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been based on these theories. See ref.[18] for more information about this subject.
82 Chapter 6. Analytical tools
x
t
x3
t t3 2 t1 t0
x0
x1
x2 f (t)
Figure 6.5: Unstable solution using the Euler method for a stiff system.
x
t
x3
tt 3 2 t1 t0
x0
x1 x2
f (t)
Figure 6.6: Stable solution using the backward Euler method for a stiff system.
6.1. Numerical integration methods 83
k : 2 3 4 5 6
b0
23
6
11
12
25
60
137
60
147
a1
43
18
11
48
25
300
137
360
147
a2 −13
− 9
11 −36
25 −300
137 −450
147
a3
2
11
16
25
200
137
400
147
a4 − 3
25 − 75
137 −225
147
a5
12
137
72
147
a6 − 10
147
Table 6.4: Coefficients for stiffly stable integrator (Gear method)
6.1.4 Stiff differential equations
‘Stiffness’ of the differential equations can roughly be defined as the presence of one
or more fast decay processes in time, with a time constant that is short compared to
the time-span of interest. For stiff problems, solutions can change on a time scale that
is very short compared to the interval of integration, while the solution of interest
changes on a much longer time scale.
The time constant is defined as the time in which a solution to a differential equation
decays by a factor 1e
. In a physical system, different elements often have different
time constants, which means that some solutions to differential equations decay
much faster than others. In such cases the signals containing fast dynamics will determine
the stability of the integration method, even when these components may
quickly decay to insignificant levels.
Methods not designed for stiff problems must use time steps small enough to resolve
the fastest possible changes, which makes them rather ineffective on intervals
where the solution changes slowly. Figure 6.5 shows what happens when we apply
the Euler method to a stiff problem, using too large a step-size: although the ODE
itself is stable, the numerical solution can be seen to diverge rapidly. The only way
to prevent this numerical instability is to reduce the step-size, but eventually roundoff
and discretisation errors will accumulate enough to result in another instability.
Notice that the transient part of the solution, which decays very fast, prevents an increase
in step-size, even though the solution is very smooth after only a few seconds.
With such small steps, computation time will soon become critical.
Figure 6.6 shows the same problem solved by the backward Euler method, which is
an implicit method:
xn+1 = xn + hn · f (xn+1, tn+1) (6.20)
This method is stable, although the accuracy of the transient part of the solution is
rather poor. The accuracy can be improved by using multistep backward differen84
Chapter 6. Analytical tools
tiation formulas, such as the following method, first implemented by C.W. Gear [18]:
xn =
kå
i=1
ai xn−i + hn b0 fn (6.21)
The coefficients ai and b0 have been listed in table 6.4. This ‘Gear method’ differs
from the Adams method in the way in which the implicit expression is solved. The
algorithm for the SIMULINK solver ode15s is related to this method.
6.2 System linearisation
The FDC toolbox includes a linearisation utility ACLIN, which can be used to extract a
linearized aircraft model from the SIMULINK implementation of the aircraft dynamics.
Such models are invaluable for the application of most control system design
methods. The tool calls the SIMULINK function LINMOD for the actual linearisation
process. Here we will briefly discuss the theoretical backgrounds.
We start with the nonlinear state equation:
x˙ (t) = f( x(t), u(t) ) (6.22)
(the disturbance vector v has been neglected this time; for this discussion we assume
disturbances to be included in the inputvector u). This expression can be expanded
in a Taylor-series about an operating point (x0, u0). Keeping only the first-order terms,
we find:
x˙ (t)
¶f
¶x
(x − x0) + ¶f
¶u
(u − u0) + x˙ 0 (6.23)
where x˙ 0 = f( x0, u0 ) and both partial derivative matrices are determined for the
operating point (x0, u0). Moving ˙x0 to the left-hand side of the equation yields:
x˙ − x˙ 0 = ¶f
¶x
(x − x0) + ¶f
¶u
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