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Application of Taguchi Design in System Identification: A simple, generallyapplicable and powerful method

Somayeh Norouzi Ghazbi, Alireza Akbarzadeh, Mohammad-R Akbarzadeh-T

PII: S0263-2241(19)30736-5DOI: https://doi.org/10.1016/j.measurement.2019.106879Reference: MEASUR 106879

To appear in: Measurement

Received Date: 18 February 2019Revised Date: 1 July 2019Accepted Date: 29 July 2019

Please cite this article as: S. Norouzi Ghazbi, A. Akbarzadeh, M-R. Akbarzadeh-T, Application of Taguchi Designin System Identification: A simple, generally applicable and powerful method, Measurement (2019), doi: https://doi.org/10.1016/j.measurement.2019.106879

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© 2019 Published by Elsevier Ltd.

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Application of Taguchi Design in System Identification: A simple, generally applicable and powerful method

Somayeh Norouzi Ghazbib,1,2, Alireza Akbarzadeha,1,3*, Mohammad-R Akbarzadeh-Tc,2,3

1Mechanical Engineering Department, Ferdowsi University of Mashhad, Mashhad, Iran2Electrical Engineering Department, Ferdowsi University of Mashhad, Mashhad, Iran3Center of Excellence on Soft Computing and Intelligent Information Processing (SCIP),

Ferdowsi University of Mashhad, Iran

a: *Corresponding author: Alireza Akbarzadeh

Address: Mechanical Engineering Department, Ferdowsi University of Mashhad,

P.O. Box: 91775-1119.

Zip code: 9177948944

Mashhad, Iran

Tel: +98-915-521-0253

E-mail addresses: ali_akbarzadeh@um.ac.ir (Alireza Akbarzadeh).

b: E-mail: somayeh.norouzi@mail.um.ac.ir

c: E-mail: Akbarzadeh@eee.org (Mohammad-R Akbarzadeh-T).

AbstractThis paper is a development of the authors’ recently published idea on a simple and generally applicable system identification algorithm based on Taguchi method. In the previous paper, the effectiveness of the algorithm in identifying nonlinear systems was shown. This paper presents the modified algorithm with the capability of identifying Multi-Input Multi-Output (MIMO) systems, and systems with unknown parameters of quite different value-ranges. Regarding the former, a contribution analysis is introduced and utilized to determine that the model between which of the input-output pairs should be identified first. The results proved that this idea significantly results in less time-cost. Moreover, the study develops the identification algorithm by adding a step of breaking down the identification process to multiple steps, each identifying a group of parameters that are in about the same value-range. Utilizing Taguchi method that includes human intelligence in the loop, the presented method effectively saved the process time and led to a more accurate estimation. In the method presented, the algorithm identifies a group of parameters at each step starting with the most contributor ones rather than identifying all of them at once. To group parameters, a contribution analysis is employed. To prove the

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effectiveness of the method, without loss of generality, the algorithm is applied on a multi-input single-output system. The experimental test-setup used in this study is a custom-made linear series elastic actuator that is designed for lower limb exoskeletons in Ferdowsi University of Mashhad (FUM)-Robotic Laboratory. Keywords: System identification, Taguchi, Multi-Input Multi-Output system, MIMO, nonlinear systems, non-parametric system

1. Introduction To find a general, fast and low cost system identification method has been an ongoing challenge for decades [1, 2, 3]. System identification is defined as identification of a mathematical model with capability of mapping the desired outputs to the inputs of interest. Up to now, various identification methods have presented that each is specific to identifying certain types of systems. For instance, methods to identify linear systems [4,5,6], nonlinear systems [7, 8], parametric and nonparametric models [9, 10], methods specific to the time domain [11, 12, 13, 14], the frequency domain [15, 16], and structural identification methods [17, 18] could be mentioned. Despite the advances in the field, there are still some drawbacks need to be addressed:

