Haddad W.M. Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach
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NonlinearBook10pt November 20, 2007
STRUCTURED PARAMETRIC UNCERTAINTY 745
control u(·) = φ(x(·)) minimizes J(x
0
, u(·)) in the sense th at
J(x
0
, φ(x(·))) = min
u(·)∈S(x
0
)
J(x
0
, u(·)). (12.120)
Finally, if D = R
n
, U = R
m
, and
V (x) → ∞ as kxk → ∞, (12.121)
then the zero solution x(t) ≡ 0 of the closed-loop system (12.108) is globally
asymptotically stable for all F (·) ∈ F.
Proof. Local and global asymptotic stability are a direct consequence
of (12.110)–(12.114) and (12.121) by applying Theorem 12.4 to the closed-
loop system (12.108). Furthermore, using (12.115), condition (12.118) is
a restatement of (12.43) as applied to the closed-loop s ys tem. Next, let
u(·) ∈ S(x
0
) and let x(·) be the solution of (12.107) with F (·, ·) = F
0
(·, ·).
Then it follows that
0 = −
˙
V (x(t)) + V
′
(x(t))F
0
(x(t), u(t)).
Hence
L(x(t), u(t)) + Γ(x(t), u(t))
= −
˙
V (x(t)) + L(x(t), u(t)) + V
′
(x(t))F
0
(x(t), u(t)) + Γ(x(t), u(t))
= −
˙
V (x(t)) + H(x(t), u(t)) + V
′
∆f
(x(t))F
0
(x(t), u(t)).
Now, using (12.117) and (12.119) and the fact that u(·) ∈ S(x
0
), it follows
that
J(x
0
, u(·)) =
Z
∞
0
[−
˙
V (x(t)) + H(x(t), u(t))]dt
= − lim
t→∞
V (x(t)) + V (x
0
) +
Z
∞
0
H(x(t), u(t))dt
= V (x
0
) +
Z
∞
0
H(x(t), u(t))dt
≥ V (x
0
)
= J(x
0
, φ(x(·)),
which yields (12.120).
If V
∆f
(x) = 0 and F consists of only the nominal nonlinear closed-loop
system F
0
(·, ·), then Γ(x, u) = 0 for all x ∈ D and u ∈ U satisfies (12.113),
and h en ce, J
f
(x
0
, u(·)) = J(x
0
, u(·)). I n this case, Theorem 12.7 s pecializes
to Theorem 8.2. Alternatively, setting V
∆f
(x) = 0 and allowing F (·, ·) ∈ F,
Theorem 12.7 specializes to Theorem 11.2.
Next, we specialize T heorem 12.7 to linear uncertain systems and pro-
vide connections to the parameter-dependent Lyapunov bounding synthesis
NonlinearBook10pt November 20, 2007
746 CHAPTER 12
framework d eveloped by Haddad and Bernstein [145, 151]. Specifically, in
this case we consider F to be the set of uncertain linear systems given by
F = {(A + ∆A)x + Bu : x ∈ R
n
, A ∈ R
n×n
, B ∈ R
n×m
, ∆A ∈ ∆
A
},
where ∆
A
⊂ R
n×n
is a given bounded uncertainty set of the uncertain
perturbation ∆A of the nominal system A such that 0 ∈ ∆
A
. For the
following result let R
1
∈ P
n
and R
2
∈ P
m
be given.
