_
(22.17.6)
Problem 22.9. A discrete time noise measurement, y[k],of an unknown constant
satisfies the model
y[k] = + v[k] (22.17.7)
where v[k] is an uncorrelated stationary process with variance 2.
Section 22.17. Problems for the Reader 717
22.9.1 Find

the open RHP. This means
[Bi(s)]jj =
rj
k=1
s + z
jk
s z
jk
(24.7.2)
where zjk, k = 1, 2, . . . , rj , are the NMP zeros of the jth element of vector To(s)gi.
We also define a column vector gi(s), where
gi(s) = Bi(s)gi =
[Bi(s)]11gi1(s)
.
[Bi(s)]mmgim(s)

120
100
80
60
Frequency [rad/s]
Magnitude [dB]
Open loop
Closed loop
Figure 22.14. Open loop and closed loop frequency responses of the beam.
Section 22.15. Summary 711
Due to the complexities of multivariable systems, criterion based synthesis
(briefly

d()
Figure 23.14. One-degree-of-freedom MPC architecture
23.6.2 Integral action
An important observation is that the architecture described in Figure 23.14 gives a
form of integral action. In particular y is taken to the set-point ys irrespective of
the

stdt
(24.6.9)
The result follows on evaluating (24.6.9) at s = zo and using (24.5.5).
Remark 24.1. When zo and o are in the RHP,p arts (ii) and (iii) of Lemma 24.3
hold for any input whose Laplace Transform converges for _(s) > 0
Comparing Lemma 24.3 with

_
(24.7.11)
The result follows on using (24.7.11) in (24.7.6),an d using the property of the
Poisson kernel in (24.7.7).
24.8 Poisson Integral Constraints on MIMO Sensitivity
When the plant has NMP zeros, a similar result to that in Theorem 24.1 on the
pa

0.115e120s
1984s+ 1
0.00253
s
0
0.00299
s
U(s) (23.10.13)
Chapter 24
FUNDAMENTAL
LIMITATIONS IN MIMO
CONTROL
24.1 Preview
Arguably, the best way to learn about real design issues is to become involved in
practical applications. Hopefully, the reader may h

726 Model Predictive Control Chapter 23
where
U = cfw_u(0), u(1), . . . , u(N 1) (23.3.9)
VN(x,U) =
N_1
_=0
L(x(), u() + F(x(N) (23.3.10)
subject to the appropriate constraints. Standard optimization methods are used to
solve the above problem.
Let the mi

Y
(z) (25.4.48)
q
25.4.2 Approximate inverses
We next show how interactors can be used to construct approximate inverses accounting
for relative degree.
A crucial property of L(s) and R(s) is that
R(s) _= Go(s)R(s); and L(s) _= L(s)Go(s) (25.4.49)
are bo

Problem 23.7. Consider a quadratic cost function
J = ( U)T ( U) (23.10.6)
where and U are vectors and is an N M matrix.
23.7.1 Show that minimum of J is achieved (when U is unconstrained) by
U = Uo = (T)1T (23.10.7)
23.7.2 Say that U is required to satisf

for some static mapping h(). What remains is to give a characterization of the
mapping h(). For general constrained problems, it will be difficult to give a simple
parameterization to h(). However, we can think of the anti-windup strategies of
Chapter 11

chute
pressure
feed
rolls
crusher
rolls
shredder
and
entry
Figure 24.2. A sugar Milling Train
The sugar mill unit under consideration constitutes one of multiple stages in the
overall process. A schematic diagram of the Milling Train is shown in Figure 24

From Chapter 20 we observe that good nominal tracking is, as in the SISO case,
connected to the issue of having low sensitivity in certain frequency bands. On
examining this requirement we see that it can be met if we can make
[I +Go(j)Co(j)]1Go(j)Co(j) I

Kucera, V. (1972). The discrete Riccati equation of optimal control. Kibernetika,
8(5):430447.
Lancaster, P. and Rodman, L. (1995). Algebraic Riccati equations. OxfordUniversity
Press, Oxford.
Willems, J. (1970). Stability Theory of Dynamical Systems. T.

the next chapter.
Excess outputs
Here we assume that p > m. In this case we cannot hope to independently control
each of the measured outputs at all times. We investigate three alternative
strategies.
Squaring down
Although all the measurements should be

