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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.17 No.1 pp.62-71
DOI : https://doi.org/10.7232/iems.2018.17.1.062

Optimization of Super Twisting Sliding Mode Control Gains using Taguchi Method

Chiew Tsung Heng, Zamberi Jamaludin*, Ahmad Yusairi Bani Hashim, Nur Aidawaty Rafan, Lokman Abdullah
Department of Robotics and Automation, Faculty of Manufacturing Engineering, Universiti Teknikal Malaysia Melaka, Malaysia
Corresponding Author, zamberi@utem.edu.my
June 23, 2016 December 31, 2016 January 31, 2018

ABSTRACT


This paper focuses on optimization of super twisting controller gains using Taguchi method with objective to minimize tracking error and the chattering effect. Two gain parameters in super twisting algorithm, that is L and W were identified as two factors with three levels respectively. The optimization method applied a L9 orthogonal array and the performance index used was root mean square of tracking error and Fast Fourier Transform of control inputs. The optimized super twisting controller with traditional sliding surface and the continuous control action laws was validated on a single axis direct driven linear motor. Analyses of variance and main effect plots were performed on the effect of gains variation on performance index. Values of L and W were chosen as 0.00002 and 0.08 respectively and were confirmed through confirmation test based on calculated confidence interval. Experimental results with 95% confidence level identified gain L as the significant factor in minimizing chattering effect while both gains L and W were responsible in minimizing tracking error in optimum condition. Optimized algorithm achieved 9.3% of reduction in root mean square of tracking error and 38.4% of reduction in chattering experimentally.



초록


    1. INTRODUCTION

    High demand on accuracy and precision in machine tool applications associated with high degree of development in technologies induced great challenge towards control engineers and machine tools developer. A comprehensive controller design is desired to ensure good performances in accuracy and precision of machine tool applications. A robust nonlinear controller, known as sliding mode control (SMC) was introduced by Utkin (1977) to meet this high demand as classical controllers such as cascade controllers are no longer able to meet. SMC was attractive due to its high disturbance rejection and robustness properties that enforced good tracking performances on motion drive system. A geometrical locus with boundaries known as sliding surface is defined as switching function in SMC where the desired response is achieved when the system is slides on the surface. Control laws which typically consist of signum function are set to attract system states (error) towards the sliding surface and to ensure the states slides along the surface. However, the SMC comes with a critical drawback known as chattering effect which might reduce the life-span of drive system due to the high frequency oscillation occurred in signum function. In fact, chattering is an undesired phenomenon especially for machine tool applications such as computer numerical control (CNC) machine that commonly used in manufacturing field. The occurrence of chattering effect greatly affects the performances of drive system in terms of accuracy and precision as well as the outcome of production. Fortunately, higher order sliding mode control (HOSMC), an extension of SMC, was introduced in 1980s to address the issue of chattering effect normally associated with traditional sliding mode control (SMC) as well as to enhance robustness and disturbance rejection properties (Fridman et al., 2011; Furat and Eker, 2014). Specifically, second order sliding mode control (SOSMC), a branch of HOSMC was popularized by Levant (1993) and Bartolini et al. (1998) as a solution to counteract chattering effect. Among the SOSMC, super twisting sliding mode control (ST-SMC) has high popularity due to its simple concept in application and its ability to suppress chattering effect while maintaining the performance level of the systems in terms of accuracy and precision (Levant, 2007; Liu and Han, 2014). Unlike the traditional chattering suppression approaches such as boundary layer method (Lee and Utkin, 2007), a continuous control action is enforced in ST-SMC to suppress chattering due to the discontinuous input (Utkin, 2013).

    Many applications of ST-SMC had been recorded and demonstrated in literature. Most works were published by the HOSMC founder himself (Levant, 2007; 2001; 2011; Levant and Michael, 2009; Bartolini et al., 1998; Bartolini et al., 2009; Bartolini and Punta, 2000; Bartolini et al., 1997), who popularized the concept of sub-optimal algorithm. ST-SMC was applied in wide range of applications including aerospace (Levant et al., 2000), marine engineering (Ismail and Putranti, 2015), diesel engines (Khan et al., 2003), power system (Valenciaga and Puleston, 2008) and robotics system (Merheb et al., 2015). In addition, a survey of SOSMC on mechanical system (Bartolini et al., 2003) was performed in 2003 with main focus on development of twisting and suboptimal algorithms. The application of ST-SMC as an observer in mechanical system was only then published by Davila et al. (2005). Other research activities related to ST-SMC are focused on finite time convergence and stability. These include the work on Lyapunov function (Moreno and Osorio, 2012), time domain analysis (Utkin, 2013) and point-to-point method for state trajectories (Davila et al., 2005; Levant, 2001).

