Du C., Xie L. Modeling and Control of Vibration in Mechanical Systems
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94 Modeling and Control of Vibration in Mechanical Systems
FIGURE 5.1
Configuration of standard optimal control.
where
F = −(D
T
12
D
12
)
−1
(D
T
12
C
1
+ B
T
2
X
2
),
K = −(Y
2
C
T
2
+ B
1
D
T
21
)(D
21
D
T
21
)
−1
. (5.33 )
The minimal H
2
norm of the transfer function T
zw
is given by
kT
zw
k
2
=
q
T race(B
T
1
X
2
B
1
) + T race[(A
T
X
2
+ X
2
A + C
T
1
C
1
)Y
2
].(5.34)
If the conditions (1)−(4) in Assumption 5.1 are not satisfied, the so-called pertur-
bation method is applied [7] so that the above d esign method to find an appropr iate
controller is still appl icable.
5.3.2 Discrete-time case
Consider t he discrete-time linear time-invariant system P (z) with the following state-
space representat ion:
x(k + 1) = Ax(k) + B
1
w(k) + B
2
u(k), (5.35)
z(k) = C
1
x(k) + D
11
w(k) + D
12
u(k), (5.36 )
y(k) = C
2
x(k) + D
21
w(k) + D
22
u(k), (5.37 )
where x ∈ R
n
x
is the state, y ∈ R
n
y
is the measurement output, z ∈ R
n
z
is the
controlled output, w ∈ R
n
w
is the disturbance input, u ∈ R
n
u
is the control input ,
and A, B
1
, B
2
, C
1
, D
11
, D
12
, C
2
, and D
21
are of appropri at e dimensions. D
22
= 0
is also assumed.
Introduction to Optimal an d Robust Control 95
Intr oduce the following dynamic output feedback controller C(z):
x
c
(k + 1) = A
c
x
c
(k) + B
c
y(k), (5.38)
u(k) = C
c
x
c
(k) + D
c
y(k). (5.39 )
Denote ξ = [x
T
x
T
c
]
T
. From (5.35)−(5.37) and (5.38)−(5.39), the closed-loop
system is given by
ξ(k + 1 ) =
¯
Aξ(k) +
¯
Bw(k), (5.40)
z(k) =
¯
Cξ(k) +
¯
Dw(k), (5.41 )
where
¯
A =
A + B
2
D
c
C
2
B
2
C
c
B
c
C
2
A
c
,
¯
B =
B
2
D
c
D
21
+ B
1
B
c
D
21
, (5.42)
¯
C = [C
1
+ D
12
D
c
C
2
D
12
C
c
] ,
¯
D = D
12
D
c
D
21
+ D
11
. (5.43)
The discrete-time H
2
control problem i s to find a proper, real rational cont roller
C(z) that stabilizes P (z) internally and minimi zes the H
2
norm of the transfer func-
tion matrix T
zw
(z) from w to z of the closed-loop system (5.40) −(5.41).
The counterpart of the Riccati equations (5.30)−(5.31) for discrete-time systems
is as foll ows.
A
T
X
2
A −(A
T
X
2
B
2
+ C
T
1
D
12
)(D
T
12
D
12
+ B
T
2
X
2
B
2
)
−1
(A
T
X
2
B
2
+ C
T
1
D
12
)
T
+C
T
1
C
1
= 0, (5.44)
AY
2
A
T
− (AY
2
C
T
2
+ B
1
D
T
21
)(D
21
D
T
21
+ C
2
Y
2
C
T
2
)
−1
(AY
2
C
T
2
+ B
1
D
T
21
)
T
+B
1
B
T
1
= 0 . (5.45 )
A discrete-time H
2
optimal controller can then be obtained as (5.32). And the
minimal H
2
norm of the transfer fun ction T
zw
is given b y
kT
zw
k
2
=
q
T race(B
T
1
X
2
B
1
) + T race[(A
T
X
2
A + C
T
1
C
1
)Y
2
]. (5.46)
A parametrizati on of all H
2
controllers is developed i n terms of LMIs as in the
following theorem which linearizes th e H
2
norm conditions ( 5.14)−(5.15) for syn-
thesis.
