Y (s)
_
=
_
GD(s) CN(s)
GN(s) CD(s)
_
V
(s)
(s)
_
(20.3.40)
where V
(s) = L[v
(t)], (s) = L[(t)]. Also,we have
_
V
(s)
(s)
_
=
_
CD(s) CN(s)
GN(s) GD(s)
_
U(s)
Y (s)
_
(20.3.41)
The result follows.
600 Analysis of MIMO Control Loops Chapter 20
20.3.5 Pole

Frequency [rad/s]
Singular values [dB]
1
()
2
()
Figure 20.5. Singular values of MME matrix
Chemical process example: yield, and throughput must be regulated
and; thermal energy, valve actuators and various utilities are available
as control variables.

0.4
0.6
0.8
1
1.2
1.4
Frequency [rad/s]
Singular values
1
()
2
()
Figure 20.4. Singular values of the sensitivity function
We can now compute the value of at which (So(j) is maximal. The singular
values of So(j) are shown in Figure 20.4.
From here,we can

(20.3.47)
which clearly has rank 1.
Example 20.4 (Quadruple tank apparatus). A very interesting piece of laboratory
equipment based on four coupled tanks has recently been described by Karl
Hendrix Johansson (see references given at the end of the chapte

G(s) =
3.71
62s + 1
3.7(1 2)
(23s + 1)(62s+ 1)
4.7(1 1)
(30s + 1)(90s+ 1)
4.72
90s + 1
(21.3.6)
We also recall that choice of 1 and 2 could change one of the system zeros
from the left to the right half plane. The RGA for this system is
=
_
1
1
_
where =

an inherently MIMO problem to a set of SISO problems.
Section 21.6. Converting MIMO Problems to SISO Problems 643
21.6 Converting MIMO Problems to SISO Problems
Many MIMO problems can be modified so that decentralized control becomes a
more viable (or att

Gii(s)
Gji
ff (s)
Gij (s)
Figure 21.6. Feedforward action in decentralized control
The feedforward gain Gji
ff (s) should be chosen in such a way that the coupling
from the j th loop to the i th loop is compensated in a particular, problem
dependent, freq

and G(s),r espectively. Assume that they are related by (20.8.1). Also assume
that a controller C(s) achieves nominal internal stability and that Go(s)C(s) and
G(s)C(s) have the same number, P,of unstable poles. Then a sufficient condition
for stability o

and Design. Wiley, New York.
626 Analysis of MIMO Control Loops Chapter 20
Robustness in MIMO systems
Glover, K. (1986). Robust stabilization of linear multivariable systems. International
Journal of Control, 43(3):741766.
Green, M. and Limebeer, D. (1995

50s
_
(21.3.12)
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.5
0
0.5
1
1.5
2
2.5
Time [s]
Plant outputs and ref.
y
2
(t)
y
1
(t)
r
1
(t)
r
2
(t)
Figure 21.4. Decentralized control of a non-minimum phase four tank system
Comparing the results in Figur

k21 = 1.
We see that a sufficient condition for robust stability of the decentralized control,
w ith the pairing (u1, y1), (u2, y2),is that |k12| < 1 and |k21| < 1. We see that
this is conservative,bu t consistent with the performance results presented ab

on the web-page
Actually, almost identical control problems to the above may be found in many
alternative industrial situations where there are longitudinal and traverse effects.
To quote one other example, similar issues arise in paper making.
An interes

z z 0.6
z 0.4 z + 0.2
_
(20.11.10)
Assume that both references are sinusoidal signals. Find the worst direction and
frequency of the reference vector (see solved problem 20.6).
Chapter 21
EXPLOITING SISO
TECHNIQUES IN MIMO
CONTROL
21.1 Preview
In the case

20.5.1 Compute its singular values for = 0.1, = 0.5 and = 5.
20.5.2 For each case,c ompare the results with the magnitude of the frequency response
of the diagonal terms.
Problem 20.6. In a feedback control loop ,w e have that the open loop transfer
funct

i=1
arg_(1 + i(j)
(20.5.16)
Thus,an y change in the angle of Fo(j) results from the combination of phase
changes in the terms 1+i(j). Hence,en circlements of the origin in the complex
plane Fo can be computed from the encirclements of (1; 0) by the combin

