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Keywords: Genetic Algorithm, Closed-Loop System. Identification, Frequency-Domain System Identification,. Weighted Least Squares, Active Magnetic Bearing.
2014 11th IEEE International Conference on Control & Automation (ICCA) June 18-20, 2014. Taichung, Taiwan

Genetic Algorithm-based System Identification of Active Magnetic Bearing System: A Frequency-domain Approach A. Noshadi, Student Member, IEEE, J. Shi, Member, IEEE, W. S. Lee, Member, IEEE, P. Shi, Senior Member, IEEE, and A. Kalam, Member, IEEE a

Abstractβ€”The main focus of this paper is on system identification of an active magnetic bearing system (AMB) using genetic algorithm (GA) for optimal controller design purpose. In the first step, an analytical model of the system is derived using principle of physics and taking into account both the rigid body and bending body modes of the system. In the next step, as AMB system is inherently open-loop unstable, a closed-loop system identification approach is adopted. The actual frequency response data are collected under closed-loop condition. As it is expected from the analytical model, the system has two dominant resonant frequencies which have to be accurately identified. To fit the frequency response of the system into a desired order transfer function, weight vectors are used to emphasise the resonant frequencies. Subsequently, GA is employed to search the optimal values of the required weight vectors and their corresponding scaling factors automatically in order to best fit the measured data. For verification of the proposed method, the model obtained from GA is compared with some well-known methods such as prediction error method (PEM) and subspace state space system identification (N4SID) method. Eventually, a PID controller and two notch filters are designed based on the obtained model and implemented on the actual system and the performance of the designed controller is compared with the on-board analogue controller.

Keywords: Genetic Algorithm, Closed-Loop System Identification, Frequency-Domain System Identification, Weighted Least Squares, Active Magnetic Bearing. I. INTRODUCTION Active magnetic bearings (AMB) employ electromagnetic force to suspend the rotor of machines and have various advantages over conventional mechanical and hydrostatic bearings. These advantages are zero frictional wear and efficient operation at extremely high speed. AMBs are ideal for clean environments where no lubrication is required. They can work in harsh environments such as high temperature, heavy load and high humidity. These special A. Noshadi is a Ph.D. student in the College of Engineering and Science, Victoria University, P O Box 14428, MCMC, Melbourne, 8001, Australia (e-mail: [email protected]). J. Shi is with the College of Engineering and Science, Victoria University, P O Box 14428, MCMC, Melbourne, 8001, Australia (e-mail: [email protected]). W. S. Lee is with the College of Engineering and Science, Victoria University, P O Box 14428, MCMC, Melbourne, 8001, Australia (e-mail: [email protected]). P. Shi is with the School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, SA, 5005, Australia (e-mail: [email protected]). A. Kalam is with the College of Engineering and Science, Victoria University, P O Box 14428, MCMC, Melbourne, 8001, Australia (e-mail: [email protected]). 978-1-4799-2837-8/14/$31.00 Β©2014 IEEE

characteristics have attracted significant attention and interests amongst the researchers in the past decade. AMBs can be utilised in many industrial applications, where fast and precise operations are desired such as energy storage flywheels, artificial heart, high speed turbines and jet engines [1], [2]. The main obstacle that has restricted the widespread commercial and industrial application of AMBs is their highly nonlinear, multi-input multi-output (MIMO) and open-loop unstable characteristics. The control problem is further complicated by the existence of high frequency resonance, rotor unbalance and demanding performance objectives in terms of robustness and stability improvements. Therefore, better modelling and sophisticated high performance controllers are needed to address the problem. Before an effective stabiliser can be designed for AMBs, sufficient knowledge of the system to be controlled is required. The designer begins by collecting information about the system from all available sources and then represents this information in the form of an identified model. There are basically two types of system identification methods, namely, open-loop and closed-loop system identification. Well-developed techniques in the literature are based on open-loop system identification. The most significant techniques for open-loop identification of systems are prediction error methods (PEM), instrumental variable method (IVM) and output error methods (OE). These methods can also be used for identification of openloop unstable AMB systems [3], [4]. However, these techniques may fail in search for the global optimum if the search space is not differentiable or linear in the parameters. Recently, subspace methods have attracted the attention of many researchers. Some popular approaches in subspace system identification family are the canonical variate analysis (CVA) [5], multi variable output error state space (MOESP) [6] and subspace state space system identification (N4SID) [7], [8], [9]. Although these methods have shown to provide satisfactory results in system identification of MIMO systems, their efficiency on identification of openloop unstable AMBs needs to be further investigated. In recent years, artificial intelligence (AI) techniques have become suitable candidates for the identification of nonlinear open-loop unstable plants. One of the most powerful AI techniques is genetic algorithm (GA) and has been widely used in optimisation problems with several local minima where conventional search algorithms fail. Kristinsson and Dumont [10] proposed a method for poleszeros identification of both continuous and discrete time systems using GA. The results proved that the GA was able

