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Chapter 8
Reference Introduction – Integral
Control
Reference Input – Zero Design
Motivation
A controller obtained by combining a control law with an
estimator is essentially a regulator design : the charac-
teristic equations of the controller and the estimator are
basically chosen for good disturbance rejection. However,
it does not lead to tracking, which is evidenced by a good
transient response of the combined system to command
changes. A good tracking performance is obtained by
properly introducing the reference input into the system.
This is equivalent to design proper zeros from the reference
input to the output.
ESAT–SCD–SISTA CACSD pag. 217
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Reference input – full state feedback
Discrete-time :The reference signal rk is typically the signal that the out-
put yk is supposed to follow. To ensure zero steady-state
error to a step input rk, the feedback control law has to be
modified.
Modification of the control law :
• Calculate the steady-state values xss and uss of the state
xk and the output yk for the step reference rss (=the
steady-state of step reference rk) :
xss = Axss +Buss
rss = Cxss +Duss
Let xss = Nxrss and uss = Nurss, then[
A− I B
C D
][
Nx
Nu
]
=
[
0
I
]
⇒ [
Nx
Nu
]
=
[
A− I B
C D
]−1 [
0
I
]
ESAT–SCD–SISTA CACSD pag. 218
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• Modify the control law:
uk = Nurk−K(xk−Nxrk) = −Kxk+(Nu +KNx)︸ ︷︷ ︸
N
rk
In this way the steady-state error to a step input will be 0.
Proof :
1. Verify that the closed-loop system from rk to yk is given
by[
xk+1
xk+1
]
=
[
A −BKLC A−BK − LC
][
xk
xk
]
+
[
BN
BN
]
rk,
yk =[
C −DK][
xk
xk
]
+DNrk.
2. If |eig(A−BK)| < 1 and |eig(A−LC)| < 1 we obtain
the following steady-state equations :
xss = Axss −BKxss + BNrss
xss = xss
yss = (C −DK)xss +DNrss
uss = −Kxss + Nrss
⇓yss = Cxss +D(−Kxss + Nrss) = Cxss +Duss = rss
ESAT–SCD–SISTA CACSD pag. 219
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So the transfer matrix relating y and r is a unity matrix
at DC ⇒ zero steady–state tracking error, steady–state
decoupling.
Note that:
• rk is an exogenous signal, the reference introduction will
NOT affect the poles of the closed-loop system.
•[
A− I B
C D
]−1
must exist, and thus for MIMO
number of references = number of outputs
• also for MIMO, reference introduction implies a steady-
state decoupling between different reference and output
pairs. This means that yss = rss.
• some properties of this controller are discussed on page
227.
Continuous-time :Try to verify that in this case
[
Nx
Nu
]
=
[
A B
C D
]−1 [
0
I
]
ESAT–SCD–SISTA CACSD pag. 220
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There are two types of interconnections for reference input
introduction with full state-feedback :
Type I: uk = Nurk −K(xk −Nxrk)
Type II: uk = −Kxk + (Nu +KNx)︸ ︷︷ ︸
N
rk
-
+-
+
-
+
EstimatorEstimator
PlantPlant
xx
rr uu y y
K K
Nu
Nx
N
For a type II interconnection, the control lawK used in the
feedback (uk = −Kxk) and in the reference feedforward
(N = Nu + KNx) should be exactly the same, otherwise
there is a steady-state error. There is no such problem in
type I.
⇒Type I is more ROBUST to parameter errors than Type
II.
ESAT–SCD–SISTA CACSD pag. 221
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Reference Input - General Compensator
Plant and compensator model :
Plant : xk+1 = Axk +Buk,
yk = Cxk +Duk;
Compensator : xk+1 = (A−BK − LC + LDK)xk
+Lyk,
uk = −KxkThe structure of a general compensator with reference in-
put r :
N
−K
M
yu
r +
+
xEstimator
Process
ESAT–SCD–SISTA CACSD pag. 222
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The general compensator is defined by the following closed-
loop equations from rk to yk :[
xk+1
xk+1
]
=
[
A −BKLC A−BK − LC
][
xk
xk
]
+
[
BN
M
]
rk,
yk =[
C −DK][
xk
xk
]
+DNrk.
