Autopilot Design for Flexible Tactical Aerospace Vehicle Using Parameter Plane Technique

Third International Conference on Advances in Control and Optimization of Dynamical Systems March 13-15, 2014. Kanpur, India

Autopilot Design for Flexible Tactical Aerospace Vehicle Using Parameter Plane Technique Siddhardha, Kedarisetty * Narayanan, Vignesh** Halder, Pulak *** 

*Birla Institute of Technology, Mesra, India (e-mail: siddhardhak2010@ gmail.com). **National Institute of technology, Kurukshetra, India (e-mail: [email protected]) *** Research Centre Imarat, Hyderabad, India, (e-mail: [email protected])} Abstract: This paper describes an application of parameter plane technique for autopilot design of flexible tactical aerospace vehicle (TAV). In literature, it is shown that parameter plane analysis simplifies the autopilot design for TAV considering rigid body dynamics only. However in the recent era, most of the TAV’s are structurally flexible in nature because of high l/d ratio. This is practiced for minimizing radar cross section (RCS) to avoid detection. The flexible nature of TAV causes structural oscillations which can be stabilized using compensators. The significant contribution of this paper is development of generalized algorithm for parameter plane technique. This algorithm can be used for both rigid as well as flexible body dynamics which was not available in literature. Stabilizing techniques such as gain stabilization and phase stabilization with any compensator dynamics can be used without modifying the algorithm. Distinct region of control gains is obtained for desired stability margins using this generalized parameter plane algorithm for three loop latax autopilot with flexible body dynamics and different stabilization techniques. Keywords: TAVs, Lateral acceleration autopilot, Parameter plane, Gain stabilization, Phase stabilization. 

increased by rate gyro feedback. Ideally, the structural oscillations are assumed to be absent, i.e. only rigid body dynamics are considered. For satisfactory performance with rigid body dynamics, the control gains for the autopilot are obtained by the parameter plane method (Sandip Ghosh et al. (2006)). However, most of the TAV’s in recent era are structurally flexible in nature because of high l/d ratio. This sort of airframe is preferred to avoid radar detection as they have small radar cross section (RCS). Here the rate gyro senses the structural oscillations in addition to the body rate and feeds them to the actuator, introducing the possibility of instability (Nesline, F., Nesline, L.M. (1985), Kabunde, Richard R. (1985)). Thus the control gains obtained by the parameter plane method for rigid body dynamics may fail to yield satisfactory stability margins when structural oscillations are taken into account. If the flexibility of the airframe is considered then some stabilizing techniques have to be used and the existing algorithms for parameter plane have to be modified. In this paper, a generalized algorithm for parameter plane technique which can include the effect of flexibility and compensation techniques to design three loop latax autopilot is developed. The order of the airframe transfer functions and compensators also can be varied as per user specifications. Thus the design procedure is made easy and user friendly.

1. INTRODUCTION The autopilot is a subsystem of the TAV which steers the vehicle in a desired path given by the guidance subsystem and maintain stability. There are several control techniques, which have been tried to obtain a robust autopilot with high accuracy. Despite numerous research work providing sufficient proof that modern control methodology improves performance of autopilots, two loop and three loop autopilots are still popular. The control design for the two loop and three loop autopilots is done by considering the dynamics of the vehicle as linear time invariant. The process of choosing the control gains for these autopilot configurations can be done using various design techniques such as pole placement technique (Kadam, N.V.(2006), Zarchan, P. (1998)), parameter plane technique (Sandip Ghosh et al. (2006)) etc. The basic design procedure involves piecewise linearization of the flight trajectory. The complete flight path is split into a number of segments and within each segment parameter like position, velocity, dynamic pressure etc. are assumed to be constant (Siouris, G.M(2004)). In each of this time slice a controller is designed. Design of controller for each time slice that satisfies the stability criteria is tedious. If the effect of structural oscillations are considered the procedure will become even more strenuous.

The paper is organized as follows: in the next section rigid body, flexible body dynamics of TAV are briefly discussed and introduction to parameter plane is also given. In the 2nd

Aerodynamic damping of the airframe dynamics for a TAV is very less (Garnell, P., East, D.J. (1977)). So the damping is

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section algorithm for generalized program is explained. In section 3 parameter plane technique for three loop lateral acceleration autopilot with rigid and flexible body dynamics are discussed. In section 4 gain stabilization of structural vibrations for three loop latax autopilot is discussed. Phase stabilization of three loop autopilot is implemented in section 5. The conclusions follow in section 6.

