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

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DISCRETE-TIME NONLINEAR CONTROL 885

robust performance here refers to a guaranteed bound on the worst-case

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

set. For the following result, let D ∈ R

be an open set, assum e 0 ∈ D, let

L : D → R, and let F ⊂ {f : D → D : f (0) = 0} denote the class of

uncertain nonlinear systems with f

(·) ∈ F deﬁ ning the nominal nonlinear

system.

Theorem 14.12. Consider the nonlinear uncertain system

x(k + 1) = f(x(k)), x(0) = x

, k ∈ Z

, (14.232)

where f (·) ∈ F, with performance functional

J(x

)

△

∞

k=0

L(x(k)). (14.233)

Assume there exist functions Γ : D → R and V : D → R such that V (·) is

continuous and

V (0) = 0, (14.234)

V (x) > 0, x ∈ D, x 6= 0, (14.235)

V (f (x)) ≤ V (f

(x)) + Γ(x), x ∈ D, f(·) ∈ F,

(14.236)

V (f

(x)) − V (x) + Γ(x) < 0, x ∈ D, x 6= 0, (14.237)

L(x) + V (f

(x)) − V (x) + Γ(x) = 0, x ∈ D, (14.238)

where f

(·) ∈ F deﬁnes the nominal n on linear system. Then the zero

solution x(k) ≡ 0 of (14.232) is locally asymptotically stable for all f(·) ∈ F

and there exists a neighborhood D

⊆ D of the origin such that the

performance functional (14.233) satisﬁes

sup

f(·)∈F

J(x

) ≤ J(x

) = V (x

), x

∈ D

, (14.239)

where

J(x

)

△

∞

k=0

[L(x(k)) + Γ(x(k))], (14.240)

and where x(k), k ∈ Z

, is the solution to (14.232) with f (x(k)) = f

(x(k)).

Finally, if D = R

, and

V (x) → ∞ as kxk → ∞, (14.241)

then the zero solution x(k) = 0, k ∈ Z

, of (14.232) is globally

asymptotically stable.

NonlinearBook10pt November 20, 2007

886 CHAPTER 14

Proof. Let f (·) ∈ F and x(k), k ∈ Z

, satisfy (14.232). Then

∆V (x(k))

△

= V (x(k + 1)) − V (x(k)) = V (f(x(k))) − V (x(k)), k ∈ Z

(14.242)

Hence, from (14.236) and (14.237), it follows that

∆V (x(k)) < 0, k ∈ Z

, x(k) 6= 0. (14.243)

Thus , from (14.234) and (14.235), and (14.243) it follows that V (·) is a

Lyapunov function f or (14.244), wh ich proves local asymptotic stability of

the zero solution x(k) ≡ 0 for all f (·) ∈ F. Consequently, x(k) → 0 as

k → ∞ for all initial cond itions x

∈ D

for some neighborhood D

⊆ D of

the origin.

Next, (14.242) implies that

0 = −∆V (x(k)) + V (f (x(k))) − V (x(k)), k ∈ Z

and h en ce, using (14.236) and (14.238),

L(x(k)) = − ∆V (x(k)) + L(x(k)) + V (f(x(k))) − V (x(k))

≤ −∆V (x(k)) + L(x(k)) + V (f

(x(k))) − V (x(k)) + Γ(x(k))

= −∆V (x(k)).

Now, summing over [0, k] yields

i=0

L(x(i)) ≤ − V (x(k + 1)) + V (x

Letting k → ∞ and noting that V (x(k)) → 0 as k → ∞ f or all x

∈ D

and

f(·) ∈ F yields J(x

) ≤ V (x

Next, let x(k), k ∈ Z

, s atisfy (14.232) with f(x(k)) = f

(x(k)). Then,

it follows fr om (14.238) that

L(x(k)) + Γ(x(k))

= −∆V (x(k)) + L(x(k)) + V (f

(x(k))) − V (x(k)) + Γ(x(k))

= −∆V (x(k)).

