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If we now assign state variables to the integrator outputs, working from right to
left and starting with the rightmost integrator, we can define the state vector for this
transfer function block as:
x(s) = V(s)
2
66666664
1
s
s2
s3
...
sn−1
3
77777775 (
6.66)
By observing the simulation diagram, we can obtain the familiar linear state and
output equations (taking notice of the fact that u and v are scalars, and x is a vector
of length n):
˙x = Ax + bu
y = cx + du (6.67)
98 Chapter 6. Analytical tools
The state matrices then become:
A =
2
66666664
0 1 0 . . . 0
0 0 1 . . . 0
0 0 0 . . . 0
...
...
...
...
0 0 0 . . . 1
−a0 −a1 −a2 . . . −an−1
3
77777775
b =
2
66666664
0
0
0...
0
1
3
77777775
(6.68)
c =
b0 b1 b2 . . . bn−1
d = bn
This form of the system matrix A is known as a companion-matrix: all elements are
zero, except for the superdiagonal 1’s and the nonzero bottom row. The simplicity of
this form makes it useful for theoretical derivations, but its numerical computation
properties are not particularly good. However, this is not a problem when dealing
with systems of low order [33].
For the FDC toolbox, the structure from figure 6.11 has been applied for the implementation
of the Dryden turbulence filters, which are defined by transfer functions
with airspeed-dependent coefficients, and for a filter in the Altitude Select mode of
the Beaver autopilot, which also uses non-constant coefficients. Obviously, the standard
Transfer Fcn block from SIMULINK is more convenient for the implementation
of transfer functions with constant coefficients.
Chapter 7
Getting started with the FDC toolbox
Having treated the underlying theory, it is now time to introduce the Flight Dynamics
and Control Toolbox software itself. This chapter explains how to install the
toolbox on your computer, and allows you to get acquainted with the software by
exploring the FDC directories and libraries. It also explains the modular and layered
architecture of the SIMULINK models and outlines the relation between models and
libraries. A detailed description of all individual FDC models and programs will
be provided later in chapters 8 to 11, while the practical use of the software will be
demonstrated in chapter 13.
7.1 Objectives of the FDC toolbox
Let’s start by summarizing the purpose of this software. The Flight Dynamics and
Control toolbox (‘FDC toolbox’) for MATLAB and SIMULINK was conceived as an
Open Source software environment for flight simulation, open-loop analysis of aircraft
dynamics, and closed-loop analysis of flight control systems. It has been constructed
around a nonlinear dynamics model of the DeHavilland DHC-2 Beaver
aircraft, which is characterized by comprehensive, yet still straightforward model
equations. The model has been implemented in a layered, modular SIMULINK block
diagram structure, which can be adapted for other airplanes with relative ease.
The toolbox includes SIMULINK models of the airplane dynamics, wind and turbulence,
radio-navigation signals, and the control laws of the Beaver autopilot, as
developed at Delft University of Technology [28]. It also provides analytical MATLAB
routines for the computation of steady-state flight conditions and linearized
flight models, and several help-utilities that simplify system handling. The individual
models, subsystems, and blocks have been systematically collected in a series of
SIMULINK libraries.
The toolbox was originally created as part of an autopilot design project. The
current version 1.4, when applied in the context of the MATLAB platform and its
multitude of systems analysis and control design tools, can be considered an advance
‘proof of concept’ of an integrated software environment for flight control system
design, which can help streamline the design cycle discussed in chapter 1. However,
its application is not limited to FCS design only – the toolbox can provide a starting
point for a wide range of stability and control related research projects and courses.
100 Chapter 7. Getting started with the FDC toolbox
7.2 System requirements
Version 1.4 of the FDC toolbox was developed forMATLAB Release 11 (MATLAB 5.3 /
SIMULINK 3.0), but it has been reported to work correctly with laterMATLAB releases
as well. Although it has been built and tested on an MS WINDOWS system, the toolbox
is intended to be system-independent. While this has not been tested extensively,
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FDC 1.4 – A SIMULINK Toolbox for Flight Dynamics and Contro(51)
