A First Course in FUZZY and NEURAL CONTROL

38 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

where

eAt=

X∞

k=0

tk

k!

Ak

withA^0 the identityn◊nmatrix andx 0 =x(0).Now

eAt=

Xm

k=1

h

Bn 1 +Bn 2 t+∑∑∑+Bnαktαk−^1

i

eλkt (2.32)

where theλks are eigenvalues ofA,theαksarecoefficients of the minimum
polynomial ofA,andtheBns are constant matrices determined solely byA.
Thus,
∞
∞eAt

∞

∞ ≤

Xm

k=1

Xαk

i=1

ti−^1

∞

∞eλkt

∞

∞kBnik

=

Xm

k=1

Xαk

i=1

ti−^1 eRe(λk)tkBnik

whereRe(λk)denotes the real part ofλk.
Thus, ifRe(λk)< 0 for allk,then
lim
t→∞
kx(t)k≤lim
t→∞
kx 0 k

∞

∞eAt

∞

∞=0

so that the origin is asymptotically stable.
Conversely, suppose the origin is asymptotically stable. ThenRe(λk)< 0
for allk, for if there exists someλksuch thatRe(λk)> 0 ,thenweseefrom
Equation 2.32 thatlimt→∞kx(t)k=∞so that the origin is unstable.
Such a matrix, or its characteristic polynomial, is said to bestable.

Example 2.2The solution of a second-order constant-coefficient differential
equation of the form
d^2 y
dt^2

+a

dy

dt

+by=0

is stable if the real parts of the roots

s 1 = −

1

2

≥

a−

p

a^2 − 4 b

¥

s 2 = −

1

2

≥

a+

p

a^2 − 4 b

¥

of the characteristic polynomials^2 +as+blie in the left-halfs-plane. In practice,
the characteristic polynomial is oftenfound by taking the Laplace transform to
get the transfer function

L(y)=

y^0 (0) + (a+s)y(0)

s^2 +as+b

The roots ofs^2 +as+bare called the poles of the rational functionL(y).

If bounded inputs provide bounded outputs, this is called BIBO stability. If
a linear system is asymptotically stable, then the associated controlled system
is BIBO stable.