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

Подождите немного. Документ загружается.

NonlinearBook10pt November 20, 2007

STRUCTURED PARAMETRIC UNCERTAINTY 735

Proposition 12.3. Let H ∈ H

and N ∈ N

be such th at R

(x) > 0,

x ∈ R

. Then the functions

Γ(x) =

′

(x)G

(x) +

(x)]R

−1

(x)[V

′

(x)G

(x) +

(x)]

−m

(x))Hm

(x)), (12.63)

∆f

(x) =

(x)

(y)Ndy, (12.64)

satisfy (12.52) with F given by (12.58) and ∆ given by (12.59).

Proof. First, note that if H ∈ H

then it follows that

[δ(y) − m

(y)]

H[δ(y) − m

(y)] ≤ 0, y ∈ R

, δ(·) ∈ ∆

Next, since δ

(y)N is a gradient vector of a real-value function the line

integral V

∆f

(x) =

(x)

(y)Ndy is well deﬁned, and hence, V

′

∆f

(x) =

(x))Nh

′

(x). Now,

0 ≤

′

(x)G

(x) +

(x) − R

(x)δ(h

(x))]R

−1

(x)

·[V

′

(x)G

(x) +

(x) − R

(x)δ(h

(x))]

= Γ(x) + m

(x))Hm

(x)) − V

′

(x)G

(x)δ(h

(x))

−

(x)δ(h

(x)) +

(x))R

(x)δ(h

(x))

= Γ(x) −V

′

(x)∆f(x) −δ

(x))Nh

′

(x)[f

(x) + G

(x)δ(h

(x))]

+[δ(h

(x)) −m

(x))]

H[δ(h

(x)) −m

(x))]

≤ Γ(x) −V

′

(x)∆f(x) −V

′

∆f

(x)[f

(x) + G

(x)δ(h

(x))],

which proves (12.52) with F given by (12.58) and ∆ given by (12.59).

Next, consider the nonlinear uncertain dynamical system (12.48) and

assume that the uncertainty set ∆ is given by (12.58) with ∆ given by

∆ = {δ(·) ∈ ∆

: δ

(y)Qδ(y) + 2δ

(y)Sy + y

Ry ≤ 0, y ∈ R

}, (12.65)

where Q ∈ P

, −R ∈ N

, and S ∈ R

×p

. For this uncertainty

characterization, the bounding function Γ(·) satisfying (12.52) can be given a

concrete form . For the following result deﬁne

h(x)

△

= Nh

′

(x)f

(x)−2Sh

(x)

and R

△

= 2Q − Nh

′

(x)G

(x) − G

(x)h

′T

(x)N.

Proposition 12.4. Let N ∈ N

be such that R

(x) > 0, x ∈ R

. Then

the f unctions

Γ(x) =

′

(x)G

(x) +

(x)]R

−1

′

(x)G

(x) +

(x)]

−h

(x)Rh

(x), (12.66)

∆f

(x) =

(x)

(y)Ndy, (12.67)

NonlinearBook10pt November 20, 2007

736 CHAPTER 12

satisfy (12.52) with F given by (12.58) and ∆ given by (12.65).

Proof. Since δ

(y)N is a gradient vector of a real-value function the

line integral V

∆f

(x) =

(x)

(y)Ndy is well deﬁned, and hence, V

′

∆f

(x) =

(x))Nh

′

(x). Now,

0 ≤

′

(x)G

(x) +

(x) − R

(x)δ(h

(x))]R

−1

(x)

·[V

′

(x)G

(x) +

(x) − R

(x)δ(h

(x))]

= Γ(x) + h

(x)Rh

(x) − V

′

(x)G

(x)δ(h

(x))

−

(x)δ(h

(x)) +

(x))R

(x)δ(h

(x))

= Γ(x) −V

′

(x)∆f(x) −δ

(x))Nh

′

(x)[f

(x) + G

(x)δ(h

(x))]

+δ

(x))Qδ(h

(x)) + 2δ

(x))Sh

(x) + h

(x)Rh

(x)

≤ Γ(x) −V

′

(x)∆f(x) −V

′

∆f

(x)[f

(x) + G

(x)δ(h

(x))],

which proves (12.52) with F given by (12.58) and ∆ given by (12.65).

