Application of Adaptive Minimum-variance Control Technique for Cyclic Production Machine

Adaptation and Learning in Control and Signal Processing Adaptation and Learning in Control and Signal Processing June 29 - July 1, 2016. Eindhoven, The Netherlands June 29 - July 1, 2016. Eindhoven, The Netherlands 12th IFAC International Workshop on Available online at www.sciencedirect.com Adaptation and Learning in Control and Signal Processing June 29 - July 1, 2016. Eindhoven, The Netherlands

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IFAC-PapersOnLine 49-13 (2016) 146–151

Application of Adaptive Minimum-variance Control Technique for Cyclic Production Machine Application of Adaptive Minimum-variance Control Technique for Cyclic Production Machine Naoki Naoki MIZUNO MIZUNO and and Daeyong Daeyong KIM KIM

Dept. Dept. of of Scientific Scientific and and Engineering Engineering Simulation Simulation Nagoya Institute Technology Naoki MIZUNO and KIM Nagoya Institute of of Daeyong Technology Nagoya, Japan Dept. of Scientific and Engineering Simulation Nagoya, Japan [email protected] Nagoya Institute of Technology [email protected] Nagoya, Japan [email protected]

Abstract— Abstract— In In this this paper, paper, we we consider consider the the control control problem problem of of production machine. There are many mechanical cyclic cyclic production machine. There are many mechanical parts parts produced by production machine. For example, springs produced by cyclic cyclic production machine. springs are are Abstract— In this paper, we considerFor theexample, control problem of usually made stock which is fed cyclic usuallyproduction made of of long long stock wire wire which fed into into cyclic forming forming cyclic machine. There are ismany mechanical parts machine. spring forming can force wire aa coiled machine. A A forming machine machine canFor force wire into into coiled produced byspring cyclic production machine. example, springs are shape during the cyclic motion, but the length of the produced shape during the cyclic motion, but the length of the produced usually made of long stock wire which is fed into cyclic forming spring in cycle is according to internal stress of spring in each each cycleforming is varying varying according to the thewire internal of machine. A spring machine can force into stress a coiled wire, mechanical uncertainty of the forming machine etc. wire, mechanical uncertainty thethe forming etc. To To shape during the cyclic motion,ofbut length machine of the produced improve the accuracy of length produced by improve of free freeaccording length of ofto spring spring produced spring in the eachaccuracy cycle is varying the internal stress by of forming machine, we propose an adaptive minimum-variance forming machine, uncertainty we propose ofanthe adaptive wire, mechanical formingminimum-variance machine etc. To control strategy for production system and successfully control for cyclic cyclic production andproduced successfully improvestrategy the accuracy of free length system of spring by applied to commercial spring forming machine. applied commercial forming machine. formingtomachine, we spring propose an adaptive minimum-variance control strategy for cyclic production system and successfully © 2016, IFAC (International Federation of Automatic Control) Keywords— cyclic production machine; deviation Keywords— cyclic production machine; deviation of of products products applied to commercial spring forming machine. Hosting by Elsevier Ltd. All rights reserved. quality ; adaptive minimum variance control; spring quality ; adaptive minimum variance control; spring forming forming machine machine Keywords— cyclic production machine; deviation of products quality ; adaptive minimum variance control; spring forming I. I. IINTRODUCTION NTRODUCTION machine