Then,L emma 20.1 on page 598 shows that Acl(s) = I,an d hence,t he control loop
is stable. Actually,t his can also be seen from (20.3.38) if we set (s) = 0. Then
(20.3.40) defines a relationship between U(s) and Y (s) (i.e. a feedback controller)
of the f

B2 left null space of C1 (I A1)1 B1.
Proof
The steps to show this parallel those for the observability case presented above.
The pole-zero cancellation issues addressed by the two lemmas above are illustrated
in the following example.
Example 20.7. Consid

i,k
[]ik () m max
i,k
[]ik
sv10 If = I, the identity matrix, then () = () = 1.
Some of the above properties result from the fact that the maximum singular
value is a norm of the matrix, and others originate from the fact that singular values
are the squar

(20.3.15)
where n
11(s) . . . n
mm(s) are polynomials. Again, GD(s) and GN(s) are stable
proper transfer functions.
Equations (20.3.13), (20.3.14), and (20.3.15) describe a special form of Right
Matrix Fraction Description (RMFD) for G(s).
20.3.4 Connect

|G|= sup
|U|_=0
|GU|
|U| (20.7.3)
611
We call |G| the induced norm on G corresponding to the vector norm |U|.
For example, when the vector norm is chosen to be the Euclidean norm, i.e.
|x| =
xHx (20.7.4)
then we have the induced spectral norm for G define

the non interacting loops,when interaction appears in the plant model, instability
or unacceptable performance may arise.
The reader is encouraged to use the SIMULINK schematic in file mimo1.mdl
to explore other interaction dynamics. We see that a SISO vi

tracking case shows that input disturbance rejection is normally less demanding,
since (Go(j) is usually low pass and hence, Go itself will provide preliminary
attenuation of the disturbance.
20.7.4 Measurement noise rejection
The effect of measurement no

Clearly all matrix transfer descriptions comprise elements having numerator and
denominator polynomials. These matrices of rational functions of polynomials can
be factorized in various ways.
Consider an (n m) transfer function matrix G(s). Let dri
(s) de

fraction descriptions (MFDs).
594 Analysis of MIMO Control Loops Chapter 20
20.3.1 State space models revisited
Linear MIMO systems can be described using the state space ideas presented in
Chapter 17. The only change is the extension of the dimensions of

Verify this with the system used in 21.1
Problem 21.4. Assume that you have a plant having nominal model given by
Go(s) =
2e0.5t
s2 + 3s + 2
0
0.5
(s + 2)(s + 1)
6
s2 + 5s + 6
(21.10.1)
21.4.1 Design,in dependently, SISO PI controllers for the two diagona

dt
= f(x(t), u(t), t) (22.3.1)
where x(t) Rn, together with the following optimization problem,
Problem (General optimal control problem). Find an optimal input
uo(t),for t [to, tf ],s uch that
uo(t) = argmin
u(t)
!_
tf
to
V(x, u, t)dt + g(x(tf )
"
(22.3.

; Cp =
_
0010
0100
_
(22.2.7)
and
Ad = 0; Bd = 0; Cd = I2 (22.2.8)
659
where I2 is the identity matrix in R22.
The augmented state space model, (A,B,C, 0) is then given by
A=
_
Ap BpCd
0 Ad
_
=
_
Ap Bp
00
_
B=
_
Bp
Bd
_
=
_
Bp
0
_
C=
_
Cp 0
Section 22.2.

in the Figure 21.9.
In this configuration, numerous cooling sprays are located across the roll, and
the flow through each spray is controlled by a valve. The cool water sprayed on the
roll reduces the thermal expansion. The interesting thing is that each

The result follows immediately from Lemma 20.6 on page 619.
The final architecture of the control system would then appear as in Figure 22.2.
y(t)
u(t) z(t)
x(t)
Observer
Plant
parallel
integrators
Feedback
gain
+
y(t) e(t)
z(t) = e(t)
Figure 22.2. Integr