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

Подождите немного. Документ загружается.

NonlinearBook10pt November 20, 2007

ROBUST NONLINEAR CONTROL 705

the zero solution x

≡ 0 to (11.235) is globally asymptotically stable (in the

sense of Problem 3.65) for all τ

≥ 0 if |||DG(s)D

−1

|||

∞

< 1, where D is a

positive-deﬁnite matrix and

G(s) ∼



A A

I 0



Furth ermore, sh ow that this problem can be represented as a feedback prob-

lem involving an uncertain diagonal operator ∆(s) satisfying |||∆(s)|||

∞

≤ 1.

Problem 11.23. Let D ⊆ R

, f : D → R

such that f(0) = 0, G :

D → R

n×m

, and R

∈ P

. Consider the nonlinear uncertain system

˙x(t) = f (x(t)) + G(x(t))δ(x(t)) + G(x(t))u(t), x(0) = x

, t ≥ 0,

(11.236)

with performance functional

, u(·))

△

∞

(t)R

x(t)dt, (11.237)

where u(·) is an admissible control and δ(·) ∈ ∆

△

= {δ : D → R

: δ(0) =

0, δ

(x)R

δ(x) ≤ δ

max

(x), x ∈ D}, where R

∈ P

and δ

max

: D → R such

that δ(0) = 0 are given. Assume that there exist a continuously diﬀerentiable

function V : D → R and a control law φ(·) such that (11.96)–(11.98) hold

and

x + V

′

(x)(f(x) + G(x)φ(x)) + δ

max

(x) + φ

(x)R

φ(x) = 0, x ∈ D.

(11.238)

Show that th ere exists a neighborhood of the origin D

⊆ D such that if

∈ D

, then th e closed-loop system of (11.236) with the feedback control

law φ(x)

△

= −

−1

(x)V

′T

(x) is locally asymptotically stable for all δ(·) ∈

∆

, and

sup

δ(·)∈∆

, φ(x(·))) ≤ J(x

, φ(x(·))) = V (x

), (11.239)

where

J(x

, u(·))

△

∞

(t)R

x(t) + u

(t)R

u(t) + δ

max

(x(t))]dt, (11.240)

and where u(·) is admissible and x(t), t ≥ 0 solves (11.236) with δ(x) ≡ 0.

Furth ermore, if x

∈ D

, show that

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), (11.241)

where S(x

) is the set of r egulation controllers for the nominal system

˙x(t) = f (x(t)) + G(x(t))u(t). (11.242)

NonlinearBook10pt November 20, 2007

706 CHAPTER 11

Problem 11.24. Let D ⊆ R

, f

: D → R

such that f

(0) = 0,

: D → R

n×m

, G : D → R

n×m

, R

∈ P

, and R

∈ P

. Consider the

nonlinear uncertain system

˙x(t) = f

(x(t)) + G

(x(t))δ(x(t)) + G(x(t))u(t), x(0) = x

, t ≥ 0,

(11.243)

with performance functional

, u(·))

△

∞

(t)R

x(t) + u

(t)R

u(t)]dt, (11.244)

where u(·) is admissible control and δ(·) ∈ ∆

△

= {δ : D → R

: δ(0) =

0, δ

(x)R

δ(x) ≤ δ

max

(x), kR

1/2

†

(x)G

(x)δ(x)k

≤ γ

max

(x), x ∈ D},

where δ

max

, γ

max

: D → R su ch that δ

max

(0) = 0, γ

max

(0) = 0, are given.

