LFRT version 2.0: modelling a missile



This page considers a classical missile model. The continuum of linearized models is derived in symbolic form and converted to LFR-form. The same script can be used for modelling directly in LFR-form (only two lines must be modified).

Matlab scripts of this page, symbolic approach or LFR approach.


The nonlinear model

Using standard notations (α angle of attack, q pitch rate, Ma Mach number, δp tail plane deflection), the nonlinear longitudinal equation is


in which


The input is δp. There are two measurements, the load factor (η) and the pitch rate (q)

The numerical values are defined here. Note that only α (denoted Al), q, Ma, δp (denoted dp) do not have a numerical value (other numerical values are initialized by invoking the script  missiledata, this file is in the directory demo).

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LFT representation of the continuum of linearized models

In the first part of this script we use the symbolic toolbox in order to compute linearized models. The LFRT toolbox can also be used in a similar way (the functions diff and eval exist also for LFR-objects). But using the symbolic toolbox permits us to take advantage of the structured tree decomposition for realization (symtreed). Using symtreed and then minlfr as proposed here, is often the most efficient approach to low order modelling.

% Load numerical data
missiledata

% Define symbolic objects
syms Al q Ma dp   % Can be replaced by  lfrs Al q Ma dp

% Build differential equations
Cz = z3*Al^3 + z2*Al^2 + z1*( 2 -(1/3)*Ma)*Al + z0*dp;
Cm = m3*Al^3 + m2*Al^2 + m1*(-7 +(8/3)*Ma)*Al + m0*dp;

A1 = q+K1*Ma*Cz*(1-Al^2/2);%+Al^4/24);
A2 = K2*Ma^2*Cm;
C1 = K3*Ma^2*Cz;

F = [A1;A2;C1]

% Differentiate for obtaining linearized models
ABCD = [diff(F,'Al') diff(F,'q') diff(F,'dp')];

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The equilibrium surface

The equilibrium surface is given by

This form of dp is substituted into the state-space matrix by invoking the function eval.
% PLug equilibrium surface into ABCD
dp = -(m3*Al^3 + m2*Al^2 + m1*(-7 +(8/3)*Ma)*Al)/m0;
ABCD = eval(ABCD);
ABCD doesn't depend any more on dp.

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Realization

An LFR-realization of the system matrix ABCD is performed using the function symtreed (tree decomposition). Skip this step if "syms Al q Ma dp" was replaced by "lfrs Al q Ma dp" (first line of the script).

ABCD = symtreed(ABCD);

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Input/output form, order reduction and normalization

The corresponding input/output form is computed using abcd2lfr. The result is further reduced using minlfr. Finally, the parameter variations are normalized using the function normalizelfr (the angle of attack is assumed to vary in [0 0.24], and the Mach number in [2 4]).

% Input/output form
sys = abcd2lfr(ABCD,2);

% Order reduction after realization
sys = minlfr(sys,1000*eps);

% Normalisation 
sys = normalizelfr(sys,{'Al','Ma'},[0  2],[0.24  4]);

With the symbolic approach (and symtreed) the order of the final system is 9, using the LFR approach, the order is 11.

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