In In many many production production process, process, the the quality quality of of products products is is varying variation varying depending depending on onI. the theINTRODUCTION variation of of materials, materials, uncertainty uncertainty of machine, and so In many production process, disturbances the quality of of production production machine, unknown unknown disturbances andproducts so on. on. is varying depending on the variation of materials, uncertainty For this kind of problem, by formulating the variation of For this kind of problem, by formulating theand variation of production machine, unknown disturbances so on. of the products quality as the output of system with stochastic the products quality as the output of system with stochastic disturbance, the control strategies like minimumFor this kind of problem, by formulating of disturbance, the stochastic stochastic control strategies the likevariation minimumvariance control are successfully applied to the continuous the products quality as the output of system with stochastic variance control are successfully applied to the continuous production disturbance,system. the stochastic control strategies like minimumproduction system. variance controlinareprocess successfully applied to the continuous Especially, Especially, in process industry, industry, the the adaptive adaptive control control production system. technique are very useful because the characteristics technique are very useful because the characteristics of of the the system may vary during the operation [1][2]. Especially, process industry, [1][2]. the adaptive control system may varyinduring the operation technique are very useful because the characteristics of the In in the production process, In contrast, contrast, induring the cyclic cyclic production process, the the control control system may vary the operation [1][2]. system is usually designed to achieve the desired system is usually designed to achieve the desired performance the In contrast,during in the each cyclicproduction production cycle, process,because the control performance during each production cycle, because the quality on in system usually depend designed to physical achieve process the desired quality of ofis products products depend on the the physical process in the the cycle performance cycle time. time. during each production cycle, because the quality of products dependeffective on the physical process in the This kind This kind of of control control is is effective to to reduce reduce the the quality quality of of cycle time. each product in each production cycle. However, each product in each production cycle. However, the the variation of production process is considered. This kind ofcyclic control is effective to reduce quality of variation of the the cyclic production process is not notthe considered. eachForproduct in each production cycle. However, the this problem, the authors have proposed a For this problem, the authors have proposed a new new idea idea variation of the cyclic production process is not considered. to to design design the the control control system system for for cyclic cyclic production production system. system. For this problem, the authors have proposed a new idea In the proposed control system, the variation of In thethe proposed controlfor system, the variation of the the to design control system cyclic production system. product product for for successive successive production production cycles cycles is is modeled modeled by by the the output time system stochastic disturbance [3]. In of thediscrete proposed system, the variation of the output of discrete timecontrol system with with stochastic disturbance [3]. product for successive production cycles is modeled by the output of discrete time system with stochastic disturbance [3]. 1 Copyright © 2016 IFAC Copyright © 2016 IFAC 1

Hiroshi Hiroshi SUGIYAMA SUGIYAMA and and Ryota Ryota FUKATSU FUKATSU nd 2 2nd Research Research and and Development Development Department Department Asahi-Seiki Manufacturing Co., Ltd. Hiroshi SUGIYAMA and Ryota FUKATSU Asahi-Seiki Manufacturing Co., Ltd. Japan 2nd ResearchOwariasahi, and Development Department Owariasahi, Japan Asahi-Seiki Manufacturing Co., Ltd. Owariasahi, Japan

In this this case, case, the the variation variation of of the the product product for for each each In production cycle can be reduced by applying appropriate production cycle can be reduced by applying appropriate stochastic In thiscontrol case, strategy. the variation of the product for each stochastic control strategy. production cycle can be reducedadaptive by applying appropriate In our research, we In our research, we propose propose an an adaptive minimum-variance minimum-variance stochastic control strategy. control strategy for cyclic production control strategy for cyclic production system system considering considering the the characteristics variation of production machine In our research, we propose an adaptive minimum-variance characteristics variation of production machine and and auto-tuning auto-tuning function for the forming machine. control strategy for of cyclic production system considering the function for start-up start-up of the spring spring forming machine. characteristics variation of production machine and auto-tuning The of adaptive control system to The application application of the adaptive control system to the the spring spring function for start-up of spring forming machine. forming machine have been proposed in the literature. forming machine have been proposed in the literature. The application of considered adaptive control system ofto mechanical the spring However, it not the However, it is ishave not been considered theinvariation variation of mechanical forming machine proposed theproduction literature. characteristics of forming machine for each characteristics of forming machine for each production cycle cycle [4]. [4]. However, it is not considered the variation of mechanical In improve the of length spring In this this paper, paper,ofto toforming improvemachine the accuracy accuracy of free free length of of spring characteristics for each production cycle [4]. produced by forming machine, we propose an adaptive produced by forming machine, we propose an adaptive minimum-variance control the cyclic In this paper, to improve accuracybased of free on length minimum-variance controlthe system system based on theof spring cyclic dynamics of the forming machine. The model structure produced by forming machine, we propose an adaptive dynamics of the forming machine. The model structure is is determined considering variation minimum-variance control the system based of on mechanical the cyclic determined considering the variation of mechanical characteristics for each The dynamics of the machine.cycle. The model structure is characteristics for forming each production production cycle. The experimental experimental results using commercial spring forming machine show determined considering the variation of results using commercial spring forming machinemechanical show the the effectiveness of the proposed control algorithm compared characteristics for each production cycle. The experimental effectiveness of the proposed control algorithm compared with with the previous control algorithm. results usingembedded commercial spring forming machine show the the previous embedded control algorithm. effectiveness of the proposed control algorithm compared with the previous embedded control algorithm. II. II. M MECHANISM ECHANISM OF OF S SPRING PRING F FORMING ORMING M MACHINE ACHINE II. MECHANISM OF SPRING FORMING MACHINE A A spring spring is is made made by by forming forming steel steel wire wire between between some some outer dies and a pitch tool, which give outer dies and a pitch tool, which give geometrically geometrically orthogonal to as Fig.1. A springforces is made bywire forming steelin between some orthogonal forces to the the wire as shown shown inwire Fig.1. outer dies and a pitch tool, which give geometrically orthogonal forces to the wire as shown in Fig.1. Outer dies Outer dies

