Automatic control and mixed sensitivity H∞ control

Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Automatic control and mixed sensitivity H∞
control
René Galindo Orozco
FIME - UANL
CINVESTAV-Monterrey, 20 de Febrero de 2008

Contents
1 Automatic control
1 Basic notions
2 Degree of mechanization
3 History
4 Class of systems
2 Mixed sensitivity H∞ control
1 Motivation
2 Basic notions
3 Background
4 Loop-shaping
5 Mixed sensitivity
6 Standard solutions
7 Parity interlacing property
1 Mixed sensitivity H∞ control in a
non conventional scheme
1 A mixed sensitivity problem
2 Direct solutions
3 Controllers
4 Tuning procedure
5 Benchmark of a mechanical
system
6 Conclusions

Basic notions
Automatic control [Wikipedia]
A research area and theoretical base for mechanization and
automation , employing methods from mathematics and
engineering
Mechanization
Machinery to assist human operators with the physical
requirements
Automation (ancient Greek: = self dictated) [Salvat
encyclopedia]
Control systems for industrial machinery and
processes, replacing human operators
Reduces the need for human sensory and mental
requirements

Basic notions
Automatic control [Wikipedia]
A research area and theoretical base for mechanization and
automation , employing methods from mathematics and
engineering
Theory that deals with inﬂuencing the behavior of dynamic
systems
Mechanization
Machinery to assist human operators with the physical
requirements
Automation (ancient Greek: = self dictated) [Salvat
encyclopedia]
Control systems for industrial machinery and
processes, replacing human operators
Reduces the need for human sensory and mental
requirements

Basic notions
Components [Wikipedia]
u (t) -
f ( )
O& O
O
%
#&
O
"
O
"
-y (t)
?
d (t)
System , set of interacting
entities, real or abstract,
forming an integrated whole
Sensor , measure some physical state
Controller , manipulates u (t) to obtain the desired y (t)
Actuator effect a response under the command of the controller
Reference or set point , a desired y (t)

Controller improved by J. Watt

Basic notions
Control objectives
1 Regulation. Ex. controller improved by Watt
lim
t!∞
x (t) ! 0, or lim
t!∞
[x (t) xd] ! 0, xd 2 <
2 Tracking or servo. Ex. radar
lim
t!∞
x (t) ! xd (t)
3 Model matching
4 Input / output decoupling
5 Disturbance rejection or attenuation, etc.

Basic notions
Type of controller
Open-loop controller
-r
K(s) -L
di
?u- P∆ (s) -L
do
? -y
Can not compensate d (t), ∆ and noise
Closed-loop controller
r
-
L
-
e
K(s) -
L
di
?u
- P∆ (s) -
L
do
?
-
y
?L dm
6
1
6
Feedback on the performance allows the controller to dynamically
compensate for d (t), ∆ and noise

Basic notions
Feedback [Wikipedia]
Basic mechanism by which systems maintain their equilibrium or
homeostasis
Types of feedback
Negative, tends to reduce output,
Positive, tends to increase output, or
Bipolar.

por ﬂotante
4.pdf

Kalman decomposition

Degree of mechanization
[Salvat encyclopedia]

History
[Wikipedia]
Ktesibios, -270 Float regulator for a water clock
Philon, -250 Keep a constant level of oil in a lamp
In China, 12th cen-
tury
South-pointing chariot used for navigational
purposes
14th century Mechanical clock
1588 Mill-hopper, a device which regulated the ﬂow
of grain in a mill
C. Drebbel, 1624 An automatic temperature control system for a
furnace
P. de Fermat, 1600’s Minimum-time principle in optics
D. Papin, 1681 A safety valve for a pressure cooker
Bernoulli, 1696
Principle of Optimality in connection with the
Brachistochrone Problem
T. Newcomen, 1712 Steam engine
E. Lee, 1745 Fantail for a windmill

curve
7.jpg
t12 =
R P2
P1
s
1 + (y0)2
2gy
dx
Brachistochrone curve

J. Brindley, 1758 Float valve regulator in a steam boiler
I.I. Polzunov, 1765 A ﬂoat regulator for a steam engine
Pontryagin, Boltyan-
sky, Gamkrelidze, and
Mishchenko 1962
On/off relay control as optimal control
L. Euler (1707-1783)
Calculus of variations . System moves in such
a way as to minimize the time integral of the
difference between the kinetic and potential
energies
W. Henry, 1771 Sentinel register
Bonnemain, 1777 A temperature regulator suitable for indus-
trial use
J. Watt, 1788,
#Industrial revolu-
tion
Centrifugal ﬂyball governor
A.-L. Breguet, 1793 A closed-loop feedback system to synchro-
nize pocket watches