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to converge to the actual values of the parameters. However, the convergence of the proposed method was slow in some cases like identification of zeros due to insensitivity of the objective function to the small changes in zeros. Later, Ghaffari et al. [11] utilised GA to identify and control a power plant. The results showed a successful identification of the high order de-superheating process and hence improvement in the performance of the steam temperature controller. Tavakolpour et al. [12] employed GA to identify the parameters of the transfer function of a rectangular flexible plate system. They used a novel algorithm for truncation-based selection which showed a faster convergence to the global optimum solution compared to conventional methods. However, increasing the model order will subsequently increase the number of parameters to be identified by GA. In this paper, GA is combined with weighted least squares (WLS) method to find the optimal weighting vector which will result in the optimal representation of the system. Hence, the number of variables to be optimised by GA is fixed regardless of the model order. Frequency weighting has been used to emphasise particular ranges of frequencies which are more of interest in controller design stage. The proposed method is verified and compared with well-known PEM and N4SID methods available in MATLAB using system identification Toolbox.

on the horizontal or x direction motion at two ends of the rotor. The analytical model of the system considering the rigid and bending body modes, current amplifier, and the displacement sensor is obtained and the resulting Bode diagram of the obtained analytical model is depicted in Fig. 2. As it is shown in Fig. 2, two resonant modes are expected at 750Hz and 2069Hz. The final state-space representation of the system is also shown in (1)-(3). The procedure is omitted here due to page limitation. 𝐀=

0 ⎑ 3.33𝑒4 ⎒ 0 ⎒ 0 ⎒ 0 ⎒ 0 βŽ’βˆ’3.83𝑒4 ⎒ 0 ⎒ 0 ⎣ 0

1 0 0 0 0 0 0 0 0 0 0 0 βˆ’3.96𝑒4 0 0 0 13.31 13.31 ⎀ βŽ₯ 0 0 1 0 0 0 0 0 0 0 6.86𝑒4 0 0 3.7𝑒5 0 0 βˆ’243.48 243.48 βŽ₯ βŽ₯ 0 0 0 0 0 1 0 0 0 βŽ₯ (1) 0 0 0 0 0 0 1 0 0 0 0 0 βˆ’2.22𝑒7 0 0 0 βˆ’15.31 βˆ’15.31 βŽ₯ 0 2.24𝑒3 0 0 βˆ’1.69𝑒8 0 0 βˆ’8.12 8.12 βŽ₯ βŽ₯ 0 0 0 0 0 0 0 βˆ’4.54𝑒3 0 0 0 0 0 0 0 0 0 βˆ’4.54𝑒3⎦ 0 οΏ½T 𝐁 = οΏ½0 0 0 0 0 0 0 0 1.14e3 0 0 0 0 0 0 0 0 0 1.14e3

(2)

0 0 𝐂 = οΏ½5000 0 658.5 0 βˆ’9.69𝑒3 9.18𝑒3 0 0 0 0οΏ½ ,𝐃 = οΏ½ οΏ½ (3) 5000 0 βˆ’658.5 0 βˆ’9.69𝑒3 βˆ’9.18𝑒3 0 0 0 0 0 0

