d(s)
; Gb(s) = s + a
d(s)
(8.12.10)
where a R+ and d(s) is a stable polynomial. Further assume that these plants
have to be under feedback control with dominant closed loop poles with real part
smaller than a
Compare the fundamental design limitations for
x[k] +
Bq
u[k] (12.8.11)
y[k] = Cqx[k] + Dqu[k]
This model is as in (12.8.9), (12.8.10) where
A =
Aq I
; B =
Bq
; C = Cq; D = Dq (12.8.12)
Example 12.8. For the PI controller of example 12.7 on the facing page we have
A = 0, B = c1
, C = 1, D = c2 (12.8
#
C
ln g(z)dz +
_N
i=1
#
C
ln z + pi
z pi
dz (9.2.12)
The first integral on the right hand side can be expressed as
#
C
ln g(z)dz = 2j
_
0
ln |g(j)|d +
_
C
ln g(z)dz (9.2.13)
where,u sing Example ? on page ?
_
C
ln g(z)dz =
_
0 for nr > 1
j for nr = 1 whe
(z 1)(z ea0)
12.14 Using Continuous State Space Models
Next we show how to derive a discrete state space model when a zero hold input is
applied to a continuous time plant (described in state space form). Thus consider
a continuous time plant where the in
= c2 (e[k + 1] e[k]) + c1e[k]
In the notation of (12.4.1) we have a0 = 1, b1 = c2, b0 = c1c2. Note that the
extra term c2e[k+1] appears on the right hand side since,in this case,t he controller
is biproper (the left and right hand sides of (12.4.5) have t
Problem 10.8. Closed loop control has to be synthesized for a plant having nominal
model Go(s) = s+4
(s+1)(s+4) ,t o achieve the following goals:
(i) Zero steady state errors to a constant reference input.
(ii) Zero steady state errors for a sinewave dist
Y (s) = Go2(s)So2(s)Dg(s) + C2(s)Go(s)So2(s)U1(s); Go(s) = Go1(s)Go2(s)
(10.9.1)
where So2(s) is the sensitivity function for the secondary loop. Note also that,
if we denote by To2(s) the complementary sensitivity in the secondary loop, then
equation (10
given by
Cz(s) = cz
1 + czHz(s)
; Hz(s) = [Cz(s)]1 c
1
(11.4.4)
Both controllers are designed so as to achieve satisfactory performance in the
control of y(t) and z(t) respectively. We also assume that Cy(s) and Cz(s)
are minimum phase biproper transfer f
of strip thickness reduction. Hence reduced tension implies an increase in exit
thickness which impedes the original reduction i.e. the thickness response holds up.
The associated dynamics depend on the effective strip spring between uncoiler and
mill. Ma
difference equation in delta form (12.8.8) we obtain
A()Y() = B()U() + f(, xo) (12.10.1)
where Y(), U() are the Delta-transforms of the sequences cfw_y[k] and cfw_u[k]
respectively;
334 Models for Sampled Data Systems Chapter 12
A() = n + a
n1n1
+a
(12.10
the left hand side of (12.14.15) could be thought of as a finite difference approximation
to the continuous derivative. These intuitions are lost in the formulation of
(12.14.11) since
lim
0
Aq = I lim
0
Bq = 0 (12.14.21)
Other advantages of the delta for
(ck, l)
[| ln |Bp(ck)| + |(ln &)(ck, l)|] (9.4.16)
where we have used the fact that |Bp(ck)| < 1 ( ln(|Bp(ck)|) < 0 for every
ck > 0.
Discussion
(i) Consider the plot of sensitivity versus frequency shown in Figure 9.2. Say we
were to require the closed
the Cauchy Integral theorem ? on page ? to this function. We use the contour
shown in Figure ? on page ? with an infinitesimal right circular indentation, C_,
at the origin. We then have that
Section 9.3. Integral Constraints on Complementary Sensitivity
C
ln To(s)
s2 ds +
M_
i=1
_
C
1
s2 ln
_
s + ci
s ci
_
ds (9.3.13)
The first integral on the right hand side vanishes,an d the second one can be
computed from
_
C
1
s2 ln
_
s + ci
s ci
_
ds = 1
ci
_
C
ln
_
1+
1
_
d (9.3.14)
where s = ci
and C is an infini
Then,t he closed loop poles are at (1;1) and the controller has a zero at
s = 0.5. Equation (8.6.12) correctly predicts overshoot for the one d.o.f. design.
