A First Course in FUZZY and NEURAL CONTROL

50 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

or
λ^2 +3λ+2=0

Comparing the coefficients, we get(− 1 −k 1 −k 2 )=3and(− 2. 0 −k 1 −k 2 )=2,
which yields two identical equations(k 1 +k 2 )=− 4. While this results in a
nonunique solution for the feedback gainsk 1 andk 2 , we see that any set of
parameters satisfying(k 1 +k 2 )=− 4 will shift the pole originally located in the
right-halfs-plane, namely(λ−2) = 0,totheleft-halfs-plane at(λ+2)=0.

2.6.2 Higher-ordersystems.....................

Implementing state-variable feedback control for higher-order systems is sim-
plified byfirst converting the given system to a controller canonical form in
which the elements of the characteristic matrix have a prescribed structure. For
example, given the characteristic equation of a system as

Q(λ)=λn+αn− 1 λn−^1 +∑∑∑+α 1 λ+α 0

the controller canonical form of the characteristic matrix is

Ac=

        

01 0

001 0

00

..

. 10

01

−α 0 −α 1 ∑∑∑ −αn− 2 −αn− 1

        

TheentriesinthelastrowoftheAcmatrix are the negatives of the coefficients
of the characteristic polynomial. In addition, the input conditioning matrixBc
for the controller canonical realization is of the form

Bc=[0 0 ∑∑∑ 01]T

Now, if we choose the feedback matrix

K=[k 1 k 2 ∑∑∑ kn− 1 kn]

the resulting matrix obtained from[A+BK]can be written as:

Acf=

        

01

01

0

..

. 1

01

−α 0 +k 1 −α 1 +k 2 ∑∑∑ −αn− 2 +kn− 1 −αn− 1 +kn

        

Comparing the coefficients of the given system with the desired system yields
the feedback gains. Therefore, the canonical form of system representation is
more convenient for determining the feedback gains when dealing with higher-
order systems.