THEOREM 5.1
[90] Consider system (5.35)−(5.37). There exists a controller (5.38)−(5.39)
such that k T
zw
k
2
2
< µ if and only if the followi ng linear matrix inequalities
and equali ty admit a solution:
T race(Π) < µ, (5.47 )
96 Modeling and Control of Vibration in Mechanical Systems
Π C
1
X + D
12
E C
1
+ D
12
D
c
C
2
∗ X + X
T
−P
2
I + Z
T
−J
∗ ∗ Y + Y
T
− H
> 0, (5.48 )
P
2
J AX + B
2
E A + B
2
D
c
C
2
B
1
+ B
2
D
c
D
21
∗ H U Y A + W C
2
Y B
1
+ W D
21
∗ ∗ X + X
′
−P
2
I + Z
′
−J 0
∗ ∗ ∗ Y + Y
T
− H 0
∗ ∗ ∗ ∗ I
> 0, (5.49)
and
D
11
+ D
12
D
c
D
21
= 0, (5 .50)
where ∗ denotes an entr y t hat can be deduced from the symmetry of the matrix,
the matrices X, E, Y , W , U, D
c
, Z, J, and the symmet ric matri ces P
2
, H
and Π are the variables. A feasible H
2
controller is obtained by choosing N
1
and M
1
nonsingular such that N
1
M
1
= Z − Y X and calculating
C
c
= (E − D
c
C
2
X)M
−1
1
, D
c
= D
c
, (5.51 )
B
c
= N
−1
1
(W − Y B
2
D
c
), (5.52 )
A
c
= N
−1
1
[U −Y (A + B
2
D
c
C
2
)X − N
1
B
c
C
2
X − Y B
2
C
c
M
1
]M
−1
1
. (5.53)
5.4 H
∞
control
5.4.1 Continuous-time case
We consider the continuous-time system P described b y (5.21)−(5.23), and the class
of causal, linear, time-invariant and finite-dimensional controllers that internally sta-
bilize P , or namely, all admissible controllers for P . Our aim is to find an admissible
controller C such that the closed-loop sy stem T
zw
satisfies
kT
zw
k
∞
< γ. (5.54 )
Assumption 5.2
(1). The pair (A, B
2
) is stabilizable and th e pair (A, C
2
) is detectable.
(2). The matrices D
12
and D
21
satisfy D
T
12
D
12
= I
n
u
and D
21
D
T
21
= I
n
y
.
(3).
rank
A − jωI B
2
C
1
D
12
= n
x
+ n
u
, for all real ω. (5.55)
(4).
rank
A − jωI B
1
C
2
D
21
= n
x
+ n
y
, for all real ω. (5.56 )
Introduction to Optimal an d Robust Control 97
The assumptio n that (A, B
2
, C
2
) is stabilizable and detectable is necessary and
sufficient for the existence of admissible controllers. The full rank assumptions (3)
and (4 ) are necessary for the existence of stabilizing solutions to t he Riccati equations
that are used to obtain the solution to th e H
∞
control problem.
Define the matri ces
ˆ
A,
ˆ
B,
˜
A and
˜
C as
ˆ
A = A − B
1
D
T
21
C
2
,
ˆ
B
ˆ
B
T
= B
1
(I − D
T
21
D
21
)B
T
1
, (5.57)
˜
A = A − B
2
D
T
12
C
1
,
˜
C
T
˜
C = C
T
1
(I − D
12
D
T
12
)C
1
. (5.58)
Suppose that U
y
maps y to u and has a minimal realization (A
u
, B
u
, C
u
, D
u
)
satisfyi ng det(I − D
22
D
u
) 6= 0 for a well-posed closed loop. Then a realization in
terms of LFT is given by
LF T (P, U
y
) =
A + B
2
D
u
MC
2
B
2
(I + D
u
MD
22
)C
u
B
1
+ B
2
D
u
MD
21
B
u
MC
2
A
u
+ B
u
MD
22
C
u
B
u
MD
21
C
1
+ D
12
D
u
MC
2
D
12
(I + D
u
MD
22
)C
u
D
11
+ D
12
D
u
MD
21
, (5.59)
with M = (I − D
22
D
u
)
−1
.
THEOREM 5.2
[71] Consider the system (5.21)−(5.23) satisfying Assumption 5.2. There
exists an admissible C such that the closed-loop system (5.26)−(5.27) satisfies
(5.54) if and only if
1. There is a solution X
∞
≥ 0 t o the algebraic Riccati equation
X
∞
˜
A +
˜
A
T
X
∞
− X
∞
(B
2
B
T
2
− γ
−2
B
1
B
T
1
)X
∞
+
˜
C
T
˜
C = 0, (5.60)
such that
˜
A −(B
2
B
T
2
− γ
−2
B
1
B
T
1
)X
∞
is asymptotically stable.
