1 2013 Nirma University International Conference on Engineering (NUiCONE)
Design of Fractional Order Controller for Undamped Control System Pritesh Shah1, S D Agashe2, Abhaya Pal Singh3 1. Symbiosis Institute of Technology, Symbiosis International University, Pune Email Id:
[email protected] 2. College of Engineering, Pune, India Email Id:
[email protected] 3. Symbiosis Institute of Technology, Symbiosis International University, Pune Email Id:
[email protected]
Abstract--This paper deals with design of fractional order controller (PIȜDȝ) for undamped control system, where order of integrator and derivative are fractional. A more robust and optimal controller design can be achieved by using fractional order controller because there are two more parameters compare with classical PID controller. For tuning of fractional order controller, optimization method called nelder mead is used. Simulation result shows that fractional order controller can achieve better result compare with classical PID controller. Index Terms--Fractional Order Controller, Undamped Control System, Fractional Calculus, PID Controller I.
A
INTRODUCTION
fractional calculus is almost three centuries old topic as classical calculus. From the last few decades, many scientists and engineers are finding many more applications of fractional calculus in different areas as modelling of physical system, signal and systems, physic, control system, etc. [1]-[3].[14] In Fraction Calculus, order of Integration and differential are real numbers i.e., ܴ א ݎ݁݀ݎା . There are two most commonly used definitions for the general fractional integrodifferential operator. They are the Grünwald-Letnikov (GL) and the Riemann-Liouville (RL) definitions for fractional Calculus [11]. The GL definition is given by ሾ୲ି୯Ȁ୦ሿ ୯ ୲ ሺሻ
ൌ ୦՜
ି
ሺെͳሻ୨ ቀ ቁ ሺ െ ሻ ୨ୀ
Where, ሾǤ ሿ implies the integral part. And RL Definition is given by ୲ ͳ ሺɒሻ ୯ න ɒ ୲ ሺሻ ൌ Ȟሺ െ Ƚሻ ୟ ሺ െ ɒሻ୯ି୬ାଵ Where , െ ͳ ൏ ൏ ݊ܽ݊݀ȞሺǤ ሻ is the Eular's gamma function. Also, M. Caputo, also given definition of fractional Integration. ୲ ͳ ሺ୬ሻ ሺɒሻ න ɒ ୲ ሺሻ ൌ Ȟሺ െ Ƚሻ ୟ ሺ െ ɒሻି୬ାଵ The main advantage of Caputo’s approach is that the initial conditions for the fractional differential equation with the 978-1-4799-0727-4/13/$31.00 ©2013 IEEE
Caputo derivatives take on the same form as for integer-order differential equations. [5], [13]. An undamped system in which damping ratio (ȗ) is zero. It has characteristics of sustain oscillation. Using Fractional PID Controller, we get better control performance compare with classical PID Controller. In this paper, design of fractional PID Controller is presented for undamping control system. Also for robust control design, tuning of fractional PID Controller is achieved using optimization method called nelder mead. Results are compared with classical PID Controller. This paper is organized in different sections. First Section covers introduction of the work. Second Section briefly reviews literature of Fractional Calculus, Fractional PID Controller and undamping system. The design and tuning of Fractional Order Controller are described in section 3 with experimental work for undamped control system followed by Conclusion, acknowledgement and references. II. FRACTIONAL CALCULUS AND FRACTIONAL PID CONTROLLER
In most of industry Proportional–integral–derivative (PID) controllers have been used for several decades for process control applications. The reason for their wide popularity lies in the simplicity of design and good performance including low percentage overshoot and small settling time for slow process plants [27]. In fractional-order proportional–integral– derivative (FOPID) controller, I and D operations are usually of fractional order; therefore, besides setting the proportional, derivative and integral constants Kp, Kd, Ki [4].we have two more parameters: the order of fractional integration Ȝ and that of fractional derivative μ [2]. Finding an optimal set of values for Kp, Ki, Kd, Ȝ and μ to meet the user specifications for a given process plant calls for real parameter optimization in five-dimensional hyperspace [10]. There are following advantages of Fractional order controller [5] • No or very less Steady State error • Attain Gain Margins and Phase cross over frequency specification
2 • • • •
Attain Phase Margins and gain crosss over frequency specification Robustness against high frequency noiise Robustness against variation in the plaant gain Good output disturbance rejection duue to five tuning parameter(Kp, Ki, Kd, Ȝ and μ)
The significance of fractional order contrrol is that it is a generalization of classical integral order control theory, which can lead to more accurate and moree robust control performance [8]. In general, there is no sysstematic way of designing proper fractional order controllerr for the control systems. The generalized transfer function of this coontroller is given by [6] ݇ ܥሺܵሻ ൌ ܭ ఒ ݇ௗ ܵఓ ǡ ߣǡ ߤ ܴ אା ܵ Where, C(S) is controller output, Kp is proportional constant gain Ki is integration constant gain Kd is derivative constant gain Ȝ is Order of integration μ is Order of differentiator.
