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Real Time Implementation of Optimized I-PD Controller for the Magnetic Levitation System using Jaya Algorithm
5th International Conference on Advances in Control and 5th International Conference on Advances in Control and Optimization of Dynamical Systems Available onlineand at www.sciencedirect.com 5th International Conference on Advances in Control Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India 5th International Conference on Advances in Control and February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India February 18-22, 2018. Hyderabad, India
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IFAC PapersOnLine 51-1 (2018) 106–111
Real Time Implementation of Optimized I-PD Controller for the Real Time Implementation Optimized the Magnetic Levitation of System using I-PD Jaya Controller Algorithm for Real Time Implementation of Optimized I-PD Controller for the Real Time Implementation Optimized Magnetic Levitation of System using I-PD Jaya Controller Algorithm for the Magnetic Levitation Levitation System using Jaya Algorithm Algorithm * ** Kumar Swain, Magnetic System Jaya *Debdoot Sain, **Subrat using * ** *** Debdoot Sain, Subrat Kumar Swain,
Kumar Mishra * *Debdoot **Subrat ***Sudhansu *** Sain, ** Kumar * ** Kumar Sudhansu Mishra Swain, Debdoot Sain, Subrat Kumar Swain, *** ***Sudhansu Kumar Mishra of Technology, Electrical *** Engineering, Indian Institute
* Department of Kharagpur, Sudhansu Kumar Mishra of Technology, Kharagpur, * Department of Electrical Engineering, Indian Institute +91-9955624206; e-mail: [email protected]) Pin: 721302, India (Tel: of Electrical Engineering, Indian Institute of Technology, Kharagpur, **,*** * Department +91-9955624206; e-mail: [email protected]) Pin:of721302, India Electrical and(Tel: Electronics Engineering, Birla of Technology, * Department of Electrical Engineering, Indian Institute of Institute Technology, Kharagpur,Mesra, **,***Department **,*** (Tel: +91-9955624206; e-mail: [email protected]) Pin:of721302, India ** Department Electrical and Electronics Engineering, Birla Institute of Technology, Mesra, (e-mail: [email protected], ***[email protected]) Ranchi, Pin: 835215, India. (Tel: +91-9955624206; e-mail: [email protected]) Pin: 721302, India **,*** **,***Department of Electrical and Electronics ** ** Engineering, Birla Institute of Technology, Mesra, **,*** (e-mail: [email protected], ***[email protected]) Ranchi, Pin: 835215, India.and Department of Electrical Electronics Engineering, Birla Institute of Technology, Mesra, Ranchi, Pin: 835215, India. (e-mail: ******[email protected], ***[email protected]) Ranchi, Pin: 835215, India. (e-mail: [email protected], ***[email protected])
Abstract: Designing a suitable controller for a nonlinear and unstable plant is always very challenging to Abstract: a suitable controller for a nonlinear unstable plant is always very challenging to the controlDesigning system practitioners. In this article, Integral and - Proportional Derivative (I-PD) controller has Abstract: Designing a suitable controller for a nonlinear and unstable plant is always very challenging to the control system practitioners. In this article, Integral Proportional Derivative (I-PD) controller has been designed and implemented in simulation and real time for the Magnetic levitation (Maglev) system Abstract: Designing a suitable controller for a nonlinear and unstable plant is always very challenging to the control system practitioners. in Insimulation this article,and Integral - Proportional Derivative (I-PD) controller has been designed and implemented real time for the Magnetic levitation (Maglev) system which is both nonlinear and unstable in article, nature. Integral The nonlinear Maglev plant has been linearized around the control system practitioners. In this - Proportional Derivative (I-PD) controller has been designed and implemented in simulation and real time for Maglev the Magnetic levitation (Maglev) system which is both nonlinear and unstable in nature. The nonlinear plant has been linearized around the equilibrium point to obtain the model transfer function. The objective function, which has been designed and implemented in linearized simulation and real time for the Magnetic levitation (Maglev) system which is both nonlinear and unstable in nature. The transfer nonlinear MaglevThe plant has been linearized around the point to obtain the linearized model function. objective function, has beenequilibrium formulated by taking modulus of the characteristic polynomial ofhas thebeen plant alongwhich with the which is both nonlinear and the unstable in nature. The nonlinear Maglev plant linearized around the equilibrium point to obtain the linearized model transfer function. The objective function, which has been formulated by taking the modulus of the characteristic polynomial of the plant along with the controller at dominant location, has beenmodel minimized using the recently evolved Jaya algorithm. the equilibrium point topole obtain the linearized transfer function. The objective function, which The has been formulated by taking the modulus of the characteristic polynomial of the plant along with The the controller at dominant pole location, has been minimized using the recently evolved Jaya algorithm. performance of the controller has been with thatpolynomial of the 1-DOF Integer Order been formulated byI-PD taking the modulus of compared the characteristic of and the 2-DOF plant along with the controller at dominant pole location, has been minimized using the recently evolved Jaya algorithm. The performance of the I-PD controller that of recently the 1-DOF 2-DOF Integer Order (IO) and Fractional Order (FO) PIDhas controller (Swain etwith al., 2017). The result ofand comparison reveals that controller at dominant pole location, hasbeen beencompared minimized using the evolved Jaya algorithm. The performance of the I-PD controller been compared of The the 1-DOF 2-DOF Integer Order (IO) and Fractional Order (FO)using PIDhas controller (Swainoutperforms etwith al., that 2017). result ofand comparison reveals that the I-PD controller optimized Jaya algorithm controllers in terms ofOrder time performance of the I-PD controller has been compared with that ofthe theother 1-DOF and 2-DOF Integer (IO) and Fractional Order (FO)using PID controller (Swainoutperforms et al., 2017).the Theother result of comparison reveals that the I-PD controller optimized Jaya algorithm controllers in terms of time domain and frequency domain specifications except the phase margin criteria. The robustness analysis (IO) and Fractional Order (FO) PID controller (Swain et al., 2017). The result of comparison reveals that the I-PDand controller optimized using Jaya algorithm outperforms the other controllers in terms analysis of time domain frequency domain specifications except the phase margin criteria. The robustness has incorporated in this study to show the robust behaviour the I-PD controller. the also I-PDbeen controller optimized using Jaya algorithm outperforms theofother controllers in terms of time domain domain the phase margin criteria. The robustness analysis has also and beenfrequency incorporated in thisspecifications study to showexcept the robust behaviour of the I-PD controller. domain and frequency domain specifications except the phase margin criteria. The robustness analysis Keywords: Maglev, PID, I-PD, Optimization, Jaya. has also IFAC been incorporated this studyoftoAutomatic show the robust of Elsevier the I-PDLtd. controller. © 2018, (InternationalinFederation Control)behaviour Hosting by All rights reserved. has also been incorporated in this study to show the robust behaviour of the I-PD controller. Keywords: Maglev, PID, I-PD, Optimization, Jaya. Keywords: Maglev, PID, I-PD, Optimization, Jaya. Keywords: Maglev, PID, I-PD, Optimization, Jaya. linearized model transfer function is used for the design of 1. INTRODUCTION linearized model transfer function is used the design of the I-PD controller. Recently developed Jayaforalgorithm (Rao, 1. INTRODUCTION linearized model transfer function is used foralgorithm the design of the I-PD controller. Recently developed Jaya (Rao, is model employed to function suitably isidentify Simple structure and1.easy implementation has made the PID 2016) INTRODUCTION linearized transfer used forthe the controller design of the I-PD controller. Recently developed Jaya algorithm (Rao, 1. INTRODUCTION 2016) employed to suitably identify the controller Simple structure and easy hasformade thetime. PID parameters by optimizing the objective function which is controller as the choice of implementation control engineers a long the I-PDiscontroller. Recently developed Jaya algorithm (Rao, 2016) is employed to suitably identify the controller Simple structure and easy implementation hasformade thetime. PID parameters by optimizing the objective function which is controller as the choice of control engineers a long formulated by taking the modulus of characteristic Till now PID controller is the most common controller used 2016) is employed to suitably identify the controller Simple structure and easy implementation has made the PID parameters by optimizing the objective function which is controller as the choice of control engineers for a long time. formulated by taking the modulus of characteristic Till now PID controller is the most common controller used polynomial by of optimizing the plant along with the function controller at the in industryas and most of controllers PID time. type. parameters the objective which is controller the choice of the control engineersareforofa long Till now PIDand controller most common used formulated by the modulus characteristic polynomial of thetaking plantAsalong with the of controller at are the in industry most satisfactory ofis controllers arecontroller of with PID good type. dominant pole the singularities of the system Though, it provides performance bylocation. taking the modulus of characteristic Till now PID controller is the the most common controller used formulated of location. the plantAsalong with the controller at are the in industry and most satisfactory of the controllers are of with PID good type. polynomial pole the of the system Though, itcriteria; provides performance usually complex nature,along thesingularities modulus hasatbeen robustness still of there a possibility getting polynomial of thein plant with the operator controller the in industry and most theiscontrollers areofof PID more type. dominant pole location. As the singularities of the system are Though, itcriteria; provides satisfactory performance with more good dominant usually complex in nature, the modulus operator has been robustness still there is a possibility of getting taken for pole getting a realAsfunctional value of of the thesystem objective improved itperformance by slightly modifying the structure of dominant location. the singularities