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This paper addresses a digital controller for a real time magnetic levitation system using series expansion of pulse transfer function, which achieves desired closed loop response. The proposed digital controller designed, based on series expansion of pulse transfer function by solving a linear equation using the method of least squares, which improves the transient performance and step tracking capability of the compensated system. The designed algorithm used for the control input is not iterative, so the calculation is very fast. The proposed control scheme has successfully applied on maglev system and also validated through the simulation and hardware experimental results.

Magnetic Levitation technology has received tremendous innovation in various engineering fields and is being utilized in various automation applications [

The magnetic levitation system is nonlinear and unstable. There are various control strategies [

The major finding from the above existing control strategies is that, the transient response (settling time and peak overshoots) of magnetic levitation system is not up to mark. Digital control provides flexibility and easy implementation of wide range of control algorithms over there analog counter parts and also achieves deadbeat response [

Looking into the advantages of digital control, the present work proposes a control scheme for magnetic levitation system, which is based on series expansion of pulse transfer function [

The rest of the article is organized as following. The mathematical modeling and control system of magnetic levitation system is given in Section 2. The brief background of series expansion method of controller design algorithm is given in Section 3 and the designing steps of controller for magnetic levitation system are given in Section 4. The simulation and experimental results are obtained in Section 5. The conclusion of the work is given in Section 6 and then after references.

The systematic diagram of magnetic levitation system is shown in

The maglev system mainly consists of four major parts: suspended steel ball, position Infra Red (IR) sensors, controller and actuator (including electro magnet and power amplifier). The steel ball is mainly controlled through current i, as clearly indicated in

The non-linear model of magnetic levitation system [

where C is a constant value which depends on the parameters of coil,

The magnetic levitation system expressed by (1) is nonlinear in nature. For easy analysis and design of controller, the system is linearized about the equilibrium point (

The linearized model of magnetic levitation system is obtained as:

By calculating the partial derivative and taking Laplace transform on both side of Equation (3) we get the transfer function as

where

In electrical equivalent circuit of magnetic levitation system in

where

Now, the transfer function can be written as:

where

The transfer function of magnetic levitation system with sensor system is obtained as:

where

Using given values in the

The Maglev system (9) has two poles at ±46.69. It is seen that one pole lies in right half of complex s-plane so system is unstable. Hence, the aim is to design a controller, which leads to overall stable system.

Let the pulse transfer function of the plant (Maglev System) and controller be P(z) and C(z) respectively [

Description of Parameters | Value with Unit |
---|---|

mass of the steel ball (m) | 0.02 kg |

Acceleration due to gravity (g) | 9.81 m/s^{2} |

Equilibrium value of current (i_{0}) | 0.8 A |

Equilibrium value of position (h_{0}) | 0.009 m |

Control voltage to coil current gain (C_{1}) | 1.05 A/V |

IR sensor gain (C_{2}), offset | 143.48 V/m, −2.8 V |

Control input voltage level (v) | ±5 V |

Sensor output voltage level (h_{v}) | +1.25 to −3.75 V |

The open loop pulse transfer function O(z) can be written as:

From (10), (11) and (12) the coefficient

(13)

The closed loop pulse transfer function for the above system can be expanded as:

Employing (13), the coefficients

Thus, the series expansion coefficient of closed loop pulse transfer function is expressed in terms of series expansion coefficient of open loop pulse transfer function and these series coefficients are arbitraterely chosen for obtaining the desired performance. The proposed control scheme is basically design on the basis of number of series coefficient of plant (10) and number of series coefficient of controller (11) on their expansion that are taken as m and n respectively during the design procedure that are discussed in next section.

Let the pulse sequence

Step 1: First specify the desired pulse response sequence

Step 2: Using the (15) solve for

(16)

Step 3: Now substitute

where

Step 4: Now, solve Equation (17) by the method of least squares and the solution of the calculated controller coefficient

Step 5: The controller coefficients are expressed as:

The (22) is the designed controller for a specific value of series coefficients of plant and controller as m and n respectively.

Note 1: If the response of closed loop system obtained from (22) along with (10) does not track the desired trajectory, then value of series coefficients of pant m and controller n are increased and all the above five steps are repeated.

The next section presents the simulation at different sampling times and various inputs as well as the hardware results for sinusoidal input when controller given by (22) is applied on maglev system (9).

The simulation diagram of proposed digital control algorithm for the controlling of maglev system is given in

The simulations are carried out for two cases, one at sampling time Ts = 0.0001 second and secondly at 0.001 second.

Case 1: Plots at sampling time (Ts) = 0.0001 second

Let the sampling time (Ts) is 0.0001 second and the desired pulse sequence is W = [0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1 1 1 ×××]. Now for step input we have to design a controller so that it can track the step input. Once the controller is designed with the help of series expansion of pulse transfer function subjected to step input then it is also effective for all type of inputs. The performance of designed controller depends upon number of series coefficients m and n considered for plant and controller respectively and plots are given for following conditions that are given below.

The simulation results for various combination of number of considered series coefficient of plant and controller are discussed below

1) For m = 25 & n = 2

In this case the controller series coefficient is obtained as

The eigen values of (23) lie at 0.9911, 0.8391, 0.3362 and −0.1825 and all are within the unit circle. Hence, system is stable. The simulation results are plotted for different inputs such as step, square wave and sinusoidal in Figures 4(a)-(c).

2) For m = 25 & n = 3

In this case, the proposed controller coefficient is obtained as

It is found that, all the eigen values of (24) lie within the unit circle. The simulation results are plotted for different inputs such as step, square wave and sinusoidal from Figures 5(a)-(c).

