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

Подождите немного. Документ загружается.

NonlinearBook10pt November 20, 2007

DISTURBANCE REJECTION CONTROL 625

V (x) > 0, x ∈ R

, x 6= 0, (10.138)

′

(x)[f(x) −

G(x)R

−1

(x)(L

(x) + G

(x)V

′T

(x) + 2J

(x)h(x))]

+Γ(x, φ(x)) < 0, x ∈ R

, x 6= 0, (10.139)

and

V (x) → ∞ as kxk → ∞, (10.140)

where R

(x)

△

= R

(x)+J

(x)J(x), φ(x) = −

−1

(x)[L

(x)+G

(x)V

′T

(x)

+2J

(x)h(x)], and

Γ(x, u) =

4γ

′

(x)J

(x)V

′T

(x) + [h(x) + J(x)u]

[h(x) + J(x)u],

(10.141)

where γ > 0, u(·) is admissible, and x(t), t ≥ 0, solves (10.118) with w(t) ≡

0. Then the zero solution x(t) ≡ 0 of the undisturbed (w(t) ≡ 0) closed-loop

system

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

, t ≥ 0, (10.142)

is globally asymptotically stable with feedback control law

φ(x) = −

−1

(x)[L

(x) + G

(x)V

′T

(x) + 2J

(x)h(x)]. (10.143)

Furth ermore, the performance functional (10.135) satisﬁes

J(x

, φ(x(·))) ≤ J(x

, φ(x(·))) = V (x

), (10.144)

where

J(x

, u(·))

△

∞

[L(x(t), u(t)) + Γ(x(t), u(t))]dt. (10.145)

In addition, the performance functional (10.145), with

(x) = φ

(x)R

(x)φ(x) − V

′

(x)f(x) −h

(x)h(x)

−

4γ

′

(x)J

(x)V

′T

(x), (10.146)

is minimized in the sense that

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)). (10.147)

Finally, with u(·) = φ(x(·)), the solution x(t), t ≥ 0, of the closed-loop

system (10.118) satisﬁes the nonexpansivity constraint

(t)z(t)dt ≤ γ

(t)w(t)dt + V (x

), w(·) ∈ L

, T ≥ 0.

(10.148)

Proof. The result is a direct consequence of Theorem 10.4 with

F (x, u) = f (x) + G(x)u, z = h(x) + J(x)u, L(x, u) = L

(x) + L

(x)u +

(x)u, J

(x) = 0, Γ(x, u) given by (10.141), D = R

, and U = R

NonlinearBook10pt November 20, 2007

626 CHAPTER 10

Speciﬁcally, with (10.118), (10.134), and (10.141), the Hamiltonian has the

form

H(x, u) = L

(x) + L

(x)u + u

(x)u + V

′

(x)(f(x) + G(x)u)

4γ

′

(x)J

(x)V

′T

(x) + [h(x) + J(x)u]

[h(x) + J(x)u].

Now, the proof follows as the proof of Corollary 10.5.

Note that since Γ(x, φ(x)) ≥ 0, x ∈ R

, (10.139) implies that

V (x)

△

= V

′

(x)[f(x) + G(x)φ(x)] < 0, x ∈ R

, x 6= 0, (10.149)

with φ(x) given by (10.143). Furthermore, (10.136), (10.138), and (10.149)

ensure th at V (·) is a Lyapunov function for the undisturbed closed-loop

system (10.142). In addition, with L

(x) given by (10.146) and φ(x) given

by (10.143), L(x, u) + Γ(x, u) can be expressed as

L(x, u) + Γ(x, u)

= [u − φ(x)]

(x)[u − φ(x)] −V

′

(x)[f(x) + G(x)u]

= [u +

−1

(x)(L

(x) + 2J

(x)h(x)]

(x)

[u +

−1

(x)(L

(x) + 2J

(x)h(x)]

−V

′

(x)[f(x) + G(x)φ(x)] −

′

(x)G(x)R

−1

(x)G

(x)V

′T

(x).