Wire guide Wire guide

Outer dies

Wire guide

Wire Wire Wire

Pitch tool Pitch tool Pitch tool Fig. 1 Mechanism of spring forming machine Fig. 1 Mechanism of spring forming machine Fig. 1 Mechanism of spring forming machine

2405-8963 © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. Copyright © 2016 IFAC 1 10.1016/j.ifacol.2016.07.942

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147

In Fig.1, the steel wire is fed into the wire guide by feed wheels. By forcing to press the wire to the outer dies, the wire bends and forms the ring like shape with desired diameter while the pitch tool controls the length of spring.

The above processes are executed in each production cycle. That is, the spring forming process can be modeled by the discrete time process synchronized

To control the free length of the spring in each production cycle, we decide to adjust the extrude position of the pitch tool for each cycle as the input.

order n corresponds the number of past data y1 ,, yn which affect the current measurement and ai is its coefficient. To determine the order and parameters of the AR model, we adopt some information criteria. For example, the criteria called AIC (Akaike’s Information Criteria) is used.

III. CARACTERISTICS OF FLUCTUATION IN SPRING PRODUCTION AND ITS MODELING The aim of spring forming machine is to produce springs of a guaranteed quality with minimum production cycle.

𝐴𝐴𝐴𝐴𝐴𝐴 = −2 log 𝐿𝐿(𝜃𝜃̃) + 2(𝑛𝑛 + 1)

(2)

Where, L  is the likelihood defined by using the parameter  and data number N as follows.

First, we consider the characteristics of the deviation in spring production

The procedure to determine the best model can be easily performed by using “System Identification Toolbox” in MATLAB [4].

For example, typical result for one fabrication condition (Table 1) is presented.

For this data, AIC gives the model (AR5) with the order 5 and the parameters a1=-0.2969，a2=-0.5453，a3=0.09458， a4=-0.1436，a5=0.04322.

Table 1 Fabrication condition Free length[mm] 40.8 Wire diameter[mm] 1.6 Outside diameter[mm] 14.9 Production speed[spm] 120

Figure 3 shows the comparison between the observed data and the output of the AR model determined by AIC. AR5 observed data

0.06

length[mm]

In this table, the production speed [spm] means the number of spring produced in 1 minute. Figure 2 shows the variation of free length of the manufactured spring under the above fabrication condition.

0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 450

500

550

600

650

700

750

800

850

900

sample number

deviation[mm ]

deviation[mm]

Fig. 3 Observed data and the output of model AR5.

From this figure, we can conclude that the future value of the deviation can be predicted by this model. Moreover, all models obtained by other data under different fabrication condition with same or different information criterion have the optimal order n  1 . This means that the production machine has some kind of cyclic dynamics.

no control

0.15 0

-0.15

0

50

100

-0.3

-0.15

200

250

300

sample number

Fig. 0.15 0

150

0

2 Typical variation of free length of the manufactured spring conventional control 50

100

150

200

250

300

IV. CONTROL INPUT AND SYSTEM MODEL FOR CONTROLLER DESIGN

-0.3 The deviation of springsample length number in Fig. 2 may be caused by some disturbances or uncertainties in forming machine, and can be regarded as random error.

In this case, we investigate that the sequence of the free length of the spring can be predicted by linear time-series model of the signal or not. If it can be modeled by such type of model, the stochastic control theory is applicable. In order to check this condition, we describe the characteristic of the time-series by AR (Auto-Regressive) model described by the following equation. n

y k   ai y k  i   k

(1)

Fig.4 Mechanism of pitch forming tool

i 1

In the previous section, we can confirm that the output sequence of the system can be predicted by linear model. This satisfies the necessary condition for stochastic control system. To construct the controller for spring forming machine, we need

Where, y k denotes measured data at sampling instant k and  k is the white noise with average 0 and variance  2 . The model

2

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the system model with appropriate input. For this purpose, it is very important to select the control input which most efficiently affects the free length of the spring within each production cycle. For this purpose, we investigate the mechanism of pitch forming tool as shown in Fig.4 and decide to adjust the extrude position of the pitch tool (ΔY in Fig. 4) for each cycle as the input u.

