Hardware Reference
In-Depth Information
does not automatically include integral action. Two possible augmentations of
the state-space design can be used to nullify the effect of input bias:
• Add an integrator and augment the plant model by considering the out-
put of integrator as a new state, and
• Augment the state space model of the open loop plant by taking input
bias as a new state.
These methods are briefly explained below and realizations of PTOS with these
augmentations are shown.
Realization of PTOS with Integral Control
The first of the two methods for rejection of input disturbance includes an
integrator in the feedback path to generate a new state, the integral of error.
For the discrete-time state space model of equation 2.48, the error is
e(k)=y
r
(k) −y(k),
(2.52)
where, y
r
(k) is the reference input. Let us define the integral of this error as
anewstatex
I
(k). Then according to the discrete realization of integrator
x
I
(k +1)=x
I
(k)+y
r
(k)−y(k).
(2.53)
This is equivalent to,
x
I
(k +1)=x
I
(k)−Hx(k)+y
r
(k).
(2.54)
Combining equations 2.48 and 2.54, we get the augmented state space equa-
tion,
∙
¸
∙
¸∙
¸
∙
¸
∙
¸
x(k +1)
x
I
(k +1)
Φ 0
−H 1
x(k)
x
I
(k)
Γ
0
0
1
=
+
u(k)+
y
r
(k),
∙
¸
x(k)
x
I
(k)
y(k)=(H 0)]
.
(2.55)
The state feedback gain K can be selected such that the eigenvalues of the
following matrix satisfy the desired design specifications,
∙
¸
∙
¸
Φ 0
−H 1
Γ
0
−
K.
(2.56)
Realization of PTOS with integral control included is shown in Figure 2.32.
