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If the system of Eq. ( 4.25 ) can be linearized by a diffeomorphism z D .x/ and
a static state feedback u D ˛.x/ C ˇ.x/ v into the following form
z i;j D z iC1;j for 1jm and 1i v j 1
z v i;j D v j
(4.26)
with P jD1 v j D n, then y j D z 1;j for 1jm are the 0-flat outputs which can
be written as functions of only the elements of the state vector x (when the flat
output of a system is only function of its states x, then this is called 0-flat). To
define conditions for transforming the system of Eq. ( 4.25 ) into the canonical form
described in Eq. ( 4.26 ) the following theorem holds [ 25 ].
Theorem. For the nonlinear systems described by Eq. ( 4.25 ) the following vari-
ables are defined: (i) G 0 D span Œg 1 ; ;g m , (ii) G 1 D span Œg 1 ; ;g m ;
ad f g 1 ; ; ad f g m , (k) G k D span f ad f g i for 0jk; 1img (where ad i f g
stands for a Lie Bracket). Then, the linearization problem for the system of Eq. ( 4.25 )
can be solved if and only if: (1). The dimension of G i ;iD 1; ;k is constant for
x2XR n and for 1in 1 , (2). The dimension of G n1 is of order n , (3). The
distribution G k is involutive for each 1kn 2 .
4.8.3
Transformation of the Neurons' Model into the Linear
Canonical Form
It is assumed now that after defining the flat outputs of the initial multi-input multi-
output (MIMO) nonlinear system (this approach will be also shown to hold for the
neuron model), and after expressing the system state variables and control inputs
as functions of the flat output and of the associated derivatives, the system can be
transformed in the Brunovsky canonical form:
x 1 D x 2
x r 1 1 D x r 1
x r 1 D f 1 .x/ C P jD1 g 1 j .x/ u j C d 1
x r 1 C1 D x r 1 C2
x p1 D x p
(4.27)
x p D f p .x/ C P jD1 g p j .x/ u j C d p
y 1 D x 1
y p D x nr p C1
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