R. Delap, M. Murray, 1799 Pressure regulator
Boulton, Watt, 1803 Combined a pressure regulator with a
float regulator
I. Newton (1642-1727),
G.W. Leibniz (1646-1716),
brothers Bernoulli (late
1600’s, early 1700’s), J.F.
Riccati (1676-1754)
Infinitesimal calculus
G.B. Airy, 1840 A feedback device for pointing a tele-
scope. Discuss the instability of closed-loop
systems . Analysis using differential equa-
tions
J.L. Lagrange (1736-
1813), W.R. Hamilton
(1805-1865)
Motion of dynamical systems using differ-
ential equations
C. Babbage, 1830 Computer principles
J.C. Maxwell, 1868, I.I.
Vishnegradsky, 1877,
"Prehistory
Analyzed the stability of Watt’s flyball
governor, Re froots (G (s))g < 0

E.J. Routh, 1877, A.
Hurwitz, 1895
Determining when a characteristic equation
has stable roots. Generalized the results of
Maxwell for linear systems
A.B. Stodola, 1893 Included the delay of the actuating mecha-
nism. System time constant
Lyapunov, 1892 Stability of nonlinear differential equations
using a generalized notion of energy
O. Heaviside, 1892-
1898
Operational calculus. The transfer function
P.-S. de Laplace
(1749-1827), J. Fourier
(1768-1830), A.L.
Cauchy (1789-1857)
Frequency domain approach
Wright Brothers, 1903,
"Primitive period
Successful test ﬂights
C.R. Darwin Feedback over long time periods is responsi-
ble for the evolution of species

E.A. Sperry, 1910
Gyroscope
J. Groszkowski
Describing function approach
H. S. Black, 1927 Apply negative feedback to electrical ampliﬁers
H. Nyquist, 1930s Regeneration theory for the design of stable am-
pliﬁers. Nyquist stability criterion for feedback
systems
A. Einstein The motion of systems occurs in such a way as
to maximize the time, in 4-D space-time
H.W. Bode, 1938
Magnitude and phase frequency response plots.
Closed-loop stability using gain and phase margin
N. Minorsky, 1922 Proportional-integral-derivative (PID ) con-
troller. Nonlinear effects in the closed-loop
system
A. Rosenblueth
and N. Wiener,
1943
Set the basis for cybernetics

plot
9.jpg
Nyquist plot

V. Volterra, 1931 Explained the balance between two populations
of ﬁsh using feedback
H.L. Házen, 1934 Theory of Servomechanisms
A.N. Kolmogorov,
1941
Theory for discrete-time stationary stochastic
processes
N. Wiener, 1942 Statistically optimal ﬁlter for stationary
continuous-time signals
A.C. Hall, 1946 Confront noise effects in frequency-domain
N.B. Nichols, 1947
Nichols Chart
Ivachenko, 1948
Relay control
W.R. Evans, 1948
Root locus technique
J. R. Ragazzini,
1950s Digital control and the z-transform
Tsypkin, 1955 Phase plane for nonlinear controls

J. von Neumann, 1948 Construction of the IAS stored-program
Sperry Rand, 1950 Commercial data processing machine, UNI-
VAC I
R. Bellman, 1957 Dynamic programming to the optimal con-
trol of discrete-time systems
L.S. Pontryagin, 1958 Maximum principle
C.S. Draper, 1960,
"Classical period
#Modern period
Inertial navigation system
V.M. Popov, 1961
Circle criterion for nonlinear stability analysis
Kalman, 1960’s Linear quadratic regulator (LQR ). Discrete
and continuous Kalman ﬁlter . Linear algebra
and matrices. Internal system state
G. Zames, 1966, I.W.
Sandberg, K.S. Naren-
dra, Goldwyn, 1964,
C.A. Desoer, 1965
Nonlinear stability