II. DESCRIPTION OF THE ACTIVE MAGNETIC BEARING SYSTEM A. Analytical Model The system under study is an MBC 500 magnetic bearing system. It consists of two active radial magnetic bearings which support both ends of a rotor and levitate the rotor by using electromagnetic force. Fig. 1 illustrates the top and front views of the experimental setup. The rotor shaft has four degrees of freedom (DOF), being translational, namely x1 and x2 in the horizontal direction and y1 and y2 in the vertical direction. The system employs four linear current amplifiers and four decentralised analog lead compensators to actively control the bearing axes.

Figure 1. Active magnetic bearing experiment.

For analytical modelling of the system, motion in the x and y directions are assumed to be independent. However, the coupling effect cannot be ignored if the rotor was spinning. An additional constant force of gravity acts in the y direction which cannot be modelled by a linear system model and hence is neglected. The derivation here is focused

Figure 2. Bode diagram of the analytical model.

B. Frequency Domain System Identification Robustness and stability analyses of systems with uncertainties is usually studied based on the accuracy of the model as a function of frequency [13]. Indeed, identification of systems to match the input-output relationship as accurately as possible in certain frequency ranges is crucial for successful controller design. In the case of periodic excitations, we favour the use of frequency-domain data over time-domain data, because frequency-domain data have many advantages that cannot be found in time-domain data. These are: 1) Noisy data can be detected and eliminated. 2) It is easy to combine data from different experiments of various frequency ranges. 3) Using multi-frequency excitations, the system behaviour in large range of frequencies can be observed. 4) No initial state estimation of the system is required. By using discrete Fourier transform (DFT), a large number of time-domain data samples are replaced by spectral lines [14]. As AMBs are inherently open-loop unstable, closed-loop system identification must be performed instead of more common open-loop system identification techniques [15], [16]. Fig. 3 shows the blockdiagram of the setup used to collect the data for identification. The data were taken while the bearing was stabilized with four on-board analogue controllers and assuming that there is no disturbance on the system.

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Figure 3. Closed-loop system identification.

Assuming that the closed-loop system is stable, the relationship between the plant output and the reference input r and noise n is shown in (4): 𝑦 = (𝐼 + 𝑃𝐢)βˆ’1 π‘ƒπ‘Ÿ + (𝐼 + 𝑃𝐢)βˆ’1 𝑛

(4)

π‘‡π‘¦π‘Ÿ (𝑠) β‰… (𝐼 + 𝑃𝐢)βˆ’1 𝑃 ≝ 𝑃(𝐼 + 𝐢𝑃)βˆ’1

(5)

𝑒 = (𝐼 + 𝐢𝑃)βˆ’1 π‘Ÿ βˆ’ (𝐼 + 𝐢𝑃)βˆ’1 𝐢𝑛

(6)

π‘‡π‘’π‘Ÿ (𝑠) β‰… (𝐼 + 𝐢𝑃)βˆ’1

(7)

𝑃(𝑠) = π‘‡π‘’π‘Ÿ βˆ’1 π‘‡π‘¦π‘Ÿ

(8)

Since the signals r and n are uncorrelated and the noise signal n has zero mean, the system response can be averaged and the transfer function between y and r can be obtained as (5). For matrices of appropriate dimensions, (5) should follow the push-through rule: Similarly, the relationship between the control output u, the reference input r, and the noise n can be written as in (6). Averaging the system response gives the transfer function between u and r as (7).

