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In current scenario, the study of fractional order PID controller tuning rules of robust control system for first order plus time delay systems have been developed. In this paper, on the basis of computational scheme, a controller is designed to satisfy the robustness property with respect to gain variation and desired phase margin criteria. In this study, numerical computation of tuning formulae and relationship between design specification and design parameter are both discussed by taking an example of ceramic infrared heating system.
In the design specification , the controller parameters and the plant conditions, a fair comparison with an optimal design integer order PID(IOPID) controller done via simulation and experimental tests to shows the controllers dynamic performance , stability and robustness when the parameter change.
In, the recent year the applications of fractional calculus have been attracting more and more researchers in the field of engineering and science. The orders of fractional calculus are real number. Today, many researchers have focused on fractional order PID controllers and have obtained some useful results. The fractional order PID controller () was proposed in as a generalization of PID controller, where expanding of the derivative and integrals to fractional orders which are adjusted to frequency response of the control system directly and continuously. This paper presents a mathematical computational tuning scheme of FOC for certain temperature system used in industry.
The main contribution of this paper include
Fractional order PID controller with and as unity is proposed (IOPID) with mathematical computation for first order plus time delay system is presented.
Fractional order PID controller is proposed with mathematical computation for first order plus time delay system is presented.
According to the systematic design and simulation, a fair comparison of control performance with IOPID controller.
From the simulation results, it can be seen that FOPID controller outperforms the IOPID controller.
Design Specification of Control plant and controllers
Because of small delay time in large number of temperature system plant, so a typical first order plus time delay plant discussed in this paper is
Which can be a approximately model of a large number of industrial plants. For the ceramic infrared heating system transfer function with the value of gain (k) variation of 3.96 to 4.2, time constant of 140 sec and lag time of 7 sec. So, the typical FOPLT plant for ceramic infrared heater taken as
The fractional order proportional integral derivative controller (FOPID) has the following form
Clearly, this is a specific form of the most common controller which includes an integrator of order and a differentiator of order.
By considering the value of and, the controller form become the IOPID in the following expression as
Where Kp,Ki and Kd represent proportional , integral and derivative gain respectively.
By considering, the tuning method present by Monje, Vinagre and their colleague used in the paper. Monje and Vinagre et. al, consider the five design criteria algorithm for design specification. These design criteria obtain by getting the value of required phase margin, critical frequency point on the Nyquist curve of plant at which
and gain margin as
By getting the phase margin () and critical frequency () , five design criteria of Monje - Vinagre et. al method are given as follow.
Phase Margin and Gain Crossover Frequency
The two frequency domain specifications are used to measure the robustness i.e gain margin and phase margin. The phase margin is related to the damping of the system, thus the following equation should be satisfied
Where is a gain crossover frequency.
Robustness due to variation in the gain of Plant
The phase is forced to be flat at and the phase plot is almost constant within the interval around to satisfy the following constraint
As, per the phase plot around the specified frequency is locally flat, which implies that the system will be more robust to variation of gain and step response is almost constant within the interval with constant overshoots.
High Frequency Noise Rejection
The following condition must be satisfy to the robustness due to high frequency noise
Where A is the desired value of the noise attenuation for frequency is rad/sec.
The following constraint must be satisfied to ensure a good output disturbance rejection.
Where B is the desirable value of sensitivity function for which the frequency is rad/sec.
Design of IOPID controller
By considering the FOPLD system for ceramic infrared heater, whose open loop transfer function is
The frequency response for ceramic IR heater system as
Where K=3.96 to 4.2 , T=140 sec , L=7 sec
The gain and phase of the plant are as follow
As per the FOPID controller, the value of integrator order () and differentiator order ( ) are taken as unity respectively then IOPID controller obtain as
In this study , a method has been proposed to obtain the proportional gain constant ( ) , the constant of integral gain ( ) and the constant of derivative gain( ).Let the be the required phase margin and be the frequency of the critical point on the Nyquist curve of plant at which and define gain margin as
Then, in order to make the phase margin of the system equal to and , the following equation must satisfied.
According to IOPID controller transfer function  , we can get the frequency response as
The gain and phase of controller are as follow,
The open loop frequency response given as
The gain and phase of the open loop frequency response as follows
According to the design specification (i) and (ii), the robustness to gain variation in the plant, we can establish an equation about as
From, the Nyquist curve, we set the gain margin and phase margin as follow by taking Nyquist and bode plot of system
By solving the equation (a), (b) and (c), we get, and directly
=2.825, =0.0855, =9.074
The IOPID controller obtained as
Output response with step input
Design of Controller
This section represents the development of a tuning method of controller for first order plus time delay system with gain parameter uncertainty structure. All parameters of the controller are calculated to satisfy the performance of the plant. Five unknown parameters of the controller are estimated solving five non-linear equations that satisfy five design criteria.
Bode plot of FOPTD systems with gain parameter uncertainty structure are successfully combined with five design criteria to obtain the controller.
The phase and amplitude of the plant in frequency domain can be drive from equation ( ) by,
FOPID Controller design
From fractional order PID controller transfer function ( ), we can get its frequency response as follows,
According to specification (1) , the phase value
According to specification (2) we get the magnitude of as
According to specification (3) we get
As per the specification (3) we get the high frequency noise rejection as
As per the specification (4) we get Good disturbance rejection as
Steady state gain k does not have any effect on phase plot of the plant. In order to design robust controller should be satisfied with transfer function of FOPID, namely must be taken at the point 'x'. The constraint of phase margin and gain margin should be satisfied at point 'y' shows the minimum phase margin.
Equation ( ) Five unknown parameter can be solved by using FMINCON optimization toolbox of MatLab. Equation ( ) is considered as a main equation and other equation are taken as non-linear constraints for optimization. Value of the all five unknown parameter get calculated to obtain the controller to control the ceramic IR heater as
Step response of C(s)G(s) are obtained by using the 'nintblocks' of MatLab shown in fig.
The step response of the system shows that the system is more robust to gain change and overshoot of the step responses is almost constant. Bode plot, Magnitude plots of T(s) and S(s) of the system obtained in MatLab. It shows that phase of the system are almost flat and almost constant within an interval around with specified constraints.
From the figure of bode plot, T(s) and S(s) , one can conclude that the controller satisfies the robust performance of the system.
In this paper, two methods for tuning of controller have been proposed. The first method is based on the idea of using unity power for the integrator and derivative function of . By solving the equation obtained by taking consideration of the constraint, we get the value of the different three parameters optimized to achieve better step response.
The proposed robust tuning method for a controller to control first order plus time delay with parameter uncertainty structure designed. The five design constraints benefits of the Monje-Vinagre et. al.method were used to derive five non-linear equations. Value of unknown parameter of phase extremum of bode envelopes of the plant is used to satisfy robust performance of the system.
The simulation results show that the proposed method of controller has better step response than IOPID controller for ceramic IR heater.
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