(1) A method capable of identifying a variety of systems is still demanded. Most of the identification methods and tools that have already presented are limited to identification of particular types of systems. To address this problem, employing optimization algorithms as identification tools are introduced and investigated. Structural and system-type independencies and simplicities are marked advantages associated with such approaches. Identification of hydraulic turbine governing system using improved gravitational search algorithm is investigated by Li et.al [19]. Panda et.al [20] used Cat Swarm Optimization (CSO) to identify infinite-impulse-response systems. Genetic Algorithm (GA) is utilized by Zagrouba et.al [21] as an identification tool for PV solar cells and modules parameters. A map reduced based parallel Niche GA is introduced by Hu et.al [22] to solve identification problem of contaminant source. Although intelligent optimization algorithms have shown promises for identification of diverse systems, to the best of authors’ knowledge, they have not employed for identification of MIMO systems yet.

(2) Additionally, although system identification has been grown to be a necessary part of various fields, from mechanical engineering to economics [23,24, 25, 26], still there is lack of approaches that can be easily implemented for every user without need to extensive background on system identification.

The paper at hand introduces a simple and yet powerful method of system identification based on Taguchi optimization algorithm. Taguchi design method is a powerful statistical optimization

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technique that is well-known for its capability of identifying the optimized parameters via a reduced number of iterations. Unique characteristics of the Taguchi method has nominated it as a powerful practical gain tuning tool. Norouzi-Gh et al. [27,28], studied gain-tuning of a Fractional PID control systems’ parameters using the Taguchi method. Yuce et al. [29] used a combination of neural network and the Taguchi method to optimize the control quality of a wood manufacturing firm; however, in this project, Taguchi algorithm is utilized as a system identification tool.

The authors, firstly, introduced the method in [30], where they implemented the proposed method to identify a Single-Input Single-Output (SISO) system. In this study, the idea has developed to MIMO systems. Moreover, one step of contribution analysis is added to the method to effectively improve the identification algorithm performance.

The benchmark for the study is a custom made Linear Series Elastic Actuator (LSEA) consisting of two subsystems with each having different levels of contribution to the system output. The actuator is constructed in FUM-Robotic Laboratory[31]. It is a variable stiffness actuator offering increased efficiency due to energy storage, low and adjustable output impedance, increased stability, low friction and impact resistance. Mechanical structure of SEAs includes a spring placed between the output load and actuator’s motor. If spring stiffness is selected properly, it will improve the structure tolerance to mechanical shocks and unwanted collisions of the output link [32,33].

2. Modeling

The linear series elastic actuator used as the benchmark for this study is shown on the left side of Figure (1), where different parts of the actuator including its mechanical and electrical parts are introduced. A graphical view of the simplified structure is also shown on the right side.

Figure 1: Left: the LSEA used in the present study, right: a graphical view of the simplified struture

As illustrated in Figure (1), the system includes a servo motor, a belt-pulley mechanism, and a ball screw system that converts the rotational motion to linear displacement. The ball screw movement actuates the nut and results in deflection in the elastic elements that consequently applies forces to the output link. Figure (2) also provides a physical representation of the system.

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Figure 2: A schematic diagram of the LSEA

In the above figure, denotes the mass of the nut block and the ball screw, stands for the 𝑚1 𝑚2

mass the out-put block, represents the spring stiffness, , and are damping coefficients 𝐾𝑠 𝐶𝑑 𝐶1 𝐶2

of the spring, viscous friction coefficient between the output link and guide rods, and viscous friction coefficient between the belt-pulley mechanism and the motor shaft, respectively. Also, 𝐹𝑜

is the actuator’s output force, denotes the displacement of the nut, and is the displacement of 𝑥 𝑥𝑜

the output link.