2. There is a solution Y
∞
≥ 0 to the algebraic Riccati equation
ˆ
AY
∞
+ Y
∞
ˆ
A
T
−Y
∞
(C
T
2
C
2
− γ
−2
C
T
1
C
1
)Y
∞
+
ˆ
B
ˆ
B
T
= 0, (5.61)
such that
ˆ
A −Y
∞
(C
T
2
C
2
−γ
−2
C
T
1
C
1
) is asymptotically s table.
3. ρ(X
∞
Y
∞
) < γ
2
.
In the case when these conditions hold, C i s an admissible controller satis-
fying (5.54) if and only if C is given by the LFT
C = LF T (C
a
, U
y
), kU
y
k
∞
< γ, (5.62)
where U
y
is a stable transfer function. The generator C
a
is given by
C
a
=
A
k
B
k1
B
k2
C
k1
0 I
C
k2
I 0
, (5.63)
98 Modeling and Control of Vibration in Mechanical Systems
where
A
k
= A + γ
−2
B
1
B
T
1
X
∞
− B
2
F
∞
− B
k1
C
2z
, (5.64)
B
k1
B
k2
=
B
1
D
T
21
+ Z
∞
C
T
2z
B
2
+ γ
−2
Z
∞
F
T
∞
, (5.65)
C
k1
C
k2
=
−F
∞
−C
2z
, (5.66 )
C
2z
= C
2
+ γ
−2
D
21
B
T
1
X
∞
, F
∞
= D
T
12
C
1
+ B
T
2
X
∞
, (5.67 )
Z
∞
= Y
∞
(I − γ
−2
X
∞
Y
∞
)
−1
= (I − γ
−2
Y
∞
X
∞
)
−1
Y
∞
. (5.68 )
A point given by Theorem 5.2 is that a solution to the H
∞
generalized regulator
problem exists if and on ly if there exist stabilizing, nonnegative definite solutions
X
∞
and Y
∞
to the algebraic Riccati equations associated with the full information
H
∞
control problem and the H
∞
estimation of C
1
x such that the coupling condition
ρ(X
∞
Y
∞
) < γ
2
is satisfied.
The optimal H
∞
control prob lem is to fin d an i nternally stabil izing controller
C(s) such that kT
zw
k
∞
of the closed-loop system (5.26)−(5.27) is minimized.
However, in practice it is often not necessary to design an optimal controller, and
it is usually appropriate to o btain a controller that gives ri se to an H
∞
norm of the
closed-loo p system less than a prescribed value. More specifically, a suboptimal H
∞
control pro blem is that given γ > 0, find an admissible controller C, if there is any,
such that kT
zw
k
∞
< γ.
The following theorem gives a design method for a suboptimal H
∞
output feed-
back controller.
THEOREM 5.3
[90] Consider system (5.21)−(5.23). Given a scalar γ > 0, there exists an
output feedback controller (5.24)−(5.25) such that kT
zw
k
2
∞
< γ if the following
LMI admits a solution (E, W, U, D
c
, X, Y ):
AX + XA
T
+ B
2
E + (B
2
E)
T
U
T
+ A + B
2
D
c
C
2
∗ A
T
Y + Y A + W C
2
+ (W C
2
)
T
∗ ∗
∗ ∗
,
B
1
+ B
2
W D
c
D
21
(C
1
X + D
12
E)
T
Y B
1
+ W D
21
(C
1
+ D
12
D
c
C
2
)
T
−γI (D
11
+ D
12
D
c
D
21
)
T
∗ −γI
< 0, (5.69)
X I
I Y
> 0 . (5.70)
In this case, a feasible H
∞
controller i s obtained from (5.51)−(5.53), where
N
1
M
1
= I − Y X.
Introduction to Optimal an d Robust Control 99
5.4.2 Discrete-time case
Assume th at the time-invariant discrete-time system (5.35)−(5.37) satisfies:
Assumption 5.3
(1). (A, B
2
, C
2
) is stabilizable and detectable.
(2). D
T
12
D
12
> 0 and D
21
D
T
21
> 0.
(3).
rank
A − e
jθ
I B
2
C
1
D
12
= n
x
+ n
u
, for all θ ∈ (−π, π]. (5.71)
(4).
rank
A −e
jθ
I B
1
C
2
D
21
= n
x
+ n
y
, for all θ ∈ (−π, π]. (5.72)
We seek a causal, linear, t ime-invariant and finite-dimensional controller C(z)
such that the closed-loop system (5.40)−(5.41) is stable and
kT
zw
k
∞
< γ, (5.73 )
or equivalently, under zero initial conditions,
kzk
2
2
− γ
2
kwk
2
2
≤ −εkwk
2
2
, (5.74)
for all w ∈ ℓ
2
[0, ∞) and some ε > 0.