However,if we first prefilter the reference by H(s) = 1
2s+1 ,t hen no overshoot occurs
in respons
feedforward
cascade control
two-degree of freedom architectures
Signal models
284 Architectural Issues in SISO Control Chapter 10
Certain classes of reference or disturbance signals can be modeled explicitly
by their Laplace transform:
Section 10.10.
288
Architectural Issues in SISO Control Chapter 10
10.12 Problems for the reader
Problem 10.1. Compute the disturbance generating polynomial, d(s),for each
of the following signals.
(a) 3 + t (b) 2 cos(0.1t + /7)
(c) 3
_
sin 1.5t
_3 (d) e0.1t cos(0.2t)
P
u = wou
vo
i hoi
[c1vo
0(t) + c2vo
ehi8t) vo
ihi(t)] (10.8.7)
279
Figure 10.6. Feedforward controller for reversing mill
Now a simple model for the uncoiler dynamics is
Ju
du(t)
dt
= Kmiu(t) (10.8.8)
where iu(t) is the uncoiler current, Ju is the uncoiler
sensitivity to parametric modeling errors
sensitivity to structural modeling errors
Rather, tuning for one of these properties automatically impacts on the others.
For example, irrespectively of how a controller is synthesized and tuned, if the
effect
C(s)
+
Gf (s)
Go1(s)
R(s) U(s) Y (s)
Figure 10.2. Disturbance feedforward scheme
From Figure 10.2 we observe that the model output and the controller output
response, due to the disturbance, are given by:
Yd(s) = So(s)Go2(s)(1 + Go1Gf (s)Dg(s) (10.7.1)
Ud
One of the general conclusions which can be drawn from the above analysis is
that the design problem becomes more difficult for large unstable open loop poles
and small NMP zeros. The notions of large and small are relative to the magnitude
of the require
ln g(z)dz = 0 (9.2.5)
where C = Ci C is the contour defined in Figure ? on page ?.
Then
#
C
ln g(z)dz = j
_
ln g(j)d
_
C
ln(1 + l(z)dz (9.2.6)
For the first integral on the right hand side of equation (9.2.6),we use the
conjugate symmetry of g(z) to obta
0.8
1
1.2
1.4
Time [s]
Plant output
Figure 10.5. First and third d.o.f. performance
277
Remark 10.2. Disturbance feedforward is sometimes referred to as the third degree
of freedom in a control loop.
Section 10.8. Industrial Applications of Feedforward Co
90(s + 1)(s + 2)(s + 4)
16s(s2 + 15s + 59)
; Cz(s) =
16(3s + 10)(s + 4)
s(s + 14)
(11.4.9)
The basic guidelines used to develop the above designs is to have the secondary
control loop faster than the primary control loop,s o that the state z(t) can be qui
down on the water bed in one area, raises it somewhere else).
These trade-offs show that systems become increasingly difficult to control as
Unstable zeros become slower
Unstable poles become faster
Time delays get bigger
9.8 Further Reading
Bodes ori
dGf _dg(t)
dt
+ Gf _dg(t)
_
d[dg(t)]
dt
+ 2dg(t)
_
1
2+ 0.2dg(t)
(10.7.10)
where _ 1.
=
275
The operator defined in equation (10.7.10) is used to compute the disturbance
feedforward error, ef (t),d efined as
ef (t) _= dg(t) + w(t) = dg(t) + Go1 Gf _dg(t)
We assume (as will almost always be the case) that the plant operates in continuous
time whereas the controller is implemented in digital form. Having the
controller implemented in digital form introduces several constraints into the problem:
(a) the cont
Saturating actuators
Bernstein, D. and Michel, A. (1995). A chronological bibliography on saturation
actuation. Int. J. of Robust and Nonlinear Control, 5:375380.
Hanus, R., Kinnaert, M., and Henrotle, J. (1987). Conditioning technique, a general
anti-win
_
Sat_C_e
+ ce C e
(11.3.9)
This can be represented as in Figure 11.9.
e(t)
c Sat_.
e(t)
+
+
usat(t)
C(s)
+
c
1
Figure 11.9. Conditioning equivalent for the anti wind-up controller
To show that this is equivalent to the previous design, we note that in F