A First Course in FUZZY and NEURAL CONTROL

2.6. STATE-VARIABLE FEEDBACK CONTROL 51

Example 2.6Consider an unstable third-order system given by the set of state
and output equations as





x ̇ 1

x ̇ 2

x ̇ 3



 =





16 − 3

− 1 − 11

− 220









x 1

x 2

x 3



+





1

1

1



u

y =

£

001

§





x 1

x 2

x 3





We notice that the system is not in canonical form and proceed to develop the
controller canonical realization. To do this, wefirst obtain the transfer function
of the given system as

Y(s)

U(s)

=C[sI−A]−^1 B+D

ThiscanbedoneveryeasilyusingMatlabby specifying theA,B,C,andD
matrices and using theMatlabfunction ìss2tfî to obtain the transfer function
ofthesystem.ThefollowingMatlabcode illustrates this process:

a=[1 6 -3;-1 -1 1;-2 2 0]; %SpecifytheAmatrix

b=[1;1;1]; %Specify the B matrix

c=[0 0 1]; %Specify the C matrix

d=0; %Specify the D matrix

[num,den]=ss2tf(a,b,c,d,1); %Note: the 1 indicates a single-input system

The resulting transfer function is

Y(s)

U(s)

=

s^2 − 13

s^3 − 3 s+2

The denominator of the transfer function factors as(s+2)(s−1)^2 ,display-
ing the roots of the characteristic equation and clearly indicating an unstable
system. From the method discussed above, the characteristic matrix for the
controller canonical realization may be obtained directly as

Ac=





010

001

− 230





Suppose the desired set of eigenvalues for the system is

{λ 1 ,λ 2 =− 1 ±j, λ 3 =− 4 }

which gives the characteristic polynomial

Q(λ)=λ^3 +6λ^2 +10λ+8