plant and the output generated by the internal model included in the controller. In the case in which the internal model completely reflects the dynamics of the actual plant, implementing the model of a plant to IMC yields the desired tracking property. Conversely, in the case in which we do not know a mathematical model of a plant, the achievement of the desired output by some sort of method based on the direct use of the data enables us to identify the plant as the internal model in IMC. Moreover, since the data has a fruitful information of the plant, the direct utilization of the data yields more desirable IMC controller in order to achieve a given specification with reflecting the actual dynamics of the plant. From these ideas, we propose the data-driven approach to IMC with Fictitious Reference Iterative Tuning (which is abbreviated as FRIT) as one of the controller design methods without using a mathematical model of a plant. We show that the minimization of the cost function in FRIT for IMC directly leads to both the optimal controller for achievement of a desired response and a mathematical model reflecting the dynamics of the actual plant, as will become apparent afterwards. Keywords: Data-Driven Approach, Fictitious Reference Iterative Tuning, Internal Model Controller 1. INTRODUCTION Recently, synthesis of a controller with the direct utilization of the data attracts as one of the effective ways for achievement of the desired specification. The reason for this is that a data has fruitful information of a dynamical system. From such a standpoint, there are some studies on the controller synthesis of the direct use of the data, what is called data-driven approaches, e.g., (Fujisaki et al. (2005)), (Kaneko (2008)), (Markovsky and Rapisarda (2008)), (Park and Bitmead (2008)), (Safonov and Tsao (1997)), (Yamamoto and Oakano (2006)) and so on. Particularly, in the cases in which a controller with a tunable parameter has already been implemented, the data-driven approach is also regarded as data-driven parameter tuning. As representative works, we can refer Iterative Feedback Tuning (which is abbreviated as IFT, cf. Hjalmarsson et al. (1998)), Virtual Reference Feedback Tuning (which abbreviated as VRFT, cf. Campi et al. (2002), Sala and Esparza (2005)), and Fictitious Reference Iterative Tuning (FRIT) (cf. Kaneko et al. (2005), Souma et al. (2004), Masuda et al. (2009), Kaneko et al. (2010b)). IFT is the tuning method that iteratively updates the variable parameter of an implemented controller so as to ⋆ This paper was not presented in any IFAC meeting
ISBN 978-3-902661-85-2/11/$20.00 © 2010 IFAC
minimize a performance index, e.g., the sum of squared error signal between the desired reference signal and the actual output, by using the input/output data obtained in the iterative closed loop experiments. This minimization can be computed as a non-linear optimization technique like Gauss-Newton method in which required quantities (gradient, Hessian, Jacobian, and so on) consist of the experimental data. This also means that IFT requires many experiments in order to update the parameter of controllers so as to achieve the minimization of the performance index. Thus IFT spends considerable expense and time, which is a crucial problem with respect to practical points of view. At this point, VRFT and FRIT require only one-shot experiment for the achievement of a desired specification, which means that these two methods have great advantage compared with IFT in the sense that the time and cost for obtaining the optimal parameter are drastically reduced. Particularly, FRIT is easily applicable for the case in which the initial experiment is performed in the closed loop. Moreover, as explained in this paper, FRIT considers the minimization of the error between the fictitious output and the actual one while VRFT focus on the error between virtual (fictitious) input and the actual one, so FRIT is intuitively understandable for the case in which the specification is given for the achievement of the desired output.
1
we can obtain the discrete time frequency response G(ejω ). The output of G(z) with respect to the input time series whose z transformed representation is u(z) is described by G(z)u(z). For a w(z), we use the norm defined by v uN ¡1 ∑ 1u wk2 (2) ∥w(z)∥N = t N
By the way, internal model controller (which is abbreviated as IMC in the following) is widely well-known as one of the easily-understandable controllers from the practical points of view (cf. Garcia and Morari (1982), Morari et al. (1984), Morari and Zafiriou (1989), Abe (1999)). In fact, there are many cases in which IMC structure has already been implemented in the operated in the closed loop. The reason for this is that IMC has a simple structure which performs to decrease the error between the output of the actual plant and the output generated by the internal model included in the controller. In the case in which the mathematical model completely reflects the dynamics of the actual plant, implementing the model of a plant to IMC completely leads to the desired tracking property. Conversely, if we do not know a mathematical model of a plant, it is expected that the achievement of the desired output by some sort of method based on the direct use of the data enables us to identify the plant as the internal mathematical model in IMC. Moreover, since the data has a fruitful information of the plant, the direct utilization of the data yields more desirable IMC controller in order to achieve a given specification with reflecting the actual dynamics of the plant.
k=0
which was introduced in Yamamoto and Oakano (2006). 2.2 Problem formulation Consider IMC structure illustrated in Fig.1 (cf. Morari and Zafiriou (1989), Garcia and Morari (1982)). In this u(‰)
r +
y(‰)
P
CIMC −
−
PM
From these ideas, we propose the data-driven approach to IMC with FRIT as one of the controller design methods without using a mathematical model of a plant. Particularly, we show that the minimization of the cost function in FRIT for IMC directly leads to both the optimal controller for achievement of a desired response and a mathematical model reflecting the dynamics of the actual plant, as will become apparent afterwards. For this very reason, we here utilize FRIT for IMC.
+
Fig. 1. Internal Model Control paper, we assume that a plant is a linear, time invariant, and minimum-phase system. In main part of this paper, though a system is regarded in discrete time, the same discussion is applicable to the continuous time case. We also assume that we know the structure of the transfer function of a plant P described by ∑m i i=0 ‰(n + i)z P (‰, z) = (3) ∑ n¡1 z n + i=0 ‰(i)z i with unknown parameter vector ‰ := [‰(0) ‰(2) · · · ‰(n + m)]T ∈ Rn+m+1 . We also assume that the relative degree n−m is known. In Fig.1, CIM C (z) is the IMC-controller and PM (z) denote the internal model of a plant which included in the controller.
This paper is organized as follows. In Section 2, we give some required preliminaries and the problem formulation. In Section 3, we give an brief explanation of FRIT. In Section 4, the application of FRIT to IMC is presented. In this section, we also explain how FRIT is embedded into the IMC framework and then show that performing FRIT leads to obtain the optimal controller and to identify the mathematical model of a plant. In Section 5, in order to the validity of our results, we show an experimental example. In Section 6, we summarize the concluding remarks.