Applying Newton’s law, the relations between , , ( motor force) and are obtained as 𝑥𝑜 𝑥 𝐹𝑚 𝐹𝑜

following,

1 0 0 1( ) ( )m s dm x F K x x C x x C x (1)

2 2( ) ( )o o s o d o om x F K x x C x x C x (2)

By taking Laplace transform, two transfer functions relating , , and , , are obtained 𝑋 𝐹𝑚 𝑋𝑜 𝑋 𝐹𝑜 𝑋𝑜

as,

02 21 1 1 1

1( ) ( ) ( )( ) ( )

d sm

d s d s

C s KX s X s F sm s C C s K m s C C s K

(3)2

2 2( ) 1( ) ( ) ( )d so o

s d s d

m s C C s KX s X s F sK C s K C s

(4)

Define , , and as,𝐴 𝐵 𝐶

21 1( )d sA m s C C s K

22 2( )d sB m s C C s K

d sC C s K (5)

Then, by substituting Eq. (4) for in Eq. (3), the transfer function relating Fo , Fm , and Xo can Xbe found as,

2

( ) ( ) ( ) ( )o m oC CF s F s B X sA A

(6)

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Eq. (6) shows how is distributed while the output load is moving. The block diagram of the 𝐹𝑜

system is shown in Figure (3). Coefficients of the A, B, and C polynomials are the parameters needed to be identified.

Figure 3: The block diagram of the system

3. Identification ProcedureThe identification problem is to identify the unknown parameters in Eq. (6). The system, as shown in Figure (3), is a MISO system including a feed-forward-term in the transfer function between Fo,

and Xo. The two mentioned characteristics make the system different than a simple linear SISO system.

To identify the system, Taguchi-based identification method which is introduced in the authors’ recent publication on the topic [26] is used. The necessary steps of the algorithm are listed as following:

Step I: Feed the actual system by a sufficiently exciting input and measure the system output.Step II: For each unknown parameter in the model, set a rough range of variation and break it into

a desired number of levels. The final ranges of variations can be totally different than the initial settings. Within

the identification process, the Taguchi algorithm will help the range to converge to the proper range.

In determining the number of levels care should be taken to ensure the array remains orthogonal. Therefore, it is important to select the level numbers for each parameter based on what the Taguchi’s system of experimental design offers[34].

Step III: According to the number of levels and the values set for variation of each parameter, form a Taguchi table of orthogonal arrays and complete the required simulations.

Step IV: Calculate Signal-to-Noise Ratio (SNR) for each simulation and analyze the SNRs diagrams; one of the following three cases may possibly happen. Case I: There are some parameters with their correspondent SNR diagram that

significantly change over different levels.

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o What to do: Choose the level that gives the maximum SNR value as the intermediate level of the next variation range of the parameter and redo step III. The variation range of some parameters may need to be expanded or tightened based on the values set as the intermediate levels.

Case II: If some SNR diagrams show significant changes and some are flat, choose different levels for only those parameters with significant variable SNR diagrams and go to step III.

Case III: If all SNR diagrams are obtained almost flat and the level differences are sufficiently small, go to step V.

Step V: Select the best levels as optimum values for each parameter and stop the procedure.

In this method, as illustrated, the only experimental step is step I meaning that the Taguchi table in step III is formed by the results of the simulated model. Figure (4) represents the above identification procedure for SISO systems, graphically.

Figure 4: Identification procedure for a SISO system

To adjust the method to the MISO system at hand, the following modifications are implemented.

Step i: To investigate the contribution of each of the inputs to the output. If the contribution values are to be similar or sufficiently close, the identification can begin with identifying each of the subsystems, otherwise, it is suggested to start with the subsystem that maps the main contributor to the output. The MISO system here can be seen as two SISO systems and the

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outputs of each can be superposed to build the final output. The idea in this part is gotten from the idea of controlling MIMO systems by “sequential loop closing approach” [35]. According to this approach, if the subsystems were coupled, it would require finding the main contributor input to each of the system outputs. Then, the whole system could be considered as a multiple MISO system and the identification could start by identifying the subsystem with a stronger relation between the main contributor input and the correspondent output.