Set Point
Fractional PID Controller
Output Plant or System
Fig 2: General Block Diagram of Frractional PID Controller
According to the desired gain margin Am and phase margin߮ , the designed fractionaal order PID controller should meet the stability robustness of the feedback control loop. From the basic definitions of gaain and phase margin, the oller ܩ ሺݏሻ should satisfy dynamic system ܩ ሺݏሻand the contro the following [15],[16]: ߮ ൌ ܽ݃ݎൣܩ ሺ݆߱ ሻܩ ሺ݆߱ ሻ൧ ߨ ͳ ܣ ൌ หܩ ሺ݆߱ ሻܩ ሺ݆߱ ሻห Where ߱ is given by ሻ ൌͳ หܩ ሺ݆߱ ሻܩ ሺ݆߱ ሻห And ߱ is given by ൣܩ ൫݆߱ ൯ܩ ൫݆߱ ൯൧ ൌ െߨ Also, Fractional PID Controller is designed to satisfied of gain plant following properties. Robustness to variations v ݀ሺሺܨሺݏሻሻሻ ൰ ൌͲ ൬ ݀߱ ఠୀఠ and High frequency noise rejection n ቤܶሺ݆߱ሻ ൌ
ܥሺ݆߱ሻܩሺ݆߱ሻ ቤ ܤ݀ܺ ا ͳ ܥሺ݆߱ሻܩሺ݆߱ ߱ሻ ௗ
݀ܽݎ ֜ ȁܶሺ݆߱௧ ሻȁௗ ൌ ܺ݀ܤ ݏ With X the desired noise attenuattion for frequencies߱ ب ௗ . ߱௧ ௦ To ensure a good output disturbancce rejection: ned as follow Sensitivity function S can be defin ͳ ฬ ܤܻ݀ ا ฬܵሺ݆߱ሻ ൌ ߱ሻ ௗ ͳ ܥሺ݆߱ሻܩሺ݆߱ ݀ܽݎ ሻ ௗ ൌ ܻ݀ܤ ߱ ا௦ ֜ ȁܵሺ݆߱௦ ሻȁ ݏ with Y the desired value of sensitivity function for ௗ [17] frequencies ߱ ߱ ا௦ ߱ ب ߱௧
Fig 1: Fractional PID Controller Convverge
All these classical PID Controllers are thee particular case of the fractional controller where Ȝ and ȝ aare equal to one (Figure: 1) [9]. Since these fractional contrrollers have two parameters more than the conventional PID D controller, two more specifications can be met, thus we ccan improve the performance of the overall system [20]. It ccan be expected that the PIȜDȝ controller may enhance the systems control performance as in [25], [26]. In many references, researches have used different modified method, methods of tuning like Ziegler Nichols m particle swarm optimization, neural network eetc. [18]-[23]. III. EXPERIMENTAL WORK Figure 2 shows general block diagram of fracttional order controller.