1.2 Parameter plane analysis Parameter plane analysis is a graphical tool which is used to find different combinations of control gains such that the system desired frequency domain specifications are satisfied. Parameter plane analysis in frequency domain is shown. Consider an open loop transfer function shown below.

1.1 Airframe dynamics

(6)

The airframe dynamics used in this paper are derived for a skid to turn, tail-controlled TAV and the aerodynamic coefficients are obtained from (Nesline, F., Nesline, L.M. (1985)).

Where,

The rigid body angular rate dynamics and acceleration dynamics of the airframe in pitch plane is given by,

Let A be the gain margin of the system and margin of the system then

B=Amplitude or gain of the system =Phase lag of the system

(1)

be the phase (7)

(2) (8) = aerodynamic rate loop gain from control surface deflection to body angular rate

Where, is known as gain-phase margin tester function (Sandip Ghosh et al. (2006)).

= aerodynamic turning rate time constant

The variables used to obtain the parameter plane are the control gains. Every point on the parameter plane gives a set of two control gains, which are x, y co-ordinates of the point.

= natural aerodynamic frequency of the TAV = aerodynamic damping ratio of the TAV

If one selects A=1 and =0 then the curve for critical stability margin between two control gains can be drawn for different crossover frequencies. This curve splits the parameter plane in two regions one where the system is stable and the other is unstable. If A=1 and desired is selected then curve for desired phase margin can be drawn as the frequency (gain cross over frequency) is varied. Similarly if desired A and =0 is selected then curve for desired gain margin can be drawn as the frequency (phase cross over frequency) is varied. There will be an intersection of three regions where the desired specifications are satisfied. This region contains all the various combinations of the control gains which will satisfy the desired stability margins.

= aerodynamic acceleration gain = aerodynamic acceleration transfer function numerator coefficients = Rigid body rotation rate = Acceleration achieved perpendicular to TAV body The flexible body dynamics of the airframe are given by (3) Kfb, ωfb = Gain, Quadratic zero of structural path of bending mode ζ1, ω1=Damping, Vibrational frequency of the bending mode

2. GENERALIZED ALGORITHM

The flexible body transfer function will be in parallel to the body angular rate dynamics because the rate gyro picks up structural vibrations in addition to the rigid body dynamics.

In literature (Sandip Ghosh et al. (2006)) the algorithm for parameter plane analysis is proposed for three loop latax autopilot design considering rigid body dynamics of TAV. However, if the vehicle dynamics are to be changed, by including the effect of flexibility, then the algorithm has to be re written. In this paper a generalized algorithm for three loop latax autopilot is proposed. This algorithm works for rigid body dynamics, flexible body dynamics, flexible body with gain stabilization and also flexible body with phase stabilization. There is no limitation on order of the transfer functions and number of compensators can be used to generate parameter plane plot without changing the algorithm.

Actuator and rate gyro dynamics are taken as second order and their transfer functions respectively are (

)

(4) (5)

ωa, ζa = natural frequency and damping of actuator ωr, ζr = natural frequency and damping of gyro

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Step 3. Calculate margin tester function for open loop transfer functions using equation 8 choosing k1 (outer loop control gain) and k2 (middle loop control gain) as unknowns. For example outer loop open loop margin tester function is given in equation (11)

Fig. 1. Generalized three loop latax autopilot block diagram (11)

Step1: Choose the type of dynamics (rigid body or flexible body) and type of stabilization required (gain or phase stabilization). Then enter the corresponding transfer functions. Accordingly the generalized transfer functions of the blocks shown in Fig. 1 will be calculated.

In the equation (11) k1 and k2 are unknowns. Thus writing it in a simplified format we obtain (12)

Table 1 Transfer functions according to chosen option Generalized transfer functions

Rigid body

Where,

Flexible body

No stabilizatio n

Gain stabilizatio n

Phase stabilizatio n

Gf1

Gi

Integrator

Integrator

Integrator

Gf2

Ga

Ga

Ga Gn

Ga Glag

Gf3

Gz

Gz

Gz

Gz

Gfb1

GqGr

(Gq+Gfb)Gr

(Gq+Gfb)Gr

(Gq+Gfb)Gr

Gfb2

1

1

1

1

Po =

(13)

Qo =

(14)

Ro= (15) Separating real and imaginary parts yields equations (16) and (17) RPo

+ RQo

+ RRo = 0

(16)

IPo

+ IQo

+ IRo = 0

(17)

Where, RPo, RQo, RRo =Real part of Po, Qo, Ro IPo, IQo, IRo =Imaginary part of Po, Qo, Ro Step 4. Solving (16) and (17) we obtain k1 and k2 for different values of . Thus plotting k1 vs k2 we obtain a curve on the parameter plane for desired stability margins.