V. DESIGN OF MINIMUM-VARIANCE CONTROLLER FOR SPRING FORMING MACHINE There are many design method of stochastic control system, minimum-variance control is most simple and effective control strategy for systems with stochastic environment. In the design of minimum-variance controller, the system is assumed to be described by the following discrete-time model.

In this case, the ARX model (Auto-Regressive Exogenous model) of the system is described as follows. n

m

i 1

i 1

𝐴𝐴(𝑧𝑧 −1 )𝑦𝑦(𝑘𝑘) = 𝑧𝑧 −𝑑𝑑 𝐵𝐵(𝑧𝑧 −1 )𝑢𝑢(𝑘𝑘) + 𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 )𝑤𝑤(𝑘𝑘) (6)

(4)

y k   ai y k i   bi u k i   k

Where the polynomials 𝐴𝐴(𝑧𝑧 −1 ), 𝐵𝐵(𝑧𝑧 −1 ) and 𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 ) are described as follows.

Where, y k and u k denote the output and input of the system respectively, at sampling instant k and  k is the white noise. The best model for controller design is also determined based on some information criteria like AIC.

𝐴𝐴(𝑧𝑧 −1 ) = 1 − 𝑎𝑎1 𝑧𝑧 −1 − ⋯ − 𝑎𝑎𝑛𝑛 𝑧𝑧 −𝑛𝑛 𝐵𝐵(𝑧𝑧 −1 ) = 𝑏𝑏0 + 𝑏𝑏1 𝑧𝑧 −1 + ⋯ + 𝑏𝑏𝑚𝑚 𝑧𝑧 −𝑚𝑚 𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 ) = 1 − 𝑐𝑐𝑤𝑤1 𝑧𝑧 −1 − ⋯ − 𝑐𝑐𝑤𝑤𝑤𝑤 𝑧𝑧 −𝑛𝑛

For this system, we make the following assumptions.

The typical results when the input for spring forming machine is driven by the Pseudo Random Binary Signal (Mseries) are shown in Fig. 5 and 6. deviation[mm]

1) 𝐴𝐴(𝑧𝑧 −1 ) , 𝐵𝐵(𝑧𝑧 −1 ) and 𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 ) are coprime and the orders of n, m and the dead time d are known. 2) 𝐵𝐵(𝑧𝑧 −1 ) and 𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 ) are asimptotically stable.

ARX561 observed data

0.2 0.1 0

In these assumptions, the coprimeness of polynomials are usually satisfied but the knowledge of orders may not be obtained without a priori information about the controlled system. The stability of B(z −1 ) depends on the characteristics of the controlled system and sampling period, but Cw (z −1 ) (which describes the characteristics of noise) is usually stable.

-0.1 -0.2 100

110

120

130

140

150

160

170

180

190

200

sample number

Fig. 5 Observed data and the output of ARX561 model

In this case, the model which gives minimum squared error between model output and observed value is ARX561 with the order n=5, m=6 and the parameters

Under above assumptions, the control system which minimize the variance E[{𝑦𝑦(𝑘𝑘) − 𝑦𝑦 ∗ (𝑘𝑘)}2 ] between the output of the system 𝑦𝑦(𝑘𝑘) and the reference signal 𝑦𝑦 ∗ (𝑘𝑘) can be constructed as follows.

a1=-0.6751，a2=0.01029，a3=-0.06032，a4=0.1437，a5=-0.199 b1=0.2879，b2=-0.1548，b3=0.001676，b4=0.04808，b5=0.03314, b6=-0.04851. (5)

First, we consider the Diophantine equation of the form as 𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 ) = 𝐴𝐴(𝑧𝑧 −1 )𝑅𝑅(𝑧𝑧 −1 ) + 𝑧𝑧 −𝑑𝑑 𝑆𝑆(𝑧𝑧 −1 )

On the other hand, AIC and MDL gives the same model (ARX121) with the order n=1, m=2 and the parameters

𝑅𝑅(𝑧𝑧 −1 ) = 1 + 𝑟𝑟1 𝑧𝑧 −1 + ⋯ + 𝑟𝑟𝑑𝑑−1 𝑧𝑧 −(𝑑𝑑−1) 𝑆𝑆(𝑧𝑧 −1 ) = 𝑠𝑠0 + 𝑠𝑠1 𝑧𝑧 −1 + ⋯ + 𝑠𝑠𝑛𝑛−1 𝑧𝑧 −(𝑛𝑛−1)

a1=-0.6453，b1=0.2877，b2=-0.1442.

