Available online at www.sciencedirect.com Procedia Technology 4 (212 ) 74 81 C3IT-212 Compensator Design for Helicopter Stabilization Raghupati Goswami a,sourish Sanyal b, Amar Nath Sanyal c a Chairman, Ideal Institute of Engineering, Kalyani Shilpanchal,Kalyani, Nadia,West Bengal, India, raghupatigoswami@rediffmail.com b Assistant Professor, College of Engineering and Management, Kolaghat,Purba Midnapur,W.B.,India, sourish27_may@yahoo.co.in c Professor, & Former HOD of Elect Engg, Jadavpur Univ.Academy of Technology, Aedconagar,Hoogly, West Bengal,India, ansanyal@yahoo.co.in 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. 211 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of C3IT Open access under CC BY-NC-ND license. 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 2212-173 212 Published by Elsevier Ltd. doi:1.116/j.protcy.212.5.9 Open access under CC BY-NC-ND license.
Raghupati Goswami et al. / Procedia Technology 4 ( 212 ) 74 81 75 do not exercise any control, the switch S 1 remains open. The model of the pilot s transfer function G 1 (s) contains a gain factor and anticipation time constant of 1 sec. and an error-smoothing time constant of 1 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 4 db and the phase margin must be at least 3 o. 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 5 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 K p, & K d. 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: K p = 5.45 and K d = 5.45. The characteristic equation for the system is given below: s 4 + 9.15 s 3 + 11.325 s 2 + 112.12 s + 3.945 = The roots of the characteristic equation: -1.66; -.365 ; -4.238 j 9.2316 Damping factor of the inner loop is found to be.3996. against 2 nd order approximation, the % overshoot is 25.4%. The phase margin is found to be 47.4 o 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.3 only could be achieved. The chosen value of the forward path gain is 1.175. The characteristic equation is given below:
76 Raghupati Goswami et al. / Procedia Technology 4 ( 212 ) 74 81 s 5 + 9.25 s 4 + 97.74 s 3 + 317.4 s 2 + 228.786 s + 6.6285 = The roots of the characteristic equation are: -3.18; -.9618 -.32; -2.5786 ± j 8.1779 Against 2 nd 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: K p = 8.45; K d =14. The closed loop transfer function of the inner loop with these values of gains is given as: 2 2s 18.6s 5.4 CLTFinner 4 3 2 s 9.15s 17.3s 284.9s 9.75 The unit step response of the system with only inner loop operative is given in fig. 2 and the Bode plot in fig..7 System: systemfb Peak amplitude:.639 Overshoot (%): 7.33 At time (sec): 3.66 Step Response.6.5 System: systemfb Rise Time (sec):.811 System: systemfb Settling Time (sec): 44.2 Amplitude.4.3.2.1 1 2 3 4 5 6 7 8 Time (sec) Fig. 2 Step Response with only inner loop operative The % overshoot is now only 7.33, the rise time is.81 sec, peak time 3.66 sec and the settling time is 44.2 sec. The Bode plot shows a gain margin of - 4.3 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 3.2 o.