Summing over [0, k] yields

i=0

[L(x(i)) + Γ(x(i))] = −V (x(k + 1)) + V (x

Now, letting k → ∞ yields J(x

) = V (x

). Finally, for D = R

global

asymptotic stability of the zero solution x(k) = 0, k ∈ Z

, is a direct

consequence of the radially unbounded condition (14.241) on V (x), x ∈ R

NonlinearBook10pt November 20, 2007

DISCRETE-TIME NONLINEAR CONTROL 887

Next, we specialize Theorem 14.12 to nonlinear un certain systems of

the f orm

x(k + 1) = f

(x(k)) + ∆f(x(k)), x(0) = x

, k ∈ Z

, (14.244)

where f

: D → R

is such that f

(0) = 0, and f

+ ∆f ∈ F. Here, F is

such that

F ⊂ {f

+ ∆f : D → D : ∆f ∈ ∆},

where ∆ is a given nonlinear uncertainty set of nonlinear perturbations ∆f

of the n ominal system dynamics f

(·) ∈ F. Since F ⊂ {f : D → D : f(0) =

0} it follows that ∆f (0) = 0.

Corollary 14.10. Consider the n onlinear uncertain system given by

(14.244) with performance functional (14.233). Assume there exist functions

Γ : D → R, P

: D → R

1×n

, P

: D → N

, and V : D → R such that V (·)

is continuous, (14.234), (14.235), (14.237), and (14.238) hold and

(0) = 0, (14.245)

∆f

(x)P

(x) + P

(x)∆f(x) + ∆f

(x)P

(x)∆f(x) ≤ Γ(x),

x ∈ D, ∆f(·) ∈ ∆, (14.246)

V (f

(x) + ∆f(x)) = V (f

(x)) + ∆f

(x)P

(x) + P

(x)∆f(x)

+∆f

(x)P

(x)∆f(x), x ∈ D, ∆f(·) ∈ ∆. (14.247)

Then then the zero solution x(k) ≡ 0 of (14.244) is locally asymptotically

stable f or all ∆f(·) ∈ ∆ and there exists a neighborh ood D

⊆ D of the

origin such that the performance functional (14.233) satisﬁes

sup

∆f(·)∈∆

J(x

) ≤ J(x

) = V (x

), x

∈ D

, (14.248)

where

J(x

)

△

∞

k=0

[L(x(k)) + Γ(x(k))], (14.249)

where x(k), k ∈ Z

, solves (14.244) with ∆f(k) ≡ 0. Finally, if D = R

and V (x), x ∈ R

, satisﬁes (14.241), then th e solution x(k) = 0, k ∈ Z

, of

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

Proof. The result is a direct consequence of T heorem 14.12 with

f(x) = f

(x) + ∆f (x) and V (f(x)) given by (14.247). Speciﬁcally, in this

case it follows from (14.246) and (14.247) that V (f(x)) ≤ V (f

(x)) + Γ(x)

for all x ∈ R

and ∆f(·) ∈ ∆. Hence, all conditions of Theorem 14.12 are

satisﬁed.

Having established the theoretical basis to our approach, we now give

a concrete structure for the bounding fun ction Γ(x) for the structure F as

speciﬁed by (11.23) with ∆ satisfying (11.24) and (11.27), respectively.

NonlinearBook10pt November 20, 2007

888 CHAPTER 14

Proposition 14.2. Let P

: R

→ R

1×n

be such that P

(0) = 0 and

let P

: R

→ N

be s uch that

− g

(x)P

(x)g

(x) > 0, x ∈ R

. (14.250)

Then the function

Γ(x) = P

(x)g

(x)[I

−g

(x)P

(x)g

(x)]

−1

(x)P

(x)

(x))m(h

(x)) (14.251)

satisﬁes (14.246) with F given by (11.23) and ∆ given by (11.24).

Proof. Note that for all x ∈ D,

0 ≤ {[P

(x)g

(x) − δ

(x))][I

− g

(x)P

(x)g

(x)]}

·[I

− g

(x)P

(x)g

(x)]

−1

{[I

− g

(x)P

(x)g

(x)]

·[P

(x)g

(x) − δ

(x))]

}

= P

(x)(g

(x)[I

− g

(x)P

(x)g

(x)]

−1

(x)P

(x)

−P

(x)g

(x)δ(h

(x)) − δ

(x))g

(x)P

(x)

+δ

(x))δ(h

(x)) − δ

(x))g

(x)P

(x)g

(x)δ(h

(x)

≤ Γ(x) −[∆f

(x)P

(x) + P

(x)∆f(x) + ∆f

(x)P

(x)∆f(x)],

which proves (14.246) with F given by (11.23) and ∆ given by (11.27).