Corollary 12.3. Consider the linear uncertain controlled dynamical
system
˙x(t) = (A + ∆A)x(t) + Bu(t), x(0) = x
0
, t ≥ 0, (12.122)
with performance functional
J
∆A
(x
0
, u(·))
△
=
Z
∞
0
[x
T
(t)R
1
x(t) + u
T
(t)R
2
u(t)]dt, (12.123)
where u(·) is admissible and ∆A ∈ ∆
A
. Furthermore, assume there exist
P ∈ P
n
, Ω
xx
: P
n
→ N
n
, Ω
xu
: P
n
→ R
n×m
, Ω
uu
: P
n
→ N
m
, and P
0
: ∆
A
→
S
n
such that
∆A
T
P + P ∆A ≤ Ω
xx
(P ) − [(A + ∆A − BR
−1
2a
P
a
)
T
P
0
(∆A)
+P
0
(∆A)(A + ∆A − BR
−1
2a
P
a
)] −P
T
a
R
−1
2a
Ω
T
xu
(P )
−Ω
xu
(P )R
−1
2a
P
a
+ P
T
a
R
−1
2a
Ω
uu
(P )R
−1
2a
P
a
, (12.124)
and
0 = A
T
P + P A + R
1
+ Ω
xx
(P ) − P
T
a
R
−1
2a
P
a
, (12.125)
where R
2a
△
= R
2
+ Ω
uu
(P ) an d P
a
△
= B
T
P + Ω
T
xu
(P ). Then the zero solution
x(t) ≡ 0 to (12.122) is globally asymptotically stable for all x
0
∈ R
n
and
∆A ∈ ∆
A
, with the feedback control u = φ(x)
△
= −R
−1
2a
P
a
x, and
sup
∆A∈∆
A
J
∆A
(x
0
, φ(x(·))) ≤ sup
∆A∈∆
A
J(x
0
, φ(x(·)))
= x
T
0
P x
0
+ su p
∆A∈∆
A
x
T
0
P
0
(∆A)x
0
, (12.126)
where
J(x
0
, u(·))
△
=
Z
∞
0
[x
T
(t)(R
1
+ Ω
xx
(P ) − A
T
P
0
(∆A) − P
0
(∆A)A)x(t)
+u
T
(t)R
2
u(t)]dt, (12.127)
and where u(·) is admissible and x(t), t ≥ 0, solves (12.122) with ∆A = 0.
Furth ermore,
J(x
0
, φ(x(·)) = min
u(·)∈S(x
0
)
J(x
0
, u(·)), (12.128)
where S(x
0
) is the set of regulation controllers for the nominal system and
x
0
∈ R
n
. If, in addition, there exists P
0
∈ S
n
such that P
0
(∆A) ≤ P
0
,
NonlinearBook10pt November 20, 2007
STRUCTURED PARAMETRIC UNCERTAINTY 747
∆A ∈ ∆
A
, then
sup
∆A∈∆
A
J
∆A
(x
0
, φ(x(·))) ≤ x
T
0
(P + P
0
)x
0
. (12.129)
Proof. The result is a direct consequence of Theorem 12.7 with
F (x, u) = (A + ∆A)x+ (B +∆B)u, F
0
(x, u) = Ax+ Bu, L(x, u) = x
T
R
1
x+
u
T
R
2
u, V
I
(x) = x
T
P x, V
∆f
(x) = x
T
P
0
(∆A)x, Γ(x, u) = x
T
Ω
xx
(P )x +
2x
T
Ω
xu
(P )u + u
T
Ω
uu
(P )u, D = R
n
, and U = R
m
. Specifically, conditions
(12.110) and (12.111) are trivially satisfied. Now, forming x
T
(12.124)x
it follows that, after some algebraic manipulation, V
′
(x)F (x, φ(x)) ≤
V
′
I
(x)F
0
(x, φ(x))+Γ(x, φ(x)), for all ∆A ∈ ∆
A
. Furthermore, it follows from
(12.125) that H(x, φ(x)) = 0, and hence, V
′
I
(x)F
0
(x, φ(x)) + Γ(x, φ(x)) < 0
for all x 6= 0. Thus , H(x, u) = H(x, u) − H(x, φ(x)) = [u − φ(x)]
T
R
2a
[u −
φ(x)] ≥ 0 so that all the conditions of Theorem 12.7 are satisfied.
Finally, since V (x) is radially unbounded (12.122), with u(t) = φ(x(t)) =
−R
−1
2a
P
a
x(t), is globally asymptotically stable for all ∆A ∈ ∆
A
.
The optimal f eedback control law φ(x) in Corollary 12.3 is derived
using the properties of H(x, u) as defin ed in Theorem 12.7. Specifically,
since H(x, u) = x
T
(A
T
P + P A + R
1
+ Ω
xx
(P ))x + u
T
R
2a
u + 2x
T
P
T
a
u it
follows that ∂
2
H/∂u
2
= R
2a
> 0. Now, ∂H/∂u = 2R
2a
u + 2P
a
x = 0 gives
the uniqu e global minimum of H(x, u). Hence, since φ(x) min imizes H(x, u)
it follows that φ(x) s atisfies ∂H/∂u = 0 or, equivalently, φ(x) = −R
−1
2a
P
a
x.