Step ii: To perform contribution analysis for unknown parameters of each subsystem. Indeed, it should be determined that which parameter/s in a subsystem contribute more in forming the map between the subsystem input and output. If the parameters have significantly different levels of contributions, categorizing them into two or three groups and identifying the parameters of each group in one phase would noticeably reduce the identification time and will result in a more accurate estimation. As would be shown later, in the absence of this step, the identification procedure would not converge or would be very time-consuming.

Step iii: To identify the unknown parameters based on the algorithm offered for SISO systems.

Flow chart (1) depicts the system identification procedure for a MISO system, graphically.

Flow chart 1. System identification procedure for a MISO system

In the next section, the identification procedure is implemented on the LSEA, and also some tricks that need to be taken care of in practical implementation are presented.

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4. Experiment

4.1.Test set-up

The experimental test set-up consists of a single-axis linear stage (shown in Figure (5)) and LSEA that is assembled upon the single-axis linear stage, Figure (6).

Servo motor of the test setup Moving block Load cells

Guide shafts Ball-Screw

Figure 5. The linear stage test bed for evaluation of the control algorithm

Servo motor of FUM-LSEA Output Link of LSEA

Figure 6. FUM-LSEA placed on the linear stage test bed

The single-axis stage consists of a fixed and a moving platform. LSEA is placed on the fixed platform of the linear stage and its output link is connected to the stage moving part. The moving block is derived from a ball-screw mechanism actuated by a 200W AC servo motor. The only degree of freedom that it can provide is a linear translation for the output link of LSEA. If the servo motor be turned off, the moving block of the LSEA would be fixed and the system would be fed by only force trajectory. If arbitrary movement of the output link of LSEA is of interest, the servo motor of the linear stage needs to be programmed appropriately. In this scenario, the LSEA inputs might be both of motion and force trajectories, and the motion trajectory is commanded to the LSEA by the moving platform of the single-axis linear stage.

The system is also equipped by two CMM2 load cells from DACELL Co. that are mounted between the output link of LSEA and the stage measuring the output force of LSEA. A motion control card by tsPishro Co. is also used to command the force trajectories to the servo motor driver of the system, and to read the output force from load cells amplifiers. Finally, the internal communication between MATLAB/Simulink desktop real-Time toolbox and the motion control board is provided by an Ethernet network. Real-time data are sampled at the rate of 1 kHz.

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4.2. Performance evaluation of the test bed

As discussed in the last section, in the presence of both motion and force trajectories, the linear stage commands motion trajectories to the LSEA. Therefore, evaluation of the stage performance is required to ensure that the stage performance is sufficiently reliable for the purposes of the study. Figure (7.a) shows the desired and the actual motion of the output link of LSEA. As illustrated in Figure (7.b), the error signal is less than mm which is in the range of expectancy.1

a) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (Sec.)

0

1

2

3

4

Xo(c

m)

Actual XoDesired Xo

b)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (Sec.)

-0.1

-0.05

0

0.05

0.1

Erro

r (cm

)

Figure 7. Evaluation of the setup accuracy; (a): desired vs. actual motion of the setup Xo, (b): Position error.

4.2.1. Data measurement

In this section, at first, a set of sufficiently enough exciting inputs are selected. Then within two experiments that in one, the system is only fed by the motion trajectory and in the other, it is commanded by only force trajectory, a sense of the contribution of each of the inputs to the output force is obtained.

4.2.2. The system fed by only force trajectoryThis experiment corresponds to identification of the model between and . To this purpose, 𝐹𝑚 𝐹𝑜

the moving block of the stage is locked; and LSEA needs to be excited by a sufficiently exciting trajectory (excitation signal). The system at hand is a LSEA that is built to assist people in the 𝐹𝑚

range of 22-60 years old in walking. Based on the defined target, considering the frequency spectrum of 1 to 5 Hz for the exciting signal seems to be enough [36]. Therefore, a sequence of sinusoidal waive with frequencies in the mentioned range are selected as the exciting signals. The

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output force signal is measured by two load cells that can only measure compression forces. Therefore, the force trajectory is obtained as , where and refer to the forces 𝐹𝑜 = 𝐹𝑜𝑓 ‒ 𝐹𝑜𝑟 𝐹𝑜𝑓 𝐹𝑜𝑟

measured by the front and the rear load cells, respectively. The system input and output for this experiment are shown in Figure (8).