W. Hoff, 1969 Microprocessor
J.R. Ragazzini, G.
Franklin, L.A. Zadeh,
C.E. Shannon, 1950’s, E.I.
Jury, 1960, B.C. Kuo, 1963
Theory of sampled data systems
Åström, Wittenmark,
1984
Industrial process control
Gelb, 1974 Digital ﬁltering theory
H.H. Rosenbrock, 1974,
A.G.J. MacFarlane, I.
Postlethwaite, 1977
Extend frequency-domain techniques to
multivariable systems. Characteristic lo-
cus, diagonal dominance and the inverse
Nyquist array
I. Horowitz, 1970’s Quantitative feedback theory
J. Doyle, G. Stein, M.G.
Safonov, A.J. Laub, G.L.
Hartmann, 1981
Singular value plots in robust multivariable
design

Class of systems
8
>>>>>>>>>>>>>><
>>>>>>>>>>>>>>:
Non-causal or
anticipative or predictive
f
Causal or
non-anticipative
8
>>>>>>>>>><
>>>>>>>>>>:
Stochastic
with noise
f
Deterministic
8
>>>>>><
>>>>>>:
Static or
without memory
f
Dynamic
8
>><
>>:
Distributed
Parameters
f
Lumped
Parameters
f
8
>><
>>:
Non-linear f
Linear
8
<
:
Discrete f
Continuous
Time varying
Time invariant

Motivation
Robust : some property is preserved
Uncertainty ∆
x (t0)
u (t)
; y (t)
∆ always exists due to frequency dependent elements,
unmodeled dynamics and failures
Inﬁnity norm kargk∞
kargk∞ := sup
w
σ (arg)

Motivation
Uncertainty ∆
x (t0)
u (t)
; y (t)
Zames in 1981 describes ∆ (s) in the frequency domain as
classical control
kargk∞ := sup
w
σ (arg)

Motivation
Uncertainty ∆
x (t0)
u (t)
; y (t)
Zames in 1981 describes ∆ (s) in the frequency domain as
classical control
kargk∞ := sup
w
σ (arg)
kargk∞ is “good” for specifying the ∆ level and the effect of
kd (t)k2 < ∞ over ky (t)k2

Basic notions
Robust H∞ control objective
rLe
K(s)
-
w- G (s)
z-
- e∆ (s)
?
1
?
Uncertainty and kw (t)k2 attenuation
in a bandwidth, over kz (t)k2,
guaranteeing stability
Minimize,
J := kz (t)k2 :=
Z ∞
∞
z2
(t) dt
1/2
For linear time invariant systems, minimize
J := kTzew (s)k∞ := supew(t):kew(t)k 1 σ (Tzew), or J := σ (Tzew (s))

Basic notions
Uncertainty models8
>>>>>>>>>>>>>>>>>><
>>>>>>>>>>>>>>>>>>:
Structured or parametric
Finite number of uncertainty parameters
Ex. diag k∆1 (s)k∞ , ..., ∆q (s) ∞
diag m1, ..., mq
Non structured
The frequency response is in a set 8w
Ex.: a) phase and gain margins
b) k∆ (s)k∞ m
u - P(s) -L
-y
d
?
- ∆a

D.C. Youla, H. A.
Jabr, J.J. Bongiorno;
1976
Explicit formula for the optimal controller based
on a least-square Wiener-Hopf minimization of a
cost functional
C.A. Desoer, R. Liu,
J. Murray, R. Saeks;
1980
Controllers placing the feedback system in a ring
of operators with the prescribed properties. The
plant is modeled as a ratio of two operators in that
ring
M. Vidyasagar, H.
Schneider, B.A.
Francis, 1982
Necessary and sufﬁcient conditions for a given
transfer function matrix to have a coprime factor-
ization . Characterization of all stabilizing compen-
sators
C.N. Nett, C.A. Ja-
cobson, M. J. Balas;
1984
Give explicit formulas for a doubly coprime frac-
tional representation of the transfer function in
state-space
K. Glover, D. Mc-
Farlane, 1989
An optimal stability margin. Characterization
of all controllers satisfying a suboptimal stabil-
ity margin, in state-space

