See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/318227338
Feedback Linearization based Optimal Controller Design for Electromagnetic Levitation System Conference Paper · July 2017 DOI: 10.1109/ICCICCT.2016.7987916
CITATIONS
READS
0
53
2 authors: Ravi V. Gandhi
Dipak Adhyaru
Nirma University
Institute of Technology, Nirma University
4 PUBLICATIONS 1 CITATION
39 PUBLICATIONS 139 CITATIONS
SEE PROFILE
SEE PROFILE
Some of the authors of this publication are also working on these related projects:
Intelligent Magnetic Levitation Controller View project
Analysis of Model based Pendulum System View project
All content following this page was uploaded by Ravi V. Gandhi on 11 October 2017. The user has requested enhancement of the downloaded file.
2016 International Conference on Control, Instrumentation, Communication and Computational Technologies (ICCICCT)
Controller Design for Feedback Linearization based Optimal Controller Electrom magnetic agnetic Levitation System Ravi V.Gandhi
Dipak M.Adhyaru M.A
Ph.D Scholar, Instrumentation & Control Department Institute of Technology, Nirma University Ahmedabad, India
[email protected]
Head, Instrumentation & Control Department Institute of Technology, Nirma University Ahmedabad, India
[email protected] [email protected]
Abstract— This paper presents the analysis of feedback linearization based optimal controller design for ElectroMagnetic Levitation (EML) system,, which is a class of nonlinear and unstable system. Using feedback linearization method, nonlinear model is first converted into controllable linearized model in large operating range under certain conditions. conditions Using linearized model, optimal controller is designed for stabilizing and tracking control of EML system. Feedback linearization based optimal controller shows kind of smooth, stabilizing and tracking response under different trajectories and bounded load variations due to variation of parameters. Simulation results and conditions under which mentioned approach is applicable are presented to validate the contribution in the paper. Keywords— Electromagnetic Levitation System, Optimal Control, Feedback linearization, Nonlinear near control, control Stabilizing control, Tracking control, Stability analysis.
I.
INTRODUCTION
Now a day, demand and need of mass transportation is on the peak, there should be the availability of such kind of mass transportation system which should be convenient, convenient environment-friendly, friendly, low maintenance, compact, light-weight, light frictionless. Magnetic levitation (Maglev) train is one of the potential candidates to satisfy those requirements [12] which work as per the principle of Electromagnetic Suspension (EMS) system. Development of such maglev system was first patented in 1934 by Hermann Kemper of Germany [1, [ 4, 12]. Prof. E. R. Laithwaite in 1965 described levitation and different mechanism related to magnetic levitation in [1]. In, 1960’s maglev system became thee favorite topic of research. Maglev phenomena using RLC based tuned circuits with detailed analysis of stability were described in [2, 3, 4]. 4] Transfer function model for maglev system using perturbation method as changes in control voltage to changes in airgap was developed by Henn in [3], to analyze the effect of parameters variations. Applications, control ontrol algorithms, types with different structures,, guidance, levitation, propulsion and working of maglev trains were reviewed very well in [5, [5 12]. Wong in 1986, proposed laboratory based small setup of linearized maglev system to stabilize spherical ball near desired position using lead compensator [6]. Dynamics of a maglev system are usually unstable and complicated [22]. Linearized model developed around nd operating conditions does not work properly under disturbance variations, large operating range, model and parameter uncertainty [7,, 8, 8 9]. In the mid of 1990’s, nonlinear feedback controllers for maglev systems were developed by researches to avoid difficulties fficulties of linearized models. Robustness for maglev system can be achieved using exact feedback linearization approach proposed in [7].
In addition to above work, vertical velocity for spherical ball was estimated using nonlinear state observer [8, 15, 25]. Digital implementation of feedback linearization based nonlinear control for airgap control with dual electromagnets for maglev system was proposed in [9]. Hall effect based position sensor with dual magnets enhances the stability of maglev system with 2-DOF control [25, 27]. Sinha in [10], proposed DSP based adaptive full state feedback control (SFC) for maglev system using stable maximum descent criterion. Mathematical modeling of maglev system on the basis differential geometry and real time nonlinear inear control based on feedback linearization with force-distance distance characteristics estimation were designed in [11]. Henly [13], proposed extended Kalman-filter Kalman based feedback linearization method for maglev system in his thesis. Coupling effect of guideway and maglev affects the levitation; nonlinear feedback linearization based controllers were developed in [14] to reduce the coupling effect on maglev. Torres and his team [16]] developed zero order Takagi-Sugeno Takagi based nonlinear controller with feedback linearization method to enhance the overall stability of maglev system. Fuzzy logic based controller algorithm with feedback linearization method to control guided elevator based on maglev concept was designed in [17, 18]. In, [24] Type-1, Type Type-2 based fuzzy PID controllers were designed for magnetic levitation system. Modeling and exact input-output output based feedback linearization for maglev system was designed using simulation in [19, 22]. Stabilizing and sinusoidal trajectory tracking control based on feedback linearization method was presented in [20]. Optimal PID controller for trajectory tracking for maglev system was presented in [23]. II.
MATHEMATICAL ATHEMATIC MODEL
In this section, nonlinear mathematical model for EML system is presented with Schematic diagram as shown in fig. 1 [23],
Figure 1. Schematic of EML system
38 978-1-5090-5240-0/16/$31.00 ©2016 IEEE
2016 International Conference on Control, Instrumentation, Communication and Computational Technologies (ICCICCT)
" =% +
Parameters used in modeling of EML system are shown in Table.1 [23], Table.1 EML System Parameters Symbol Lc
Description Coil inductance
Value 412.5 mH
Rc
Coil resistance
10Ω
Rs
Sensing resistance
1Ω
Mb
Steel ball mass
0.068Kg
Gi
Electromagnet force
6.5308E-005 N.m2/ A2
constant
9.81 m/S
Tb
Steel ball total travel
0.014 m
Above EML system is a combination of mechanical and electrical subsystems. Main purpose of the system is to maintain the position (xb = Tb - x) of steel ball which is the actual airgap by lifting the steel ball upward direction against the gravity. Using Newton’s law of motion and force body diagram, the force balance equation can be express as, +
(1)
Where, F and F are accordingly the upward lifting force by coil and equivalent disturbance in downward directions. Lifting force ( ) depends on ball position (xb) and coil current (i), as describe by following equation [23, 24], = +
−
(3)
Now, expression of coil voltage using KVL is described by, =
+
(4) can be rewrite as, =
(
1
)−
(
for EML coil loop +
(
+
)
(4) )
(5)
Using above differential equations, nonlinear state space model for EML system is developed by considering ball upward position , upward velocity and coil current as three state variables as x1 = xb, x2 = " , x3 = i and coil voltage (Vc) as controlled input (u) with ball position (Airgap) as output variable (y). Using (3) and (5) detailed nonlinear state space model is described by (6),
2
-. -/
, *2 =
0 3 + 4 0 7 8, y = *6
0 +
, *1 =
, *6 =
;. ;