are Though, provides satisfactory performance with good complex ina nature, the modulus has been robustness criteria; stillby there is a possibility of more taken for real functional valueoperator of the should objective improved performance slightly modifying thegetting structure of usually function. Ingetting an ideal case, functional value be the PID controller. In this paper, I-PD controller has more been usually complex in nature, the modulus operator has been robustness criteria; still there is a possibility of getting taken for getting a real functional value of the objective improved performance by slightly modifying the structure of Ingetting an ideal case, the functional be the PID controller. In by this paper, I-PD has been equal toforzero. designed for the Maglev system to modifying show controller the improvement in function. taken a real functional value value of the should objective improved performance slightly the structure of function. In an ideal case, the functional value should be the PID controller. In this paper, I-PD controller has been equal to zero. designed for the Maglev system to show the improvement in the system performance as compared to thecontroller PID controller. In an idealdesign case, isthecomplete, functional should be PID controller. In this paper, I-PD has been function. Once to thezero. controller thevalue performance of equal designed the Maglevassystem to show improvement the systemfor performance compared to thethe PID controller. in equal to zero. designed for the Maglev system to show the improvement in Once the controller design is complete, the performance of Because inherent non linearity and unstable open loop the I-PD controller is compared with that of the 1-DOF and the systemofperformance as compared to the PID controller. Once the controller controller isdesign is complete, the performance of the system performance as compared to the PID controller. the I-PD compared with that of the 1-DOF and Because of inherent non linearity and unstable open loop 2-DOF IOPID (three parameters to tune) and FOPID (five behaviour, it is of prime importance to design a suitable Once the controller design is complete, the performance of the I-PD controller is compared with that of the 1-DOF and Because of inherent non linearity and unstable open loop 2-DOF IOPID (three parameters to tune) and FOPID (five behaviour, itinherent is of prime importance design open aissuitable parameters to tune) iscontroller discussed al. controller the Maglev as and its to application widely I-PD controller comparedaswith that of by theSwain 1-DOFetand Because offor non system linearity unstable loop the 2-DOF IOPID (threecontroller parameters tune) and FOPID behaviour, it the is of primesystem importance to design aissuitable parameters to tune) as to discussed by Swain et(five al. controller for Maglev as its application widely The result of comparison reveals that the I-PD controller spread in different fields of research which include high2-DOF IOPID (three parameters to tune) and FOPID (five behaviour, it is of prime importance to design a suitable parameters to tune) controller as discussed by Swain et al. controller for the Maglev system as its application is widely The result of comparison reveals that the I-PD controller spread in different fields of research which include highJayacontroller algorithm as outperforms controllers speed systems (Holmer, is widely 2003), optimized parametersusing to tune) discussed other by Swain et al. controller transportation for the Maglev system as its application The resultusing of comparison reveals that the other I-PD controllers controller spread in transportation different fields ofsystems research which include 2003), high- in speed (Holmer, Jaya algorithm outperforms terms of time domain andreveals frequency specifications photolithography semiconductor manufacturing The result of comparison thatdomain the I-PD controller spread in differentdevices fields for of research which include high- optimized Jaya algorithm outperforms other controllers speed transportation systems (Holmer, 2003), optimized in termsthe ofusing time domain and frequency domain specifications photolithography semiconductor manufacturing except phase margin criteria. The robustness analysis has (kim 1998),devices seismic for attenuators for(Holmer, gravitational wave optimized using Jaya algorithm outperforms other controllers speedet al.,transportation systems 2003), in termsthe of phase time domain and frequency domain specifications photolithography devices for semiconductor manufacturing except margin criteria. The robustness analysis has (kim et al., 1998), seismic attenuators for gravitational wave been incorporated in this study domain to show the robust antennas (Varvelladevices et al., for 2004), self-bearing manufacturing blood pumps also in terms of time domain and frequency specifications photolithography semiconductor the phase margin criteria. The robustness analysis has (kim et al., 1998), seismic attenuators for gravitational wave except also been incorporated in this study to show the robust antennas (Varvella et al., 2004), self-bearing blood pumps behaviour the I-PD controller. (Masuzawa et al., seismic 2003) for the use in heartswave etc. except the of phase margin criteria. The robustness analysis has (kim et al., 1998), attenuators forartificial gravitational also been incorporated in this study to show the robust antennas (Varvella et al., 2004), self-bearing blood pumps the I-PD controller. (Masuzawa et al., for thebyuse artificial etc. Several efforts have2003) been the in engineers tohearts design an behaviour also been ofis incorporated in five this sections. study to Section show the antennas (Varvella et al.,made 2004), self-bearing blood pumps This paper organized into 1 isrobust about behaviour of the controller. (Masuzawa et al., 2003) for the use in artificial hearts etc. Several efforts have been made by the engineers to design an This paper is organized into fiveThe sections. Section 1 is about efficient controller for the Maglev system using different of the I-PD I-PD controller. (Masuzawa et al., 2003) for the use in artificial hearts etc. behaviour the introduction of the paper. schematic diagram and Several efforts have been made by engineers to efficient controller for the system using different paper is organized fiveThe sections. Section 1 is about the introduction of the into paper. schematic diagram and control techniques al.,Maglev 2007, et al., 2016, Lin an et This Several efforts have(Lin beenet made by the theSain engineers to design design an calculation oforganized linearized model transfer function ofabout the This paper is into five sections. Section 1 is efficient controller the system using different introduction of the paper. Thetransfer schematic diagram and control techniques al.,Maglev 2007, Sain et al., 2016, Lin et et the calculation of linearized model function of the al., 2005, Chopade (Lin etfor 2016, Ghosh et al., 2014, Swain efficient controller foral.,et the Maglev system using different Maglev system isof provided in section 2. Section diagram 3 deals with the introduction the paper. The schematic and control techniques (Lin al., 2007, Sain et 2016, Lin of is linearized model transfer function of with the al., Chopade et 2017). al.,et Ghosh et still al., 2014, Swain et Maglev system provided in section 2. Section 3 deals al., 2005, 2017, Sain et al., there is a possibility control techniques (Lin et 2016, al.,But 2007, Sain et al., al., 2016, Lin to et calculation the design of I-PD controller for the Maglev system. Section calculation of linearized model transfer function of the al., 2005, Chopade et al., 2016, Ghosh et al., 2014, Swain et Maglev system is provided in section 2. Section 3 deals with al., 2017, Sain et al., 2017). But there is still a possibility to the design of I-PD controller for the Maglev system. Section improve performance the Maglev by modifying al., 2005,the Chopade et al., of 2016, Ghosh etsystem al., 2014, Swain et 4 is about the response of theinsystem the I-PD controller, system is provided sectionwith 2. Section 3 deals with al., 2017,the Sain et al., 2017). ButMaglev there issystem still a possibility to Maglev design of I-PD controller for the with Maglev system. Section improve performance of the 4 is about the response of the system the I-PD controller, the 2017, traditional control structure themodifying efficient al., Sain et al., 2017). But and thereemploying is still a by possibility to the performance comparison and robustness analysis. In Section section the design of I-PD controller for the Maglev system. improve the performance of the Maglev system by modifying is about the comparison response of the system with analysis. the I-PD controller, the traditional control structure and employing themodifying efficient andscope robustness In section optimization algorithms. Maglev plant has 4 improve the performance ofThe the nonlinear Maglev system by 5,isconclusion the further ofwith research is highlighted. 4performance about the and response of the system the I-PD controller, the traditionalalgorithms. control structure and employing theplant efficient performance comparison and robustness analysis. section optimization The nonlinear Maglev has 5, conclusion and the further scope of research is highlighted. beentraditional linearizedcontrol around the equilibrium point the performance comparison and robustness analysis. In the structure and employing the and efficient In section optimization algorithms. The nonlinear Maglev plant has 5, conclusion and the further scope of research is highlighted. been linearized around the equilibrium point and the optimization algorithms. The nonlinear Maglev plant has 5, conclusion and the further scope of research is highlighted. been been linearized linearized around around the the equilibrium equilibrium point point and and the the 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 IFAC 106 Peer review©under of International Federation of Automatic Copyright 2018 responsibility IFAC 106Control. 10.1016/j.ifacol.2018.05.018 Copyright © 2018 IFAC 106 Copyright © 2018 IFAC 106
5th International Conference on Advances in Control and Optimization of Dynamical Systems Debdoot Sain et al. / IFAC PapersOnLine 51-1 (2018) 106–111 February 18-22, 2018. Hyderabad, India
107
f (i, x ) .. f (i, x ) (2) x i x i x i0 , x 0 i0 , x 0 where, x and i is the small deviation from the equilibrium
2. MAGLEV SYSTEM The schematic diagram of the Maglev system (Ghosh et al., 2014) is depicted in Fig. 1. In this study, the Maglev system with Model no. 33-210 from Feedback Instruments is considered. The coil gets magnetized when current flows through it and attracts the ball in the upward direction. At the same time, the gravitational force of earth pulls it in the downward direction. There is an infrared sensor present in the system which continuously monitors the position of the ball.
