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limited so the process will stop if the solution does not converge.
6.4 Miscellaneous simulation issues
In the previous sections, we have discussed most analytical functions from figure 6.1,
focussing on numerical integration, linearisation and trimming methods. In this section,
some other issues which may be encountered during numerical nonlinear simulations
will be highlighted. These topics are relevant for the FDC toolbox, because
they shed some light on the practical difficulties one may encounter, and provide
background information for some analytical functions. The information from section
6.4.2 has been applied in practice to create some FDC submodels.
6.4.1 Algebraic loops
Continuous systems are often expressed as block diagrams, which divide the system
into logical modules, and suggest a certain order of computations. The latter,
though convenient for digital simulations, is not in accordance with reality: in the
actual system, all variables will change simultaneously, not sequentially. A suitable
calculation sequence is needed for digital simulations, but we should realize that this
is an artificial construct – a sequential representation of what is, in effect, a parallel
system.
If feedback is applied to a system, it may not be possible to find a suitable computation
sequence for digital simulations. This situation generally occurs when blocks
having direct feedthrough of their input signals form a feedback-loop: the block outputs
cannot be computed without knowing the values of the signals entering the
blocks, while the block inputs cannot be computed without knowing the signals leaving
the blocks. This is called an algebraic loop.
A simple example of this situation can be seen in figure 6.8, which shows a system
consisting of a gain K with negative unity feedback. Mathematically, this loop
implies that the output of the summation junction is an algebraic state e, constrained
to equal the first input u minus the output y, which equals K · e:
y = K · e = K(u − y) (6.54)
The solution of this simple loop is:
y =

K
1 + K

u (6.55)
but most algebraic loops cannot be solved so easily [4].
94 Chapter 6. Analytical tools
K
u e y
+_
Figure 6.8: Gain with unity feedback: an algebraic loop
0 1 2 3 4 5
-2
-1
0
1
2
3
4
t [ s ]
y( t ) / K Input: unit step at t = 0
KKK === 011...055 Figure 6.9: Dynamics, introduced by the algebraic loop from figure 6.8
Algebraic
Equations
ALB
x g(x)
Figure 6.10: Iterative solution of an algebraic loop
6.4. Miscellaneous simulation issues 95
It is interesting to see what will happen if we try to integrate this system with a
numerical integration method, using a step-size hn, by introducing an artificial computation
sequence as follows:
en = un − yn−1
yn = Ken = K(un − yn−1) (6.56)
where yn  y(n · hn) and yn−1  y((n − 1)·hn−1) for an integer value n. Taking the
Z-transform of these equations yields:
Y(z)
U(z)
= Kz
z + K
(6.57)
which indicates that inadvertent additional dynamics have been introduced as a consequence
of trying to calculate the responses of a parallel system using sequential calculations.
Figure 6.9 visualizes unit step-responses for the Z-transfer function (6.57)
for several values of the gain K.
When SIMULINK detects an algebraic loop it will calls a loop solving routine at
each time step. The loop solver tries to determine a solution by means of Newton-
Rhapson iterations [4]. First, a new block ALB is added to the system containing the
implicit algebraic equations from the algebraic loop, see figure 6.10. This block has
an input g(x) and an output x. The block attempts to find a value of x such that
g(x) = x, which means that it tries to solve the following nonlinear equation:
G(x)  g(x) − x = 0 (6.58)
This is done by applying the Newton-Rhapson method [8]:
xi+1 = xi −
G(xi)
G0(xi)
(6.59)
where:
G0(xi) = G(xi + Dx)
Dx
(6.60)
for small values of Dx. The subscript i in these equations denotes the iteration number,
not the time-step. In some situations, this method will not converge to a solution
within a reasonable number of iterations. SIMULINK will display an error message if
it is unable to solve the algebraic loop within 200 iterations (it is possible and recommended
to specify an initial guess; see ref.[4] for more information).
Since the Newton-Rhapson iterations have to be carried out for every time-step,
the presence of one or multiple algebraic loops will slow down simulations. One
　
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