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

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NonlinearBook10pt November 20, 2007

DISTURBANCE REJECTION CONTROL 635

given by

V (x) = x

P x +

k=2

and L(x, u) given by (10.192) becomes

L(x, u) = x

k=2

k−1

k=2

k−1

(S − γ

−2

)

k=2

k−1

+ u

Furth ermore, if

, k = 3, . . . , r, then M

= M

, k = 3, . . . , r, satisﬁes

(10.198). In this case, we require the solution of only one modiﬁed Riccati

equation in (10.198). Proposition 10.1 generalizes the deterministic version

of the s tochastic nonlinear-nonquadratic optimal control problem considered

in [411] to the disturbance rejection setting. Furthermore, unlike the results

of [411], this result is not limited to sixth-order cost functionals and cubic

nonlinear controllers since it addresses a polynomial nonlinear performance

criterion.

Next, we consider the linear s y s tem (10.101) controlled by nonlinear

controllers where r(z, w) = 2z

w. For the statement of the next result

recall the deﬁnitions of R

, S

, R

, A

, B

and let ℓ : R

→ R be a

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

. Furthermore, assume that

≤ S

and p = d.

Proposition 10.2. Consider the linear dynamical system (10.101) and

(10.112) 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

+ P DR

P − P S

P, (10.199)

and

0 = p

′

(x)[A

− (S

− DR

)P ]x + ℓ(x). (10.200)

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)) + E

x), and the

performance functional (10.135) with R

(x) ≡ R

, L

(x) ≡ 0, and

(x) = x

x + ℓ(x) +

′

(x)(S

− DR

′

(x), (10.201)

NonlinearBook10pt November 20, 2007

636 CHAPTER 10

satisﬁes

J(x

, φ(x(·))) ≤ J(x

, φ(x(·))) = x

P x

+ p(x

), (10.202)

where

J(x

, u(·)) =

∞

[L(x(t), u(t)) + Γ(x(t), u(t))]dt (10.203)

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

Γ(x, u) = [D

(P x +

′

(x)) −E

x − E

·[D

(P x +

′

(x)) − E

x − E

u]. (10.204)

Furth ermore,

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), (10.205)

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 passivity constraint

(t)z(t) + V (x

) ≥ 0, w(·) ∈ L

, T ≥ 0. (10.206)

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

− DR

)

′

(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

(10.199)x + (10.200), it follows that, after some algebraic manipulations,

′

(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 (10.199) and (10.200) 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)))+

x(t)), is globally asymptotically stable.

Since A

− (S

− DR

)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 (10.200) is satisﬁed.

Finally, if p(x), x ∈ R

, is a polynomial function of the form

k=2

, then it follows from (10.200) that ℓ(x) =

k=2

k−1

x, where M

∈ N

, k = 2, . . . , r, an d M

satisﬁes

0 = [A

−(S

−DR

P )]

−(S

−DR

P )]+

, k = 2, . . . , r,

(10.207)

NonlinearBook10pt November 20, 2007

DISTURBANCE REJECTION CONTROL 637

and where P satisﬁes (10.199). In this case, the optimal control law φ(x) is

given by

φ(x) = −R

−1

P x +

k=2

k−1

+ E

the corresponding Lyapunov function guaranteeing closed-loop stability is

given by

V (x) = x

P x +

k=2

and L(x, u) given by (10.201) becomes

L(x, u) = x

k=2

k−1

x + u

k=2

k−1

− D

)

k=2

k−1

10.9 Problems

Problem 10.1. Consider the linear dynamical system given by (10.68)

and (10.69), where A is Hurwitz. Show that if (A, D) is controllable and

= A +

, α > 0, then the solution to the Riccati equation (10.76) is

given by (10.77).