point x0 and i0 respectively. Evaluating partial derivatives and taking Laplace transform on both side of (2), the transfer function can be obtained as x i
ki
s
2
(3)
kx
where, k i
2g i0
and k x
2g x0
As x and i are proportional to x v and u, the transfer function can be modified to the form x v / u (Ghosh et al., 2014) and given by G p (s)
xv u
b
s
2
p
2
3518 . 85 s
2
(4)
2180
where, x v is sensor output and u is the input to current amplifier. Fig. 1. Schematic diagram of Maglev system
The poles of the Maglev system are located at 46.69. Because of the presence of one pole on the right half of ‘s’ plane, the system becomes highly unstable. To achieve a stable and controlled behaviour of the system, it becomes extremely important to design suitable controller.
The mechanical, electrical unit along with connectioninterface panel that assembles into a complete control system setup is provided in Fig. 2
Table 1. The system parameters of the Maglev system (Ghosh et al., 2014)
Fig. 2. Maglev control system In terms of ball position x and electromagnetic coil current i, the simplest nonlinear model (Ghosh et al., 2014) of the Maglev system is given by ..
m x mg k
i x
2 2
(1)
Parameter
Notation
Value
Mass of the steel ball
m
0.02 kg
Acceleration due to gravity
g
9.81 m/s2
Equilibrium value of current
i0
0.8 A
Equilibrium value of position
x0
0.009 m
Control voltage to coil current gain
k1
1.05 A/V
Sensor gain, offset
k2 ,
143.48 V/m, -2.8 V
Control voltage input level
U
Sensor output voltage level
xv
where, m represents the mass of the ball, g is the gravitational constant and k is dependent on the coil parameters.
±
5V
+ 1.25 V to -3.75V
3. DESIGN OF I-PD CONTROLLER
For the linearization of the nonlinear Maglev plant, the calculation of equilibrium point is mandatory. The equilibrium point of the current and position is calculated by
3.1. I-PD Controller Design using Jaya
..
3.1.1. Dominant Pole calculation
equating x 0 and found to be at 0.8A and 0.009 m (-1.5 V, when expressed in volts) respectively. The nonlinear system (1) around the obtained equilibrium point can be linearized to
For this particular study, the design specifications have been considered as
107
5th International Conference on Advances in Control and 108 Debdoot Sain et al. / IFAC PapersOnLine 51-1 (2018) 106–111 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
Damping ratio = 0.8 and Settling time ts =
4
f I PD min 1
2 sec
3518 . 85 2 s 2180
K
P
K
I
K
s
D
s
n
Using = 0.8 and ts = 2 sec, the value of n has been found as 2.5 rad/sec and the dominant poles have been obtained by solving the standard second order characteristic equation and given by s 1, 2 2 1 . 5 i .
(8) s s1
3.1.4 Objective Function Optimization using Jaya The Jaya algorithm (Rao et al., 2016) is a modern and recently emerged optimization algorithm inspired by the popular Teaching Learning Based Optimization (TLBO) technique. Unlike TLBO, the algorithmic steps of Jaya include only a single phase and the computational execution time is reduced as compared to TLBO where two phases (teaching and learning) are involved. Both algorithms do not require any algorithm-specific parameters, but at the same time, the working patterns of both the algorithms are different from each other. However, in terms of implementation and performance, the Jaya algorithm is simple as well as effective. In case of Jaya algorithm, the objective function f obj ( x ) , to be minimized at any iteration
3.1.2. I-PD Controller The I-PD controller is a simple modification of conventional the PID controller. The integral (I) block is present on the forward path while the proportional and derivative (P-D) blocks are kept in the feedback path. As different signal paths are present for the set-point and process outputs, the I-PD controller has got more flexibility to satisfy the design specifications accurately. The structure of the I-PD controller is shown in Fig. 3.
t comprises of ‘m’ design variables and ‘n’ candidate solutions. Among all the possible candidate solutions, the best candidate will provide an optimal value of f obj ( x ) . If X
p ,q ,r
be the value of pth variable for qth candidate during the
rth iteration, then the updated state equation is represented by the following equation X
new
X p , q , r k1, p , r X p , best , r X p , q , r
p ,q,r
Fig. 3. Control Structure of an I-PD controller
Characteristics equation of the system with I-PD controller for unity feedback is given by
1 G p (s) G c (s) G c I
PD
(s) 0
K I K P K D s 0 s
i.e 1 G p ( s )
where, X
X
(5)
X
(6)
new p ,q ,r
is the value of ‘p’ for the best candidate,
is the value of ‘p’ for the worst candidate, and
is the updated value of the term X and
k 2, p ,r
p ,q ,r
. Whereas,
updated random numbers for the pth
variables during rth iteration in the range [0, 1]. The term
k 1, p , r X
With Maglev system the characteristics equation becomes, 3518 . 85 K I 1 K P K D s 0 2 s 2180 s
p , worst , r
k 1, p , r
where, K P , K I and K D represent proportional, integral and derivative gain respectively.
p , best , r
(9)
k 2 , p , r X p , worst , r X p , q , r
p , best , r
X
p ,q ,r
is responsible for making the
solution closer to the best one, whereas the term is responsible for the k 2 , p , r X p , worst , r X p , q , r
(7)
elimination of the worst solutions. If the updated value
Interestingly the characteristics equation of the plant with PID and I-PD controller is identical for the same set of controller parameter values. But the removal of inappropriate zeros from the closed loop transfer function, in case of the I-PD controller, improves the response of the system as compared to the PID controller which will be evident in the results and discussion section.
X
new
provides better optimal functional value then it will be
p ,q ,r
accepted as the input to next iteration. The objective function considered over here has three unknowns i.e. K P , K I and K D . As the range of unknowns affects the optimality of the solution, in the beginning, a wider solution space is considered. After getting the initial solution, in the subsequent steps, the solution space has been shortened. The range of these parameters taken for writing the MATLAB code is provided in Table 2. It can be noted that all the gains are negative and the reason can easily be verified from the Routh–Hurwitz stability criterion.
3.1.3 Objective Function Formulation The objective function f I PD considered for obtaining the value of K P , K I and K D has been formulated by taking the modulus of the characteristic polynomial of the plant along with the controller at dominant pole location and given by
108
5th International Conference on Advances in Control and Optimization of Dynamical Systems Debdoot Sain et al. / IFAC PapersOnLine 51-1 (2018) 106–111 February 18-22, 2018. Hyderabad, India
6
Table 2. Range of controller parameters considered for writing the MATLAB code
0
0
10
20 30 time ( sec )
40
50
Fig. 4. Simulink response of Maglev system with different controllers The performance of the Maglev system with I-PD controller has been compared with the 1-DOF and 2-DOF IOPID and FOPID controller (Swain et al., 2017) and summarized in Table 4 and Table 5. 6 ball position ( V )
KI -6.8
2
-2
Table 3. I-PD controller parameter values after optimization KP -2.5
Reference 1-DOF IOPID 1-DOF FOPID 2-DOF IOPID 2-DOF FOPID I-PD using Jaya
4
ball position ( V )
KI KD Parameter KP Lower range -2.5 -8 -0.165 Upper range 0 -6.8 -0.1 After optimizing the objective function through Jaya algorithm within the mentioned range of parameters with population size 50 and number of iteration 10, the values of K P , K I and K D have been found and given in Table 3. With these set of controller parameters, the objective function value is 2.4564, which is not equal to zero but as all the closed loop poles of Maglev system with I-PD controller lie on the left half of ‘s’ plane, the system is guaranteed to be stable. It has been observed that using the Genetic algorithm available in Optimization Tool in MATLAB, it requires 51 iterations to reach the optimum values, whereas, Jaya takes very few iterations to produce the same optima. It has also been seen that the optima reached using Jaya is on the constraint boundaries and it is expected to change whenever the boundary is changed, but as we are getting satisfactory performance with these set of controller parameters, we have not explored further.