As one of the candidates for CIM C (z), we apply 2. PRELIMINARIES
CIM C (z) = Td (z)PM (z)¡1 . (4) In Eq.(4), Td (z) is the desired transfer function from the reference r to the output y, which is given for the purpose of the achievement of the desired response property. Since the relative degree of Td (z) is normally fixed equal or greater than n − m, Eq.(4) can be realized as a feedback controller. Throughout this paper, we apply IMC controller described by Eq.(4) and the internal model PM (‰, z), as illustrated in Fig.2. The transfer function from r to the output is denoted with T (‰), which is described by
2.1 Notations Let R and Rn denote the set of real numbers and that of real vectors of size n, respectively. Let z denote the z−operator. Let R(z) denote the set of real coefficient rational functions with the indeterminate z. Let R(z)n denote the set of the vectors of size n whose elements are included in R(z). If w is the discrete time signal, wt denote the value at the time t. For a discrete time signal w = {w0 , w1 , · · · , wk , · · · }, its z-transformed representation is described by w(z) =
∞ ∑
wk z ¡k .
T (‰, z) =
(1)
k=0
1
Td (z) P (z) 1¡Td (z) PM (‰;z) Td (z) P (z) + 1¡T d (z) PM (‰;z)
.
(5)
The input and the output also depend on a controller with tunable parameter ‰, so we denote them u(‰) and y(‰)(:= T (‰, z)r(z)), respectively.
Let G(z) ∈ R(z) denote a discrete time transfer function of a linear time-invariant system. By substituting ejω into z,
2
u(‰)
r +
Td PM (‰) −
+
−
PM (‰)
obtain the initial data u0 and y 0 . Here we also assume that C(‰0 , z) tentatively stabilize the closed loop so as to yield the bounded input and output. By using them, we compute the fictitious reference signal r˜(‰) (which was introduced by Safonov and Tsao (1997) in the unfalsified control framework) described by
y(‰)
P
¡1
r˜(‰, z) = C(‰, z)¡1 u0 (z) + y 0 (z). (9) Next, we introduce the following cost function in the fictitious domain:
Fig. 2. IMC with a tunable parameter
JF (‰) = ∥y 0 (z) − Td (z)˜ r(‰, z)∥N (10) And then we minimize JF (‰) and implement ‰˜⁄ := arg min‰ JF (‰) to the controller. Note that Eq.(10) with the fictitious reference r˜(‰) in Eq.(9) requires only u0 and y 0 . This means that the minimization of Eq.(10) can be performed off-line by using only one-shot experimental data. As for the relationship between the minimization of Eq.(10) and that of Eq.(6), we obtain the following result. Proposition 3.1. For a parameter ‰˜⁄ , J(˜ ‰⁄ ) = 0 holds if ⁄ and only if JF (˜ ‰ ) = 0 holds. 2
In order to obtain the parameter for the desired tracking property, we introduce the cost function described by J(‰) := ∥y(‰, z) − Td (z)r(z)∥N . (6) Under these settings, the problem we consider here is formulated as follows. Problem 2.1. For a given desired transfer function Td (z), consider the IMC structure with the tunable parameter ‰ ∈ Rn+m+1 illustrated in Fig.2. Set the initial parameter ‰0 . Perform one-shot experiment data u0 := u(‰0 ) and y 0 := y(‰0 ), respectively. Then, the problem is to find the optimal parameter: ‰⁄ := arg min J(‰)
See Theorem 3.1 in Souma et al. (2004) for the detailed proof and discussions. Although it is impossible to completely minimize the cost function JF (‰), this relation implicitly means that the minimization of JF (‰) is deeply related to that of J(‰). Moreover, as explained later in this paper, if FRIT is applied to IMC, the decreasing of the cost function in FRIT leads to that of the original cost function to be minimized.
(7)
‰
with the property that PM (‰⁄ , z) is also closer to the actual plant P (z). 2 Notice that the last problem of Problem 2.1 related to the issue on identificaction. Thus, Problem 2.1 requires the optimal parameter not only for the desired tracking property but also for the model reflecting the dynamics of the actual plant simultaneously.
4. FICTITIOUS REFERENCE ITERATIVE TUNING FOR IMC 4.1 Basic idea
3. FICTITIOUS REFERENCE ITERATIVE TUNING
Firstly, we provide the following theorem as one of the key results in the proposed method. Theorem 4.1. Consider IMC structure in Fig.2 with a tunable parameter. Then, T (‰⁄ , z) = Td (z) is achieved if and only if P (‰, z) = P (z) holds. 2
In this section, we review the fictitious reference iterative tuning (FRIT) based on the references (Kaneko et al. (2005), Souma et al. (2004)). Fig.3 illustrates a conventional feedback control system that consists of a plant and a controller C(‰) described by C(‰, z) =
Poof. It follows from the basic concept of IMC (cf. Morari and Zafiriou (1989)) that the ‘If’ part clearly holds. Thus, we focus on the ‘only if ’ part. From Eq.(5), we see
∑ν
‰(ν + i)z i ∑ ν¡1 z ν + i=0 ‰(i)z i i=0
(8)
Td (z)PM (‰⁄ , z)¡1 P (z) . (11) 1 − Td (z) + Td (z)PM (‰⁄ , z)¡1 P (z) Since the left hand side is equal to Td , Eq.(11) yields 1 − Td (z) + Td (z)PM (‰⁄ , z)¡1 P (z) = PM (‰⁄ , z)P (z). Thus, PM (‰⁄ ) = P holds. 2 T (‰⁄ , z) =
with a tunable parameter ‰ := [‰(0), · · · , ‰(2ν)] ∈ R . For simplicity, we assume that the relative degree of T
u(ρi )
r C(ρ) +
P
2ν+1
y(ρ)
In the case in which P is unknown, Theorem 4.1 implies that we can also obtain the mathematical model of the actual plant by achieving the desired output.
−
By rewriting Fig.2 as Fig.3, we see that the controller is described by
Fig. 3. A conventional feedback control system the controller C(‰, z) is zero (this assumption can be eliminated).