In these cases, the role of assigning model name is [model type][the order of auto-regression n][the order of moving average m][dead time d].

deviation[mm]

(10) (11) (12)

By multiplying 𝑦𝑦(𝑘𝑘) to the both sides of Eq. (10), we obtain

𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 )𝑦𝑦(𝑘𝑘) = 𝐴𝐴(𝑧𝑧 −1 )𝑅𝑅(𝑧𝑧 −1 )𝑦𝑦(𝑘𝑘) + 𝑧𝑧 −𝑑𝑑 𝑆𝑆(𝑧𝑧 −1 )𝑦𝑦(𝑘𝑘) = 𝐵𝐵(𝑧𝑧 −1 )𝑅𝑅(𝑧𝑧 −1 )𝑢𝑢(𝑘𝑘 − 𝑑𝑑) + 𝑆𝑆(𝑧𝑧 −1 )𝑦𝑦(𝑘𝑘 − 𝑑𝑑) + 𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 )𝑅𝑅(𝑧𝑧 −1 )𝑤𝑤(𝑘𝑘) = 𝐵𝐵𝑅𝑅 (𝑧𝑧 −1 )𝑢𝑢(𝑘𝑘 − 𝑑𝑑) + 𝑆𝑆(𝑧𝑧 −1 )𝑦𝑦(𝑘𝑘 − 𝑑𝑑) + 𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 )𝑅𝑅(𝑧𝑧 −1 )𝑤𝑤(𝑘𝑘) (13) From the above result, the error equation of the system becomes as follows.

Figure 6 shows the comparison between the observed data and the output of the ARX model determined by AIC/MDL, respectively. ARX121 observed data

0.2

(7) (8) (9)

0.1 0 -0.1 -0.2 100

120

140

160

180

200

= 𝐵𝐵𝑅𝑅

sample number

Fig. 6 Observed data and the output of model ARX121.

These figures show that both model may be used for the controller design. Moreover, rather low order model gibes the good estimate for However, from the practical point of view, the adaptive model is better than the fixed model.

𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 )[𝑦𝑦(𝑘𝑘) −1 (𝑧𝑧 )𝑢𝑢(𝑘𝑘

− 𝑦𝑦 ∗ (𝑘𝑘)] = 𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 )𝑒𝑒(𝑘𝑘) − 𝑑𝑑) + 𝑆𝑆(𝑧𝑧 −1 )𝑦𝑦(𝑘𝑘 − 𝑑𝑑) − 𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 )𝑦𝑦 ∗ (𝑘𝑘) + 𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 )𝐵𝐵𝑅𝑅 (𝑧𝑧 −1 )w(k) (14)

By defining the parameter vector 𝜃𝜃 andsignal vector 𝜁𝜁(𝑘𝑘 − 𝑑𝑑) as

In case of adaptive model, from the implementation point of view, the simple model is better than the complicate model. 3

𝜃𝜃 𝑇𝑇 = [𝑏𝑏0 , 𝜃𝜃0𝑇𝑇 ]，𝑏𝑏𝑅𝑅0 = 𝑏𝑏0 = [𝑏𝑏0 , 𝑏𝑏𝑅𝑅1 , ⋯ , 𝑏𝑏𝑅𝑅𝑅𝑅+𝑑𝑑−𝑎𝑎 , 𝑠𝑠0 , ⋯ , 𝑠𝑠𝑛𝑛−1 , 𝑐𝑐𝑤𝑤1 , ⋯ , 𝑐𝑐𝑤𝑤𝑤𝑤 ]𝑇𝑇 (15) 𝜁𝜁 𝑇𝑇 (𝑘𝑘 − 𝑑𝑑) = [𝑢𝑢(𝑘𝑘 − 𝑑𝑑), 𝜁𝜁0𝑇𝑇 (𝑘𝑘 − 𝑑𝑑)]

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From the above result, it can be seen that the influence of the random disturbance is reduced by minimum variance control.

