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

Подождите немного. Документ загружается.

NonlinearBook10pt November 20, 2007

OPTIMAL INTEGRATOR BACKSTEPPING CONTROL 565

˙x

(t) = u(t), x

(0) = x

. (9.48)

We start by considering the system

˙x

(t) = x

(t) − x

(t) + x

(t), x

(0) = x

, t ≥ 0, (9.49)

˙x

(t) = x

(t), x

(0) = x

, (9.50)

which has the form of (9.1) and (9.2) with x = x

, ˆx = x

, u = x

, f(x

) =

− x

, and G(x) = 1. To apply Proposition 9.1 we require a stabilizing

feedback for the subsystem (9.49) and a corresponding Lyapunov function

sub

) such that (9.14)–(9.17) are satisﬁed. For th e nonlinear subsystem

(9.49) we choose the Lyapunov function candidate V

sub

) =

and the

stabilizing feedback control α(x

) = −x

−x

. It is straightforward to show

that V

sub

) and α(x

) given above satisfy the conditions of Proposition

9.1. Hence, it follows from Proposition 9.1 that

= −(x

+ x

) −x

− (4x

+ 1)(x

− x

+ x

) (9.51)

is a globally stabilizing feedback control law for (9.49) and (9.50) w ith

Lyapunov function

, x

) = V

sub

) +

+ x

)

. (9.52)

Next, we view (9.46)–(9.48) as (9.1) and (9.2) with x = [x

]

ˆx = x

, f(x

, x

) = [x

− x

+ x

, and G(x

, x

) = [0 1]

. Now, with

, x

) = −(x

+ x

) −x

− (4x

+ 1)(x

− x

+ x

), (9.53)

and V

, x

) given by (9.52) it follows that all the conditions of Proposition

9.1 are satisﬁed. Hence, it follows from Proposition 9.1 that

u = −(x

− α

, x

)) − G

, x

) + α

′

, x

)

·[f(x

, x

) + G(x

, x

] (9.54)

globally stabilizes (9.46)–(9.49) with Lyapunov function

V (x

, x

) = V

, x

) +

− α

, x

))

. (9.55)

△

As shown in this section, backstepping control provides a systematic

procedure for ﬁnding a Lyapunov function f or cascade and block cascade

systems while choosing a stabilizing feedback control law. Since the control

designer has a signiﬁcant amount of freedom in designing the controller,

it is natural to ask the question: Does there exist analytical measures of

performance or notions of optimality for backstepping controllers? This

question is addressed in the remainder of this chapter.

NonlinearBook10pt November 20, 2007

566 CHAPTER 9

9.3 Optimal Integrator Backstepping Controllers

In this section, we develop an optimality-based framework f or backstepping

controllers. The key motivation for developing an optimal nonlinear back-

stepping control theory is that it provides a family of candidate backstepping

controllers parameterized by the cost functional that is minimized. In order

to ad dress the optimality-based backstepping nonlinear control problem

we use the nonlinear-nonquadratic optimal control framework developed in

Chapter 8. For our ﬁ rst result, deﬁne

L(x, ˆx, u)

△

= L

(x, ˆx) + L

(x, ˆx)u + u

(x, ˆx)u, (9.56)

where L

: R

×R

→ R, L

: R

×R

→ R

1×m

, and R

: R

×R

→ P

and d eﬁ ne

S(x

, ˆx

)

△

= {u(·) : u(·) ∈ U and (x(·), ˆx(·)) given by

(9.1) and (9.2) satisﬁes (x(t), ˆx(t)) → 0 as t → ∞}.