Parameter Value
109
KD -0.165
Reference 1-DOF IOPID 1-DOF FOPID 2-DOF IOPID 2-DOF FOPID I-PD using Jaya
4 2 0 -2
4. RESULTS AND DISCUSSION
0
For this study, a square wave with mean -1.55 V has been considered as the input signal. The reason behind choosing the input signal with mean -1.55 V is to ensure that the operating point of the system is around the equilibrium point. The simulation has been carried out for 50 seconds and the response of simulation and real-time with I-PD controller optimized using Jaya has been provided in Fig. 4 and 5. For the real time simulation, first 5 seconds have been used for compensating the disturbance encountered, due to hand positioning of the ball as well as inherent nonlinearities present in the system. The control signal associated with the I-PD controller for the real time control of Maglev plant has also been provided in Fig. 6. Once the ball gets stabilized, it can be observed that the control signal remains in the range of ± 5 V.
10
20 30 time ( sec )
40
50
Fig. 5. Real-time response of Maglev system with different controllers
control signal ( V )
5 control signal ( I-PD)
0
-5
0
10
20 30 time ( sec )
40
50
Fig. 6. Control signal associated with I-PD controller (real time) Table 4. Time domain performance comparison between different controllers
109
Controller
Maximum overshoot (%)
*1-DOF IOPID
57.03
Peak time (sec) 0.56
Settling time (sec) 2.83
5th International Conference on Advances in Control and 110 Debdoot Sain et al. / IFAC PapersOnLine 51-1 (2018) 106–111 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
*1-DOF FOPID *2-DOF IOPID *2-DOF FOPID I-PD using Jaya
30.66 1.86 2.4 0.39
0.13 3.6 0.63 0.55
average of the input signal) at interval of 10 sec each has been applied as output disturbance to show the robust behaviour of the plant with I-PD controller. Also hand held disturbance has been applied between 25 sec to 30 sec to check whether the system is capable to withstand such disturbance or not. The response with pulse and hand held disturbance is provided in Fig. 8 and 9.
1.54 3.06 0.85 0.66
Table 5. Bode analysis of Maglev system with different controllers Gain Margin (dB) -11.7 -15.4 -18.5 -17.2 44.1
Controller *1-DOF IOPID *1-DOF FOPID *2-DOF IOPID *2-DOF FOPID I-PD using Jaya
Phase Margin (degrees) 89.8 88.6 89.3 82 73.1
Moreover, Control voltage to coil current gain (𝑘𝑘1 ) and sensor gain (𝑘𝑘2 ) have been varied between -20% to +20%. It can be observed from Fig. 10 that the Maglev system with the I-PD controller satisfies the iso-damping property around the gain cross-over frequency and ensures good loop robustness to model uncertainties.
Closed loop stability yes yes yes yes yes
* Swain et al., 2017 (Bold indicates best result)
6
ball position ( V )
From the data available in Table 4 and 5, it is clear that I-PD controller provides better performance as compared to other mentioned controllers in terms of time domain and frequency domain specifications except the phase margin criteria. The bode plot associated with the Maglev system with I-PD controller is shown in Fig. 7. Bode Diagram Gm = 44.1 dB (at 81.3 rad/s) , Pm = 73.1 deg (at 3.47 rad/s) 50
4 2 0 -2
Magnitude (dB)
0
0
10
20 30 time ( sec )
-50
50
6
ball position ( V )
-200 -90 -135 -180 -225 -270
40
Fig. 8. Maglev response with I-PD controller in presence of pulse disturbance
-100 -150
Phase (deg)
Reference Maglev response with I-PD controller in presence of pulse disturbance
-1
10
0
10
1
10
2
10
3
10
Reference Maglev response with I-PD controller in presence of hand held disturbance
4 2 0
4
10
-2
Frequency (rad/s)
0
Fig. 7. Bode plot of Maglev system with I-PD controller The positive gain margin and phase margin ensure the closed loop stability of the Maglev system with I-PD controller. The closed loop stability of the system can also be verified by observing the location of closed loop poles, which has been discussed already.
10
20 30 time ( sec )
40
50
Fig. 9. Maglev response with I-PD controller in presence of hand held disturbance
4.1. Robustness Analysis A system is said to be robust when it can hold its stability in presence of some noise, disturbance and parameter variation. In this paper, a pulse of magnitude 0.2325 V (i.e. 15% of the 110
5th International Conference on Advances in Control and Optimization of Dynamical Systems Debdoot Sain et al. / IFAC PapersOnLine 51-1 (2018) 106–111 February 18-22, 2018. Hyderabad, India Bode Diagram Gm = 39.1 dB (at 61.1 rad/s) , Pm = 67.3 deg (at 3.99 rad/s)
Kim, W. J. and Trumper, D. L. 1998. High-precision magnetic levitation stage for photolithography. Precision Engineering, Vol. 22, No. 2, pp. 66-77.
50
Magnitude (dB)
0 -50
Varvella, M. et al. 2004. Feasibility of magnetic suspension for second generation gravitational wave interferometers. Astroparticle Physics, Vol. 21, No. 3, pp. 325-335.
-100 -150 -200 -250 -90
Phase (deg)
111
Masuzawa, T. et al. 2003. Magnetically suspended centrifugal blood pump with an axially levitated motor. Artificial Organs, Vol. 27, No. 7, pp. 631-638.
k1 and k2 not varied k1 and k2 varied by +20%
-135
k1 and k2 varied by -20% -180
10
Lin, F.J., Teng, L.T. and Shieh, P.H. 2007. Intelligent slidingmode control using RBFN for magnetic levitation system. IEEE Transactions on Industrial Electronics, Vol. 54, no. 3, pp. 1752-1762.
Fig. 10. Bode plot of Maglev system with I-PD controller by varying k1 and k2 from -20 % to +20%
Sain, D., Swain, S. K. and Mishra, S. K. 2016. TID and I-TD controller design for magnetic levitation system using genetic algorithm. Perspectives in Science, vol. 8, pp. 370–373.
-225 -270
-1
10
0
10
1
10
2
10
3
10
4
10
5
Frequency (rad/s)
Lin, F. J. et al. 2005. Hybrid controller with recurrent neural network for magnetic levitation system. IEEE transactions on magnetics, Vol. 41, No. 7, pp. 2260–2269. 5. CONCLUSION
Chopade, S. et al. 2016. Design and implementation of digital fractional order PID controller using optimal pole-zero approximation method for magnetic levitation system. IEEE/CAA Journal of Automatica Sinica, Vol. PP, No. 99, pp. 1-12.
In this paper, an attempt has been made to design the I-PD controller for the Maglev system in both simulation and real time. The controller parameters are identified by optimizing the objective function using recently evolved Jaya algorithm. The performance of the controller has been evaluated and compared with other PID controllers (Swain et al., 2017). The result of comparison shows that by using proper control structure and with the help of proper optimization algorithm, three term controller like I-PD has also the capability to provide better response as compared to the five term controller like FOPID. The usual sluggish behaviour of the I-PD controller has been avoided in this design. The robustness analysis has also been incorporated in this study to show the robust behaviour of the I-PD controller. This particular approach of designing the I-PD controller optimized using Jaya algorithm can be extended to other class of plants to improve the performance of the plant or process. Moreover, as a further scope of research IOI-PD and FOI-PD controller may be designed using different optimization algorithms and the performance may be compared.
Ghosh, A. et al. 2014. Design and implementation of a 2-DOF PID compensation for magnetic levitation systems. ISA Transactions, Vol. 53, pp. 1216-1222. Swain, S.K. et al. 2017. Real time implementation of fractional order PID controllers for a magnetic levitation plant. International Journal of Electronics and Communications, Vol. 78, pp. 141-156. Sain, D., Swain, S. K. and Mishra, S. K. 2017. Real-time Implementation of Robust Set-point Weighted PID Controller for Magnetic Levitation System. International Journal on Electrical Engineering and Informatics, Vol. 9, No. 2, pp. 272-283. Rao, R. V. 2016. Jaya: A simple and new optimization algorithm for solving constrained and unconstrained optimization problems. International Journal of Industrial Engineering Computations, Vol. 7, pp. 19–34.
