- Email: [email protected]
Fictitious Reference Iterative Tuning of Disturbance Observers for Attenuation of the Effect of Periodic Unknown Exogenous Signals
11th IFAC International Workshop on Adaptation and Learning in Control and Signal Processing July 3-5, 2013. Caen, France
ThS6T2.3
Fictitious Reference Iterative Tuning of Disturbance Observers for Attenuation of the Effect of Periodic Unknown Exogenous Signals ⋆ Fumiaki Uozumi ∗ Osamu Kaneko ∗∗ ShigeruYamamoto ∗∗ ∗
Graduate School of Natural Science and Technology, Kanazawa University, Kakumamachi, Kanazawa, Ishikawa 920-1192, Japan (Tel: +81-76-264-6332; e-mail: [email protected]). ∗∗ Institute of Science and Engineering, Kanazawa University, Kakumamachi, Kanazawa, Ishikawa 920-1192, Japan (e-mail: o-kaneko, [email protected]) Abstract: In this paper, we develop a data-driven controller tuning for the disturbance observers. Particularly, we focus on the disturbance attenuation problem for a periodic unknown disturbance. However, in the case where it is difficult to construct inverse mathematical model of a plant which is implemented in the disturbance observer, we have to tale much time and costs for identification. In order to overcome such a difficulty, we utilize fictitious reference iterative tuning (FRIT), which is data-driven controller tuning with only one-shot experimental data, for tuning of parameters of controllers to eliminate the effect of the disturbance. Keywords: Data-driven parameter tuning, Fictitious Reference Iterative Tuning (FRIT), Disturbance observer, Disturbance attenuation 1. INTRODUCTION It is usual that an unknown disturbance causes the unexpected phenomena in the control system. For example, a periodic disturbance caused by the motor, vibration of the structure, and so on, is one of the representative problems in many practical situations. Normally, it is difficult to measure such a disturbance directly. Thus, we should estimate the disturbance or its effect in one way or another. For this purpose, the disturbance observer, which was proposed and studied in Ohnishi [1987], Hori [1993], Umeno and Hori [1991], and Mita et al. [1998], is one of the effective methods to estimate or to eliminate the effect of the disturbance. In addition, there are also many studies on disturbance observers from the practical points of view, e.g., Choi et al. [2003] and Kempf and Kobayashi [1999]. In the case where the control system does not contain the architecture of the disturbance observer, we should implement it and re-construct whole of the control systems. On the other hand, there are many cases in which such an implementation or reconstructing of the control system spends much cost. In these cases, it is prefer to only add the off-line mechanism for the estimation of the disturbance. Moreover, it is fundamental that the architecture of the disturbance observer needs the inverse model of the plant. To obtain such a model, we also need a mathematical model of a plant, which requires experiment for the ⋆ This work was partially supported by JSPS Grant in Aid for Scienti c Research (C) 22560442, (B) 23360183, and (C) 25420432 (The third one partially supports this work in the revision process of the nal manuscript.)
978-3-902823-37-3/2013 © IFAC
576
identification. On the other hand, there are many cases in which it is difficult to apply a persistently exciting signal for identifying the dynamics in broader frequency band from the view points of the safety. In addition, there are also many cases where it is impossible to take much cost for experiment from the view points of the scheduling of industry. In these cases, the direct use of the data is expected as one of the rational approaches. As representative works, iterative feedback tuning (IFT) in Hjalmarsson et al. [1998], virtual reference iterative tuning (VRFT) in Campi et al. [2002], the non-iterative version of correlation based tuning in Karimi et al. [2007], and the fictitious reference iterative tuning (FRIT) in Souma et al. [2004] and Kaneko et al. [2005] were proposed and studied. Among of them, IFT is the most rational approach because the cost function to be minimized in IFT directly represents the purpose of the tuning of a controller. However, IFT requires many experiments, which is a crucial drawback from the practical points of view. The later three methods enables us to obtain the desired parameter with only one shot experiment. In Kaneko [2013] (to appear in this workshop), the comparisons of these methods are presented, so Kaneko [2013] for more details. Here we take FRIT approach. Because this method is intuitively understandable since the output is focused in this method for off-line minimization. From these backgrounds, we address a data-driven approach to the attenuation of exogenous periodic disturbance. Particularly, we use the idea of the fictitious reference iterative tuning (which is abbreviated as FRIT), which was proposed and developed in Souma et al. [2004], 10.3182/20130703-3-FR-4038.00140
11th IFAC ALCOSP July 3-5, 2013. Caen, France
Kaneko et al. [2005], Kaneko et al. [2010], Kaneko et al. [2011], and Kaneko et al. [2012]. In this paper, we provide a novel off-line mechanism for disturbance observer to be tuned based on the direct use of the experimental data. In Yasuda et al. [2009] and Masuda and Dong [2010], the disturbance attenuation by utilizing FRIT were studied. In these studies, the exogenous disturbance is assumed to be known. Compared with them, our setting does not assume that the exogenous disturbance is known.
We assume that C(ρini ) with the initial parameter ρini tentatively stabilizes the closed loop so as to obtain the bounded input uini := u(ρini ) and output yini = y(ρini ).
Contents of this paper is the following. In Section 2, we formulate the problem to be addressed in this paper. In Section 3, we give brief reviews on some preliminaries on the disturbance observer and FRIT. In Section 4, the main result of this paper is presented, particularly, a novel off-line architecture for tuning of a controller for the disturbance attenuation. In order to validate our proposed method, we show some numerical examples in Section 5. Finally, concluding remarks are given in the Section 6.
