Design and application of discrete wavelet packet transform based multiresolution controller for liquid level system

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Research article

Design and application of discrete wavelet packet transform based multiresolution controller for liquid level system Rimi Paul a,n, Anindita Sengupta b a b

EE department, Aliah University, New Town, Action Area II, A/27, Kolkata 700156, West Bengal, India EE department, Indian Institute of Engineering Science and Technology, Shibpur, Botanic Garden, Howrah 711103, West Bengal, India

art ic l e i nf o

a b s t r a c t

Article history: Received 24 August 2015 Received in revised form 19 May 2017 Accepted 31 July 2017

A new controller based on discrete wavelet packet transform (DWPT) for liquid level system (LLS) has been presented here. This controller generates control signal using node coefficients of the error signal which interprets many implicit phenomena such as process dynamics, measurement noise and effect of external disturbances. Through simulation results on LLS problem, this controller is shown to perform faster than both the discrete wavelet transform based controller and conventional proportional integral controller. Also, it is more efficient in terms of its ability to provide better noise rejection. To overcome the wind up phenomenon by considering the saturation due to presence of actuator, anti-wind up technique is applied to the conventional PI controller and compared to the wavelet packet transform based controller. In this case also, packet controller is found better than the other ones. This similar work has been extended for analogous first order RC plant as well as second order plant also. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Liquid level system Proportional-Integral controller Minimum description length Discrete wavelet transform based PI controller Discrete wavelet packet transform based PI controller Anti-wind up PI controller

1. Introduction The applications of wavelet filter bank have been well established in processing signals of diverse applications. The wavelet analysis accommodates non-uniform bandwidths to make it possible to implement through different levels of decimation in filter bank [1]. A discrete wavelet analysis, an asymmetric decomposition method, decomposes a signal into high frequency demesne (contingent signal) and low frequency demesne (average signal). The low frequency demesne is then decomposed into second level approximation and detailed signal and the process are iterated. Discrete Wavelet Packet Transform (DWPT) method, a symmetric extension of filter bank technique, is an inscrutable wavelet analysis that offers more detail information for signal analysis [2]. Wavelet packets are the linear combination of wavelets which form bases that retain many of the orthogonality, smoothness and localization attributes of their parent wavelets. The performance of WPT is appreciable while comparing with the DWT decomposition technique since WPT analysis can provide more precise frequency resolution than the other counter parts of wavelet analysis. The potency of WPT provides simultaneous time-frequency n

Corresponding author. E-mail addresses: [email protected] (R. Paul), [email protected] (A. Sengupta).

representation of non-stationary and non-periodic signals through its decomposed coefficients to focus on short time intervals for high frequency components and longtime intervals for low frequency components [2,3]. Several researchers have already applied packet transform for signal analysis and denoising [4,5]. Paul et al. applied packet transform to remove noise from the output response of a liquid level system [6]. It is seen that the application of packet transform partially helps to improve the performance of the control scheme also. The frequency analysis can be taken as an effective technique to analyze and classify signals with complex characteristics. Here, minute frequency analysis ability of packet transform has been applied directly to generate the control signal. Most of the process industries involved with liquids [6–8] contains proportional integral controller. An analytical design method for a conventional proportional-integral (PI) controller is developed for an optimal control of the liquid level loop to explicitly handle the control specifications [9]. Boonsrimuang et al. presented design methodology of auto-adjustable PI controller using MRAC technique in [10]. The design of PI and PID controllers involve the tuning of controller gains. The Ziegler– Nichols (Z-N) tuning formula is most well-known tuning technique for PI and PID controllers [11,12]. A relay tuning feedback method, popularized in 1987, also known as auto tune variation (ATV) method has been adopted with adaptive PID controller for identification with a recursive parameter estimation of second order system with delay

http://dx.doi.org/10.1016/j.isatra.2017.07.030 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Paul R, Sengupta A. Design and application of discrete wavelet packet transform based multiresolution controller for liquid level system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.030i