ACKNOWLEDGMENTS The authors would like to heartily thank anonymous referees for their constructive review comments, which led to definite improvements in the quality of the manuscript. REFERENCES Holmer, P. 2003. Faster than a speeding bullet train. IEEE Spectrum, Vol. 40, No. 8, pp. 30-34.
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IFAC PapersOnLine 51-1 (2018) 106–111
Real Time Implementation of Optimized I-PD Controller for the Real Time Implementation Optimized the Magnetic Levitation of System using I-PD Jaya Controller Algorithm for Real Time Implementation of Optimized I-PD Controller for the Real Time Implementation Optimized Magnetic Levitation of System using I-PD Jaya Controller Algorithm for the Magnetic Levitation Levitation System using Jaya Algorithm Algorithm * ** Kumar Swain, Magnetic System Jaya *Debdoot Sain, **Subrat using * ** *** Debdoot Sain, Subrat Kumar Swain,
Kumar Mishra * *Debdoot **Subrat ***Sudhansu *** Sain, ** Kumar * ** Kumar Sudhansu Mishra Swain, Debdoot Sain, Subrat Kumar Swain, *** ***Sudhansu Kumar Mishra of Technology, Electrical *** Engineering, Indian Institute
* Department of Kharagpur, Sudhansu Kumar Mishra of Technology, Kharagpur, * Department of Electrical Engineering, Indian Institute +91-9955624206; e-mail: [email protected]) Pin: 721302, India (Tel: of Electrical Engineering, Indian Institute of Technology, Kharagpur, **,*** * Department +91-9955624206; e-mail: [email protected]) Pin:of721302, India Electrical and(Tel: Electronics Engineering, Birla of Technology, * Department of Electrical Engineering, Indian Institute of Institute Technology, Kharagpur,Mesra, **,***Department **,*** (Tel: +91-9955624206; e-mail: [email protected]) Pin:of721302, India ** Department Electrical and Electronics Engineering, Birla Institute of Technology, Mesra, (e-mail: [email protected], ***[email protected]) Ranchi, Pin: 835215, India. (Tel: +91-9955624206; e-mail: [email protected]) Pin: 721302, India **,*** **,***Department of Electrical and Electronics ** ** Engineering, Birla Institute of Technology, Mesra, **,*** (e-mail: [email protected], ***[email protected]) Ranchi, Pin: 835215, India.and Department of Electrical Electronics Engineering, Birla Institute of Technology, Mesra, Ranchi, Pin: 835215, India. (e-mail: ******[email protected], ***[email protected]) Ranchi, Pin: 835215, India. (e-mail: [email protected], ***[email protected])
Abstract: Designing a suitable controller for a nonlinear and unstable plant is always very challenging to Abstract: a suitable controller for a nonlinear unstable plant is always very challenging to the controlDesigning system practitioners. In this article, Integral and - Proportional Derivative (I-PD) controller has Abstract: Designing a suitable controller for a nonlinear and unstable plant is always very challenging to the control system practitioners. In this article, Integral Proportional Derivative (I-PD) controller has been designed and implemented in simulation and real time for the Magnetic levitation (Maglev) system Abstract: Designing a suitable controller for a nonlinear and unstable plant is always very challenging to the control system practitioners. in Insimulation this article,and Integral - Proportional Derivative (I-PD) controller has been designed and implemented real time for the Magnetic levitation (Maglev) system which is both nonlinear and unstable in article, nature. Integral The nonlinear Maglev plant has been linearized around the control system practitioners. In this - Proportional Derivative (I-PD) controller has been designed and implemented in simulation and real time for Maglev the Magnetic levitation (Maglev) system which is both nonlinear and unstable in nature. The nonlinear plant has been linearized around the equilibrium point to obtain the model transfer function. The objective function, which has been designed and implemented in linearized simulation and real time for the Magnetic levitation (Maglev) system which is both nonlinear and unstable in nature. The transfer nonlinear MaglevThe plant has been linearized around the point to obtain the linearized model function. objective function, has beenequilibrium formulated by taking modulus of the characteristic polynomial ofhas thebeen plant alongwhich with the which is both nonlinear and the unstable in nature. The nonlinear Maglev plant linearized around the equilibrium point to obtain the linearized model transfer function. The objective function, which has been formulated by taking the modulus of the characteristic polynomial of the plant along with the controller at dominant location, has beenmodel minimized using the recently evolved Jaya algorithm. the equilibrium point topole obtain the linearized transfer function. The objective function, which The has been formulated by taking the modulus of the characteristic polynomial of the plant along with The the controller at dominant pole location, has been minimized using the recently evolved Jaya algorithm. performance of the controller has been with thatpolynomial of the 1-DOF Integer Order been formulated byI-PD taking the modulus of compared the characteristic of and the 2-DOF plant along with the controller at dominant pole location, has been minimized using the recently evolved Jaya algorithm. The performance of the I-PD controller that of recently the 1-DOF 2-DOF Integer Order (IO) and Fractional Order (FO) PIDhas controller (Swain etwith al., 2017). The result ofand comparison reveals that controller at dominant pole location, hasbeen beencompared minimized using the evolved Jaya algorithm. The performance of the I-PD controller been compared of The the 1-DOF 2-DOF Integer Order (IO) and Fractional Order (FO)using PIDhas controller (Swainoutperforms etwith al., that 2017). result ofand comparison reveals that the I-PD controller optimized Jaya algorithm controllers in terms ofOrder time performance of the I-PD controller has been compared with that ofthe theother 1-DOF and 2-DOF Integer (IO) and Fractional Order (FO)using PID controller (Swainoutperforms et al., 2017).the Theother result of comparison reveals that the I-PD controller optimized Jaya algorithm controllers in terms of time domain and frequency domain specifications except the phase margin criteria. The robustness analysis (IO) and Fractional Order (FO) PID controller (Swain et al., 2017). The result of comparison reveals that the I-PDand controller optimized using Jaya algorithm outperforms the other controllers in terms analysis of time domain frequency domain specifications except the phase margin criteria. The robustness has incorporated in this study to show the robust behaviour the I-PD controller. the also I-PDbeen controller optimized using Jaya algorithm outperforms theofother controllers in terms of time domain domain the phase margin criteria. The robustness analysis has also and beenfrequency incorporated in thisspecifications study to showexcept the robust behaviour of the I-PD controller. domain and frequency domain specifications except the phase margin criteria. The robustness analysis Keywords: Maglev, PID, I-PD, Optimization, Jaya. has also IFAC been incorporated this studyoftoAutomatic show the robust of Elsevier the I-PDLtd. controller. © 2018, (InternationalinFederation Control)behaviour Hosting by All rights reserved. has also been incorporated in this study to show the robust behaviour of the I-PD controller. Keywords: Maglev, PID, I-PD, Optimization, Jaya. Keywords: Maglev, PID, I-PD, Optimization, Jaya. Keywords: Maglev, PID, I-PD, Optimization, Jaya. linearized model transfer function is used for the design of 1. INTRODUCTION linearized model transfer function is used the design of the I-PD controller. Recently developed Jayaforalgorithm (Rao, 1. INTRODUCTION linearized model transfer function is used foralgorithm the design of the I-PD controller. Recently developed Jaya (Rao, is model employed to function suitably isidentify Simple structure and1.easy implementation has made the PID 2016) INTRODUCTION linearized transfer used forthe the controller design of the I-PD controller. Recently developed Jaya algorithm (Rao, 1. INTRODUCTION 2016) employed to suitably identify the controller Simple structure and easy hasformade thetime. PID parameters by optimizing the objective function which is controller as the choice of implementation control engineers a long the I-PDiscontroller. Recently developed Jaya algorithm (Rao, 2016) is employed to suitably identify the controller Simple structure and easy implementation hasformade thetime. PID parameters by optimizing the objective function which is controller as the choice of control engineers a long formulated by taking the modulus of characteristic Till now PID controller is the most common controller used 2016) is employed to suitably identify the controller Simple structure and easy implementation has made the PID parameters by optimizing the objective function which is controller as the choice of control engineers for a long time. formulated by taking the modulus of characteristic Till now PID controller is the most common controller used polynomial by of optimizing the plant along with the function controller at the in industryas and most of controllers PID time. type. parameters the objective which is controller the choice of the control engineersareforofa long Till now PIDand controller most common used formulated by the modulus characteristic polynomial of thetaking plantAsalong with the of controller at are the in industry most satisfactory ofis controllers arecontroller of with PID good type. dominant pole the singularities of the system Though, it provides performance bylocation. taking the modulus of characteristic Till now PID controller is the the most common controller used formulated of location. the plantAsalong with the controller at are the in industry and most satisfactory of the controllers are of with PID good type. polynomial pole the of the system Though, itcriteria; provides performance usually complex nature,along thesingularities modulus hasatbeen robustness still of there a possibility getting polynomial of thein plant with the operator controller the in industry and most theiscontrollers areofof PID more type. dominant pole location. As the singularities of the system are Though, itcriteria; provides satisfactory performance with more good dominant usually complex in nature, the modulus operator has been robustness still there is a possibility of getting taken for pole getting a realAsfunctional value of of the thesystem objective improved itperformance by slightly modifying the structure of dominant location. the singularities are Though, provides satisfactory performance with good complex ina nature, the modulus has been robustness criteria; stillby there is a possibility of more taken for