Let
B =
B
1
B
2
, C
d
=
C
1
0
, (5.75 )
D
d
=
D
11
D
12
I
n
w
0
, (5.76 )
J
s
=
I
n
z
0
0 −γ
2
I
n
w
, (5.77)
J
t
=
I
n
w
0
0 −γ
2
I
n
u
. (5.78)
With the assumptions (1)−(4), a causal, linear, finite-dimensional stabil izing con-
troller that leads to kT
zw
k
∞
< γ exists if and only if the following two conditions
hold [71].
1. There exists a solution to the Riccati equation
X
∞
= C
T
d
J
s
C
d
+ A
T
X
∞
A − W
T
R
−1
W, (5.79 )
with
R =
R
1
R
T
2
R
2
R
3
= D
T
d
J
s
D
d
+ B
T
X
∞
B, (5.80 )
W =
W
11
W
21
= D
T
d
J
s
C
d
+ B
T
X
∞
A, W
21
∈ R
n
u
×n
x
, (5.81)
100 Modeling and Control of Vibratio n in Mechanical Systems
such that A − BR
−1
W is asymptotically stable and
X
∞
≥ 0 , (5.82 )
R
1
−R
T
2
R
−1
3
R
2
< 0, R
1
∈ R
n
w
×n
w
, R
3
∈ R
n
u
×n
u
. (5.83)
Denote
∇ = R
1
− R
T
2
R
−1
3
R
2
, W
∇
= W
11
− R
T
2
R
−1
3
W
21
, (5.84 )
and let E
1
be an n
u
×n
u
matrix such that
E
T
1
E
1
= R
3
, (5.85 )
and E
2
be an n
w
× n
w
matrix such that
E
T
2
E
2
= −γ
−2
(R
1
−R
T
2
R
−1
3
R
2
) = −γ
−2
∇. (5.86)
Define the syst em
A
t
B
t
C
t
D
t
=
A −B
1
∇
−1
W
∇
B
1
E
−1
2
0
E
1
R
−1
3
(W − R
2
∇
−1
W
∇
) E
1
R
−1
3
R
2
E
−1
2
I
C
2
− D
21
∇
−1
W
∇
D
21
E
−1
2
0
. (5.87)
2. There exists a solution to the Riccati equation
Y
∞
= B
t
J
t
B
T
t
+ A
t
Y
∞
A
T
t
−M
t
S
−1
t
M
T
t
, (5.88 )
where
S
t
= D
t
J
t
D
T
t
+ C
t
Y
∞
C
T
t
=
S
1
S
2
S
T
2
S
3
, (5. 89)
M
t
= B
t
J
t
D
T
t
+ A
t
Y
∞
C
T
t
=
M
t1
M
t2
, (5.90 )
such that A
t
− M
t
S
−1
t
C
t
is asymptotically stable and
Y
∞
≥ 0, (5.91 )
S
1
−S
2
S
−1
3
S
T
2
< 0. (5.92 )
Note that C
t
in (5.87) is p artitioned as
C
t
=
C
t1
C
t2
=
E
1
R
−1
3
(W − R
2
∇
−1
W
∇
)
C
2
− D
21
∇
−1
W
∇
. (5. 93)
A contro ller that achieves the objective (5.73) is given by
x
c
(k + 1) = A
t
x
c
(k) + B
2
u(k) + M
t2
S
−1
3
(y(k) − C
t2
x
c
(k)), (5.9 4)
E
1
u(k) = −C
t1
x
c
(k)1 − S
2
S
−1
3
(y(k) − C
t2
x
c
(k)). (5.95)
Introduction to Optimal an d Robust Control 101
All controllers that achieve the obj ective (5.73) are generated by the LFT C =
LF T (C
a
, U
c
), where U
c
is a l inear causal system such that kU
c
k
∞
< γ , and the
generator C
a
is given b y
I − B
2
−M
t2
(X
T
2
)
−1
0 E
1
S
2
(X
T
2
)
−1
0 0 X
2
x
c
(k + 1)
u
(
k)
η(k)
=
A
t
0 (M
t1
−M
t2
S
−1
3
S
2
)(γ
2
X
T
1
)
−1
−C
t1
0 X
1
−C
t2
I 0
x
c
(k)
y(k)
φ(k)
, (5.96 )
with
X
2
X
T
2
= S
3
, (5.97 )
X
1
X
T
1
= −γ
−2
(S
1
− S
2
S
−1
3
S
T
2
). (5. 98)
The following theorem g ives one parametrization approach of all suboptimal discrete-
time H
∞
output feedback controllers.