(10.150)

Since R

(x) ≥ R

(x) > 0 for all x ∈ R

the ﬁrst term of the right-hand

side of (10.150) is nonnegative, while (10.149) implies that the second term

is nonnegative. Thus, we have

L(x, u) + Γ(x, u) ≥ −

′

(x)G(x)R

−1

(x)G

(x)V

′T

(x), (10.151)

which shows that L(x, u) + Γ(x, u) may be negative. As a result, there

may exist a control in put u for which the auxiliary performance functional

J(x

, u) is negative. However, if the disturbance rejection control u is a

regulation controller, that is, u ∈ S(x

), th en it f ollows from (10.144) and

(10.147) that

J(x

, u(·)) ≥ V (x

) ≥ 0, x

∈ R

, u(·) ∈ S(x

Furth ermore, in this case substituting u = φ(x) into (10.150) yields

L(x, φ(x)) + Γ(x, φ(x)) = −V

′

(x)[f(x) + G(x)φ(x)],

which, by (10.149), is positive.

Next, we specialize Theorem 10.4 to the p assivity case. Speciﬁcally,

we consider th e case where p = d and r(z, w) = 2z

w. For the following

result we consider performance variables

z(t) = h(x(t)) + J(x(t))u(t) + J

(x(t))w(t), (10.152)

NonlinearBook10pt November 20, 2007

DISTURBANCE REJECTION CONTROL 627

where J : R

→ R

p×m

, J

: R

→ R

p×p

satisﬁes J

(x) + J

(x) > 0, x ∈ R

and h : R

→ R

satisﬁes h(0) = 0. Furthermore, we consider performance

integrands L(x, u) of the form given by (10.134).

Corollary 10.7. Consid er the nonlinear controlled dynamical system

(10.118) and (10.152) with perform ance fun ctional (10.135). Assume that

there exist a continuously diﬀerentiable function V : R

→ R and a function

: R

→ R

1×m

such that

V (0) = 0, (10.153)

(0) = 0, (10.154)

V (x) > 0, x ∈ R

, x 6= 0, (10.155)

′

(x)[f(x) −

G(x)R

−1

(x)(L

(x) + J

(x)R

(x)[2h(x) − J

(x)V

′T

(x)])]

+Γ(x, φ(x)) < 0, x ∈ R

, x 6= 0, (10.156)

and

V (x) → ∞ as kxk → ∞, (10.157)

where R

(x)

△

= (J

(x) + J

(x))

−1

, R

(x)

△

= R

(x) + J

(x)R

(x)J(x),

φ(x) = −

−1

(x)[G

(x)V

′T

(x) + L

(x) + J

(x)R

(x)(2h(x)

−J

(x)V

′T

(x))],

and

Γ(x, u) =



(x)V

′T

(x) − (h(x) + J(x)u)



(x)



(x)V

′T

(x) − (h(x) + J(x)u)



, (10.158)

where u(·) is admissible, and x(t), t ≥ 0, solves (10.118) with w(t) ≡ 0.

Then the zero solution x(t) ≡ 0 of the undisturbed (w(t) ≡ 0) closed-loop

system (10.142) is globally asymptotically stable with feedback control law

φ(x). Furthermore, the performance functional (10.135) satisﬁes

J(x

, φ(x(·))) ≤ J(x

, φ(x(·))) = V (x

), (10.159)

where

J(x

, u(·))

△

∞

[L(x(t), u(t)) + Γ(x(t), u(t))]dt. (10.160)

In addition, the performance functional (10.160), with

(x) = φ

(x)R

(x)φ(x) −V

′

(x)f(x) −[

(x)V

′T

(x) − h(x)]

(x)

·[

(x)V

′T

(x) − h(x)]h

(x)h(x) −

4γ

′

(x)J

(x)V

′T

(x),

(10.161)

is minimized in the sense that

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)). (10.162)

NonlinearBook10pt November 20, 2007

628 CHAPTER 10

Finally, with u(·) = φ(x(·)), the solution x(t), t ≥ 0, of the closed-loop

system (10.118) satisﬁes the positivity constraint

(t)w(t)dt + V (x

) ≥ 0, w(·) ∈ L

, T ≥ 0. (10.163)

Proof. The result is a direct consequ en ce of Theorem 10.4 with

F (x, u) = f(x) + G(x)u, z = h(x) + J(x)u + J

(x)w, L(x, u) = L

(x) +

(x)u + u

(x)u, J

(x) = 0, Γ(x, u) given by (10.158), D = R

, and

U = R

. Speciﬁcally, with (10.134) and (10.158), the Hamiltonian has the

form

H(x, u) = L

(x) + L

(x)u + u

(x)u + V

′

(x)(f(x) + G(x)u) + Γ(x, u).