    To-date, design and applications of ST-SMC in machine tools area are scarcely reported. Also, to the best knowledge of the authors, analysis and optimization of the gain parameters of ST-SMC have not been extensively investigated in literature. Only limited number of tuning methods were documented for design of the gain parameters such as the uncommon bio-inspired optimization technique, Ecological System Algorithm (ESA) by Merheb et al. (2015), method involving complex calculation and algebraic manipulation (Valenciaga and Puleston, 2008), and a heuristic-like tuning method proposed by Khan et al. (2003). However, there is no record on design methodology of ST-SMC control law gain parameters that strike a balance between requirement on tracking control performance and chattering suppression. Therefore, there exists a need for a systematic guideline on best design strategy that form a trade-off between precision control and stability.

    For optimization purposes, one attractive systematic approach commonly applied in industry is the Taguchi method. It is an extension of the traditional design of experiment (DOE) method using less number of experiments or trials for the same number of parameters (Vlachogiannis and Roy, 2005; Mori, 2011). It is also a popular optimization approach due to its time efficient in experiment compared to other DOE approach such as full factorial approach (Roy, 2001; Dewantoro, 2015). However, literature reviews have showed that Taguchi method was primarily applied for design of classical controllers in control engineering field with limited records observed for its application in synthesis of advanced controllers (Dewantoro, 2015; Vlachogiannis and Roy, 2005; Hasanien and Muyeen, 2013). Classical controllers that are often PID-based, are more commonly applied in industrial applications compared to advanced controllers; thus limiting potential application of Taguchi method as supporting tool in synthesis of advanced controllers for machine tools application in the industry. In view of the strength of ST-SMC, there exists an urgent need for active industrial application of ST-SMC designed using Taguchi method for optimization of its control law gain parameters. This systematic methodology could serve as a supplement in knowledge regarding design of ST-SMC control gain parameters that would enhance the applicability of STSMC in machine tool applications.

    In this paper, a ST-SMC was designed and implemented on a direct drive single axis positioning system to promote the application of ST-SMC on machine tools which was limitedly found in literature. ST-SMC is beneficial to be applied on machine tool applications as it provides high robustness and suppression of chattering effect normally associated with traditional SMC. In addition, in this paper, the gain parameters of ST-SMC was optimized using well-known Taguchi method and validated experimentally to improve the performance of controller and the overall system, that is, to minimize chattering as well as the tracking error. Furthermore, significant factors (gain parameters) that affect tracking error reduction and chattering effect were identified through analysis of variance using Taguchi method.

    This paper is organized as follows. Section 2 presents the system identification of the single axis direct driven positioning system and the experimental setup for the overall experiment. Transfer function represented by the positioning system was determined through system identification process prior to the design of controller. Section 3 shows the design steps for the ST-SMC while Section 4 describes the optimization method using Taguchi method. Section 5 discusses on experimental results obtained between optimized ST-SMC and nonoptimized ST-SMC using single axis positioning system. Lastly, Section 5 concludes the findings with future recommendations.

    2. SYSTEM IDENTIFICATION

    The experimental setup applied is a direct driven single axis positioning system equipped with a 4 μm-resolution linear encoder. A dSPACE DS 1104 controller board is used as the data acquisition unit that interfaced the host computer with the single axis positioning system. The host computer is equipped with MATLAB/SIMULINK and ControlDesk software for data monitoring and control design purposes. The single axis positioning system is connected to the amplifier that linked to the data acquisition unit. Figure 1 shows the overall schematic diagram of the experimental setup.

