Available online at www.sciencedirect.com
Procedia Technology 4 (2012) 74 – 81
C3IT-2012
Compensator Design for Helicopter Stabilization Raghupati Goswamia,Sourish Sanyalb, Amar Nath Sanyalc a
Chairman, Ideal Institute of Engineering, Kalyani Shilpanchal,Kalyani, Nadia,West Bengal, India,
[email protected] b Assistant Professor, College of Engineering and Management, Kolaghat,Purba Midnapur,W.B.,India,
[email protected] c Professor, & Former HOD of Elect Engg, Jadavpur Univ.Academy of Technology, Aedconagar,Hoogly, West Bengal,India,
[email protected]
Abstract Helicopters are inherently unstable unlike aeroplanes. They are to be stabilized using appropriate feedback loops. A control structure with an inner and an outer loop is best-suited for them. The paper describes a design procedure for such a control scheme aiming at fulfilment of the design specs. The problem has been treated analytically with selfbuilt programmes at first. Then recourse has been made to MATLAB control system window for refinement of the design. Finally SISOTOOL has been used to find out the best possible design configuration under given constraints. The design methodology yields a stable control system which has to be kept operative under flying conditions.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of C3IT Keywords: Helicopter; Feedback Control;Stabilization;Compensation
1. Introduction The helicopter is a type of rotorcraft in which lift and thrust are supplied by one or more engine driven rotors. In contrast with fixed-wing aircraft, this allows the helicopter to take off and land vertically, to hover, and to fly forwards, backwards and laterally. These attributes allow helicopters to be used in congested or isolated areas where fixed-wing aircraft would not be able to take off or land. The capability to efficiently hover for extended periods of time allows a helicopter to accomplish tasks that fixed-wing aircraft and other forms of vertical takeoff and landing aircraft cannot perform Fixed wing aircrafts possess a moderate degree of inherent stability but the helicopters do not have this property. They need be stabilized using feedback loops. The control is generally effected by an inner automatic stabilization loop and an outer loop manually controlled by the pilot. 1.1 The system description The block diagram of the system is given in fig. 1. The automated inner loop contains the helicopter dynamics in the forward path and a PD-type feedback. The outer loop is switched on by the pilot as and when required. He inserts commands into it based on the attitude error displayed to him. When the pilot
2212-0173 © 2012 Published by Elsevier Ltd. doi:10.1016/j.protcy.2012.05.009
Raghupati Goswami et al. / Procedia Technology 4 (2012) 74 – 81
do not exercise any control, the switch S1 remains open. The model of the pilot’s transfer function G1(s) contains a gain factor and anticipation time constant of 1 sec. and an error-smoothing time constant of 10 sec. 1.2 The design specifications The system must be closed loop stable. Under automated condition (without intervention by the pilot, while only the inner loop is operative), the overshoot has to be limited to 8%, and the settling time has to be limited to 45 sec. The gain margin must be at least 40 db and the phase margin must be at least 30o.
Fig. 1 Helicopter stabilization: block diagram Construction of reference
While the outer loop is operative, the system is desired to be only slightly underdamped, the peak overshoot being within 1% and the settling time within 15 sec. The gain margin must be at least 50 db and the phase margin more. 2. The Design Methodology The usual way is to use root locus technique to meet the specifications. But root locus is a graphical method, from which it is difficult to accurately meet the design specs. At first, analytical approach has been made to find out appropriate values of the feedback gain Kp , & Kd . 2.1. Analytical method The system is considered to be automated and the pilot control loop to be open. The feedback gains cannot be chosen so as to satisfy the design specs. However stability can be ensured by relaxing the design constraints. The chosen valuea are: Kp = 5.45 and Kd= 5.45. The characteristic equation for the system is given below: s4 + 9.15 s3 + 110.325 s2 + 112.12 s + 3.945 = 0 The roots of the characteristic equation: -1.0660; -0.0365 ; -4.0238േ j 9.2316 Damping factor of the inner loop is found to be 0.3996. against 2nd order approximation, the % overshoot is 25.4%. The phase margin is found to be 47.4o and the gain margin 35.3 db. Then, keeping the inner loop feedback gain unchanged, the forward path gain of the outer loop was varied. A damping factor of 0.3 only could be achieved. The chosen value of the forward path gain is 1.175. The characteristic equation is given below:
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s5 + 9.25 s4 + 97.74 s3 + 317.04 s2 + 228.786 s + 6.6285 = 0 The roots of the characteristic equation are: -3.1008; -0.9618 -0.0302; -2.5786 ± j 8.1779 Against 2nd order approximation, the % overshoot is found to be 37.3%. As the design has to be discarded, the stability margins are not found out. The characteristic equation. has been formed using Mason’s gain formula and the roots of the characteristic equation have been found out by a combination of Regula-Falsi and Newton-Raphson method. Though the system has been made stable, the design specification has not been fulfilled.
2.2. Design with MATLAB The control system window of MATLAB is now being used for refinement of the design. In this case the design constraints are more stringent. The value of the feedback path gains have been fixed up to: Kp= 8.45; Kd=14 . The closed loop transfer function of the inner loop with these values of gains is given as:
CLTFinner
20 s 2 180.6 s 5.4 s 4 9.15s 3 170.3s 2 284.9 s 9.075
The unit step response of the system with only inner loop operative is given in fig. 2 and the Bode plot in fig. System: systemfb Peak amplitude: 0.639 Overshoot (%): 7.33 At time (sec): 3.66
0.7
Step Response
0.6 System: systemfb Settling Time (sec): 44.2
Amplitude
0.5
System: systemfb Rise Time (sec): 0.811
0.4
0.3
0.2
0.1
0 0
10
20
30
40
50
60
70
80
Time (sec)
Fig. 2 Step Response with only inner loop operative
The % overshoot is now only 7.33, the rise time is 0.81 sec, peak time 3.66 sec and the settling time is 44.2 sec. The Bode plot shows a gain margin of - 40.3 db. That the system is stable is evident from the Nyquist plot of the loop gain. So the negative sign has to be ignored. The phase margin has been found out to be 30.2o.