Hence, the equations defining the compensator are
xk+1 = (A−BK − LC + LDK)xk + Lyk
+(M − LDN)rk,
uk = −Kxk + Nrk
where M ∈ Rn×m and N ∈ R
p×m.
The estimator error dynamics are
xk+1 = (A− LC)xk +BNrk −Mrk.
ESAT–SCD–SISTA CACSD pag. 223
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Poles:
Characteristic equation:
det
(
zI −[
A −BKLC A−BK − LC
])
= 0.
This is the same characteristic equation as without ref-
erence introduction. So introducing references will NOT
change the poles.
Zeros :
The equations for a transmission zero are (see page 82)
det
ζI − A BK −BN−LC ζI − A +BK + LC −MC −DK DN
= 0
⇔
ζI − A BK −BN−LC ζI − A +BK + LC −MC −DK DN
u
v
w
︸ ︷︷ ︸6=0
= 0
⇔
ESAT–SCD–SISTA CACSD pag. 224
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det
[
ζI − A −BC D
]
det
[
ζI − A +BK + LC −M−K N
]
= 0
The first term determines the transmission zeros of the open
loop system while the second term corresponds to the trans-
mission zeros of the compensator from rk to uk:
xk+1 = (A−BK − LC + LDK)xk + (M − LDN)rk,
uk = −Kxk + Nrk
These transmission zeros are designed via reference intro-
duction.
ESAT–SCD–SISTA CACSD pag. 225
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Autonomous estimator (cfr. pag. 218-220) :
Select M and N such that the state estimator error equa-
tion is independent of r ⇒
M = BN
where N is determined by the method for introducing the
reference input with full state feedback.
−K
y
r +
xEstimator
Process
+
u
N
ESAT–SCD–SISTA CACSD pag. 226
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Zeros :
The transmission zeros from rk to uk in this case are deter-
mined by
det(ζI − A + LC) = 0
which is the characteristic equation for the estimator, hence
the transmission zeros from rk to uk cancel out the poles
of the state estimator.
Properties :
• The compensator is in the feedback path. The refer-
ence signal rk goes directly into both the plant and the
estimator.
• Because of the pole-zero cancelation which causes “un-
controllability” of the estimator modes, the poles of the
transfer function from rk to yk consist only of the state
feedback controller poles (the roots of det(sI − A +
BK) = 0).
• The nonlinearity in the input (saturation) cancels out in
the estimator since in this case the state estimator error
equation is independent of u (xk+1 = (A− LC)xk)
ESAT–SCD–SISTA CACSD pag. 227
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Tracking–error estimator
Select M and N such that only the tracking error, ek =
(rk − yk), is used in the controller.
⇒ N = 0, M = −L
−K
y
xEstimator
Processu
r+
−−e
The control designer is sometimes forced to use a tracking–
error estimator, for instance when the sensor measures only
the output error. For example, some radar tracking sys-
tems have a reading that is proportional to the pointing
error, and this error signal alone must be used for feedback
control.
ESAT–SCD–SISTA CACSD pag. 228
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Zeros :
The transmission zeros from rk to yk are determined by
det
[
ζI − A −BC D
]
det
[
ζI − A +BK + LC L
−K 0
]
= 0
⇔
det
[
ζI − A −BC D
]
det
[
ζI − A L
−K 0
]
= 0.
Once K and L are fixed by the control and estimator de-
sign, so are the zeros. So there is no way to choose the
zeros.
Properties :
• The compensator is in the feedforward path. The ref-
erence signal r enters the estimator directly only. The
closed-loop poles corresponding to the response from rk
to yk are the control poles AND the estimator poles (the
roots of det(sI − A + BK) det(sI − A + LC) = 0).
• In general for a step response there will be a steady-state
error and there will exist a static coupling between the
input-output pairs.
• Used when only the output error ek is available.
ESAT–SCD–SISTA CACSD pag. 229
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Zero-assignment estimator (SISO) :
Select M and N such that n of the zeros of the overall
transfer function are placed at desired positions. This
method provides the designer with the maximum flexibil-
ity in satisfying transient-response and steady-state gain
constraints. The previous two methods are special cases of
this method.