Gf1, Gf2, Gf3, Gfb1, Gfb2 varies according to the type of dynamics and type stabilization chosen by the user. The flowchart of the program is shown in appendix B. Also choose various values of A and for desired phase and gain margins. Fix the value of inner loop gain (k3) such that the inner loop bandwidth is approximately ¼ th the bandwidth of actuator.

Step 6. Steps 2 to 5 are repeated for middle and inner most loop to obtain region of control gains which satisfy desired stability margins. 3. THREE LOOP LATERAL ACCELERATION AUTOPILOT

Step 2. Calculate all the three loops open loop transfer functions and disintegrate Gf1, Gf2, Gf3, Gfb1, and Gfb2 into their corresponding numerators and denominators. For example, outer loop open loop transfer function is shown below.

3.1 Parameter plane analysis for rigid body dynamics In this paper, the focus is on the design of lateral acceleration autopilots for a tail-controlled, skid-to-turn aerial vehicle configuration. The three loop lateral acceleration TAV autopilot configuration shown in Fig. 2 (Zarchan, P. (1998)).

(9)

(10) The values of in Nf1 (jω), Df1 (jω), Nf2 (jω), Df2 (jω), Nf3 (jω), Df3 (jω), Nfb1 (jω), Dfb1 (jω), Nfb2 (jω), Dfb2 (jω) for various frequencies are calculated in MATLAB.

Fig. 2. Three loop lateral acceleration autopilot with rigid body dynamics

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For most of the TAV’s three loop autopilot is required (Garnell, P., East, D.J. (1977), Parijat Bhoumick and Dr. Gourhari Das (2012)) because of the following reasons.

Table 2: Frequency domain information for three loop latax autopilot with rigid body dynamics

It has direct control over the body rate because the body rate error is made zero in three loop, also the maximum body rate can be limited. For homing TAV body rate error must be zero for proper tracking of the target. Bandwidth of the system is improved. The under shoot is reduced, thus less wastage of control energy. The value of kr is chosen constant such that the inner loop bandwidth is approximately 1/4 th of the actuators bandwidth (kr=0.33835) (Kadam, N.V. (2006), Garnell, P., East, D.J. (1977)). Parameter plane analysis for three loop autopilot with rigid body dynamics is shown in Fig. 3. Desired gain and phase margins for all the three loops are considered as 6 dB and 40 degrees.

Loop

Gain margin

Phase margin

Gain crossover

Phase crossover

Inner

8.09db

44deg

71.7rad/sec

161 rad/sec

Inter

11.2db

58.4deg

15.7rad/sec

69.1rad/sec

Outer

12.5db

70.7deg

8.33rad/sec

42.9rad/sec

Fig. 5. Bode response for three loop autopilot with rigid body dynamics Fig. 3. Parameter plane for three loop autopilot with rigid body dynamics

We can see that all the gain and phase margins for all the loops are satisfied. If the time domain specification are not satisfied then we can choose other point in the shaded region. Thus the design of the autopilot is much easier using parameter plane.

Considering a point within the region i.e. ka=-0.0055, ki= 11, kr=0.3385. For this set of control gains, time and frequency domain responses are shown in Fig. 4 and Fig. 5 respectively.

3.2 Parameter plane analysis with flexible body dynamics Rise time=0.131 sec; Settling Time=0.226 sec; %Peak over shoot=1.89%; Undershoot=11 %.

Three loop lateral acceleration autopilot scheme with flexible body dynamics is shown in Fig. 6. The bending mode mode transfer function is shown in parallel because the rate gyro detects the vibrations along with the body rate (Nesline, F., Nesline, L.M. (1985)).

Fig. 6. Three loop lateral acceleration autopilot with flexible body dynamics

Fig. 4. Step response for three loop autopilot with rigid body dynamics

From the parameter plane method the obtained region which satisfies the gain margin to be 6db for outer, inter and inner loops is shown in Fig. 7.

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Fig. 9. Bode response for three loop autopilot with flexible body dynamics

Fig. 7. Parameter plane for three loop autopilot with flexible body dynamics

We can see that the system is stable. However step response is highly oscillatory and none of the margins are satisfactory. This sort of response is not at all acceptable. The peak at the bending mode frequency in the Fig. 9 indicates that the system is susceptible for noise at these high frequencies. Thus some stabilizing technique has to be used to make system stable with satisfactory performance.

