Airframe Trim and Linearize

This example uses:

This example shows how to trim and linearize an airframe using Simulink® Control Design™.

Designing an autopilot using classical design techniques requires linear models of the airframe pitch dynamics for a number of trimmed flight conditions. MATLAB® can determine the trim conditions and derive linear state-space models directly from the nonlinear Simulink® and Aerospace Blockset™ model. This saves time and helps to validate the model. The functions provided by Simulink Control Design allow you to visualize the behavior of the airframe in terms of open-loop frequency (or time) responses.

Initialize Guidance Model

The first problem is to find the elevator deflection, and the resulting trimmed body rate (q), which will generate a given incidence value when the missile is traveling at a set speed. Once the trim condition is found, a linear model can be derived for the dynamics of the perturbations in the states around the trim condition.

open_system('AirframeTrim');

Define State Values

hInitial     = 10000/m2ft;      % Trim Height [m]
MInitial     = 3;               % Trim Mach Number
alphaInitial = -10*d2r;         % Trim Incidence [rad]
thetaInitial = 0*d2r;           % Trim Flightpath Angle [rad]
vInitial     = MInitial*(340+(295-340)*hInitial/11000); 	% Total Velocity [m/s]
qInitial     = 0;               % Initial pitch Body Rate [rad/sec]

Set Operating Point and State Specifications

The first state specifications are Position states, the second state specification is Theta. Both are known, but not at steady state. The third state specifications are body axis angular rates of which the variable w is at steady state.

opspec = operspec('AirframeTrim');
opspec.States(1).Known = [1;1];
opspec.States(1).SteadyState = [0;0];
opspec.States(2).Known = 1;
opspec.States(2).SteadyState = 0;
opspec.States(3).Known = [1 1];
opspec.States(3).SteadyState = [0 1];

Search for Operating Point, Set I/O, Then Linearize

op = findop('AirframeTrim',opspec);

io(1) = linio('AirframeTrim/Fin Deflection',1,'input');
io(2) = linio('AirframeTrim/Selector',1,'output');
io(3) = linio(sprintf(['AirframeTrim/Aerodynamics &\n', ...
                    'Equations of Motion']),3,'output');

sys = linearize('AirframeTrim',op,io);

 Operating point search report:
---------------------------------

opreport = 


 Operating point search report for the Model AirframeTrim.
 (Time-Varying Components Evaluated at time t=0)

Operating point specifications were successfully met.
States: 
----------
   Min         x         Max       dxMin        dx       dxMax   
__________ __________ __________ __________ __________ __________
                                                                 
(1.) AirframeTrim/Aerodynamics & Equations of Motion/3DOF (Body Axes)/Position
    0          0          0         -Inf     967.6649     Inf    
-3047.9999 -3047.9999 -3047.9999    -Inf    -170.6254     Inf    
(2.) AirframeTrim/Aerodynamics & Equations of Motion/3DOF (Body Axes)/Theta
    0          0          0         -Inf     -0.21604     Inf    
(3.) AirframeTrim/Aerodynamics & Equations of Motion/3DOF (Body Axes)/U,w
 967.6649   967.6649   967.6649     -Inf     -14.0977     Inf    
-170.6254  -170.6254  -170.6254      0      -7.439e-08     0     
(4.) AirframeTrim/Aerodynamics & Equations of Motion/3DOF (Body Axes)/q
   -Inf     -0.21604     Inf         0      3.3582e-08     0     

Inputs: 
----------
  Min      u      Max  
_______ _______ _______
                       
(1.) AirframeTrim/Fin Deflection
 -Inf   0.13615   Inf  

Outputs: 
----------
   Min         y         Max    
__________ __________ __________
                                
(1.) AirframeTrim/q
   -Inf     -0.21604     Inf    
(2.) AirframeTrim/Az
   -Inf    -7.439e-08    Inf

Select Trimmed States, Create LTI Object, and Plot Bode Response

airframe = ss(sys.A(3:4,3:4),sys.B(3:4,:),sys.C(:,3:4),sys.D);

set(airframe,'inputname',{'Elevator'}, ...
             'outputname',[{'az'} {'q'}]);

ltiview('bode',airframe);

bdclose('AirframeTrim');