F-14 Longitudinal Model

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F-14 Longitudinal Model

NOTE: You will need to have an installed and functioning version of MATLAB® and Simulink® to run this example.

Section Layout

Import the System

Simulate the System

Import the Subsystem

Initialization

$with (BlockImporter)$

$[BuildDE, Import, PrintSummary, SimplifyModel]$

(1.1)

Import the System

We import the model with the following command. We need to specify the name of the model to import, as well as a MATLAB® script that initializes the variable names.

$datadir ≔ BlockImporter :- DataDirectory () &colon;$

$model1 ≔ Import (m_f14, path = datadir, init = f14_init, inplace) &colon;$

Using the Summary command, we can view the model that we have imported.

$PrintSummary (model1)$

$[K_{0, Ka} = 0.677000000000000046, K_{0, Kf} = - 1.74600000000000000, K_{0, Ki} = - 3.86399999999999988, K_{0, Kq} = 0.815599999999999992, K_{0, Md} = - 6.88469999999999960, K_{0, Mq} = - 0.657100000000000017, K_{0, Mw} = - 0.00591999999999999992, K_{0, Swg} = 3., K_{0, Ta} = 0.0500000000000000028, K_{0, Tal} = 0.395899999999999974, K_{0, Ts} = 0.100000000000000006, K_{0, Uo} = 689.399999999999977, K_{0, Vto} = 690.399999999999977, K_{0, W1} = 2.97100000000000009, K_{0, W2} = 4.14400000000000013, K_{0, Zd} = - 63.9979000000000013, K_{0, Zw} = - 0.638499999999999956, K_{0, a} = 2.53480000000000016, K_{0, b} = 64.1299999999999955]$

(2.1)

We can now simplify the model to reduce the number of equations.

$smodel1 ≔ SimplifyModel (model1) &colon;$

$PrintSummary (smodel1)$

(2.2)

Simulate the System

First we take the system and build a set of differential equations, and assign the result to the variable sys.

$sys1 ≔ BuildDE (smodel1)$

$record (equations, initialeqs, known, outputvars, parameters, sourceeqs)$

(3.1)

The following table illustrates the different components of the variable sys1.

Differential Equations

$sys1 &colon; - equations$

$[\frac{&DifferentialD;}{&DifferentialD; t} x_{3, 1} (t) = \frac{K_{0, Kf} (\frac{&DifferentialD;}{&DifferentialD; t} x_{33, 1} (t)) K_{0, Ta} + x_{33, 1} (t) K_{0, Ki} K_{0, Ta} - x_{3, 1} (t)}{K_{0, Ta}}, \frac{&DifferentialD;}{&DifferentialD; t} x_{15, 1} (t) = \frac{- \sqrt{3} K_{0, Swg} K_{0, Ta} K_{0, Zw} K_{0, a} x_{26, 2} (t) + K_{0, Uo} x_{16, 1} (t) \sqrt{K_{0, a}^{3}} K_{0, Ta} + K_{0, Zw} x_{15, 1} (t) \sqrt{K_{0, a}^{3}} K_{0, Ta} - K_{0, Swg} K_{0, Ta} K_{0, Zw} x_{26, 1} (t) + K_{0, Zd} x_{3, 1} (t) \sqrt{K_{0, a}^{3}}}{\sqrt{K_{0, a}^{3}} K_{0, Ta}}, \frac{&DifferentialD;}{&DifferentialD; t} x_{16, 1} (t) = \frac{1}{16} \frac{16 K_{0, Mq} x_{16, 1} (t) K_{0, b}^{2} \sqrt{K_{0, a}^{3}} K_{0, Ta} + 16 K_{0, Mw} x_{15, 1} (t) K_{0, b}^{2} \sqrt{K_{0, a}^{3}} K_{0, Ta} - 4 \sqrt{3} π K_{0, Mq} K_{0, Swg} K_{0, Ta} K_{0, a} K_{0, b} x_{26, 2} (t) + \sqrt{K_{0, a}^{3}} π^{2} K_{0, Mq} K_{0, Ta} K_{0, Vto} x_{25, 1} (t) - 4 π K_{0, Mq} K_{0, Swg} K_{0, Ta} K_{0, b} x_{26, 1} (t) - 16 \sqrt{3} K_{0, Mw} K_{0, Swg} K_{0, Ta} K_{0, a} K_{0, b}^{2} x_{26, 2} (t) - 16 K_{0, Mw} K_{0, Swg} K_{0, Ta} K_{0, b}^{2} x_{26, 1} (t) + 16 K_{0, Md} x_{3, 1} (t) K_{0, b}^{2} \sqrt{K_{0, a}^{3}}}{K_{0, b}^{2} \sqrt{K_{0, a}^{3}} K_{0, Ta}}, \frac{&DifferentialD;}{&DifferentialD; t} x_{25, 1} (t) = - \frac{1}{4} \frac{- 4 x_{26, 2} (t) \sqrt{3} K_{0, Swg} K_{0, a} K_{0, b} + π K_{0, Vto} x_{25, 1} (t) \sqrt{K_{0, a}^{3}} - 4 x_{26, 1} (t) K_{0, Swg} K_{0, b}}{\sqrt{K_{0, a}^{3}} K_{0, b}}, \frac{&DifferentialD;}{&DifferentialD; t} x_{26, 1} (t) = x_{26, 2} (t), \frac{&DifferentialD;}{&DifferentialD; t} x_{26, 2} (t) = \frac{u_{1, 1, 1} (t) K_{0, a}^{2} - 2 x_{26, 2} (t) K_{0, a} - x_{26, 1} (t)}{K_{0, a}^{2}}, \frac{&DifferentialD;}{&DifferentialD; t} x_{32, 1} (t) = - K_{0, W2} x_{32, 1} (t) + x_{16, 1} (t), {Sink}_{Scope, 34, 1, 1} (t) = \frac{x_{15, 1} (t)}{K_{0, Uo}}, {Sink}_{Scope, 34, 2, 1} (t) = x_{16, 1} (t), \frac{&DifferentialD;}{&DifferentialD; t} x_{35, 1} (t) = \frac{x_{15, 1} (t) K_{0, Tal} - x_{35, 1} (t) K_{0, Uo}}{K_{0, Uo} K_{0, Tal}}, \frac{&DifferentialD;}{&DifferentialD; t} x_{37, 1} (t) = \frac{{Source}_{SigGen, 36, 1} (t) K_{0, Ts} - x_{37, 1} (t)}{K_{0, Ts}}, \frac{&DifferentialD;}{&DifferentialD; t} x_{33, 1} (t) = - \frac{K_{0, Kq} K_{0, Tal} K_{0, Ts} K_{0, W1} x_{32, 1} (t) - K_{0, Kq} K_{0, Tal} K_{0, Ts} K_{0, W2} x_{32, 1} (t) + K_{0, Kq} K_{0, Tal} K_{0, Ts} x_{16, 1} (t) + K_{0, Ka} x_{35, 1} (t) K_{0, Ts} - x_{37, 1} (t) K_{0, Tal}}{K_{0, Ts} K_{0, Tal}}]$