It should also be noted that the amplitudes of the signals are to be chosen in such a way that the assumption of linearity is maintained. The MIMO measurement is taken in 20 individual experiments. In each experiment, five different chirp signals with different amplitudes and frequencies (shown in Table I) are sent to one input channel and the responses of four output channels are collected. The timedomain response of the system is collected using dSPACE ControlDesk software package and then exported to MATLAB for discrete Fourier transform (DFT) analysis. After the aforementioned procedures, the experimental frequency responses for the AMB system are obtained as shown in Fig. 4. The diagonal terms in Fig. 4 represent the frequency response data for x-axis and y-axis, respectively. The off-diagonal terms on the other hand represent the crosscoupling effect of each input on the other three outputs. For better comparison, the signal averaging is used and the response of each channel is also depicted in Fig. 4 in red. From the results, it can be seen that the magnitude of offdiagonal terms are very small. It agrees that the original MIMO system can be decoupled into four single-input single-output (SISO) subsystems for system identification purpose. In this paper, only SISO identification of the first channel (x1-axis) is studied. However, similar approach can be used to identify the model of other three diagonal channels.

By using (5) by (7), the open-loop unstable transfer function of the magnetic bearing system can be estimated from the collected data via two step closed-loop system identification as: III. DATA ACQUISITION OF SYSTEM RESPONSE Chirp signals are employed as inputs to the system while the outputs from the system are collected via different experiments. Initial frequency, target time and the frequency at the target time of the chirp signals are chosen in such a way that the frequency does not increase too fast so that the system has enough time to obtain its steady-state response. The fixed step size of 0.000049 seconds is used. A smaller step size could result in aliasing because of high frequency excitations. Due to the presence of resonant frequencies, different amplitudes of inputs are necessary to obtain accurate results. This problem can be solved by taking several measurements with different amplitudes and frequencies and combining them in MATLAB. The amplitude and frequency range of each chirp signal is depicted in Table I. TABLE I. Measurement Chirp Start f (Hz) Chirp End f (Hz) Amplitudes (Volts)

Figure 4. Frequency response data for the multi-input multi-output channels.

IV. EXPERIMENTAL RESULTS The main goal here is to identify a model of (8) from the obtained frequency response data over the frequency range Ο‰. Generally, transfer function of a linear dynamic system can be written as a ratio of two frequency-dependent polynomials of:

Low Freq 1

First Resonance 610

Middle Freq 975

Second Resonance 1880

High Freq 2220

610

975

1880

2220

5000

0.6

0.4

0.6

0.1

0.2

2

π‘˜

𝑛 +𝑛 (π‘—πœ”)+𝑛 (π‘—πœ”) +β‹―+π‘›π‘˜ (π‘—πœ”) 𝑁(π‘—πœ”) 𝐺� (π‘—πœ”) = 0 1 (π‘—πœ”)+𝑑2 (π‘—πœ”)2 = (π‘—πœ”) 𝑙

CHIRP SIGNAL INPUTS

𝑑0 +𝑑1

2

+β‹―+𝑑𝑙

𝐷(π‘—πœ”)

(9)

The desired order of the numerator (k) and denominator (l) are pre-assigned. The problem here is to fit the obtained data P(jω) from (8) as a ratio of two frequency-dependent polynomials in the form of (9). The error in the fitting can be represented as:

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𝑁(π‘—πœ”) 𝑒(π‘—πœ”) = 𝑃(π‘—πœ”) βˆ’ 𝐺� (π‘—πœ”) = 𝑃(π‘—πœ”) βˆ’ 𝐷(π‘—πœ”)

(10)

The coefficients of the fitted function (𝑛0 , 𝑛1 , 𝑛2 , … and 𝑑0 , 𝑑1 , 𝑑2 , … ) can be evaluated by minimising the weighted sum of the squared error between the fitted polynomial and the experimental complex data at all experimental points 𝑃(π‘—πœ”) as shown in (11). Note that the weight function Wk(Ο‰k) is actually a vector with the same length as collected data P(jΟ‰). min βˆ‘π‘›π‘˜=1|𝑒(π‘—πœ”π‘˜ )|2

π‘Šπ‘˜ (πœ”π‘˜ )

(11)

𝑁(π‘—πœ”π‘˜ ) 2 οΏ½ 𝐷(π‘—πœ”π‘˜ )