Theorem 9.1. Consider th e nonlinear cascade system (9.1) and (9.2)

with performance functional

J(x

, ˆx

, u(·))

△

∞

L(x(t), ˆx(t), u(t))dt, (9.57)

where u(·) is admissible, (x(t), ˆx(t)), t ≥ 0, satisﬁes (9.1) and (9.2),

and L(x, ˆx, u) is given by (9.56). Assume th at there exist continuously

diﬀerentiable functions α : R

→ R

and V

sub

: R

→ R such that

α(0) = 0, (9.58)

sub

(0) = 0, (9.59)

sub

(x) > 0, x ∈ R

, x 6= 0, (9.60)

′

sub

(x)[f(x) + G(x)α(x)] < 0, x ∈ R

, x 6= 0. (9.61)

Furth ermore, let L

: R

× R

→ R

1×m

be such that L

(0, 0) = 0 and

(ˆx −α(x))



(x)V

′T

sub

(x) − 2

′

(x)(f(x) + G(x)ˆx)

−1

(x, ˆx)[

P (ˆx − α(x)) +

(x, ˆx)]



< 0, (x, ˆx) ∈ R

× R

, ˆx 6= α(x),

(9.62)

where

P ∈ R

m×m

is an arbitrary positive-deﬁnite matrix. Then the zero

solution (x(t), ˆx(t)) ≡ (0, 0) of the cascade system (9.1) and (9.2) is globally

asymptotically stable with the feedback control law

u = φ(x, ˆx) = −R

−1

(x, ˆx)

P [ˆx − α(x)] −

−1

(x, ˆx)L

(x, ˆx). (9.63)

NonlinearBook10pt November 20, 2007

OPTIMAL INTEGRATOR BACKSTEPPING CONTROL 567

Furth ermore,

J(x

, ˆx

, φ(x(·), ˆx(·))) = V (x

, ˆx

), (x

, ˆx

) ∈ R

× R

, (9.64)

where

V (x, ˆx) = V

sub

(x) + [ˆx −α(x)]

P [ˆx −α(x)], (9.65)

and the performan ce functional (9.57), with

(x, ˆx) = φ

(x, ˆx)R

(x, ˆx)φ(x, ˆx) −V

′

sub

(x)[f(x) + G(x)ˆx]

+2[ˆx − α(x)]

P α

′

(x)[f(x) + G(x)ˆx], (9.66)

is minimized in the sense that

J(x

, ˆx

, φ(x(·), ˆx(·))) = min

u(·)∈S(x

,ˆx

)

J(x

, ˆx

, u(·)). (9.67)

Proof. The result follows as a direct consequence of Theorem 8.3

applied to the system

˜x(t) =

f(˜x(t)) +

G(˜x(t))u(t), ˜x(0) = ˜x

, t ≥ 0, (9.68)

where

˜x =



ˆx



f(˜x) =



f(x) + G(x)ˆx



G(˜x) =





Speciﬁcally, conditions (8.59)–(8.62) are trivially satisﬁed by (9.58)–(9.62)

and (9.65). Next, using (9.63) and (9.68), φ(x, ˆx) can be written as

φ(˜x) = −

−1

(˜x)[L

(˜x) +

(˜x)V

′T

(˜x)] (9.69)

so that (8.65) is satisﬁed. Finally, using (9.65) and (9.68) it follows that

(9.66) or, equivalently,

(˜x) = φ

(˜x)R

(˜x)φ(˜x) − V

′

(˜x)

f(˜x), (9.70)

satisﬁes (8.66).

Note th at the only restrictions on the choice of L

(·, ·) are the

conditions L

(0, 0) = 0 and (9.62) which must be satisﬁed. Therefore, a

signiﬁcant amount of freedom with regard to the form of the controller

is available to the control designer. Furthermore, since L

(x, ˆx) appears

explicitly in both the feedback control law φ(x, ˆx) and the cost functional

J(x

, ˆx

, u), it ties together the particular choice of control and the form

of the performance criterion for which the control is optimal. A particular

choice of L

(x, ˆx) satisfying condition (9.62) is given by

(x, ˆx) =

′

sub

(x)G(x)

−1

− 2[f (x) + G(x)ˆx]

′

(x)

(x, ˆx). (9.71)

In this case, for u(t) ∈ R, t ≥ 0, the feedback control law given by (9.63)

specializes to the integrator backstepp ing controller given by Lemma 2.8

NonlinearBook10pt November 20, 2007

568 CHAPTER 9

of [247] by setting

P =

and R

(x, ˆx) = 1/c.

