GARNELL’S PITCH AUTOPILOT DESIGN - AN ANALYSIS

Introduction

1. An autopilot is a closed loop system and it is a minor loop inside the main guidance loop. Broadly speaking autopilots either control the motion in the pitch and yaw planes, in which they are called lateral autopilots, or they control the motion about the fore and aft axis in which case they are called roll autopilots. In aircraft autopilots, those designed to control the motion in the pitch plane are called longitudinal autopilots and only those to control the motion in yaw are called lateral autopilots. For a symmetrical cruciform missile however pitch and yaw autopilots are often identical; one injects a g bias in the vertical plane to offset the effect of gravity but this does not affect the design of the autopilot. Lateral “g” autopilots are designed to enable a missile to achieve a high and consistent “g” response to a command. They are particularly relevant to SAMs and AAMs.

2. The block diagram of a lateral autopilot is as shown in figure above. The general working of the blocks is as given below:-

(a) An accelerometer is placed in the yaw plane of the missile, to sense the sideways acceleration of the missile. This accelerometer produces a voltage proportional to the linear acceleration.

(b) This measured acceleration is compared with the ‘demanded’ acceleration.

(c) The error is then fed to the fin servos, which actuate the rudders to move the missile in the desired direction.

(d) This closed loop system does not have an amplifier to amplify the error. This is because of the small static margin in the missiles and even a small error (unamplified) provides large airframe movement.

Objective

3. Thus the primary objective in the design of a missile autopilot is to force the missile to track a desired acceleration command generated by an outer loop, namely the guidance loop. The autopilot loop is subject to uncertainties in aerodynamic parameters, cross coupling effects, nonlinearities and measurement inaccuracies. Stability has to be maintained throughout the flight envelope while meeting the required performance which is a challenging task. In case of a tail controlled missile, the job becomes further difficult due to the non minimum phase characteristics of the lateral acceleration dynamics.

Classical Control Approach

4. In classical control approach, the missile dynamics is linearized around certain finite number of operating points in the flight envelope and then controllers are designed for each fixed operating point so as to deliver the desired performance characteristics. Gain scheduling, is then employed to deliver the performance in the complete flight envelope of the missile. Such linear control techniques have dominated missile autopilot design over the past several decades [1]–[3].

5. Lateral Autopilot Using One Accelerometer and One Rate Gyro. The design used in [1] is further elaborated below. An arrangement whereby an accelerometer provides the main feedback and a rate gyro is used to act as a damper is common in many high performance command and homing missiles. The diagram below shows the arrangement in a simplified form for a missile with rear controls.

6. Assumptions. The simplifications are as follows: -

(a) The dynamic lags of the rate gyro and accelerometer have been omitted as their bandwidth is usually more than 80 Hz and hence the phase lags they introduce in the frequencies of interest are negligible.

(b) It is assumed that the fin servos are adequately described by a quadratic lag.

(c) The small numerator terms in the transfer function fy/ζ have been omitted. For clarity this transfer function has been expressed as a steady state gain kae and a quadratic lag (i.e., the weathercock frequency ωnae and a damping ratio).

(d) Also, a stable missile with rear controls has a negative steady state gain.

(e) Similarly, if we assume that the gain of the feedback instruments are positive and that their outputs are subtracted from the input demand then a negative feedback situation will be achieved only if the servo gain is shown as negative i.e., a positive voltage input produces a negative rudder deflection.

7. Analysis. The autopilot shown in the diagram is a Type 0 closed loop system.

(a) The mean open loop steady state gain must be 10 or more to make the closed loop gain relatively insensitive to variations in aerodynamic gain; this open loop gain is ks*kae*(ka+kg/U).

(b) Gain and feedback will reduce the steady state gain and raise the bandwidth of the system. Assuming that the open loop gain cross over frequency approximates to the fundamental closed natural frequency, let us see the requirement of servo loop bandwidth when we are aiming for a minimum autopilot bandwidth of say 40 rad/s.

(i) Since the open loop gain cross over frequency will be at least 2 or 3 times the open loop weathercock frequency we can regard the lightly damped airframe as producing very nearly 180 deg phase lag at gain crossover.