Assume that there exist a continuously diﬀerentiable f unction V : D → R

and control laws φ

(·), φ

(·) such that φ

(0) = 0, φ

(0) = 0, (11.96) and

(11.97) hold, and

0 = x

x + V

′

(x)[f

(x) + G(x)φ

(x) + (I

− G(x)G

†

(x))G

(x)φ

(x)]

+δ

max

(x) + γ

max

(x) + φ

(x), x ∈ D, (11.245)

where

= ρR

, ρ > 0, 2ρ

kφ

(x)k

≤ λ

min

)kxk

, x ∈ D, and

min

) < ρ. Show th at there exits a neighborhood of the origin D

⊆ D

such that if x

∈ D

, then the closed-loop system (11.243) with feedback

control law u = φ

(x) = −

−1

(x)V

′T

(x) is locally asymptotically stable

for all δ(·) ∈ ∆

, and

sup

δ(·)∈∆

, φ

(x(·)) ≤ J(x

, φ

(·), φ

(·)) = V (x

), (11.246)

where

J(x

, u

(·), u

(·))

△

∞

(t)R

x(t) + u

(t) + u

(t)

+ρ

max

(x(t)) + γ

max

(x(t))]dt, (11.247)

and where u

(·), u

(·) are admissible and x(t), t ≥ 0, s olves

˙x(t) = f

(x(t)) + G(x(t))u

(t) + [I

− G(x(t))G

†

(x(t))]G

(x(t))u

(t),

x(0) = x

, t ≥ 0. (11.248)

(Hint: Decompose the uncertainty G

(x)δ(x) into the sum of a matched

component and an unm atched component by projecting G

(x)δ(x) onto the

range of G(x), that is,

(x)δ(x) = G(x)G

†

(x)G

(x)δ(x) −(I

−G(x)G

†

(x))G

(x)δ(x).) (11.249)

Problem 11.25. Show that the robust stability and performance

conditions given in Problem 11.24 still hold if G(x(t))u(t) in (11.243) is

NonlinearBook10pt November 20, 2007

ROBUST NONLINEAR CONTROL 707

replaced by G(x(t))(u(t)+ J

(x(t))u(t)), where J

: D → R

m×m

is uncertain

and satisﬁes J

(x) ≥ 0, x ∈ D.

Problem 11.26. Consider the nonlinear uncertain dynamical system

(11.125) and let (∆f, ∆G) ∈ ∆

× ∆

, where ∆

× ∆

is given by

∆

× ∆

△

= {(∆f, ∆G) : ∆f (x) = G

(x)∆(x)h

(x),

∆G(x) = G

(x)∆(x)J

(x), x ∈ R

, ∆(·) ∈ ∆},

where ∆ satisﬁes

∆ = {∆ : R

→ R

×p

: ∆

(x)∆(x) ≤ M

(x)M(x), x ∈ R

}, (11.250)

and where G

: R

→ R

n×m

, h

: R

→ R

, and J

: R

→ R

×m

are

ﬁxed functions denoting the structure of the uncertainty, ∆ : R

→ R

×p

is an uncertain matrix function, and M : R

→ R

×p

is a given matrix

function. Show that condition (11.162) is satisﬁed with

(x) =

′

(x)G

(x)g

(x)V

′T

(x) + h

(x)M

(x)M(x)h

(x),

(11.251)

(x) = 2h

(x)M

(x)M(x)J

(x), (11.252)

(x) = J

(x)M

(x)M(x)J

(x), (11.253)

and the robust stabilizing control law (11.146) is given by

φ(x) = −

−1

(x)[L

(x) + V

′

(x)G

(x) + 2h

(x)M

(x)M(x)J

(x)]

(11.254)

where R

(x)

△

= R

(x) + J

(x)M

(x)M(x)J

(x).

Problem 11.27. Consider the nonlinear uncertain disturbed system

˙x(t) = Ax(t) + ∆f (x(t)) + Dw(t), x(0) = x

, t ≥ 0, (11.255)

z(t) = Ex(t), (11.256)

where x ∈ R

, z ∈ R

, w(·) ∈ L

, ∆ f (·) ∈ ∆

, where

∆

△

= {∆f : R

→ R

: ∆f (x) = B

δ(C

x), x ∈ R

, δ(·) ∈ ∆}, (11.257)

and

∆

△

= {δ : R

→ R

: δ(0) = 0, kδ(y

≤ γ

−1

, y

= C

x ∈ R

(11.258)

where B

∈ R

n×m

and C

∈ R

×n

are ﬁxed matrices denoting the structure

of the uncertainty, δ(·) is an uncertain f unction, and γ > 0. Show that if

there exist n × n matrices P > 0 and R > 0, and scalars γ > 0 and γ

> 0

such that

0 = A

P + P A + γ

−2

P DD

P + γ

−2

P B

P + C

+ R, (11.259)