real functional valueoperator of the should objective improved performance slightly modifying thegetting structure of usually function. Ingetting an ideal case, functional value be the PID controller. In this paper, I-PD controller has more been usually complex in nature, the modulus operator has been robustness criteria; still there is a possibility of getting taken for getting a real functional value of the objective improved performance by slightly modifying the structure of Ingetting an ideal case, the functional be the PID controller. In by this paper, I-PD has been equal toforzero. designed for the Maglev system to modifying show controller the improvement in function. taken a real functional value value of the should objective improved performance slightly the structure of function. In an ideal case, the functional value should be the PID controller. In this paper, I-PD controller has been equal to zero. designed for the Maglev system to show the improvement in the system performance as compared to thecontroller PID controller. In an idealdesign case, isthecomplete, functional should be PID controller. In this paper, I-PD has been function. Once to thezero. controller thevalue performance of equal designed the Maglevassystem to show improvement the systemfor performance compared to thethe PID controller. in equal to zero. designed for the Maglev system to show the improvement in Once the controller design is complete, the performance of Because inherent non linearity and unstable open loop the I-PD controller is compared with that of the 1-DOF and the systemofperformance as compared to the PID controller. Once the controller controller isdesign is complete, the performance of the system performance as compared to the PID controller. the I-PD compared with that of the 1-DOF and Because of inherent non linearity and unstable open loop 2-DOF IOPID (three parameters to tune) and FOPID (five behaviour, it is of prime importance to design a suitable Once the controller design is complete, the performance of the I-PD controller is compared with that of the 1-DOF and Because of inherent non linearity and unstable open loop 2-DOF IOPID (three parameters to tune) and FOPID (five behaviour, itinherent is of prime importance design open aissuitable parameters to tune) iscontroller discussed al. controller the Maglev as and its to application widely I-PD controller comparedaswith that of by theSwain 1-DOFetand Because offor non system linearity unstable loop the 2-DOF IOPID (threecontroller parameters tune) and FOPID behaviour, it the is of primesystem importance to design aissuitable parameters to tune) as to discussed by Swain et(five al. controller for Maglev as its application widely The result of comparison reveals that the I-PD controller spread in different fields of research which include high2-DOF IOPID (three parameters to tune) and FOPID (five behaviour, it is of prime importance to design a suitable parameters to tune) controller as discussed by Swain et al. controller for the Maglev system as its application is widely The result of comparison reveals that the I-PD controller spread in different fields of research which include highJayacontroller algorithm as outperforms controllers speed systems (Holmer, is widely 2003), optimized parametersusing to tune) discussed other by Swain et al. controller transportation for the Maglev system as its application The resultusing of comparison reveals that the other I-PD controllers controller spread in transportation different fields ofsystems research which include 2003), high- in speed (Holmer, Jaya algorithm outperforms terms of time domain andreveals frequency specifications photolithography semiconductor manufacturing The result of comparison thatdomain the I-PD controller spread in differentdevices fields for of research which include high- optimized Jaya algorithm outperforms other controllers speed transportation systems (Holmer, 2003), optimized in termsthe ofusing time domain and frequency domain specifications photolithography semiconductor manufacturing except phase margin criteria. The robustness analysis has (kim 1998),devices seismic for attenuators for(Holmer, gravitational wave optimized using Jaya algorithm outperforms other controllers speedet al.,transportation systems 2003), in termsthe of phase time domain and frequency domain specifications photolithography devices for semiconductor manufacturing except margin criteria. The robustness analysis has (kim et al., 1998), seismic attenuators for gravitational wave been incorporated in this study domain to show the robust antennas (Varvelladevices et al., for 2004), self-bearing manufacturing blood pumps also in terms of time domain and frequency specifications photolithography semiconductor the phase margin criteria. The robustness analysis has (kim et al., 1998), seismic attenuators for gravitational wave except also been incorporated in this study to show the robust antennas (Varvella et al., 2004), self-bearing blood pumps behaviour the I-PD controller. (Masuzawa et al., seismic 2003) for the use in heartswave etc. except the of phase margin criteria. The robustness analysis has (kim et al., 1998), attenuators forartificial gravitational also been incorporated in this study to show the robust antennas (Varvella et al., 2004), self-bearing blood pumps the I-PD controller. (Masuzawa et al., for thebyuse artificial etc. Several efforts have2003) been the in engineers tohearts design an behaviour also been ofis incorporated in five this sections. study to Section show the antennas (Varvella et al.,made 2004), self-bearing blood pumps This paper organized into 1 isrobust about behaviour of the controller. (Masuzawa et al., 2003) for the use in artificial hearts etc. Several efforts have been made by the engineers to design an This paper is organized into fiveThe sections. Section 1 is about efficient controller for the Maglev system using different of the I-PD I-PD controller. (Masuzawa et al., 2003) for the use in artificial hearts etc. behaviour the introduction of the paper. schematic diagram and Several efforts have been made by engineers to efficient controller for the system using different paper is organized fiveThe sections. Section 1 is about the introduction of the into paper. schematic diagram and control techniques al.,Maglev 2007, et al., 2016, Lin an et This Several efforts have(Lin beenet made by the theSain engineers to design design an calculation oforganized linearized model transfer function ofabout the This paper is into five sections. Section 1 is efficient controller the system using different introduction of the paper. Thetransfer schematic diagram and control techniques al.,Maglev 2007, Sain et al., 2016, Lin et et the calculation of linearized model function of the al., 2005, Chopade (Lin etfor 2016, Ghosh et al., 2014, Swain efficient controller foral.,et the Maglev system using different Maglev system isof provided in section 2. Section diagram 3 deals with the introduction the paper. The schematic and control techniques (Lin al., 2007, Sain et 2016, Lin of is linearized model transfer function of with the al., Chopade et 2017). al.,et Ghosh et still al., 2014, Swain et Maglev system provided in section 2. Section 3 deals al., 2005, 2017, Sain et al., there is a possibility control techniques (Lin et 2016, al.,But 2007, Sain et al., al., 2016, Lin to et calculation the design of I-PD controller for the Maglev system. Section calculation of linearized model transfer function of the al., 2005, Chopade et al., 2016, Ghosh et al., 2014, Swain et Maglev system is provided in section 2. Section 3 deals with al., 2017, Sain et al., 2017). But there is still a possibility to the design of I-PD controller for the Maglev system. Section improve performance the Maglev by modifying al., 2005,the Chopade et al., of 2016, Ghosh etsystem al., 2014, Swain et 4 is about the response of theinsystem the I-PD controller, system is provided sectionwith 2. Section 3 deals with al., 2017,the Sain et al., 2017). ButMaglev there issystem still a possibility to Maglev design of I-PD controller for the with Maglev system. Section improve performance of the 4 is about the response of the system the I-PD controller, the 2017, traditional control structure themodifying efficient al., Sain et al., 2017). But and thereemploying is still a by possibility to the performance comparison and robustness analysis. In Section section the design of I-PD controller for the Maglev system. improve the performance of the Maglev system by modifying is about the comparison response of the system with analysis. the I-PD controller, the traditional control structure and employing themodifying efficient andscope robustness In section optimization algorithms. Maglev plant has 4 improve the performance ofThe the nonlinear Maglev system by 5,isconclusion the further ofwith research is highlighted. 4performance about the and response of the system the I-PD controller, the traditionalalgorithms. control structure and employing theplant efficient performance comparison and robustness analysis. section optimization The nonlinear Maglev has 5, conclusion and the further scope of research is highlighted. beentraditional linearizedcontrol around the equilibrium point the performance comparison and robustness analysis. In the structure and employing the and efficient In section optimization algorithms. The nonlinear Maglev plant has 5, conclusion and the further scope of research is highlighted. been linearized around the equilibrium point and the optimization algorithms. The nonlinear Maglev plant has 5, conclusion and the further scope of research is highlighted. been been linearized linearized around around the the equilibrium equilibrium point point and and the the 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 IFAC 106 Peer review©under of International Federation of Automatic Copyright 2018 responsibility IFAC 106Control. 10.1016/j.ifacol.2018.05.018 Copyright © 2018 IFAC 106 Copyright © 2018 IFAC 106
5th International Conference on Advances in Control and Optimization of Dynamical Systems Debdoot Sain et al. / IFAC PapersOnLine 51-1 (2018) 106–111 February 18-22, 2018. Hyderabad, India
107
f (i, x ) .. f (i, x ) (2) x i x i x i0 , x 0 i0 , x 0 where, x and i is the small deviation from the equilibrium
2. MAGLEV SYSTEM The schematic diagram of the Maglev system (Ghosh et al., 2014) is depicted in Fig. 1. In this study, the Maglev system with Model no. 33-210 from Feedback Instruments is considered. The coil gets magnetized when current flows through it and attracts the ball in the upward direction. At the same time, the gravitational force of earth pulls it in the downward direction. There is an infrared sensor present in the system which continuously monitors the position of the ball.