In order to make explicit connections with linear robust control, we
now assign explicit stru ctur e to the set ∆
A
and the bounding functions
Ω
xx
(·), Ω
xu
(·), and Ω
uu
(·). Specifically, the uncertainty set ∆
A
is assumed
to be of the form
∆
A
△
= {∆A : ∆A = B
0
F C
0
, F ∈ ∆}, (12.130)
where ∆ satisfies
∆ ⊆ ∆
bs
= {F ∈ R
s×s
: M
1
≤ F ≤ M
2
},
and where B
0
∈ R
n×s
, C
0
∈ R
s×n
, are fixed matrices denoting the structure
of the uncertainty, F ∈ S
s
is a symmetric uncertain matrix, and M
1
, M
2
∈ S
s
are the uncertainty bounds such th at M
△
= M
2
− M
1
∈ P
s
.
Next, let
Ω
xx
(P ) = (
˜
C + B
T
0
P )
T
R
−1
0
(
˜
C + B
T
0
P ) + C
T
0
M
1
B
T
0
P + P B
0
M
1
C
0
,
Ω
xu
(P ) = (
˜
C + B
T
0
P )
T
R
−1
0
NC
0
B,
Ω
uu
(P ) = B
T
C
T
0
N
T
R
−1
0
NC
0
B,
where
˜
C
△
= HC
0
+ N C
0
(A + B
0
M
1
C
0
) and R
0
△
= (HM
−1
− NC
0
B
0
) +
NonlinearBook10pt November 20, 2007
748 CHAPTER 12
(HM
−1
− N C
0
B
0
)
T
. Furthermore, let P
0
(F ) = C
T
0
F
s
NC
0
, w here F
s
△
=
F − M
1
. Now, note that for all H ∈ H
p
and N ∈ N
s
,
0 ≤ [(
˜
C + B
T
0
P − BR
−1
2a
P
a
) − R
0
F
s
C
0
]
T
R
−1
0
[(
˜
C + B
T
0
P − BR
−1
2a
P
a
)
−R
0
F
s
C
0
] + 2C
T
0
(F
s
− F
s
M
−1
F
s
)C
0
,
which further implies
0 ≤ Ω
xx
(P ) − P
T
a
R
−1
2a
Ω
T
xu
(P ) − Ω
T
xu
(P )R
−1
2a
P
a
+ P
T
a
R
−1
2a
Ω
uu
(P )R
−1
2a
P
a
−A
T
C
T
0
N
T
F
s
C
0
− C
T
0
F
s
NC
0
A −C
T
0
M
1
B
T
0
C
T
0
N
T
F
s
C
0
−C
T
0
F
s
NC
0
B
0
M
1
C
0
+ P
T
a
R
−1
2a
B
T
C
T
0
N
T
F
s
C
0
+ C
T
0
F
s
NC
0
BR
−1
2a
P
a
−C
T
0
F B
T
0
P − P B
0
F C
0
− C
T
0
F
s
NC
0
B
0
F
s
C
0
− C
T
0
F
s
B
T
0
C
T
0
N
T
F
s
C
0
= Ω
xx
(P ) − [(A + ∆A − BR
−1
2a
P
a
)
T
P
0
(F ) + P
0
(F )(A + ∆A − BR
−1
2a
P
a
)]
−P
T
a
R
−1
2a
Ω
T
xu
(P ) − Ω
xu
(P )R
−1
2a
P
a
+ P
T
a
R
−1
2a
Ω
uu
(P )R
−1
2a
P
a
−(∆A
T
P + P ∆A), (12.131)
and h en ce, (12.124) holds. Fu rthermore, (12.125) specializes to
0 = A
T
P
P + P A
P
+ R
1
+
˜
C
T
R
−1
0
˜
C + P B
0
R
−1
0
B
T
0
P − P
T
a
R
−1
2a
P
a
, (12.132)
where A
P
△
= A+B
0
M
1
C
0
+BR
−1
0
˜
C and P
a
and R
2a
specialize to P
a
= B
T
P +
B
T
C
T
0
N
T
R
−1
0
(
˜
C+B
T
0
P ) and R
2a
= R
2
+B
T
C
T
0
N
T
R
−1
0
NC
0
B, respectively.
