Haddad W.M. Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach

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NonlinearBook10pt November 20, 2007

ROBUST NONLINEAR CONTROL 655

There are many alternative deﬁnitions for the bound Γ(·). To illustrate

some of these alternatives consider the nonlinear uncertain dynamical system

(11.16) and assume the uncertainty set F to be of th e form

F = {f

+ ∆f : R

→ R

: ∆f (x) = G

(x)δ(h

(x)), x ∈ R

, δ(·) ∈ ∆},

(11.23)

where

∆ = {δ : R

→ R

: δ(0) = 0, δ

(y)δ(y) ≤ m

(y)m(y), y ∈ R

(11.24)

: R

→ R

n×m

and h

: R

→ R

satisfying h

(0) = 0 are ﬁxed functions

denoting the stru cture of the uncertainty, δ : R

→ R

is an uncertain

function, and m : R

→ R

is a given function such that m(0) = 0. The

special case m(y) = γ

−1

y, where γ > 0, is worth noting. Speciﬁcally, in this

case, (11.24) specializes to

∆ = {δ : R

→ R

: δ(0) = 0, δ

(y)δ(y) ≤ γ

−2

y, y ∈ R

}, (11.25)

which corresponds to a nonlinear s mall gain-type norm bounded uncertainty

characterization. For the structure of F as speciﬁed by (11.23) and ∆ given

by (11.24), the bounding function Γ(·) satisfying (11.18) can now be given

a concrete form.

Proposition 11.1. The function

Γ(x) =

′

(x)G

(x)V

′T

(x) + m

(x))m(h

(x)) (11.26)

satisﬁes (11.18) with F given by (11.23) and ∆ given by (11.24).

Proof. Note that

0 ≤ [δ

(x)) −

′

(x)G

(x)][δ

(x)) −

′

(x)G

(x)]

= δ

(x))δ(h

(x)) +

′

(x)G

(x)V

′T

(x) − V

′

(x)G

(x)δ(h

(x))

≤ m

(x))m(h

(x)) +

′

(x)G

(x)V

′T

(x) − V

′

(x)G

(x)δ(h

(x))

= Γ(x) −V

′

(x)∆f(x),

which proves (11.18) with F given by (11.23) and ∆ given by (11.24).

Alternatively, consider the nonlinear un certain dynamical s y s tem given

by (11.16) and assume the uncertainty s et F is given by (11.23) with ∆ given

∆ = {δ : R

→ R

: δ(0) = 0, [δ(y) − m

(y)]

[δ(y) − m

(y)] ≤ 0,

y ∈ R

}, (11.27)

where m

, m

: R

→ R

are given functions such that m

(0) = 0, m

(0) =

0, and m

(y)m

(y) ≤ 0, y ∈ R

. For the structure of F as speciﬁed

by (11.23) with ∆ given by (11.27), the bounding function Γ(·) satisfying

NonlinearBook10pt November 20, 2007

656 CHAPTER 11

(11.18) can now be given a concrete form. For this result deﬁne m(y)

△

(y) − m

(y).

Proposition 11.2. The function

Γ(x) =

[m(h

(x)) + G

(x)V

′T

(x)]

[m(h

(x)) + G

(x)V

′T

(x)]

′

(x)G

(x)m

(x)) (11.28)

satisﬁes (11.18) with F given by (11.23) and ∆ given by (11.27).

Proof. Note that

0 ≤ [

m(h

(x)) +

(x)V

′T

(x) − (δ(h

(x)) − m

(x)))]

·[

m(h

(x)) +

(x)V

′T

(x) − (δ(h

(x)) − m

(x)))]

= [δ(h

(x)) − m

(x))]

[δ(h

(x)) − m

(x))] −V

′

(x)G

(x)δ(h

(x))

[m(h

(x)) + G

(x)V

′T

(x)]

[m(h

(x)) + G

(x)V

′T

(x)]

′

(x)G

(x)m

(x))

≤ Γ(x) −V

′

(x)∆f(x),

which proves (11.18) with F given by (11.23) and ∆ given by (11.27).

Finally, consider the nonlinear uncertain d ynamical system (11.16) and

assume the uncertainty s et F is given by (11.23) with ∆ given by

∆ = {δ : R

→ R

: δ(0) = 0, δ

(y)Qδ(y) + 2δ

(y)Sy + y

Ry ≤ 0,

y ∈ R

}, (11.29)

where Q ∈ P

, −R ∈ N

, and S ∈ R

×p

. For this uncertainty

characterization, the bounding fun ction Γ(·) satisfying (11.18) can now be

given a concrete form.