Under these settings, in this paper, we obtain the optimal parameter such that
[Notation] We denote the set of the real numbers as R. For a time signal w, let w(t) denote the value of w at time t. For a time signal w, let Φw (jω) denote the power spectrum density in the frequency domain. We introduce √ ∑ N 1 ||w||(N,∆) := N t=1 ω(t∆)2 with sampling period ∆. In terms of the enhancement of the readability, we often omit the notation ‘s’ from a transfer function G(s) if it is clear from the context. In addition, the output of a transfer function G with respect to the input u is denoted with y = Gu. 2. PROBLEM FORMULATION In Fig. 1, we illustrate the conventional control system with a tunable controller with respect to a parameter ρ. Assume that a plant G is single-input and single output, linear, time-invariant, and minimum phase. We assume that G is unknown while the structure of G is known. Let Gn denote a nominal model of a plant. Let d denote an unknown periodic disturbance. A feedback controller C(ρ) is with a tunable parameter vector ρ. For example, C(ρ) is parameterized by ρ1 sn + ρ2 sn−1 + · · · + ρn s + 1 , (1) ρn+1 sn + ρn+2 sn−1 + · · · + ρ2n s + ρ2n+1 using parameter vector ρ := [ρ1 ρ2 · · · ρ2n ρ2n+1 ]. The closed loop can be also regarded as the function of ρ, so we denote G T (ρ) := 1 + GC(ρ) as the transfer function from d to y. Moreover, we also denote the input and output of the closed loop by y(ρ) and u(ρ), respectively. C(ρ) =
Fig. 1. A closed loop control system contains the unknown disturbance 577
We are also given a desired property for attenuation of the disturbance as the reference model from d to y as Td . We set Td such that Td (ejωd ) ≃ 0 holds where ωd is one of the frequency modes of the periodic disturbance to be attenuated.
2
J(ρ) = ∥y(ρ) − Td d∥(N,∆)
(2)
is minimized with only using the data uini and yini . Moreover, if necessary, we consider new architecture for the estimation of the disturbance. 3. PRELIMINARIES In this section, we give brief reviews on the disturbance observer and FRIT. 3.1 Disturbance Observer We give a brief review of disturbance observer based on Ohnishi [1987] and Hori [1993]. The disturbance observer is one of the novel mechanisms for the estimation of an unknown disturbance. By using the estimated disturbance, we can compensate the degradation of the performance of the closed loop due to the disturbance. The structure of the disturbance observer is illustrated in Fig. 2. The output of the inverse nominal plant G−1 n is applied to the low pass filter Q. The observed disturbance dest is generated by u and the output of the filter Q. We can calculate dest in Fig. 2 as −1 dest = QG−1 (3) n Gd + Q(Gn G − 1)u. We assume that the nominal plant Gn is close to the plant G in the frequency band of Q. Thus, it is possible to estimate dest ≃ d in the frequency range where we can regard QG−1 n G ≃ 1. 3.2 Fictitious Reference Iterative Tuning (FRIT) We give a brief review of FRIT based on Souma et al. [2004] and Kaneko et al. [2005]. Figure 3 illustrates a conventional feedback control system that consists of a plant and a controller C(ρ) with a tunable parameter ρ. Let M (ρ) denote the transfer function from r to y. And consider the model reference problem where the purpose of the tuning of controller is that the control system yields
Fig. 2. The conventional disturbance observer
11th IFAC ALCOSP July 3-5, 2013. Caen, France
By defining Tn (ρ) := we have
Fig. 3. A closed loop control system the desired output Md r, where Md is a given reference model from r to y. By using the initial parameter ρini , perform the initial experiment on the closed loop system with C(ρini ). We then obtain the initial data uini := u(ρini ) and yini := y(ρini ). We also assume that C(ρini ) tentatively stabilize the closed loop so as to yield the bounded input and output. By using the initial data, we compute the fictitious reference signal r˜(ρ) (which was introduced by Safonov and Tsao [1997] in the unfalsified control framework) described by r˜(ρ) = C(ρ)−1 uini + yini .
(4)
JF (ρ) = ∥yini −
(5)
And then we minimize JF (ρ) and implement ρ˜∗ := arg minρ JF (ρ) to the controller. Note that (5) with the fictitious reference r˜(ρ) in (4) requires only uini and yini . This means that the minimization of (5) can be performed off-line by using only one-shot experimental data. The relationship between the minimization of (5) and that of (2) can be given by rewriting the cost function JF (ρ) as
(
2 )
Md
JF (ρ) = 1 − yini .
M (ρ) (N,∆)
d˜est (ρ) = QTn (ρ)−1 yini . (8) Since the role of Q is to compensate the non-properness of G−1 n , it is possible to give Q such that the frequency pass-band of Q is higher than the frequency range of the exogenous disturbance. Thus, in the frequency range where we can regard Qyini ≃ yini , the estimated observation is approximated by d˜est (ρ) = Tn (ρ)−1 yini . (9) which is equivalent to Tn (ρ)d˜est (ρ) = yini .
(10)
Now we introduce a new cost function to be minimized
2
JF D (ρ) = yini − Td d˜est (ρ) . (11) (N,∆)
Next, we introduce the cost function described by Md r˜(ρ)∥2(N,∆) .
Gn , 1 + Gn C(ρ)
(6)
From (6), we see that the minimization of JF (ρ) in (5) leads to the minimization of the relative error between the desired reference model and the actual closed loop with the parameter ρ under the effect of the initial output. This is an explanation on how FRIT can achieve the desired output.
From (9), JF D can be rewritten as
(
2 )
Td
JF D (ρ) = 1 − Qyini
Tn (ρ)
.
(12)
(N,∆)
If JF D (ρ) can be completely minimized, namely JF D (ρ∗ ) = 0 at ρ∗ := arg min JF D (ρ), then the relation Gn 1 + Gn C(ρ∗ ) generically holds. This implies that Td = Tn (ρ∗ ) =
Tn (ρ∗ )d˜est (ρ∗ ) = Td d˜est (ρ∗ )
(13)
(14)
holds. On the other hand, from Fig. 4, we also see that T (ρ)d = y(ρ) = Tn (ρ)d˜est (ρ)
(15)
holds for any parameter ρ. From (14) and (15), we obtain y(ρ∗ ) = T (ρ∗ )d = Td d˜est (ρ∗ ).
(16)
4. MAIN RESULT 4.1 Basic Idea We introduce a new off-line mechanism for the estimation of the disturbance by using the observed output. Consider the block diagram illustrated in Fig. 1. The initial control output yini is obtained in the experiment using initial controller parameter ρini , which is affected by the disturbance d. Now, we consider a new off-line architecture illustrated in Fig. 4 where we add the inverse of the nominal model of the plant G−1 n . At this point, since the nominal model Gn is strictly proper, the inverse nominal plant is nonproper. To solve this problem, we use a low-pass filter Q to estimate the disturbance d, which is from the concept of the disturbance observer. In Fig. 4, the estimated disturbance which is denoted with d˜est and also depends on the parameter ρ, is generated by ( ) 1 + Gn C(ρ) d˜est (ρ) = Q G−1 yini . (7) n + C(ρ) yini = Q Gn 578
Fig. 4. The off-line estimation for the estimated disturbance d˜est (ρ)
Fig. 5. Td d˜est (ρ) and the nominal closed loop
11th IFAC ALCOSP July 3-5, 2013. Caen, France
Under the assumption that the involved signals are with the ergodic property, the Parseval theorem enables us to transform (16) into the frequency domain as Φy(ρ∗ ) (jω) = Td (jω)Φd˜est (ρ∗ ) (jω).
