Hardware Reference
In-Depth Information
tional single rate control design methodologies can be applied. Successive loop
closure (SLC), pole placement, the singular perturbation method, the Linear
Quadratic Gaussian (LQG) method, parameter optimization methods [13] are
few of these methods that have been used for multi-rate systems.
As recounted in [5], multi-rate control can increase servo bandwidth when
the PES sampling rate is relatively low compared to the open-loop crossover
frequency. Multi-rate design is also useful in designing sharp notch filters
to combat the actuator resonances by discretizing the notch filters at faster
rates than the PES sampling rate [204]. Design of controller with multi-rate
discretization of state space model is discussed in this section. This design
follows the one presented in [56] where a perfect tracking control based on
multirate feedforward control is discussed.
Let us consider a continuous-time nth-order single input single output
(SISO) plant P
c
(s) described by
x(t)=A
c
x(t)+B
c
u(t),
y(t)=C
c
x(t)+D
c
u(t).
(3.110)
The discrete-time model of the plant P [z
s
] discretized by single rate sampling
period T
y
(= T
u
) becomes
x[k +1] = A
s
x[k]+B
s
u[k],
(3.111)
y[k]=C
s
x[k]+D
s
u[k],
(3.112)
R
where x[k]=x(kT
y
), z
s
= e
sT
y
and A
s
=e
A
c
T
N
, B
s
=
T
f
/N
0
e
A
c
τ
B
c
dτ.
Figure 3.51: Intersample of multirate sampling.
The discrete-time plant P [z] discretized by generalized multirate sampling
control as shown in Figure 3.51 can be represented by
x[i +1] = Ax[i]+Bu[i],
(3.113)
y[i]=Cx[i]+Du[i],
(3.114)