௦
od for approximation of We have used Oustaloup's metho fractional order system. This meethod is based on the approximation of a function of the form ܪሺݏሻ ൌ ݏఓ ǡ ߤ߳Թା It is given by ே
ሺݏሻ ൌ ܥෑ ܪ ୀିே
ͳ ݏȀ߱݇ ͳ ݏȀ߱Ԣ݇
Where, N is order of the finite TF approximation. [20], [24] Let's say one example,
3
Oustaloup-Recursive-Approximation for s^0.5
Bode Diagram
30 150
10
100
0
Magnitude (dB)
M a g n it u d e ( d B )
20
-10 -20 -30 60
50
0
P h as e (de g)
-50
-100 0
30
-4
10
10
-3
-2
10
10
-1
0
10
10
1
2
10
10
3
4
10
5
6
10
10
Frequency (rad/s)
Fig 3: Approximation of s^0.5 for and order 10 using Oustaloup's method
In figure 3, we have drawn diagram for μ =0.5 and order of approximation is 10.
Phase (deg)
-45
0
-90
-135
-180 -1
10
0
1
10
10
2
10
Frequency (rad/s)
(b) Frequency Response of Undamped System
Designing and Tuning of fractional order controller is shown below for undamping control system. An undamping system in which damping ratio (ȗ) is zero. Undamping system has a characteristics of sustain oscillations. We have taken following undamped control system ͳ ܩሺݏሻ ൌ ଶ ݏ ͳ Where, damping ratio (ȗ) is zero and natural damping frequency is 4 rad/sec. In Figure 4 (a) and (b), step response and frequency response of above system is shown.
Fig 4: (a) and (b) different response of undamped System
For comparison of result of fractional PID controller with classical PID controller, simulation is done for classical PID as shown in figure 5 below.
Step Response 0.14
0.12
0.1
Fig 5: Simulation of Classical PID Controller
Amplitude
0.08
Now, we will do design of fractional PID controller.
0.06
0.04
In all optimization, we have an IAE (Integral of the Absolute value of Error) as objective parameters. [7]
0.02
T
IAE = ³ e(t ) dt
0
-0.02
0
5
10
15
20
25
30
Time (seconds)
(a) Step Response of Undamped System
35
40
0
4 Step Response of Undamped System
1 Step Response of Fractional PID Controller Step Response of Classical PID Controller
Output
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50 60 Time (Sec)
70
80
90
100
Fig 7: Step Response of Fractional and Classical PID Controller
IV. CONCLUSION
Fig 6: Optimization Toolbox from FOMCON
Fractional order PID controller, ܩ ሺݏሻ ൌ െͳǤͷͺͳͻ
ͳʹǤ͵ͳ ͳǤͳ͵Ͳͺ ݏǤ଼ଵଽସଽ ݏǤଽହସ଼
Table I Values of various parameters fro Fractional PID Controller
Sr. No 1 2 3 4 5
Parameter
Value
Proportional Gain (Kp) Integral Gain (Ki) Order of Integration (Ȝ) Derivative Gain (Kd) Order of Differential (ȝ)
-1.5819 12.3761 0.90548 1.1308 0.81949
Proposed Fractional Controller design method gives better result in simulation for undamping control system. Simulation results also show that the requirements are totally fulfilled for undamping control system. Thus, advantage has been taken of the fractional orders Ȝ and ȝ to full fill additional specifications of design, ensuring a robust performance of the controlled system to gain changes, disturbances and noise. The optimization tool is used for tuning of Fractional PID Controller for better performance. A result of fractional PID Controller is compared with classical PID Controller. ACKNOWLEDGMENT We are thankful of FOMCON (“Fractional-order Modeling and Control”) group for their toolbox in MATLAB. It is an interdisciplinary project dedicated to research and development of applications of fractional calculus to modeling and control of complex dynamic system. REFERENCES [1] [2]
Overall transfer function of system ܥሺݏሻ ͳǤͳ͵Ͳͺ ݏ כଵǤଶହ െ ͳǤͷͺͳͻ ݏ כǤଽହସ଼ ͳʹǤ͵ͳ ൌ ଶǤଽହହ ͳǤͳ͵Ͳͺ ݏ כଵǤଶହ ͳͶǤͶͳͺͳ ݏ כǤଽହସ଼ ͳʹǤ͵ͳ ܴሺݏሻ ݏ
Figure 7 shows comparison of step response of Fractional PID controller with classical PID Controller.
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[4]
[5] [6] [7]
[8]
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