We now combin e the results of Corollary 12.1 and Propositions 12.3

and 12.4 to obtain conditions guaranteeing robust stability and performance

for the nonlinear system (12.48). For the statement of the next result

deﬁne

h(x)

△

= H[m

(x)) + m

(x))] + Nh

′

(x)f

(x) and R

(x)

△

2H − Nh

′

(x)G

(x) − G

(x)h

′T

(x)N.

Proposition 12.5. Consider th e nonlinear uncertain system (12.48).

Let L(x) > 0, x ∈ R

, let H ∈ H

, let N ∈ N

, and let R

(x) > 0, x ∈ R

Assume th at there exists a continuously diﬀerentiable fun ction V

: R

→ R

such that V

(0) = 0, V

(x) > 0, x ∈ R

, x 6= 0, V

(x) → ∞ as kxk → ∞, and

0 = V

′

(x)f

(x) +

′

(x)G

(x) +

h(x)]R

−1

(x)[V

′

(x)G

(x) +

h(x)]

−m

(x))Hm

(x)) + L(x), x ∈ R

. (12.68)

Then the zero solution x(t) ≡ 0 to (12.48) is globally asymptotically stable

for all ∆f(·) ∈ ∆, with ∆ given by (12.59), and

∆f

) ≤ J(x

) = V (x

), ∆f(·) ∈ ∆, (12.69)

where V (x) = V

(x) + V

∆f

(x) with V

∆f

(x) =

(x)

(y)Ndy, and

J(x

)

△

∞

[L(x(t)) + Γ(x(t)) −V

′

∆f

(x(t))f

(x(t))]dt, (12.70)

and where Γ(x) is given by (12.63) and x(t), t ≥ 0, is the solution to (12.48)

with ∆f (x) ≡ 0.

For the following result d eﬁ ne

h(x)

△

= Nh

′

(x)f

(x) − 2Sh

(x) and

△

= 2Q − Nh

′

(x)G

(x) − G

(x)h

′T

(x)N.

NonlinearBook10pt November 20, 2007

STRUCTURED PARAMETRIC UNCERTAINTY 737

Proposition 12.6. Consider th e nonlinear un certain system (12.48).

Let L(x) > 0, x ∈ R

, let N ∈ N

, and let R

(x) > 0, x ∈ R

. Assume that

there exists a continuously d iﬀerentiable function V

: R

→ R such that

(0) = 0, V

(x) > 0, x ∈ R

, x 6= 0, V

(x) → ∞ as kxk → ∞, and

0 = V

′

(x)f

(x) +

′

(x)G

(x) +

h(x)]R

−1

(x)[V

′

(x)G

(x) +

h(x)]

−h

(x)Rh

(x) + L(x), x ∈ R

. (12.71)

Then the zero solution x(t) ≡ 0 to (12.48) is globally asymptotically stable

for all ∆f(·) ∈ ∆, with ∆ given by (12.65), and

∆f

) ≤ J(x

) = V (x

), ∆f (·) ∈ ∆, (12.72)

where V (x) = V

(x) + V

∆f

(x) with V

∆f

(x) =

(x)

(y)Ndy, and

J(x

)

△

∞

[L(x(t)) + Γ(x(t)) −V

′

∆f

(x(t))f

(x(t))]dt, (12.73)

and where Γ(x) is given by (12.66) and x(t), t ≥ 0, is the solution to (12.48)

with ∆f (x) ≡ 0.

The following corollary specializes Theorem 12.4 to a class of linear

uncertain systems which connects the framework of Theorem 12.4 to

the parameter-dependent Lyapunov bounding framework of Haddad and

Bernstein [145,151]. Speciﬁcally, in this case we consider ∆ 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 bou nded uncertainty set of uncertain

perturbations ∆A of the nominal system matrix A such that 0 ∈ ∆

Corollary 12.2. Let R ∈ P

. Consider the linear uncertain dynamical

system

˙x(t) = (A + ∆A)x(t), x(0) = x

, t ≥ 0, (12.74)

with performance functional

∆A

)

△

∞

(t)Rx(t)dt, (12.75)

where ∆A ∈ ∆

. Let Ω

: N ⊆ S

→ S

and P

: ∆

→ S

be s uch that

∆A

P + P ∆A ≤ Ω

(P ) − [(A + ∆A)

(∆A) + P

(∆A)(A + ∆A)],

∆A ∈ ∆

, P ∈ N. (12.76)

Furth ermore, suppose there exists P ∈ P

satisfying

0 = A

P + P A + Ω

(P ) + R (12.77)

