MATLAB SIMULATIONS FOR GARNELL's ROLL AUTOPILOT

MATLAB SIMULATIONS FOR GARNELL’S DESIGN OF ROLL AUTOPILOT

Introduction

1. An autopilot is a closed loop system and it is a minor loop inside the main guidance loop; not all missile systems require an autopilot. In a roll stabilised missile, the function of the roll autopilot is to maintain the roll position constant or zero so that roll rate is zero. This simplifies the design of pitch and yaw autopilot.

(a) The roll position demand (d), in the case of Twist and Steer control, is compared with the actual roll position (), sensed by the roll gyro.

(b) The error is amplified and fed to the servos, which in turn move the ailerons.

(c) The movement of the ailerons, results in the change in the roll orientation of the missile airframe.

(d) The changes in the airframe orientation due to external disturbances, biases etc are also shown in the achieved roll position.

(e) The controlling action (feed back) continues till the demanded roll orientation is achieved.

CLASSICAL CONTROL APPROACH

2. Now consider an air-to-air homing missile which is roll position stabilized and travels at a velocity in the range M=1.4 to M=2.8.

DYNAMICS OF THE ROLL AUTOPILOT

3. Roll Rate/Aileron . This is the simplest aerodynamic transfer function.

(a) Let us first consider that no disturbance is expected in the roll channel.

1/

( 1)p

a

l

T s

l1

s( )s

( )s

where can be regarded as a steady state gain and and can be regarded as an aerodynamic constant.

(a) The block diagram representation of this transfer function will be as follows: -

4. Dynamics of Actuator. The actuator dynamics is represented by the second order transfer function as given below: -

The same can be represented in time constant form as follows: -

5. Dynamics of Roll Gyro. The roll gyro also will take some time to come to its steady state value after some change has been applied. However the time constant of roll gyro is quite small when compared to the time constant of the rest of the open loop system; hence it can be assumed that the roll gyro comes to steady state in no time at all and gives an output amplified by the steady state gain Kg.

6. Considering that a disturbance L is expected to be applied on the missile body, the revised block diagram representation including actuator (servo) and feedback (gyro) dynamics will be as follows: -

7. The transfer function for this block diagram will be : -

ANALYSIS OF THE DYNAMICS OF ROLL AUTOPILOT

8. In order to design the roll loop one must know the maximum anticipated induced rolling moment and the desired roll position accuracy.

SL NO

PARAMETER VALUE FOR M=2.8

1 13,500

2 37.3

3 0.0257

4 362

(a) The aerodynamicist estimates that the largest rolling moments will occur at M=2.8 due to unequal incidence in pitch and yaw and will have a maximum value of 1000 Nm.

(b) If the maximum missile roll angle permissible is 1/20 rad then the stiffness of the loop must be not less than 1000 x 20 = 20,000 Nm/rad.

(c) This means that in order to balance this disturbing moment we have to use 1000/13,500 rad aileron, and this is approximately 4.2˚.

(d) The block diagram above shows the roll position control loop with demanded roll position equal to zero.

(e) The actual servo steady state gain – ks has to be negative in order to ensure a negative feedback system. Since the steady state roll angle φOSS for a constant disturbing torque L is given by

it follows that ks*kg must be not less than 20000/13500 = 1.48.

(f) If kg is set at unity then ks must be 1.48. The open loop gain is now fixed at

=535.

9. Unmodelled Dynamics. When we ignore the dynamics of a particular system which is part of a larger control loop on the basis of negligible time constant, for e.g., gyro which is actually a second order system can be equated just to a amplifier with a certain amount of gain, then we call such an analysis as unmodelled dynamics. Ignoring the actuator dynamics, the loop transfer function is given by

(g) The corresponding frequency response function will be

(i) The gain of the system will be given by the modulus of this transfer function GH i.e.,

(ii) The gain cross-over frequency is the frequency at which the gain is unity i.e.,

(iii) The phase margin is calculated from the phase angle at the gain cross-over frequency i.e.,

Thus phase margin by definition is given by

By choosing Ks with phase of fifteen degrees, we can make the system stable.

10. If the dynamics of the servo are included and we are given the values of ωns = 180 rad/sec and μs = 0.5, the loop transfer function will be suitably modified. Considering the phase margin at gain cross-over frequency alone since the gain cross-over frequency will not be altered much, the phase of the transfer function at gain cross-over frequency is calculated as: -

Hence Phase Margin (PM) = 180-230 = -50˚

Thus it is seen that by ignoring the servo dynamics, PM was +15 and when servo dynamics was included, the PM has gone negative and that too, by a large value, -50. Hence in a practical system, the servo dynamics is going to make the closed loop system unstable when included.

When this happens, compensators will have to be incorporated.

For the example being considered let us introduce a lag compensator with the transfer function

where β=15 and Tb = 0.05.

(a) The phase margin now improves to +10.2 deg from – 50 deg thus making the system stable.

(b) But the gain cross over frequency has reduced to 32.6 rad/sec thus reducing the bandwidth of the system.

(c) Also a phase margin of 10 degrees is not acceptable for missile control systems since the complex dynamics involved will cover up this phase margin driving it negative again. A margin of +40 to 50 degrees is normally acceptable.