Problem 10.2. Let X ∈ R

n×n

be a n on negative-deﬁnite matrix. Show

that lim

q→∞

[tr (I

◦ X)

]

1/q

= d

max

(X) and lim

q→∞

[tr X

]

1/q

= λ

max

(X),

where ◦ denotes the Hadamard product (i.e., entry-by-entry prod uct),

max

(X) denotes the m aximum diagonal entry of X, and λ

max

(X) denotes

the maximum eigenvalue of X. Also show that d

max

(X) ≤ λ

max

(X) ≤

tr(X). Use the above results to obtain diﬀerentiable bounds for |||G|||

given

by (10.78).

Problem 10.3. Consider the linear time-invariant system given by

(10.68) and (10.69). Assume A is Hurwitz. Show that

sup

ω ∈R

max

(G(ω)) = sup

w(·)∈L

|||z|||

2,2

|||w|||

2,2

, (10.208)

where G(s) = E(sI − A)

−1

D + E

∞

NonlinearBook10pt November 20, 2007

638 CHAPTER 10

Problem 10.4. Consider the linear dynamical system

G(s) ∼



A D

E 0



where G ∈ RH

∞

. Deﬁne the entropy of G at inﬁnity by

I(G, γ)

△

= lim

→∞



−

2π

∞

−∞



det(I

− γ

−2

∗

(ω)G(ω))





+ ω



dω



(10.209)

and assume there exists a positive-deﬁnite matrix P ∈ R

n×n

satisfying

0 = A

P + P A + γ

−2

P DD

P + E

E, (10.210)

where γ > 0. Show that the following statements hold:

i) The transfer function G satisﬁes |||G|||

∞

≤ γ.

ii) If |||G|||

∞

< γ, then I(G, γ) ≤ tr D

P D.

iii) The H

norm of G satisﬁes |||G|||

≤ I(G, γ).

iv) All real symmetric solutions to (10.210) are non negative deﬁnite.

v) There exists a (unique) minimal solution to (10.210) in th e class of

real symmetric solutions.

vi) P is the minimal solution to (10.210) if and only if α(A+γ

−2

P ) <

vii) |||G|||

∞

< γ if and only if A + γ

−2

P is Hurwitz, wh ere P is the

minimal solution to (10.210).

viii) If P is th e minimal solution to (10.210) and |||G|||

∞

< γ, then I(G, γ) =

tr D

P D.

Problem 10.5. Consider the nonlinear dynamical system G given by

(10.3) and (10.4) with J

(x) ≡ 0 and p = d. Show that if there exists a

continuously diﬀerentiable, positive-deﬁnite function W : D → R such that

′

(x)f(x) ≤ −δh

(x)h(x), δ > 0, (10.211)

′

(x)J

(x) = h

(x), (10.212)

then the nonlinear system G is n on exp ansive with gain 1/δ.

Problem 10.6. A nonlinear dynamical system G given by (10.1) and

(10.2) is said to be input-to-output stable if for every x

∈ R

and every

continuous bounded input w(t) ∈ R

, t ≥ 0, the solution x(t), t ≥ 0, to

(10.1) exists and the output satisﬁes

kz(t)k ≤ η(kx

k, t) + γ



sup

0≤τ≤t

kw(τ)k



, t ≥ 0, (10.213)

NonlinearBook10pt November 20, 2007

DISTURBANCE REJECTION CONTROL 639

where η(s, t), s > 0, is a class KL f unction and γ(s), s > 0, is a class K

function. Show that if G is input-to-state stable, then G is input-to-outp ut

stable. Alternatively, show th at if

kh(x) + J

(x)wk ≤ α

(kxk) + α

(kwk), (10.214)

where α

(·) and α

(·) are class K functions, then G is input-to-output stable.

Problem 10.7. Consider the linear dynamical system with state delay

given by

˙x(t) = Ax(t) + A

x(t −τ

), x(θ) = φ(θ), −τ

≤ θ ≤ 0, (10.215)

where x(t) ∈ R

, A ∈ R

n×n

, A

∈ R

n×n

, and φ : [−τ

, 0] → R

is a

continuous vector-valued function specifying the initial state of the system.