Td (z)PM (‰, z)¡1 . (12) 1 − Td (z) In this case, the fictitious reference in Eq.(9) is described by C(‰, z) =
First, by using the initial parameter ‰0 , perform the first experiment on the closed loop system with C(‰0 , z) and
3
r˜(‰, z) =
1 − Td (z) PM (‰, z)u0 (z) + y 0 (z). Td (z)
0. Prepare the initial parameter ‰0 so as to obtain the closed loop data, and give the desired transfer function Td . 1. In Fig.2, we implement ‰0 . 2. Perform one-shot experiment and obtain the data u0 and y 0 . 3. Perform FRIT: construct the cost function JF (‰) by using the fictitious reference r˜(‰) described by Eq.(13) and minimize it by non-linear optimization off-line. 4. We obtain the optimal parameter ‰⁄ := arg min‰ JF (‰) which yields the desired controller and the mathematical model of the plant.
(13)
By substituting this r˜(‰) into JF (‰) in Eq.(10), the rest to be done is only to minimize JF (‰). As for the relationship between the optimal parameter obtained in FRIT and the desired parameter for the achievement of control and modeling, we firstly provide the following theorem. Theorem 4.2. Consider the cost function JF (‰) described by Eq.(10) for the IMC controller described by Eq.(12). Then, for a parameter ‰˜⁄ , the following properties: 1) JF (˜ ‰⁄ ) = 0 ⇔ J(˜ ‰⁄ ) = 0, ⁄ 2) JF (˜ ‰ ) = 0 ⇔ PM (‰⁄ ) = P
4.2 Some comments
hold. 2
Remark 4.1. The actual measured data u0 and y 0 include the noise. If it is difficult to neglect the effect of noise, we repeat the experiment with respect to the same controller C0 (q) twice under the assumption that the noises in the different experiments are uncorrelated each other. This technique and the assumption are also taken by IFT or VRFT ( cf. Hjalmarsson et al. (1998) and Campi et al. (2002)). We denote the first experimental data with (1) 0(1) (1) 0(1) {yn := y 0(1) + ny , un = u0(1) + nu } and the second 0(2) (2) 0(2) = := y 0(2) + ny , un experimental data with {yn (i) (i) (2) 0(2) u + nu }, respectively. Here, ny and nu denote the noise in the i-th experiment on the input and the output, respectively. y 0(i) and u0(i) denote the pure signal required in this method. The experiment is performed in the closed 0(1) (1) loop, the correlation between e.g., ny and un can not be neglected. However, the two experiments are performed (i) in the different time, it is possible to assume that ny and 0(j) 0(j) (j) (j) (i) nu have no correlation with ny , nu , yn and un , where i, j = (1, 2) or (2, 1). Thus, by modifying the cost function JF (‰) as
In this theorem, the first equivalent condition is an immediate consequence of Proposition 3.1. This implies that the complete minimization of JF (‰), i.e., JF (˜ ‰⁄ ) = 0, is equivalent to the achievement of the desired response. The second condition is an immediate consequence of Theorem 4.1, which implies that JF (˜ ‰⁄ ) = 0 is equivalent to that the actual dynamics of a plant can be also completely obtained. Thus, Theorem 4.2 implicitly relates the minimization of JF (‰) in FRIT to the achievement of the desired response and the identification of the actual plant. Actually, it is impossible to completely minimize the cost function JF (‰) = 0. However, the following theorem relates the minimization of JF (‰) with IMC controller to both the desired controller parameter and the actual parameter of a plant. Theorem 4.3. Let ‰˜ be the parameter such that JF (˜ ‰) < JF (‰0 ).
(14)
Then, the following properties: ∫
¯2 ¯2 ¯ ∫ π ¯ ¯ ¯ ¯ PM (‰0 , ejω ) ¯ PM (˜ ‰, ejω ) ¯ ¯ ¯ ¯ ¯ P (ejω ) − 1¯ dω (15) ¯ P (ejω ) − 1¯ dω < ¡π ¡π
( )T ( ) J˜F (‰) = yn 0(1) − Td (z)˜ r(1) (‰) yn 0(2) − Td (z)˜ r(2) (‰)
π
˜ ¡1 un 0i + yn 0i , i = 1, 2), we can (where, r˜(i) (‰) := C(‰) approximate the cost function so as to eliminate the effect of noise. 2 Remark 4.2. The discussions of this paper have been done in the discrete time case. In the continuous time case, a conventional discretization of involved transfer functions, e.g., Td , PM , C(‰) and so on, enables us to apply the same algorithm. Or, it is useful and valid to utilize an approximate computation of the output of the continuous time transfer function with respect to the discrete time input signal, e.g., Td r˜, C(‰)¡1 u0 , and so on 1 . This is another way for dealing with the continuous time case from the practical points of view.
and ¯ ¯2 ¯2 ∫ π ¯ jω ¯ T (˜ ¯ ¯ T (‰0 , ejω ) ¯ ¯ ‰, e ) − 1¯ dω < ¯ ¯ ¯ Td (ejω ) ¯ ¯ Td (ejω ) − 1¯ dω (16) ¡π ¡π
∫
π
hold. 2 The detailed proof will appear in the reference (Kaneko et al. (2010a)). From the inequality in Eq.(15) of Theorem 4.3, as long as the value of JF (˜ ‰) is less than the initial value JF (‰0 ), the total of the power spectrum density of the relative error between the actual plant P and the obtained model PM (˜ ‰) can be less than that between P and the initial plant model PM (‰0 ). Similarly, from Eq.(16), as long as the value of JF (˜ ‰) is less than the initial value JF (‰0 ), the total of the power spectrum density of the error between the desired transfer function Td and the closed loop with the obtained parameter ‰˜ is less than that between Td and the initial closed loop Td (‰0 ) . Thus, utilizing FRIT for IMC enables us to obtain both the desired controller and the model of a plant with only one-shot experiment.
5. EXPERIMENTAL EXAMPLES In this section, we give experimental examples in order to show the validity of our results. The system we address here is illustrated in Fig. 4. This is the cart-positioning 1
For example, the library function lsim included in Matlab can be used for the computation of the output of the continuous time transfer function. Matlab is the registered trademark of The Mathworks Inc.
We summarize the algorithm of the proposed method.