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model [13]. The guidelines for PI and PID tuning of SISO processes are provided. Though many tuning methods have been proposed for PID controller by several researchers, so far, no scheme has replaced the simple Z-N tuning rules in terms of familiarity and satisfaction of performances. The conventional fixed-gain PI and PID controllers are quite sensitive to step change of command speed, parameter variations and load disturbances [14]. Therefore, the motivation of this work is to design a suitable control scheme with an ability to overcome measurement noise and disturbances exist in liquid level system due to presence of sensor. A wavelet based multiresolution PID controller is developed and implemented in [14] for motion control systems. The proposed multiresolution controller is used to generate control signal by adding the wavelet transformed coefficients of error signal of different frequency sub bands of the discrete wavelet transform (DWT) tree after scaled by their respective gains. The design of a multiresolution controller for speed control of a travelling wave ultrasonic motor is presented in the work by Mitronikas and Tatakis [15]. Khan et al. developed a new wavelet based multiresolution PID controller to control the temperature of thermal power plant in [16]. Wang et al. [17] developed a DSPbased adjustable speed control system and a position control system for the micro-PMSM to improve the performance of closed-loop control system. Multiresolution wavelet controller for regulating the DC-link voltage of a fuel cell powered electric drive system is presented in [18]. The proposed controller was designed to achieve improved control dynamics and good steady state performance. Due to its application for the analysis and synthesis of time signals [19], several researchers applied WPT algorithm to separate harmonic components of power system waveforms and measure the rms value and power of each harmonic component [2]. The application area of WPT impels the idea to develop high performance symmetrical wavelet packet controller with good denoising technique than asymmetrical wavelet controller. The real-time systems are usually accompanied by number of sensors to sense and transmit the measured variable to control as well as display units. The output response of a system containing sensors represents the cumulative effect of many underlying phenomena such as process dynamics, measurement noise, parameter variations, effects of external disturbances revealed on various scales [14]. Noise consists of high frequency components and is thus localized in detailed levels of the transform. In these levels, the important coefficients, corresponding to true signal information resides with the lower magnitude coefficients. Discarding the high frequency noise using PID control may result removal of the important information disguised with noise. In case of DWT controller, it is possible to remove the high frequency noise in detailed level by lowering the value of coefficients to zero. But it also discards important information hidden with the noisy coefficients. Applying packet transform method, it is possible to remove the noise without affecting the useful information. Compared to the normal wavelet analysis, it has special abilities to achieve higher discrimination by analyzing the high frequency domain of a signal. The frequency domain divided by the wavelet packet can be easily selected and classified according to the characteristics of the analyzed signal. Therefore, in case of wavelet packet transform, it is possible to separate important information from high frequency useful signal. The application of wavelet transform for the design of controller requires optimal selection of mother wavelet and proper number of levels of resolution of wavelet transform. Here, minimum description length (MDL) [20,21,23] and entropy criterion [22,23] are used to select suitable wavelet function and levels of decomposition of the error signal. The main focus of this paper is to generate the control signal using discrete wavelet packet

transform for liquid level system and its analogous first order and second order model to achieve higher performance quotients than conventional PI Controller. Since the conventional PI controller is designed in a linear region avoiding the saturation-type nonlinearity which may occur due to presence in actuator in LLS, the closed-loop performance will be significantly deteriorated with respect to the desired linear performance. This performance degradation is referred to as windup phenomenon [25,26], which creates large overshoot, slow settling time [27,28]. To overcome the windup phenomenon, an anti-windup controller based on the conditioning technique is proposed in the presence of the nonlinearities by Hanus et al. [27], and its usefulness is compared with other anti-windup controllers through a computer simulation [26]. In this paper, the performance of this controller is also compared to the conventional controller and the wavelet controller also. In the Sections 2 and 3, brief descriptions of liquid level system and discrete wavelet and wavelet packet transform are described. In Section 4, discrete wavelet and wavelet packet controller is described. In Section 5, anti-wind up technique is demonstrated. Comparison of performances of wavelet packet controller and other controllers is elaborated in Section 6. The performance of developed controller in presence of noise is also explained in Section 6.

2. Liquid level system The existing liquid level system consists of a reservoir, process tank, constant speed motor pump, valve with actuator and differential pressure transmitter (DPT) [4]. The actuating system takes the control signal from existing PI controller, converts into equivalent pressure signal to be transmitted to the valve. The valve position dictates the amount of flow passing through the valve into the tank [4]. The transfer function of laboratory scale LLS is

G=

R 5379.82 e−τds = e−0.5s RCs + 1 (204.43s + 1)

where R ¼ 5379.82 s/m2, C ¼ 0.038 m2 and τd ¼ 0.5 calculated from physical measurement. The physical structure of the actual liquid level system is shown in Fig. 1. Physical and design parameters of process tank have been tabulated in Table 1.

3. Discrete wavelet transform and discrete wavelet packet transform Discrete wavelet transform (DWT) is an effective technique to decompose temporal non-stationary signals [1] into summation of

Fig. 1. Laboratory scale Liquid Level System (LLS).