    The frequency domain identification method (Pintelon and Schoukens, 2012) was used to estimate the dynamics behavior of the single input single output (SISO) model. The SISO frequency response function (FRF) was then obtained using the H1 estimator based on the measured input voltage and output position with band-limited white noise as the excitation signal at sampling frequency of 5000 Hz. The measured FRF was then fitted by a parametric model using nonlinear least square frequency domain method (Pintelon and Schoukens, 2012) to yield a second order model with time delay, Td of 0.00045 seconds as shown in Equation (1). The time domain representation of the system is given by Equation (2).

    Y ( s ) U ( s ) = A s ( s + B ) e s T d ,
    (1)
    y ¨ ( t ) = B y ˙ ( t ) + A u ( t ) ,
    (2)

    where A = 7.5 × 108 μm/V·s2, and B = 3622 s-1. U(s) and Y(s) are the control input voltage in volt, and the output position in micrometer respectively.

    3. CONTROL STRUCTURE OF SUPER TWISTING ALGORITHM

    Both ST-SMC and the traditional SMC share similar control structure. There are two main components to be considered in design of super twisting algorithm, namely; the switching function and the control laws (Utkin et al., 2009; Edwards and Spurgeon, 1998).

    3.1 Switching Function

    The traditional sliding surface, s(t) was used as the switching function as shown in Equation (3) and (4). It relates the tracking error, e(t) and the first derivative of tracking error, ė(t) with positive constant λ.

    s ( t ) = ( λ + d d t ) n 1 e ( t ) ,
    (3)
    e ( t ) = y ( t ) r ( t ) ,
    (4)

    where n represents the order of the uncontrolled system while both r(t) and y(t) are the desired position and output position respectively. By using n = 2, both second order derivatives of tracking error, ё(t) and first order derivative of the sliding surface, (t) are obtained via simple derivations as shown in Equations (5) and (6) respectively.

    e ¨ ( t ) = y ¨ ( t ) r ¨ ( t ) ,
    (5)
    s ˙ ( t ) = λ e ˙ ( t ) + e ¨ ( t ) ,
    (6)

    By substituting Equations (2) and (5) into Equation (6), the outcome was as shown in Equation (7).

    s ˙ ( t ) = λ e ˙ ( t ) + A u ( t ) B y ˙ ( t ) r ¨ ( t ) .
    (7)

    3.2 Control Laws

    The control laws of ST-SMC consist of three parts, namely; the equivalent control, a continuous state function, and an integrated discontinuous input, u1(t) as shown in Equations (8) and (9) (Fridman et al., 2011). The continuous state function and the integration of discontinuous input mark the significant differences between ST-SMC and traditional SMC. The two additional functions contribute in providing additional chattering suppressing effect; a factor that is lacking in SMC.

    u ( t ) = u e q ( t ) L | s ( t ) | 0.5 s i g n ( s ( t ) ) + u 1 ( t ) ,
    (8)
    u ˙ 1 ( t ) = W s i g n ( s ( t ) ) ,
    (9)

    where both L and W are tunable positive gains. When (t) = 0, the equivalent control, ueq(t) denoted in Equation (10) was formed based on Equation (7).

    u e q ( t ) = 1 A ( B y ˙ ( t ) + r ¨ ( t ) λ e ˙ ( t ) ) .
    (10)

    Figure 2 shows the general control scheme of the ST-SMC. In the absence of a speed sensor, a Kalman- Bucy filter was designed to estimate the velocity signal from the measured output position signal. This approach has the advantage in system noise reduction (Stengel, 1986). Stability was ensured using Lyapunov stability criterion as demonstrated extensively in (Moreno and Osorio, 2012; Polyakov and Poznyak, 2009).

    4. PARAMETERS OPTIMIZATION USING THE TAGUCHI METHOD

    Taguchi method is a systematic approach for optimization problem and it was used to optimize the two gain parameters of designed ST-SMC in control laws; that are gains L and W (refer to Equations (8) and (9)). This is because the performance of the controller was mainly contributed by these two parameters (Fridman et al., 2011). Moreover, the chattering phenomenon in ST-SMC was mainly due to the imperfection during switching originating from the signum function in the control laws (Utkin, 2013). This then becomes the primary motivation for optimal selection of gains L and W of the control laws. The major steps of Taguchi method (Hasanien and Muyeen, 2013; Sorgdrager et al., 2015; Dewantoro, 2015) are listed below:

    • (i) The determination of orthogonal array based on number of factors and levels.