point x0 and i0 respectively. Evaluating partial derivatives and taking Laplace transform on both side of (2), the transfer function can be obtained as x i
ki
s
2
(3)
kx
where, k i
2g i0
and k x
2g x0
As x and i are proportional to x v and u, the transfer function can be modified to the form x v / u (Ghosh et al., 2014) and given by G p (s)
xv u
b
s
2
p
2
3518 . 85 s
2
(4)
2180
where, x v is sensor output and u is the input to current amplifier. Fig. 1. Schematic diagram of Maglev system
The poles of the Maglev system are located at 46.69. Because of the presence of one pole on the right half of ‘s’ plane, the system becomes highly unstable. To achieve a stable and controlled behaviour of the system, it becomes extremely important to design suitable controller.
The mechanical, electrical unit along with connectioninterface panel that assembles into a complete control system setup is provided in Fig. 2
Table 1. The system parameters of the Maglev system (Ghosh et al., 2014)
Fig. 2. Maglev control system In terms of ball position x and electromagnetic coil current i, the simplest nonlinear model (Ghosh et al., 2014) of the Maglev system is given by ..
m x mg k
i x
2 2
(1)
Parameter
Notation
Value
Mass of the steel ball
m
0.02 kg
Acceleration due to gravity
g
9.81 m/s2
Equilibrium value of current
i0
0.8 A
Equilibrium value of position
x0
0.009 m
Control voltage to coil current gain
k1
1.05 A/V
Sensor gain, offset
k2 ,
143.48 V/m, -2.8 V
Control voltage input level
U
Sensor output voltage level
xv
where, m represents the mass of the ball, g is the gravitational constant and k is dependent on the coil parameters.
±
5V
+ 1.25 V to -3.75V
3. DESIGN OF I-PD CONTROLLER
For the linearization of the nonlinear Maglev plant, the calculation of equilibrium point is mandatory. The equilibrium point of the current and position is calculated by
3.1. I-PD Controller Design using Jaya
..
3.1.1. Dominant Pole calculation
equating x 0 and found to be at 0.8A and 0.009 m (-1.5 V, when expressed in volts) respectively. The nonlinear system (1) around the obtained equilibrium point can be linearized to
For this particular study, the design specifications have been considered as
107
5th International Conference on Advances in Control and 108 Debdoot Sain et al. / IFAC PapersOnLine 51-1 (2018) 106–111 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
Damping ratio = 0.8 and Settling time ts =
4
f I PD min 1
2 sec
3518 . 85 2 s 2180
K
P
K
I
K
s
D
s
n
Using = 0.8 and ts = 2 sec, the value of n has been found as 2.5 rad/sec and the dominant poles have been obtained by solving the standard second order characteristic equation and given by s 1, 2 2 1 . 5 i .
(8) s s1
3.1.4 Objective Function Optimization using Jaya The Jaya algorithm (Rao et al., 2016) is a modern and recently emerged optimization algorithm inspired by the popular Teaching Learning Based Optimization (TLBO) technique. Unlike TLBO, the algorithmic steps of Jaya include only a single phase and the computational execution time is reduced as compared to TLBO where two phases (teaching and learning) are involved. Both algorithms do not require any algorithm-specific parameters, but at the same time, the working patterns of both the algorithms are different from each other. However, in terms of implementation and performance, the Jaya algorithm is simple as well as effective. In case of Jaya algorithm, the objective function f obj ( x ) , to be minimized at any iteration
3.1.2. I-PD Controller The I-PD controller is a simple modification of conventional the PID controller. The integral (I) block is present on the forward path while the proportional and derivative (P-D) blocks are kept in the feedback path. As different signal paths are present for the set-point and process outputs, the I-PD controller has got more flexibility to satisfy the design specifications accurately. The structure of the I-PD controller is shown in Fig. 3.
t comprises of ‘m’ design variables and ‘n’ candidate solutions. Among all the possible candidate solutions, the best candidate will provide an optimal value of f obj ( x ) . If X
p ,q ,r
be the value of pth variable for qth candidate during the
rth iteration, then the updated state equation is represented by the following equation X
new
X p , q , r k1, p , r X p , best , r X p , q , r
p ,q,r
Fig. 3. Control Structure of an I-PD controller
Characteristics equation of the system with I-PD controller for unity feedback is given by
1 G p (s) G c (s) G c I
PD
(s) 0
K I K P K D s 0 s
i.e 1 G p ( s )
where, X
X
(5)
X
(6)
new p ,q ,r
is the value of ‘p’ for the best candidate,
is the value of ‘p’ for the worst candidate, and
is the updated value of the term X and
k 2, p ,r
p ,q ,r
. Whereas,
updated random numbers for the pth
variables during rth iteration in the range [0, 1]. The term
k 1, p , r X
With Maglev system the characteristics equation becomes, 3518 . 85 K I 1 K P K D s 0 2 s 2180 s
p , worst , r
k 1, p , r
where, K P , K I and K D represent proportional, integral and derivative gain respectively.
p , best , r
(9)
k 2 , p , r X p , worst , r X p , q , r
p , best , r
X
p ,q ,r
is responsible for making the
solution closer to the best one, whereas the term is responsible for the k 2 , p , r X p , worst , r X p , q , r
(7)
elimination of the worst solutions. If the updated value
Interestingly the characteristics equation of the plant with PID and I-PD controller is identical for the same set of controller parameter values. But the removal of inappropriate zeros from the closed loop transfer function, in case of the I-PD controller, improves the response of the system as compared to the PID controller which will be evident in the results and discussion section.
X
new
provides better optimal functional value then it will be
p ,q ,r
accepted as the input to next iteration. The objective function considered over here has three unknowns i.e. K P , K I and K D . As the range of unknowns affects the optimality of the solution, in the beginning, a wider solution space is considered. After getting the initial solution, in the subsequent steps, the solution space has been shortened. The range of these parameters taken for writing the MATLAB code is provided in Table 2. It can be noted that all the gains are negative and the reason can easily be verified from the Routh–Hurwitz stability criterion.
3.1.3 Objective Function Formulation The objective function f I PD considered for obtaining the value of K P , K I and K D has been formulated by taking the modulus of the characteristic polynomial of the plant along with the controller at dominant pole location and given by
108
5th International Conference on Advances in Control and Optimization of Dynamical Systems Debdoot Sain et al. / IFAC PapersOnLine 51-1 (2018) 106–111 February 18-22, 2018. Hyderabad, India
6
Table 2. Range of controller parameters considered for writing the MATLAB code
0
0
10
20 30 time ( sec )
40
50
Fig. 4. Simulink response of Maglev system with different controllers The performance of the Maglev system with I-PD controller has been compared with the 1-DOF and 2-DOF IOPID and FOPID controller (Swain et al., 2017) and summarized in Table 4 and Table 5. 6 ball position ( V )
KI -6.8
2
-2
Table 3. I-PD controller parameter values after optimization KP -2.5
Reference 1-DOF IOPID 1-DOF FOPID 2-DOF IOPID 2-DOF FOPID I-PD using Jaya
4
ball position ( V )
KI KD Parameter KP Lower range -2.5 -8 -0.165 Upper range 0 -6.8 -0.1 After optimizing the objective function through Jaya algorithm within the mentioned range of parameters with population size 50 and number of iteration 10, the values of K P , K I and K D have been found and given in Table 3. With these set of controller parameters, the objective function value is 2.4564, which is not equal to zero but as all the closed loop poles of Maglev system with I-PD controller lie on the left half of ‘s’ plane, the system is guaranteed to be stable. It has been observed that using the Genetic algorithm available in Optimization Tool in MATLAB, it requires 51 iterations to reach the optimum values, whereas, Jaya takes very few iterations to produce the same optima. It has also been seen that the optima reached using Jaya is on the constraint boundaries and it is expected to change whenever the boundary is changed, but as we are getting satisfactory performance with these set of controller parameters, we have not explored further.
