MATLAB SIMULATIONS FOR GARNELL's PITCH AUTOPILOT
MATLAB SIMULATIONS FOR GARNELL's PITCH AUTOPILOT
MATLAB SIMULATIONS FOR GARNELL's PITCH AUTOPILOT
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.
(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.
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.
(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.
(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.
(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.
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 U2, were considered
where U=500. The denominator coefficients of the closed loop system are
chosen as per the STANDARD ITAE model.
%----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;
den1=[a4 a3 a2 a1 a0];
sys=tf(num1,den1);
% Transfer function:
subplot(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;
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;
sys3=tf(num3,den3);
subplot(2,2,3)
step(num3,den3)
The linearised model for the pitch plane of a roll stabilised missile is given
by the following equations:-
αt=zααt+zqqt+zδδt+q(t)
qt=mααt+mqqt+mδδ(t)
δt=-1τδt+1τδc(t)
azt=U[zααt+zqqt+zδδt]