Please cite this article as: Paul R, Sengupta A. Design and application of discrete wavelet packet transform based multiresolution controller for liquid level system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.030i

R. Paul, A. Sengupta / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Table 1 Physical and design parameters of the process tank. Physical Parameters Values Motor Process tank Tank capacity DP Transmitter Actuator Reservoir Exit pipe resistance

220 V,50 Hz, 0.37KW/0.5HP, 2.5 A, 25  25 m, 2800 rpm, head:20 m, Capacitor:10μF,440 V Height: 620 cm, Dia: 22.2 cm, Maximum Fill Level: 600 mm 0.038 m2 I/p:0-600mm,4–20 mA D.C, O/P power:1.2 W, supply: 11-42.4 V D.C, Max temp. 93 °C 24 V D.C,12.5 VA, Class 2 3/4-inch stroke 33  58  58 cm 5379.82 s/m2

time domain basis function of various frequency resolutions. It has an ability to employ long time intervals for low frequency information and shorter time regions for high frequency information. In case of DWT, the frequency divisions are in octave bands. Each level of the transform represents a frequency range half as wide as the level above it and twice as wide as that of the level below it. Conversely, the time scale on each level is twice that of the level below it and half that of the level above it. This characteristic of the DWT poses problems while attempting to localize higher frequencies. Discrimination of frequency is sacrificed for time localization at higher levels in the transform. Therefore, a far more versatile transform, the wavelet packet transform (WPT), developed by Dr. Ronald A. Coifman of Yale University, generalizes the time-frequency analysis of the wavelet transform. For discrete wavelet packet transform (DWPT), an extension of DWT [2], provides a more complex and flexible analysis. For this decomposition, both detail and approximation coefficients can be decomposed and 2n different sets of coefficients (or nodes) are produced for n levels of decomposition. The frequency bins of WPT are of equal width due to its both layers decomposition method. Generally, its decomposition divides the frequency space into various components and provides better frequency localization of signals [2–5]. In other words, with increasing octave number, the time resolution is reduced [2]. The decomposition tree structure of DWPT is shown in Fig. 2.

controller works on the actuating signal (e) as its input and acts on the error to generate a control output (uc), as shown in the following equation:

uc (t) = Kpe( t ) + Ki

∫ e( t )d( t ) + Kd dd(t ) e( t )

(1)

In terms of frequency information, the proportional and integral acts as a low pass filter and derivative captivates the high frequency components (high pass filter) of the signal [14]. In [14,23], discrete wavelet transform was used to develop a multiresolution controller to achieve better transient performances than traditional one. The block diagram of multiresolution controller based on DWT decomposition connected with appropriate gain is shown in Fig. 3. In this Fig. 3, up to third levels of decomposition is shown. At first level, sampled data are decomposed at approximate and detail level. In the next level, only approximate coefficients are decomposed to get next level approximate and detail coefficients. Finally, at level 3, proportional (kp3) and integral gain (ki3) is connected to approximate coefficients (a) and detail gain kd3 is connected to detail level (d). At level 1 and level 2, detail gains (kd2 and kd1) are connected to their corresponding detail level (d). The advantages of DWTPI controller over conventional one are: 1. Possible to represent a transparent time-frequency representation of signal in different layers. 2. Different types of frequency i.e. low, medium and high frequency are available at different detailed layers (kd1, kd2, kd3) of decomposition (Fig. 3). In this section, based on multiresolution property of wavelet, a new wavelet packet controller is developed which follow the PID principle in frequency domain. The structure of DWPT is flexible but complex in nature. The reasons behind to develop this new packet based controller over DWT based PI controller are described here:

 The variety of orthonormal bases which can be formed by the 

4. Wavelet packet controller As liquid level control is an important aspect widely used in industrial process control, it is required to design a controller which can control the level of the liquid at its set point value and it must accept the variable disturbances, noise on the plant. Generally, a PID

3

 

WPT algorithm are coupled with infinite number of wavelet and scaling functions yields a very efficacious analysis tool. The WPT allows tailoring of the wavelet analysis to selectively localize spectral bands in input data as well as to correlate the signal to the wavelet. It can split the high pass filter which means by improving the resolution of frequency. It can effectively be extracting the specific frequency band. By adaptive selection of frequency band of WPT makes the

Fig. 2. Discrete wavelet packet based decomposition tree structure.

Please cite this article as: Paul R, Sengupta A. Design and application of discrete wavelet packet transform based multiresolution controller for liquid level system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.030i

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R. Paul, A. Sengupta / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Fig. 3. Level three decompositions of error signal using discrete wavelet transform.

 





 

signal spectrum matched and improve the time-frequency resolution. Not only can the best wavelet be opted to analyze a specific signal but the best orthonormal basis can as well. The best basis search algorithm of wavelet packets uses a minimum entropy criterion and for a signal gives the most concise description for the dictionary in hand. The application of best basis search to the wavelet packet dictionary is equivalent to an optimal filtering of the signal. Wavelet packet atoms are waveforms indexed by three naturally interpreted parameters: position, scale (as in wavelet decomposition), and frequency. For a given orthogonal wavelet function, a library of bases can be generated called wavelet packet bases. Each of these bases offers a specific way of coding signals, preserving global energy, and reconstructing exact features. For DWT, removal of noise means discarding whole useful coefficients disguised with noise at highest level of detail part of the decomposed signal. But in case of DWPT, removal of particular node at highest detail level is enough to do the same operation. Therefore, more useful information can be preserved for further analysis of system response. A symmetrical and balanced tree structure based frequency band of signal is possible. It is complex but flexible analysis.