    • (ii) Conduct of experiments.

    • (iii) Data transformation using signal-to-noise ratio.

    • (iv) Analysis of results based on main effect plot and analysis of variance.

    • (v) Confirmation test.

    An L9 orthogonal array was considered for the two factors (L and W), with three levels based on degree of freedom approach (Roy, 2001; Vlachogiannis and Roy, 2005). The L9 orthogonal array was deemed sufficient for the scope of this work as reviews (Vlachogiannis and Roy, 2005; Taguchi et al., 2005) have shown that it is sufficient for the application on system with a number of factors up to four and number of levels up to three (34). The upper and lower limits of both gain parameters were obtained based on heuristic method (Khan et al., 2003). Initially, both gain L and W was set to a small value. Then, the tuning was initiated by increasing W gradually until the occurrence of chattering that limits the system performance. The value obtained before the deterioration of performance was used as the upper limit. The lower limit value, on the other hand, was obtained when the chattering was started. The same method was then used to identify the upper and lower limits of L. Table 1 shows the identified values of three levels for both control factors while Table 2 shows the orthogonal array of L9.

    A total of 9 experiments were conducted based on L9 orthogonal array as for any pairs of columns, all combinations of factor levels occur, and occur with equal number of times (Roy, 2001). In these 9 experiments, all combination of factors and levels was covered as there are only two tunable factors (L and W) in control laws. Two performance indexes were considered for the effect of parameters variation, namely; the root mean square of tracking error (RMSE) and the chattering amplitude at 2000 Hz. The chattering amplitude at 2000 Hz was used as it showed a significant spike in the spectrum analysis (refer to Figure 6). The RMSE was obtained from the measured tracking error (measured values - reference values) while the chattering amplitude was obtained using the Fast Fourier Transform (FFT) of the measured input voltage to the system (signal u in Figure 2). The quality characteristic of smaller-the-better was chosen to evaluate the performance indexes for each parameter combinations. This is due to the sole objective of the controller design that is to minimize tracking error and reducing the chattering effect. The signal-to-noise (S/N) ratio was shown in Equation (11):

    S / N = 10 log ( 1 q i = 1 q p i 2 ) .
    (11)

    where q is the total number of measurements and pi is the measurement of performance indexes. Table 3 tabulates results of the S/N ratio results for each experiment performed using single axis positioning system mentioned in Section 2 and control scheme as shown in Figure 2 with a sinusoidal reference input of amplitude 20,000 μm and frequency 0.5 Hz.

    4.1 Analysis of the Main Effect

    The main effect analysis was conducted in order to identify the effective level of each control factors towards the performance indexes (Mori, 2011). The factor level average, Fi or the mean S/N ratio of each level of factor was calculated using Equation (12) and the results are tabulated in Table 4.

    F i = 1 m j = 1 m v i j ,
    (12)

    where m is the number of experiment with level i and vij is the S/N ratio of each experiment with level i. Figure 3 and Figure 4 showed the graphical representative of results tabulated in Table 4 that are the main effect plots.

    Results in Table 4, Figure 3 and Figure 4 showed that L1 and W3 resulted in highest S/N ratio for both RMSE and chattering amplitude. Therefore, both L1 and W3 are identified as the most suitable candidates for optimum output. The main effect plot for RMSE also showed that both L and W produced similar S/N ratio for the RMSE response. However, it was observed that gain L has slightly stronger effect compared to W due to the facts that gain L having a wider range of factor level average value than W. In contrast, L acted as the main contributor in chattering amplitude response as its range of factor level average value was much wider compared to the range of factor level average value of W as shown in Figure 4. It was also clearly observed that gain W was not a significant factor for chattering amplitude. The results also displayed that low value of gain L and high value of gain W minimized the RMSE as they produced high S/N as shown in Figure 3. Similarly, low value of L is preferable to minimize the chattering amplitude as suggested in Figure 4. The following Section (4.2) looks into the analysis of variance that decides the significant factor for each performance indexes.

