Airframe Demo (Requires Aerospace Blockset)

This is a demonstration of how to trim and linearize a non-linear Simulink airframe model. The 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 travelling at a set speed. Once the trim condition is found, a linear model can be derived for the dynamics of the states around the trim condition.

        Fixed parameters :
                            Incidence (Theta)
                            Body Attitude (U)
                            Position
        Trimmed steady state parameters :
                            Elevator deflection (w)
                            Body rate (q)

The model can be opened using the command

scdairframe

Generation of operating points

To get operating point specification object use the command

opspec = operspec('scdairframe')

 Operating Specificaton for the Model scdairframe.
 (Time-Varying Components Evaluated at time t=0)

States: 
----------
(1.) scdairframe/EOM/ Equations of Motion (Body Axes)/Position
      spec:  dx = 0,  initial guess:             0
      spec:  dx = 0,  initial guess:    -3.05e+003
(2.) scdairframe/EOM/ Equations of Motion (Body Axes)/Theta
      spec:  dx = 0,  initial guess:             0
(3.) scdairframe/EOM/ Equations of Motion (Body Axes)/U,w
      spec:  dx = 0,  initial guess:           984
      spec:  dx = 0,  initial guess:             0
(4.) scdairframe/EOM/ Equations of Motion (Body Axes)/q
      spec:  dx = 0,  initial guess:             0
 
Inputs: 
-----------
(1.) scdairframe/Fin Deflection
      initial guess: 0            
 
Outputs: 
-----------
(1.) scdairframe/q
      spec:  none
(2.) scdairframe/az
      spec:  none

The first state specifications to set are the Position states which are known but not at steady state:

opspec.States(1).Known = [1;1];
opspec.States(1).SteadyState = [0;0];

The second state specification is Theta which is known but not at steady state:

opspec.States(2).Known = 1;
opspec.States(2).SteadyState = 0;

The third state specification are the body axis angular rates where the variable w is at steady state:

opspec.States(3).Known = [1 1];
opspec.States(3).SteadyState = [0 1];

Now search for the operating point that meets this specification

op = findop('scdairframe',opspec);

 Operating Point Search Report:
---------------------------------

 Operating Point Search Report for the Model scdairframe.
 (Time-Varying Components Evaluated at time t=0)

Operating condition specifications were successully met.

States: 
----------
(1.) scdairframe/EOM/ Equations of Motion (Body Axes)/Position
      x:             0      dx:           984 
      x:    -3.05e+003      dx:             0 
(2.) scdairframe/EOM/ Equations of Motion (Body Axes)/Theta
      x:             0      dx:      -0.00972 
(3.) scdairframe/EOM/ Equations of Motion (Body Axes)/U,w
      x:           984      dx:          22.7 
      x:             0      dx:     6.29e-010 (0)
(4.) scdairframe/EOM/ Equations of Motion (Body Axes)/q
      x:      -0.00972      dx:     5.74e-017 (0)
 
Inputs: 
-----------
(1.) scdairframe/Fin Deflection
      u:       0.00142    [-Inf Inf]
 
Outputs: 
-----------
(1.) scdairframe/q
      y:      -0.00972    [-Inf Inf]
(2.) scdairframe/az
      y:        -0.242    [-Inf Inf]

Model Linearization

The operating points are now ready for linearization. First specify the input and output points using the commands:

io(1) = linio('scdairframe/Fin Deflection',1,'in');
io(2) = linio('scdairframe/EOM',3,'out');
io(3) = linio('scdairframe/Selector',1,'out');

Linearize the model and plot the Bode magniture response for each condition.

sys = linearize('scdairframe',op,io);
bodemag(sys)