(17)
As stated above, since Td is given by the designer, the magnitude of Td at the frequency mode ω 0 at which the disturbance should be attenuated can be restrained as Td (jω 0 ) ≃ 0. As a result, the effect of d on y(ρ∗ ) is also attenuated. That is, by implementing ρ∗ to the controller, we can obtain the closed loop that attenuates the effect of the disturbance d. This is an intuitive explanation of the proposed method. 4.2 Analysis for the case of JF D (ρ) ̸= 0 We consider the case where JF D (ρ) = 0 can not be achieved. Notice the cost function JF D (ρ) can be rewritten as (12). From this, we see that the minimization of JFD corresponds to that of the relative error between the desired property Td and the nominal closed loop which can be computed in the off-line. Assume that we obtain the optimal parameter ρ∗ := arg min JF D (ρ). Then, since there exists a relative error, between Td and Tn (ρ∗ ), these two transfer function can be related as Tn (ρ∗ ) = (1 + δ(ρ∗ ))Td
(18)
∗
∗
where δ(ρ ) is a relative error. By implementing ρ to the actual closed loop, it follows from the architecture of Fig. 4 that we obtain y(ρ∗ ) = T (ρ∗ )d = Tn (ρ∗ )d˜est (ρ∗ ).
(19)
From (18), we also obtain Tn (ρ∗ )d˜est (ρ∗ ) = (1 + δ(ρ∗ ))Td d˜est (ρ∗ ).
(20)
Thus, we see y(ρ∗ ) = (1 + δ(ρ∗ ))Td d˜est (ρ∗ ) = Td d˜′est (ρ∗ ),
(21)
where d˜′est (ρ∗ ) := (1 + δ(ρ∗ ))d˜est (ρ∗ ). Again, from the Parseval theorem, we obtain Φy(ρ∗ ) (jω) = Td (jω)Φd˜′
est (ρ
∗)
(jω).
(22)
Due to the same reason in the ideal case stated in the previous subsection, we see that the effect of the disturbance can be attenuated by using the parameter ρ∗ .
In the step 3), we can utilize some conventional nonlinear optimization methods, e.g., Gauss-Newton methods, CMA-ES in Hansen and Ostermeier [2001], and so on. Notice that we do not have to reconstruct the control system and it is enough to add a simple off-line architecture as Fig. 4. Remark 1. It is normal that there exists a difference between the actual plant G and the nominal model Gn . Thus, we should check the stability of T (ρ∗ ) by using that of Tn (ρ∗ ) before implementing of the obtained parameter ρ∗ . Generally, the stability check in direct data-driven approaches is one of the open problems to be clarified. For the strategy to this problem, the utilization of the Kharitonov theorem in Kharitonov [1978] and Anderson et al. [1987] is one of the rational approaches. This theorem is known as a useful tool for checking whether the roots of a polynomial with unknown coefficients whose bounds are known lie in the open left half plane. In this case, we can calculate such bounds of each coefficients of the denominator of T (ρ) if we know each interval in which each coefficient of the numerator and the denominator of G takes the value is known. And then we can utilize Kharitonov theorem to check the stability of T (ρ). If the answer is that T (ρ) is unstable, we should search an alternative parameter in the non-linear optimization. The detailed discussions will be in the Workshop and our forthcoming papers. For more generalized situations, this is important open problem, which is to be clarified in the data-driven setting. 5. NUMERICAL EXAMPLE In order to show the validity of the proposed method, we show two examples, because it is better to show that our method is applicable for the various disturbances. We consider 12s2 + 47s + 8 G= 3 9s + 62s2 + 73s + 5 as the actual plant and 12.8573s2 + 49.5832s + 8.3643 Gn = 9.6919s3 + 66.7601s2 + 75.0548s + 5.4394 as its nominal model. Of course we assume that we do not know the exact coefficients of G. We also use the tunable feedback controller ρ1 s2 + ρ2 s + 1 C(ρ) = , ρ = [ ρ1 ρ2 ρ3 ρ4 ρ5 ]. ρ3 s2 + ρ4 s + ρ5 We define the initial parameter as ρini = [1 4 1 5 6]. We set the sampling period ∆ = 0.001 sec. 5.1 The case where the disturbance is the combination of some sine waves
4.3 Algorithm Here we summarize the proposed algorithm. 1) Set the initial parameter ρini . 2) Perform the initial experiment with ρini in Fig. 1 and obtain the data uini and yini . 3) Minimize the cost function JF D (ρ) in (11) with the “fictitious” estimated disturbance d˜est (ρ) in (9) offline. 4) Implement ρ˜∗ in Fig. 1 and perform the experiment again. 579
First, we consider the case where the unknown periodic disturbance d is the combination of some sine-waves. We use d(t) = sin 2t + 0.6 sin 5t + 0.8 sin 0.8t + w(t), where w(t) is the white noise with the variance 0.01. With the initial parameter ρini , we perform a one-shot experimentation and obtain the data. The disturbance d and the initial output yini are illustrated by the broken line
11th IFAC ALCOSP July 3-5, 2013. Caen, France
Fig. 6. The initial output yini (the broken line), the output after the optimal parameter y(ρ∗ ) (the solid line), and the disturbance d (the chained line)
Fig. 8. The initial output yini (the broken line), the output after the optimal parameter y(ρ∗ ) (the solid line), and the disturbance d (the chained line)
Fig. 7. Frequency responses of the desired function Td (dotted line), the initial closed loop system T (ρini ) (the chained line), and the closed loop with the optimal parameter T (ρ∗ ) (the solid line)
Fig. 9. Frequency responses of the desired function Td (the broken line), the initial closed loop system T (ρini ) (the chained line), and the closed loop with the optimal parameter T (ρ∗ ) (the solid line)
and the chained line, respectively in Fig. 6. By using the initial experimental data, we apply our proposed method. Here, we give the desired function Td as 10s + 1 Td = 10s2 + 1000s + 1 so as to eliminate the effect of the disturbance over a sufficiently large frequency range. In the optimization process, we also check the stability of the closed loop by using Kharitonov theorem. As a result, we obtain the optimal parameter ρ∗ = [44.27 100 0.46 1.06 0.13]. By implementing ρ∗ to the controller in Fig. 1, we again perform the experiment. The output y(ρ∗ ) is also illustrated in Fig. 6 as the solid line. Compared with the initial output yini , we see that the effect of the disturbance can be sufficiently attenuated in Fig. 6.
line, respectively. From Fig. 7, we also see the frequency characteristics of Td and T (ρ∗ ) are almost the same in the low frequency region compared with the initial closed loop T (ρini ).