The schematic diagram of a plant with DWPTPI controller is shown in Fig. 4.a. In case of DWPT, it works on the actuating signal in this way so that lucid frequency information of the signal is made available. The decomposition structure of packet based controller is shown in Fig. 4.b.

Fig. 5. The block diagram of anti-wind up controller.

In case of DWPT method, coefficients are available at nodes. The wavelet packet method is a generalization of wavelet decomposition that offers a richer range of possibilities for signal analysis. Noise is available at different nodes of decomposed detail levels. Each component in this wavelet packet tree can be viewed as a filtered component with bandwidth decreasing but with increasing level of decomposition and the whole tree can be viewed as a filter bank. In this Fig. 5., sampled data is decomposed and up to fifth node of third levels is shown. Here, node 1 is decomposed to get node 2 and node 3. In packet transform, both node 2 and node 3 should be decomposed. Here, only node 2 is decomposed to obtain node 4 and node 5. In third level, both node 4 and node 5 are decomposed to connect proper gain to these decomposed coefficients. The discrete wavelet and wavelet packet based PI controller works in similar manner of PI controller. For packet controller, each of the components of node is weighted by their respective gain and summed up together to generate control signal. As the levels of decomposition as well as mother wavelet affects the performance of the controller, entropy and MDL methods [20–23] are used to obtain suitable levels and mother wavelet for multiresolution controllers. The three orthogonal, compact support wavelet filters such as Daubechies, Symlet and Coiflet has been considered to decompose the actuating signal. Among these three wavelets, selection of optimal wavelet has been taken based on minimum value of MDL. The control signals u1(t) of DWTPID and u2(t) of DWPTPID controller are given below:

u1(t) = kd1ed1 + kd2ed2 +……kdnedn + k aiean + k apean kd1,

kd2

kdn

Here, …. are derivative gains and portional and integral gain respectively in (2).

(2) kap

and

kai

are pro-

 In this equation, by connecting suitable gain with approximate 

 

signal at highest level ( ean) , transient performance can be achieved. Commands and disturbance signals of the LLS plant are low frequency signal; therefore, the gain connected to approximate error coefficients ( ean in in (2)) is increased to improve transient response and to remove disturbance Controlling the gain connected to the detailed level at highest level ( edn) adds damping to the system, is increased to improve steady state response of the system [14]. The denoising can be achieved by controlling gain connected to detailed level error coefficients ( ed1, ed2,. .edn − 1) to remove the effect of sensor noise

u2(t) = A aLa k p + A aLa ki + A aLd kd + A dLa k pori + A dLd kd Fig. 4. a. Block diagram of closed loop SISO system with discrete wavelet packet controller. b. Best tree structure of decomposed error signal using discrete wavelet packet transform.

+DaLak pori + DaLd kd + DdLak pori + DdLd kd In (3), ‘A’ and ‘D’ represents the approximate and the detail

Please cite this article as: Paul R, Sengupta A. Design and application of discrete wavelet packet transform based multiresolution controller for liquid level system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.030i

(3)

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signal at first level of decomposition respectively. Here, levels L ¼ 2, 3……n-1 where, number of samples are 2n and n is number of decompositions. After that both ‘A’ and ‘D’ signals again split up. Here, k p and k i are proportional and integral gain and kd is derivative gain. By proper tuning associated with approximate part of the approximate level side ( AaLa , AaLd ) as well as detail level side ( DaLa , DdLa ), transient performances can be improved. Also, by careful handling of detailed part of approximate and detail level reduces the noisy coefficients by picking up different nodes of same layer or different layers of decomposition. Z-N tuning rule is used to find out the proportional and integral gain of the new controller [12]. The symbols used in (3) are. AaLa ¼ Approximated coefficients of decomposed approximate signal at approximate level (A). AaLd ¼ Detailed coefficients of decomposed approximate signal at approximate level (A). AdLa ¼ Approximated coefficients of decomposed detail signal at approximate level (A). AdLd ¼ Detailed coefficients of decomposed detail signal at approximate level (A). DaLa ¼ Approximated coefficients of decomposed approximate signal at detail level (D). DaLd ¼ Detailed coefficients of decomposed approximate signal at detail level (D). DdLa ¼ Approximated coefficients of decomposed detail signal at detail level (D). DdLd ¼ Detailed coefficients of decomposed detail signal at detail level (D). The steps followed to implement new controller are: Step1: In NI LABVIEW software environment, simulated plant with PC based discrete wavelet, wavelet packet and PI controllers are implemented. Step2: The error coefficients have been taken for selection of suitable Table numbers of decomposition and mother wavelet. Step3: According to this number of levels of decomposition, the new controller has been constructed and using the selected wavelet, the performance of this controller is tabulated and compared with the conventional controller. Z-N tuning rule is used to find out the gain value. Step 4: After successful implementation, performance of new controller for analogous single tank liquid level system using RC series circuit has been tested in simulation. This similar work has been extended up to general second order plant. Step 5: In the design of a control system, consideration must be made for the physical system with noise during operation. Therefore, in presence of noise, the performance of three controllers have been investigated and compared.