    4.2 Analysis of Variance (ANOVA)

    The analysis of variance (ANOVA) (Montgomery, 2012) for S/N ratio was performed using MINITAB software at 95% confidence level to identify the statistical significance of the parameters. The generated ANOVA table is shown in Tables 5 and 6.

    EQ

    The degree of freedom of a factor, DOF = N - 1, where N is the total number of levels. Equation (13) denotes the sum of squared of a factor SS through the mathematical manipulation on Equation (12) and the grand average of respective S/N ratioT .

    S S = i = 1 k [ m ( F i T ¯ ) 2 ] ;
    (13)

    where k is the total number of levels for a factor. The mean of SS, represented by MSS was calculated by dividing the SS with DOF, as shown in Equation (14).

    M S S = S S / D O F ;
    (14)

    After obtaining the MSS, the F-ratio of a factor was calculated using Equation (15).

    F r a t i o = S S f a c t o r / M S S e ;
    (15)

    The percentage of contribution of a factor, PC in the last column of ANOVA table can be calculated using Equation (16) after considering Equations (13) and (14).

    P C = S S f a c t o r D O F f a c t o r × M S S e S S T o t a l × 100 % ;
    (16)

    According to Table 5, both L and W contributed to the RMSE responses with 39.95% and 24.46% respectively. Result of ANOVA table was in line with the results presented previously in Figure 3 where both gains contributed to the RMSE response. On the other hand, result in Table 6 showed that the main factor that affected the chattering amplitude was factor L and which is consistent with the results of the main effect plot (refer to Figure 4).

    Based on analyses on main effect plots and ANOVA, appropriate tuning on both gains L and W were required to produce low RMSE values. For the purpose of low chattering amplitude, gain L was evidently the significant factor to be considered in the tuning process. Results from main effect plot and ANOVA suggested the use of gain L1 and W3 for optimal performance on RMSE reduction and chattering attenuation.

    In further analysis, the expected mean of performances, S / N in optimum condition was derived using Equation (17) with the Fi of significant factors and grand average S/N ratio of respective responses.

    S / N = T ¯ + ( L 1 T ¯ ) + ( W 3 T ¯ ) ;
    (17)

    Next, the confidence interval, CI was obtained using Equations (18) and (19) by considering number of measured results neff. Results were summarized and tabulated in Table 7.

    C I = F ( 1 , α , D O F e ) M S S e n e f f ;
    (18)
    n e f f = n 1 + D O F f ;
    (19)

    where F(1, α, DOFe) is the F-ratio of risk α which could be obtained from F-distribution table (Roy, 2001), α is the risk, and 1-α is the confidence level. DOFe indicates the degree of freedom for error while DOFf represents the degree of freedom of all factors included in estimating the mean performance at optimum condition.

    The confirmation of results is proven when the responses in RMSE and chattering amplitude with optimum control factors lies between the expected range (Mori, 2011; Vlachogiannis and Roy, 2005). The obtained experimental results of RMSE were 2.0312 μm or -6.1549 dB whereas chattering amplitude obtained are 1.5667 × 10-6 or 116.1005 dB in conjunction with the optimized gains L = 2 × 10-5 and W = 0.08 (refer to Table 3). These results showed that the optimized gains values were confirmed as the obtained results fall within the expected range.

    5. EXPERIMENTAL TESTING AND RESULTS

    In order to show the effectiveness of optimization on ST-SMC, experiment was performed to compare the RMSE and chattering responses on single axis positioning system using optimized ST-SMC (gains L = 2 × 10-5 and W = 0.08) and non-optimized ST-SMC (gains L = 3 × 10-5 and W = 0.06). The experiment was conducted based on considered setup shown in Figure 1 and control scheme shown in Figure 2. A sinusoidal signal of amplitude 20 000 μm and frequency of 0.5 Hz was inserted as reference signal, r(t) to the test setup. The measured tracking error e(t), input voltage signal u, and reference signal r(t) were captured and recorded. Figure 5 shows the results of reference signal and corresponding measured tracking error while Figure 6 shows the chattering amplitude at 2000 Hz obtained through the FFT of input voltage u.