The step performance of proposed controller is summarized in

Remark 1: Looking at the

m & n | Rise Time (second) | Settling Time (second) | Overshoot | Peak | Peak Time (second) |
---|---|---|---|---|---|

m = 25 & n = 2 | 0.001 | 0.0111 | 3.65% | 1.05 | 0.0038 |

m = 25 & n = 3 | 0.001 | 0.0057 | 2.19% | 1.04 | 0.0039 |

is reduced from 0.0111 second to 0 .0057 second and overshoot is also decreased from 3.65% to 2.19%. The simulation results for various input clearly state that the proposed algorithms is gives the better tracking response whatever the input such as step, square and sinusoidal.

Note 2: Experimental results cannot be verified for sampling time Ts = 0.0001 second because the magnetic levitation provided by Feedback Instrument is manufactured for sampling time Ts = 0.001 second.

Now, the performance of designed controller is tested at sampling time 0.001 second through simulation as well as on the system hardware (Magnetic Levitation System 33 - 210, Feedback Instruments) which is presented through Case 2.

Case 2: Plots at sampling time (Ts) = 0.001 second

The similar steps are carried out as in Case 1 for designing of controller. The simulation and hardware experimental results are plotted for following conditions as in Section A and Section B respectively in below.

A. Simulation results for various inputs at sampling time Ts = 0.001 second

1) For m = 7 & n = 2

In this case the controller coefficient is obtained as

and all the eigen values of (25) lie within the unit circle. The simulation results are plotted for various inputs as shown in Figures 6(a)-(c).

2) For m = 12 & n = 3

In this case the controller coefficients are obtained as

Here, also all the eigen values of (26) lie within unit circle. The results are plotted for various inputs as shown in Figures 7(a)-(c).

Remark 2: The simulation results for Case 2 of m = 7 & n = 2 and m = 12 & n = 3 are plotted in Figures 6(a)-(c) and Figures 7(a)-(c), from the above Figures it is clear that, tracking is almost achieved for desired trajectories such as step, square and sinusoidal. The step performances for this case are summarized in

It is seen from

B. Hardware experimental results

The effectiveness of proposed controller is verified on setup of maglev system (33-942S) provided by feedback instrument. The maglev setup has two PCI port as PCI1711 Lab I/O ADC port is configured for plant output and PCI1711 Lab I/O DAC port is dedicated for input to the maglev system. The maglev system is manufactured

m & n | Rise Time second) | Settling Time (second) | Overshoot | Peak | Peak Time (second) |
---|---|---|---|---|---|

m = 7 & n = 2 | 0.01 | 0.07 | 13.43 | 1.66 | 0.04 |

m = 12 & n = 3 | 0.01 | 0.07 | 7.39 | 1.66 | 0.04 |

for sampling time Ts = 0.001 second. The proposed hardware experimental diagram is given in

The hardware results are tested for all two cases as discussed in Section A for Case 2 and hardware result is plotted for sinusoidal input at sampling tine 0.001 second. The position of ball to reference input is presented in voltage [V] as well as in meter (m) along with control effort in voltage [V].

1) For m = 7 & n = 2

For this case, hardware experimental result is shown from

2) For m = 12 & n = 3

For this case, the hardware experimental result is shown from

Remark 3: From the hardware experimental results as shown in

It could be remarked here that on increasing the series coefficients of plant and controller, the transient and steady state behavior of system have been improved.

Comparison

The designed control strategy is quite useful for complex system and it can be easily implemented on any real time system via computer-programmed algorithm where as conventional continuous control scheme may suffer during real time implementation of linear or nonlinear control algorithms. To show the effectiveness of proposed control strategy, a comparative simulation result of designed control scheme (m = 25 & n = 3 at sampling time 0.0001 second) with conventional PID (

The comparative results analysis with conventional PID and FOPID controller are given in

From the

magnetic levitation provided by Feedback Instrument is manufactured for sampling time Ts = 0.001 second.

The designed controller lies in z-domain and it will bypass the requirement of higher sampling rate. Another beauty of this design algorithm is that it is applicable to any higher order system also.

Type of Control | Rise Time (second) | Settling Time (second) | Overshoot | Peak | Peak Time (second) |
---|---|---|---|---|---|

Proposed control scheme m = 25 & n = 3 | 0.001 | 0.0057 | 2.19% | 1.040 | 0.0039 |

With Conventional PID control [_{ } | 0.0030 | 0.9211 | 15.07% | 1.1507 | 0.1327 |

With FOPID Control [ | 0.0034 | 0.9712 | 37.6215 | 1.3767 | 0.0086 |

An algorithm for digital controller design has been proposed and implemented for a magnetic levitation system. The proposed digital controller is designed based on series expansion of pulse transfer function by solving a linear equation using the method of least squares. The simulation and hardware experimental results are given to show the applicability of proposed controller. The designed controller provides better tracking and transient response (settling time and peak overshoots etc.) as number of series coefficient of plant and controller is increased. The designed algorithm used for the control input is not iterative so the calculation is very fast. The proposed control technique is also compared with convention PID and FOPID control scheme. In this method the reliability criterion for a controller should be satisfied when the desired pulse response sequence is known. This method can be used for stable plant as well as unstable plant. Furthermore, it is possible to extend the method for multi input multi output (MIMO) system also.

Pati, A., Verma, V.K., Negi, R. and Nagar, S.K. (2016) Real Time Implementation of Series Expansion Based Digital Controller for Magnetic Levitation System. Intelligent Control and Automation, 7, 110-128. http://dx.doi.org/10.4236/ica.2016.74011