Proposition 14.3. Let P

: R

→ R

1×n

be such that P

(0) = 0 and

let P

: R

→ N

be s uch that

− g

(x)P

(x)g

(x) > 0, x ∈ R

. (14.252)

Then the function

Γ(x) = [P

(x)g

(x) + m

(x)) + m

(x))][2I

− g

(x)P

(x)g

(x)]

−1

·[P

(x)g

(x) + m

(x)) + m

(x))]

+ m

(x))m

(x))

(x))m

(x)) (14.253)

satisﬁes (14.246) with F given by (11.23) with ∆ given by (11.27).

Proof. The pr oof is identical to that of Proposition 14.2 and, hence,

is left as an exercise for the reader.

We n ow combine the results of Corollary 14.10 and Propositions 14.2

and 14.3 to obtain a series of suﬃcient conditions guaranteeing robust

stability and performance for the nonlinear u ncertain discrete-time system

(14.244).

Proposition 14.4. Consider the nonlinear uncertain system (14.244).

NonlinearBook10pt November 20, 2007

DISCRETE-TIME NONLINEAR CONTROL 889

Let L(x) > 0, x ∈ R

, and suppose there exist functions P

: R

→ R

1×n

: R

→ N

, and a continuous radially unbounded function V : R

→ R

such that (14.234), (14.235), and (14.247) hold and

V (x) = V (f

(x)) + P

(x)g

(x)[I

− g

(x)P

(x)g

(x)]

−1

(x)P

(x)

(x))m(h

(x)) + L(x), x ∈ R

. (14.254)

Then the zero solution x(t) ≡ 0 to (14.244) is globally asymptotically stable

for all ∆f(·) ∈ ∆ with ∆ given by (11.24), and th e performance function

(14.233) satisﬁes

sup

∆f(·)∈∆

J(x

) ≤ J(x

) = V (x

), (14.255)

where

J(x

)

△

∞

k=0

[L(x(k)) + P

(x(k))g

(x(k))[I

− g

(x(k))P

(x(k))g

(x(k))]

−1

·g

(x(k))P

(x(k)) + m

(x(k)))m(h

(x(k))), (14.256)

where x(k), k ∈ Z

, is the solution to (14.244) with ∆f (x) ≡ 0.

Proposition 14.5. Consider the nonlinear uncertain system (14.244).

Let L(x) > 0, x ∈ R

, and suppose there exist functions P

: R

→ R

1×n

: R

→ N

, and a continuous radially unbounded function V : R

→ R

such that (14.234), (14.235), and (14.247) hold and

V (x) = V (f

(x)) + [P

(x)g

(x) + m

(x)) + m

(x))][2I

−g

(x)P

(x)g

(x)]

−1

(x)g

(x) + m

(x)) + m

(x))]

(x))m

(x)) + m

(x))m

(x)) + L(x), x ∈ R

(14.257)

Then the zero solution x(t) ≡ 0 to (14.244) is globally asymptotically stable

for all ∆f(·) ∈ ∆ with ∆ given by (11.27), and th e performance function

(14.233) satisﬁes

sup

∆f(·)∈∆

J(x

) ≤ J(x

) = V (x

), (14.258)

where

J(x

)

△

∞

k=0

[L(x(k)) + [P

(x(k))g

(x(k)) + m

(x(k))) + m

(x(k)))]

·[2I

−g

(x(k))P

(x(k))g

(x(k))]

−1

(x(k))g

(x(k))

(x(k))) + m

(x(k)))]

+ m

(x(k)))m

(x(k)))

(x(k)))m

(x(k))), (14.259)

where x(k), k ∈ Z

, is the solution to (14.244) with ∆f (x) ≡ 0.