THEOREM 5.4
[90] Consider system (5.35)−(5.37). Given a scalar γ > 0, there exists an
output feedback controller (5.38)−(5.39) such that kT
zw
k
2
∞
< γ if the following
LMI admits a solution:
P
∞
J AX + B
2
E A + B
2
D
c
C
2
B
1
+ B
2
D
c
D
21
0
∗ H U Y A + V C
2
Y B
1
+ V D
21
0
∗ ∗ X + X
′
−P
∞
I + Z
′
− J 0 X
T
C
T
1
+ E
T
D
T
12
∗ ∗ ∗ Y + Y
T
−H 0 C
T
1
+ C
T
2
D
T
c
D
T
12
∗ ∗ ∗ ∗ I D
T
11
+ D
T
21
D
T
c
D
T
12
∗ ∗ ∗ ∗ ∗ γI
> 0, (5.99)
where the matrices X, E, Y , V , U, D
c
, Z, J, and the symmetric matrices P
∞
and H are t he variables. A feasible H
∞
controller is obtained from (5. 51)-
(5.53).
5.5 Robust control
The H
∞
norm is used to test robust stability of a nominally stable system under
unstructured pertu rbations. The fo llowing so-called small gain t heorem is the basis
for robust stability analysis.
102 Modeling and Control of Vibratio n in Mechanical Systems
THEOREM 5.5
Smal l Gain Theorem Consider a proper and stable transfer function matrix
T (s). Suppose that a stabl e ∆(s) is connected from the output of T (s) to the
input of T (s) as shown in Figure 5.2. Then the closed-loop system given in
Figure 5.2 is internally stable if
¯σ[∆(jω)]¯σ[T (jω)] < 1, ∀ω ∈ R
[
∞. (5.100)
FIGURE 5.2
A closed-loop system wit h uncertainty.
The small gain condition is sufficient to guarantee internal stabilit y of the closed-
loop system even if ∆ is nonlinear and time-varying. The small gain theorem tells us
that an H
∞
norm bound on T implies closed-loop stability in the presence of certain
H
∞
norm bounded system uncertainties. The H
∞
norm bound implies a certain
stabil ity robustness. Note from (5.100) that the size of tolerable uncertainty vari es
inversely proportional to th e H
∞
norm bo und of T , which means that the robustness
increases as the H
∞
norm bound decreases.
In what follows, the small gain theorem will be used to test robust stability under
model uncertainties. The modeling error ∆ is assumed to be stable and suitably
scaled with weigh ting functions W
1
and W
2
, i.e., the uncertainty can b e represented
as W
1
∆W
2
.
Additive uncertainty
We assume that the model uncertainty can be represented by an additive p erturba-
tion:
Π = P + W
1
∆W
2
, (5.101)
Introduction to Optimal an d Robust Control 103
which is shown in Figure 5.3.
FIGURE 5.3
A closed-loop system wit h additive uncertainty for robust stability analysis.
THEOREM 5.6
[70] Let Π = P + W
1
∆W
2
, and C be a stabilizing controller for the nominal
plant P . Then the closed-loop system is well-posed and internally stable for
all k∆k
∞
< 1 i f and only if kW
2
CSW
1
k
∞
≤ 1 , where
S =
1
1 + P C
. (5.10 2)
Similarly, the closed-loop sy stem is stable f or all stable ∆ with k∆k
∞
≤ 1 if and
only if kW
2
CSW
1
k
∞
< 1.
Multiplicative uncertainty
The system model is described by the fo llowing multiplicative pertur bation:
Π = (I + W
1
∆W
2
)P (5 .103)
where W
1
, W
2
and ∆ are stable. Consider the feedback system shown in Figure 5.4.
THEOREM 5.7
[70] Let Π = (I + W
1
∆W
2
)P , C be a stabilizing controller f or the nominal
plant P , and T = 1 − S. Then
(i) the closed-loop system is well-posed and internally stable for all stable
∆ with k∆k
∞
< 1 if and only if kW
2
T W
1
k
∞
≤ 1.
(ii) the closed-loop system is well-posed and internally stable for all stable
∆ with k∆k
∞
≤ 1 if kW
2
T W
1
k
∞
< 1.
(iii) the robust stability of the closed-loop system for all stable ∆ with
k∆k
∞
≤ 1 does not necessarily imply kW
2
T W
1
k
∞
< 1 .