In this case, the optimal feedback law is given by φ(x) = −R
−1
2a
P
a
x. This
corresponds to the results obtained by Haddad and Bernstein [145,151] for
full-state feedback control.
12.5 Rob u st Control for Nonlinear Uncertain Affine Systems
In this section, we specialize Theorem 12.7 to affine (in the control) uncertain
systems of the form
˙x(t) = f
0
(x(t))+ ∆f(x(t))+ G
0
(x(t))u(t), x(0) = x
0
, t ≥ 0, (12.133)
where f
0
: R
n
→ R
n
satisfies f
0
(0) = 0, G
0
: R
n
→ R
n×m
, D = R
n
, U = R
m
,
and ∆f ∈ ∆. Furthermore, we consider performance integrands L(x, u) of
the form
L(x, u) = L
1
(x) + u
T
R
2
(x)u, (12.134)
where L
1
: R
n
→ R and R
2
: R
n
→ P
m
so that (12.109) becomes
J(x
0
, u(·)) =
Z
∞
0
[L
1
(x(t)) + u
T
(t)R
2
(x)u(t)]dt. (12.135)
Corollary 12.4. Consider the nonlinear uncertain controlled affine
system (12.133) with performance functional (12.135). Assume there exist
functions V
I
, V
∆f
, V : R
n
→ R, Γ
xx
: R
n
→ R, Γ
xu
: R
n
→ R
1×m
, and
NonlinearBook10pt November 20, 2007
STRUCTURED PARAMETRIC UNCERTAINTY 749
Γ
uu
: R
n
→ N
m
, where V
I
(·), V
∆f
(·) are continuously differentiable such
that V
I
(x) + V
∆f
(x) = V (x), x ∈ R
n
, and
V (0) = 0, (12.136)
Γ
xu
(0) = 0, (12.137)
V (x) > 0, x ∈ R
n
, x 6= 0, (12.138)
V
′
(x)∆f(x) −V
′
∆f
(x)[f
0
(x) −
1
2
G(x)R
−1
2a
(x)V
a
(x)] ≤ Γ
xx
(x)
−
1
2
Γ
xu
(x)R
−1
2a
(x)V
a
(x) +
1
4
V
T
a
(x)R
−1
2a
(x)Γ
uu
(x)R
−1
2a
(x)V
a
(x), ∆f(·) ∈ ∆,
(12.139)
V
′
I
(x)[f
0
(x) −
1
2
G(x)R
−1
2a
(x)V
a
(x)] + Γ
xx
(x) −
1
2
Γ
xu
(x)R
−1
2a
(x)V
a
(x)
+
1
4
V
T
a
(x)R
−1
2a
(x)Γ
uu
(x)R
−1
2a
(x)V
a
(x) < 0, x ∈ R
n
, x 6= 0, (12.140)
and
V (x) → ∞ as kxk → ∞, (12.141)
where R
2a
(x)
△
= R
2
(x) + Γ
uu
(x) and V
a
(x)
△
= [Γ
xu
(x) + V
′
I
(x)G(x)]
T
. Then
the zero solution x(t) ≡ 0 of the nonlinear uncertain system (12.133) is
globally asymptotically stable for all ∆f (·) ∈ ∆ with the feedb ack control
law
φ(x) = −
1
2
R
−1
2a
(x)V
a
(x), (12.142)
and the performan ce functional (12.135) satisfies
sup
∆f∈∆
J(x
0
, φ(x(·))) ≤ J(x
0
, φ(x(·))) = V (x
0
), (12.143)
where
J(x
0
, u(·))
△
=
Z
∞
0
[L(x(t), u(t)) + Γ(x(t), u(t))
−V
′
∆f
(x(t))(f
0
(x(t)) + G(x(t))u(t))]dt, (12.144)
and
Γ(x, u) = Γ
xx
(x) + Γ
xu
(x)u + u
T
Γ
uu
(x)u, (12.145)
where u(·) is admissible, and x(t), t ≥ 0, solves (12.133) with ∆f (x) = 0.