Let R > 0 and suppose there exist an n ×n positive-deﬁnite matrix P and

a scalar α > 0 such that

o = A

P + P A + α

+ α

−2

P A

P + R. (10.216)

Show that th e zero solution x

≡ 0 to (10.215) is globally asymp totically

stable (in the sense of Problem 3.65) for all τ

≥ 0. Furthermore, show that

this problem can be formulated as a feedback involving the operator ∆x(t)

△

x(t− τ

) satisfying the nonexpansivity constraint k∆x(t)k

≤ kx(t)k

. (Hint:

Use the Lyapunov-Krasovskii functional candidate given by

V (ψ) = ψ

(0)P ψ(0) + α

−τ

(θ)ψ(θ)dθ, ψ ∈ C([−τ

, 0], R

).)

(10.217)

Problem 10. 8. Consider the linear controlled system with state delay

given by

˙x(t) = Ax(t) + A

x(t − τ

) + Bu(t), x(θ) =

φ(θ), −τ

≤ θ ≤ 0,

(10.218)

where x(t) ∈ R

, u(t) ∈ R

, A, A

∈ R

n×n

, B ∈ R

n×m

, and

φ : [−τ

, 0] →

is a continuous vector-valued function specifying the initial state of the

system. Let α > 0, R

> 0, and R

> 0. Show that the zero solution x

≡ 0

to (10.218) is globally asymptotically stable (in the sense of Problem 3.65)

for all τ

≥ 0 with the feedback control φ(x) = −R

−1

P x, where P > 0

satisﬁes

0 = A

P + P A + R

+ α

−2

P A

P − P BR

−1

P. (10.219)

Problem 10.9. Consider the nonlinear controlled system with state

delay given by

˙x(t) = Ax(t) + f

(x(t −τ

)) + Bu(t), x(θ) =

φ(θ), −τ

≤ θ ≤ 0,

NonlinearBook10pt November 20, 2007

640 CHAPTER 10

(10.220)

where x(t) ∈ R

, u(t) ∈ R

, A ∈ R

n×n

, B ∈ R

n×m

, f

: R

→ R

satisﬁes

(0) = 0, and

φ : [−τ

, 0] → R

is a continuous vector-valued function

specifying the initial state of the system. Let α > 0, R

> 0, R

> 0, and

assume that kf

(x)k ≤ γkxk

, x ∈ R

. Show that the zero solution x

≡ 0

to (10.218) is globally asymptotically stable (in the sense of Problem 3.65)

for all τ

≥ 0 with the feedback control φ(x) = −R

−1

P x, where P > 0

satisﬁes

0 = A

P + P A + R

+ α

−2

− 2P BR

−1

P. (10.221)

(Hint: Use the Lyapunov-Krasovskii functional candidate given by

V (ψ) = ψ

(0)P ψ(0) + α

−τ

(ψ(θ))f

(ψ(θ))dθ, ψ ∈ C([−τ

, 0], R

).)

(10.222)

Problem 10.10. Consider the nonlinear scalar dynamical s y s tem

˙x(t) = −x

(t) + u(t) + x(t)w(t), x(0) = x

, t ≥ 0, (10.223)

where x(t), u(t) ∈ R and w(t) ∈ W = [−1, 1]. Find a stabilizing feedback

controller u(t) = φ(x(t)) that minimizes

J(x

, u(·)) =

∞

(t) + u

(t)]dt. (10.224)

Compare your controller to the feedback linearization controller u

(t) =

(t) −2x(t) by plotting u(t) versus x(t).

Problem 10.11. Consider the port-controlled Hamiltonian system giv-

en in Problem 8.21. Find an asymptotically stabilizing feedback control

law of the form u = φ(x) = −α(γ)G

(x)



∂H

∂x

(x)



, where α(γ) > 0 and

α(γ)



∂H

∂x

(x)



satisﬁes the Hamilton-Jacobi-Bellman equation.