Proposition 11.3. The function

Γ(x) = [

′

(x)G

(x) − h

(x)S

−1

[

′

(x)G

(x) − h

(x)S

]

−h

(x)Rh

(x) (11.30)

satisﬁes (11.18) with F given by (11.23) and ∆ given by (11.29).

Proof. Note that

0 ≤ [Q

1/2

δ(h

(x)) + Q

−1/2

(Sh

(x) −

(x)V

′T

(x))]

·[Q

1/2

δ(h

(x)) + Q

−1/2

(Sh

(x) −

(x)V

′T

(x))]

= δ

(x))Qδ(h

(x)) + 2δ

(x))Sh

(x) − V

′

(x)G

(x)δ(h

(x))

′

(x)G

(x) − h

(x)S

−1

[

′

(x)G

(x) − h

(x)S

]

= δ

(x))Qδ(h

(x)) + 2δ

(x))Sh

(x) + h

(x)Rh

(x)

NonlinearBook10pt November 20, 2007

ROBUST NONLINEAR CONTROL 657

−V

′

(x)∆f(x) + Γ(x)

≤ Γ(x) −V

′

(x)∆f(x),

which proves (11.18) with F given by (11.23) and ∆ given by (11.29).

Example 11. 1. Consider the n onlinear u ncertain dy namical sy s tem

given by

˙x

(t) = x

(t) + δ

(t) cos(x

(t) + δ

) + δ

(t) sin(δ

(t)x

(t)),

(0) = x

, t ≥ 0, (11.31)

˙x

(t) = x

(t) sin(x

(t)), x

(0) = x

, (11.32)

where δ

∈ [−1, 1], δ

∈ [−5, 50], δ

∈ [0, 1], and δ

∈ [−50, 0]. Now, with

x = [x

, x

]

, f

(x) = [x

, x

sin x

]

, G

(x) = [1, 0]

, h

(x) = x,

and δ(x) = δ

cos(x

+ δ

) + δ

sin(δ

), (11.31) and (11.31) can be

written in the f orm of (11.16) with F given by (11.23). Furthermore, s ince

(x) ≤ x

+ x

it follows th at δ(·) ∈ ∆, where ∆ is given by (11.24) with

m(x) =

+ x

. △

Example 11.2. Consider the nonlinear uncertain m atrix second-order

dynamical system of an n-link robot discussed in Example 5.2 given by

M(q(t))¨q(t) + W (q(t), ˙q(t)) = 0, q(0) = q

, ˙q(0) = ˙q

, t ≥ 0, (11.33)

where q, ˙q, ¨q ∈ R

represent generalized position, velocity, and acceleration

co ordinates, respectively, M(q) is a positive-deﬁnite inertia matrix function

for all q ∈ R

, and W (q, ˙q) is a vector function lu mping the centrifugal and

Coriolis f orces, dissipation forces due to friction, and gravitational forces.

Here, we assume that M

(q) ≤ M(q) ≤ M

(q), where M

(q) > 0, q ∈

, and kW (q, ˙q)k

≤ w(q, ˙q). Now, with x = [x

, x

]

, where x

= q

and x

= ˙q, f

(x) = [x

, 0]

, G

(x) = [0, I

]

, h

(x) = x, and δ(x) =

−1

)W (x

, x

), (11.33) can be written in the form of (11.16) with F

given by (11.23). Furthermore, since kδ(x)k

= kM

−1

)W (x

, x

≤

max

−1

))kW (x

, x

≤ σ

−1

max

))w(x

, x

) it follows that δ(·) ∈

∆, w here ∆ is given by (11.24) with km(x)k

= σ

−1

max

))w(x

, x

). △

Example 11. 3. Consider the n onlinear u ncertain dy namical sy s tem

given by

˙x

(t) = x

(t) + x

(t), x

(0) = x

, t ≥ 0, (11.34)

˙x

(t) = x

(t) + δx

(t)[0.55 + 0.44δ cos(x

(t))], x

(0) = x

, (11.35)

where δ ∈ [−1, 1]. Now, with x = [x

, x

]

, f

(x) = [x

, x

]

, G

(x) =

[0, 1]

, h

(x) = x

, and δ(x

) = δx

(0.55+0.44δ cos(x

), (11.34) and (11.35)

can be written in the form of (11.16) with F given by (11.23). Furth ermore,

since δ

(0.55 + 0.44δ cos(x

))

+ 0.1x

≤ 1.1δx

(0.55 + 0.44δ cos(x

)) it

follows that δ(·) ∈ ∆, wh ere ∆ is given by (11.27) with m

) = 0.1x

and

NonlinearBook10pt November 20, 2007

658 CHAPTER 11

) = x

. △

We n ow combine the results of Corollary 11.1 and Propositions 11.1–

11.3 to obtain a series of conditions guaranteeing robust stability and

performance for the nonlinear uncertain system (11.16).

