Page 770 - Mechanical Engineers' Handbook (Volume 2)

2 The Pole Placement Design Method 761

consequence of these limitations of the pole placement method is that the design process
involves considerable trial and error.

2.2 Modification for Constant Reference and Disturbance Inputs
The pole placement method described is appropriate for regulation problems. For the case
of nonzero reference inputs that may be constant or varying with time, the system outputs
are required to follow the reference inputs. The control law, Eq. (1), needs to be modified
for such problems. If the output vector y and the input vector u have the same dimension
and if the D matrix is zero in Eqs. (7) and (13) in Chapter 17, the modified control law has
the form
u Kx Ny (11)
d
where y is a vector of reference inputs. For constant reference inputs y , the error y y
d
d
d
can be reduced to zero under steady-state conditions by selecting
1
N [C( A BK) B] 1 (12)
for continuous-time systems and
1
N [C(I F GK) G] 1 (13)
n
for discrete-time systems. The matrices to be inverted on the right-hand sides of the preceding
equations exist if and only if the corresponding open-loop transfer matrices [C(sI A) 1
n
1
3
B] and [C(zI F) G] have no zeros at the origin and at z 1, respectively. The K
n
matrix in Eq. (11) is chosen to give the desired closed-loop poles as before. It should be
noted that the gain matrix N is outside the feedback loop. Hence, the controlled system
performance, particularly the steady-state error, would be sensitive to modeling error or error
in the elements of the system matrices.
It is well known from classical control theory that integral controller action on the error
has the effect of reducing the steady-state error to reference and disturbance inputs. In par-
ticular, the steady-state error is reduced to zero for constant reference and disturbance inputs.
3
A similar result can be obtained within the framework of state-variable feedback and will
be described for the case where the y and u vectors have the same dimension and the D
matrix is zero in Eqs. (7) and (13) in Chapter 17.
Consider the case of constant but unknown disturbance inputs:
˙ x(t) Ax(t) Bu(t) w(t) (14)
y(t) Cx(t) (15)
for continuous-time systems and
x(k 1) Fx(k) Gu(k) w(k) (16)
y(k) Cx(k) (17)

for discrete-time systems. The state-space equations are augmented by
˙ q (t) y(t) Cx(t) (18)
e
for continuous-time systems and
q (k 1) q (k) y(k) (19)
e
e
q (k) Cx(k)
e