In addition, the performance functional (12.144), with
L
1
(x) = φ
T
(x)R
2a
(x)φ(x) −V
′
I
(x)f
0
(x) − Γ
xx
(x), (12.146)
is minimized in the sense that
J(x
0
, φ(x(·))) = min
u(·)∈S(x
0
)
J(x
0
, u(·)). (12.147)
Proof. The result is a direct consequence of Theorem 12.7 with D =
R
n
, U = R
m
, F
0
(x, u) = f
0
(x) + G
0
(x)u, F (x, u) = f
0
(x) + ∆f(x) + G(x)u,
L(x, u) given by (12.134), and Γ(x, u) given by (12.145). Specifically, with
NonlinearBook10pt November 20, 2007
750 CHAPTER 12
(12.133), (12.134), and (12.145), the Hamiltonian has the form
H(x, u) = L
1
(x) + u
T
R
2
(x)u + V
′
I
(x)(f
0
(x) + G(x)u)
+Γ
xx
(x) + Γ
xu
(x)u + u
T
Γ
uu
(x)u.
Now, the proof follows as in the p roof of Corollary 11.4.
12.6 Rob u st Nonlinear Controllers with Polynomial
Performance Criteria
In this section, we specialize the results of Section 12.4 to linear uncertain
systems controlled by nonlinear controllers that minimize a polynomial cost
functional. Specifically, assume F to be the set of uncertain systems given
by
F = {(A + ∆A)x + Bu : x ∈ R
n
, u ∈ R
m
, A ∈ R
n×n
, B ∈ R
n×m
, ∆A ∈ ∆
A
},
where ∆
A
⊂ R
n×n
is a given bounded uncertainty set of the uncertain
perturbation ∆A of the nominal system A such that 0 ∈ ∆
A
. For the
following result let R
1
∈ P
n
, R
2
∈ P
m
, and
ˆ
R
k
∈ N
n
, k = 2, . . . , r, be given
where r is a positive integer.
Theorem 12.8. Consider the linear uncertain controlled system
˙x(t) = (A + ∆A)x(t) + Bu(t), x(0) = x
0
, t ≥ 0, (12.148)
where u(·) is admissible and ∆A ∈ ∆
A
. Assume there exist functions Ω :
N
n
→ N
n
, Ω
xx
: P
n
→ N
n
, Ω
xu
: P
n
→ R
n×m
, Ω
uu
: P
n
→ N
m
, and
P
0
: ∆
A
→ S
n
, such that
∆A
T
P + P ∆A ≤ Ω(P ), ∆A ∈ ∆
A
, P ∈ N
n
, (12.149)
and assume there exist P ∈ P
n
and Y
k
∈ N
n
, k = 2, . . . , r, such that
∆A
T
P + P ∆A ≤ Ω
xx
(P ) − [(A + ∆A − BR
−1
2a
P
a
)
T
P
0
(∆A)
+P
0
(∆A)(A + ∆A − BR
−1
2a
P
a
)] −P
T
a
R
−1
2a
Ω
T
xu
(P )
−Ω
xu
(P )R
−1
2a
P
a
+ P
T
a
R
−1
2a
Ω
uu
(P )R
−1
2a
P
a
, (12.150)
0 = A
T
P + P A + R
1
+ Ω
xx
(P ) − P
T
a
R
−1
2a
P
a
, (12.151)
and
0 = (A − BR
−1
2a
P
a
)
T
Y
k
+ Y
k
(A −BR
−1
2a
P
a
) +
ˆ
R
k
+ Ω(Y
k
), k = 2, . . . , r.