Loop-shaping
d (t), usually in w < wl
σ (P∆ (s)) " and any phase of P∆ (s), in w > wh =)worst ∆ (s) in
w > wh
σ (arg) gives a measure of the “gain” of ∆ (s) or d (t)
Stabilize the system under ∆ (s) ,
¯σ (To (s)) # ()
_
σ (Lo (s)) # , in w > wh
Regulation of y (t) under d (t) ,
_
σ(So (s)) # () σ (Lo (s)) " , in w < wl
9
>>>>>>>>=
>>>>>>>>;
Loop-shaping

Loop-shaping
W2 (s) and W1 (s) high and low pass weightings
wl and wh depends on
the speciﬁc application
the knowledge of d (t) and ∆ (s)
the Bode Phase-Gain Relation

Mixed sensitivity
r
-
L
-
e
K(s) -
L
di
?u
- P∆ (s) -
L
do
?
-
y
?L dm
6
1
6
Robust stability
By the small gain theorem if e∆ (s)
∞
< 1, stability is guaranteed if,
W2 (s) Tu∆y∆
(s) ∞
< 1
Robust performance
kW1 (s) So (s)k∞ < 1, minK(s) ke (t)k2
So (s) = Tdoy (s) = Ter (s) = (I + P (s) K (s)) 1
: output sensitivity
P∆ (s) : uncertain plant

Mixed sensitivity
Minimize
kW1 (s) So (s)k∞ and W2 (s) Tu∆y∆
(s) ∞
in the frequency range in which kd (t)k2 and k∆ (s)k∞ are
signiﬁcative by a stable K (s) designed for P (s), guaranteeing robust
performance and stability, i.e. minimize,
J1 :=
W1 (s) So (s)
W2 (s) Tu∆y∆
(s) ∞
Uncertainty model Tu∆y∆
(s)
Additive K (s) So (s)
Multiplicative at the output To (s) := So (s) P (s) K (s)
Feedback at the input So (s) P (s)

Standard solutions
General Standard H∞ Optimal Problem
[Doyle, 1981], [Glover, 1984], [Francis, Doyle, 1987], [Chiang, Safonov,
1997]
K (s) stabilizing P (s), and minimizing, J := sup
w:kwk2
2 k
kz (t)k2,
J = kTzw (s)k∞
[Nett, Jacobson, Balas, 1984]
Formula for the YJBK-parametrization,
using static state feedback to stabilize P (s)
=)
Recursive
procedures

Parity interlacing property
P (s) 2 L
pxm
∞ is strongly stabilizable ()
the unstable poles of P (s) between every
even real and unstable zeros of P (s), is even
9
=
;
) 9K (s) 2 RH∞
Strong stability )
8
<
:
For loop breaking
For closed-loop bandwidth "

Problem
J1 is transformed into [Galindo, Malabre, Kuˇcera, 2004]:
J2 :=
Sol
Tu∆y∆h ∞
R (s) is ﬁxed solving a MSP without an augmented system
Sol and Tu∆y∆h becomes real matrices
J2 involves the simultaneous minimization of kSolk∞ and Tu∆y∆h ∞
,
min
K(s)
kSolk∞
subject to kSolk∞ = Tu∆y∆h ∞
that is equivalent to minimize the Lagrange function [Galindo,
Herrera, Martínez, 2000],
f := kSolk∞ η kSolk∞ Tu∆y∆h ∞
η Lagrange multiplier

Direct solutions
[Galindo, Sanchez, Herrera, 2002]
Suppose that det f(s + a) In R (s)g is a Hurwitz polynomial, R (s) 2 <H∞
Deﬁne
X (s) = eX (s) = aIn + A 2 <H∞
Y (s) = eY (s) = In 2 <H∞
eNp (s) = Np (s) =
1
s + a
In 2 <H∞
eDp (s) = Dp (s) =
1
s + a
(sIn A) 2 <H∞
NpD 1
p =
1
s + a
1
s + a
(sIn A)
1
XNp + YDp = (aIn + A)
1
s + a
+
1
s + a
(sIn A) = In
eNp (s), Np (s), eDp (s) and Dp (s) are of low order ) less
computational effort