4
0.15
pulley
servo motor
cart Output[m]
0.1
belt
0.05
0
Fig. 4. The cart system -0.05 0.15
0
2
4
6
8
10
Time[s]
Fig. 6. The initial output y 0 (the solid line) and the desired output yd (the dotted line)
Input[m]
0.1
0.05
0.15 0
0.1 0
2
4
6
8
10
Output[m]
-0.05
Time[s]
Fig. 5. The initial input u
0
0
servo system. The cart is attached to the belt which is moving by the rolling of the servo motor attached in the left side. The location y (output) [m] from the initial position of the cart is measured by the potentiometer attached in the right side. The reference command [m] is applied to this system as the input u. Thus, this system is described by the following transfer function: P =
1 ‰1 s2 + ‰2 s + 1
-0.05
0
2
4
6
8
10
Time[s]
Fig. 7. The output with the optimal parameter y(‰⁄ ) (the solid line) and the desired output yd (the dotted line)
(17)
with unknown parameter ‰ = [‰1 ‰2 ]T in the continuous time case. the sampling time is 1.0 × 10¡3 [sec]. For this system, we implement IMC with unknown parameter ‰, which illustrated in Fig.2.
0.15
0.1 Output[m]
We give the desired transfer function Td as 1 Td = (0.5s + 1)2
0.05
(18)
0.05
0
with the step reference r = 0.1 [m] and the initial parameter ‰0 = [0.1 0.5]T . Under this setting, we perform one-shot experiment and obtain the data. The initial input and output are illustrated in Fig.5 and Fig.6, respectively. In Fig.6, we also plot the desired output yd as the dotted line. Next, by using u0 and y0 , we apply the proposed method for IMC controller. As a result, we obtain the optimal parameter ‰⁄ = [0.0328 1.5874]T . We implement ‰⁄ to the controller and we again perform an experiment, which is plotted in Fig. 7 as the solid line. (In this figure, we also plot yd as the dotted line). Since yd and y(‰⁄ ) are almost the same in Fig.7, we see that the proposed method yields the optimal parameter for the achievement of the desired response.
-0.05
0
2
4
6
8
10
Time[s]
Fig. 8. The validation of mathematical model with the obtained parameter ‰⁄ (the solid line: the experiment with Cv , the dotted line:the simulation of the output of the closed loop consisting PM (‰⁄ ) and Cv ) closed loop experiment with another controller, which is a conventional PI controller Cv := 1 + 4s . In Fig.8, we illustrate the experimental data and the closed loop simulation by using PM (‰⁄ ). From this figure, we also see that the obtained parameter ‰⁄ can reflect the actual dynamics of this plant.
Next, we examine whether PM (‰⁄ ) reflect the actual dynamics of the plant. In order to do this, we perform a
5
6. CONCLUSION
Morari, M., Skogestad, S., and Rivera, D. (1984). Implications of internal model control for pid controllers. Proceedings of American Control Conference, 661–666. Morari, M. and Zafiriou, E. (1989). Robust Process Control. Prentice-Hall, Inc., New Jersey. Park, J. and Bitmead, R.R. (2008). Controller certification. Automatica, 44, 167–176. Safonov, M. and Tsao, T. (1997). The unfalsified control concept and learning. IEEE Transactions on Automatic Control, 42, 843–847. Sala, A. and Esparza, A. (2005). Extentions to “virtual reference feedback tuning”: A direct method for the design of feedback controllers. Automatica, 41, 1473– 1476. Souma, S., Kaneko, O., and Fujii, T. (2004). A new method of controller parameter tuning based on inputoutput data, –fictitious reference iterative tunining. IFAC Workshop on Adaptation and Learning in Control and Signal Processing (ALCOSP 04), 789–794. Yamamoto, S. and Oakano, K. (2006). Direct controller tuning based on data matching. Proceedings of the SICE-ICCAS Joint Conference, 4028–4031.
In this paper, we have proposed the data-driven approach to IMC with FRIT. We have explained how FRIT can be utilized for IMC, and observed how the optimality of the obtained parameter plays a crucial role in the achievement of the desired output and the identification of the plant. Moreover, in order to the validity of our results, we have shown an experimental example. As future studies, we are now trying to see the case in which the length of time delay is unknown, which will be presented in this workshop. Moreover, studies on the effect of the noise and the stability issue are also important directions of this research. We hope that it will be clarified in near future.
REFERENCES Abe, N. (1999). Closed-loop identification and imc design for system with an input time delay. Proceedings of European Control Conference 1999, CD-ROM. Bruyne, F.D. (2003). Iterative feedback tuning for internal mode controllers. Control Engineerinf Practice, 11, 1043–1048. Campi, M., Lecchini, A., and Savaresi, S. (2002). Virtual reference feedback tuning: a direct method for the design of feedback controllers. Automtica, 38, 1337–1446. Fujisaki, Y., Duan, Y., and Ikeda, M. (2005). System representation and optimal tracking in data space. Preprints of The 16th IFAC World Congress, CD–ROM. Garcia, E. and Morari, M. (1982). Internal model control: 1. a unifying review and some results. Industrial and Engineering Chemistry Process Design and Development, 21, 308–323. Hjalmarsson, H., Gevers, M., Gunnarsson, S., and Lequin, O. (1998). Iterative feedback tuning:theory and applications. IEEE control systems magazine, 1, 26–41. Kaneko, O. (2008). On linear canonical controllers within the unfalsified control framework. Preprints of The 17th IFAC World Congress, 12279–12784. Kaneko, O., Souma, S., and Fujii, T. (2005). A fictitious reference iterative tuning (frit) in the two-degree of freedom control scheme and its application to closed loop system identification. Preprints of The 16th IFAC World Congress, CD-ROM. Kaneko, O., Wadagaki, Y., and Yamamoto, S. (2010a). Fictitious reference iterative tuning for internal model controllers - a simultaneous design of controller and mathematical model of a plant. submitted to Journal. Kaneko, O., Yamashina, Y., and Yamamoto, S. (2010b). Fictitious reference tuning for the optimal parameter of a feedfoward controller in the two-degree-of-freedom control system. Proceedings of IEEE Multi Conference on Systems and Control (appear in). Markovsky, I. and Rapisarda, P. (2008). Data driven simulation and control. International Journal on Control, 81, 1946–1959. Masuda, S., Kano, M., and Yasuda, Y. (2009). A fictitious reference iterative method with simultaneous delay parameter tuning of the reference model. Proceedings of IEEE International Conference on Networking, sensingm and Control, 422–427.