There are four regions shown in the Fig. 7. The region shaded in dark blue is the stability region. If any point from other than this region is considered to be control gains, the system will be unstable. It can be seen that other shaded regions which satisfy gain margin stabilities are not in intersection with the stability region. Thus there is no possible combination of control gains that can achieve required stability margins while keeping the system stable. The time domain and frequency domain plots are shown in Fig. 8 and 9 respectively. Considering a point (ka = -0.02, ki = 12, kr = 0.33835) within the stable region (shaded in blue).

4. GAIN STABILIZATION Gain stabilization is used to attenuate the gain at the modal frequencies using notch filters so that they will not cause destabilizing effect on the TAV. The notch filters used must have gain attenuation in a small region around the modal frequencies instead of attenuation only at the modal frequencies. The symmetric notch filters is in the form of

Rise time=0.0124 sec; Settling Time=1.86sec; %Peak over shoot=152%; Undershoot=56%

(18) =damping of the filter =attenuation frequency Bode response of the symmetric notch filter used is shown in Fig. 10. Fig. 8. Step response for three loop autopilot with flexible body dynamics Table 3: Frequency domain information for three loop latax autopilot with flexible body dynamics. Loop

Gain margin

Phase margin

Gain crossover

Phase crossover

Inner

0.11db

1.59deg

262rad/sec

261rad/sec

Inter

0.576db

4.34deg

40.6rad/sec

44.7rad/sec

Outer

0.532db

7.29deg

38.7rad/sec

43.8rad/sec

Fig. 10. Bode response for the symmetric notch filter

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Table 4: Frequency domain information after gain stabilization with symmetric notch filter

Fig. 11. Three loop lateral acceleration autopilot with flexible body dynamics with notch filter

Loop

Gain margin

Phase margin

Gain crossover

Phase crossover

Inner

6.5db

32.7deg

68.4rad/sec

117rad/sec

Inter

9.32db

58.3deg

15.6rad/sec

64.5rad/sec

Outer

11.9db

70.3deg

8.3rad/sec

43.2rad/sec

As shown in Fig. 11 there will be a symmetrical notch filter placed in the inner loop forward path to attenuate the gain at modal frequency. Parameter plane for gain stabilized flexible body TAV is shown in Fig. 12. The satisfactory inner loop phase margin is considered as 30 deg. Any set of control gains picked up from the shaded region will satisfy the desired stability margins.

Fig. 14. Bode response for gain stabilized (symmetric notch) three loop autopilot with flexible body dynamics It can be observed that due to addition of notch filter the gain at the modal frequency is attenuated and the phase margin of the inner loop has degraded by 11.3o. However 34o of phase margin is satisfactory. Bandwidth of the inner loop is reduced by 3.3 rad/sec only which is allowable.

Fig. 12. Parameter plane for gain stabilized (symmetric notch) three loop autopilot with flexible body dynamics Step and bode plots for gains ka=-0.0055, ki=11, kr=0.33835 are shown in Fig. 13 and Fig. 14 respectively.

5. PHASE STABILIZATION Phase stabilization is a stabilization process in which extra phase lag is induced in the system to shift the phase crossover frequency such that system becomes stable and required stability margins are achieved. In Nesline, F., Nesline, L.M.(1985), phase stabilization is done by reducing the natural frequency of the actuator. By reducing actuators natural frequency additional phase lag is induced. However this process of changing the actuator specifications is not practically feasible. Instead of varying the actuator dynamics, lag compensator is used to produce required phase lag. Lag compensator used is of the form

Rise time=0.13 sec; Settling Time=0.226 sec; %Peak over shoot=2.14%; Undershoot=12.4%

(19)

The bode response of the lag compensator used is shown in Fig. 15.

Fig. 13. Step response for gain stabilized (symmetric notch) three loop autopilot with flexible body dynamics

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Table 5: Frequency domain information after phase stabilization Loop

Gain margin

Phase margin

Gain crossover

Phase crossover

Inner

6.32db

34.5deg

66.9rad/sec

113rad/sec

Inter

10.3db

45.5deg

14rad/sec

54.1rad/sec

Outer

10.7db

51.6deg

10.8rad/sec

37.6rad/sec

Fig. 15. Second order lag compensator Parameter plane for satisfactory margins for phase stabilization with second order lag compensator is shown in Fig. 16. The satisfactory inner loop gain margin and inner loop phase margin is considered as 6db and 30 deg respectively.