NonlinearBook10pt November 20, 2007

708 CHAPTER 11

then th e zero solution x(t) ≡ 0 of the undisturbed (w(t) ≡ 0) system (11.255)

is globally asymptotically stable for all ∆f(·) ∈ ∆

. Furthermore, show

that the solution x(t), t ≥ 0, of the d isturbed system (11.255) satisﬁes the

disturbance rejection constraint

(s)z(s)ds < γ

(s)w(s)ds + x

P x

, w(·) ∈ L

, T ≥ 0.

(11.260)

Problem 11.28. Consider the nonlinear uncertain cascade s y s tem

˙x(t) = f

(x(t)) + ∆f (x) + G

(x(t))ˆx(t),

x(0) = x

, f

+ ∆f(·) ∈ F, t ≥ 0, (11.261)

ˆx(t) = u(t), ˆx(0) = ˆx

, (11.262)

where F satisﬁes (11.23) and ∆ in (11.23) satisﬁes (11.24) with performance

functional

J(x

, ˆx

, u(·))

△

∞

L(x(t), ˆx(t), u(t))dt, (11.263)

where u(·) is admissible and (x(t), ˆx(t)), t ≥ 0, solves (11.261) and (11.262)

and where

L(x, ˆx, u)

△

= L

(x, ˆx) + L

(x, ˆx)u + u

(x, ˆx)u, (11.264)

where L

: R

×R

→ R, L

: R

×R

→ R

1×m

, and R

: R

×R

→ P

Assume there exist continuously diﬀerentiable functions α : R

→ R

and

sub

: R

→ R such that

α(0) = 0, (11.265)

sub

(0) = 0, (11.266)

sub

(x) > 0, x ∈ R

, x 6= 0, (11.267)

′

sub

(x)[f

(x) + G

(x)α(x)] +

′

sub

(x)G

(x)V

′T

sub

(x)

(x))m(h

(x)) < 0, x ∈ R

, x 6= 0. (11.268)

Furth ermore, let L

: R

× R

→ R

1×m

and

P ∈ R

m×m

P > 0, be such

that L

(0, 0) = 0 and

(ˆx − α(x))



−1

(x)V

′T

sub

(x) − 2α

′

(x)(f

(x) + G

(x)ˆx)

−R

−1

(x, ˆx)[2

P (ˆx − α(x)) + L

(x, ˆx)] −α

′

(x)G

(x)V

′T

sub

(x)

+α

′

(x)G

(x)α

′

(x)

P (ˆx −α(x))



< 0, (x, ˆx) ∈ R

× R

, ˆx 6= α(x).

(11.269)

Show that the zero solution (x(t), ˆx(t)) ≡ (0, 0) of the nonlinear uncertain

cascade system (11.261) and (11.262) is globally asymptotically stable for

NonlinearBook10pt November 20, 2007

ROBUST NONLINEAR CONTROL 709

all δ(·) ∈ ∆, w here ∆ is given by (11.24), with feedback control law

φ(x, ˆx) = −R

−1

(x, ˆx)[

(x, ˆx) +

P (ˆx − α(x))]. (11.270)

Furth ermore, show that the performance functional (11.263) satisﬁes

sup

δ(·)∈∆

J(x

, ˆx

, φ(x, ˆx))≤J(x

, ˆx

, φ(x, ˆx))= V (x

, ˆx

), (x

, ˆx

) ∈ R

× R

where

J(x

, ˆx

, u(·))

△

∞

(x, ˆx) + L

(x, ˆx)u + u

(x, ˆx)u

(x))m(h

(x))