We also show the effectiveness of our proposed method in the frequency domain. In Fig. 7, frequency responses of the desired function Td , the initial closed loop system T (ρini ), and the closed loop with the optimal parameter T (ρ∗ ) are drawn by the dotted line, the chained line, and the solid 580
5.2 The case where the disturbance is the sawtooth wave Here, we treat the sawtooth typed disturbance d with time period 10 with the white noise w with the variance 0.01. The disturbance d and the initial output yini are illustrated by the broken line and the chained line, respectively in Fig. 8. By using the initial experimental data, we apply our proposed method. Here, we give the desired function Td as 0.1s2 + 0.06 Td = . 3 0.1s + 1.1s2 + 1.06s + 0.63 This is the notch filter so as to eliminate the effect of such a periodic disturbance. In the optimization process, we also check the stability by using the Kharitonov theorem. As a result, we obtain the optimal parameter ρ∗ = [1.76 1.85 0.19 0.0 0.12]. By implementing ρ∗ to the
11th IFAC ALCOSP July 3-5, 2013. Caen, France
controller in Fig. 1, we again perform the experiment. The output y(ρ∗ ) is also illustrated in Fig. 8 as the solid line. Compared with the initial output yini , we see that the effect of the disturbance can be sufficiently attenuated in this figure. We also illustrate the effectiveness of our proposed method in the frequency domain. In Fig. 9, frequency responses of the desired function Td , the initial closed loop system T (ρini ), and the closed loop with the optimal parameter T (ρ∗ ) are drawn by the broken line, the chained line, and the solid line, respectively. From Fig. 9, we also see the frequency characteristics of Td and T (ρ∗ ) are almost the same in the low frequency region compared with the initial closed loop T (ρini ). 6. CONCLUDING REMARKS In this paper, we have proposed a data-driven approach to the attenuation of unknown periodic disturbance. In this paper, we have provided a novel off-line mechanism for a disturbance observer to be tuned with the basis of the experimental data directly. In addition, by applying the idea of FRIT, we have proposed an off-line tuning algorithm for the parameter of a controller achieving disturbance attenuation. REFERENCES B. D. O. Anderson, E. I. Jury, and M. Mansour. On rubust Hurwitz polynomial. IEEE Transactions on Automatic Control, 32:909–913, 1987. M. C. Campi, A. Lecchini, and S. M. Savaresi. Virtual reference feedback tuning: a direct method for the design of feedback controllers. Automtica, 38:1337–1446, 2002. Y. Choi, K. Yang, W. K. Chung, H. R. Kim, and I. H. Suh. On the robustness and performance of disturbance observers for second order systems. IEEE Transactions on Automatic Control, 48:315–320, 2003. N. Hansen and A. Ostermeier. Completely derandomized self-adaptation in evolution strategies. Evolutionary computation, 9:75–101, 2001. H. Hjalmarsson, M. Gevers, S. Gunnarsson, and O. Lequin. Iterative feedback tuning:theory and applications. IEEE control systems magazine, 18:26–41, 1998. Y. Hori. High performance control of robot manipulator without using inverse dynamics. Control Engineering Practice, 1:529–538, 1993. O. Kaneko. Data-driven controller tuning: FRIT approach. IFAC 11th Workshop on Adaptation and Learning in Control and Signal Processing, 2013. O. Kaneko, S.Souma, and T. Fujii. 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, 2005. O. Kaneko, Y. Wadagaki, and S. Yamamoto. Fictitious reference iterative tuning for internal model controllers. Proceedings of the 10th IFAC Workshop on Adaptation and Learning in Control and Signal Processing, 2010. O. Kaneko, Y. Wadagaki, and S. Yamamoto. Frit based pid parameter tuning for linear time delay systems simultaneous attainment of models and controllers -. Proceedings of IFAC Conference on Advances in PID Control, 2012. 581
O. Kaneko, S. Yamamoto, and Y. Wadagaki. Simultaneous attainment of model and controller for linear time delay systems with the data-driven Smith compensator. Preprints of the 18th IFAC World Congress, pages 7684– 7689, 2011. A. Karimi, K. van Heuden, and D. Bonvin. Noniterative data-driven controller tuning using the correlation approach. Proceedings of European Control Conference, pages 5189–5195, 2007. C. J. Kempf and S. Kobayashi. Disturbance observer and feedforward design for high-speed direct drive positioning table. IEEE Transactions on Control System Technology, 7:513–526, 1999. V. L. Kharitonov. Asymptotic stability of an equilibrium position of a family of systems of differential equations. Differentsialnye uravneniya, 14:2086–2088, 1978. S. Masuda and L. G. Dong. An FRIT method for disturbance attenuation using input-output data for disturbance responses. Proceedings of the 10th IFAC Workshop on Adaptation and Learning in Control and Signal Processing, CD-ROM, 2010. T. Mita, M. Hirata, K. Murara, and H. Zhang. Hinfinity control versus disturbance observer based control. IEEE Transactions on Industrial Electronics, 45: 488–495, 1998. K. Ohnishi. A new servo method in mechatronics. Transactions Japanese Society of Electric Engineering, 107-D: 83–86, 1987. M.G. Safonov and T.C. Tsao. The unfalsified control concept and learning. IEEE Transactions on Automatic Control, 42:843–847, 1997. S. Souma, O. Kaneko, and T. Fujii. A new method of controller parameter tuning based on input-output data, –Fictitious Reference Iterative Tunining. IFAC Workshop on Adaptation and Learning in Control and Signal Processing (ALCOSP 04), pages 789–794, 2004. T. Umeno and Y. Hori. Robust speed control of dc servomotors using two degrees-of-freedom controller design. IEEE Transactions on Industrial Electronics, 38:363– 368, 1991. Y. Yasuda, S. Masuda, and M. Kano. PID gain tuning for disturbance attenuation using FRIT method. Proceedings of ICROS-SICE International Joint Conference 2009, pages 941–944, 2009.