(ii) A glance at the instrument feedback shows that the rate gyro produces some monitoring feedback equal to kg/U and some first derivative of output equal to kgTi/U. It is this first derivative component which is so useful in promoting closed loop stability.

(iii) If now the accelerometer is placed at a distance c ahead of the c.g., the total acceleration it sees is equal to the acceleration of the c.g.(fy) plus the angular acceleration (r dot) times this distance c. This total is fy(1+cs/U+cTis2/U). Thus if c is positive, we have from the two instruments some monitoring feedback plus some first and second derivative of the feedback, all negative feedback.

(iv) Thus it appears that we may be able to achieve 70 deg or more phase advance in the feedback path with this arrangement.

(v) If this is so, we can allow the servo to produce say 20-25 deg phase lag at gain cross over frequency in order to achieve 50 deg open loop phase margin.

(vi) This means that the servo bandwidth must be 3 or 4 times greater than the desired autopilot bandwidth, say a minimum of 150 rad/s for an autopilot bandwidth of 40 rad/s.

MATLAB SIMULATIONS FOR GARNELL’S PITCH AUTOPILOT

Simulations were carried out for the above block diagram model using the numerical values as given in Garnell’s book on “Guided Missile Control Systems”. Three different velocities of the missile namely U/√2, U and U√2, were considered where U=500. The denominator coefficients of the closed loop system are chosen as per the STANDARD ITAE model.

%----Lateral Autopilot Design----%----unmodelled dynamics----%----Aerodynamic Tf is -kae/(s^2/wnae^2+2*muae*s/wnae+1)-----kae=10;wnae=180;%----fy/fyd=(-ypsi*s^2+ypsi*nr*s+U(npsi*yv-nv*ypsi))/(a4*s^4+a3*s^3+a2*s^2+%a1*s+a0)----U=500;ypsi=+180;yv=-3.0;npsi=-500;nv=+1.0;c=0.5;nr=-3.0;%----FOR U=500--a0=6.39*10^5;a1=1.636*10^4;a2=2.06*10^2;a3=0.82;a4=4.41*10^-3;num1=[-ypsi ypsi*nr U*(npsi*yv-nv*ypsi)];den1=[a4 a3 a2 a1 a0];sys=tf(num1,den1);% Transfer function:% -180 s^2 - 540 s + 660000% -------------------------------------------------------% 0.00441 s^4 + 0.82 s^3 + 206 s^2 + 1.636e004 s + 639000subplot(2,2,1)step(num1,den1)%----FOR U=500/sqrt(2)--a10=1.821*10^5;a11=8.291*10^4;a12=1.746*10^2;a13=0.806;a14=4.41*10^-3;num2=[-ypsi ypsi*nr U*(npsi*yv-nv*ypsi)/sqrt(2)];den2=[a14 a13 a12 a11 a10];sys2=tf(num2,den2);subplot(2,2,2)step(num2,den2)%----FOR U=500*sqrt(2)--a20=2.37*10^6;a21=3.204*10^4;a22=2.675*10^2;a23=0.839;a24=4.41*10^-3;num3=[-ypsi ypsi*nr U*(npsi*yv-nv*ypsi)*sqrt(2)];den3=[a24 a23 a22 a21 a20];sys3=tf(num3,den3);subplot(2,2,3)step(num3,den3)

RESULTS OF MATLAB SIMULATIONS

(a) With U=500 as in the first diagram, the acceleration output settles to the commanded acceleration value with the required transient and steady state error values. The system is found to be non-minimum phase in its characteristics since the transfer function is non-minimum phase.

(b) In case of variation of velocity to U/√2 caused the system to go unstable as shown in the second diagram.

(c) In case of variation of velocity to U√2 , the system became underdamped with oscillations as shown in the third diagram.

PITCH AUTOPILOT DYNAMICS – MATHEMATICAL LINEARISED MODEL

The linearised model for the pitch plane of a roll stabilised missile is given by the following equations:-

α̇ (t )=zα α (t )+zqq ( t )+zδ δ (t )+q (t)

q̇ (t )=mα α ( t )+mqq ( t )+mδ δ(t )

δ̇ (t )=−1τδ (t )+ 1

τδ c (t)

az ( t )=U [ zαα (t )+zqq (t )+zδ δ ( t )]