= 𝑒(π‘—πœ”π‘˜ )𝑒 βˆ— (π‘—πœ”π‘˜ ) (12)

𝑒 βˆ— (π‘—πœ”π‘˜ ) = 𝑃 βˆ— (π‘—πœ”π‘˜ ) βˆ’

π‘βˆ— (π‘—πœ”π‘˜ ) π·βˆ— (π‘—πœ”π‘˜ )

π‘Šπ‘˜ (πœ”) ∈ [1, … ,1, Ξ±Ο‰1start , … , Ξ±Ο‰1end , 1, … ,1, Ξ²Ο‰2start , … , Ξ²Ο‰2end , 1, … ,1]

(15)

Convergence of the GA is depicted in Fig. 5. It shows that changing the Wk(Ο‰k) minimises the difference between the collected frequency response data and the fitted model. After some trials, the initial population and number of generations were chosen as 20 in the GA algorithm.

where the squared of error in (11) is: |𝑒(π‘—πœ”π‘˜ )|2 = �𝑃(π‘—πœ”π‘˜ ) βˆ’

variables are defined and to be optimised by GA in order to scale up the error in (11) around the two resonant modes. These parameters are mainly the lower and upper frequency ranges and the corresponding scaling factors as shown in (15).

(13)

Where (*) represents the complex conjugate operator. To minimise the error between the experimental frequency response and the fitted model, (12) can be differentiated with respect to each unknown coefficients in N(Ο‰) and D(Ο‰). πœ•π‘’

πœ•π‘› οΏ½ πœ•π‘’π‘˜ πœ•π‘‘π‘˜

π‘’βˆ— + 𝑒 π‘’βˆ— + 𝑒

πœ•π‘’ βˆ—

πœ•π‘›π‘˜ πœ•π‘’ βˆ— πœ•π‘‘π‘˜

= 0, π‘˜ = 1, … , 𝑛 = 0, π‘˜ = 1, … , 𝑛

(14)

Equation (14) can be solved by creating a set of linear simultaneous algebraic equations, for which a least-squares (LS) solution is to be found [17], [18]. Iterative approach known as iterative re-weighted linear least squares (IRLS) has been suggested in the literature for finding the weight function Wk(Ο‰k) in (11) to overcome the problem in [17] which often leads to models well fitted to high-frequency data but poorly fitted to low-frequency data [19]. However, in this paper, GA is employed to automatically find the optimal values of the required weight function Wk(Ο‰k) to emphasise particular ranges of frequencies which are more of interest in controller design stage. In IRLS method, when the desired order is specified, the steps to obtain the optimal solution are 1) to initialise the οΏ½(π‘—πœ”), 3) to use the residuals weight function, 2) to estimate 𝐺 to re-compute the weight, and 4) to repeat the iteration with the updated weight until the minimum solution is obtained. The advantage of IRLS over convex programming is that it can be used with numerical algorithms such as LevenbergMarquardt, Gauss-Newton and Gradient Methods [20], [21]. The weight function can also be interpreted as weighting function in spectral analysis which is often called β€œfrequency window”. The desired action is to use these weight functions to emphasise particular ranges of frequencies [22]. In this paper, the weighted least squares (WLS) method is combined with GA to update the weight function automatically. Therefore, in each iteration 1) GA generates a population of weight vectors Wk(Ο‰k), 2) WLS will be solved explicitly by utilising the constructed weight vector, 3) GA will alter the weight vector Wk(Ο‰k) by changing its range and its corresponding scaling factor, and 4) the iteration will continue until the minimum difference between the frequency response of the measured data and the fitted model is achieved.

Figure 5. Convergence of GA.

Table II summarises the outputs of GA. It can be inferred from the results that scaling up the frequency range of πœ”1 ∈ [604Hz βˆ’ 806Hz] by factor of 12 and similarly the frequency range of πœ”2 ∈ [1694Hz – 2166Hz] by 23 will result in the optimal fitted model of the system. Fig. 6 shows one step of the optimisation process and the effect of frequency scaling on the explicit solution of the least squares and hence on the bode magnitude of the fitted model.