′

sub

(x) − 2(ˆx − α(x))

P α

′

(x)]G

(x)

·G

(x)[V

′

sub

(x) − 2(ˆx −α(x))

P α

′

(x)]

]dt,

(11.271)

where u(·) is admissible, and (x(t), ˆx(t)), t ≥ 0, solves (11.261) and (11.262)

with δ(h

(x)) ≡ 0. In addition, show that the performance functional

(11.271), with

(x, ˆx) = φ

(x, ˆx)R

(x, ˆx)φ(x, ˆx) − [V

′

sub

(x) − 2(ˆx −α(x))

P α

′

(x)]

·((f

(x) + G

(x)ˆx)) −

′

sub

(x) − 2(ˆx −α(x))

P α

′

(x)]G

(x)

·G

(x)[V

′

sub

(x) − 2(ˆx − α(x))

P α

′

(x)]

− m

(x))m(h

(x)),

(11.272)

is minimized in the sense that

J(x

, ˆx

, φ(x(·))) = min

u(·)∈S(x

,ˆx

)

J(x

, ˆx

, u(·)), (11.273)

where S(x

, ˆx

) is the set of regulation controllers for the nonlinear system

(11.261) and (11.262) with ∆f(x) ≡ 0 by

S(x

, ˆx

)

△

= {u(·) : u(·) is admissible and (x(·), ˆx(·)) given by (11.261) and

(11.262) satisﬁes (x(t), ˆx(t)) → 0 as t → ∞ with ∆f(x) ≡ 0}.

Finally, show that a p articular choice of L

(x, ˆx) satisfying (11.269) is given

(x, ˆx) =



′

sub

(x)G

(x)

−1

− 2[f

(x) + G

(x)ˆx)]

′

(x)

−V

′

sub

(x)G

(x)α

′

(x)

+(ˆx −α(x))

P α

′

(x)G

(x)α

′

(x)



(x, ˆx), (11.274)

yielding the feedback control law

φ(x, ˆx) = −[R

−1

(x, ˆx) +

′

(x)G

(x)g

(x)α

′

(x)]

P (ˆx − α(x))

NonlinearBook10pt November 20, 2007

710 CHAPTER 11

+α

′

(x)[f

(x) + G

(x)ˆx] +

[α

′

(x)G

(x)g

(x)

−

−1

(x)]V

′T

sub

(x). (11.275)

Problem 11.29. Consider the nonlinear uncertain cascade system

(11.261) and (11.262) where F satisﬁes (11.23) and ∆ in (11.23) satisﬁes

(11.27), with performance functional (11.263). Assume there exist contin-

uously diﬀerentiable functions α : R

→ R

and V

sub

: R

→ R such that

(11.265)–(11.267) are satisﬁed and

′

sub

(x)[f

(x) + G

(x)α(x)] + V

′

sub

(x)G

(x)m

(x))

[m(h

(x)) + G

(x)V

′T

sub

(x)]

[m(h

(x)) + G

(x)V

′T

sub

(x)] < 0, x 6= 0.

(11.276)

Furth ermore, let L

: R

× R

→ R

1×m

and

P ∈ R

m×m

P > 0, be such

that L

(0, 0) = 0 and

(ˆx − α(x))



−1

(x)V

′T

sub

(x) − 2α

′

(x)(f

(x)+G

(x)ˆx)

−R

−1

(x, ˆx)[2

P (ˆx − α(x))] + L

(x, ˆx)

−α

′

(x)G

(x)[m

(x)) + m

(x)) + G

(x)V

′T

sub

(x)]

+α

′

(x)G

(x)α

′

(x)

P (ˆx −α(x))



< 0, (x, ˆx) ∈ R

× R

, ˆx 6= α(x).