0 1 2 3 4 5 6 7 8 9 10Time (Sec.)

-300

-200

-100

0

100

200

300

Mag

nitu

de (N

)

System Output ( Fo)System Input (Fm)

Figure 8 .Force Trajectories; (a): Applied motor force trajectory, (b): Measured output force trajectory.

4.2.3. The system fed by only motion trajectory

In the second experiment, the moving block of the stage is programmed to move and command the LSEA output link. The input motion trajectory and the output force of the system are depicted in Figure (9). The motion trajectory is selected to be similar to the walking trajectory at hip joint.

0 1 2 3 4 5 6 7 8 9 10Time (Sec.)

-20

-10

0

10

20

Mag

nitu

de

System Output (Fo)System Input (Xo)

Figure 9. Output force (N) in response to the motion trajectory, Xo (cm)

Comparing the obtained range of the input force from the first and the second experiments that are N and N respectively, it is resulted that the main contributor to the output signal is ± 150 ± 15

force trajectory. And since here, the MISO system can be seen as two decoupled subsystems, the identification process is done within two phases, beginning with identification of the unknown parameters in the transfer function between and .𝐹𝑚 𝐹𝑜

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4.3. Taguchi-based identification, 1st phase

Remembering from Eq. (5), there are 4 parameters that need to be identified in the first phase of

identification. The parameters are and the coefficient of at function that is 1, ,d sC K m s A 1.dC C

The parameters have mechanical natures, as a result, the design team, already have a rough estimation of the order of the values. Therefore, for each parameter, a range of variation consisting three discrete values is selected, Table (1). In the case of dealing with a wider range of values for each parameter, selection of a greater number of levels would be helpful.

Table 1: Factor levels for the 1st iteration

LevelParameters 1 2 3𝐶𝑑 0.01 0.01 0.10𝐾𝑠 10.0 50.0 100𝑚1 0.50 0.75 1.00𝐶𝑐 0.01 0.01 0.10

4.3.1. Contribution analysis of unknown parameters

After selection of the levels for each parameter, a contribution analysis needs to be done to determine the contribution of the parameters in the subsystem of . In the absence of this 𝐹𝑜 𝑡𝑜 𝐹𝑚

step, the identification process will proceed with identifying the main contributor parameters which are dominant sources of the identification error at each iteration. When the main contributor parameters are identified, the identification error would converge to a constant value which could be seen as a corresponding error to the local optimized values for the unknown parameters. To reduce the error more, one solution is to try new value-ranges for each parameter which would be much time-consuming.

Therefore, a variation range including three values of three different orders is set for each unknown parameter. The sets of simulations are designed as at each three runs only one parameter changes. Then, for each simulation, a cost function that is a summation of the squared error between the actual output and the result of the simulation is calculated. Finally, the percentages of error variations for each parameter are calculated as the difference between the maximum and the minimum cost function corresponding to only the changes of that parameter divided by the total average of the cost values. Table (2) shows the cost function’s values for each simulation and the percentage of the error variation for each set of three simulations.

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Table 2: Contribution analysis

Parameters value Parameters value

# 𝐶𝑑 𝐾𝑠 𝑚1 𝐶𝑐

Cost Function

% of Error

variation # 𝐶𝑑 𝐾𝑠 𝑚1 𝐶𝑐

Cost Function

% of Error

variation

1 0.001 50 0.75 0.01 147 7 0.01 10 0.50 0.01 032

2 0.01 50 0.75 0.01 140 8 0.01 10 0.75 0.01 140

3 0.1 50 0.75 0.01 091

48%

9 0.01 10 1.00 0.01 196

142%

4 0.01 010 0.75 0.01 86 10 0.01 50 0.75 0.001 147

5 0.01 050 0.75 0.01 140 11 0.01 50 0.75 0.010 140

6 0.01 100 0.75 0.01 031

94%

12 0.01 50 0.75 0. 10 093

46%

To provide a sample calculation, the % of error variation for changes of is calculated as 𝑚1

following,

% of error variation for changes of =𝑚1

1 1

12

1 100, 7,8,9max( ) min( )

ii

j jm m

CostFunctionj

CostFunction CostFunction

(7)