Proposition 11.4. Consider th e nonlinear uncertain system (11.16).

Let L(x) > 0, x ∈ R

, and suppose there exists a continuously diﬀerentiable

radially unbounded function V : R

→ R such that (11.6) and (11.7) hold

and

0 = V

′

(x)f

(x) +

′

(x)G

(x)V

′T

(x) + m

(x))m(h

(x))

+L(x), x ∈ R

. (11.36)

Then the zero solution x(t) ≡ 0 to (11.16) is globally asymptotically stable

for all ∆f(·) ∈ ∆ w ith ∆ given by (11.24), and the performance functional

(11.5) satisﬁes

sup

∆f(·)∈∆

∆f

) ≤ J(x

) = V (x

), (11.37)

where

J(x

) =

∞

[L(x(t)) +

′

(x(t))G

(x(t))V

′T

(x(t))

(x(t)))m(h

(x(t)))]dt, (11.38)

and x(t), t ≥ 0, is the solution of (11.16) with ∆f (x) ≡ 0.

Proposition 11.5. Consider th e nonlinear uncertain system (11.16).

Let L(x) > 0, x ∈ R

, and suppose there exists a continuously diﬀerentiable

radially unbounded function V : R

→ R such that (11.6) and (11.7) hold

and

0 = V

′

(x)f

(x) +

[m(h

(x)) + G

(x)V

′T

(x)]

[m(h

(x)) + G

(x)V

′T

(x)]

′

(x)G

(x)m

(x)) + L(x), x ∈ R

. (11.39)

Then the zero solution x(t) ≡ 0 to (11.16) is globally asymptotically stable

for all ∆f(·) ∈ ∆ w ith ∆ given by (11.27), and the performance functional

(11.5) satisﬁes

sup

∆f(·)∈∆

∆f

) ≤ J(x

) = V (x

), (11.40)

where

J(x

) =

∞

[L(x(t)) +

[m(h

(x(t))) + G

(x(t))V

′T

(x(t))]

·[m(h

(x(t))) + G

(x(t))V

′T

(x(t))]

′

(x(t))G

(x(t))m

(x(t)))]dt, (11.41)

and x(t), t ≥ 0, is the solution of (11.16) with ∆f (x) ≡ 0.

NonlinearBook10pt November 20, 2007

ROBUST NONLINEAR CONTROL 659

Proposition 11.6. Consider th e nonlinear un certain system (11.16).

Let L(x) > 0, x ∈ R

, and suppose there exists a continuously diﬀerentiable

radially unbounded function V : R

→ R such that (11.6) and (11.7) hold

and

0 = V

′

(x)f

(x) + [

′

(x)G

(x) − h

(x)S

−1

[

′

(x)G

(x) − h

(x)S

]

−h

(x)Rh

(x) + L(x), x ∈ R

. (11.42)

Then the zero solution x(t) ≡ 0 to (11.16) is globally asymptotically stable

for all ∆f(·) ∈ ∆ w ith ∆ given by (11.29), and the performance functional

(11.5) satisﬁes

sup

∆f(·)∈∆

∆f

) ≤ J(x

) = V (x

), (11.43)

where

J(x

) =

∞

[L(x(t)) + [

′

(x(t))G

(x(t)) −h

(x(t))S

−1

·[

′

(x(t))G

(x(t)) − h

(x(t))S

]

−h

(x(t))Rh

(x(t))]dt, (11.44)

and x(t), t ≥ 0, is the solution of (11.16) with ∆f (x) ≡ 0.