The next result provides inverse optimal controllers for cascade

systems with guaranteed sector margins.

Theorem 9.2. Consider th e nonlinear cascade sys tem (9.1) and (9.2).

Assume that there exist continuously diﬀerentiable functions α : R

→ R

and V

sub

: R

→ R such that

α(0) = 0, (9.72)

sub

(0) = 0, (9.73)

sub

(x) > 0, x ∈ R

, x 6= 0, (9.74)

′

sub

(x)[f(x) + G(x)α(x)] < 0, x ∈ R

, x 6= 0. (9.75)

Then, with the f eedback s tabilizing control law given by

φ(x, ˆx) =



−(c

+ ρ(x, ˆx))β(x, ˆx), ˆx 6= α(x),

0, ˆx = α(x),

(9.76)

where β(x, ˆx)

△

= 2

P (ˆx −α(x)), µ(x, ˆx)

△

= 2α

′

(x)[f(x) + G(x)ˆx] −

−1

(x)

·V

′T

sub

(x),

P ∈ P

ρ(x, ˆx)

△

(β

(x, ˆx)µ(x, ˆx))

+ (β

(x, ˆx)β(x, ˆx))

− β

(x, ˆx)µ(x, ˆx)

(x, ˆx)β(x, ˆx)

and c

> 0, the cascade system (9.1) and (9.2) has a sector (and, hen ce, gain)

margin (

, ∞). Furthermore, with the feedback control law u = φ(x, ˆx) the

performance functional

J(x

, ˆx

, u(·)) =

∞

[β

(x(t), ˆx(t))µ(x(t), ˆx(t)) −V

′

sub

(x)[f(x)

+G(x)α(x)] −

η(x(t),ˆx (t))

(x(t), ˆx(t))β(x(t), ˆx(t))

2η(x(t),ˆx(t))

(t)u(t)]dt, (9.77)

where

η(x, ˆx)

△



+ ρ(x, ˆx), ˆx 6= α(x),

, ˆx = α(x),

(9.78)

is minimized in the sense that

J(x

, ˆx

, φ(x(·))) = min

u(·)∈S(x

,ˆx

)

J(x

, ˆx

, u(·)), (x

, ˆx

) ∈ R

× R

(9.79)

Proof. The result is a direct consequence of Corollary 8.8 and Theorem

8.10 with R

(x, ˆx) =

2η(x,ˆx)

and L

(x) = β

(x, ˆx)µ(x, ˆx)−V

′

sub

(x)[f(x)+

G(x)α(x)] +

η(x,ˆx)

(x, ˆx)β(x, ˆx). Speciﬁcally, it follows from (9.78) that

NonlinearBook10pt November 20, 2007

OPTIMAL INTEGRATOR BACKSTEPPING CONTROL 569

(x, ˆx) > 0, x ∈ R

, ˆx ∈ R

, and

(x, ˆx) = β

(x, ˆx)µ(x, ˆx) −V

′

sub

(x)[f(x) + G(x)α(x)]

η(x,ˆx)

(x, ˆx)β(x, ˆx)







−V

′

sub

(x)[f(x) + G(x)α(x)]

(x, ˆx)β(x, ˆx)(c

+ ρ(x, ˆx)), β(x) 6= 0,

−V

′

sub

(x)[f(x) + G(x)α(x)], β(x) = 0.

(9.80)

Now, it follows from (9.75) and (9.80) that L

(x, ˆx) ≥ 0, (x, ˆx) ∈ R

× R

so that all conditions of Corollary 8.8 are satisﬁed.

9.4 Optimal Linear Block Backstepping Controllers

In this section, we generalize the results of Section 9.2 to cascade systems

where the input subsystem is characterized by a linear time-invariant system.