= [𝑢𝑢(𝑘𝑘 − 𝑑𝑑), 𝑢𝑢(𝑘𝑘 − 𝑑𝑑 − 1), ⋯ , 𝑢𝑢(𝑘𝑘 − 𝑚𝑚 − 2𝑑𝑑 + 1), 𝑦𝑦(𝑘𝑘 − 𝑑𝑑), ⋯, (16) 𝑦𝑦(𝑘𝑘 − 𝑑𝑑 − 𝑛𝑛 + 1), 𝑦𝑦 ∗ (𝑘𝑘 − 1), ⋯ , 𝑦𝑦 ∗ (𝑘𝑘 − 𝑑𝑑 − 𝑛𝑛)] 𝑒𝑒(𝑘𝑘) = 𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 )−1 [𝜃𝜃 𝑇𝑇 𝜁𝜁(𝑘𝑘 − 𝑑𝑑) − 𝑦𝑦 ∗ (𝑘𝑘)] + 𝑅𝑅(𝑧𝑧 −1 )𝑤𝑤(𝑘𝑘) = 𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 )−1 [𝑏𝑏0 𝑢𝑢(𝑘𝑘 − 𝑑𝑑) + 𝜃𝜃0𝑇𝑇 𝜁𝜁(𝑘𝑘 − 𝑑𝑑) − 𝑦𝑦 ∗ (𝑘𝑘)] + 𝑅𝑅(𝑧𝑧 −1 )𝑤𝑤(𝑘𝑘) (17) In this case, we determine the control input so as to suffice

VII. IMPLEMENT TO COMMERCIAL SPRING FORMING MACHINE We implement the proposed control algorithm to the commercial spring forming machine as shown in Photo1. In the implementation of the minimum-variance controller, we simply assume that the stochastic disturbance in Eq. (6) is the white noise ( Cw (z −1 ) = 1 ) and the dead time of the system unity (d=1).

(18)

𝜃𝜃 𝑇𝑇 𝜁𝜁(𝑘𝑘) = 𝑦𝑦 ∗ (𝑘𝑘 + 𝑑𝑑)

Or 𝑢𝑢(𝑘𝑘) = [𝑦𝑦 ∗ (𝑘𝑘 + 𝑑𝑑) + 𝜃𝜃0𝑇𝑇 (𝑘𝑘 + 𝑑𝑑)𝜁𝜁0 (𝑘𝑘)]⁄𝑏𝑏0

(19) (20)

𝑢𝑢(𝑘𝑘) = 𝐵𝐵𝑅𝑅 (𝑧𝑧 −1 )−1 [𝐶𝐶𝑤𝑤 (𝑧𝑧 −1 )𝑦𝑦 ∗ (𝑘𝑘 + 𝑑𝑑) − 𝑆𝑆(𝑧𝑧 −1 )𝑦𝑦(𝑘𝑘)]

149

Then the error becomes

(21)

𝑒𝑒(𝑘𝑘) = 𝑅𝑅(𝑧𝑧 −1 )𝑤𝑤(𝑘𝑘)

This means the achievement of the minimum variance of the error. Figure 7 shows the block diagram of the minimum-variance control system.

Photo 1. Spring forming machine CFX-2

We have used 2 types of ARX model determined based on the MDL and AIC. The parameters of each model are indicated below. ARX120：a1=-0.6534，b1=0.288，b2=-0.1473 ARX560：a1=-0.682，a 2=0.1891，a 3=-0.037 a4=0.1499，a5=-0.2006

b1=0.2899，b2=-0.1597，b3=0.009677， b4=0.03483，b5=0.03325, b6=-0.0503

Fig.7 Block diagram of minimum variance control system

Figure 9 shows the comparison of experimental results obtained by different types of control algorithms.

VI. PRELIMINARY EVALUATION BY COMPUTER SIMULATIONS

length[mm]

0 -0.05 0

20

40

0 -0.6 -1.2 0

20

40

60

80

100

-3 0

20

40

60

80

100

sample number

length[mm]

Fig. 8 (b) Output of minimum variance controller (Input to machine model) 1.2 0.6 0 -0.6 -1.2 0

20

40

60

80

80

100

120

140

160

180

200

sample number

0.16 0.12 0.08 0.04 0 -0.04 0 -0.08 -0.12

existing offset

20

40

60

80

100

120

140

160

ARX120

180

200

sample number

3 2 1 0 -1 -2

length[mm]

input[deg]

Fig. 8 (a) Output deviation without control.

60

-0.1

1.2 0.6

ARX120

0.1 0.05

-0.15

sample number

deviation[mm]

no offset

0.15

length[mm]

deviation[mm]

Before applying the proposed control scheme for spring production, computer simulations are performed using estimated ARX models and controller designed by the same model. Figure 8 shows the typical simulation results of minimum variance control under pseud random disturbance using model ARX561.