Zeros of the system from rk to uk:
det
[
ζI − A +BK + LC −M−K N
]
= 0
⇓ M∆= MN−1
λ(ζ)∆= det(ζI − A + BK + LC − MK) = 0
ESAT–SCD–SISTA CACSD pag. 230
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Solution :
Determine M using a estimator pole-placement strategy
for “system” (Az, Cz), with
Az = A− BK − LC, Cz = K,
N is determined such that the DC gain from rk to yk is
unity.
For instance, in the case of a SISO system in continuous
time, for which D = 0
N = − 1
C(A−BK)−1B[1−K(A− LC)−1(B − M)]
and finally M = MN .
ESAT–SCD–SISTA CACSD pag. 231
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Example Tape drive control - reference introduction
Autonomous Estimator :
Consider the model of the tape drive on page 39. From the
pole-placement design example on page 114, K is known.[
A B
C D
]−1 [
0
I
]
=
[
1 0 1 0 0 0 0 0
−2.5 0 2.5 0 −0.67 0.67 −0.67 0.67
]T
.
Thus,
Nx =
1 −2.5
0 0
1 2.5
0 0
0 −0.67
0 0.67
, Nu =
[
0 −0.67
0 0.67
]
.
ESAT–SCD–SISTA CACSD pag. 232
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Let M = BN . The control law is
u = −Kx + (Nu +KNx)r
= −Kx +
[
0.8666 −1.6514
1.2779 2.1706
]
r.
Let L be the matrix from the pole placement estimator
design example on page 172. Then the closed-loop system
from r to y is[
x˙x
]
=
[
A −BKLC A− BK − LC
][
x
x
]
+
[
BN
BN
]
r,
y =[
C 0][
x
x
]
.
This system is NOT controllable, as was expected (Try to
prove it!).
ESAT–SCD–SISTA CACSD pag. 233
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Step responses from the reference r to the output y.
0 2 4 6 8 10 12 14 16 18 20−0.5
0
0.5
1
1.5
Time (secs)
Am
plitu
de
0 2 4 6 8 10 12 14 16 18 20−0.5
0
0.5
1
1.5
Time (secs)
Am
plitu
de
T
T
p3
p3
Step input to reference 1
Step input to reference 2
Output 1 (p3) follows a step input to reference 1 while
output 2 (T ) is zero in steady state.
Output 2 (T ) follows a step input to reference 2 while
output 1 (p3) is zero in steady state.
⇒ steady-state decoupling.
ESAT–SCD–SISTA CACSD pag. 234
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Integral Control and Robust Tracking
Motivation:The choice of N will result in a step response with a zero
steady-state error (see page 218). But the result is not
robust because any change in the parameters will cause the
error to be nonzero. Integral control is needed to obtain
robust tracking of step inputs.
A more general method for robust tracking, called the error
space approach (see page 238), can solve a broader class of
tracking problems, i.e. tracking signals that do not go to
zero in steady-state (a step, ramp, or sinusoidal signal).
ESAT–SCD–SISTA CACSD pag. 235
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Integral control
Augment the plant
xk+1 = Axk +Buk,
yk = Cxk +Duk
with extra states integrating the output error ek = yk− rkxIk+1
= xIk+ Cxk +Duk − rk︸ ︷︷ ︸
ek
.
The augmented state equations become[
xIk+1
xk+1
]
=
[
I C
0 A
][
xIk
xk
]
+
[
D
B
]
uk −[
I
0
]
rk.
What are the equivalent equations in continuous-time ?
We now close the loop to stabilize the system. The feedback
law is
uk = −[
K1 K0
]
︸ ︷︷ ︸K
[
xIk
xk
]
.
ESAT–SCD–SISTA CACSD pag. 236
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Use pole placement or LQR methods to design the control
feedback gain K. Once the closed-loop is stable, the track-
ing error e goes to zero even if some parameters change.
Processr y
+
−+
+−K1u
x
1z−1
−K0
The states of the plant xk are estimated using a state es-
timator. The estimator gain L is determined using pole
placement or Kalman filtering techniques. The integrator
states xIk need not to be estimated as they are being com-
puted explicitly.