Speciﬁcally, we consider a nonlinear system w ith a linear square input

subsystem having the form

˙x(t) = f(x(t)) + G(x(t))y(t), x(0) = x

, t ≥ 0, (9.81)

ˆx(t) = Aˆx(t) + Bu(t), ˆx(0) = ˆx

, (9.82)

y(t) = C ˆx(t), (9.83)

where ˆx ∈ R

, u, y ∈ R

, A ∈ R

q×q

, B ∈ R

q×m

, and C ∈ R

m×q

First, we consider the case in which the linear input subsystem (9.82)

is feedback strict positive real, that is, there exist matrices K ∈ R

m×q

P ∈ P

, and

Q ∈ N

such that

0 = (A + BK)

P +

P (A + BK) +

Q, (9.84)

0 = B

P − C. (9.85)

Recall that it follows from Theorem 6.12 that the input subsystem (9.82)

and (9.83) is feedback strict positive real if and only if d et(CB) 6= 0 and

(9.82) and (9.83) is minimum phase.

Theorem 9.3. Consider the nonlinear cascade system (9.81)–(9.83)

with performance functional (9.57) where L(x, ˆx, u) is given by (9.56).

Assume that the triple (A, B, C) is feedback strict positive real and the

nonlinear subsystem (9.81) has a globally stable equ ilibr ium at x = 0 with

y = 0 and Lyapunov function V

sub

: R

→ R so that

′

sub

(x)f(x) < 0, x ∈ R

, x 6= 0. (9.86)

Furth ermore, let L

: R

× R

→ R

1×m

be s uch that L

(0, 0) = 0 and

(x)V

′T

sub

(x) − R

−1

(x, ˆx)L

(x, ˆx) − 2Kˆx] ≤ 0, (x, ˆx) ∈ R

× R

(9.87)

NonlinearBook10pt November 20, 2007

570 CHAPTER 9

Then the zero solution (x(t), ˆx(t)) ≡ (0, 0) of the cascade system (9.81)–

(9.83) is globally asymptotically stable with the feedback control law

u = φ(x, ˆx) = −

−1

(x, ˆx)[2B

P ˆx + L

(x, ˆx)], (9.88)

where

P ∈ P

satisﬁes (9.84) and (9.85). Furthermore,

J(x

, ˆx

, φ(x(·), ˆx(·))) = V (x

, ˆx

), (x

, ˆx

) ∈ R

×R

, (9.89)

where

V (x, ˆx) = V

sub

(x) + ˆx

P ˆx, (9.90)

and the performan ce functional (9.57), with

(x, ˆx) = φ

(x, ˆx)R

(x, ˆx)φ(x, ˆx) + ˆx

P B(2K ˆx −G

(x)V

′T

sub

(x))

+ˆx

Qˆx − V

′

sub

(x)f(x), (9.91)

is minimized in the sense that

J(x

, ˆx

, φ(x(·), ˆx(·)) = min

u(·)∈S(x

,ˆx

)

J(x

, ˆx

, u(·)), (x

, ˆx

) ∈ R

× R

(9.92)

Proof. The result follows as a direct consequence of Theorem 8.3

applied to the system

˜x(t) =

f(˜x(t)) +

G(˜x(t))u(t), ˜x(0) = ˜x

, t ≥ 0, (9.93)

where

˜x =



ˆx



f(˜x) =



f(x) + G(x)y

Aˆx



G(˜x) =





Speciﬁcally, conditions (8.59)–(8.62) are trivially satisﬁed by (9.86), (9.87)

and (9.90). Next, using (9.88) and (9.93), φ(x, ˆx) can be written as

φ(˜x) = −

−1

(˜x)[L

(˜x) +

(˜x)V

′T

(˜x)] (9.94)

so that (8.65) is satisﬁed. Finally, using (9.90) and (9.93) it follows that

(9.91) or, equivalently,

(˜x) = φ

(˜x)R

(˜x)φ(˜x) − V

′

(˜x)

f(˜x) (9.95)

satisﬁes (8.66).