100

sample time

0.15 0.1 0.05 0 -0.05 0 -0.1 -0.15 -0.2

0.16 0.12 0.08 0.04 0 -0.04 0 -0.08 -0.12

no offset 20

40

60

80

100

120

140

160

ARX560 180

200

sample number existing offset

20

40

60

80

100

120

140

160

ARX560

180

200

sample number

Fig. 9. Comparison of experimental results with different types of compensation

Fig. 8 (c) Output deviation with minimum variance control

4

IFAC ALCOSP 2016 June 29 - July 1, 2016. Eindhoven, The Netherlands 150 Naoki MIZUNO et al. / IFAC-PapersOnLine 49-13 (2016) 146–151

Table 2 shows the comparison of standard deviation for two

we investigate the property of recursive parameter estimation algorithm using real data of spring forming machine. First, we adopt the lowest model order obtained by AIC/MDL and evaluate the characteristics of the parameter estimation. In this case, the same data for the ARX modeling is used and the orders in Eq. (6) are assigned n=1 and m=2 respectively. Figure 10 shows the trend of the estimated parameters and Table 3 indicate some measures for comparison of several measures for estimated parameters.

types control systems.

Table 2 Comparison of standard deviation for experimental results No offset Standard deviation

0.054

Existing offset

ARX120

ARX560

0.047

0.039

0.037

From the results in Fig. 9 and Table 2, we conclude that the minimum-variance control using the fixed model has better performance than that of existing compensation.

coefficient

1

VIII. DESIGN OF ADAPTIVE MINIMUM VARIANCE CONTLOLLER

 (k )  (k ) 

1   ( k ) T  ( k  1) ( k )

0

20

40

60

80

100

120

140

160

180

200

sample number

coefficient

1

b1

0.5 0 -0.5

However, from the practical point of view, it is expected that the system model for control depends the type of forming machine, the material of wire, the fabrication condition and so on. For this situation, we investigate the adaptive minimum variance control algorithm for spring forming machine [5]. In this case, the control input Eq. (19) should be calculated based on the estimated parameters of the model Eq. (6). Moreover, for simplicity, we assume that the disturbance is pure white noise ( Cw (z −1 ) = 1) and the parameters in the model are estimated by the following recursive algorithm.

 ( k  1) T  ( k )  y ( k )

0

-0.5

Based on the results mentioned in the previous sections, we conclude that the minimum-variance control based on the ARX model for cyclic dynamics of the spring forming machine has better performance than that of existing compensation.

ˆ(k )  ˆ(k  1)  (k - 1) (k ) (k)

a1

0.5

0

20

40

60

80

100

120

140

160

180

200

sample number

coefficient

1

b2

0.5 0 -0.5

0

20

40

60

80

100

120

140

160

180

200

sample number

Fig. 10 Trend of estimated parameters Table 3

Comparison of several measures for estimated parameters

(22) (23)

2 (k )(k - 1) (k ) (k )T (k  1)  (24) 1  (k  1)  1 (k )  1 (k )  2 (k ) (k )T (k  1) (k ) 

In Fig.10, we can see that the estimated parameters converge its steady state values within 50 cycles and the converged values are rather near values determined by ARX modeling. This means that the model order n=1 and m=2 are appropriate for full adaptive control. Notice that the initial values for estimated parameters are all set at 1 and in the estimated algorithm, (0)  1000 I ，

Where 0   1 ( k )  1 , 0   2 ( k )  2 , (0)   T (0)  0 . To achieve the theoretical minimum-variance control property, we should assign 1 (k )  1 .

1 (k )  0.91 (k  1)  0.1 ， 1 (0)  0.95 ， 2 (k )  1 are used.

IX. APPLICATION OF ADAPTIVE MINIMUM-VARIANCE CONTROL ALGORITM TO COMMERCIAL SPRING FORMING MACHINE

B. Evaluation of Proposed Adaptive Minimum-Variance Control Algorithm for Commercial Spring Forming Machine

In the application of the adaptive control algorithm to real systems, the selection of the model structure is very important to obtain stable and high control performance. In general, the model order in the adaptive system should be low for fast convergence unless the modeling error becomes large. To select the rational model order for real system, we need the information about the dynamics of the system.

We applied the adaptive minimum-variance control scheme combining the parameter estimation algorithm introduced in the above subsection and the control algorithm described in section III to the CFX-2 spring forming machine.

A. Evaluation of Recursive Parameter Estimation using Real Data of Spring Forming Machine Before implementing the full adaptive minimum-variance control algorithm to the commercial spring forming machine,

Figure 11 shows the reference comparison of experimental results obtained by no compensation and conventional compensation. In this case, the fabrication condition is same as shown in Table 1. 5

IFAC ALCOSP 2016 June 29 - July 1, 2016. Eindhoven, The Netherlands Naoki MIZUNO et al. / IFAC-PapersOnLine 49-13 (2016) 146–151

deviation[mm]

deviation[mm]

These results are the standard for the evaluation of the proposed algorithms.