Fig. 18. Bode response for phase stabilized three loop autopilot with flexible body dynamics By using the lag compensator it is evident that we can obtain the required stability margin. Large peaks at the modal frequency in the inner loop gain plot can be seen in the Fig. 29. This indicates that at the modal frequency the system is prone to noise. The noise generated at those frequencies can result in excessive fin rates and heating of the actuator. If there is uncertainty in modal frequency, the designed lag compensator may not produce enough lag. This could result in different crossover frequency which would result in lower stability margins or worse. Due to the above reasons if the bending mode frequency is not in the range of autopilot bandwidth then gain stabilization is preferred.

Fig. 16. Parameter plane for phase stabilized three loop autopilot with flexible body dynamics Considering a point within the shaded region, ka=-0.01046, ki=6.293, kr=0.33835. The step response and bode plots for the chosen control gains are shown in Fig. 17 and Fig. 18 respectively. Rise time=0.0872 sec; Settling Time=0.379 sec; %Peak over shoot=16.5%; Undershoot=14.5%

6. CONCLUSIONS A generalized algorithm to obtain the control gains through parameter plane analysis for multiple configurations is developed. The virtue of this algorithm is that it can consider the practical problem of structural modes into account, and allows designers to use stabilization techniques such as gain and phase stabilization. This algorithm also allows the designer to change the vehicle dynamics (rigid or flexible body), specifications of the notch filter in gain stabilization and compensator in phase stabilization. Based on the requirement, the user can obtain the control gains and directly use it in the design process. Thus the design process of autopilot is made easy and user friendly. Presence of structural vibrations in three loop lateral acceleration autopilot designed for rigid body dynamics may destabilize the system. From parameter plane analysis it is

Fig. 17. Step response for phase stabilized three loop autopilot with flexible body dynamics

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clear that there is very small region where the system can be stable after introducing flexibility, however those region of control gains will not give satisfactory margins. All possible combinations of control gains required for the satisfactory performance of the autopilot are found out from the parameter plane analysis after stabilizing the system using gain stabilization and phase stabilization techniques. The optimal point of the control gain in the satisfactory region can be chosen using a cost function and optimizing technique may be the future scope of work.

Structural mode parameters kfb = 0.00134 ωzfb= 345 rad/sec ξ1 = 0.015 ω1 = 259 rad/sec Structural filter parameters ωoi = 259rad/sec ξoi = 0.35 mki= 5 Lag compensator parameters ξl1 = 0.4 ωl1 = 300 rad/sec ξl2 = 0.65 ωl2 = 200 rad/sec

REFERENCES Garnell, P., East, D.J.(1977). Guided weapon systems. Pergamon press.

Appendix B. FLOWCHART

Kabunde, Richard R. (1985). A state space approach to body bending compensation for missile autopilot design. American control conference, Boston, 341-344. Kadam, N.V. (2009). A practical design of flight control systems, Allied publishers Pvt. Ltd. Nesline F, Jr., Nesline, L. Mark. (1985). Phase vs gain stabilization of structural feedback oscillations in homing missile autopilots. American control conference, Boston, 323-329. Parijat Bhoumick and Dr. Gourhari Das, Three loop Lateral Missile Autopilot Design in pitch plane using State feedback & Reduced Order Observer(DGO), International Journal of Engineering Research and Development (IJERD) , Volume 1, Issue 8(June 2012), PP 12-17. Sandip Ghosh, Kalyankumar Datta, Shyamal Kumar Goswami, Samar Bhattacharya. A Parameter Plane Design Methodology for Three Loop Missile Autopilot, Journal of Institute of Engineers(India) vol-86 Jan2006 Page no. 214219. Siouris, G.M. (2004). Missile guidance and control systems, Springer-verlog, New York. Zarchan, P. Tactical and Strategic Missile Guidance American institute of Aeronotics and Astronomics 3rd Ed. Aug 1998. Appendix A. PARMETERS AND COEFFICIENTS Actuator and rate gyro parameters ωa = 250 rad/sec ξa = 0.7 ωg = 250 rad/sec ξg = 0.7 Aerodynamic parameters ωd = 22.4 rad/sec ξd = 0.052 k3 = 0.6477 Tα = 0.676 A11= 0.001054 A12= -0.00081 k1 = -1116.5

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