6
Fictitious Reference Iterative Tuning for Internal Model Controller Osamu Kaneko ⁄ Yusuke Wadagaki ⁄⁄ Shigeru Yamamoto ⁄ ⁄ Institute of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa, 920-1192, Japan (e-mail: {o-kaneko, shigeru}@t.kanazawa-u.ac.jp). ⁄⁄ Graduate School of Natural Science and Technology, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa, 920-1192 Japan (e-mail: [email protected])
Abstract: Internal model controller (which is abbreviated as IMC in the following ) is widely well-known as one of the easily-understandable controllers from the practical points of view. The reason for this is that IMC has a simple structure which works to decrease the error between the output of the actual plant and the output generated by the internal model included in the controller. In the case in which the internal model completely reflects the dynamics of the actual plant, implementing the model of a plant to IMC yields the desired tracking property. Conversely, in the case in which we do not know a mathematical model of a plant, the achievement of the desired output by some sort of method based on the direct use of the data enables us to identify the plant as the internal model in IMC. Moreover, since the data has a fruitful information of the plant, the direct utilization of the data yields more desirable IMC controller in order to achieve a given specification with reflecting the actual dynamics of the plant. From these ideas, we propose the data-driven approach to IMC with Fictitious Reference Iterative Tuning (which is abbreviated as FRIT) as one of the controller design methods without using a mathematical model of a plant. We show that the minimization of the cost function in FRIT for IMC directly leads to both the optimal controller for achievement of a desired response and a mathematical model reflecting the dynamics of the actual plant, as will become apparent afterwards. Keywords: Data-Driven Approach, Fictitious Reference Iterative Tuning, Internal Model Controller 1. INTRODUCTION Recently, synthesis of a controller with the direct utilization of the data attracts as one of the effective ways for achievement of the desired specification. The reason for this is that a data has fruitful information of a dynamical system. From such a standpoint, there are some studies on the controller synthesis of the direct use of the data, what is called data-driven approaches, e.g., (Fujisaki et al. (2005)), (Kaneko (2008)), (Markovsky and Rapisarda (2008)), (Park and Bitmead (2008)), (Safonov and Tsao (1997)), (Yamamoto and Oakano (2006)) and so on. Particularly, in the cases in which a controller with a tunable parameter has already been implemented, the data-driven approach is also regarded as data-driven parameter tuning. As representative works, we can refer Iterative Feedback Tuning (which is abbreviated as IFT, cf. Hjalmarsson et al. (1998)), Virtual Reference Feedback Tuning (which abbreviated as VRFT, cf. Campi et al. (2002), Sala and Esparza (2005)), and Fictitious Reference Iterative Tuning (FRIT) (cf. Kaneko et al. (2005), Souma et al. (2004), Masuda et al. (2009), Kaneko et al. (2010b)). IFT is the tuning method that iteratively updates the variable parameter of an implemented controller so as to ⋆ This paper was not presented in any IFAC meeting
ISBN 978-3-902661-85-2/11/$20.00 © 2010 IFAC
minimize a performance index, e.g., the sum of squared error signal between the desired reference signal and the actual output, by using the input/output data obtained in the iterative closed loop experiments. This minimization can be computed as a non-linear optimization technique like Gauss-Newton method in which required quantities (gradient, Hessian, Jacobian, and so on) consist of the experimental data. This also means that IFT requires many experiments in order to update the parameter of controllers so as to achieve the minimization of the performance index. Thus IFT spends considerable expense and time, which is a crucial problem with respect to practical points of view. At this point, VRFT and FRIT require only one-shot experiment for the achievement of a desired specification, which means that these two methods have great advantage compared with IFT in the sense that the time and cost for obtaining the optimal parameter are drastically reduced. Particularly, FRIT is easily applicable for the case in which the initial experiment is performed in the closed loop. Moreover, as explained in this paper, FRIT considers the minimization of the error between the fictitious output and the actual one while VRFT focus on the error between virtual (fictitious) input and the actual one, so FRIT is intuitively understandable for the case in which the specification is given for the achievement of the desired output.
1
we can obtain the discrete time frequency response G(ejω ). The output of G(z) with respect to the input time series whose z transformed representation is u(z) is described by G(z)u(z). For a w(z), we use the norm defined by v uN ¡1 ∑ 1u wk2 (2) ∥w(z)∥N = t N
By the way, internal model controller (which is abbreviated as IMC in the following) is widely well-known as one of the easily-understandable controllers from the practical points of view (cf. Garcia and Morari (1982), Morari et al. (1984), Morari and Zafiriou (1989), Abe (1999)). In fact, there are many cases in which IMC structure has already been implemented in the operated in the closed loop. The reason for this is that IMC has a simple structure which performs to decrease the error between the output of the actual plant and the output generated by the internal model included in the controller. In the case in which the mathematical model completely reflects the dynamics of the actual plant, implementing the model of a plant to IMC completely leads to the desired tracking property. Conversely, if we do not know a mathematical model of a plant, it is expected that the achievement of the desired output by some sort of method based on the direct use of the data enables us to identify the plant as the internal mathematical model in IMC. Moreover, since the data has a fruitful information of the plant, the direct utilization of the data yields more desirable IMC controller in order to achieve a given specification with reflecting the actual dynamics of the plant.
k=0
which was introduced in Yamamoto and Oakano (2006). 2.2 Problem formulation Consider IMC structure illustrated in Fig.1 (cf. Morari and Zafiriou (1989), Garcia and Morari (1982)). In this u(‰)
r +
y(‰)
P
CIMC −
−
PM
From these ideas, we propose the data-driven approach to IMC with FRIT as one of the controller design methods without using a mathematical model of a plant. Particularly, we show that the minimization of the cost function in FRIT for IMC directly leads to both the optimal controller for achievement of a desired response and a mathematical model reflecting the dynamics of the actual plant, as will become apparent afterwards. For this very reason, we here utilize FRIT for IMC.
+
Fig. 1. Internal Model Control paper, we assume that a plant is a linear, time invariant, and minimum-phase system. In main part of this paper, though a system is regarded in discrete time, the same discussion is applicable to the continuous time case. We also assume that we know the structure of the transfer function of a plant P described by ∑m i i=0 ‰(n + i)z P (‰, z) = (3) ∑ n¡1 z n + i=0 ‰(i)z i with unknown parameter vector ‰ := [‰(0) ‰(2) · · · ‰(n + m)]T ∈ Rn+m+1 . We also assume that the relative degree n−m is known. In Fig.1, CIM C (z) is the IMC-controller and PM (z) denote the internal model of a plant which included in the controller.