Direct solutions
[Galindo, Sanchez, Herrera, 2002]
Then, a proper stabilizing K (s) 2RH∞ is:
K (s) = A + [(s + a) In R (s)] 1
[(s + a) aIn + R (s) s]
and,
kSolk∞ = 1
a2 k(aIn Rl) Ak∞
kKhSohk∞ = kA + aIn + Rhk∞
kTohk∞ = 1
wh
kA + aIn + Rhk∞
kSohPhk∞ = 1
wh
kSolk∞ # by a ", and kTohk∞ # by wh "
For P (s) strictly proper Toh = Loh
Select rIn for Rh and Rl, and r < a,
kSolk∞ =
a r
a2
kAk∞
A solution of kTohk∞ = kSolk∞ for Tu∆y∆h = SohPh is,
re = a 1
a
wh kAk∞
(1)

Direct Solutions
9 [ a, a] in which kTohk∞ # (") and kSolk∞ " (#) as linear functions of
r [Galindo, Malabre, Kuˇcera, 2004],
-
a
6
1
wh
kA + aInk∞
1
wh
kA + 2aInk∞
1
a kAk∞
@
@
@
@
@
@
@
@
@
@
@
@
a r
kSolk∞
kTohk∞

Direct solutions
[Galindo, Malabre, Kucera, 2004]
Let rIn be for R (s) 2 <H∞
Then, a value for r is:
r = a 1
γmina
(wh + 1) kAk∞
where
γmin = [1 + λmax (YX)]1/2
being Y and X the solutions of the Riccati equations
ATX + XA X2 + In = 0
AY + YAT Y2 + In = 0
The optimal value for r lies in,
r 2 [rb, a]
and a lower bound rb for r is:
rb =
a (wh a) kAk∞ a2
wh kAk∞ + a2
lim
wh !0
rb = (a + kAk∞), lim
wh !∞
rb = a

Let rIn be for R (s) 2 <H∞
Then, an optimal value for r is:
re =
b2 b1
m1 m2
where
b1 :=
1
a
kAk∞ , m1 :=
1
a2
kAk∞
b2 :=
1
wh
kA + aInk∞
m2 :=
1
awh
(kA + 2aInk∞ kA + aInk∞)
Moreover
kSolk∞ =
kA + 2aInk∞ kAk∞
wh kAk∞ + a (kA + 2aInk∞ kA + aInk∞)

Controllers
˙x (t) = Ax (t) + Bu (t)
Deﬁne v1 (t) := Bu (t) ) ˙x (t) = Ax (t) + v1 (t)
+
u (t) = BLv1 (t) ) ˙x (t) = Ax (t) + BBLv1 (t)
) E1 := BBL In
xd-L
- K1(s)
v1- BL -u
(sIn A) 1
B -L
d1
? -x
?L d2
6
1
6
A 2 <n n, B 2 <n m, C 2 <p n
v1 (t) the output of the precompensator K1 (s)
BL a left inverse of B
xd (t) the input reference

Controllers
Let A, BBL, C ,
u (s) = BL
K1 (s) (xd (s) x (s))
Dual system AT, CT, BBL T
,
u (s) = CT
L
KT
2 (s) (y (s) by (s))
In original coordinates,
u (s) = K2 (s) CR
(y (s) by (s))
x
-
L ξ
- C - CR - K2(s) - BL -
v2
(sIn A) 1
B -
bx
6
1
6
CR a right inverse of C
v2 (t) the output of K2 (s)
bx (t) and by (t), the estimated state and output

Controllers
(
bx (t) = Abx (t) + v2 (t)
by (t) = Cbx (t)
Deﬁne
bφ (t) := bx (t)
)
(
bx (t) = Abx (t) + v2 (t)
bφ (t) = bx (t)
+
bx (t) = CRby (t)
)
(
bx (t) = Abx (t) + v2 (t)
bφ (t) = CRCbx (t)
) E2 := CRC In

Controllers
Nominal plant P (s)
(A, B, C) a stabilizable and detectable realization of P (s) satisfying
the parity interlacing property
A =
A11 A12
A21 A22
B =
0
B1
C = C eC
B1 2 <m m non-singular
For P (s) proper, transform quadruples into and extended triples
[Basile, Marro, 1992]