NonlinearBook10pt November 20, 2007

890 CHAPTER 14

The following corollary specializes Theorem 14.12 to linear uncertain

systems and connects the framework of Th eorem 14.12 to the quadratic

Lyapunov bounding framework of Haddad et al. [148]. Speciﬁcally, in this

case we consider F 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 bound ed uncertainty set of uncertain

perturbations ∆A of the nominal system matrix such th at 0 ∈ ∆

. For

the statement of this result let R ∈ P

Corollary 14.11. Consider the linear uncertain system

x(k + 1) = (A + ∆A)x(k), x(0) = x

, k ∈ Z

, (14.260)

with performance functional

∆

)

△

∞

k=0

(k)Rx(k). (14.261)

where ∆A ∈ ∆

. Let Ω : N ⊆ S

→ N

be such that

P ∆A + ∆A

P A + ∆A

P ∆A ≤ Ω(P ), ∆ A ∈ ∆

, P ∈ N. (14.262)

Furth ermore, suppose there exists P ∈ P

satisfying

P = A

P A + Ω(P ) + R. (14.263)

Then the zero solution x(k) ≡ 0 of (14.260) is globally asymptotically stable

for all ∆A ∈ ∆

, and

sup

∆A∈∆

∆

) ≤ J(x

) = x

P x

, x

∈ R

, (14.264)

where

J(x

) =

∞

k=0

(k)[R + Ω(P )]x(k), (14.265)

and where x(k), k ∈ Z

, solves (14.260) with ∆ A = 0.

Proof. The result is a direct consequence of T heorem 14.12 with

f(x) = (A + ∆A)x, f

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

Rx, V (x) = x

P x, Γ(x) =

Ω(P )x, and D = R

. Speciﬁcally, conditions (14.234) and (14.235)

are trivially satisﬁed. Now, V (f(x)) = x

P Ax + x

(∆A

P + P ∆A +

∆A

P ∆A)x, and hence, it follows from (14.262) that V (f(x)) ≤ V (f

(x))+

Γ(x) = x

P A + Ω(P ))x, for all x ∈ R

and ∆A ∈ ∆

. Furthermore,

it follows from (14.263) that L(x) + V (f

(x)) − V (x) + Γ(x) = 0, x ∈ R

and hence, V (f

(x)) − V (x) + Γ(x) < 0, for all x ∈ R

, x 6= 0, so that all

conditions of Theorem 14.12 are satisﬁed. Finally, since V (x) is radially

unbounded the zero solution x(k) ≡ 0 of (14.260) is globally asymptotically

stable for all ∆A ∈ ∆

NonlinearBook10pt November 20, 2007

DISCRETE-TIME NONLINEAR CONTROL 891

Corollary 14.11 is the deterministic version of Theorem 4.1 of [173]

involving quadratic Lyapunov bounds for addr essin g robust stability and

performance analysis of linear uncertain discrete-time systems. To demon-

strate the applicability of Corollary 14.11 consider the u ncertain system

(14.260) with ∆

given by (11.59). For th is uncertainty characterization

the bound Ω(·) satisfying (14.262) can now be given a concrete form.

Proposition 14.6. Let M ⊆ S

denote the set of P ∈ P

such that

△

= {P ∈ N

: I

− B

P B

> 0}. (14.266)

Then the function

Ω(P ) = A

P B

− B

P B

)

−1

P A + C

(14.267)

satisﬁes (14.262) with ∆

given by (11.59).

Proof. The proof is a direct consequence of Proposition 14.2 with

(x) = x

P , P

(x) = P , and m(h

(x) = N

1/2

Finally, as in the nonlinear case, we now combine the results of

Corollary 14.11 and P roposition 14.6 to obtain a suﬃcient condition for

guaranteeing robust stability and performance for the uncertain discrete-

time system (14.260).

Proposition 14.7. Consider th e linear uncertain s ystem (14.260). Let

R ∈ P

, N ∈ N

, and suppose there exists P ∈ M satisfying

P = A

P A + A

P B

−B

P B

)

−1

P A + C

+ R. (14.268)

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

given by (11.59),

and

sup

∆A∈∆

∆A

) ≤ x

P x

, x

∈ R

. (14.269)

14.13 Problems

Problem 14.1. Consider the system (14.1) and (14.2) with F (x, u, k)

= Ax + Bu and L(x, u, k) = x

x + u

u, where A ∈ R

n×n

, B ∈ R

n×m

∈ R

n×n

, and R

∈ R

m×m

, such that R

≥ 0 and R

> 0. Show that the

optimal control law characterized by the Bellman equation (14.10) is given

u(k) = −[R

+ B

P (k + 1)B]

−1

P (k + 1)Ax(k), (14.270)

where P (k

+ 1) = 0 and

P (k) = A

P (k+1)A+R

−A

P (k+1)B[R

P (k+1)B]

−1

P (k+1)A.