This paper is organized as follows. In Section 2, we give some required preliminaries and the problem formulation. In Section 3, we give an brief explanation of FRIT. In Section 4, the application of FRIT to IMC is presented. In this section, we also explain how FRIT is embedded into the IMC framework and then show that performing FRIT leads to obtain the optimal controller and to identify the mathematical model of a plant. In Section 5, in order to the validity of our results, we show an experimental example. In Section 6, we summarize the concluding remarks.
As one of the candidates for CIM C (z), we apply 2. PRELIMINARIES
CIM C (z) = Td (z)PM (z)¡1 . (4) In Eq.(4), Td (z) is the desired transfer function from the reference r to the output y, which is given for the purpose of the achievement of the desired response property. Since the relative degree of Td (z) is normally fixed equal or greater than n − m, Eq.(4) can be realized as a feedback controller. Throughout this paper, we apply IMC controller described by Eq.(4) and the internal model PM (‰, z), as illustrated in Fig.2. The transfer function from r to the output is denoted with T (‰), which is described by
2.1 Notations Let R and Rn denote the set of real numbers and that of real vectors of size n, respectively. Let z denote the z−operator. Let R(z) denote the set of real coefficient rational functions with the indeterminate z. Let R(z)n denote the set of the vectors of size n whose elements are included in R(z). If w is the discrete time signal, wt denote the value at the time t. For a discrete time signal w = {w0 , w1 , · · · , wk , · · · }, its z-transformed representation is described by w(z) =
∞ ∑
wk z ¡k .
T (‰, z) =
(1)
k=0
1
Td (z) P (z) 1¡Td (z) PM (‰;z) Td (z) P (z) + 1¡T d (z) PM (‰;z)
.
(5)
The input and the output also depend on a controller with tunable parameter ‰, so we denote them u(‰) and y(‰)(:= T (‰, z)r(z)), respectively.
Let G(z) ∈ R(z) denote a discrete time transfer function of a linear time-invariant system. By substituting ejω into z,
2
u(‰)
r +
Td PM (‰) −
+
−
PM (‰)
obtain the initial data u0 and y 0 . Here we also assume that C(‰0 , z) tentatively stabilize the closed loop so as to yield the bounded input and output. By using them, we compute the fictitious reference signal r˜(‰) (which was introduced by Safonov and Tsao (1997) in the unfalsified control framework) described by
y(‰)
P
¡1
r˜(‰, z) = C(‰, z)¡1 u0 (z) + y 0 (z). (9) Next, we introduce the following cost function in the fictitious domain:
Fig. 2. IMC with a tunable parameter
JF (‰) = ∥y 0 (z) − Td (z)˜ r(‰, z)∥N (10) And then we minimize JF (‰) and implement ‰˜⁄ := arg min‰ JF (‰) to the controller. Note that Eq.(10) with the fictitious reference r˜(‰) in Eq.(9) requires only u0 and y 0 . This means that the minimization of Eq.(10) can be performed off-line by using only one-shot experimental data. As for the relationship between the minimization of Eq.(10) and that of Eq.(6), we obtain the following result. Proposition 3.1. For a parameter ‰˜⁄ , J(˜ ‰⁄ ) = 0 holds if ⁄ and only if JF (˜ ‰ ) = 0 holds. 2
In order to obtain the parameter for the desired tracking property, we introduce the cost function described by J(‰) := ∥y(‰, z) − Td (z)r(z)∥N . (6) Under these settings, the problem we consider here is formulated as follows. Problem 2.1. For a given desired transfer function Td (z), consider the IMC structure with the tunable parameter ‰ ∈ Rn+m+1 illustrated in Fig.2. Set the initial parameter ‰0 . Perform one-shot experiment data u0 := u(‰0 ) and y 0 := y(‰0 ), respectively. Then, the problem is to find the optimal parameter: ‰⁄ := arg min J(‰)
See Theorem 3.1 in Souma et al. (2004) for the detailed proof and discussions. Although it is impossible to completely minimize the cost function JF (‰), this relation implicitly means that the minimization of JF (‰) is deeply related to that of J(‰). Moreover, as explained later in this paper, if FRIT is applied to IMC, the decreasing of the cost function in FRIT leads to that of the original cost function to be minimized.
(7)
‰
with the property that PM (‰⁄ , z) is also closer to the actual plant P (z). 2 Notice that the last problem of Problem 2.1 related to the issue on identificaction. Thus, Problem 2.1 requires the optimal parameter not only for the desired tracking property but also for the model reflecting the dynamics of the actual plant simultaneously.
4. FICTITIOUS REFERENCE ITERATIVE TUNING FOR IMC 4.1 Basic idea
3. FICTITIOUS REFERENCE ITERATIVE TUNING
Firstly, we provide the following theorem as one of the key results in the proposed method. Theorem 4.1. Consider IMC structure in Fig.2 with a tunable parameter. Then, T (‰⁄ , z) = Td (z) is achieved if and only if P (‰, z) = P (z) holds. 2
In this section, we review the fictitious reference iterative tuning (FRIT) based on the references (Kaneko et al. (2005), Souma et al. (2004)). Fig.3 illustrates a conventional feedback control system that consists of a plant and a controller C(‰) described by C(‰, z) =
Poof. It follows from the basic concept of IMC (cf. Morari and Zafiriou (1989)) that the ‘If’ part clearly holds. Thus, we focus on the ‘only if ’ part. From Eq.(5), we see
∑ν
‰(ν + i)z i ∑ ν¡1 z ν + i=0 ‰(i)z i i=0
(8)
Td (z)PM (‰⁄ , z)¡1 P (z) . (11) 1 − Td (z) + Td (z)PM (‰⁄ , z)¡1 P (z) Since the left hand side is equal to Td , Eq.(11) yields 1 − Td (z) + Td (z)PM (‰⁄ , z)¡1 P (z) = PM (‰⁄ , z)P (z). Thus, PM (‰⁄ ) = P holds. 2 T (‰⁄ , z) =
with a tunable parameter ‰ := [‰(0), · · · , ‰(2ν)] ∈ R . For simplicity, we assume that the relative degree of T
u(ρi )
r C(ρ) +
P
2ν+1
y(ρ)
In the case in which P is unknown, Theorem 4.1 implies that we can also obtain the mathematical model of the actual plant by achieving the desired output.
−
By rewriting Fig.2 as Fig.3, we see that the controller is described by
Fig. 3. A conventional feedback control system the controller C(‰, z) is zero (this assumption can be eliminated).