(12.152)
Then, with the f eedback control law
u = φ(x)
△
= −R
−1
2a
P
a
+
r
X
k=2
(x
T
Y
k
x)
k−1
B
T
Y
k
!
x,
NonlinearBook10pt November 20, 2007
STRUCTURED PARAMETRIC UNCERTAINTY 751
the zero solution of the uncertain system (12.148) is globally asymptotically
stable for all x
0
∈ R
n
and ∆A ∈ ∆
A
, and the performance functional
(12.144) satisfies
J
∆A
(x
0
, φ(x(·))) ≤ J(x
0
, φ(x(·)))
= x
T
0
(P + P
0
(∆A))x
0
+
r
X
k=2
1
k
(x
T
0
Y
k
x
0
)
k
, ∆A ∈ ∆
A
,
(12.153)
where
J(x
0
, u(·))
△
=
Z
∞
0
[L(x, u) + Γ(x, u) −x
T
(t)(A
T
P
0
(∆A) + P
0
(∆A)A)x(t)]dt,
(12.154)
and where u(·) is admissible, and x(t), t ≥ 0, solves (12.148) with ∆A = 0
and
Γ(x, u) = x
T
Ω
xx
(P ) +
r
X
k=2
(x
T
Y
k
x)
k−1
Ω(Y
k
)
!
x
+2x
T
Ω
xu
(P )u + u
T
Ω
uu
(P )u,
where u(·) is admissible and ∆A ∈ ∆
A
. In addition, the performance
functional (12.144), with R
2
(x) = R
2
and
L
1
(x) = x
T
R
1
+
r
X
k=2
(x
T
Y
k
x)
k−1
ˆ
R
k
+
"
r
X
k=2
(x
T
Y
k
x)
k−1
B
T
Y
k
#
T
R
−1
2a
·
"
r
X
k=2
(x
T
Y
k
x)
k−1
B
T
Y
k
#!
x,
is minimized in the sense that
J(x
0
, φ(x(·)) = min
u(·)∈S(x
0
)
J(x
0
, u(·)), (12.155)
where S(x
0
) is the set of regulation controllers for the nominal system and
x
0
∈ R
n
.
Proof. The result is a direct consequence of Corollary 12.4 with
f
0
(x) = Ax, ∆f (x) = ∆Ax, G
0
(x) = B, V
I
(x) = x
T
P x+
P
r
k=2
(
1
k
)(x
T
Y
k
x)
k
,
and V
∆f
(x) = x
T
P
0
(∆A)x.
The Lyapunov function that establishes robust stability in Theorem
12.8 is given by
V (x) = x
T
P x +
r
X
k=2
1
k
(x
T
Y
k
x)
k
+ x
T
P
0
(∆A)x
NonlinearBook10pt November 20, 2007
752 CHAPTER 12
and, hence, is explicitly depend ent on the uncertain parameters ∆A. In the
terminology of [145,151], this is a parameter-dependent Lyapunov function.
As discussed in S ection 12.1 and [145, 151] the ability of such Lyapunov
functions to guarantee stability with respect to time-varying parameter
variations is curtailed, thus reducing conservatism with respect to constant
real parameter uncertainty.
Theorem 12.8 requires the solutions of r −1 modified Riccati equations
in (12.152) to obtain the optimal robust controller. However, if
ˆ
R
k
=
ˆ
R
2
,
k = 3, . . . , r, then Y
k
= Y
2
, k = 3, . . . , r, satisfies (12.152). In this case,
we require the solution of one modified Riccati equation in (12.152). This
special case is considered in Corollary 12.5 below.
As in Chapters 10 and 11 the performance functional can be written
as
J
∆A
(x
0
, u(·)) =
Z
∞
0
"
x
T
(R
1
+
r
X
k=2
(x
T
Y
k
x)
k−1
ˆ
R
k
)x + u
T
R
2
u + φ
T
NL
(x)R
2a
φ
NL
(x)
#
dt,
where φ
NL
(x) is the nonlinear part of the optimal feedback control
φ(x) = φ
L
(x) + φ
NL
(x),
where φ
L
(x)
△
= −R
−1
2a
P
a
x and φ
NL
△
= −R
−1
2a
B
T
P
r
k=2
(x
T
Y
k
x)
k−1
Y
k
x.