The proof now follows as in the pr oof of Corollary 10.5.

10.7 Stability Margins, Meaningful Inverse Optimality, and

Nonexpansive Control Lyapunov Functions

In this section, we specialize the results of Section 10.6 to the case wh ere

L(x, u) is n onnegative for all (x, u) ∈ R

×R

. Here, we assume L

(x) ≡ 0

and L

(x) = h

(x)h(x), x ∈ R

. We begin by specializing Corollary 10.5 to

aﬃne systems of the form (10.118) with performance variables (10.119) and

we assume h

(x)J(x) ≡ 0 and R

(x)

△

= J

(x)J(x) > 0, x ∈ R

, so that the

performance functional (10.121) becomes

J(x

, u(·)) =

∞

(x(t)) + u

(t)R

(x(t))u(t)]dt. (10.164)

Corollary 10.8. Consider the nonlinear controlled dynamical system

(10.118) and (10.119) with perform ance functional (10.164). Assume that

there exists a continuously d iﬀerentiable function V : R

→ R such that

V (0) = 0, (10.165)

V (x) > 0, x ∈ R

, x 6= 0, (10.166)

0 = V

′

(x)f(x) + h

(x)h(x) +

4γ

′

(x)J

(x)V

′T

(x)

−

′

(x)G(x)R

−1

(x)G

(x)V

′T

(x), x ∈ R

, (10.167)

and

V (x) → ∞ as kxk → ∞, (10.168)

where γ > 0. Furthermore, assum e that the system (10.118) and (10.119)

is zero-state obs ervable. Th en the zero solution x(t) ≡ 0 of the undisturbed

(w(t) ≡ 0) closed-loop system

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

, t ≥ 0, (10.169)

NonlinearBook10pt November 20, 2007

DISTURBANCE REJECTION CONTROL 629

is globally asymptotically stable with feedback control law

φ(x) = −

−1

(x)G

(x)V

′T

(x). (10.170)

Furth ermore, the performance functional (10.164) satisﬁes

J(x

, φ(x(·))) ≤ J(x

, φ(x(·))) = V (x

), (10.171)

where

J(x

, u(·))

△

∞

[L(x(t), u(t)) + Γ(x(t), u(t))]dt, (10.172)

Γ(x, u) =

4γ

′

(x)J

(x)V

′T

(x), (10.173)

and γ > 0. In addition, the perf ormance functional (10.172) is minimized

in the sense that

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)). (10.174)

Finally, with u(·) = φ(x(·)), the solution x(t), t ≥ 0, of the closed-loop

system (10.118) satisﬁes the nonexpansivity constraint

(t)z(t)dt ≤ γ

(t)w(t)dt + V (x

), w(·) ∈ L

, T ≥ 0.

(10.175)

Proof. The resu lt follows as a direct consequence of Corollary 10.5.

Next, we provide sector and gain margins for the nonlinear dynamical

system G given by (10.118) and (10.119). To consider relative stability mar-

gins for nonlinear nonexpansive regulators consider the nonlinear dynamical

system given by (10.118) and (10.119), along with the output

y(t) = −φ(x(t)), (10.176)

where φ(·) is such that the input-output map from w to z is nonexpans ive

with u = φ(x). Furthermore, assume that (10.118) and (10.176) is zero-state

observable. For the next result deﬁne

△

inf

x∈R

min

(x)J

(x))

sup

x∈R

max

(G(x)R

−1

(x)G

(x))

. (10.177)

Theorem 10.5. Let γ > 0 and ρ ∈ (0, 1]. Consider the nonlinear

dynamical system G given by (10.118) and (10.119) where φ(x) is a

nonexpansive feedback control law given by (10.170) and where V (x),

x ∈ R

, satisﬁes

V (0) = 0, (10.178)

V (x) > 0, x ∈ R

, x 6= 0, (10.179)

NonlinearBook10pt November 20, 2007

630 CHAPTER 10

0 = V

′

(x)f(x) + h

(x)h(x) +

4γ

′

(x)J

(x)V

′T

(x)