In order to alter the weight vector Wk(Ο‰k), first a vector of all ones with the same length as the actual collected data is initialised. It implies no scaling at any frequency. Six 1284

TABLE II.

OPTIMUM WEIGHT FREQUENCIES AND FACTORS

Ο‰1 Start frequency of the weight vector (Ο‰start) End frequency of the weight vector (Ο‰end) Weight Factors (Ξ±,Ξ²)

Ο‰2

604 Hz

1694 Hz

806 Hz

2166 Hz

12

23

Figure 6. A snapshot of GA optimisation.

The desired order of the identified model has to be defined before the optimisation procedure is initiated. Since the total order of four is required to model the two resonant modes, it can be deduced from Fig.7 that models with order less than five fail to accurately model the actual plant. Furthermore, the higher the order model, the more difficult to design the controller (the best model obtained by GA was 11th order, see Fig. 7). The ultimate goal here is to find a model of the real system that is as simple as possible and yet capable of capturing all of the important characteristics of the plant. Therefore, a 6th order model is chosen in order to model both the rigid-body and bending-body modes with a 2nd order and 4th order transfer function, respectively. 𝑃(𝑠) =

βˆ’0.0026695 (𝑠+3.595𝑒04) (π‘ βˆ’1200) (𝑠 2 βˆ’ 1240𝑠 + 2.127𝑒07) (𝑠 2 βˆ’ 1140𝑠 + 1.631𝑒08) (s+389.8) (sβˆ’212.5) (s2 + 46.56s + 2.344e07) (s2 + 49.37s + 1.673e08)

V. CONTROLLER DESIGN From the open-loop bode plot of the system shown in Fig. 8, it is obvious that a PID controller alone cannot stabilise the system. First, two notch filters need to be designed in order to notch out the resonant frequencies. Then, a model reduction method can be used to reduce the order of the model by removing the resonant modes from the plant transfer function. Note that the RHP zero must be retained in the reduced order model. Otherwise, the designed controller will not be reliable, because RHP zeros introduce limitation on system performance and closed-loop bandwidth. The following structure is used to design the notch filters.

(16)

It is important to note that the system under study is noncollocated. It means that the position of sensors and actuators are at different points along the shaft. Noncollocation causes the zeros of the transfer function to be very sensitive to the sensor location. The system is unstable (pole at s = 212.5) and non-minimum phase. A nonminimum phase system consists of a right half plane (RHP) zero (s = 1200) which introduces limitation on the closedloop bandwidth [23]. For verification of the proposed method, the 6th order model identified via GA-based method is compared with the 6th order model obtained from the wellknown methods such as prediction error method (PEM) and state space subspace system identification (N4SID). The results are depicted and compared in Fig. 8. The results show the effectiveness of the proposed method in identifying systems with various resonant modes where more attention needs to be paid around these resonant frequencies.

𝑁(𝑠) =

𝑠 2 +πœ‰π‘πœ”π‘› 𝑠+πœ”π‘› 2

(17)

𝑠 2 +π‘πœ”π‘› 𝑠+πœ”π‘› 2

Where Ο‰n is the notch frequency, ΞΎ is the damping ratio and b bandwidth of the filter. The two designed notch filters are as follows: π‘π‘œπ‘‘π‘β„Ž1 (𝑠) =

𝑠 2 +24.2𝑠+2.35Γ—107

𝑠 2 +1940𝑠+2.35Γ—107

(18)

𝑠 2 +40.7𝑠+1.67Γ—108

(19)

βˆ’0.0023616 (𝑠+3.595𝑒04) (π‘ βˆ’1200)

(20)