A particular choice of L

(x, ˆx) satisfying condition (9.87) is given by

(x, ˆx) =



′

sub

(x)G(x) − 2ˆx



(x, ˆx). (9.96)

In this case, the feedback control φ(x, ˆx) is given by

φ(x, ˆx) = K ˆx −R

−1

(x, ˆx)B

P ˆx −

(x)V

′T

sub

(x). (9.97)

NonlinearBook10pt November 20, 2007

OPTIMAL INTEGRATOR BACKSTEPPING CONTROL 571

Alternatively, choosing L

(x, ˆx) satisfying condition (9.87) by

(x, ˆx) = V

′

sub

(x)G(x)R

(x, ˆx) −2ˆx

(x, ˆx) +

P B), (9.98)

the f eedback control φ(x, ˆx) given by (9.88) specializes to

φ(x, ˆx) = K ˆx −

(x)V

′T

sub

(x). (9.99)

In the case where m = 1, (9.99) specializes to the control law obtained in

Lemma 2.13 of [247].

Next, we consider a class of systems in which the su bsystem (9.81) is

assumed to be globally stabilizable at x = 0 through y instead of having a

globally asymptotically stable equilibrium at x = 0 w ith y = 0.

Theorem 9.4. Consider the nonlinear cascade system (9.81)–(9.83)

with performance functional

J(x

, ˆx

, u) =

∞

L(x(t), ˆx(t), u(t))dt, (9.100)

where L(x, ˆx, u) is given by

L(x, ˆx, u) = L

(x, ˆx) + L

(x, ˆx)(CAˆx + CBu)

+(CAˆx + CBu)

(x, ˆx)(CAˆx + CBu), (9.101)

where u ∈ R

, L

: R

×R

→ R, L

: R

×R

→ R

1×m

satisﬁes L

(0, 0) = 0,

and R

: R

×R

→ P

. Assume that there exist continuously diﬀerentiable

functions α : R

→ R

and V

sub

: R

→ R such that

α(0) = 0, (9.102)

sub

(0) = 0, (9.103)

sub

(x) > 0, x ∈ R

, x 6= 0, (9.104)

′

sub

(x)[f(x) + G(x)α(x)] < 0, x ∈ R

, x 6= 0. (9.105)

Furth ermore, let L

(x, ˆx) satisfy

(y − α(x))



(x)V

′T

sub

(x) − 2

′

(x)(f(x) + G(x)y)

−1

(x, ˆx)[

P (y − α(x)) +

(x, ˆx)]



< 0, (x, y) ∈ R

× R

, y 6= α(x),

(9.106)

where

P ∈ R

m×m

is an arbitrary positive-deﬁnite matrix, and assume

that the linear subsystem (9.82) is m inimum phase and has relative degree

{1, 1, . . . , 1}. Th en the zero solution (x(t), ˆx(t)) ≡ (0, 0) of th e cascade

system (9.81)–(9.83) is globally asymptotically stable with the feedback

control law

u = (CB)

−1

(v − CAˆx), (9.107)

NonlinearBook10pt November 20, 2007

572 CHAPTER 9

where

v = φ(x, ˆx) = −R

−1

(x, ˆx)

P (y − α(x)) −

−1

(x, ˆx)L

(x, ˆx). (9.108)

Furth ermore,

J(x

, ˆx

, (CB)

−1

(φ(x, ˆx) −CAˆx)) = V (x

, C ˆx

), (x

, ˆx

) ∈ R

× R

(9.109)

where

V (x, y) = V

sub

(x) + [y − α(x)]

P [y − α(x)], (9.110)

and the performan ce functional (9.100), with

(x, ˆx) = φ

(x, ˆx)R

(x, ˆx)φ(x, ˆx) − V

′

sub

(x)(f(x) + G(x)y)

+2(y − α(x))