NonlinearBook10pt November 20, 2007

738 CHAPTER 12

and such that P +P

(∆A) ∈ P

, ∆ A ∈ ∆

. Then the zero solution x(t) ≡ 0

to (12.74) is globally asymptotically stable for all ∆A ∈ ∆

, and

sup

∆A∈∆

∆A

) ≤ sup

∆A∈∆

J(x

) = x

P x

+ su p

∆A∈∆

(∆A)x

, x

∈ R

(12.78)

where

J(x

)

△

∞

(t)[R + Ω

(P ) −A

(∆A) −P

(∆A)A]x(t)dt, (12.79)

and where x(t), t ≥ 0, solves (12.74) with ∆A = 0. If, in addition, there

exists P

∈ S

such that P

(∆A) ≤ P

, ∆ A ∈ ∆

, then

sup

∆A∈∆

∆A

) ≤ x

(P + P

. (12.80)

Proof. The result is a direct consequ en ce of Theorem 12.4 with

f(x) = (A + ∆A)x, f

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

Rx, V

(x) = x

P x,

∆f

(x) = x

(∆A)x, Γ(x) = x

Ω

(P )x, and D = R

. Speciﬁcally,

conditions (12.38) and (12.39) are trivially satisﬁed. Now, V

′

(x)f(x) =

[(A + ∆A)

(P + P

(∆A)) + (P + P

(∆A))(A + ∆A)]x, and hence, it

follows from (12.76) 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 (12.77) 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 12.4 are satisﬁed. Finally, since

V (x) is radially unbounded (12.74) is globally asymptotically stable for all

∆A ∈ ∆

Corollary 12.2 is the deterministic version of Theorem 3.1 of [151]

involving parameter-dependent Lyapunov fun ctions for addressing rob ust

stability and performance analysis of linear uncertain systems with constant

real parameter uncertainty.

Next, we illustrate two bounding fun ctions for two diﬀerent uncer-

tainty characterizations. Speciﬁcally, we assign explicit structure to the

uncertainty set ∆ and the function Ω

(·) and P

(·) satisfying (12.76). First,

we assume that the uncertainty set ∆

is of the form

∆

= {∆A : ∆A = B

F C

, F ∈ ∆} (12.81)

where ∆ satisﬁes

∆ ⊂ ∆

= {F ∈ S

: M

≤ F ≤ M

}, (12.82)

and where B

∈ R

n×s

, C

∈ R

s×n

, are ﬁxed matrices denoting the structure

of the uncertainty, F ∈ S

is a symmetric uncertainty matrix, and M

, M

∈

∆

are uncertainty bounds such that M

△

= M

−M

∈ P

. Note that ∆ may

be equal to ∆

, although, for generality, ∆ may be a speciﬁed proper subset

NonlinearBook10pt November 20, 2007

STRUCTURED PARAMETRIC UNCERTAINTY 739

of ∆

. For example, ∆

may consist of block-structured matrices F =

block−diag[I

ℓ

⊗ F

, . . . , I

ℓ

⊗ F

]. Note that if F = block−diag[F

, . . . , F

= block−diag[M

, . . . , M

], and M

= block−diag[M

, . . . , M

], then

≤ F

≤ M

, i = 1, . . . , r. Furth ermore, we assume that 0 ∈ ∆ and

, M

∈ ∆. Finally, we deﬁne the sets H

, N

, and N

such that the

product of every matrix in H

and N

and every matrix in ∆

is symmetric

and the p roduct of every matrix in N

and F − M

, where F ∈ ∆

, is

nonnegative deﬁnite by

△

= {H ∈ P

: F H = HF, F ∈ ∆

△

= {N ∈ S

: F N = NF, F ∈ ∆

and

△

= {N ∈ N

: (F − M

)N ≥ 0, F ∈ ∆

The cond ition F N = NF , F ∈ ∆

, is analogous to the commuting

assumption between the D-scales and ∆ uncertainty b locks in µ-analysis

which accounts for structure in the uncertainty ∆. For the statement

of the next result deﬁne

△

[H(M

+ M

+ NC

A] and R

△

H −

[NC

− B

N].

Proposition 12.7. Let M

, M

∈ S

, H ∈ H

, N ∈ N

, and R

> 0.