The results indicate that the variation of and have more significant effects on the changes of 𝑚1 𝐾𝑠

the system output; thereafter, the parameters are categorized into two sets (one include and , 𝑚1 𝐾𝑠

and the other includes the rest two) and each set of parameters are identified in a separate phase.

4.3.2. Taguchi- based identification, and 𝐦𝟏 𝐊𝐬

The required simulations for the purpose of identification are designed by L9 Taguchi orthogonal array. L9 is an appropriate Taguchi array for a process with two factors each including three levels. Therefore, a total number of 9 simulations have to be done at each identification iteration. Usually, not more than four to five iterations are required while using Taguchi designs. The initial levels for and are selected similarly to the ones reported in Table (1). Table (3) shows the L9 𝑚1 𝐾𝑠

orthogonal array and also the costs and SNRs calculated for each simulation.

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“The-smaller-the-better” characteristic is used to calculate the SNRs. Then, the sum of SNRs at each level and for each parameter is calculated as illustrated in Figure (8). It is always of interest to have a larger SNR, then the correspondent levels to the SNR with the highest ratio is selected. In the case that there are two simulations with close SNRs. For example, for the problem in this study, the authors first continued with the levels in simulation number 7, however, since the results did not meet the expectations, the simulation reconducted based on the levels of simulation number 4. The selected levels for iterations 2 and 3 are shown in Table (4). SNRs and costs also reported in Table (5), and finally Figures (10) and (11) show the main effect plots, and the actual and estimated responses, respectively.

Table 3: L9 Orthogonal array and costs for 1st iteration

Levels Step 1# of Run 𝐾𝑠 𝑚1 CF SNR

1 1 1 1993 -105.92 1 2 3251 -90.203 1 3 2477 -87.874 2 1 5380 -74.605 2 2 9200 -79.276 2 3 1430 -83.107 3 1 4930 -73.808 3 2 5620 -74.909 3 3 5560 -74.90

Table 4: Parameter levels for 2nd and 3rd iterations based on run 7

Step 2- Levels Step 3-LevelParameter 1 2 3 Parameter 1 2 3

𝐾𝑠 80 100 120 𝐾𝑠 110 120 130

𝑚1 0.3 0.5 0.7 𝑚1 0.2 0.3 0.4

Table 5: CF values and SNRs for 2nd and 3rd iterations based on run 7

Levels Step 2 Step 3# of Run 𝐾𝑠 𝑚1 CF SNR CF SNR1 1 1 4631 -73.3 4486 -73.02 1 2 5088 -74.1 4588 -73.23 1 3 5533 -74.8 4721 -73.44 2 1 4581 -73.2 4466 -72.95 2 2 4927 -73.8 4540 -73.16 2 3 5213 -74.3 4667 -73.37 3 1 4540 -73.1 4470 -73.08 3 2 4766 -73.5 4541 -73.19 3 3 4983 -93.9 4623 -73.2

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Figure 10: Main effects plot for SNRs

0 1 2 3 4 5 6 7 8 9 10Time(Sec)

-300

-200

-100

0

100

200

300

Out

put (

N)

Estimated outputActual output

Figure 11: The actual and estimated outputs in 3rd iteration

Since the amount of CF is still large, the simulations reconducted based on the levels obtained from run 4 in Table (3). Tables (6) and (7) and Figure (11) show the same values and graphs as Tables (2) and (3) and Figure (10) for the reconducted simulations based on the values in run 4.