Problem 10. 12. Consider the nonlinear dynamical system G given by

˙x(t) = f(x(t)) + G(x(t))u(t) + J

(x(t))w(t), x(0) = x

, w(·) ∈ L

, t ≥ 0,

(10.225)

y(t) = h(x(t)), (10.226)

with performance variables

z(t) = E

1∞

(x(t)) + E

2∞

u(t), (10.227)

where x ∈ R

, u, y ∈ R

, w ∈ R

, z ∈ R

, f : R

→ R

satisﬁes f (0) = 0,

G : R

→ R

n×m

, J

: R

→ R

n×d

, h : R

→ R

satisﬁes h(0) = 0,

1∞

: R

→ R

, E

2∞

∈ R

p×m

, R

△

= E

2∞

> 0, and E

2∞

1∞

(x) = 0,

NonlinearBook10pt November 20, 2007

DISTURBANCE REJECTION CONTROL 641

x ∈ R

. Assume that, with w(t) ≡ 0, (10.225) and (10.226) is passive, zero-

state observable, and completely reachable with a continuously diﬀerentiable

radially unbounded storage fu nction V

: R

→ R. Furtherm ore, let

V : R

→ R be a continuously diﬀerentiable function such that V (0) = 0,

V (x) > 0, x ∈ R

, x 6= 0, V (x) → ∞ as kxk → ∞, and

0 = L

(x) + V

′

(x)f(x) −

′

(x)G(x)R

−1

(x)V

′T

(x)

4γ

′

(x)J

(x)V

′T

(x), x ∈ R

, (10.228)

where γ > 0, L

(x) = ℓ

(x)ℓ(x) + h

(x)R

−1

h(x) −

4γ

′

(x)J

(x)

·V

′T

(x) ≥ 0, and ℓ(·) satisﬁes (5.145). Show that the output feedback

controller u(t) = −R

−1

y(t) asymptotically stabilizes the undisturbed

(w(t) ≡ 0) nonlinear sys tem (10.225) and minimizes the performance

criterion

J(x

, u(·)) =

∞

[ℓ

(x(t))ℓ(x(t)) + y

(t)R

−1

y(t) + u

(t)R

−1

u(t)]dt,

(10.229)

where x(t), t ≥ 0, solves (10.225) with w(t) ≡ 0, in the sense that

J(x

, φ(y(·))) = min

u(·)∈S(x

)

J(x

, u(·)), x

∈ R

. (10.230)

Furth ermore, show that with u(t) = −R

−1

y(t), the solution x(t), t ≥ 0, of

the closed-loop system (10.225) satisﬁes the nonexpansivity constraint

(t)z(t)dt ≤ γ

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

), w(·) ∈ L

, T ≥ 0.

(10.231)

Problem 10.13. Consider the nonlinear cascade dynamical system

˙x(t) = f (x(t)) + G(x(t))ˆx(t) + J

(x)w(t), x(0) = x

, w(·) ∈ L

, t ≥ 0,

(10.232)

ˆx(t) = u(t) + J

(ˆx)w(t), ˆx(0) = ˆx

, (10.233)

z(t) = h(x(t), ˆx(t)) + J(x(t), ˆx(t))u(t), (10.234)

with performance functional

J(x

, ˆx

, u(·))

△

∞

L(x(t), ˆx(t), u(t))dt, (10.235)

where u(·) is admissible and (x(t), ˆx(t)), t ≥ 0, s olves (10.232) and (10.233)

and where

L(x, ˆx, u)

△

= L

(x, ˆx) + L

(x, ˆx)u + u

(x, ˆx)u, (10.236)

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

NonlinearBook10pt November 20, 2007

642 CHAPTER 10

sub

: R

→ R such that for all x ∈ R

and ˆx ∈ R

α(0) = 0, (10.237)

sub

(0) = 0, (10.238)

sub

(x) > 0, x 6= 0, (10.239)

′

sub

(x)[f(x) + G(x)α(x)] +

4γ

′

sub

(x)J

(x)V

′T

sub

(x) < 0, x 6= 0,

(10.240)

where R

(x, ˆx)