ThS6T2.3
Fictitious Reference Iterative Tuning of Disturbance Observers for Attenuation of the Effect of Periodic Unknown Exogenous Signals ⋆ Fumiaki Uozumi ∗ Osamu Kaneko ∗∗ ShigeruYamamoto ∗∗ ∗
Graduate School of Natural Science and Technology, Kanazawa University, Kakumamachi, Kanazawa, Ishikawa 920-1192, Japan (Tel: +81-76-264-6332; e-mail: [email protected]). ∗∗ Institute of Science and Engineering, Kanazawa University, Kakumamachi, Kanazawa, Ishikawa 920-1192, Japan (e-mail: o-kaneko, [email protected]) Abstract: In this paper, we develop a data-driven controller tuning for the disturbance observers. Particularly, we focus on the disturbance attenuation problem for a periodic unknown disturbance. However, in the case where it is difficult to construct inverse mathematical model of a plant which is implemented in the disturbance observer, we have to tale much time and costs for identification. In order to overcome such a difficulty, we utilize fictitious reference iterative tuning (FRIT), which is data-driven controller tuning with only one-shot experimental data, for tuning of parameters of controllers to eliminate the effect of the disturbance. Keywords: Data-driven parameter tuning, Fictitious Reference Iterative Tuning (FRIT), Disturbance observer, Disturbance attenuation 1. INTRODUCTION It is usual that an unknown disturbance causes the unexpected phenomena in the control system. For example, a periodic disturbance caused by the motor, vibration of the structure, and so on, is one of the representative problems in many practical situations. Normally, it is difficult to measure such a disturbance directly. Thus, we should estimate the disturbance or its effect in one way or another. For this purpose, the disturbance observer, which was proposed and studied in Ohnishi [1987], Hori [1993], Umeno and Hori [1991], and Mita et al. [1998], is one of the effective methods to estimate or to eliminate the effect of the disturbance. In addition, there are also many studies on disturbance observers from the practical points of view, e.g., Choi et al. [2003] and Kempf and Kobayashi [1999]. In the case where the control system does not contain the architecture of the disturbance observer, we should implement it and re-construct whole of the control systems. On the other hand, there are many cases in which such an implementation or reconstructing of the control system spends much cost. In these cases, it is prefer to only add the off-line mechanism for the estimation of the disturbance. Moreover, it is fundamental that the architecture of the disturbance observer needs the inverse model of the plant. To obtain such a model, we also need a mathematical model of a plant, which requires experiment for the ⋆ This work was partially supported by JSPS Grant in Aid for Scienti c Research (C) 22560442, (B) 23360183, and (C) 25420432 (The third one partially supports this work in the revision process of the nal manuscript.)
978-3-902823-37-3/2013 © IFAC
576
identification. On the other hand, there are many cases in which it is difficult to apply a persistently exciting signal for identifying the dynamics in broader frequency band from the view points of the safety. In addition, there are also many cases where it is impossible to take much cost for experiment from the view points of the scheduling of industry. In these cases, the direct use of the data is expected as one of the rational approaches. As representative works, iterative feedback tuning (IFT) in Hjalmarsson et al. [1998], virtual reference iterative tuning (VRFT) in Campi et al. [2002], the non-iterative version of correlation based tuning in Karimi et al. [2007], and the fictitious reference iterative tuning (FRIT) in Souma et al. [2004] and Kaneko et al. [2005] were proposed and studied. Among of them, IFT is the most rational approach because the cost function to be minimized in IFT directly represents the purpose of the tuning of a controller. However, IFT requires many experiments, which is a crucial drawback from the practical points of view. The later three methods enables us to obtain the desired parameter with only one shot experiment. In Kaneko [2013] (to appear in this workshop), the comparisons of these methods are presented, so Kaneko [2013] for more details. Here we take FRIT approach. Because this method is intuitively understandable since the output is focused in this method for off-line minimization. From these backgrounds, we address a data-driven approach to the attenuation of exogenous periodic disturbance. Particularly, we use the idea of the fictitious reference iterative tuning (which is abbreviated as FRIT), which was proposed and developed in Souma et al. [2004], 10.3182/20130703-3-FR-4038.00140
11th IFAC ALCOSP July 3-5, 2013. Caen, France
Kaneko et al. [2005], Kaneko et al. [2010], Kaneko et al. [2011], and Kaneko et al. [2012]. In this paper, we provide a novel off-line mechanism for disturbance observer to be tuned based on the direct use of the experimental data. In Yasuda et al. [2009] and Masuda and Dong [2010], the disturbance attenuation by utilizing FRIT were studied. In these studies, the exogenous disturbance is assumed to be known. Compared with them, our setting does not assume that the exogenous disturbance is known.
We assume that C(ρini ) with the initial parameter ρini tentatively stabilizes the closed loop so as to obtain the bounded input uini := u(ρini ) and output yini = y(ρini ).
Contents of this paper is the following. In Section 2, we formulate the problem to be addressed in this paper. In Section 3, we give brief reviews on some preliminaries on the disturbance observer and FRIT. In Section 4, the main result of this paper is presented, particularly, a novel off-line architecture for tuning of a controller for the disturbance attenuation. In order to validate our proposed method, we show some numerical examples in Section 5. Finally, concluding remarks are given in the Section 6.
Under these settings, in this paper, we obtain the optimal parameter such that
[Notation] We denote the set of the real numbers as R. For a time signal w, let w(t) denote the value of w at time t. For a time signal w, let Φw (jω) denote the power spectrum density in the frequency domain. We introduce √ ∑ N 1 ||w||(N,∆) := N t=1 ω(t∆)2 with sampling period ∆. In terms of the enhancement of the readability, we often omit the notation ‘s’ from a transfer function G(s) if it is clear from the context. In addition, the output of a transfer function G with respect to the input u is denoted with y = Gu. 2. PROBLEM FORMULATION In Fig. 1, we illustrate the conventional control system with a tunable controller with respect to a parameter ρ. Assume that a plant G is single-input and single output, linear, time-invariant, and minimum phase. We assume that G is unknown while the structure of G is known. Let Gn denote a nominal model of a plant. Let d denote an unknown periodic disturbance. A feedback controller C(ρ) is with a tunable parameter vector ρ. For example, C(ρ) is parameterized by ρ1 sn + ρ2 sn−1 + · · · + ρn s + 1 , (1) ρn+1 sn + ρn+2 sn−1 + · · · + ρ2n s + ρ2n+1 using parameter vector ρ := [ρ1 ρ2 · · · ρ2n ρ2n+1 ]. The closed loop can be also regarded as the function of ρ, so we denote G T (ρ) := 1 + GC(ρ) as the transfer function from d to y. Moreover, we also denote the input and output of the closed loop by y(ρ) and u(ρ), respectively. C(ρ) =
Fig. 1. A closed loop control system contains the unknown disturbance 577
We are also given a desired property for attenuation of the disturbance as the reference model from d to y as Td . We set Td such that Td (ejωd ) ≃ 0 holds where ωd is one of the frequency modes of the periodic disturbance to be attenuated.
