A First Course in FUZZY and NEURAL CONTROL

48 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

2.6 State-variablefeedbackcontrol

Here we address a controller design technique known as state-variable feedback
control. The concept behind state-variable feedback control is to determine the
poles of the transfer function of the closed-loop system and make changes so that
new poles are assigned that meet some design criteria. We will see that the basic
solution is to place poles in the controller such that the root loci of the overall
system is moved into the left-halfs-plane. As mentioned earlier, the objective in
control systems design is to stabilize an otherwise unstable plant. The method
by which an inherently unstable system can be made stable by reassigning its
poles to lie in the left-halfs-plane is referred to aspole placement.Thetheory
behind this technique comes from the result in Theorem 2.1 on page 37 that
states that the linear systemx ̇=Axis asymptotically stable if and only if all
eigenvalues of the matrixAhave negative real parts, and the fact that these
eigenvalues are the poles of the transfer function.
It is not necessary that only unstable systems be considered for such con-
trol. Even stable systems that do not meet the criteria for a suitable transient
response can be candidates for state-variable feedback control. However, it is
necessary that all states of the original system be accessible and that the system
be completely controllable.

2.6.1 Second-ordersystems

We can discuss state-variable feedback control with some simple examples.
These examples are chosen only to demonstrate how the behavior of a system
can be modified and are not intended to provide a complete theoretical eval-
uation of state-variable feedback control. There are excellent references that
provide full details of this type of control, such as [16] and [28], and the reader
is strongly urged to refer to the literature on this topic.

Example 2.5Consider a second-order system represented by the state and
output equations as follows:
∑
x ̇ 1
x ̇ 2

∏

=

∑

− 13

02

∏∑

x 1

x 2

∏

+

∑

1

1

∏

u

y =

£

10

§

∑

x 1

x 2

∏

These equations are of the formx ̇=Ax+Bu, the state equation, andy=Cx,
the output equation. The eigenvalues of this system can be determined by
consideringdet [λI−A]=0, the characteristic polynomial of the system. This
gives

det

μ

λ

∑

10

01

∏

−

∑

− 13

02

∏∂

=0

that simplifies to

det

μ∑

λ+1 − 3

0 λ− 2

∏∂

=(λ+1)(λ−2) = 0