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(u − u0) (6.24)
Now define:
x0 = x − x0, a vector of length n
u0 = u − u0, a vector of length m
and:
A = ¶f
¶x

(x0,u0)
B = ¶f
¶u

(x0,u0)
Substitution into equation (6.24) yields the small-perturbations formula:
x˙ 0 = Ax0 + Bu0 (6.25)
which is the desired linear state equation, with x0 and u0 being perturbations of the
operating point values of the state and control vectors, respectively.
In practice, the operating point (x0, u0) is chosen to be a singular point or equilibrium
point. The system is ‘at rest’ when all of the state derivatives are identically
zero, and we can examine the behaviour of the system near the equilibrium point by
6.2. System linearisation 85
slightly perturbing some of the variables. For instance, if the state trajectory departs
rapidly from the singular point in response to a small perturbation, the human pilot
is unlikely to be able to control the aircraft [33]. In section 6.3 the computation of
steady-state equilibrium flight conditions will be explained.
The matrices A and B can be evolved analytically by evaluating the wind-axes
force equations, the moment equations, the kinematic equations, and the position
equations presented in section 2.5. This analysis involves a large series of partial
derivatives of the forces and moments with respect to other variables, the so-called
stability and control derivatives. Refs.[14] and [33] explain how these derivatives can
be determined; the first reference contains an comprehensive list of derivatives, including
derivatives for an large number of observation variables.
While this analytical linearisation provides useful insight in the force and moment
build-up (aerodynamic forces and moments in particular), we don’t need to
perform this complete analysis just to find the numerical values of the system matrices
A and B. Since the nonlinear aircraft model has been presented in state-space
form also, it is actually more convenient to bypass the stability derivatives and calculate
the system matrices directly. This is done by perturbing the state and control
variables from the steady-state condition, and numerically evaluating the partial derivatives.
In this report we will consider the latter method only; analytical linearisation
tools will be kept in mind for possible future FDC versions.
The matrix A can be written out as follows:
A =
2
664
¶ f1
¶x1
. . . ¶ f1
¶xn ...
...
¶ fn
¶x1
. . . ¶ fn
¶xn
3
775
(6.26)
where the partial derivatives are valid for the equilibrium point. This matrix can be
approximated by:
A 
2
664
Df1
Dx1
. . . Df1
Dxn ...
...
Dfn
Dx1
. . . Dfn
Dxn
3
775
=
2
664
f1(x0+Dx1,u0)−f1(x0,u0)
Dx1
. . . f1(x0+Dxn,u0)−f1(x0,u0)
Dxn ...
fn(x0+Dx1,u0)−fn(x0,u0)
Dx1
. . . fn(x0+Dxn,u0)−fn(x0,u0)
Dxn
3
775
(6.27)
with:
Dxi = Dxi ·
2
6664
di,1
di,2
...
di,n
3
7775
di,j =

0 if i 6= j
1 if i = j
(6.28)
The columns of A can be written as vectors, yielding:
A 
h
f(x0+Dx1,u0)−f(x0,u0)
Dx1
. . . f(x0+Dxn,u0)−f(x0,u0)
Dxn
i
=
h
x˙ x1−x˙ 0
Dx1
. . . ˙xxn−˙x0
Dxn
i
(6.29)
In this equation we used the definition:
˙xxi  f(x0 + Dxi, u0) (6.30)
which represents the output from the state equation (6.22) in the operating point with
the ith element of the state vector being perturbed by the amount Dxi. If this perturbation
is chosen properly, an approximation of the matrix A can now be determined
by subsequently computing its columns (i = 1, . . . , n).
86 Chapter 6. Analytical tools
A similar method can be used to find the matrix B. This time, we need to perturb the
elements of the input vector u, which results in the following approximation:
B 
h
f(x0,u0+Du1)−f(x0,u0)
Du1
. . . f(x0,u0+Dum)−f(x0,u0)
Dum
i
=
h
x˙ u1−x˙ 0
Du1
. . . x˙ um−x˙ 0
Dum
i
(6.31)
with:
Dui = Dui ·
2
6664
di,1
di,2
...
di,m
3
7775
di,j =

0 if i 6= j
1 if i = j
(6.32)
and:
x˙ ui  f(x0, u0 + Dui) (6.33)
　
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