(11.277)

Show that the zero solution (x(t), ˆx(t)) ≡ (0, 0) of the nonlinear uncertain

cascade system (11.261) and (11.262) is globally asymptotically stable for

all δ(·) ∈ ∆, w here ∆ is given by (11.27), with feedback control law

φ(x, ˆx) = −R

−1

(x, ˆx)[

(x, ˆx) +

P (ˆx − α(x))]. (11.278)

Furth ermore, show that the performance functional (11.263) satisﬁes

sup

δ(·)∈∆

J(x

, ˆx

, φ(x, ˆx)) ≤ J(x

, ˆx

φ(x, ˆx))

= V (x

, ˆx

), (x

, ˆx

) ∈ R

× R

, (11.279)

where

J(x

, ˆx

, u(·))

△

∞



(x, ˆx) + L

(x, ˆx)u + u

(x, ˆx)u



′

sub

(x) − 2(ˆx −α(x))

P α

′

(x)]G

(x)m

(x))

[m(h

(x)) + G

(x){V

′T

sub

(x) − 2α

′

(x)P (ˆx −α(x))}]

·[m(h

(x)) + G

(x){V

′T

sub

(x) − 2α

′

(x)P (ˆx −α(x))}]



dt, (11.280)

NonlinearBook10pt November 20, 2007

ROBUST NONLINEAR CONTROL 711

where u(·) is admissible, and (x(t), ˆx(t)), t ≥ 0, solves (11.261) and (11.262)

with δ(h

(x)) ≡ 0. In addition, show that the performance functional

(11.280), with

(x, ˆx) = φ

(x, ˆx)R

(x, ˆx)φ(x, ˆx)

−[V

′

sub

(x) − 2(ˆx − α(x))

P α

′

(x)](f

(x) + G

(x)ˆx)

−

[m(h

(x)) + G

(x){V

′T

sub

(x) − 2α

′

(x)P (ˆx −α(x))}]

·[m(h

(x)) + G

(x){V

′T

sub

(x) − 2α

′

(x)P (ˆx − α(x))}]

−[V

′

sub

(x) − 2(ˆx − α(x))

P α

′

(x)]G

(x)m

(x)), (11.281)

is minimized in the sense that

J(x

, ˆx

, φ(x(·))) = min

u(·)∈S(x

,ˆx

)

J(x

, ˆx

, u(·)). (11.282)

Problem 11.30. Consider the nonlinear uncertain block cascade sys-

tem

˙x(t) = f

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

, t ≥ 0, (11.283)

ˆx(t) =

(ˆx(t)) + ∆

f(ˆx(t)) +

G(ˆx(t))u(t), ˆx(0) = ˆx

, (11.284)

y(t) = h(ˆx(t)), (11.285)

where (11.128) has been augmented by a nonlinear input subsystem with

∆G(x) ≡ 0, ˆx ∈ R

f : R

→ R

satisﬁes

(0) = 0, h : R

→ R

satisﬁes

h(0) = 0,

G : R

→ R

q×m

, and

(·)+∆

f(·) ∈

F, where

F is assumed to have

the same form as F deﬁned by (11.23). Assume that the input subsystem

(11.284) and (11.285) is f eedback strictly passive for all

(·) + ∆

f(·) ∈

such that there exist a positive-deﬁnite storage function V

: R

→ R and

functions Γ

: R

→ R and k : R

→ R

such that

0 ≥ V

′

(ˆx)∆

f(ˆx) − Γ

(ˆx),

(·) + ∆

f(·) ∈

F, (11.286)

0 > V

′

(ˆx)

(ˆx) + V

′

(ˆx)

G(ˆx)k(ˆx) + Γ

(ˆx), ˆx ∈ R

, ˆx 6= 0, (11.287)

0 =

(ˆx)V

′T

(ˆx) − h(ˆx). (11.288)

Also, assume that the subsystem (11.283) has a globally stable equilibrium

at x = 0 with y = 0 for all ∆f (·) ∈ ∆ and Lyapunov function V

sub

(x),

x ∈ R

, so that

′

sub

(x)∆f(x) ≤ Γ

sub

(x), ∆f (·) ∈ ∆, (11.289)

