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should try to avoid this whenever possible. An algebraic loop can be ‘broken’ by
including an additional dynamic element in the feedback loop, e.g. a filter or an
artificial time-delay. However, the effect of such measures needs to be weighted
carefully – remember the inadvertent dynamics from figure 6.9.
Sometimes, it is possible to write out the system equations in an explicit form.
For instance, the algebraic loop from figure 6.8 can be eliminated by replacing this
diagram with a single block representing equation (6.55). Another example was provided
in section 3.4, where an algebraic loop in the equation for the aircraft’s sideslip
angle was averted by re-writing the differential equation in an explicit form.
96 Chapter 6. Analytical tools
U(s) s V n-1
1/s
s V n-2 s V V(s) Y(s)
++
++
++
bn-1
bn-2
b1
b0
++
++
++
+-
an-1
an-2
a1
a0
bn
++
s V n
1/s 1/s
Figure 6.11: Block-diagram equivalent of a transfer function
6.4.2 Obtaining state-models from transfer functions
Linear systems are often represented in terms of transfer functions, which can be implemented
in SIMULINK block-diagrams by means of Transfer Fcn blocks. However,
since the numerical integrators cannot be applied directly to such transfer functions,
a conversion to state-space format is necessary. Obviously, SIMULINK will handle
this conversion for Transfer Fcn blocks, but we will discuss this topic in a little more
detail nevertheless, as this knowledge will prove to be useful for the implementation
of transfer functions with non-constant coefficients.
There exist a variety of techniques to convert transfer functions into linear state
models. Refs.[8, 33] and [36] contain some examples, and ref.[10] includes a more
fundamental discussion of this topic. Regardless of the method, the resulting state
models will not be unique, as they depend on the choice of state variables. Here we
will focus on what is called the ‘controllable canonical form realisation’ [10], which
will generally be adequate for our needs.
For a broad class of systems the dynamic behaviour of variations about operating
point values can be approximated by an nth-order linear differential equation:
dny
dtn + an−1
dn−1y
dtn−1 + . . . + a2
d2y
dt2 + a1
dy
dt
+ a0y =
= bm
dmu
dtm + bm−1
dm−1u
dtm−1 + . . . + b2
d2u
dt2 + b1
du
dt
+ b0u (6.61)
where u(t) and y(t) represent the variations of input and output signals, ai and bi are
6.4. Miscellaneous simulation issues 97
real constants, and m and n are scalars, representing the highest-order derivatives
of the input and output signals, respectively (m  n). The corresponding transfer
function is:
H(s) 
Y(s)
U(s)
= bmsm + bm−1sm−1 + . . . + b2s2 + b1s + b0
sn + an−1sn−1 + . . . + a2s2 + a1s + a0
(6.62)
If there are no input derivatives present in the differential equation (i.e. if m = 0), the
task of finding a state representation is trivial: state variables can simply be assigned
to the output y and its derivatives up to order n − 1, and n state equations can be
written down immediately.
To derive the state equations in the general case where input derivatives are present,
we will first re-write this transfer function, introducing a help function V(s):
V(s) 
Y(s)
bmsm + . . . + b1s + b0
= U(s)
sn + . . . + a1s + a0
(6.63)
This yields the following equations:
Y(s) = (bmsm + . . . + b1s + b0) V(s) (6.64)
snV(s) = U(s) − an−1sn−1V(s) − . . . − a1sV(s) − a0V(s) (6.65)
These equations can be represented in a ‘simulation diagram’, as illustrated in figure
6.11. This diagram is constructed from a series of cascaded first-order integrals
(the 1s
blocks) and simple gain blocks; V(s) is the output of the last integrator and
the number of integrators is equal to the degree of the denominator polynomial of
transfer function (6.62).
Equation (6.65) relates the input of the first integrator to the transfer function input
U(s) and the feedback signals from later integrators. This equation corresponds
to the lower half of figure 6.11. Equation (6.64) establishes the resulting output signal
Y(s); the output connections have been drawn in the upper half of the figure. Notice
that the figure assumes the highest order of the input and output derivatives to be
equal, i.e. m = n. For m < n, some of the bi coefficients are zero.
　
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