5. Anti-windup controller The control variable may reach to the actuator limits due to different range of operating conditions in control system [26]. The integral controller integrates the error, i.e. that the integral term may become very large or it winds up [25]. Furthermore, the most significant effects of the integrator windup take place when the process is of low order. Briefly, any controller with integral action may give large transients when the actuator saturates. The effect of windup is in the form of significant performance deterioration; large overshoots, long settling time in the output and sometimeseven instability [28]. A more common approach in practice is to add extra feedback compensation at the stage of control implementation. It can be implemented to avoid integrator windup. An alternative approach to

5

conditional integration known as back-calculation consists of recomposing the integral term once the controller saturates [27]. In particular, the integral value is reduced by feeding back the difference of the saturated and unsaturated control signal, as shown in Fig. 5. where,Tt is called tracking time constant. Fig. 5. shows the block diagram for back calculation method, where the integral is recomputed so that its new value gives an output at the saturation limit when the controller output saturates. Here, the control system has an extra feedback path that is generated by measuring the difference between the actual actuator output (v) and the controller output (u) and forming an error signal es. This error signal is then fed back to the integrator through gain 1/Tt. The signal is zero when there is no saturation. Thus, it will not have any effect on the normal operation when the actuator does not saturate. When the actuator saturates, the signal es is different from zero. It is advantageous not to reset the integrator instantaneously but dynamically with a time constant Tt. Tt governs how quickly the integral term is reset. Tt can be chosen equal to the [1/ 2*integral time constant (Ti)]. Small values for Tt decrease the saturation time of the controller output and also the settling time of the process. In that case, the process response will be slow and will not produce an overshoot. Big values (with respect to the required value) chosen for Tt provide a long saturation time at the controller output and it will also produce large overshoot in the system response but gives faster response.

Table 2 Selections of nodes of decomposition and optimal mother wavelet for DWPTPI controller. Wavelet filters

Nodes of decomposition

MDL

Db2 Db3 Db4 Db5 Db6 Db7 Db8 Db9 Db10 Sym4 Sym5 Sym6 Sym7 Sym8 Sym9 Coif1 Coif2 Coif3 Coif4 Coif5

5 4 3 4 6 5 4 4 5 6 6 4 4 4 6 6 5 4 5 5

 222.4648  194.0703  172.9731  233.8923  205.7124  201.8707  226.3860  216.6383  192.0822  208.4463  260.1632*  189.4511  232.6989  187.9995  246.1361  199.5380  208.1685  255.8906  241.6359  254.3615

Table 3 Performance analysis of three controllers of LLS. Transient performance

PI controller

DWTPI controller (Sym5)

DWPTPI controller (Sym5)

Percent overshoot Peak time(s) Rise time(s) Settling time(s) Steady state error Percent undershoot

2.55404 356 148 453 0.00000000179415 0

1.61452 192 65 106 0.0000000570321 0

0 0 62 128 0.00000023784 0

Please cite this article as: Paul R, Sengupta A. Design and application of discrete wavelet packet transform based multiresolution controller for liquid level system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.030i

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6. Experimental results The developed packet controller has been implemented in PC using NI LABVIEW software for existing LLS plant in simulation. After that, its performance has been tested for two different analogous first order and second order plant models. Case 1: Performance analysis of controller for LLS Discrete wavelet transform based PID controller for LLS plant has already been published in [23]. Here, discrete wavelet packet transform based PI (DWTPI) controller for liquid level system has been compared to the other two controllers. It is seen that performance of PI controller is better than PID controller for existing LAB scale plant. To find out proper levels and optimal mother wavelet for packet decomposition, twenty mother wavelets and one hundred twenty eight data samples has been taken. The result of application of MDL criterion and logarithmic energy based level

selection has been tabulated in Table 2. Sym5 up to level four has been found as an optimal choice for discrete wavelet controller [23]. This similar work has been extended for wavelet packet based controller also. Here, Table 2 shows that sym5 up to sixth node is an optimal choice for packet based controller. Generally, for conventional controller, three tuning parameters can be used to generate control signal u(t). But for wavelet and wavelet packet controller, more than three tuning parameters are utilized to control the system response. As proportional and integral captivates the low frequency and derivative captures the high frequency of signal, these gains are connected to their respective level. Here, derivative gain of conventional PI controller as well as detailed level gain of multiresolution controller has been taken zero. Keeping the same proportional and integral gain of approximate signal at level four for discrete wavelet controller and at node six at approximate level for discrete wavelet packet controller, the same performance can be achieved. But the additional

Fig. 6. a. The output response of three controllers connected to LLS. b. The output response of four controllers connected to LLS.