    Based on Figure 5, it was clearly observed that the overall tracking error of optimized ST-SMC (Figure 5b) was lower compared to non-optimized ST-SMC. This showed the effectiveness of optimized ST-SMC in better tracking error reduction where less noise was found in the measured tracking error. Quantitatively, the RMSE of non-optimized ST-SMC was reduced from 2.2403 μm to 2.0312 μm after the optimization process using Taguchi method, which is an improvement of 9.3%. On the other hand, the peak amplitude of non-optimized ST-SMC was greatly reduced after the optimization process, that is from 2.6×10-6 V (Figure 6a) to 1.6×10-6 V (Figure 6b), an improvement of 38.4%. The great chattering attenuation was achieved through optimization of gain parameters and this contributed positively in terms of tracking error reduction by reducing noises in overall system. These results showed the superiority of optimized STSMC against non-optimized ST-SMC in terms of RMSE reduction and chattering attenuation, thus, validating the optimization process performed. Table 8 summarizes the results of RMSE and peak amplitude from FFT of input voltage signal.

    6. CONCLUSION AND FUTURE RECOMMENDATIONS

    This paper demonstrates design and implementation of ST-SMC on a direct drive single axis positioning system. The control laws gain parameters of ST-SMC were optimized using Taguchi method and validated experimentally. The optimized ST-SMC produced a control performance that is a trade-off between precise tracking and suppression of chattering commonly associated with SMC application. This promotes the applicability of STSMC for machine tool applications. Furthermore, significant factors that affect tracking error reduction and chattering were identified using analysis of variance based on the Taguchi method applied, thus provides an insight in design of the gains parameters of ST-SMC that meet desired requirement of users. This also serves as a systematic design methodology of ST-SMC supported with proper statistical analysis such as ANOVA and main effect plots. The main effect plots and ANOVA results both showed that gains L and W both affected the RMSE values of the measured tracking errors of the system while only gain L significantly affected the response to chattering amplitude. Both gains are considered in controller tuning process to minimize RMSE while gain L was prioritized for chattering attenuation. Conclusively, the level 1 of L (0.00002) and level 3 of W (0.08) were selected as these controller parameters results in optimum RMSE values and minimum chattering amplitude. Confirmation test with 95% confidence level had been carried out and proven the reproducibility. An improvement of 9.3% in RMSE reduction and 38.4% in chattering reduction were achieved by optimized ST-SMC after the optimization process on gains. However, prior knowledge in ST-SMC was required to identify the factors and there was no reduction of the number of experiment as the number of factors is 2 in this case. In the future, extensions on the ST-SMC can be explored as the significant factor, that is, gains for respective responses are identified. Furthermore, the effect of other factors such as sliding surface can be considered for chattering suppression especially for the case of RMSE response.

    ACKNOWLEDGEMENT

    The authors would like to acknowledge the Faculty of Manufacturing Engineering, Universiti Teknikal Malaysia Melaka for the facilities and laboratories support provided. This research work is financially supported by the Ministry of Higher Education, Malaysia with reference number FRGS/1/2015/TK03/FKP/02/F00281.

    Figure

    IEMS-17-62_F1.gif

    Schematic diagram of the overall experimental setup.

    IEMS-17-62_F2.gif

    Control scheme of ST-SMC.

    IEMS-17-62_F3.gif

    Plot of the main effect for RMSE response.

    IEMS-17-62_F4.gif

    Plot of the main effect for chattering amplitude response.

    IEMS-17-62_F5.gif

    Reference input signal with corresponding measured tracking error for (a) non-optimized ST-SMC, and (b) optimized ST-SMC.

    IEMS-17-62_F6.gif

    Chattering amplitude at 2000 Hz obtained through FFT of control input for (a) non-optimized ST-SMC, and (b) optimized ST-SMC.

    Table

    Control factors of ST-SMC with respective ranges and values at three levels

    The orthogonal array of L9.

    S/N ratio for RMSE and chattering amplitude responses

    Factor level average for RMSE and chattering amplitude responses

    ANOVA table for RMSE response

    ANOVA table for chattering amplitude response

    Summary of calculated parameters, expected mean performances and confidence intervals at optimum condition for both RMSE and chattering amplitude responses

    Summary on results of RMSE and peak amplitude from FFT of input voltage signal for optimized and non-optimized ST-SMC

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