−

′

(x)G(x)R

−1

(x)V

′T

(x), x ∈ R

. (10.180)

Furth ermore, assume R

(x) = diag[r

(x), . . . , r

(x)], where r

: R

→ R,

(x) > 0, i = 1, . . . , m. Then the undisturbed (w(t) ≡ 0) nonlinear system

G has a sector (and, hence, gain) margin (

, ∞), where

△

= 1 −

η(1 −ρ

)

Finally, with u = σ(φ(x)), th e solution x(t), t ≥ 0, of the closed-loop system

(10.118) satisﬁes the nonexpansivity constraint

(t)z(t)dt ≤ (γ/ρ)

(t)w(t)dt + V (x

), w(·) ∈ L

, T ≥ 0,

(10.181)

where σ : R

→ R

is such that σ(0) = 0 and for every u

∈ R

, σ(u

) =

[σ

), . . . , σ

)]

and αu

< 2σ

, u

6= 0, i = 1, . . . , m.

Proof. With u = σ(φ(x)), the closed-loop sys tem (10.118) and

(10.176) is given by

˙x(t) = f(x(t)) + G(x(t))σ(φ(x(t)) + J

(x(t))w(t),

x(0) = x

, w(·) ∈ L

, t ≥ 0. (10.182)

Next, consider the Lyapunov function candidate V (x), x ∈ R

, satisfying

(10.180) and let

V (x) d en ote the Lyapun ov derivative along the trajectories

of the closed-loop system (10.182). Now, it follows from (10.180) that for

all w ∈ R

V (x) = V

′

(x)f(x) + V

′

(x)G(x)σ(φ(x)) + V

′

(x)J

(x)w

≤ φ

(x)R

(x)φ(x) + V

′

(x)G(x)σ(φ(x)) + V

′

(x)J

(x)w

−

4γ

′

(x)J

(x)V

′T

(x) − h

(x)h(x)

≤ αφ

(x)R

(x)φ(x) + V

′

(x)G(x)σ(φ(x)) + V

′

(x)J

(x)w

−h

(x)h(x) −

4γ

′

(x)J

(x)V

′T

(x)

≤

i=1

(x)y

(αy

+ 2σ

(−y

)) + (γ/ρ)

w − z

≤ (γ/ρ)

w − z

which proves the nonexpansivity property. Th e proof of sector (and, hence,

gain) margin guarantees is similar to the proof of Theorem 8.7 and, hence,

is omitted.

The following specialization of Theorem 10.5 to linear dynamical

systems is immediate.

NonlinearBook10pt November 20, 2007

DISTURBANCE REJECTION CONTROL 631

Corollary 10.9. Let γ > 0 and ρ ∈ (0, 1]. Consider the linear

dynamical system G given by (10.101) and (10.102) and let P ∈ P

satisfy

(10.104). Furthermore, assume R

= diag[r

, . . . , r

], where r

> 0,

i = 1, . . . , m. Then with the nonexp an s ive f eedback control law φ(x) =

−R

−1

P x, the undisturbed (w(t) ≡ 0) linear dynamical system G has a

sector (and, hence, gain) margin (

, ∞), where

△

= 1 −

η(1 −ρ

)

Finally, with u = σ(φ(x)), the solution x(t), t ≥ 0, of the closed-loop system

satisﬁes the nonexpansivity constraint

(t)z(t)dt ≤ (γ/ρ)

(t)w(t)dt + V (x

), w(·) ∈ L

, T ≥ 0,

(10.183)

where σ : R

→ R

is such that σ(0) = 0 and for every u

∈ R

, σ(u

) =

[σ

), . . . , σ

)]

and αu

< 2σ

, u

6= 0, i = 1, . . . , m.

Next, we introduce the notion of nonexpansive control Lyapunov

functions for the nonlinear dynamical system (10.118) and (10.119).

Deﬁnition 10.1. Consider the controlled n on linear dynamical system

given by (10.118) and (10.119). A continuously diﬀerentiable positive-

deﬁnite function V : R

→ R satisfying

′

(x)f(x) + h

(x)h(x) +

4γ

′

(x)J

(x)V

′T

(x) < 0, x ∈ R,

(10.184)

where R

△

= {x ∈ R

: x 6= 0 : V

′

(x)G(x) = 0}, is called a nonexpansive

control Lyapunov function.

