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include several built-in software tools to fulfil these functions. Although it is
not necessary (and often not possible) to know the exact algorithms used by these
built-in tools, it is still useful to have at least a basic understanding of the underlying
theory. For this purpose, this chapter provides an introduction to the theory of simulation
(in particular: numerical integration), system linearisation, and some elements
from linear system analysis. It also introduces a specialized trimming algorithm, tailored
to the nonlinear aircraft model.
6.1 Numerical integration methods
The aircraft equations of motion from chapter 2, which were further elaborated in
chapter 3, express the aircraft dynamics in terms of a set of twelve nonlinear ordinary
differential equations (ODEs). To ‘simulate’ aircraft responses to control inputs or
external disturbances, we therefore need to solve an initial-value problem.
The FDC toolbox delegates this task to the built-in SIMULINK integrators, which
have been documented in the MATLAB and SIMULINK user-manuals, refs. [3] and [4],
and in ref.[32]. These documents can all be downloaded from www.mathworks.com. A
summary of the SIMULINK solvers will be given in chapter 11. Here, we will provide
general background information about numerical integration of ODEs.
74 Chapter 6. Analytical tools
Initialisation
data
Coefficient
data
Steadystate
trim
Nonlinear
simulation
Linearise
Control
system
simulation
Linear
design
techniques
Nonlinear
state-space model
x ( t ) = f ( x ( t ) , u ( t ) , t )
.
y ( t ) = g ( x ( t ) , u ( t ) , t )
x ( t ) = A x ( t ) + B u ( t )
.
y ( t ) = C x ( t ) + D u( t )
Linear statespace
model
Initialisation data
Trial
design
Figure 6.1: State-space flight models and the associated analytical tools [33]
6.1.1 The type of problems considered
The numerical integration methods considered here are used to determine time-trajectories
of state variables of continuous dynamical systems, which are characterized
by a set of coupled ODEs:
x˙ (t) = f(x(t), u(t), v(t), t); x(t0) = x0 (6.1)
where x is the state vector, u is the input vector, v is a vector of external disturbances,
f is some nonlinear function, and x0 is the initial value of the state vector at time
t0. This equation represents a so-called initial value problem. Since few differential
equations can be solved exactly, it is usually necessary to approximate the solutions
of these ODEs numerically.
If the state vector x has N elements, N constants of integration appear in the
solution of equation (6.1). A unique solution to this system can be obtained only
if the initial values of all states are specified. The techniques for solving the vector
equation (6.1) are essentially the same as the techniques used for solving scalar initial
value problems given by the equation:
x˙(t) = f (x(t), t) (6.2)
in which we have omitted the input signal u(t) and the disturbances v(t) for sake of
brevity; including the latter signals does not change the integration methods. Numer6.1.
Numerical integration methods 75
x2
x
0
t0 t1 t2
Initial value
Euler solution
x
t
x1
t3
x3
Figure 6.2: Family of solutions of an unstable ODE. The Euler approximation from equation
(6.3) has been displayed graphically.
ical integrators generate a sequence of discrete points t0, t1, t2, . . . in time, possibly
with variable spacing hn = tn+1 − tn (the step-size). At each point tn the solution
x(tn) is approximated by xn, which is computed from earlier values of x. If k earlier
values xn, xn−1, . . . , xn−k+1 are used for the computation of xn+1, the method is called
a ‘k-step method’. For instance, the Euler method is a single-step method:
xn+1 = xn + hn · f (xn, tn) (6.3)
This method is used by the SIMULINK integrator ode1.
6.1.2 Stability, errors, and order of a numerical integration method
Figure 6.2 shows a typical family of solutions of a first-order differential equation for
different initial values x0. If we select a wrong value of the initial condition, the deviation
from the desired solution will increase in time, i.e. the differential equation is
unstable. The figure demonstrates how the Euler approximation from equation (6.3)
will cross from one solution to another between subsequent time steps, due to the
numerical inaccuracy. For this unstable differential equation, the resulting error is
shown to increase in time.
Figure 6.3 shows a family of solutions for a stable differential equation. In this
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FDC 1.4 – A SIMULINK Toolbox for Flight Dynamics and Contro(39)