Example 11.4. Consider the nonlinear u ncertain system

˙x(t) = Ax(t) + B

δ(y(t)), x(0) = x

, t ≥ 0, (11.45)

y(t) = C

x(t), (11.46)

where A ∈ R

n×n

, B

∈ R

n×m

, C

∈ R

×n

, and δ(y) ∈ ∆ with ∆ given by

(11.25). To obtain suﬃcient conditions for robust stability for the nonlinear

uncertain s y s tem (11.45) and (11.46) we use Proposition 11.4 with L(x) =

Rx, wh ere R > 0, m(h

(x)) = γ

−1

y = γ

−1

x, and V (x) = x

P x, where

P > 0. In this case, (11.36) yields

0 = A

P + P A + P B

P + γ

−2

+ R, (11.47)

or, equivalently,

0 = A

P + P A + γ

−2

P B

P + C

+ R. (11.48)

Now, it follows from P roposition 11.4 that if there exists a positive-deﬁnite

P ∈ R

n×n

satisfying (11.48), th en the zero solution x(t) ≡ 0 to (11.45) is

globally asymptotically stable for all δ(·) ∈ ∆, where ∆ is given by (11.25).

Alternatively, s uppose δ(y) ∈ ∆, where ∆ is given by (11.27) with

(t) = M

y and m

(y) = M

y, where M

, M

∈ R

×m

are symmetric

matrices such that M

△

= M

−M

. In this case, to obtain suﬃcient conditions

for robust stability for the nonlinear uncertain system (11.45) and (11.46) we

use Proposition 11.5 with L(x) = x

Rx, where R > 0, m(h

(x)) = My =

NonlinearBook10pt November 20, 2007

660 CHAPTER 11

x, m

(x)) = M

y = M

x, and V (x) = x

P x, where P > 0. In

this case, (11.39) yields

0 = (A + B

)

P + P (A + B

)

+ B

P )

(

+ B

P ) + R. (11.49)

Now, it follows from Proposition 11.5 that if there exists a positive-deﬁnite

P ∈ R

n×n

satisfying (11.49), then the zero solution to (11.45) is globally

asymptotically stable for all δ(·) ∈ ∆, where ∆ is given by (11.27) with

(y) = M

y and m

(y) = M

y. △

The following corollary specializes Theorem 11.1 to a class of linear

uncertain systems which connects the framework of Theorem 11.1 to

the quadratic Lyapunov bounding (Ω-bound) framework of Bernstein and

Haddad [50]. Speciﬁcally, in this case, we consider F to be the set of

uncertain linear systems given by

F = {(A + ∆A)x : x ∈ R

, A ∈ R

n×n

, ∆A ∈ ∆

where ∆

⊂ R

n×n

is a given bound ed uncertainty set of uncertain

perturbations ∆A of the nominal asymptotically stable system matrix A

such that 0 ∈ ∆

Corollary 11.2. Let R ∈ P

. Consider the linear uncertain dynamical

system

˙x(t) = (A + ∆A)x(t), x(0) = x

, t ≥ 0, (11.50)

with performance functional

∆A

)

△

∞

(t)Rx(t)dt, (11.51)

where ∆A ∈ ∆

. Let Ω : N ⊆ S

→ N

be such that

∆A

P + P ∆A ≤ Ω(P ), ∆A ∈ ∆

, P ∈ N. (11.52)

Furth ermore, suppose there exist P ∈ P

satisfying

0 = A

P + P A + Ω(P ) + R. (11.53)

Then the zero solution x(t) ≡ 0 to (11.50) is globally asymptotically stable

for all ∆A ∈ ∆

, and

sup

∆A∈∆

∆A

) ≤ J(x

) = x

P x

, x

∈ R

, (11.54)

where

J(x

)

△

∞

(t)(Ω(P ) + R)x(t)dt, (11.55)

and where x(t), t ≥ 0, solves (11.50) with ∆A = 0.

NonlinearBook10pt November 20, 2007

ROBUST NONLINEAR CONTROL 661

Proof. The result is a direct consequence of Theorem 11.1 with f (x) =

(A + ∆A)x, f

(x) = Ax, L(x) = x

Rx, V (x) = x

P x, Γ(x) = x

Ω(P )x,

and D = R

. Speciﬁcally, conditions (11.6) and (11.7) are trivially satisﬁed.