Td (z)PM (‰, z)¡1 . (12) 1 − Td (z) In this case, the fictitious reference in Eq.(9) is described by C(‰, z) =
First, by using the initial parameter ‰0 , perform the first experiment on the closed loop system with C(‰0 , z) and
3
r˜(‰, z) =
1 − Td (z) PM (‰, z)u0 (z) + y 0 (z). Td (z)
0. Prepare the initial parameter ‰0 so as to obtain the closed loop data, and give the desired transfer function Td . 1. In Fig.2, we implement ‰0 . 2. Perform one-shot experiment and obtain the data u0 and y 0 . 3. Perform FRIT: construct the cost function JF (‰) by using the fictitious reference r˜(‰) described by Eq.(13) and minimize it by non-linear optimization off-line. 4. We obtain the optimal parameter ‰⁄ := arg min‰ JF (‰) which yields the desired controller and the mathematical model of the plant.
(13)
By substituting this r˜(‰) into JF (‰) in Eq.(10), the rest to be done is only to minimize JF (‰). As for the relationship between the optimal parameter obtained in FRIT and the desired parameter for the achievement of control and modeling, we firstly provide the following theorem. Theorem 4.2. Consider the cost function JF (‰) described by Eq.(10) for the IMC controller described by Eq.(12). Then, for a parameter ‰˜⁄ , the following properties: 1) JF (˜ ‰⁄ ) = 0 ⇔ J(˜ ‰⁄ ) = 0, ⁄ 2) JF (˜ ‰ ) = 0 ⇔ PM (‰⁄ ) = P
4.2 Some comments
hold. 2
Remark 4.1. The actual measured data u0 and y 0 include the noise. If it is difficult to neglect the effect of noise, we repeat the experiment with respect to the same controller C0 (q) twice under the assumption that the noises in the different experiments are uncorrelated each other. This technique and the assumption are also taken by IFT or VRFT ( cf. Hjalmarsson et al. (1998) and Campi et al. (2002)). We denote the first experimental data with (1) 0(1) (1) 0(1) {yn := y 0(1) + ny , un = u0(1) + nu } and the second 0(2) (2) 0(2) = := y 0(2) + ny , un experimental data with {yn (i) (i) (2) 0(2) u + nu }, respectively. Here, ny and nu denote the noise in the i-th experiment on the input and the output, respectively. y 0(i) and u0(i) denote the pure signal required in this method. The experiment is performed in the closed 0(1) (1) loop, the correlation between e.g., ny and un can not be neglected. However, the two experiments are performed (i) in the different time, it is possible to assume that ny and 0(j) 0(j) (j) (j) (i) nu have no correlation with ny , nu , yn and un , where i, j = (1, 2) or (2, 1). Thus, by modifying the cost function JF (‰) as
In this theorem, the first equivalent condition is an immediate consequence of Proposition 3.1. This implies that the complete minimization of JF (‰), i.e., JF (˜ ‰⁄ ) = 0, is equivalent to the achievement of the desired response. The second condition is an immediate consequence of Theorem 4.1, which implies that JF (˜ ‰⁄ ) = 0 is equivalent to that the actual dynamics of a plant can be also completely obtained. Thus, Theorem 4.2 implicitly relates the minimization of JF (‰) in FRIT to the achievement of the desired response and the identification of the actual plant. Actually, it is impossible to completely minimize the cost function JF (‰) = 0. However, the following theorem relates the minimization of JF (‰) with IMC controller to both the desired controller parameter and the actual parameter of a plant. Theorem 4.3. Let ‰˜ be the parameter such that JF (˜ ‰) < JF (‰0 ).
(14)
Then, the following properties: ∫
¯2 ¯2 ¯ ∫ π ¯ ¯ ¯ ¯ PM (‰0 , ejω ) ¯ PM (˜ ‰, ejω ) ¯ ¯ ¯ ¯ ¯ P (ejω ) − 1¯ dω (15) ¯ P (ejω ) − 1¯ dω < ¡π ¡π
( )T ( ) J˜F (‰) = yn 0(1) − Td (z)˜ r(1) (‰) yn 0(2) − Td (z)˜ r(2) (‰)
π
˜ ¡1 un 0i + yn 0i , i = 1, 2), we can (where, r˜(i) (‰) := C(‰) approximate the cost function so as to eliminate the effect of noise. 2 Remark 4.2. The discussions of this paper have been done in the discrete time case. In the continuous time case, a conventional discretization of involved transfer functions, e.g., Td , PM , C(‰) and so on, enables us to apply the same algorithm. Or, it is useful and valid to utilize an approximate computation of the output of the continuous time transfer function with respect to the discrete time input signal, e.g., Td r˜, C(‰)¡1 u0 , and so on 1 . This is another way for dealing with the continuous time case from the practical points of view.
and ¯ ¯2 ¯2 ∫ π ¯ jω ¯ T (˜ ¯ ¯ T (‰0 , ejω ) ¯ ¯ ‰, e ) − 1¯ dω < ¯ ¯ ¯ Td (ejω ) ¯ ¯ Td (ejω ) − 1¯ dω (16) ¡π ¡π
∫
π
hold. 2 The detailed proof will appear in the reference (Kaneko et al. (2010a)). From the inequality in Eq.(15) of Theorem 4.3, as long as the value of JF (˜ ‰) is less than the initial value JF (‰0 ), the total of the power spectrum density of the relative error between the actual plant P and the obtained model PM (˜ ‰) can be less than that between P and the initial plant model PM (‰0 ). Similarly, from Eq.(16), as long as the value of JF (˜ ‰) is less than the initial value JF (‰0 ), the total of the power spectrum density of the error between the desired transfer function Td and the closed loop with the obtained parameter ‰˜ is less than that between Td and the initial closed loop Td (‰0 ) . Thus, utilizing FRIT for IMC enables us to obtain both the desired controller and the model of a plant with only one-shot experiment.
5. EXPERIMENTAL EXAMPLES In this section, we give experimental examples in order to show the validity of our results. The system we address here is illustrated in Fig. 4. This is the cart-positioning 1
For example, the library function lsim included in Matlab can be used for the computation of the output of the continuous time transfer function. Matlab is the registered trademark of The Mathworks Inc.
We summarize the algorithm of the proposed method.