Raghupati Goswami et al. / Procedia Technology 4 ( 212 ) 74 81 77 5 Bode Diagram Gm = -4.3 db (at.532 rad/sec), Pm = 3.2 deg (at 11.6 rad/sec) Magnitude (db) -5-1 -9 Phase (deg) -18-27 -36 1-3 1-2 1-1 1 1 1 1 2 1 3 Frequency (rad/sec) Fig. 3 Bode plot with only inner loop operative 15 Nyquist Diagram db 1 5 Imaginary Axis -5 System: frd Phase Margin (deg): 3.2 Delay Margin (sec):.454 At frequency (rad/sec): 11.6 Closed Loop Stable? Yes -1-15 -14-12 -1-8 -6-4 -2 2 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 K 1 =.2 CLTF full 3 2 4s 4.12s 37.2s 1.8 5 4 3 2 s 9.25s 175.2s 284.9s 74.77s 1.988
78 Raghupati Goswami et al. / Procedia Technology 4 ( 212 ) 74 81 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..7 Step Response.6 System: sysfinal Settling Time (sec): 14.3 Amplitude.5.4.3 System: sysfinal Rise Time (sec): 8.99 System: sysfinal Peak amplitude >=.55 Overshoot (%): 1.19 At time (sec) > 25.2.1 5 1 15 2 25 Time (sec) Fig. 5 Unit step response while the outer loop is operative 2 Bode Diagram Gm = 51.7 db (at 41.3 rad/sec), Pm = 145 deg (at.795 rad/sec) Magnitude (db) -2-4 -6-8 -45 Phase (deg) -9-135 -18-225 1-3 1-2 1-1 1 1 1 1 2 Frequency (rad/sec) Fig. 6 Bode plot while the outer loop is operative
Raghupati Goswami et al. / Procedia Technology 4 ( 212 ) 74 81 79 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 Parameters Tuneable Elements Fixed Elements C F G H Gain 5.3954 1.2894 4 1 Zeroes -1.65 Nil -9;-1;-.3 nil Poles -1.8462 Nil -3.669±j11.97; -1.78; -.1; -.32 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
8 Raghupati Goswami et al. / Procedia Technology 4 ( 212 ) 74 81 Root Locus Editor for Open Loop 1 (OL1) 4.21.15.15.7.44.2 35 3.32 25 2 2 15.55 1 5.55-2 5 Bode Editor for Closed Loop 1 (CL1) 5 1 15 2 25.32 3.21.15.15.7.44.2-4 35-1 -8-6 -4-2 Open-Loop Bode Editor for Open Loop 1 (OL1) 2 1-1 -2-3 G.M.: Inf -4 Freq: Inf Stable loop -5 Requirement: GM > 5 PM > 8-45 -5-9 -135-9 P.M.: 125 deg Freq:.49 rad/sec -18-18 1-2 1-1 1 1 1 1 2 1-4 1-2 1 1 2 Frequency (rad/sec) Frequency (rad/sec) 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.524 % at a peak time of 14 sec. The settling time has reduced to 8.79 sec. The phase margin is as high as 125 o. 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. S. Dasgupta, Control system theory, Khanna Publishers 2. A.M. Law and W.D. Kelton, Simulation, modeling and analysis, Mcgraw-Hill, New York, 2 nd Edition, 1991. 3. M. Gopal, Modern control system theory, 2 nd Ed., New Age International 4. S.M. Shinners, Modern control system theory and design, John Wiley and Sons. 5. J.J. D Azzo, C.H. Houpis and S.N. Sheldon, Linear control system analysis and design with MATLAB, 5e, Marcel Dekker Inc. New York, BASEL
Raghupati Goswami et al. / Procedia Technology 4 ( 212 ) 74 81 81 6. A.J. Grace, N. Laub, J.N. Little and C. Thomson, Control system tool box for use with MATLAB, User Guide, Mathworks, 199. 7. J.M. Maciejowski, Multivariable feedback design, Addison-Wesley, Wokingham, England, 1989. ISBN -21-18243-2. 8. D.C. Youla, H.A. Jabr and C.N. Lu, Single-loop feedback stabilization of linear multivariable dynamical plants, Automatica, 1: pp. 159 173, 1974. 9. A.N. Sanyal and S. Sanyal, A generalized overview of control systems, Conf. Proc. of Modern Trends in Instrumentation and Control (MTIC1), CIEM, Kolkata, March 11-12, 21. 1. Santhanam G, Ryu SI, Yu BM, Afshar A, Shenoy KV (26) A high-performance brain computer interface. Nature 442: 195 198. 11. A. S. Krupadanam, A. M. Annaswamyy, and R. S. Mangoubiz, A Multivariable Adaptive Control Design with Applications to Autonomous Helicopters, http://web.mit.edu/aaclab/pdfs/journal.pdf