Now, V

′

(x)f(x) = x

P + P A)x + x

(∆A

P + P ∆A)x, and hence,

it follows fr om (11.52) that V

′

(x)f(x) ≤ V

′

(x)f

(x) + Γ(x) = x

P +

P A + Ω(P ))x, for all ∆A ∈ ∆

. Furthermore, it follows from (11.53) that

L(x)+V

′

(x)f

(x)+Γ(x) = 0, and hence, V

′

(x)f

(x)+Γ(x) < 0 for all x 6= 0,

so that all the conditions of Theorem 11.1 are satisﬁed. Finally, since V (x),

x ∈ R

, is rad ially unbounded, (11.50) is globally asymptotically stable for

all ∆A ∈ ∆

Corollary 11.2 is the deterministic version of Theorem 4.1 of [50]

involving quadratic Lyapunov bounds for addr essin g robust stability and

performance analysis of linear uncertain systems. For convenience we shall

say that Ω(·) bounds ∆

if (11.52) is satisﬁed. To apply Corollary 11.2,

we ﬁr st specify a function Ω(·) and an uncertainty set ∆

such that Ω(·)

bounds ∆

. If the existence of a positive-deﬁnite solution P to (11.53) can be

determined analytically or numerically, then robust stability is guaranteed

and the performance bound (11.54) can be computed. We can then enlarge

∆

, modify Ω(·), and again attempt to solve (11.53). If, however, a

positive-deﬁnite solution to (11.53) cannot be determined, then ∆

must be

decreased in size until (11.53) is solvable. For example, Ω(·) can be replaced

by εΩ(·) to bound ε∆

, where ε > 1 enlarges ∆

and ε < 1 shrinks ∆

Of course, the actual range of uncertainty that can be bounded depends on

the nominal matrix A, the function Ω(·), and the structure of ∆

Since the ord ering induced by the cone of nonnegative-deﬁnite matrices

is only a partial ordering, it should not be expected that there exists an

operator Ω(·) satisfying (11.52), which is a least upper bound. Next, we

illustrate two bounding functions for two diﬀerent uncertainty characteriza-

tions. Speciﬁcally, we assign exp licit structure to the uncertainty set ∆

and the bound Ω (·) satisfying (11.52). First, we assume that the uncertainty

set ∆

to be of the form

∆

(

∆A ∈ R

n×n

: ∆A =

i=1

≤ 1

)

, (11.56)

where for i = 1, . . . , p, A

is a ﬁxed matrix denoting the str ucture of the

parametric uncertainty; α

is a given positive number; and δ

is an uncertain

parameter. Note that the uncertain parameters δ

are assumed to lie in a

speciﬁed ellipsoidal region in R

. For the structure ∆

as speciﬁed by

(11.56), the bound Ω(·) satisfying (11.52) can now be given a concrete form.

NonlinearBook10pt November 20, 2007

662 CHAPTER 11

Proposition 11.7. Let α > 0 and N = N

. Then the function

Ω(P ) = αP + α

−1

i=1

P A

(11.57)

satisﬁes (11.52) with ∆

given by (11.56).

Proof. Note that

0 ≤

i=1

[(

1/2

− (

1/2

]

P [(

1/2

− (

1/2

]

= α

i=1

(

)P + α

−1

i=1

P A

−

i=1

P + P A

which, since

i=1

/α

≤ 1, proves (11.52) with ∆

given by (11.56).

Note that ∆

given by (11.56) includes repeated parameters without

loss of generality. For example, if δ

= δ

then discard δ

and replace A

. Furthermore, ∆

includes real full b lock uncertainty. For example

∆A =





then ∆A =

i=1

, where



1 0

0 0



and likewise for A

, A

, and A

. Finally, for i = 1, . . . , p, letting A

= B

where B

∈ R

n×q

, C

∈ R

×n

, and q

≤ n, and deﬁnin g B

△

= [B

···B

]

and C

△

= [C

···C

]

, ∆

can be wr itten as

∆



∆A ∈ R

n×n

: ∆A = B

∆C

, ∆ = block−diag[δ

, . . . , δ

i=1

≤ 1, i = 1, . . . , p



. (11.58)

Since an uncertainty set of the form (11.58) can always be written in the

form of (11.56) by partitioning B

and C

as above and deﬁning A

= B

i = 1, . . . , p, robust stability of A + ∆A for all ∆A ∈ ∆

is equ ivalent to the

robust stability of the f eedback interconnection of G(s)

△

= C

(sI − A)

−1

and ∆ given in (11.58).