2
J(ρ) = ∥y(ρ) − Td d∥(N,∆)
(2)
is minimized with only using the data uini and yini . Moreover, if necessary, we consider new architecture for the estimation of the disturbance. 3. PRELIMINARIES In this section, we give brief reviews on the disturbance observer and FRIT. 3.1 Disturbance Observer We give a brief review of disturbance observer based on Ohnishi [1987] and Hori [1993]. The disturbance observer is one of the novel mechanisms for the estimation of an unknown disturbance. By using the estimated disturbance, we can compensate the degradation of the performance of the closed loop due to the disturbance. The structure of the disturbance observer is illustrated in Fig. 2. The output of the inverse nominal plant G−1 n is applied to the low pass filter Q. The observed disturbance dest is generated by u and the output of the filter Q. We can calculate dest in Fig. 2 as −1 dest = QG−1 (3) n Gd + Q(Gn G − 1)u. We assume that the nominal plant Gn is close to the plant G in the frequency band of Q. Thus, it is possible to estimate dest ≃ d in the frequency range where we can regard QG−1 n G ≃ 1. 3.2 Fictitious Reference Iterative Tuning (FRIT) We give a brief review of FRIT based on Souma et al. [2004] and Kaneko et al. [2005]. Figure 3 illustrates a conventional feedback control system that consists of a plant and a controller C(ρ) with a tunable parameter ρ. Let M (ρ) denote the transfer function from r to y. And consider the model reference problem where the purpose of the tuning of controller is that the control system yields
Fig. 2. The conventional disturbance observer
11th IFAC ALCOSP July 3-5, 2013. Caen, France
By defining Tn (ρ) := we have
Fig. 3. A closed loop control system the desired output Md r, where Md is a given reference model from r to y. By using the initial parameter ρini , perform the initial experiment on the closed loop system with C(ρini ). We then obtain the initial data uini := u(ρini ) and yini := y(ρini ). We also assume that C(ρini ) tentatively stabilize the closed loop so as to yield the bounded input and output. By using the initial data, we compute the fictitious reference signal r˜(ρ) (which was introduced by Safonov and Tsao [1997] in the unfalsified control framework) described by r˜(ρ) = C(ρ)−1 uini + yini .
(4)
JF (ρ) = ∥yini −
(5)
And then we minimize JF (ρ) and implement ρ˜∗ := arg minρ JF (ρ) to the controller. Note that (5) with the fictitious reference r˜(ρ) in (4) requires only uini and yini . This means that the minimization of (5) can be performed off-line by using only one-shot experimental data. The relationship between the minimization of (5) and that of (2) can be given by rewriting the cost function JF (ρ) as
(
2 )
Md
JF (ρ) = 1 − yini .
M (ρ) (N,∆)
d˜est (ρ) = QTn (ρ)−1 yini . (8) Since the role of Q is to compensate the non-properness of G−1 n , it is possible to give Q such that the frequency pass-band of Q is higher than the frequency range of the exogenous disturbance. Thus, in the frequency range where we can regard Qyini ≃ yini , the estimated observation is approximated by d˜est (ρ) = Tn (ρ)−1 yini . (9) which is equivalent to Tn (ρ)d˜est (ρ) = yini .
(10)
Now we introduce a new cost function to be minimized
2
JF D (ρ) = yini − Td d˜est (ρ) . (11) (N,∆)
Next, we introduce the cost function described by Md r˜(ρ)∥2(N,∆) .
Gn , 1 + Gn C(ρ)
(6)
From (6), we see that the minimization of JF (ρ) in (5) leads to the minimization of the relative error between the desired reference model and the actual closed loop with the parameter ρ under the effect of the initial output. This is an explanation on how FRIT can achieve the desired output.
From (9), JF D can be rewritten as
(
2 )
Td
JF D (ρ) = 1 − Qyini
Tn (ρ)
.
(12)
(N,∆)
If JF D (ρ) can be completely minimized, namely JF D (ρ∗ ) = 0 at ρ∗ := arg min JF D (ρ), then the relation Gn 1 + Gn C(ρ∗ ) generically holds. This implies that Td = Tn (ρ∗ ) =
Tn (ρ∗ )d˜est (ρ∗ ) = Td d˜est (ρ∗ )
(13)
(14)
holds. On the other hand, from Fig. 4, we also see that T (ρ)d = y(ρ) = Tn (ρ)d˜est (ρ)
(15)
holds for any parameter ρ. From (14) and (15), we obtain y(ρ∗ ) = T (ρ∗ )d = Td d˜est (ρ∗ ).
(16)
4. MAIN RESULT 4.1 Basic Idea We introduce a new off-line mechanism for the estimation of the disturbance by using the observed output. Consider the block diagram illustrated in Fig. 1. The initial control output yini is obtained in the experiment using initial controller parameter ρini , which is affected by the disturbance d. Now, we consider a new off-line architecture illustrated in Fig. 4 where we add the inverse of the nominal model of the plant G−1 n . At this point, since the nominal model Gn is strictly proper, the inverse nominal plant is nonproper. To solve this problem, we use a low-pass filter Q to estimate the disturbance d, which is from the concept of the disturbance observer. In Fig. 4, the estimated disturbance which is denoted with d˜est and also depends on the parameter ρ, is generated by ( ) 1 + Gn C(ρ) d˜est (ρ) = Q G−1 yini . (7) n + C(ρ) yini = Q Gn 578
Fig. 4. The off-line estimation for the estimated disturbance d˜est (ρ)
Fig. 5. Td d˜est (ρ) and the nominal closed loop
11th IFAC ALCOSP July 3-5, 2013. Caen, France
Under the assumption that the involved signals are with the ergodic property, the Parseval theorem enables us to transform (16) into the frequency domain as Φy(ρ∗ ) (jω) = Td (jω)Φd˜est (ρ∗ ) (jω).
