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Trimming and Linearizing an Airframe

This example shows how to trim and linearize an airframe. We first need to find the elevator deflection and the resulting trimmed body rate (q) that will generate a given incidence value when the airframe is traveling at a set speed. Once we find the trim condition, we can derive a linear model 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)

Generating Operating Points

Open the model.

scdairframe

To get the operating point specification object, you use the operspec command:

opspec = operspec('scdairframe')
 Operating Specification 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+03
(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

First, we set the Position state specifications, 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 includes 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];

Next, we search for the operating point that meets this specification.

op = findop('scdairframe',opspec);
 Operating Point Search Report:
---------------------------------

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

Operating point specifications were successfully met.
States: 
----------
(1.) scdairframe/EOM/ Equations of Motion (Body Axes)/Position
      x:             0      dx:           984
      x:     -3.05e+03      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:     -1.44e-11 (0)
(4.) scdairframe/EOM/ Equations of Motion (Body Axes)/q
      x:      -0.00972      dx:      6.31e-16 (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]

Linearizing the Model

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

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

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

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

bdclose('scdairframe')

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