′

sub

(x)f

(x) + Γ

sub

(x) < 0, x 6= 0, (11.290)

where Γ

sub

: R

→ R such that Γ

sub

(0) = 0. Furthermore, assume there

exists a function L

: R

× R

→ R

1×m

such that

(0, 0) = 0, (11.291)

(ˆx)



(ˆx)V

′T

sub

(x) −

−1

(x, ˆx)L

(x, ˆx) −k(ˆx)



≤ 0. (11.292)

NonlinearBook10pt November 20, 2007

712 CHAPTER 11

Show that the zero solution (x(t), ˆx(t)) ≡ (0, 0) of the nonlinear uncertain

block cascade s ystem (11.283)–(11.285) is globally asymp totically stable for

all (f

+ ∆f,

+ ∆

f) ∈ F ×

F with the feedback control law

φ(x, ˆx) = −

−1

(x, ˆx)



(x, ˆx) + h(ˆx)



. (11.293)

Furth ermore, show that the performance functional (11.263) satisﬁes

J(x

, ˆx

, φ(x(·), ˆx(·))) ≤ J(x

, ˆx

, φ(x(·), ˆx(·))) = V

sub

) +

V (ˆx

(11.294)

where

J(x

, ˆx

, u)

△

∞



(x, ˆx) +L

(x, ˆx)u+ u

(x, ˆx)u +Γ

sub

(x)+Γ

(ˆx)



dt,

(11.295)

where u(·) is admissible and (x(t), ˆx(t)), t ≥ 0, solves (11.283) and (11.284)

with (∆f (x), ∆

f(ˆx) ≡ (0, 0). In ad dition, show that the performance

functional (11.295) with

(x, ˆx) = φ

(x, ˆx)R

(x, ˆx)φ(x, ˆx) −V

′

sub

(x)[f

(x) + G(x)h(ˆx)]

−V

′

(ˆx)

(ˆx) − Γ

sub

(x) − Γ

(ˆx), (11.296)

is minimized in the sense that

J(x

, ˆx

, φ(x(·), ˆx(·))) = min

u(·)∈S(x

,ˆx

)

J(x

, ˆx

, u(·)). (11.297)

Problem 11.31. Consider the controlled nonlinear dynamical system

˙x(t) = F (x(t), σ(u(t))), x(0) = x

, t ≥ 0, (11.298)

where u(·) ∈ U is an admissible input such that u(t) ∈ U, t ≥ 0, where the

control constraint set U is given such that 0 ∈ U and σ(·) ∈ M ⊂ {σ : U →

: σ(0) = 0} denotes an input nonlinearity. Furth ermore, consider the

performance functional

J(x

, u(·))

△

∞

L(x(t), u(t))dt, (11.299)

where L : D×U → R. Assume there exist functions V : D → R, Γ : D×U →

R, and control law φ : D → U , where V (·) is a continuously diﬀerentiable

function, such that

V (0) = 0, (11.300)

V (x) > 0, x ∈ D, x 6= 0, (11.301)

φ(0) = 0, (11.302)

′

(x)F (x, σ(φ(x))) ≤ V

′

(x)F (x, σ

(φ(x))) + Γ(x, φ(x)), x ∈ D, σ(·) ∈ M,

(11.303)

′

(x)F (x, σ

(φ(x))) + Γ(x, φ(x)) < 0, x ∈ D, x 6= 0, (11.304)

NonlinearBook10pt November 20, 2007

ROBUST NONLINEAR CONTROL 713

H(x, φ(x)) = 0, x ∈ D, (11.305)

H(x, u) ≥ 0, x ∈ D, u ∈ U, (11.306)

where σ

(·) ∈ M is a given nominal input nonlinearity and

H(x, u) , L(x, u) + V

′

(x)F (x, σ

(u)) + Γ(x, u). (11.307)