(14.271)

NonlinearBook10pt November 20, 2007

892 CHAPTER 14

Problem 14.2. Consider the cascade system

x(k + 1) = f (x(k)) + G(x(k))y(k), x(0) = x

, k ∈ Z

, (14.272)

ˆx(k + 1) = Aˆx(k) + Bu(k), ˆx(0) = ˆx

, (14.273)

y(k) = C ˆx(k) + Du(k), (14.274)

where ˆx ∈ R

, u, y ∈ R

, A ∈ R

q×q

, B ∈ R

q×m

, C ∈ R

m×q

, and D ∈ R

m×m

with performance functional

J(x

, ˆx

, u) =

∞

k=0

L(x(k), ˆx(k), u(k)), (14.275)

where (x(k), ˆx(k)), k ∈ Z

, solves (14.272) and (14.273), and L(x, ˆx, u) is

given by (14.276) and where

L(x, ˆx, u)

△

= L

(x, ˆx) + L

(x, ˆx)u + u

(x)u. (14.276)

Assume that (A, B, C, D) is strict feedback positive real, that is, there exist

a scalar ρ > 1 and matrices

P ∈ P

, L ∈ R

l×q

, W ∈ R

l×m

, and K ∈ R

m×q

such that

P = (A + BK)

P (A + BK) + L

L, (14.277)

P (A + BK) = C + DK −W

L, (14.278)

D + D

= B

P B + W

W, (14.279)

and the nonlinear subsystem (14.272) has a globally stable equilibrium at

x(k) = 0, k ∈ Z

, and Lyapunov function V

sub

(x) so that

sub

(f(x)) < V

sub

(x), x ∈ R

, x 6= 0. (14.280)

Furth ermore, assume there exist functions L

: R

× R

→ R

1×m

, P

→ R

1×m

, and P

: R

→ N

such that

(0, 0) = 0, (14.281)

(0) = 0, (14.282)

sub

(f(x) + G(x)y) = V

sub

(f(x)) + P

(x)y + y

(x)y, (14.283)

(x) + P

(x)Cˆx − 2K ˆx −(I

(x)D)R

−1

(x)

·[2(B

P A + D

(x)C)ˆx + L

(x, ˆx) + D

(x)]} ≤ 0,

(x, ˆx) ∈ R

× R

, (14.284)

where R

(x)

△

= R

(x) + B

P B + D

(x)D and

P , K satisfy (14.277)–

(14.279). Show th at the zero solution (x(k), ˆx(k)) ≡ (0, 0) of th e cascade

system (14.272) and (14.273) is globally asymptotically stable with the

feedback control law u = φ(x, ˆx), where

φ(x, ˆx) = −

−1

(x)[2(B

P A + D

(x)C)ˆx + L

(x, ˆx) + D

(x)].

(14.285)

NonlinearBook10pt November 20, 2007

DISCRETE-TIME NONLINEAR CONTROL 893

Furth ermore, show that for (x

, ˆx

) ∈ R

× R

J(x

, ˆx

, φ(x(·), ˆx(·))) = V (x

, ˆx

), (14.286)

where

V (x, ˆx) = V

sub

(x) + ˆx

P ˆx, (14.287)

and the performan ce functional (14.275), with

(x, ˆx) = φ

(x, ˆx)R

(x)φ(x, ˆx) + V

sub

(x)

−V

sub

(f(x) + G(x)Cˆx) + ˆx

(

P − A

P A)ˆx, (14.288)

is minimized in the sense that

J(x

, ˆx

, φ(x(·), ˆx(·))) = min

u∈S(x

,ˆx

)

J(x

, ˆx

, u(·)), (14.289)

where

S(x

, ˆx

)

△

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

(14.272) and (14.273) satisﬁes (x(k), ˆx(k)) → 0 as k → ∞}.

