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Equation (6.33) represents the output from the state equation (6.22) in the operating
point with the ith element of the input vector perturbed by the amount Dui.
Next, we can linearize the nonlinear output equation:
y(t) = g( x(t), u(t) ) (6.34)
Again, we start with the Taylor-series expansion of this equation about the operating
point (x0, u0):
y(t)
¶g
¶x
(x − x0) + ¶g
¶u
(u − u0) + y0 (6.35)
if we neglect the higher-order terms; y0 = g( x0, u0 ) and both partial derivative
matrices are determined for the operating point (x0, u0). This equation can be developed
into the following small-perturbation equation for the output vector y:
y0 y − y0 = Cx0 + Du0 (6.36)
with:
C = ¶g
¶x
(x0,u0)
D = ¶g
¶u
(x0,u0)
C and D can be approximated in a similar fashion as the matrices A and B from
the state equation. We can thus approximate the complete set of system matrices
(A, B, C, D) of the small perturbations model, without computing any stability and
control derivatives. With n states and m observation variables, the dimensions of
these matrices become:
A : (n × n)
B : (n × m)
C : (m × n)
D : (m × m)
For the Beaver model from the FDC toolbox, n equals 12 and m equals 89. The latter
number includes the twelve state variables themselves, see chapter 3. This numerical
linearisation method forms the basis of the SIMULINK linearisation function LINMOD,
which is called by the FDC linearisation program ACLIN.
6.3. Steady-state trimmed flight 87
6.3 Steady-state trimmed flight
The FDC toolbox contains a specialized utility ACTRIM, which determines steadystate
trimmed flight conditions that can be used as ‘operating points’ for system linearisation,
or as initial conditions for simulations. Although SIMULINK itself already
includes a generic trim function TRIM, it was decided to implement a specialized
tailor-made trim routine instead, using an algorithm from ref.[33] that has proved
to be very effective in practice. Based on that reference, this section will discuss the
underlying theory.
6.3.1 Definition of steady-state flight
In section 2.5, it was shown that the general rigid-body dynamics can be written as
a nonlinear vector equation:
˙x(t) = f(x(t), u(t), v(t), t) (6.37)
with state vector x, input vector u, and external disturbance vector v. In fact, for
the Beaver aircraft, the ˙b equation turned out to be implicit, as the aerodynamic
side-force itself was directly dependent on ˙b (similar dependencies are common in
aerodynamic models). In section 3.4 we solved this problem by collecting the linear
˙b
terms on one side of the equation, but here we will take a more general approach
by considering the original implicit differential equation:
f(x˙ (t), x(t), u(t), v(t), t) = 0 (6.38)
Notice that this equation is time variant, as it reflects the most general case, where
we have to cater for time-dependencies such as gradual weight reduction due to
fuel burn. However, in the relatively short time intervals considered in our simulations,
the system can be treated as time invariant: the states, inputs, and disturbances
change with time, but they are not directly dependent on time itself.
We can now introduce the concept of a singular point or equilibrium point of a timeinvariant
system with no external control inputs. The coordinates of the singular
point(s) of the implicit nonlinear state equations are given by the solution vector
x = xeq, which satisfies:
f(x˙ , x, u) = 0 , with: x˙ = 0 and: u = 0 or constant (6.39)
Here we have omitted the disturbance vector v. The system is ‘at rest’ when all of
the time-derivatives are identically zero, and we can examine the behaviour of the
system near the equilibrium point by slightly perturbing some of the variables, as
we have outlined in the previous section.
Steady-state trimmed flight can be defined as a condition in which all of the motion
variables are constant or zero, and all acceleration components are zero. This
definition is very restrictive, unless some simplifying assumptions are made. We
will assume the aircraft’s mass to remain constant, and restrict the analysis to the
flat-Earth equations of motion, so that the definition will allow steady wings-level
flight and steady turning flight. If the change in atmospheric density with altitude is
neglected during the trim process, a wings-level climb and a climbing turn are also
permitted as steady-state flight conditions.
88 Chapter 6. Analytical tools
One method to find a steady-state flight condition is to balance the individual forces
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