Controllers [Galindo, 2006]
exd-L e- K1(s)
v1- BL -
?
u
P(s) -L
d1
? -y
?L
?
d2
6
1
6
L
- (sIn A) 1
B
bx- C - 1 -L
CRK2(s)
v2
BL
6
bx (t) : estimated state
e1 (t) : deviation from the desired state trajectory xd (t)
exd (s) := Wr (s) xd (s) : filtered state reference
Satisfies the separation principle
Allows to get a stable H∞ compensator
bxss = xss, lim
ri!ai
xss ! xdss, Toh ! 0
The class of systems depends of the observability in closed loop
Some of the poles are fixed. For eC = 0, det (sIn m A11) must be

Controllers
[Galindo, 2007]
exd-
L e1- K1(s)
v1 -
?
BL -
u
P(s) -
L
d1
?
-
y
?L
?
d2
6
1
6
L
- (sIn A) 1 bx- C - 1 -L
CR
e2
K2(s)
v2
6
A simplified version of the one of [Galindo, 2006]
The separation principle is not satisfied
Ki (s), become PI as ri ! ai, low complexity controllers
The closed loop poles depends on the selection of the free
parameters of BL and CR, and the rest s = ai, are stable poles
Some of the poles are fixed. For eC = 0, det (sIm A22) and for

Tuning procedure
Tuning Procedure [Galindo, Malabre, Kuˇcera, 2004]
For a desired time response, attenuation of kd1 (t)k2, i.e., for a given a1
1 Select a2 = a1 > 0
2 Find the largest free parameters of BL and CR, and the lowest wh,
satisfying
a) The stationary state error specifications,
b) ri ai (1 ai), i = 1, 2,
c) and minimizing kE1 (ρ1A + In)k∞ and k(ρ2A + In) E2k∞
ρi :=
(ai ri) /a2
i if ri 6= ai
wl/a2
i if ri = ai
wl a fixed frequency in the low frequency bandwidth of Ki (s)
3 If possible select xd 2 Im B to assure that lim
ri!ai
bxss ! xss
4 If needed, use a pre-filter Wr (s) for the reference

Controllers
[Galindo, 2007]
exd
- φ(s)
?- C -L e- K2(s)CR -L
- V(s) -u
P(s) -L
d1
? -y
?L d2
6
1
6
V (s) := σ (s) BLK1 (s) Γ 1 (s), Γ (s) := In + σ (s) K2 (s) CRC,
σ (s) := s+a1 r1
(s+a1)2 , Φ (s) = sIn A
For P∆ (s), we must satisfy also the Small Gain Theorem
Tu∆y∆
(s) does not depend on exd (t), indeed Tu∆y∆
(s) becomes
K (s) So (s), To (s) := P (s) K (s) So (s), and So (s) P (s), where
K (s) = V (s) K2 (s) CR

Controllers
Assignment of part of the poles by a change of basis
Let a change of basis, [Galindo, 2007]
T =
Iq 0
T21 Iq
, T 1 =
Iq 0
T21 Iq
preserving the structure of B, where q m, and
A =
eA11
eA12
eA21
eA22
be partitioned accordingly with the block partition of T. So,
A = TAT 1 ="
eA11
eA12T21
eA12
T21
eA11 + eA21 T21
eA12 + eA22 T21 T21
eA12 + eA22
#
Then,
T21 = eAR
12
eA11 Λ11
assigns a desired dynamics Λ11 to A11, and,
T21 = Λ22
eA22
eAL
12
assigns a desired dynamics Λ22 to A22

Mixed sensitivity
[Galindo, 2007] The norm-∞ of Tu∆y∆h is,
kKhSohk∞ = 1
wh
BLD1D2CR
∞
kTohk∞ = 1
w2
h
CBBLD1D2CR
∞
kSohPhk∞ = 1
wh
kCBk∞
Di := A + (ai + ri) In, i = 1, 2.

Mixed sensitivity
[Galindo, Malabre, Kuˇcera, 2004] lim
s!0
lim
ρi!0
So (s)
∞
=
wl
a2
i
kAk∞
Robust stability is achieved
ai # , but the performance is ameliorated
wh " , but the high frequency bandwidth is decreased
Tuning Procedure
1 Look for the highest values of ai, i = 1, 2, satisfying stability
conditions, minimizing Tu∆y∆h ∞
and satisfying plant input
speciﬁcations
2 Fix the value of the free parameters of BL and CR
3 Select wh, satisfying stability conditions, stationary state error
speciﬁcations and minimizing Tu∆y∆h ∞

