Simulink Control Design 3.0

Trimming and Linearizing an Airframe

Contents

In this example, we need to find the elevator deflection and the resulting trimmed body rate (q) that will generate a given incidence value when the missile 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)

The model can be opened using the scdairframe command

scdairframe

Generating Operating Points

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+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

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+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:    -1.44e-011 (0)
(4.) scdairframe/EOM/ Equations of Motion (Body Axes)/q
      x:      -0.00972      dx:     6.31e-016 (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,'in');
io(2) = linio('scdairframe/EOM',3,'out');
io(3) = linio('scdairframe/Selector',1,'out');

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

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

bdclose('scdairframe')