## Simulink Control Design |

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')