Then the functions

Ω

(P ) = [

C + B

P ]

−1

[

C + B

P ] −

+ M

(12.83)

(F ) = C

F NC

, (12.84)

satisfy (12.76) with ∆ given by (12.82).

Proof. The proof is a direct consequence of Proposition 12.4.

An alternative to Proposition 12.7 is given below. For the statement

of the next result deﬁne

△

= HC

+ NC

(A + B

), R

△

= (HM

−1

−

) + (HM

−1

− N C

)

, and F

△

= F − M

Proposition 12.8. Let M

, M

∈ S

, H ∈ H

, N ∈ N

, and R

> 0.

Then the functions

Ω

(P ) = [

C + B

P ]

−1

[

C + B

P ] + P B

+ C

(12.85)

(F ) = C

, (12.86)

satisfy (12.76) with ∆ given by (12.82).

Proof. Since R

> 0 and by Problem 12.1 F

− F

−1

≥ 0, it

NonlinearBook10pt November 20, 2007

740 CHAPTER 12

follows that

0 ≤ [

C + B

P − R

]

−1

[

C + B

P − R

]

+2C

H[F

− F

−1

= Ω

(P ) − P B

− C

P − [

C + B

P ]

−C

[

C + B

P ] + C

+2C

H[F

− F

−1

= Ω

(P ) − A

− C

−P B

F C

−C

A − C

− C

F B

−C

− C

= Ω

(P ) − [(A + ∆A)

(F ) + P

(F )(A + ∆A)] − [∆A

P + P ∆A],

which proves (12.76) with ∆ given by (12.82).

Next, assume the uncertainty set ∆ to be of the form

∆ ⊂ ∆

= {F ∈ R

m×p

: F

QF + F

S + S

F + R ≤ 0}, (12.87)

where Q ∈ P

, S ∈ R

m×p

, and −R ∈ N

. For the statement of the next

result deﬁne

△

= {N ∈ R

p×m

: F

N = N

F ≥ 0, F ∈ ∆

△

= N C

A − SC

, and R

△

= Q −NC

− B

Proposition 12.9. Let N ∈ N

and let R

> 0. Then th e functions

Ω

(P ) = [

C + B

P ]

−1

[

C + B

P ] − C

, (12.88)

(F ) = C

, (12.89)

satisfy (12.76) with ∆ given by (12.89).

Proof. Since R

> 0 it follows that

0 ≤ [

C + B

P − R

F C

]

−1

[

C + B

P − R

F C

]

−1

C −

F C

− C

C + C

F C

= Ω

(P ) + C

− A

F C

−C

F C

− P B

F C

−C

A − C

− C

P + C

QF C

−C

F C

− C

= Ω(P ) − [(A + ∆A)

(F ) + P

(F )(A + ∆A)] − [∆A

P + P ∆A]

QF + F

S + S

F + R]C

= Ω(P ) − [(A + ∆A)

(F ) + P

(F )(A + ∆A)] − [∆A

P + P ∆A],

which proves (12.76) with ∆ given by (12.89).

NonlinearBook10pt November 20, 2007

STRUCTURED PARAMETRIC UNCERTAINTY 741

As in the nonlinear case, we now combine the results of Corollary 12.2

and Propositions 12.8 and 12.9 to obtain conditions guaranteeing robus t

stability and performance for the linear system (12.74). For the statement

of the next result deﬁne

△

= HC

+ N C

+ B

), R

△

= (HM

−1

−

) + (HM

−1

− N C

)

, and F

△

= F − M

Proposition 12.10. Consider the linear uncertain system (12.74). Let

R ∈ P

, M

∈ S

, H ∈ H

, N ∈ N

, and R

> 0. Furthermore,

suppose there exists a P ∈ P

satisfying

0 = (A + B

)

P + P (A + B

) + [

C + B

P ]

−1

[

C + B

P ] + R.

(12.90)

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

with ∆ given by

(12.82), and

sup

∆A∈∆

∆A

) ≤ x

P x + sup

∆A∈∆

≤ x

(P + P

, x

∈ R

(12.91)

where P

∈ S

is s uch that F

N ≤ P

for all F

∈ ∆.