Finally, we show that for every nonlinear dynamical system for which

a nonexpansive control Lyapunov function can be constructed there exists

an inverse optimal nonexpansive feedback control law with sector and gain

margin guarantees of at least (

, ∞).

Theorem 10.6. Let γ > 0 and ρ ∈ (0, 1]. Consider the nonlinear

dynamical system G given by (10.118) and (10.119), and let the continuously

diﬀerentiable positive-deﬁnite, radially unbounded function V : R

→ R be

a nonexpansive control Lyapunov f unction of (10.118) and (10.119), that is,

′

(x)f(x) + h

(x)h(x) +

4γ

′

(x)J

(x)V

′T

(x) < 0, x ∈ R,

(10.185)

where R

△

= {x ∈ R

: x 6= 0 : V

′

(x)G(x) = 0}. Then, with the feedback

NonlinearBook10pt November 20, 2007

632 CHAPTER 10

control law given by

φ(x) =







−



α(x)+

√

(x)+(β

(x)β(x))

(x)β(x)



β(x), β(x) 6= 0,

0, β(x) = 0,

(10.186)

where α(x)

△

= V

′

(x)f(x) + h

(x)h(x) +

4γ

′

(x)J

(x)V

′T

(x), β(x)

△

(x)V

′T

(x), and c

> 0, the nonlinear sys tem G given by (10.118) and

(10.119) has a sector (and, hence, gain) margin (

, ∞), where

△

= 1 −

η(1 −ρ

)

Finally, with u = σ(φ(x)), th e solution x(t), t ≥ 0, of the closed-loop system

(10.118) satisﬁes the nonexpansivity constraint

(t)z(t)dt ≤ (γ/ρ)

(t)w(t)dt + V (x

), w(·) ∈ L

, T ≥ 0,

(10.187)

where σ : R

→ R

is such that σ(0) = 0 and for every u

∈ R

, σ(u

) =

[σ

), . . . , σ

)]

and αu

< 2σ

, u

6= 0, i = 1, . . . , m.

Proof. The result is a direct consequence of Corollary 10.8 and

Theorem 10.5 with R

(x) =

2η(x)

and L

(x) = −α(x) +

η(x)

(x)β(x),

where

η(x) =







−



α(x)+

√

(x)+(β

(x)β(x))

(x)β(x)



, β(x) 6= 0,

0, β(x) = 0.

(10.188)

Speciﬁcally, note that R

(x) > 0, x ∈ R

, and

(x) = −α(x) +

η(x)

(x)β(x) (10.189)











−



(x)β(x) − α(x)

(x) + (β

(x)β(x))



, β(x) 6= 0,

−α(x), β(x) = 0.

Now, it follows from (10.189) that L

(x) ≥ 0, β(x) 6= 0, and s ince V (·)

is a nonexpansive control Lyapu nov function of (10.118), it f ollows that

(x) = −α(x) ≥ 0, for all x ∈ R. Hence, (10.189) yields L

(x) ≥ 0,

x ∈ R

, so that all the conditions of Corollary 10.9 are satisﬁed.

NonlinearBook10pt November 20, 2007

DISTURBANCE REJECTION CONTROL 633

10.8 Nonlinear Controllers with Multilinear and Polynomial

Performance Criteria

In this section, we specialize the r esults of Section 10.5 to linear systems con-

trolled by nonlinear controllers that minimize a multilinear cost functional.

Speciﬁcally, we consider the linear system (10.101) controlled by nonlinear

controllers. First, we consider the case in which r(z, w) = γ

w − z

where γ > 0. Recall the deﬁnitions of S, R

, and R

and let ℓ : R

→ R

be a multilinear function such that ℓ(x) ≥ 0, x ∈ R

. Furthermore, assume

that γ

−2

≤ S, w here γ > 0 is given.

