National Instruments Xmath Interactive Control Design Module ICDM - Manual (Page 90)

Chapter 11 Introduction to MIMO Design
Xmath Interactive Control Design Module 11-4 ni.com
Notice that in the SISO case, these “complementary pairs” of transfer
functions (obtained by swapping P and C) are the same. It is important to
remember that in the MIMO case they can be different; they even have
different dimensions if n
y
≠ n
u
.
In addition to these transfer functions you encounter two (complementary)
open-loop transfer functions:
• The loop transfer function L is defined as L = PC. This is the transfer
function of the loop cut at the sensor (or the error e).
• The actuator loop transfer function (or complementary loop transfer
function) L
act
is defined as L
act
= CP. This is the transfer function of the
loop cut at the actuator.
Integral Action
Integral action can be quite complicated in the MIMO setting. The simplest
case occurs when the plant is “square”, that is, n
y
= n
u
, the plant has no
poles at s = 0, and no zeros at s = 0—which in this case means debt
P(0)
≠ 0.
Suppose the controller has the form
where has no poles at s = 0 and the (constant) matrix is nonsingular
(for example, R
0
= I). Then you have T(0) = I, so that you have perfect
asymptotic decoupling and tracking of constant reference inputs. You also
have perfect asymptotic rejection of constant actuator disturbances. The
condition on C means the controller has an integrator in each of its
channels. In this case the integrators can be thought of as either acting on
the sensor signals or acting on the actuator signals.
You often have integrators associated with some of the sensors, or some of
the actuators. Then the matrix can be less than full rank, and you generally
get the benefits of integral action in only some of the I/O channels; that is,
only some of the entries of T(0) are 1 (or 0).
When things get more complicated. The rank of R
0
is at most
r = max{n
y
n
u
}. If you ask ICDM to insert more than r integrators (say,
using an integrator in each sensor channel for a plant with three actuators
and five sensors), you will have a closed-loop system that cannot be
stabilized—it will have an uncontrollable or unobservable mode at s = 0.
If this happens, ICDM will warn you. Even if you do not have excess
integrators, you should realize that you will get perfect asymptotic tracking
C s ( ) 1 s ⁄ ( )R
0
C˜ s ( )
+ =
C˜
n
y
n
u
≠