For the statement of the next result deﬁne

△

= NC

A − SC

and

△

= Q − N C

−B

Proposition 12.11. Consider the linear uncertain system (12.74). Let

R ∈ P

, N ∈ N

, and R

> 0. Furth ermore, suppose there exists a P ∈ P

satisfying

0 = A

P + P A + [

C + B

P ]

−1

[

C + B

P ] − C

R. (12.92)

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

with ∆ given by

(12.89), and

sup

∆A∈∆

∆A

) ≤ x

P x+ s up

∆A∈∆

≤ x

(P +P

, x

∈ R

(12.93)

where P

∈ S

is s uch that F

N ≤ P

for all F

∈ ∆.

Next, we use Theorem 12.2 and Corollary 12.2 to provide robust sta-

bility analysis tests for linear uncertain systems with structured uncertainty

via an algebraic Riccati equation. Consider the linear uncertain dynamical

system (12.74) with ∆A = B

F C

, where B

∈ R

n×p

, C

∈ R

p×n

denote the

structure of the uncertainty, and F ∈ ∆

⊆ R

p×p

is the uncertain matrix.

Here, ∆

is assumed to be the of the form

∆

= {F ∈ R

p×p

: F = F

, F = block−diag[δ

, . . . , δ

; ∆

r+1

, . . . ,

∆

r+q

], δ

∈ R, i = 1, . . . , r; ∆

∈ R

×p

, i = r + 1, . . . , r + q},

(12.94)

NonlinearBook10pt November 20, 2007

742 CHAPTER 12

where the integers l

, i = 1, . . . , r, p

, i = r + 1, . . . , r + q, are such that

i=1

r+q

i=r+1

= p. The following result provides a necessary and

suﬃcient condition for robust stability of the feedback interconnection of

G(s)

△

= C

(sI − A)

−1

and F for all F ∈ ∆

, where

∆

= {F ∈ ∆ : σ

max

(F ) ≤ γ

−1

}, (12.95)

and where γ > 0.

Theorem 12.5. Let γ > 0 and suppose A is Hurwitz. Then A+B

F C

is Hurwitz for all F ∈ ∆

if and only if

µ(G(ω)) < γ, ω ∈ R ∪ ∞. (12.96)

Proof. The result is a direct consequence of Theorem 12.2 with G(s) =

(sI − A)

−1

and ∆ = F .

Next, we use Corollary 12.2 to pr ovide an upper bound to the

structured singular value via an algebraic Riccati equation. For the following

result the set of compatible scaling matrices H

and N

are given by

= {H ∈ P

: HF = F H, F ∈ ∆

}, (12.97)

= {N ∈ S

: F N = N F, F N ≥ γ

−1

N, F ∈ ∆

}, (12.98)

and R

and

C are given by

= γH −NC

−C

C = HC

+N C

(A−γ

−1

). (12.99)

Theorem 12.6. Let γ > 0, R ∈ P

, H ∈ H

, N ∈ N

, and R

0. Cons ider the linear uncertain dynamical system (12.74) where ∆A =

F C

, F ∈ ∆

, with perform ance functional (12.75). Furthermore, assume

there exist P ∈ P

such that

0 = (A−γ

−1

)

P +P (A−γ

−1

)+(

C + B

P )

−1

(

C +B

P )+ R.

(12.100)

Then the zero solution x(t) ≡ 0 to (12.74) is globally asymptotically stable

for all F ∈ ∆

, and hence,

µ(G(ω)) < γ, ω ∈ R ∪ ∞. (12.101)

Furth ermore,

sup

F ∈∆

) ≤ su p

F ∈∆

J(x

) = x

P x

+ sup

F ∈∆

(F )x

, x

∈ R

(12.102)

where

J(x

)

△

∞

(t)(R + Ω

(P ) − A

(F ) − P

(F )A)x(t)dt, (12.103)

NonlinearBook10pt November 20, 2007

STRUCTURED PARAMETRIC UNCERTAINTY 743

(F )

△

= C

(F − γ

−1

I)NC

, and x(t), t ≥ 0, solves (12.74) with F = 0. If,

in addition, there exists P

∈ S

such that P

(F ) ≤ P

, F ∈ ∆

, then

sup

F ∈∆

) ≤ x

(P + P

. (12.104)

Proof. The result is a direct consequence of Corollary 12.2 and

Theorem 12.5.