Proposition 10.1. Consider the linear dynamical system (10.101) and

(10.102) and assume that there exist a positive-deﬁnite matrix P ∈ R

n×n

and a function p : R

→ R such that p(x) ≥ 0, x ∈ R

0 = A

P + P A + R

+ γ

−2

P DD

P − P SP, (10.190)

and

0 = p

′

(x)[A − (S − γ

−2

)P ]x + ℓ(x), (10.191)

where γ > 0. Then th e zero solution x(t) ≡ 0 of the undisturbed (w(t) ≡

0) system (10.101) is globally asymptotically stable for all x

∈ R

with

the feedback control law u = φ(x) = −R

−1

(P x +

′

(x)), and the

performance functional (10.135) with R

(x) ≡ R

, L

(x) ≡ 0, and

(x) = x

x + ℓ(x) +

′

(x)(S − γ

−2

′

(x), (10.192)

satisﬁes

J(x

, φ(x(·))) ≤ J(x

, φ(x(·))) = x

P x

+ p(x

), (10.193)

where

J(x

, u(·)) =

∞

[L(x(t), u(t)) + Γ(x(t), u(t))]dt (10.194)

and where u(·) is admissible, x(t), t ≥ 0, solves (10.101) with w(t) ≡ 0, and

Γ(x, u) = γ

−2

[P x +

′

(x)]

[P x +

′

(x)]. (10.195)

Furth ermore,

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), (10.196)

where S(x

) is the set of regulation controllers for the system (10.101) with

w(t) ≡ 0 and x

∈ R

. Finally, with u = φ(x), the solution x(t), t ≥ 0, of

(10.101) satisﬁes the nonexpansivity constraint

(t)z(t)dt ≤ γ

w(t)

w(t)dt + V (x

), w(·) ∈ L

, T ≥ 0.

(10.197)

NonlinearBook10pt November 20, 2007

634 CHAPTER 10

Proof. The result is a direct consequ en ce of Theorem 10.4 with

F (x, u) = Ax + Bu, J

(x) = D, L(x, u) = x

x +

′

(x)(S − γ

−2

)

′

(x) + ℓ(x) + u

u, V (x) = x

P x + p(x), D = R

, and U = R

Speciﬁcally, conditions (10.88)–(10.91) are trivially satisﬁed. Now, forming

(10.190)x + (10.191) it follows that, after some algebraic man ipulations,

′

(x)J

(x)w ≤ r(z, w) + L(x, φ(x), w) + Γ(x, φ(x), w) for all x ∈ D and w ∈

W. Furthermore, it follows from (10.190) and (10.191) that H(x, φ(x)) = 0

and H(x, u) = H(x, u) − H(x, φ(x)) = [u − φ(x)]

[u − φ(x)] ≥ 0 so that

all conditions of Theorem 10.4 are satisﬁed. Finally, since V (·) is radially

unbounded, (10.101), with u(t) = φ(x(t)) = −R

−1

P x(t) +

′

(x(t))),

is globally asymptotically stable.

Since A − (S − γ

−2

)P is Hurwitz and ℓ(x), x ∈ R

, is a

nonnegative multilinear f unction, it follows from Lemma 8.1 that there exists

a nonnegative p(x), x ∈ R

, such that (10.191) is satisﬁed. Proposition 10.1

generalizes the classical results of Bass and Webber [33] to optimal control

of nonlinear systems with bounded energy disturbances. As discussed in

Chapter 8 the performance functional (10.192) is a derived perf ormance

functional in the sen se that it cannot be arbitrarily speciﬁed. However,

this performance functional does weigh the state variables by arbitrary even

powers. Fur thermore, (10.192) has the form

J(x

, u(·)) =

∞

x + ℓ(x) + u

u −

−2

′

(x)DD

′

(x)

+φ

(x)R

(x)

dt,

where φ

(x)

△

= −

−1

′

(x) is the nonlinear part of the optimal

feedback control

φ(x) = φ

(x) + φ

(x),

where φ

(x)

△

= −R

−1

P x.

If p(x) is a polynomial function of the form p(x) =

k=2

then it follows from (10.191) that ℓ(x) =

k=2

k−1

x, where

∈ N

, k = 2, . . . , r, and M

satisﬁes

0 = [A−(S −γ

−2

P )]

[A−(S−γ

−2

P )]+

, k = 2, . . . , r,

(10.198)

where P satisﬁes (10.190). In this case, the optimal control law φ(x) is given

φ(x) = −R

−1

P x +

k=2

k−1

the corresponding Lyapunov function guaranteeing closed-loop stability is