Please cite this article as: Paul R, Sengupta A. Design and application of discrete wavelet packet transform based multiresolution controller for liquid level system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.030i

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flexibility of DWPTPI controller is that the presence of approximate coefficients (kd3p or i, kd5p or i) of decomposed detailed level. Here, controlling kd3i or kd5i gain of packet based controller (DWPTPI), it is possible to reduce the overshoot than discrete wavelet transform based PI (DWTPI) controller. The performance of DWPTPI (shown in Table 3) shows that the closeness of the response to the desired response represented by overshoot and settling time as well as the swiftness of response by rise time and the peak time is better than PI controller. The output response of the LLS with three controllers is shown in Fig. 6.a. Considering the saturation due presence of actuator and time delay of the system, the output response is observed and plotted in Fig. 6.b. To overcome the wind up problem due to saturation limit, anti-windup technique is applied to PI controller. Here, time delay is taken 40 sec. It is observed that anti-windup PI controller reduces the overshoot and settling time of the PI controller. By increasing the value of Tt and saturation limit shown in Fig. 5, it is possible to achieve faster response of the anti-windup PI controller compare to PI and DWTPI controller but it increase the overshoot higher than the other three controllers which is not desirable. Case 2: Transient analysis with noisy input and with the noise insertion before plant Effective methods to remove noise from signal using wavelet techniques have been developed [4].Wavelet based noise removal exploits the time-scale characteristics of the DWT. Generally, noise consists of high frequency components and is thus localized in detailed levels of the transform. At these levels, the important coefficients corresponding to true signal information and the lower magnitude coefficients represent the noise. Therefore, applying packet transform method, it is possible to remove noise without affecting the important information. Here, laboratory scale liquid level system consists of a differential pressure transmitter. Due to presence of sensor, high frequency noise may contaminate the output response. Continuous vibration of the wire connected to the DP transmitter generates high frequency noise [6]. Also, thermal and white noise is caused by the random motion of free electrons and vibrating ions in a conductor. In electrical systems, the shot noise due to semiconductor devices such as diodes and transistors also may add to the output response of the closed loop system. Therefore, to investigate the performance of the controllers in presence of noise [2], three different types of noise with their appearance at four positions have been considered. The noise may appear at input, before the actuator, plant as well as sensor. At all positions, introduction of uniform white noise, Gaussian white noise and pink noise contamination to the output response is considered and the performance of controllers are measured and compared. All these cases, it is seen that multiresolution controllers give faster transient response with better noise rejection potency than conventional one. In this paper, two general cases have been shown in Fig. 7.a. and b. In case of noise with input, the noise power and seeds are taken 3.515 and 8000 and in other case (noise appears before plant), it has taken 0.001 and 8000 respectively. Comparison of performance (Figs. 8 and 9) of controllers shows that transient performance of multiresolution controllers is better than the traditional one. Moreover, higher signal to noise ratio (SNR) and lower mean squared error (MSE) [4,23] indicates the better performance of multiresolution controller. Proper selection of gain either at node 5 or node 3 at detail level improves the noise rejection ability as well as closeness to the response of desired value of packet based controller compare to DWTPI controller. The key motivation for the development of this new controller is its ability to handle noise more effectively as well as to give the faster response. For two general cases, performance of three controllers is shown in Fig. 8a–c. and Fig. 9a–c.

Fig. 7. Block diagram of closed loop system with a.noisy input and b. noise appears before plant.

6.1. Performance analysis of controller for first order plant Case 1: Transient analysis with normal reference voltage The transfer function of the open loop system is

G(s) =

1 u(s) τs + 1

Here, time constant (τ) ¼ RC. The practical value of resistance (R) and capacitance (C) are 9.919KΩ, 98.13 μF respectively. From the input and output data set, the simulated open loop plant transfer function is identified using NI LabVIEW identification toolbox. The open loop plant transfer function is

G(s) =

0.987157 0.9950788s + 1

Simulation results of comparison between the PC based multiresolution controller and proportional integral (PI) controller for first order plant have been shown in Fig. 10. It is seen from Table 4 that the performance of DWPTPI controller is better than the other two controllers for first order system also. Controlling the integral gain at node five (kd5i) of DWTPI controller, overshoot as well as steady state error is reduced drastically than the other two controllers. Case 2: Transient analysis with noisy input and with the noise insertion before plant Here, the performance of controllers for the analogous RC plant has also been investigated in presence of white noise also. The simulated results have been shown in Figs. 11. and 12. From these performance graphs, it is seen that signal to noise ratio is improved for new controller. Also, it reduces overshoot, mean square error, rise time and settling time than the conventional controller. 6.2. Performance analysis of controllers for second order system: A software result The investigation of performance of new developed controller has also been extended for second order plant model. The identified transfer function of second order plant model is

Please cite this article as: Paul R, Sengupta A. Design and application of discrete wavelet packet transform based multiresolution controller for liquid level system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.030i

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Fig. 8. a. Performance graph of three controllers of LLS connected to Gaussian white noise with input. b. Performance graph of three controllers of LLS connected to Pink Noise with input. c. Performance graph of three controllers of LLS connected to Uniform White Noise with input.