π‘π‘œπ‘‘π‘β„Ž2 (𝑠) =

𝑠 2 +5160𝑠+1.67Γ—108

The reduced order model is then used to design the PID controller. The reduced order model can be obtained by removing the poles and zeros corresponding to the resonant modes from (16) and considering their DC gain in the reduced order model. π‘ƒπ‘Ÿπ‘’π‘‘π‘’π‘π‘’π‘‘ (𝑠) =

(s+389.8) (sβˆ’212.5)

A very useful PID empirical tuning formula is ZieglerNichols method. Since the frequency response of the reduced order model is available, the ultimate gain where the response of the system becomes oscillatory (Ku) and the corresponding crossover frequency (Ο‰u) can be obtained from either root-locus or Nyquist plot. Once the ultimate gain and the corresponding crossover frequency is obtained, let Tu = 2Ο€/Ο‰u. In practical application, the derivative action is usually cascaded by a first-order low pass filter in order to suppress the undesirable high frequency noise amplifications. The final PID controller parameters can be obtained from (21)-(24).

Figure 7. Bode diagram of the identified models using GA with different model orders.

𝐾𝑝 = 0.6𝐾𝑒

(21)

𝑇𝑑 = 0.125𝑇𝑒

(23)

𝑇𝑖 = 0.5𝑇𝑒

π‘ˆπ‘ƒπΌπ· (𝑠) = 𝐾𝑝 οΏ½1 +

1

𝑇𝑖 𝑠

+

(22) 𝑠𝑇𝑑

1+𝑠𝑇𝑑 πœ€

οΏ½,πœ€ β‰ͺ 1

(24)

Here Ξ΅ was chosen 0.13. Attenuation of high frequency noises by a factor of at least 100 (-40 dB/decade) is usually desired for removing the measurement noise signals above the bandwidth frequency. Therefore, another low pass-filter with cut-off frequency of 2Γ—104 (rad/s) is added in series to the PID and the two notch filters giving the following final controller design: 𝐢(𝑠) =

Figure 8. Comparison of the frequency responses of the identified model via GA, PEM and N4SID methods. 1285

3.97966𝑒5 (𝑠+295.2) (𝑠+196.8)(𝑠 2 +24.2𝑠+2.35𝑒7)(𝑠 2 +40.7𝑠+1.67𝑒7) s(s+3822)(s+2e4)(𝑠 2 +1940𝑠+2.35𝑒7)(𝑠 2 +5160𝑠+1.67𝑒8)

(25)

Ultimately, the control hardware is implemented with a dSPACE DS1104 digital control board. The controller is designed using MATLAB/Simulink. Continuous-time controller is converted to discrete-time using the Tustin approximation with a sampling frequency of 20 kHz. Two positive and negative unit step disturbances are introduced to the system in order to assess the robustness of the closedloop system in the presence of disturbances. The measurements are carried out at one end of the rotor in the xdirection. The result from our designed controller is compared with the on-board analogue controller in Fig. 9. From the results, it is clear that the designed controller on the basis of the identified model has a considerably enhanced performance.

[3] [4] [5] [6] [7] [8] [9] [10] [11]

[12]

[13] [14]

Figure 9. Performance comparison of the system under on-board analogue controller and our designed digital controller.

[15]

VI. CONCLUSION

[16]

AMBs are ideal for extremely fast and accurate operations. However, the system’s resonant modes could threaten the stability of the system and need to be accurately modelled. In this paper, GA was employed in conjunction with WLS for closed-loop identification of the AMB system in frequency-domain. Frequency windows were defined in order to improve the model around a particular range of frequencies. Compared with some widely known identification techniques, GA performed very well or even better in terms of MSE between the frequency responses of the identified model and the real system. This method is very effective in identifying system models for problems where the system has various resonant modes which have to be carefully modelled and the application of conventional algorithms is difficult. Based on the identified model, a PID controller in series with two notch filters have been successfully designed and implemented experimentally for the AMB system stabilisation. The results show the effectivenss of the designed controller based on the acquired model of the system compared to the analogue on-board controller.

[17] [18] [19] [20]

[21] [22]

[23]

VII. REFERENCES [1]

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