(17)
As stated above, since Td is given by the designer, the magnitude of Td at the frequency mode ω 0 at which the disturbance should be attenuated can be restrained as Td (jω 0 ) ≃ 0. As a result, the effect of d on y(ρ∗ ) is also attenuated. That is, by implementing ρ∗ to the controller, we can obtain the closed loop that attenuates the effect of the disturbance d. This is an intuitive explanation of the proposed method. 4.2 Analysis for the case of JF D (ρ) ̸= 0 We consider the case where JF D (ρ) = 0 can not be achieved. Notice the cost function JF D (ρ) can be rewritten as (12). From this, we see that the minimization of JFD corresponds to that of the relative error between the desired property Td and the nominal closed loop which can be computed in the off-line. Assume that we obtain the optimal parameter ρ∗ := arg min JF D (ρ). Then, since there exists a relative error, between Td and Tn (ρ∗ ), these two transfer function can be related as Tn (ρ∗ ) = (1 + δ(ρ∗ ))Td
(18)
∗
∗
where δ(ρ ) is a relative error. By implementing ρ to the actual closed loop, it follows from the architecture of Fig. 4 that we obtain y(ρ∗ ) = T (ρ∗ )d = Tn (ρ∗ )d˜est (ρ∗ ).
(19)
From (18), we also obtain Tn (ρ∗ )d˜est (ρ∗ ) = (1 + δ(ρ∗ ))Td d˜est (ρ∗ ).
(20)
Thus, we see y(ρ∗ ) = (1 + δ(ρ∗ ))Td d˜est (ρ∗ ) = Td d˜′est (ρ∗ ),
(21)
where d˜′est (ρ∗ ) := (1 + δ(ρ∗ ))d˜est (ρ∗ ). Again, from the Parseval theorem, we obtain Φy(ρ∗ ) (jω) = Td (jω)Φd˜′
est (ρ
∗)
(jω).
(22)
Due to the same reason in the ideal case stated in the previous subsection, we see that the effect of the disturbance can be attenuated by using the parameter ρ∗ .
In the step 3), we can utilize some conventional nonlinear optimization methods, e.g., Gauss-Newton methods, CMA-ES in Hansen and Ostermeier [2001], and so on. Notice that we do not have to reconstruct the control system and it is enough to add a simple off-line architecture as Fig. 4. Remark 1. It is normal that there exists a difference between the actual plant G and the nominal model Gn . Thus, we should check the stability of T (ρ∗ ) by using that of Tn (ρ∗ ) before implementing of the obtained parameter ρ∗ . Generally, the stability check in direct data-driven approaches is one of the open problems to be clarified. For the strategy to this problem, the utilization of the Kharitonov theorem in Kharitonov [1978] and Anderson et al. [1987] is one of the rational approaches. This theorem is known as a useful tool for checking whether the roots of a polynomial with unknown coefficients whose bounds are known lie in the open left half plane. In this case, we can calculate such bounds of each coefficients of the denominator of T (ρ) if we know each interval in which each coefficient of the numerator and the denominator of G takes the value is known. And then we can utilize Kharitonov theorem to check the stability of T (ρ). If the answer is that T (ρ) is unstable, we should search an alternative parameter in the non-linear optimization. The detailed discussions will be in the Workshop and our forthcoming papers. For more generalized situations, this is important open problem, which is to be clarified in the data-driven setting. 5. NUMERICAL EXAMPLE In order to show the validity of the proposed method, we show two examples, because it is better to show that our method is applicable for the various disturbances. We consider 12s2 + 47s + 8 G= 3 9s + 62s2 + 73s + 5 as the actual plant and 12.8573s2 + 49.5832s + 8.3643 Gn = 9.6919s3 + 66.7601s2 + 75.0548s + 5.4394 as its nominal model. Of course we assume that we do not know the exact coefficients of G. We also use the tunable feedback controller ρ1 s2 + ρ2 s + 1 C(ρ) = , ρ = [ ρ1 ρ2 ρ3 ρ4 ρ5 ]. ρ3 s2 + ρ4 s + ρ5 We define the initial parameter as ρini = [1 4 1 5 6]. We set the sampling period ∆ = 0.001 sec. 5.1 The case where the disturbance is the combination of some sine waves
4.3 Algorithm Here we summarize the proposed algorithm. 1) Set the initial parameter ρini . 2) Perform the initial experiment with ρini in Fig. 1 and obtain the data uini and yini . 3) Minimize the cost function JF D (ρ) in (11) with the “fictitious” estimated disturbance d˜est (ρ) in (9) offline. 4) Implement ρ˜∗ in Fig. 1 and perform the experiment again. 579
First, we consider the case where the unknown periodic disturbance d is the combination of some sine-waves. We use d(t) = sin 2t + 0.6 sin 5t + 0.8 sin 0.8t + w(t), where w(t) is the white noise with the variance 0.01. With the initial parameter ρini , we perform a one-shot experimentation and obtain the data. The disturbance d and the initial output yini are illustrated by the broken line
11th IFAC ALCOSP July 3-5, 2013. Caen, France
Fig. 6. The initial output yini (the broken line), the output after the optimal parameter y(ρ∗ ) (the solid line), and the disturbance d (the chained line)
Fig. 8. The initial output yini (the broken line), the output after the optimal parameter y(ρ∗ ) (the solid line), and the disturbance d (the chained line)
Fig. 7. Frequency responses of the desired function Td (dotted line), the initial closed loop system T (ρini ) (the chained line), and the closed loop with the optimal parameter T (ρ∗ ) (the solid line)
Fig. 9. Frequency responses of the desired function Td (the broken line), the initial closed loop system T (ρini ) (the chained line), and the closed loop with the optimal parameter T (ρ∗ ) (the solid line)
and the chained line, respectively in Fig. 6. By using the initial experimental data, we apply our proposed method. Here, we give the desired function Td as 10s + 1 Td = 10s2 + 1000s + 1 so as to eliminate the effect of the disturbance over a sufficiently large frequency range. In the optimization process, we also check the stability of the closed loop by using Kharitonov theorem. As a result, we obtain the optimal parameter ρ∗ = [44.27 100 0.46 1.06 0.13]. By implementing ρ∗ to the controller in Fig. 1, we again perform the experiment. The output y(ρ∗ ) is also illustrated in Fig. 6 as the solid line. Compared with the initial output yini , we see that the effect of the disturbance can be sufficiently attenuated in Fig. 6.
line, respectively. From Fig. 7, we also see the frequency characteristics of Td and T (ρ∗ ) are almost the same in the low frequency region compared with the initial closed loop T (ρini ).