G(s) =

1 0.99174s 2 + 1.99185s + 1

The rate of sampling frequency of controller is 100 Hz. For second order system, Db4 up to sixth node of decomposition is an optimal choice to get better performance of the controller. From

the output responses shown in Fig. 13., it is seen that the performance of DWPTPI controller in terms of rise time, peak time, settling time in sec as well as steady state error reduces than the traditional PI controller. In presence of noise also, the performance of packet controller is ameliorated with its noise rejection ability than the other two controllers (shown in Figs. 14. and 15.). It also

Please cite this article as: Paul R, Sengupta A. Design and application of discrete wavelet packet transform based multiresolution controller for liquid level system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.030i

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Fig. 9. a. Performance graph of three controllers of LLS connected to Gaussian White Noise before plant. b. Performance graph of three controllers of LLS connected to Pink Noise before plant. c. Performance graph of three controllers of LLS connected to Uniform White Noise before plant.

Please cite this article as: Paul R, Sengupta A. Design and application of discrete wavelet packet transform based multiresolution controller for liquid level system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.030i

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improves the transient as well as steady state performance than the PI controller. 6.3. Stability analysis of wavelet controller and PI controller for second order system

Fig. 10. The output response of three controllers for first order plant.

Table 4 Performance analysis of three controllers of first order plant. Transient performance

PI controller

DWTPI controller (Db4)

DWPTPI controller (Db4)

Percent overshoot Peak time(s) Rise time(s) Settling time(s) Steady state error Percent undershoot

1.00786 1.61 0.58 0.91 0.000242379 0

0.423059 1.47 0.46 0.78 0.000118835 0

0 0 0.44 0.78 0.00000437948 0

In this section, an attempt is taken to judge the stability of the system fed with the developed controller. In most of the time, the performance of controller is adjudicated based on time domain characteristics. But frequency domain characteristics are also an important criterion for measurement of system sensitivity to sensor noise and parameter variations. Correlation between time and frequency domain characteristics is very important to judge the overall stability issues. In general, the magnitude of resonant peak gives the measure of relative stability of a closed loop system. A large resonant peak corresponds to large maximum overshoot of the step response [24]. Also, increment of resonant peak decreases the damping ratio and band width decreases with an increment of damping ratio. From Figs. 14 and 15. and Table 5 it is construed that for wavelet controller, resonant peak as well as overshoot decreases. Conversely, large bandwidth corresponds to faster rise time, since higher frequency signals can more easily passed through the system. It is seen that rise time for conventional controller is slightly higher than wavelet controller and band width increases for second order plant. Therefore, the system with new controller responds slower due to larger rise time. But due to small bandwidth, only signals of relatively low frequencies are passed [24]. Also, it is also seen from Table 5 that phase margin of a system with wavelet controller is slightly higher than the system with PI controller. Increment of phase margin again indicates the increment of damping ratio which reduces the maximum

Fig. 11. Performance graph of three controllers of first order plant connected to White Noise with input.

Please cite this article as: Paul R, Sengupta A. Design and application of discrete wavelet packet transform based multiresolution controller for liquid level system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.030i

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Fig. 12. Performance graph of three controllers of first order plant connected to White Noise before plant.

 

Fig. 13. The output response of three controllers for second order plant.



overshoot and settling time. From Figs. 14 and 15, it can be concluded that wavelet controller gives better stability with noise rejection potency but at the same time, the system becomes slightly sluggish. ➣ Discussion on Experimental results The overall observations from the experimental results are summarized here:

 For DWPTPI controller, the useful information disguised with noise is available from detail levels. Decomposition of each detail level means separation of the useful information from noisy part as much as possible and the useful information can be obtained from the node of approximate part of detail level. By connecting proper gain with the node coefficients, it is possible to improve the transient performance of closed loop