Next, assume the uncertainty set ∆

to be of the form

∆

= {∆A ∈ R

n×n

: ∆A = B

F C

, F

F ≤ N }, (11.59)

NonlinearBook10pt November 20, 2007

ROBUST NONLINEAR CONTROL 663

where B

∈ R

n×r

and C

∈ R

s×n

are ﬁxed matrices denoting the structure

of the uncertainty, F ∈ R

r×s

is an uncertain matrix, and N ∈ N

is a given

uncertainty bound. The special case N = γ

−2

, where γ > 0, is worth

noting. Speciﬁcally, in this case, (11.59) specializes to

∆

= {∆A ∈ R

n×n

: ∆A = B

F C

, σ

max

(F ) ≤ γ

−1

}, (11.60)

which corresponds to a small gain norm bounded uncertainty characteriza-

tion. For this uncertainty characterization, the bound Ω(·) satisfying (11.52)

can now be given a concrete form.

Proposition 11.8. Let α > 0. Then the function

Ω(P ) = αP B

P + α

−1

(11.61)

satisﬁes (11.52) with ∆

given by (11.59).

Proof. Note that

0 ≤ [α

1/2

P −α

−1/2

F C

]

[α

1/2

P − α

−1/2

F C

]

= αP B

P + α

−1

F C

− (C

P + P B

F C

)

≤ αP B

P + α

−1

−(∆A

P + P ∆A)

= Ω(P ) − (∆A

P + P ∆A),

which proves (11.52) with ∆

given by (11.59).

Finally, assume the uncertainty set ∆

to be of the form

∆

= {∆A ∈ R

n×n

: ∆A = B

F C

, F

QF + F

S + S

F + R ≤ 0},

(11.62)

where B

∈ R

n×r

and C

∈ R

s×n

are ﬁxed matrices denoting the structure

of the uncertainty, F ∈ R

r×s

is an u ncertain matrix, Q ∈ P

, S ∈ R

r×s

, and

−R ∈ N

. For this uncertainty characterization, the bound Ω(·) satisfying

(11.52) can now be given a concrete form.

Proposition 11.9. The function

Ω(P ) = [B

P − SC

]

−1

P −SC

] − C

(11.63)

satisﬁes (11.52) with ∆

given by (11.62).

Proof. Note that

0 ≤ [Q

1/2

F C

+ Q

−1/2

(SC

− B

P )]

1/2

F C

+ Q

−1/2

(SC

− B

P )]

= C

QF + F

S + S

F )C

− P B

F C

− C

+[B

P −SC

]

−1

P −SC

]

= C

QF + F

S + S

F + R)C

−(∆A

P + P ∆A) + Ω(P )

NonlinearBook10pt November 20, 2007

664 CHAPTER 11

≤ Ω(P ) − (∆A

P + P ∆A),

which proves (11.52) with ∆

given by (11.62).

As in the nonlinear case, we now combine the results of Corollary 11.2

and Pr opositions 11.7–11.9 to obtain a series of conditions guaranteeing

robust stability and performance for the linear uncertain system (11.50).

Proposition 11.10. Consider the linear uncertain system (11.50). Let

R ∈ P

, α, α

, . . . , α

> 0, and suppose there exists P ∈ P

satisfying

0 = A

P + P A

i=1

P A

+ R, (11.64)

where A

= A+

αI. Then A+∆A is asymptotically stable for all ∆A ∈ ∆

given by (11.56), and

sup

∆A∈∆

∆A

) ≤ x

P x

, x

∈ R

. (11.65)

Proposition 11.11. Consider the linear uncertain system (11.50). Let

R ∈ P

, α > 0, N ∈ N

, and suppose there exists P ∈ P

satisfying

0 = A

P + P A + αP B

P + α

−1

+ R. (11.66)

Then A + ∆A is asymptotically stable for all ∆A ∈ ∆

given by (11.59),

and

sup

∆A∈∆

∆A

) ≤ x

P x

, x

∈ R

. (11.67)

Proposition 11.12. Consider the linear uncertain system (11.50). Let

R ∈ P

, Q ∈ P

, S ∈ R

r×s

, and −R ∈ N

, and suppose there exists P ∈ P

satisfying

0 = A

P + P A + [B

P − SC

]

−1

P − SC

] −C

R. (11.68)

Then A + ∆A is asymptotically stable for all ∆A ∈ ∆

given by (11.62),

and

sup

∆A∈∆

∆A

) ≤ x

P x

, x

∈ R

. (11.69)

We conclude this section with several observations. First, note that

sup

∆A∈∆

∆A

) pr ovides a worst-case characterization of the H

norm

of the uncertain system (11.50) with an averaged free response. Speciﬁcally,

since A + ∆ A is asymptotically stable for all ∆A ∈ ∆

it follows that

sup

∆A∈∆

E[J

∆A

)] = sup

∆A∈∆



∞

(t)Rx(t)dt