Table 4

0 0

-0.15

50

100

-0.3

150

200

250

300

sample number conventional control

0.15 0

0

-0.15

50

100

-0.3

150

200

250

Table 5

In the next experiment, the estimated parameters are updated until 50 cycles of the production, after 50 cycles, the controller parameters are fixed to the converged values of the estimation. This operation is corresponding to the “Auto-Tuning” in several commercial PID controllers but has superior performance compared with no compensation and conventional compensation (Fig. 12). input[deg] deviation[mm]

Table 6

101～200

input AMV control No.7-1

50

100

150

200

250

300

350

400

450

500

sample number

0

0

50

100

150

200

250

300

350

400

450

sample number

Fig. 12 Input and Output of Auto-tuned minimum-variance control

The next Figure 13 shows the result of full adaptive minimum-variance control case. In the experiment, during the first 50 cycles of the production, only the parameters of the cyclic dynamic model are estimated for the protection of the machine, after 50 production cycle, the cyclic production is controlled by adaptive minimum-variance control algorithm. input[deg]

deviation[mm ]

0

50

100

150

200

250

300

sample number

0 -0.15 -0.3

[2] H. Lavrič, M. Bugeza and R. Fiser, “Novel Approach to Closed - Loop Control of Wire Bending Machine” IEEE PEDS2011, Singapore, 5 - 8 December (2011)

AMV control No.7-2

0.15

0

50

100

150

200

250

0.129 -0.139 0.268 -0.005 0.037

[1] M. Rosenberger, M. Schellhorn, G. Linss, M. Schumann, P. Werner and S. Lübbecke, “QUALITY CONTROL FOR SPRINGS DURING THE PRODUCTION PROCESS BASED ON IMAGE PROCESSING TECHNOLOGIES”, http://www.tuilmenau.de/fileadmin/media /qualimess/ Veroeffentlichungen/ quality_loop_for_spring_control.pdf

0.5 0

101～301

REFERENCES

1

-0.5

0.129 -0.139 0.268 -0.008 0.038

The experimental results show the effectiveness of the proposed system under various operating conditions.

input AMV control No.7-2

1.5

201～300

0.096 -0.089 0.185 -0.003 0.036

In this paper, we propose a feasible adaptive control algorithm for cyclic production machine. The proposed algorithm is based on the discrete-time model of the cyclic dynamics of the production machine. The proposed algorithm is successfully applied to reduce the variation of free length of the spring produced by spring forming machine.

500

-0.15 -0.3

101～500 0.130 -0.093 0.223 -0.005 0.034

X. CONCLUSIONS

AMV control No.7-1

0.15

201～300 0.097 -0.083 0.180 -0.005 0.033

Based on the above tables, we conclude that the proposed method has better performance than that of existing compensation.

0.5 0

101～200 0.067 -0.081 0.148 -0.006 0.032

Comparison of several measures for AMV control

Max[mm] Min[mm] Data range[mm] Average[mm] Standard deviation

1

0

Conventional control 0.148 -0.146 0.294 -0.018 0.063

Comparison of several measures for Auto-tuned MV control

Max[mm] Min[mm] Data range[mm] Average[mm] Standard deviation

Fig. 11 Deviation of spring length without control (upper trace) and with compensation (lower trace)

-0.5

No control 0.084 -0.285 0.369 -0.100 0.070

300

sample number

1.5

Deviation of spring length with and without comparison

Max[mm] Min[mm] Data range[mm] Average[mm] Standard deviation

no control

0.15

151

300

[3] N.Mizuno et.al., “Improvement of Spring Forming Accuracy Using Minimum-Variance Control”, Proc. Of ASCC (2015)

sample number

Fig. 13 Input and Output of adaptive minimum-variance control

[4]F. Lorito, “ADAPTIVE QUALITY CONTROL FOR SPRINGS PRODUCTION,” Control Enli. Practice, Vol. 5, No.8, pp. 10431051 (1997)

From the above results, we can see that the proposed adaptive minimum-variance controller gives less variance of free length of the produced springs than those of without control and with conventional control, without off-line modeling of the system. To evaluate more quantitative comparison, we introduce several measures for experimental results.

[5] K. J. Astrom, “Introduction to Stochastic Control Theory,” Dover Publications, January 6 (2006).

6