Table 6: Parameter levels for 2nd and 3rd iterations based on run 4

Step 2- Levels Step 3-LevelParameter 1 2 3 Parameter 1 2 3

𝐾𝑠 40 50 60 𝐾𝑠 45.0 50.0 55.0

𝑚1 0.4 0.5 0.6 𝑚1 0.35 0.40 0.50

Table 7: CF values and SNRs for 2nd and 3rd iterations based on run 4

Levels Step 2 Step 3# of Run 𝐾𝑠 𝑚1 CF SNR CF SNR1 1 1 9025 -79.1 5992 -75.52 1 2 1298 -82.2 8733 -78.83 1 3 1808 -85.1 9822 -79.84 2 1 0266 -48.4 5379 -74.65 2 2 9200 -79.2 0266 -48.4

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6 2 3 0121 -41.6 7206 -77.17 3 1 5438 -74.7 5529 -74.88 3 2 4717 -73.4 7313 -77.29 3 3 7251 -77.2 6946 -76.8

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Figure 12: Main effects plot for SNRs

As illustrated in Table (8), the cost function value is dramatically decreased. Figure (13) shows the actual and estimated force responses.

0 1 2 3 4 5 6 7 8 9 10Time (Sec)

-300

-200

-100

0

100

200

300

Out

put (

N)

Estimated output

Actual output

Figure 13: The actual and estimated outputs in 3rd iteration

4.3.3. Taguchi- based identification, and 𝐂𝐝 𝐂𝟏

Similar to the previous section, identification of and has been done here. Tables (8) and (9) 𝐶𝑑 𝐶1

show the parameter levels and CFs and SNRs, respectively.

Table 8: Parameter levels

Step 1 Step 2Levels Levels

Parameter 1 2 3 1 2 3

𝐶𝑑 0.001 0.01 0.1 0.008 0.010 0.030

𝐶1 0.001 0.01 0.1 0.000 0.001 0.005

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Table 9: CF values and SNRs

Levels Step 1 Step 2# of Run 𝐶𝑑 𝐶1 CF SNR CF SNR

1 1 1 0271 -48.65 262 -48.362 1 2 0264 -48.43 262 -48.363 1 3 0622 -55.87 261 -48.334 2 1 0261 -48.33 261 -48.335 2 2 0266 -48.49 261 -48.336 2 3 0675 -56.58 262 -48.367 3 1 0602 -55.59 278 -48.888 3 2 0657 -56.35 281 -48.979 3 3 1220 -61.72 291 -49.27

In addition, the main effect plots and the estimated values are also shown in Figures (14) and (15).

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321C1

Dat

a

Cd

Panel variable: C13

Levels

Figure 14: Main effects plot for SNRs

0 1 2 3 4 5 6 7 8 9 10Time (Sec)

-300

-200

-100

0

100

200

300

Out

put (

N)

Estimated outputActual output

Figure 15: The actual and estimated outputs in 3rd iteration

The results show that after two steps of identification, the cost function value is improved from 266 to 261. Although the identification could be continued for a better value, the estimated response in Figure (15) seems to be accurate enough.

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4.4. Taguchi-based identification, 2nd phase

In the second phase of identification, parameters and need to be identified. The 𝑚2 𝐶2

identification process is similar to section 4.4. Tables (8) and (9) show the parameter levels, CFs and SNRs. Also Figures (16) illustrates the main effect plots.