11. A lead compensator can be added further to push the phase margin to a higher value. Hence let us now introduce a lead compensator with the transfer function

(a) The phase margin improves to 48 deg at 40.9 rad/sec.

(b) The gain margin is 11.2 db at 147 rad/sec i.e., gain cross over frequency also has improved as also the bandwidth.

12. The Bode plots of the above step-by-step procedure of building the roll autopilot were obtained by using MATLAB. The plots are as shown in the graphs below: -

FIG (a) UNMODELLED DYNAMICS OF ROLL AUTOPILOT

FIG (b) RESPONSE WHEN SERVO DYNAMICS INCLUDED

FIG (c) ROLL AUTOPILOT WITH PHASE LAG COMPENSATION

FIG (d) ROLL AUTOPILOT WITH PHASE LEAD AND LAG COMPENSATION

13. The time response plots for the various stages of design of roll autopilot are as shown below:-

(a) The first diagram shows the time response of the roll autopilot when unmodeled dynamics of actuator and gyro are not considered. Though the system is underdamped it meets the transient response and steady state error requirements.

(b) The second diagram shows the time response when the second actuator dynamics are included. It shows that the system becomes unstable.

(c) The third diagram shows the time response when a lead-lag compensator is included to compensated for the second order unmodeled actuator dynamics. The desired transient response characteristics are met with satisfactory settling time though the overshoot has become nearly 25.

(d) The fourth diagram shows a combination of roll autopilot without unmodeled dynamics of actuator and with a compensated system including unmodeled dynamics of

actuator.

14. The MATLAB program for Bode Plot used is as shown below:-

%---Aerodynamic tf for roll rate/aileron=p/Epsilon=lE/(s-lP)num1=[0.05 1];den1=[0.75 1];num2=[0.0257 1];den2=[0.0018 1];num3=[0 0 535]den3=[0.026 1 0];sys=tf(num3,den3);subplot(2,2,1)title('Unmodelled dynamics of roll autopilot')margin(sys)% [Gm,pm,wcp,wcg]=margin(sys);% GmdB=20*log10(Gm);% [GmdB pm wcp wcg]%----Include Servo Dynamics--omegans=180.0;mus=0.5;num4=[1.48];den4=[1/omegans^2 2*mus/omegans 1];[num5,den5]=series(num3,den3,num4,den4);sys1=tf(num5,den5);subplot(2,2,3)title('Roll Autopilot with Servo Dynamics Included')margin(sys1)% [Gm,pm,wcp,wcg]=margin(sys1);% GmdB=20*log10(Gm);% [GmdB pm wcp wcg]%----Phase Lag compensator---%[num6,den6]=feedback(num5,den5,num1,den1);[num6,den6]=series(num5,den5,num1,den1);

% [num6,den6]=parallel(num5,den5,num1,den1);sys2=tf(num6,den6);subplot(2,2,2)title('Roll Autopilot with Phase Lag Compensation')margin(sys2)% [Gm,pm,wcp,wcg]=margin(sys2);% GmdB=20*log10(Gm);% [GmdB pm wcp wcg]%------Phase Lead Compensator---%[num7,den7]=feedback(num6,den6,num2,den2);[num7,den7]=series(num6,den6,num2,den2);% [num7,den7]=parallel(num6,den6,num2,den2);sys3=tf(num7,den7);subplot(2,2,4)title('Roll Autopilot with Phase Lead&Lag Compensation')margin(sys3)[Gm,pm,wcp,wcg]=margin(sys3);%GmdB=20*log10(Gm);%[GmdB pm wcp wcg]

15. The MATLAB program for time response plots is as shown below:-

%--------------------------------------------------------------------------%SIMULATION OF ROLL AUTOPILOT GIVEN IN BOOK BY P.GARNELL%--------------------------------------------------------------------------global num1 den1 num2 den2 num3 den3 num4 den4 omegans mus ks kg Cn1 C1 C2%---Aerodynamic tf for roll rate/aileron=p/Epsilon=lE/(s-lP) num3=[535];den3=[0.026 1];sys=tf(num3,den3);%----Include Servo Dynamics--omegans=180.0;mus=0.5;kg=1.0;ks=1.48;Cn1=kg*ks*omegans^2num4=[Cn1];C1=omegans^2;C2=2*mus*omegans;den4=[1 C2 C1];%-----Include lead-lag compensatornum1=[0.05 1];den1=[0.75 1];num2=[0.0257 1];den2=[0.0018 1]; sim('rollapoct09');subplot(2,2,1)plot(t,yo)title('Roll autopilot-no unmodeled dynamics');%figuresubplot(2,2,2)plot(t,yo1)title('Roll Autopilot with Servo Dynamics Included');%figure%----Include Servo Dynamics and Phase Lead Lag Compensator--subplot(2,2,3)plot(t,yo2)title('Roll Autopilot with Servo Dynamics and Compensator Included');subplot(2,2,4);%figureplot(t,yo,'r',t,yo2,'b');title('Roll Rate plots for Unmodeled and Compensated Systems');