Benchmark of a Mechanical System
-
u
m1
7 ! x1(t)
k
b
m2
7 ! x2(t)
-
d
Consider a model of a mechanical system,
˙x (t) = Ax (t) + Bu (t) + Ψd (t)
y (t) = Cx (t)
where x (t)T
:= x1 (t) x2 (t) ˙x2 (t) ˙x1 (t) ,
A =
2
6
6
6
4
0 0 0 1
0 0 1 0
k
m2
k
m2
b
m2
b
m2
k
m1
k
m1
b
m1
b
m1
3
7
7
7
5
, B =
2
6
6
4
0
0
0
1
m1
3
7
7
5 , Ψ =
2
6
6
4
0
0
1
m2
0
3
7
7
5
C = 0 1 0 0

Non-collocated case,
The control input acts only on one uncertainty mass 0.1 m1 3
and the output is the position of m2
The nominal value of m1 = 1
d (t) unknown disturbance
k and b the elasticity and friction coefﬁcients
m1 and m2 the mass
m2 = k = b = 1

Let a := a1 = a2
So, r := r1 = r2 which implies K1 (s) = K2 (s)
(A, B, C) is a minimal realization and B has the desired structure
P (s) satisﬁes the parity interlacing property
eC = 0, det (sIm A22) = s + 1 is Hurwitz
A desired dynamics Λ22 =diagf 2, 2g, and T with q = 2 is
realized, getting,
A =
2
6
6
4
1 1 0 1
1 1 1 0
1 3 2 0
3 1 0 2
3
7
7
5 , B = B, C = C
det sIm A22 = s + 2 is Hurwitz

Select a = 2 and,
BL = g1 0 0 1
CR = g2 1 0 g3
T
g1 = 1.6, g2 = 0.3, g3 = 0.55
CB = 0 =) kTohk∞ = 0 and kSohPhk∞ = 0,
wh kKhSohk∞ kKhSohk∞ kKhSohk∞
with r with rb with re
1 1.9 stable 0.05 unstable 0.909 unstable
3 0.675 stable 0.487 unstable 0.57 unstable
5 0.412 unstable 0.35 unstable 0.377 stable
10 0.209 unstable 0.195 stable 0.201 stable
12 0.175 unstable 0.165 stable 0.169 stable
15 0.14 unstable 0.134 stable 0.136 unstable
20 0.105 unstable 0.102 unstable 0.103 unstable

Select wh = 3 rad/sec., wh = 17 rad/sec. and wh = 12 rad/sec.
=) r = 1.61, rb = 1.622, and re = 1.631
with r with rb with re
kE1 (ρ1A + In)k∞ 2.067 2.052 2.042
k(ρ2A + In) E2k∞ 1.627 1.625 1.623
The characteristic polynomials det (sI AK) of the overall
compensators are stables
Tol = CclA 1
cl Bcl =) exd (t) = (1/To2l) xd (t)
xd (t) = 0 yd 0 0
T
/2 Im B

yd = 5, under d (t) = 0.1 (sin (10t) + sin (100t))

x2 (t) tracks the reference signal with r, rb and re, under the d (t)
and the variation of the parameter m1
d (t) remains as very small oscillations at y (t)
Sinusoidal functions of frequencies over wh = 3 rad/sec.,
wh = 17 rad/sec. and wh = 12 rad/sec. for r, rb and re, are well
attenuated at y (t)
Bigger time response with re and less control energy, the contrary
with r, and rb in the middle
Smooth control energy
As m1 ", more energy is required, and the peaks and frequency
of the oscillations decrease

Conclusions
1 A methodology to design a mixed sensitivity H∞ compensator
for a LTI MIMO plant is proposed
2 A nominal compensator is designed for the nominal plant
solving a mixed sensitivity H∞ problem, in a non-conventional
observer-compensator scheme
3 A mixed sensitivity H∞ control law and necessary and sufﬁcient
stability conditions are given
4 Good performance guaranteeing stability, in spite of the
uncertainties and of the external disturbances that are attenuated
5 The controllers with r, rb and re have good performance and their
selection depends on the desired time response and the plant
input speciﬁcations
6 An analytic or a numerical method replacing the tuning
procedure is still an open problem

Thank you

Automatic control and mixed sensitivity H∞ control

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