Skogestad, S. and Postlethwaite, I. (1996). Multivariable Feedback Control: Analysis
and Design. Wiley, New York.
Sule, V. and Athani, V. (1991). Directional sensitivity trade-offs in multivariable
feedback systems. Automatica, 27(5):869872.
Sugar mill ap

state error occurs for reference step inputs in all channels. Then for a plant zero
zo with left directions hT1
, hT2
, . . . , hT
z,an d a plant pole o with right directions
g1, g2, . . . , gp satisfying _(zo) > and _(o) > ,we have
(i) For a positive uni

CD(s) = I Go(s)Q(s) CN(s) = Q(s) (25.2.7)
GoD(s) = I GoN(s) = Go(s) (25.2.8)
GoD(s) =I GoN(s) = Go(s) (25.2.9)
Actually the above choices show that (22.12.22) is satisfied and hence that closed
loop stability is guaranteed by having Q(s) stable. We will f

(24.10.23)
where
n11(s) = 0.0023(s+ 1) n12(s) = s2 0.005s 0.005 (24.10.24)
n21(s) = s + 1 n22(s) = 5(s + 1) (24.10.25)
d
(s) = s + 0.121 d+(s) = s + 0.137 (24.10.26)
Then, from (24.10.20) and (24.10.22), the controller is given by
C(s) =
(25s + 1)(n11(s)M

strictly proper,s ince its determinant vanishes for = . We use the construction
procedure outlined in the proof of Theorem 25.1.
Note that n1 = 0 with f1 =
1
6
[2 1] and n2 = 0 with f2 =
1
6
[2 1]
Section 25.5. Dealing with NMP Zeros 803
(i) We first form

B2] [D1C1 (sI A1)1 B1]
(25.5.68)
= [D1 D1C2
1
sI A2 + B2D1C2
1 B2D1] [C1 (sI A1)1 B1]
(25.5.69)
Section 25.5. Dealing with NMP Zeros 813
The reader is invited to check that this is exactly equal to [(s)]1[T]i(s)
where [(s)]1 has the state space realizatio

x(2),
. . . , uo
1), 0 (23.4.6)
Then the increment of the Lyapunov function on using the true MPC optimal
input and when moving from x to x(1) = f(x, h(x) satisfies:
hV oN
(x) _= V oN
(x(1) V oN
(x) (23.4.7)
= VN(x(1), Uo
x1) VN(x,Uo
x)
However, since Uo

3z1 z2
10
3 1
Y
(z)
=
z1 0
3z1(1 z1) z2
Y
q
(z)
(25.4.34)
We can evaluate the resultant time response of the system as follows.
A unit step in y
1[k] (the first component of y[k]) produces the following output
sequences:
cfw_y1[k] = [0 1 1 1 ] (25.4.35)
c

entries of the form
_
zo( + s)
s zo
_k
where k N and R+ (25.5.24)
[L(s)]1 and [R(s)]1 are stable triangular matrices with diagonal elements
of the form
_
s zo
zo( + s)
_k
(25.5.25)
lims L(s) = ZL and lims R(s) = ZR arematrices with finite nonzero
deter

00100
01000
(23.7.4)
B=
0.2784
0.0334
0.0894
00
(23.7.5)
This system was sampled with a zero order hold and sampling period of 0.5.
Details of the model predictive control optimization criterion of Equation (23.4.6)
were
=
_
90 0
03
_
; = 0.1 (23.7.6)
Opt

(s 0.137)p11(s)
s2l11(s)
; and M22(s) =
(s 0.137)p22(s)
s2l22(s)
(24.10.17)
where p11(s), l11(s), l22(s) and p22(s) are chosen using polynomial pole placement
techniques (see Chapter 7). For simplicity, we choose the same denominator polynomial
for T11(s)

z2(t) = C2x2(t) + Du(t) (25.5.46)
to represent the system with bi-proper transfer function L(s)Go(s). Also, for
square plants, we know from the properties of L(s) that detcfw_D _= 0.
This seems to fit the theory given in section 22.6 for Model Matching. H

_
(25.5.78)
To cast this into the problem formulation outlined above,we next reparameterize
Q(s) to force integration in the feedback loop. We thus use
Q(s) = Go(0)1 + sQ(s) =
1
15
_
14 8
42
_
+ sQ(s) (25.5.79)
and the introduction of a weighting function