Table 10. Factor levels for phase 2 of the identification, 𝑚2‚ 𝐶2

Step 1 Step 2 Step 3Levels Levels Levels

Parameter 1 2 3 1 2 3 1 2 3

𝑚2 0.050 0.10 0.20 0.200 0.300 0.400 0.150 0.200 0.250

𝐶2 0.001 0.01 0.05 0.001 0.001 0.001 0.001 0.001 0.001

Table 11. CF values and SNRs

Levels Step 1 Step 2 Step 3# of Run 𝑚2 𝐶2 CF SNR CF SNR CF SNR

1 1 1 1.3078 -2.3308 1.2385 -1.8579 1.250 -1.9382 1 2 1.3077 -2.3301 1.2383 -1.8565 1.250 -1.9383 1 3 1.3071 -2.3261 1.2380 -1.8544 1.250 -1.9384 2 1 1.2739 -2.1027 1.2485 -1.9277 1.231 -1.7985 2 2 1.2738 -2.1020 1.2482 -1.9256 1.231 -1.7986 2 3 1.2731 -2.0972 1.2479 -1.9236 1.231 -1.7987 3 1 1.2395 -1.8649 1.3031 -2.2995 1.231 -1.7988 3 2 1.2394 -1.8642 1.3027 -2.2968 1.231 -1.7989 3 3 1.2385 -1.8579 1.3023 -2.2942 1.231 -1.798

321

-1.8

-1.9

-2.0

-2.1

-2.2

-2.3

-2.4

321c2

Dat

a

m2

Panel variable: C10

Levels

Figure 16. Main effects plot for SNRs

The parameters and obtained as 200 gr and 0.001. The estimated response is shown in 𝑚2 𝐶2

Figure (17).

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0 1 2 3 4 5 6 7 8 9 10Time (Sec)

-10

-5

0

5

10

Out

put (

N)

Estimated dataActual Output

Figure 16: The actual and estimated outputs in 3rd iteration

4.5. System results in presence of both inputs

Finally, the estimated system output where it is fed by both force and motion trajectories are obtained as following,

0 1 2 3 4 5 6 7 8 9 10Time (Sec)

-300

-200

-100

0

100

200

300

Out

put (

N)

Estimated outputActual output

Figure 18: The actual and estimated outputs in presence of both of the inputs

The cost value for the final result is 301.

5. Conclusion In this paper, the authors developed their recently introduced Taguchi-based identification method. The approach is a generally applicable method that can accurately identify both linear and nonlinear systems, and is not limited to identification of only SISO systems. This paper expanded the previously presented method to MISO systems and also discussed modifications required for identification of MIMO ones. The paper firstly modified the Taguchi-based identification method by adding one important step of contribution analysis. The analysis is utilized to categorize the unknown parameters into a necessary number of classes such that the identification process could be conducted in multiple steps starting from identifying the group of parameters with more contribution in building the output. The efficacy of contribution analysis in reducing the

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identification time and resulting in more accurate estimation was also discussed. For the case-study in this paper, unknown parameters in the sub-system of were grouped into two classes, 𝐹𝑜 𝑡𝑜 𝐹𝑚

and (as main contributor parameters) in one class, and and in the other one. Also, 𝐾𝑠 𝑚1 𝐶𝑑 𝐶1

unknown parameters in the second subsystem were identified together. In addition, it was discussed that in a MISO system, contribution analysis is promising to identify which subsystem should be identified first. Identification should be started with identifying the subsystem with more contribution in building the output response of the overall system. Also, the way the algorithm could be used for identifying a MIMO system was discussed. Moreover, within the simulations and implementation of Taguchi method, it was shown that on the contrary to what Taguchi method suggest on selecting the levels corresponding to largest SNR value at each iteration, there are cases that such levels will not result in accurate identification of parameters. In such situations (as happened in this paper) the problem could be solved by reconduction of the simulations using the levels correspondent to the next larger SNR value. The case study for the tests was a custom made linear series elastic actuator. The results clearly indicated the effectiveness of the method and somehow involving the human intellect in the process of system identification.

6. AcknowledgementsThis study was financially supported by Ferdowsi University of Mashhad (Grant No21786 ).

7. References

A simple and generally applicable system identification algorithm based

on Taguchi method.

Enables identifying a variety of systems including linear and nonlinear

systems, mechanical or economic systems, etc.

Effectively increase the model accuracy while significantly reduce the time

required for the identification.

Experimental results proved the efficacy of the method proposed.

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