P α

′

(x)(f(x) + G(x)y), (9.111)

is minimized in the sense that

J(x

, ˆx

, (CB)

−1

(φ(x, ˆx) −CAˆx)) = min

u(·)∈S(x

,ˆx

)

J(x

, ˆx

, u(·)),

, ˆx

) ∈ R

× R

. (9.112)

Proof. Since the linear input subsystem (9.82) an d (9.83) has

relative degree {1, 1, . . . , 1}, it follows from Lemma 6.2 that there exists

a n on s ingular trans formation matrix T ∈ R

q×q

such that, in the coordinates





△

= T ˆx, (9.113)

the linear diﬀerential equation (9.82) is equivalent to the normal f orm

˙y(t) = CAˆx(t) + CBu(t), y(0) = C ˆx

, t ≥ 0, (9.114)

˙z(t) = A

z(t) + B

y(t), z(0) = z

, (9.115)

where A

∈ R

(q−m)×(q−m)

and B

∈ R

(q−m)×m

. Fur th ermore, the

eigenvalues of A

are the transmission zeros of the trans fer function matrix

H(s) = C(sI − A)

−1

B of the linear input subsystem (9.82) and (9.83)

(see Problem 6.39). Next, applying the feedback transformation (9.107)

to (9.114) yields the transformed cascade system

˙x(t) = f(x(t)) + G(x(t))y(t), x(0) = x

, t ≥ 0, (9.116)

˙y(t) = v(t), y(0) = C ˆx

, (9.117)

˙z(t) = A

z(t) + B

y(t), z(0) = z

. (9.118)

Initially ignoring the zero dynamics (9.118), it follows from Theorem

9.1, with ˆx replaced by y and u replaced by v, that the equilibrium x = 0,

y = 0 of (9.116) and (9.117) with the feedback control law v = φ(x, ˆx), where

φ(x, ˆx) is given by (9.108), is globally asymptotically stable. Now, since

the linear subsystem (9.82) is minimum phase, A

is Hurwitz. Since the

NonlinearBook10pt November 20, 2007

OPTIMAL INTEGRATOR BACKSTEPPING CONTROL 573

zero solution y(t) ≡ 0 to (9.117) can be made asymp totically stable, y(t) is

bounded and lim

t→∞

y(t) = 0. Hence, the zero solution z(t) ≡ 0 to (9.118) is

globally asymptotically stable. Furthermore, s ince ˆx = T

−1

]

, global

asymptotic stability of the zero solution (y(t), z(t)) ≡ (0, 0) to (9.117) and

(9.118) implies global asymptotic stability of the zero solution ˆx(t) ≡ 0 to

(9.82).

Next, with (9.116), (9.117), (9.101), and (9.110), the Hamiltonian has

the f orm

H(x, ˆx, u) = L

(x, ˆx) + L

(x, ˆx)(CAˆx + CBu)

+(CAˆx + CBu)

(x, ˆx)(CAˆx + CBu)

′

sub

(x)(f(x) + G(x)y)

+2(y −α(x))

P [CAˆx + CBu − α

′

(x)(f(x) + G(x)y)].

(9.119)

Now, the feedback control law (9.107) is obtained by setting

∂H

∂u

= 0. Using

(0, 0) = 0 it follows that φ(0, 0) = 0, which pr oves (8.43). Next, with

(x, ˆx) given by (9.111) it follows that (8.46) holds. Finally, since

H(x, ˆx, u) = [v − φ(x, ˆx)]

(x, ˆx)[v − φ(x, ˆx)],

and R

(x, ˆx) > 0, (x, ˆx) ∈ R

× R

, (8.45) holds. The result now follows as

a direct consequence of Theorem 8.3.

A particular choice of L

(x, ˆx) satisfying condition (9.106) is given by

(x, ˆx) =

−1

(x)V

′T

sub

(x) − 2α

′

(x)(f(x) + G(x)y)

(x, ˆx). (9.120)

In this case, for u(t) ∈ R, t ≥ 0, the control law given by (9.107) specializes

to the linear block backstepping controller obtained in Lemma 2.23 of [247]

by setting

P =

and R

(x, ˆx) = 1/c.