4
0.15
pulley
servo motor
cart Output[m]
0.1
belt
0.05
0
Fig. 4. The cart system -0.05 0.15
0
2
4
6
8
10
Time[s]
Fig. 6. The initial output y 0 (the solid line) and the desired output yd (the dotted line)
Input[m]
0.1
0.05
0.15 0
0.1 0
2
4
6
8
10
Output[m]
-0.05
Time[s]
Fig. 5. The initial input u
0
0
servo system. The cart is attached to the belt which is moving by the rolling of the servo motor attached in the left side. The location y (output) [m] from the initial position of the cart is measured by the potentiometer attached in the right side. The reference command [m] is applied to this system as the input u. Thus, this system is described by the following transfer function: P =
1 ‰1 s2 + ‰2 s + 1
-0.05
0
2
4
6
8
10
Time[s]
Fig. 7. The output with the optimal parameter y(‰⁄ ) (the solid line) and the desired output yd (the dotted line)
(17)
with unknown parameter ‰ = [‰1 ‰2 ]T in the continuous time case. the sampling time is 1.0 × 10¡3 [sec]. For this system, we implement IMC with unknown parameter ‰, which illustrated in Fig.2.
0.15
0.1 Output[m]
We give the desired transfer function Td as 1 Td = (0.5s + 1)2
0.05
(18)
0.05
0
with the step reference r = 0.1 [m] and the initial parameter ‰0 = [0.1 0.5]T . Under this setting, we perform one-shot experiment and obtain the data. The initial input and output are illustrated in Fig.5 and Fig.6, respectively. In Fig.6, we also plot the desired output yd as the dotted line. Next, by using u0 and y0 , we apply the proposed method for IMC controller. As a result, we obtain the optimal parameter ‰⁄ = [0.0328 1.5874]T . We implement ‰⁄ to the controller and we again perform an experiment, which is plotted in Fig. 7 as the solid line. (In this figure, we also plot yd as the dotted line). Since yd and y(‰⁄ ) are almost the same in Fig.7, we see that the proposed method yields the optimal parameter for the achievement of the desired response.
-0.05
0
2
4
6
8
10
Time[s]
Fig. 8. The validation of mathematical model with the obtained parameter ‰⁄ (the solid line: the experiment with Cv , the dotted line:the simulation of the output of the closed loop consisting PM (‰⁄ ) and Cv ) closed loop experiment with another controller, which is a conventional PI controller Cv := 1 + 4s . In Fig.8, we illustrate the experimental data and the closed loop simulation by using PM (‰⁄ ). From this figure, we also see that the obtained parameter ‰⁄ can reflect the actual dynamics of this plant.
Next, we examine whether PM (‰⁄ ) reflect the actual dynamics of the plant. In order to do this, we perform a
5
6. CONCLUSION
Morari, M., Skogestad, S., and Rivera, D. (1984). Implications of internal model control for pid controllers. Proceedings of American Control Conference, 661–666. Morari, M. and Zafiriou, E. (1989). Robust Process Control. Prentice-Hall, Inc., New Jersey. Park, J. and Bitmead, R.R. (2008). Controller certification. Automatica, 44, 167–176. Safonov, M. and Tsao, T. (1997). The unfalsified control concept and learning. IEEE Transactions on Automatic Control, 42, 843–847. Sala, A. and Esparza, A. (2005). Extentions to “virtual reference feedback tuning”: A direct method for the design of feedback controllers. Automatica, 41, 1473– 1476. Souma, S., Kaneko, O., and Fujii, T. (2004). A new method of controller parameter tuning based on inputoutput data, –fictitious reference iterative tunining. IFAC Workshop on Adaptation and Learning in Control and Signal Processing (ALCOSP 04), 789–794. Yamamoto, S. and Oakano, K. (2006). Direct controller tuning based on data matching. Proceedings of the SICE-ICCAS Joint Conference, 4028–4031.
In this paper, we have proposed the data-driven approach to IMC with FRIT. We have explained how FRIT can be utilized for IMC, and observed how the optimality of the obtained parameter plays a crucial role in the achievement of the desired output and the identification of the plant. Moreover, in order to the validity of our results, we have shown an experimental example. As future studies, we are now trying to see the case in which the length of time delay is unknown, which will be presented in this workshop. Moreover, studies on the effect of the noise and the stability issue are also important directions of this research. We hope that it will be clarified in near future.
REFERENCES Abe, N. (1999). Closed-loop identification and imc design for system with an input time delay. Proceedings of European Control Conference 1999, CD-ROM. Bruyne, F.D. (2003). Iterative feedback tuning for internal mode controllers. Control Engineerinf Practice, 11, 1043–1048. Campi, M., Lecchini, A., and Savaresi, S. (2002). Virtual reference feedback tuning: a direct method for the design of feedback controllers. Automtica, 38, 1337–1446. Fujisaki, Y., Duan, Y., and Ikeda, M. (2005). System representation and optimal tracking in data space. Preprints of The 16th IFAC World Congress, CD–ROM. Garcia, E. and Morari, M. (1982). Internal model control: 1. a unifying review and some results. Industrial and Engineering Chemistry Process Design and Development, 21, 308–323. Hjalmarsson, H., Gevers, M., Gunnarsson, S., and Lequin, O. (1998). Iterative feedback tuning:theory and applications. IEEE control systems magazine, 1, 26–41. Kaneko, O. (2008). On linear canonical controllers within the unfalsified control framework. Preprints of The 17th IFAC World Congress, 12279–12784. Kaneko, O., Souma, S., and Fujii, T. (2005). A fictitious reference iterative tuning (frit) in the two-degree of freedom control scheme and its application to closed loop system identification. Preprints of The 16th IFAC World Congress, CD-ROM. Kaneko, O., Wadagaki, Y., and Yamamoto, S. (2010a). Fictitious reference iterative tuning for internal model controllers - a simultaneous design of controller and mathematical model of a plant. submitted to Journal. Kaneko, O., Yamashina, Y., and Yamamoto, S. (2010b). Fictitious reference tuning for the optimal parameter of a feedfoward controller in the two-degree-of-freedom control system. Proceedings of IEEE Multi Conference on Systems and Control (appear in). Markovsky, I. and Rapisarda, P. (2008). Data driven simulation and control. International Journal on Control, 81, 1946–1959. Masuda, S., Kano, M., and Yasuda, Y. (2009). A fictitious reference iterative method with simultaneous delay parameter tuning of the reference model. Proceedings of IEEE International Conference on Networking, sensingm and Control, 422–427.
6