Parameter Value
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KD -0.165
Reference 1-DOF IOPID 1-DOF FOPID 2-DOF IOPID 2-DOF FOPID I-PD using Jaya
4 2 0 -2
4. RESULTS AND DISCUSSION
0
For this study, a square wave with mean -1.55 V has been considered as the input signal. The reason behind choosing the input signal with mean -1.55 V is to ensure that the operating point of the system is around the equilibrium point. The simulation has been carried out for 50 seconds and the response of simulation and real-time with I-PD controller optimized using Jaya has been provided in Fig. 4 and 5. For the real time simulation, first 5 seconds have been used for compensating the disturbance encountered, due to hand positioning of the ball as well as inherent nonlinearities present in the system. The control signal associated with the I-PD controller for the real time control of Maglev plant has also been provided in Fig. 6. Once the ball gets stabilized, it can be observed that the control signal remains in the range of ± 5 V.
10
20 30 time ( sec )
40
50
Fig. 5. Real-time response of Maglev system with different controllers
control signal ( V )
5 control signal ( I-PD)
0
-5
0
10
20 30 time ( sec )
40
50
Fig. 6. Control signal associated with I-PD controller (real time) Table 4. Time domain performance comparison between different controllers
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Controller
Maximum overshoot (%)
*1-DOF IOPID
57.03
Peak time (sec) 0.56
Settling time (sec) 2.83
5th International Conference on Advances in Control and 110 Debdoot Sain et al. / IFAC PapersOnLine 51-1 (2018) 106–111 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
*1-DOF FOPID *2-DOF IOPID *2-DOF FOPID I-PD using Jaya
30.66 1.86 2.4 0.39
0.13 3.6 0.63 0.55
average of the input signal) at interval of 10 sec each has been applied as output disturbance to show the robust behaviour of the plant with I-PD controller. Also hand held disturbance has been applied between 25 sec to 30 sec to check whether the system is capable to withstand such disturbance or not. The response with pulse and hand held disturbance is provided in Fig. 8 and 9.
1.54 3.06 0.85 0.66
Table 5. Bode analysis of Maglev system with different controllers Gain Margin (dB) -11.7 -15.4 -18.5 -17.2 44.1
Controller *1-DOF IOPID *1-DOF FOPID *2-DOF IOPID *2-DOF FOPID I-PD using Jaya
Phase Margin (degrees) 89.8 88.6 89.3 82 73.1
Moreover, Control voltage to coil current gain (𝑘𝑘1 ) and sensor gain (𝑘𝑘2 ) have been varied between -20% to +20%. It can be observed from Fig. 10 that the Maglev system with the I-PD controller satisfies the iso-damping property around the gain cross-over frequency and ensures good loop robustness to model uncertainties.
Closed loop stability yes yes yes yes yes
* Swain et al., 2017 (Bold indicates best result)
6
ball position ( V )
From the data available in Table 4 and 5, it is clear that I-PD controller provides better performance as compared to other mentioned controllers in terms of time domain and frequency domain specifications except the phase margin criteria. The bode plot associated with the Maglev system with I-PD controller is shown in Fig. 7. Bode Diagram Gm = 44.1 dB (at 81.3 rad/s) , Pm = 73.1 deg (at 3.47 rad/s) 50
4 2 0 -2
Magnitude (dB)
0
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20 30 time ( sec )
-50
50
6
ball position ( V )
-200 -90 -135 -180 -225 -270
40
Fig. 8. Maglev response with I-PD controller in presence of pulse disturbance
-100 -150
Phase (deg)
Reference Maglev response with I-PD controller in presence of pulse disturbance
-1
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0
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1
10
2
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3
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Reference Maglev response with I-PD controller in presence of hand held disturbance
4 2 0
4
10
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Frequency (rad/s)
0
Fig. 7. Bode plot of Maglev system with I-PD controller The positive gain margin and phase margin ensure the closed loop stability of the Maglev system with I-PD controller. The closed loop stability of the system can also be verified by observing the location of closed loop poles, which has been discussed already.
10
20 30 time ( sec )
40
50
Fig. 9. Maglev response with I-PD controller in presence of hand held disturbance
4.1. Robustness Analysis A system is said to be robust when it can hold its stability in presence of some noise, disturbance and parameter variation. In this paper, a pulse of magnitude 0.2325 V (i.e. 15% of the 110
5th International Conference on Advances in Control and Optimization of Dynamical Systems Debdoot Sain et al. / IFAC PapersOnLine 51-1 (2018) 106–111 February 18-22, 2018. Hyderabad, India Bode Diagram Gm = 39.1 dB (at 61.1 rad/s) , Pm = 67.3 deg (at 3.99 rad/s)
Kim, W. J. and Trumper, D. L. 1998. High-precision magnetic levitation stage for photolithography. Precision Engineering, Vol. 22, No. 2, pp. 66-77.
50
Magnitude (dB)
0 -50
Varvella, M. et al. 2004. Feasibility of magnetic suspension for second generation gravitational wave interferometers. Astroparticle Physics, Vol. 21, No. 3, pp. 325-335.
-100 -150 -200 -250 -90
Phase (deg)
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Masuzawa, T. et al. 2003. Magnetically suspended centrifugal blood pump with an axially levitated motor. Artificial Organs, Vol. 27, No. 7, pp. 631-638.
k1 and k2 not varied k1 and k2 varied by +20%
-135
k1 and k2 varied by -20% -180
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Lin, F.J., Teng, L.T. and Shieh, P.H. 2007. Intelligent slidingmode control using RBFN for magnetic levitation system. IEEE Transactions on Industrial Electronics, Vol. 54, no. 3, pp. 1752-1762.
Fig. 10. Bode plot of Maglev system with I-PD controller by varying k1 and k2 from -20 % to +20%
Sain, D., Swain, S. K. and Mishra, S. K. 2016. TID and I-TD controller design for magnetic levitation system using genetic algorithm. Perspectives in Science, vol. 8, pp. 370–373.
-225 -270
-1
10
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2
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3
10
4
10
5
Frequency (rad/s)
Lin, F. J. et al. 2005. Hybrid controller with recurrent neural network for magnetic levitation system. IEEE transactions on magnetics, Vol. 41, No. 7, pp. 2260–2269. 5. CONCLUSION
Chopade, S. et al. 2016. Design and implementation of digital fractional order PID controller using optimal pole-zero approximation method for magnetic levitation system. IEEE/CAA Journal of Automatica Sinica, Vol. PP, No. 99, pp. 1-12.
In this paper, an attempt has been made to design the I-PD controller for the Maglev system in both simulation and real time. The controller parameters are identified by optimizing the objective function using recently evolved Jaya algorithm. The performance of the controller has been evaluated and compared with other PID controllers (Swain et al., 2017). The result of comparison shows that by using proper control structure and with the help of proper optimization algorithm, three term controller like I-PD has also the capability to provide better response as compared to the five term controller like FOPID. The usual sluggish behaviour of the I-PD controller has been avoided in this design. The robustness analysis has also been incorporated in this study to show the robust behaviour of the I-PD controller. This particular approach of designing the I-PD controller optimized using Jaya algorithm can be extended to other class of plants to improve the performance of the plant or process. Moreover, as a further scope of research IOI-PD and FOI-PD controller may be designed using different optimization algorithms and the performance may be compared.
Ghosh, A. et al. 2014. Design and implementation of a 2-DOF PID compensation for magnetic levitation systems. ISA Transactions, Vol. 53, pp. 1216-1222. Swain, S.K. et al. 2017. Real time implementation of fractional order PID controllers for a magnetic levitation plant. International Journal of Electronics and Communications, Vol. 78, pp. 141-156. Sain, D., Swain, S. K. and Mishra, S. K. 2017. Real-time Implementation of Robust Set-point Weighted PID Controller for Magnetic Levitation System. International Journal on Electrical Engineering and Informatics, Vol. 9, No. 2, pp. 272-283. Rao, R. V. 2016. Jaya: A simple and new optimization algorithm for solving constrained and unconstrained optimization problems. International Journal of Industrial Engineering Computations, Vol. 7, pp. 19–34.
ACKNOWLEDGMENTS The authors would like to heartily thank anonymous referees for their constructive review comments, which led to definite improvements in the quality of the manuscript. REFERENCES Holmer, P. 2003. Faster than a speeding bullet train. IEEE Spectrum, Vol. 40, No. 8, pp. 30-34.
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