Show that, with the feedback control u(·) = φ(x(·)), there exists a

neighborhood D

⊆ D of the origin such that if x

∈ D

, the zero solution

x(t) ≡ 0 of the closed-loop system

˙x(t) = F (x(t), σ(φ(x(t)))), x(0) = x

, t ≥ 0, (11.308)

is locally asymptotically stable for all σ(

) ∈ M. Furthermore, show that

sup

σ(·)∈M

J(x

, φ(x(·))) ≤ J(x

, φ(x(·))) = V (x

), (11.309)

where

J(x

, u(·))

△

∞

[L(x(t), u(t)) + Γ(x(t), u(t))]dt, (11.310)

and where u(·) is admissible and x(t), t ≥ 0, solves (11.298) with σ(u(t)) =

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

∈ D

then show that the feedback control u(·) =

φ(x(·)) minimizes J(x

, u(·)) in the sense th at

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), (11.311)

where

S(x

)

△

= {u(·) : u(·) is admissible and x(·) given by (11.298)

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

(·)}.

Finally, if D = R

, U = R

, and

V (x) → ∞ as kxk → ∞,

show that the solution x(t) = 0, t ≥ 0, of the closed-loop system (11.308) is

globally asymptotically stable.

Problem 11.32. Consider the controlled nonlinear dynamical system

˙x(t) = f (x(t)) + G(x(t))σ(u(t)), x(0) = x

, t ≥ 0, (11.312)

where f : R

→ R

satisﬁes f (0) = 0, G : R

→ R

n×m

, D = R

, and

σ(·) ∈ M

△

= {σ : R

→ R

: σ(0) = 0,

[σ(u) −M

u] ≤ 0, u ∈ R

}, (11.313)

where M

, M

∈ R

m×m

are given diagonal matrices such that M

△

= M

−M

is positive deﬁnite, and u(t) ∈ R

for all t ≥ 0. Furthermore, consider the

NonlinearBook10pt November 20, 2007

714 CHAPTER 11

performance functional

J(x

, u(·)) =

∞

(x(t)) + L

(x(t))u(t) + u

(t)R

(x(t))u(t)]dt, (11.314)

where L

: R

→ R, L

: R

→ R

1×m

, and R

: R

→ P

. Assume there

exist a continuously diﬀerentiable function V : R

→ R and a function

: R

→ R

1×m

such that

V (0) = 0, (11.315)

(0) = 0, (11.316)

V (x) > 0, x ∈ R

, x 6= 0, (11.317)

′

(x)[f(x) −

G(x)M

−1

(x)V

(x)]

[−

−1

(x)V

(x) + G

(x)V

′T

(x)]

·[−

−1

(x)V

(x) + G

(x)V

′T

(x)] < 0, x ∈ R

, x 6= 0, (11.318)

and

V (x) → ∞ as kxk → ∞, (11.319)

where R

(x)

△

= R

(x) +

M and V

(x)

△

= [L

(x) +

′

(x)G(x)(M

)]

. Show that, with the feedback control law

φ(x) = −

−1

(x)V

(x), (11.320)

the zero solution x(t) ≡ 0 of the nonlinear system (11.312) is globally

asymptotically stable for all σ(·) ∈ M. Furthermore, show that the

performance functional (11.314) satisﬁes

sup

σ(·)∈M

J(x

, φ(x(·))) ≤ J(x

, φ(x(·))) = V (x

), (11.321)

where

J(x

, u(·))

△

∞

(x) + L

(x)u + u

(x)u +

[Mu + G

(x)V

′T

(x)]

·[Mu + G

(x)V

′T

(x)]dt, (11.322)

where u(·) is admissible, and x(t), t ≥ 0, solves (11.312) with σ(u) = M

In addition, show that the performance functional (11.322), with

(x) = φ

(x)R

(x)φ(x) −V

′

(x)f(x) −

′

(x)G(x)G

(x)V

′T

(x),

(11.323)

is minimized in the sense that

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), (11.324)

where

S(x

)

△

= {u(·) : u(·) is admissible and x(·) given by (11.312)