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Bode Diagram Gm = -40.3 dB (at 0.532 rad/sec) , Pm = 30.2 deg (at 11.6 rad/sec)
Magnitude (dB)
50
0
-50
Phase (deg)
-100 -90
-180
-270
-360 -3
-2
10
-1
10
0
10
1
10
2
10
3
10
10
Frequency (rad/sec)
Fig. 3 Bode plot with only inner loop operative
Nyquist Diagram 150 0 dB
100
Imaginary Axis
50
0 System: frd Phase Margin (deg): 30.2 Delay Margin (sec): 0.0454 At frequency (rad/sec): 11.6 Closed Loop Stable? Yes
-50
-100
-150 -140
-120
-100
-80
-60
-40
-20
0
20
Real Axis
Fig. 4 Nyquist plot while only inner loop is operative
Now considering the outer loop to be switched on and operative, the following transfer function is obtained with K1= 0.2
CLTFfull
4 s 3 40.12s 2 37.2 s 1.08 s 5 9.25s 4 175.2s 3 284.9 s 2 74.77 s 1.988
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The corresponding unit step response and the Bode plot are given in fig 5 and fig. 6. The % overshot overshoot has now reduced to 1.19, The rise time is 8.99 sec, the peak time 25 sec and the settling time 14.3 sec. So much improvement of the design has been made by using MATLAB tools.
Step Response 0.7
0.6
System: sysfinal Settling Time (sec): 14.3
0.5
Amplitude
System: sysfinal Rise Time (sec): 8.99
System: sysfinal Peak amplitude >= 0.55 Overshoot (%): 1.19 At time (sec) > 25
0.4
0.3
0.2
0.1
0 0
5
10
15
20
25
Time (sec)
Fig. 5 Unit step response while the outer loop is operative
Bode Diagram Gm = 51.7 dB (at 41.3 rad/sec) , Pm = 145 deg (at 0.0795 rad/sec) 20
Magnitude (dB)
0 -20 -40 -60 -80 0
Phase (deg)
-45 -90 -135 -180 -225 -3
10
-2
10
-1
10
0
10
Frequency (rad/sec)
Fig. 6 Bode plot while the outer loop is operative
1
10
2
10
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2.3. Application of SISOTOOL The design configuration is given in fig.7 and the tunable and fixed elements in the Table1.
Fig. 7. Block diagram of design made using SISOTOOL
Table 1. Results Found for Different Blocks Tuneable Elements
Parameters
Fixed Elements
C
F
G
H
Gain
5.3954
1.2894
4
1
Zeroes
-1.0065
Nil
-9;-1;-0.03
nil
Poles
-1.8462
Nil
-3.669±j11.97; -1.78; -0.1; 0.032
nil
In the SISOTOOL approach the block C & F are tunable, the other blocks are fixed. An appropriate gain has been used for block F and a lead compensator for block C. The closed loop step response and the open loop Bode plot using SISOTOOL are given in fig. 8 and the open loop Bode plot and the root locus in fig. 9.
Fig. 8 Step response and open loop Bode plot using SISOTOOL
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Open-Loop Bode Editor for Open Loop 1 (OL1)
Root Locus Editor f or Open Loop 1 (OL1) 40
0.21
0.15
0.32 20 0.55
20
35 0.105 0.070.0440.02 30 25 20 15 10 5
10 0
0
-10
Requirement: GM > 50 PM > 80
5 10 -20 0.55 15 -20 20 -30 25 0.32 30 G.M.: Inf 0.21 0.15 0.105 0.070.0440.02 35 -40 -40 Freq: Inf -10 -8 -6 -4 -2 0 Stable loop Bode Editor for Closed Loop 1 (CL1) -50 50 0
0
-45
-50 0
-90
-135
-90
P.M.: 125 deg Freq: 0.409 rad/sec -180 -2 10
-180 -1
10
0
1
10 10 Frequency (rad/sec)
2
10
-4
10
-2
0
10 10 Frequency (rad/sec)
2
10
Fig. 9. Open and closed loop Bode plot and root locus using SISOTOOL
Using SISOTOOL, the design has been further improved. In this case, the system is found to be slightly underdamped. The peak overshoot is only 0.524 % at a peak time of 14 sec. The settling time has reduced to 8.79 sec. The phase margin is as high as 125o. and the gain margin is Therefore the design is much better than the earlier designs. It has been noted that the MATLAB-tools are more efficient and yields improved design which satisfies all the design constraints. It has been found that SISOTOOL is still better which can be conveniently used for the design even when the constraints are stringent and the requirements controversial.
3. Conclusions Helicopters are of great importance as vehicles for traversing through relatively smaller distances. It has both military and domestic applications. These are suitable for such fields of transportation, construction, firefighting, search and rescue, and military activities, where a conventional aeroplane cannot be conveniently used. As its construction and principle of operation are substantively different from that of an aeroplane, the control strategy is also different. Unlike an aeroplane these devices are inherently unstable. The best way to stabilize them is by using a control structure with an inner and outer loop, the inner loop being automated and the outer loop switched and handled by the pilot as and when required. The design procedure for such a system has been discussed in the paper. The design has been made both analytically and by using MATLAB-tools.
References 1. 2. 3. 4. 5.
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