(3.2)

Parameter values

$sys1 :- parameters$

(3.3)

Initial conditions

$sys1 :- initialeqs$

$[x_{3, 1} (0) = 0, x_{15, 1} (0) = 0, x_{16, 1} (0) = 0, x_{25, 1} (0) = 0, x_{26, 1} (0) = 0, x_{26, 2} (0) = 0, x_{32, 1} (0) = 0, x_{33, 1} (0) = 0, x_{35, 1} (0) = 0, x_{37, 1} (0) = 0]$

(3.4)

Equations for the sources (inputs)

$sys1 :- sourceeqs$

$[u_{1, 1, 1} (t) = 0, {Source}_{SigGen, 36, 1} (t) = {\begin{array}{c} - 1 & t - 4.000000000 π trunc (\frac{0.2500000000 t}{π}) < 2.000000000 π \\ 1 & otherwise \end{array}]$

(3.5)

List of sinks (outputs)

$sys1 :- outputvars$

$[{Sink}_{Scope, 34, 1, 1} (t), {Sink}_{Scope, 34, 2, 1} (t)]$

(3.6)

Using the information in the variable sys, we construct a simulation procedure.

$sol1 ≔ dsolve ([eval (eval (sys1 :- equations, sys1 :- parameters), sys1 :- sourceeqs) [], sys1 :- initialeqs []], numeric)$

$proc (x_rkf45_dae) ... end proc$

(3.7)

Then we plot the simulation results.

$plots [odeplot] (sol1, [t, sys1 :- outputvars [1]], 0 .. 20, numpoints = 200, title = Angle-of-attack)$

$plots [odeplot] (sol1, [t, sys1 :- outputvars [2]], 0 .. 20, numpoints = 200, title = Rate of pitch)$

Import the Subsystem

First we want to close the Simulink model without saving it, so we can reload it with the subsystem. We send the command close_system('m_f14', 0) through the link to MATLAB®.

$Matlab :- evalM (close_system ('m_f14',0))$

We can import a subsystem from the model.

$model2 ≔ Import (m_f14, m_f14/Aircraft dynamics, path = datadir, init = f14_init, inplace = true) &colon;$

The set of equations are simplified and the simplified set of equations is printed using the PrintSummary command.

$smodel2 ≔ SimplifyModel (model2) &colon;$

$PrintSummary (smodel2) &colon;$

$[K_{0, Md} = - 6.88469999999999960, K_{0, Mq} = - 0.657100000000000017, K_{0, Mw} = - 0.00591999999999999992, K_{0, Uo} = 689.399999999999977, K_{0, Zd} = - 63.9979000000000013, K_{0, Zw} = - 0.638499999999999956]$

(4.1)

The differential equations are created.

$sys2 ≔ BuildDE (smodel2)$

$record (equations, initialeqs, known, outputvars, parameters, sourceeqs)$

(4.2)

The following table illustrates the different components of the variable sys2.

Differential Equations

$sys2 :- equations$

$[y_{1, 1, 1} (t) = \frac{x_{12, 1} (t)}{K_{0, Uo}}, \frac{&DifferentialD;}{&DifferentialD; t} x_{12, 1} (t) = K_{0, Uo} y_{1, 2, 1} (t) + K_{0, Zd} u_{1, 1, 1} (t) + K_{0, Zw} x_{12, 1} (t) - u_{1, 2, 1} (t), \frac{&DifferentialD;}{&DifferentialD; t} x_{13, 1} (t) = K_{0, Md} u_{1, 1, 1} (t) + K_{0, Mq} x_{13, 1} (t) + K_{0, Mw} x_{12, 1} (t) - u_{1, 3, 1} (t), y_{1, 2, 1} (t) = x_{13, 1} (t)]$

(4.3)

Parameter values

$sys2 :- parameters$

(4.4)

Initial conditions

$sys2 :- initialeqs$

$[x_{12, 1} (0) = 0, x_{13, 1} (0) = 0]$

(4.5)

Equations for the sources (inputs)

$sys2 :- sourceeqs$

$[u_{1, 1, 1} (t) = 0, u_{1, 2, 1} (t) = 0, u_{1, 3, 1} (t) = 0]$

(4.6)

List of sinks (outputs)

$sys2 :- outputvars$

$[y_{1, 1, 1} (t), y_{1, 2, 1} (t)]$

(4.7)

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