Problem 14.3. Consider the cascade system

x(k + 1) = f (x(k)) + G(x(k))y(k), x(0) = x

, k ∈ Z

, (14.290)

ˆx(k + 1) =

f(ˆx(k)) + ˆg(ˆx(k))u(k), ˆx(0) = ˆx

, (14.291)

y(k) = h(ˆx(k)) + J(ˆx(k))u(k), (14.292)

where ˆx ∈ R

, u, y ∈ R

f : R

→ R

satisﬁes

f(0) = 0, ˆg : R

→ R

q×m

h : R

→ R

satisﬁes h(0) = 0, and J : R

→ R

m×m

, with performance

functional (14.275) where (x(k), ˆx(k)), k ∈ Z

, solves (14.290) and (14.291),

and L(x, ˆx, u) is given by (14.276). Assum e that the in put sub system

(14.291) and (14.292) is feedback strictly passive and assum e the subsystem

(14.290) has a globally stable equilibrium at x(k) = 0, k ∈ Z

, and

Lyapunov function V

sub

(x) so that

sub

(f(x)) < V

sub

(x), x ∈ R

, x 6= 0. (14.293)

Furth ermore, assume there exist functions L

: R

× R

→ R

1×m

, P

→ R

1×m

, and P

: R

→ N

such that

(0, 0) = 0, (14.294)

(0) = 0, (14.295)

sub

(f(x) + G(x)y) = V

sub

(f(x)) + P

(x)y + y

(x)y, (14.296)



(x) + P

(x)h(ˆx) −2k(ˆx) −(I

(x)J(ˆx))R

−1

(x, ˆx)[

(ˆx)

+2J

(ˆx)P

(x)h(ˆx) + L

(x, ˆx) + J

(ˆx)P

(x)]



≤ 0, (x, ˆx) ∈ R

× R

(14.297)

NonlinearBook10pt November 20, 2007

894 CHAPTER 14

where R

(x, ˆx)

△

= R

(x, ˆx) +

(ˆx) + J

(ˆx)P

(x)J(ˆx) and k(ˆx) satisﬁes

0 > V

(

f(ˆx) + ˆg(ˆx)k(ˆx)) − V

(ˆx) + l

(ˆx)l(ˆx), ˆx 6= 0, (14.298)

0 =

(ˆx) + W

(ˆx)l(ˆx) − (h(ˆx) + J(ˆx)k(ˆx)) +

(ˆx)k(ˆx), (14.299)

0 =

(ˆx) + W

(ˆx)W(ˆx) − (J(ˆx) + J

(ˆx)). (14.300)

Show that the zero solution (x(k), ˆx(k)) ≡ (0, 0) of the cascade system

(14.290) and (14.291) is globally asymptotically stable with the feedback

control law u = φ(x, ˆx), where

φ(x, ˆx) = −

−1

(x, ˆx)[

(ˆx) + 2J

(ˆx)P

(x)h(ˆx) + L

(x, ˆx)

(ˆx)P

(x)]. (14.301)

Furth ermore, show that for (x

, ˆx

) ∈ R

× R

J(x

, ˆx

, φ(x(·), ˆx(·))) = V (x

, ˆx

), (14.302)

where

V (x, ˆx) = V

sub

(x) + V

(ˆx), (14.303)

and the performan ce functional (14.275), with

(x, ˆx) = φ

(x, ˆx)R

(x, ˆx)φ(x, ˆx) + V

sub

(x)

−V

sub

(f(x) + G(x)h(ˆx)) + V

(ˆx) − V

(

f(ˆx)), (14.304)

is minimized in the sense that

J(x

, ˆx

, φ(x(·), ˆx(·))) = min

u∈S(x

,ˆx

)

J(x

, ˆx

, u(·)). (14.305)

Problem 14.4. Consider the discrete-time linear dyn amical system

G(z) ∼



A D

E 0



where G ∈ RH

∞

. Deﬁne the discrete entropy of G at inﬁnity by

I(G, γ)

△

−γ

2π

−π



det(I

−γ

−2

∗

θ

)G(e

θ

))



dθ, (14.306)

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

n×n

satisfying

P = A

P A + A

P D(γ

I − D

P D)

−1

P A + E

E, (14.307)

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) ≤ −ln det(γ

I − D

P D).

iii) The H

norm of G satisﬁes |||G|||

≤ I(G, γ).

iv) All real symmetric solutions to (14.307) are non negative deﬁnite.