Next, we consider the special case in which r = 2. In this case, note
that if there exist P ∈ P
n
and Y
2
∈ N
n
such that
0 = A
T
P + P A + R
1
+ Ω
xx
(P ) − P
a
R
−1
2a
P
a
and
0 = (A − BR
−1
2a
P
a
)
T
Y
2
+ Y
2
(A − BR
−1
2a
P
a
) +
ˆ
R
2
+ Ω(Y
2
),
then (12.148), with the performance functional
J
∆A
(x
0
, u(·)) =
Z
∞
0
[x
T
R
1
x + u
T
R
2
u + (x
T
Y
2
x)(x
T
ˆ
R
2
x)
+(x
T
Y
2
x)
2
(x
T
Y
2
BR
−1
2a
B
T
Y
2
x)]dt,
is globally asymptotically stable for all x
0
∈ R
n
with the feedback control
law u = φ(x) = −R
−1
2a
(P
a
+ (x
T
Y
2
x)B
T
Y
2
)x.
Finally, using the uncertainty characterization given by (12.130) we
present a specialization of Theorem 12.8.
Corollary 12.5. Consider the uncertain controlled system
˙x(t) = (A + ∆A)x(t) + Bu(t), x(0) = x
0
, t ≥ 0, (12.156)
NonlinearBook10pt November 20, 2007
STRUCTURED PARAMETRIC UNCERTAINTY 753
where u(·) is admissible and ∆A ∈ ∆
A
, where ∆
A
is given by (12.130).
Assume H ∈ H
p
and N ∈ N
nd
and suppose there exist P ∈ P
n
and Y
2
∈ N
n
such that
0 = A
T
P
P + P A
P
+ R
1
+
˜
C
T
R
−1
0
˜
C + P B
0
R
−1
0
B
T
0
P − P
T
a
R
−1
2a
P
a
, (12.157)
0 = (A
Y
2
−BR
−1
2a
P
a
)
T
Y
2
+ Y
2
(A
Y
2
− BR
−1
2a
P
a
) +
ˆ
R
2
+
1
2
C
T
0
MC
0
+
1
2
Y
2
B
0
MB
0
Y
2
, (12.158)
where A
Y
2
△
= A +
1
2
B
0
(M
1
+ M
2
)C
0
. Then, with the feedback control law
u = φ(x) = −R
−1
2
B
T
(P +
P
r
k=2
(x
T
Y
2
x)
k−1
Y
2
)x, the zero solution x(t) ≡ 0
of th e uncertain system (12.156) is globally asymptotically stable for all
x
0
∈ R
n
and ∆A ∈ ∆
A
, and the performance functional (12.144) satisfies
J
∆A
(x
0
, φ(x(·))) ≤ J(x
0
, φ(x(·))) = x
T
0
(P + P
0
(∆A))x
0
+
r
X
k=2
1
k
(x
T
0
Y
k
x
0
)
k
, ∆A ∈ ∆
A
, (12.159)
where
J(x
0
, u(·))
△
=
Z
∞
0
[L(x, u) + Γ(x, u) −x
T
(t)(A
T
P
0
(∆A) + P
0
(∆A)A)x(t)]dt,
(12.160)
and where u(·) is admissible, and x(t), t ≥ 0, solves (12.156) with ∆A = 0
and
Γ(x, u) = x
T
"
(
˜
C + B
T
0
P )
T
R
−1
0
(
˜
C + B
T
0
P )
+
1
2
r
X
k=2
(x
T
Y
2
x)
k−1
(C
0
+ B
T
0
Y
2
)
T
M(C
0
+ B
T
0
Y
2
)
#
x
+2x
T
B
T
C
T
0
N
T
R
−1
0
(
˜
C + B
T
0
P )u + u
T
B
T
0
C
T
0
N
T
R
−1
0
NC
0
Bu.
In addition, the performance functional (12.144), with R
2
(x) = R
2
and
L
1
(x) =
x
T
R
1
+
r
X
k=2
(x
T
Y
2
x)
k−1
ˆ
R
2
+
r
X
k=2
(x
T
Y
2
x)
k−1
!
2
Y
2
BR
−1
2a
B
T
Y
2
x,
is minimized in the sense that
J(x
0
, φ(x(·)) = min
u(·)∈S(x
0
)
J(x
0
, u(·)), (12.161)
where S(x
0
) is the set of regulation controllers for the nominal system and
x
0
∈ R
n
.