△

= R

(x, ˆx) + J

(x, ˆx)J(x, ˆx) and γ > 0. Furth ermore,

assume th at there exist a function L

: R

×R

→ R

1×m

such that L

(0, 0) =

0 and a positive-deﬁnite matrix

P ∈ R

m×m

such that, for all x ∈ R

and

ˆx ∈ R

2(ˆx − α(x))



−1

(x)V

′T

sub

(x) − α

′

(x)(f(x) + G(x)ˆx)

−

−1

(x, ˆx){2

P (ˆx −α(x)) + L

(x, ˆx) + 2J

(x, ˆx)h(x, ˆx)}

2γ

{− α

′

(x)J

(x)V

′T

sub

(x)

+[α

′

(x)J

(x)α

′

(x) + J

(ˆx)J

(ˆx)]

P (ˆx − α(x))}



+[h(x, ˆx) + J(x, ˆx)φ(x, ˆx)]

[h(x, ˆx) + J(x, ˆx)φ(x, ˆx)] < 0, ˆx 6= α(x),

(10.241)

where φ(x, ˆx) = −

−1

(x, ˆx)[L

(x, ˆx) + 2

P (ˆx − α(x)) + 2J

(x, ˆx)h(x, ˆx)].

Show that, w ith the feedback control law

φ(x, ˆx) = −

−1

(x, ˆx)[L

(x, ˆx) + 2

P (ˆx −α(x)) + 2J

(x, ˆx)h(x, ˆx)],

(10.242)

the zero s olution (x(t), ˆx(t)) ≡ (0, 0) of the undisturbed (w(t) ≡ 0) cascade

system

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

, t ≥ 0, (10.243)

ˆx(t) = φ(x(t), ˆx(t)), ˆx(0) = ˆx

, (10.244)

is globally asymptotically stable. Furthermore, show that

J(x

, ˆx

, φ(x, ˆx)) ≤ J(x

, ˆx

, φ(x, ˆx)) = V (x

, ˆx

), (x

, ˆx

) ∈ R

× R

(10.245)

where

J(x

, ˆx

, u(·))

△

∞

[L(x(t), ˆx(t), u(t)) + Γ(x(t), ˆx(t), u(t))]dt, (10.246)

V (x, ˆx) = V

sub

(x) + (ˆx − α(x))

P (ˆx −α(x)), (10.247)

NonlinearBook10pt November 20, 2007

DISTURBANCE REJECTION CONTROL 643

and

Γ(x, ˆx, u) =

4γ

′

(x, ˆx)J

(x, ˆx)V

′T

(x, ˆx)

+[h(x, ˆx) + J(x, ˆx)u]

[h(x, ˆx) + J(x, ˆx)u], (10.248)

where J

(x, ˆx)

△

= [J

(x), J

(ˆx)]

and (x(t), ˆx(t)), t ≥ 0, solves (10.232) and

(10.233) with w(t) ≡ 0. In addition, show that the performance functional

(10.246), with

(x, ˆx) = φ

(x, ˆx)R

(x, ˆx)φ(x, ˆx) + 2(ˆx −α(x))

P α

′

(x)(f(x) + G(x)ˆx)

−V

′

sub

(x)(f(x) + G(x)ˆx) −h

(x, ˆx)h(x, ˆx)

−

4γ

′

(x, ˆx)J

(x, ˆx)V

′T

(x, ˆx), (10.249)

is minimized in the sense that

J(x

, ˆx

, φ(x(·))) = min

u(·)∈S(x

,ˆx

)

J(x

, ˆx

, u(·)), (10.250)

where S(x

, ˆx

) is the set of regulation controllers for the nonlinear system

(10.232) and (10.233) with w(t) ≡ 0. With u(·) = φ(x(·), ˆx(·)), show that

the solution (x(t), ˆx(t)), t ≥ 0, of the cascade system (10.232) and (10.233)

satisﬁes the nonexpansivity constraint

(t)z(t)dt ≤ γ

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

, ˆx

), w(·) ∈ L

T ≥ 0.