What will be the closed-loop response from rk to yk ? Try
to derive a state-space model.
Note that pole placement or LQR might not work since the
augmented system is NOT always stabilizable and in this
case integral control can not be used.
ESAT–SCD–SISTA CACSD pag. 237
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Tracking control - the error-space approach
Integral control is limited to step response tracking. A
more general approach, the error-space approach, gives a
control system the ability to track a non-decaying or even
a growing input such as a step, a ramp, or a sinusoid.
Suppose the external signal, the reference, is generated by a
certain dynamic system. By including the dynamic system
as a part of the formulation and solving the control problem
in an error space, the error approaches zero.
ESAT–SCD–SISTA CACSD pag. 238
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Given the plant dynamics
xk+1 = Axk +Buk,
yk = Cxk +Duk
and the reference dynamics
rk+2 + α1rk+1 + α2rk = 0,
the tracking error is defined as
ek = yk − rk.
ESAT–SCD–SISTA CACSD pag. 239
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Define the error-space state:
ξk∆= xk+2 + α1xk+1 + α2xk,
and the error-space control:
µk = uk+2 + α1uk+1 + α2uk.
Then
ek+2 + α1ek+1 + α2ek = Cξk +Dµk,
and the state equation for ξk becomes
ξk+1 = Aξk +Bµk
Combining these two equations, the final error system is
zk+1 = Aezk +Beµk
where
zk =
ek
ek+1
ξk
, Ae =
0 I 0
−α2I −α1I C
0 0 A
, Be =
0
D
B
.
ESAT–SCD–SISTA CACSD pag. 240
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Controllability of the error system:
If (A,B) is controllable and has no zero at the roots of
αe(z) = z2 + α1z + α2,
then (Ae, Be) is controllable.
Control design: Pole-placement or LQR
µk = −[
K2 K1 K0
]
ek
ek+1
ξk
= −Kzk
The actual control uk is determined by the following inter-
nal model:
(u +K0x)k+2 +2∑
i=1
αi(u +K0x)k+2−i = −2∑
i=1
Kiek+2−i.
Once the closed-loop is stable, ek and ek+1 go to zero even
if some parameters change.
ESAT–SCD–SISTA CACSD pag. 241
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Disturbance rejection
by disturbance estimation
Motivation
If the state is not available then −Kx can be replaced by
the estimate−Kx where x comes from the state estimator.
The disturbance rejection problem consists in designing an
estimator such that the error x = x− x goes to zero even
when there is a disturbance signal with known dynamics.
Suppose that the disturbance is generated by a certain
known dynamic system. The method consists in augment-
ing the estimator with the disturbance system in a way to
cancel out the disturbance effects in the estimator output.
ESAT–SCD–SISTA CACSD pag. 242
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Augmenting the disturbance system to the plant
Given a plant with a disturbance input:
xk+1 = Axk +B(uk + wk),
yk = Cxk +Duk
and the disturbance dynamics (suppose 2nd order):
wk+2 + α1wk+1 + α2wk = 0.
The final error system is
zk+1 = Adzk +Bduk
where
z =
wk
wk+1
xk
, Ad =
0 I 0
−α2I −α1I 0
B 0 A
, Bd =
0
0
B
,
Cd =[
D 0 C]
, Dd = D.
ESAT–SCD–SISTA CACSD pag. 243
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Observability:
If the plant (A,C) is observable and has no zero at any
roots of
αd(z) = z2 + α1z + α2,
then (Ad, Cd) is observable.
Estimator for the error system:
zk+1 = Adzk + Bduk + L(yk − Cdzk −Dduk).
The output uk:
uk = −Kxk + Nrk︸︷︷︸
introduce reference
− wk︸︷︷︸cancel disturbance
.
Final closed-loop system:
xk+1 = (A− BK)xk + BNrk +BKxk + Bwk.
where xk = xk − xk and wk = wk − wk.
Stable estimator ⇒ xk → 0 and wk → 0. The final state
is NOT affected by the disturbance.
ESAT–SCD–SISTA CACSD pag. 244