We also show the effectiveness of our proposed method in the frequency domain. In Fig. 7, frequency responses of the desired function Td , the initial closed loop system T (ρini ), and the closed loop with the optimal parameter T (ρ∗ ) are drawn by the dotted line, the chained line, and the solid 580
5.2 The case where the disturbance is the sawtooth wave Here, we treat the sawtooth typed disturbance d with time period 10 with the white noise w with the variance 0.01. The disturbance d and the initial output yini are illustrated by the broken line and the chained line, respectively in Fig. 8. By using the initial experimental data, we apply our proposed method. Here, we give the desired function Td as 0.1s2 + 0.06 Td = . 3 0.1s + 1.1s2 + 1.06s + 0.63 This is the notch filter so as to eliminate the effect of such a periodic disturbance. In the optimization process, we also check the stability by using the Kharitonov theorem. As a result, we obtain the optimal parameter ρ∗ = [1.76 1.85 0.19 0.0 0.12]. By implementing ρ∗ to the
11th IFAC ALCOSP July 3-5, 2013. Caen, France
controller in Fig. 1, we again perform the experiment. The output y(ρ∗ ) is also illustrated in Fig. 8 as the solid line. Compared with the initial output yini , we see that the effect of the disturbance can be sufficiently attenuated in this figure. We also illustrate the effectiveness of our proposed method in the frequency domain. In Fig. 9, frequency responses of the desired function Td , the initial closed loop system T (ρini ), and the closed loop with the optimal parameter T (ρ∗ ) are drawn by the broken line, the chained line, and the solid line, respectively. From Fig. 9, we also see the frequency characteristics of Td and T (ρ∗ ) are almost the same in the low frequency region compared with the initial closed loop T (ρini ). 6. CONCLUDING REMARKS In this paper, we have proposed a data-driven approach to the attenuation of unknown periodic disturbance. In this paper, we have provided a novel off-line mechanism for a disturbance observer to be tuned with the basis of the experimental data directly. In addition, by applying the idea of FRIT, we have proposed an off-line tuning algorithm for the parameter of a controller achieving disturbance attenuation. REFERENCES B. D. O. Anderson, E. I. Jury, and M. Mansour. On rubust Hurwitz polynomial. IEEE Transactions on Automatic Control, 32:909–913, 1987. M. C. Campi, A. Lecchini, and S. M. Savaresi. Virtual reference feedback tuning: a direct method for the design of feedback controllers. Automtica, 38:1337–1446, 2002. Y. Choi, K. Yang, W. K. Chung, H. R. Kim, and I. H. Suh. On the robustness and performance of disturbance observers for second order systems. IEEE Transactions on Automatic Control, 48:315–320, 2003. N. Hansen and A. Ostermeier. Completely derandomized self-adaptation in evolution strategies. Evolutionary computation, 9:75–101, 2001. H. Hjalmarsson, M. Gevers, S. Gunnarsson, and O. Lequin. Iterative feedback tuning:theory and applications. IEEE control systems magazine, 18:26–41, 1998. Y. Hori. High performance control of robot manipulator without using inverse dynamics. Control Engineering Practice, 1:529–538, 1993. O. Kaneko. Data-driven controller tuning: FRIT approach. IFAC 11th Workshop on Adaptation and Learning in Control and Signal Processing, 2013. O. Kaneko, S.Souma, and T. Fujii. 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, 2005. O. Kaneko, Y. Wadagaki, and S. Yamamoto. Fictitious reference iterative tuning for internal model controllers. Proceedings of the 10th IFAC Workshop on Adaptation and Learning in Control and Signal Processing, 2010. O. Kaneko, Y. Wadagaki, and S. Yamamoto. Frit based pid parameter tuning for linear time delay systems simultaneous attainment of models and controllers -. Proceedings of IFAC Conference on Advances in PID Control, 2012. 581
O. Kaneko, S. Yamamoto, and Y. Wadagaki. Simultaneous attainment of model and controller for linear time delay systems with the data-driven Smith compensator. Preprints of the 18th IFAC World Congress, pages 7684– 7689, 2011. A. Karimi, K. van Heuden, and D. Bonvin. Noniterative data-driven controller tuning using the correlation approach. Proceedings of European Control Conference, pages 5189–5195, 2007. C. J. Kempf and S. Kobayashi. Disturbance observer and feedforward design for high-speed direct drive positioning table. IEEE Transactions on Control System Technology, 7:513–526, 1999. V. L. Kharitonov. Asymptotic stability of an equilibrium position of a family of systems of differential equations. Differentsialnye uravneniya, 14:2086–2088, 1978. S. Masuda and L. G. Dong. An FRIT method for disturbance attenuation using input-output data for disturbance responses. Proceedings of the 10th IFAC Workshop on Adaptation and Learning in Control and Signal Processing, CD-ROM, 2010. T. Mita, M. Hirata, K. Murara, and H. Zhang. Hinfinity control versus disturbance observer based control. IEEE Transactions on Industrial Electronics, 45: 488–495, 1998. K. Ohnishi. A new servo method in mechatronics. Transactions Japanese Society of Electric Engineering, 107-D: 83–86, 1987. M.G. Safonov and T.C. Tsao. The unfalsified control concept and learning. IEEE Transactions on Automatic Control, 42:843–847, 1997. S. Souma, O. Kaneko, and T. Fujii. A new method of controller parameter tuning based on input-output data, –Fictitious Reference Iterative Tunining. IFAC Workshop on Adaptation and Learning in Control and Signal Processing (ALCOSP 04), pages 789–794, 2004. T. Umeno and Y. Hori. Robust speed control of dc servomotors using two degrees-of-freedom controller design. IEEE Transactions on Industrial Electronics, 38:363– 368, 1991. Y. Yasuda, S. Masuda, and M. Kano. PID gain tuning for disturbance attenuation using FRIT method. Proceedings of ICROS-SICE International Joint Conference 2009, pages 941–944, 2009.