(10.251)

Finally, for J(x, ˆx) ≡ 0 and h(x, ˆx) = E(x, ˆx)(ˆx−α(x), w here E : R

×R

→

p×m

, show th at a particular choice satisfying (10.241) is given by

(x, ˆx) = 2[

′

sub

(x)G(x)

−1

− [f (x) + G(x)ˆx]

′

(x)

−

2γ

′

sub

(x)J

(x)α

′

(x)

−(ˆx − α(x))

P [α

′

(x)J

(x)α

′

(x) + J

(ˆx)J

(ˆx)]}

(ˆx −α(x))

(x, ˆx)E(x, ˆx)

−1

(x, ˆx). (10.252)

yielding the feedback control

φ(x, ˆx) = −R

−1

(x, ˆx)

P (ˆx − α(x)) −

−1

(x, ˆx)E(x, ˆx) (ˆx − α(x))

−1

(x)V

′T

sub

(x)

+ α

′

(x)[f(x) + G(x)ˆx]

2γ

′

(x)J

(x)V

′T

sub

(x)

−

′

(x)J

(x)α

′

(x) + J

(ˆx)J

(ˆx)

P (ˆx − α(x))

Problem 10.14. Consider the nonlinear dynamical system (10.232)–

(10.234) with J(x, ˆx) ≡ 0. Assume that there exist continuously diﬀeren-

NonlinearBook10pt November 20, 2007

644 CHAPTER 10

tiable fun ctions α : R

→ R

and V

sub

: R

→ R such that

α(0) = 0, (10.253)

sub

(0) = 0, (10.254)

sub

(x) > 0, x ∈ R

, x 6= 0, (10.255)

′

sub

(x)[f(x) + G(x)α(x)] +

4γ

′

sub

(x)J

(x)V

′T

sub

(x) < 0,

x ∈ R

, x 6= 0, (10.256)

and let h(x, ˆx) = E(x, ˆx)(ˆx − α(x)), w here E : R

× R

→ R

p×m

. Show

that, with the feedback stabilizing control law given by

φ(x, ˆx) =



−(c

+ ρ(x, ˆx))β(x, ˆx), ˆx 6= α(x),

0, ˆx = α(x),

(10.257)

where β(x, ˆx)

△

= 2

P (ˆx −α(x)),

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)

µ(x, ˆx)

△

= 2α

′

(x)[f(x)+G(x)ˆx]−

−1

(x)V

′T

sub

(x),

P ∈ P

, and c

> 0, the

cascade system (10.232) and (10.233) has a s ector (and, hence, gain) margin

(

, ∞), where ρ ∈ [0, 1] and α = 1 −

η(1−ρ

)

. Finally, with u = σ(φ(x, ˆx)),

show that the solution (x(t), ˆx(t)), t ≥ 0, of the closed-loop system (10.232)

and (10.233) satisﬁes the nonexpansivity constraint

(t)z(t)dt ≤ (γ/ρ)

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

), w(·) ∈ L

, T ≥ 0,

(10.258)

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.

Problem 10.15. Consider the nonlinear cascade dynamical system

˙x(t) = f(x(t)) + G(x(t))y(t) + J

(x)w(t), x(0) = x

, w(·) ∈ L

, t ≥ 0,

(10.259)

ˆx(t) =

f(ˆx) +

G(ˆx)u(t) + J

(ˆx)w(t), ˆx(0) = ˆx

, (10.260)

y(t) =

h(ˆx), (10.261)

z(t) = h(x(t), ˆx(t)) + J(x(t), ˆx(t))u(t), (10.262)

with performance functional (10.235) wh ere L(x, ˆx, u) is given by (10.236)

and J

(x)J

(ˆx) ≡ 0. Assu me that the input subsystem (10.260) and

(10.261) is feedback strictly passive such that there exist a positive-deﬁnite

storage function V

(ˆx) and a function k : R

→ R

satisfying

0 > V

′

(ˆx)

f(ˆx) +

G(ˆx)k(ˆx)

, ˆx ∈ R

, ˆx 6= 0, (10.263)