Proof. The result is a direct consequence of Theorem 12.8 with
NonlinearBook10pt November 20, 2007
754 CHAPTER 12
Ω
xx
(P ) = (
˜
C +B
T
0
P )
T
R
−1
0
(
˜
C +B
T
0
P ) + C
T
0
M
1
B
T
0
P + P B
0
M
1
C
0
, Ω
xu
(P ) =
B
T
C
T
0
N
T
R
−1
0
(
˜
C + B
T
0
P ), Ω
uu
(P ) = B
T
C
T
0
N
T
R
−1
0
NC
0
B, Ω(Y
2
) =
1
2
(C
0
+
B
T
0
Y
2
)
T
M(C
0
+B
T
0
Y
2
)+C
T
0
M
1
B
T
0
Y
2
+Y
2
B
0
M
1
C
0
, and P
0
(F ) = C
T
0
F
s
NC
0
.
In this case, as was s hown in (12.131) and (12.150) holds. Furthermore, for
all ∆A ∈ ∆
A
it follows that
0 ≤
1
2
[(C
0
+ B
T
0
Y
2
) −2M
−1
F
s
C
0
]
T
M[(C
0
+ B
T
0
Y
2
) − 2M
−1
F
s
C
0
]
+2C
T
0
[F
s
− F
s
M
−1
F
s
]C
0
= Ω(Y
2
) −C
T
0
F B
T
0
Y
2
− Y
2
B
0
F C
0
= Ω(Y
2
) −[∆A
T
Y
2
+ Y
2
∆A],
and hence, (12.149) holds. The result now follows as a direct consequence
of Theorem 12.8.
12.7 Rob u st Nonlinear Controllers with Multilinear
Performance Criteria
In this section, we specialize the results of Section 12.5 to linear uncertain
systems controlled by nonlinear controllers that minimize a multilinear cost
functional. Specifically, we assum e F to be the set of uncertain linear
systems given by
F = {(A + ∆A)x + Bu : x ∈ R
n
, u ∈ R
m
, A ∈ R
n×n
, B ∈ R
n×m
, ∆A ∈ ∆
A
},
where ∆
A
⊂ R
n×n
is a given bounded uncertainty set of the uncertain
perturbation ∆A of the nominal system A such that 0 ∈ ∆
A
. For the
following let R
1
∈ P
n
, R
2
∈ P
m
, and
ˆ
R
2ν
∈ N
(2ν,n)
, ν = 2, . . . , r, be given
where r is a given integer.
Theorem 12.9. Consider the uncertain controlled system
˙x(t) = (A + ∆A)x(t) + Bu(t), x(0) = x
0
, t ≥ 0, (12.162)
where u(·) is admissible and ∆A ∈ ∆
A
. Assume there exist Ω
xx
: N
n
→ N
n
,
Ω
xu
: N
n
→ R
n×m
, Ω
uu
: N
n
→ N
m
, P
0
: ∆
A
→ S
n
, P ∈ P
n
,
ˆ
Ω
ν
: N
(2ν,n)
→
N
(2ν,n)
, and
ˆ
P
ν
∈ N
(2ν,n)
, ν = 2, . . . , r, such that
∆A
T
P + P ∆A ≤ Ω
xx
(P ) − [(A + ∆A − BR
−1
2a
P
a
)
T
P
0
(∆A) + P
0
(∆A)
·(A + ∆A −BR
−1
2a
P
a
)] −P
T
a
R
−1
2a
Ω
T
xu
(P ) (12.163)
−Ω
xu
(P )R
−1
2a
P
a
+ P
T
a
R
−1
2a
Ω
uu
(P )R
−1
2a
P
a
, ∆A ∈ ∆
A
,
ˆ
Ω
ν
(
ˆ
P
ν
) −
ˆ
P
ν
(
2ν
⊕ ∆A) ∈ N
(2ν,n)
, ∆A ∈ ∆
A
, ν = 2, . . . , r, (12.164)
0 = A
T
P + P A + R
1
+ Ω
xx
(P ) − P
T
a
R
−1
2a
P
a
, (12.165)
and
0 =
ˆ
P
ν
[
2ν
⊕ (A −BR
−1
2a
P
a
)] +
ˆ
R
2ν
+
ˆ
Ω
ν
(
ˆ
P
ν
), ν = 2, . . . , r. (12.166)