SISO system. Also, either by discarding nodes or connecting gains with that nodes of different levels, signal to noise ratio can also be improved. Handling of particular node means the other nodes available at this level may be kept intact. Therefore, more energy can be preserved. In simulation, a closed loop model consisting of liquid level system with PI, DWTPI, DWPTPI controller (separately), sensor and actuator in forward path has been taken to compare the performance of these three controllers. For wavelet controller, choice of mother wavelet and proper levels of decomposition are important criteria as both have direct impact on system response. Therefore, by applying suitable wavelet ‘Sym5’, found by MDL method, performance quotients have been tabulated. From Table 3, it is seen that percent overshoot, peak and rise time of DWPTPI controller are better compared to other two controllers. But it slightly degrades the settling time and steady state error compared to DWTPI controller. It is well known that design of the conventional PI controller in a linear region by avoiding the saturation-type nonlinearity due to presence in actuator in LLS may significantly degrades the closed-loop performance with respect to the desired linear performance. Therefore, Anti-wind up PI controller is taken to improve the performance of PI controller. It is also seen that considering the time delay, the response of DWPTPI controller shown in Fig. 6.b. indicates that it gives better performance in terms of overshoot, rise time, settling time, peak time and steady state error compare to the other three controllers of LLS. Keeping the same gain value of controllers, three varieties of noises such as Gaussian White Noise, Pink Noise, Uniform White Noise are incorporated at different positions of the closed loop in simulation and performance of controllers have been compared. From Fig. 8a to Fig. 9c., it is seen that signal to noise ratio has been improved as well as overshoot and MSE is reduced for DWPTPI controller compared to the other two controllers (PI and DWTPI). Higher signal to noise ratio establishes the higher potency of new controller for removing noise. It is also seen that

Please cite this article as: Paul R, Sengupta A. Design and application of discrete wavelet packet transform based multiresolution controller for liquid level system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.07.030i

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Fig. 14. Performance graph of three controllers of second order plant connected to White Noise with input.

Fig. 15. Performance graph of three controllers of second order plant connected to White Noise before plant.

R. Paul, A. Sengupta / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Table 5 Frequency domain specifications of PI and DWTPI controller for second order system.

Phase margin Gain Margin Gain cross over frequency Resonant peak for closed loop system Bandwidth

System with PI controller

System with wavelet controller

65.9560 Infinity 0.5999 0.00009849

68.0757 Infinity 0.5937 0.000027112

0.9389

0.9189

overall performance of both multiresolution controllers (swiftness) is highly improved compared to the traditional one. It is also found that the settling time and steady state errors are also improved for wavelet controllers especially in presence of uniform white noise and pink noise. The reason for such improvement is the availability of lucid frequency information of wavelet controller than conventional one. Moreover, tuning flexibility at different levels makes it faster than PI controller. Similar transient analysis has also been performed for analogous first order plant and second order plant in simulation. The results of Table 4 and Fig. 13. show that the new controller gives lower overshoot with faster transient response compared to the other controllers. The steady state performance of this controller is also improved compared to DWTPI controller as well as traditional one. For lab scale LLS with it analogous different order plant, the signal to noise ratio improvement as well as reduction of mean squared error establishes the fact that new controller can remove noise better than other two controllers. Also, reduction of overshoot for packet based controller means it can control overflowing of the tank in a better way than the PI controller. Moreover, it can settle down faster compare to traditional one. After transient analysis, stability study has been taken care of for analogous second order plant in presence of white noise. From Fig. 14, Fig. 15 and Table 5, it can be concluded that with wavelet controller,







➣ ➣ ➣ ➣

Resonant peak as well as overshoot decreases Band width increases Phase margin is higher. Rise time is larger

Therefore, the system with new controller responds slower. But due to small bandwidth, only signals of relatively low frequencies are passed. Also, it increases phase margin which corresponds to increase in damping ratio. Higher damping ratio reduces the overshoot and settling time. So, it can be concluded that the wavelet controller gives better stability with noise rejection potency but at the same time, the system becomes slightly sluggish for second order plant.

7. Conclusion A generalized multiresolution PI controller based on the discrete wavelet packet transform is developed and tested for SISO system. Just like conventional controller, this new controller is intuitive and effective. The controller gains have an explicit relationship with the characteristics of actuating signal which makes tuning of the controller insightful. DWPT method is more complex but yields flexible analysis. Its decomposition technique represents

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the error signal at different nodes in symmetrical structure, offers higher resolution than the DWTPI controller as well as conventional PI controller and is the basis of the new controller design. Compared to DWT, without disturbing the gain of approximate signal, only handling the detail part, both the improved performance as well as better denoising with smooth response can be achieved. Here, anti-wind up PI controller is taken to control the wind up problem. It is seen that DWPTPI controller gives better performance than the other three controllers in presence of delay of the system. This new controller is not only a balanced structure but also it can be used as controller as well as an efficient filter. As packet controller is an extension of wavelet controller, therefore, it may be used as general multiresolution controller also by assigning zero value of the approximated coefficients of detailed signal. Therefore, optimal choice between controllers is possible.

Acknowledgement The research is funded by the institutional fund of Indian Institute of Engineering Science and Technology, Shibpur.

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