Setting N = 0 and H = I in (12.100) yields

0 = A

P + P A + γ

−1

P B

P + γ

−1

+ R, (12.105)

or, equivalently,

0 = A

P + P A + γ

−2

P B

P + C

+ R. (12.106)

If only robust stability is of interest, then the weighting matrix R need not

have physical signiﬁcance. In th is case, one can set R = εI, where ε > 0

is arbitrarily small. Hence, it follows from the results of Chapter 10 that

if there exists a P ∈ P

such that (12.106) is satisﬁed, then |||G|||

∞

≤ γ,

where G(s) = C

(sI − A)

−1

. Thus, it follows from Theorem 12.6 that

µ(G(ω)) ≤ |||G|||

∞

12.4 Robust Optimal Control for Nonlinear Systems via

Parameter-Dependent Lyapunov Functions

In this section, we consider a control problem involving a notion of optimality

with respect to an auxiliary cost which guarantees a bound on the worst-

case value of a nonlinear-nonquadratic cost criterion over a prescribed

uncertainty set. The optimal robust feedback controllers are derived as a

direct consequence of Theorem 12.4. To address the robust optimal control

problem let D ⊂ R

be an open set and let U ⊂ R

, where 0 ∈ D and 0 ∈ U.

Furth ermore, let F ⊂ {F : D × U → R

: F (0, 0) = 0}. Next, consider the

controlled uncertain system

˙x(t) = F (x(t), u(t)), x(0) = x

, t ≥ 0, (12.107)

where F (·, ·) ∈ F and the control u(·) is restricted to the class of admissible

controls consisting of measurab le functions u(·) such that u(t) ∈ U for all

t ≥ 0, where the control constraint set U is given. We assume 0 ∈ U. Given

a control law φ(·) and a feedback control u(t) = φ(x(t)), the closed-loop

system has the form

˙x(t) = F (x(t), φ(x(t))), x(0) = x

, t ≥ 0, (12.108)

for all F (·, ·) ∈ F.

NonlinearBook10pt November 20, 2007

744 CHAPTER 12

Next, we present an extension of Theorem 11.2 for characterizing

robust feedback controllers that guarantee robust stability over a class

of nonlinear uncertain systems and min imize an auxiliary performance

functional. For the statement of this result let L : D × U → R and deﬁne

the set of r egulation controllers for the nominal nonlinear system F

(·, ·) by

S(x

)

△

= {u(·) : u(·) is admissible and x(·) given by (12.107)

satisﬁes x(t) → 0 as t → ∞ with F (·, ·) = F

(·, ·)}.

Theorem 12.7. Consider the nonlinear uncertain controlled system

(12.107) with performance functional

, u(·))

△

∞

L(x(t), u(t))dt, (12.109)

where F (·, ·) ∈ F and u(·) is an admissible control. Assum e there exist

functions V

, V

∆f

, V : D → R, Γ : D × U → R, and control law φ : D → U,

where V

(·) and V

∆f

(·) are continuously diﬀerentiable functions such that

(x) + V

∆f

(x) = V (x) for all x ∈ D and

V (0) = 0, (12.110)

V (x) > 0, x ∈ D, x 6= 0, (12.111)

φ(0) = 0, (12.112)

′

(x)F (x, φ(x)) ≤ V

′

(x)F

(x, φ(x)) + Γ(x, φ(x)), x ∈ D, F (·, ·) ∈ F,

(12.113)

′

(x)F

(x, φ(x)) + Γ(x, φ(x)) < 0, x ∈ D, x 6= 0, (12.114)

H(x, φ(x)) = 0, x ∈ D, (12.115)

H(x, u) ≥ 0, x ∈ D, u ∈ U, (12.116)

where F

(·, ·) ∈ F deﬁnes the n omin al system and

H(x, u)

△

= L(x, u) + V

′

(x)F

(x, u) + Γ(x, u). (12.117)

Then, with the feedback control u(·) = φ(x(·)), there exists a neighborhood

of th e origin D

⊆ D such that if x

∈ D

, the zero solution x(t) ≡ 0

of the closed-loop s ystem (12.108) is locally asymptotically stable for all

F (·, ·) ∈ F. Furthermore,

, φ(x(·))) ≤ J(x

, φ(x(·))) = V (x

), F (·, ·) ∈ F, (12.118)

where

J(x

, u(·))

△

∞

[L(x(t), u(t)) + Γ(x(t), u(t)) − V

′

∆f

(x(t))F

(x(t), u(t))]dt,

(12.119)

and where u(·) is admissible and x(t), t ≥ 0, solves (12.107) with

F (x(t), u(t)) = F

(x(t), u(t)). In addition, if x

∈ D

then the feedback