9.5 Optimal Nonlinear Block Backstepping Controllers

In this section, we generalize the results of Section 9.4 to the case wh ere the

input subsystem (9.82) and (9.83) is nonlinear. Speciﬁcally, consider th e

nonlinear cascade system

˙x(t) = f(x(t)) + G(x(t))y(t), x(0) = x

, t ≥ 0, (9.121)

ˆx(t) =

f(ˆx(t)) +

G(ˆx(t))u(t), ˆx(0) = ˆx

, (9.122)

y(t) = h(ˆx(t)), (9.123)

where ˆx ∈ R

, u, y ∈ R

f : R

→ R

satisﬁes

f(0, 0) = 0,

G : R

→ R

q×m

and h : R

→ R

satisﬁes h(0) = 0.

NonlinearBook10pt November 20, 2007

574 CHAPTER 9

First, we consider the case in w hich the input subsystem (9.122) and

(9.123) is feedb ack strictly passive. Speciﬁcally, we assume th ere exists a

positive-deﬁnite storage function V

: R

→ R and a function k : R

→ R

such that

0 > V

′

(ˆx)(

f(ˆx) +

G(ˆx)k(ˆx)), ˆx ∈ R

, ˆx 6= 0, (9.124)

0 =

(ˆx)V

′T

(ˆx) − h(ˆx). (9.125)

Recall that it follows from Theorem 6.12 that the nonlinear input subsystem

(9.122) and (9.123) is feedback strictly passive if and only if (9.122) and

(9.123) has relative d egree {1, 1, . . . , 1} at x = 0 and is minimum phase.

Theorem 9.5. Consider the nonlinear cascade system (9.121)–(9.123)

with performance functional (9.57) where L(x, ˆx, u) is given by (9.56).

Assume that the input su bsystem (9.122) and (9.123) is feedback strictly

passive with positive-deﬁnite storage function V

: R

→ R satisfying (9.124)

and (9.125) and the subsystem (9.121) h as a globally stable equilibrium at

x = 0 with y = 0 and Lyapunov function V

sub

: R

→ R so that

′

sub

(x)f(x) < 0, x ∈ R

, x 6= 0. (9.126)

Furth ermore, let L

: R

× R

→ R

1×m

be s uch that L

(0, 0) = 0 and

(x)V

′T

sub

(x) − R

−1

(x, ˆx)L

(x, ˆx) −2k(ˆx)] ≤ 0, (x, ˆx) ∈ R

× R

(9.127)

Then the zero solution (x(t), ˆx(t)) ≡ (0, 0) of the cascade system (9.121)–

(9.123) is globally asymptotically stable with the feedback control law

u = φ(x, ˆx) = −

−1

(x, ˆx)[

(ˆx)V

′T

(ˆx) + L

(x, ˆx)]. (9.128)

Furth ermore,

J(x

, ˆx

, φ(x(·), ˆx(·))) = V (x

, ˆx

), (x

, ˆx

) ∈ R

× R

, (9.129)

where

V (x, ˆx) = V

sub

(x) + V

(ˆx), (9.130)

and the performan ce functional (9.57), with

(x, ˆx) = φ

(x, ˆx)R

(x, ˆx)φ(x, ˆx) + V

′

(ˆx)

G(ˆx)(k(ˆx) −

(x)V

′T

sub

(x))

−V

′

(ˆx)(

f(ˆx) +

G(ˆx)k(ˆx)) −V

′

sub

(x)f(x), (9.131)

is minimized in the sense that

J(x

, ˆx

, φ(x(·), ˆx(·))) = min

u(·)∈S(x

,ˆx

)

J(x

, ˆx

, u(·)), (x

, ˆx

) ∈ R

× R

(9.132)

Proof. The result follows as a direct consequence of Theorem 8.3