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

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NonlinearBook10pt November 20, 2007

DISCRETE-TIME THEORY 815

where V

: R

→ R and d

: R

× R

→ R contain higher-order terms of

(·), d(·, ·), respectively.

Next, let f(x) = Ax + f

(x) and h(x) = Cx + h

(x), where f

(x) and

(x) contain n on linear terms of f (x) and h(x), respectively, and let G(x) =

B+G

(x) and J(x) = D+J

(x), where G

(x) and J

(x) contain nonconstant

terms of G(x) and J(x), respectively. Using (13.185) and (13.186), (13.184)

can be written as

0 = (Ax + Bu)

P (Ax + Bu) −x

P x − (x

QCx + 2x

QDu

QDu + 2x

Su + 2u

Su + u

Ru)

+(Lx + W u)

(Lx + W u) + δ(x, u), (13.187)

where δ(x, u) is such that

lim

kxk

+kuk

→0

δ(x, u)

kxk

+ kuk

= 0.

Now, viewing (13.187) as the Taylor series expansion of (13.184) about x = 0

and u = 0 it follows that f or all x ∈ R

and u ∈ R

0 = x

P A − P − C

QC + L

L)x

+2x

P B − C

S − C

QD + L

W )u

W − D

QD − D

S − S

D − R + B

P B)u. (13.188)

Now, equating coeﬃcients of equal powers in (13.188) yields (13.179)–

(13.181).

Finally, to show that P > 0 in the case w here (A, C) is observable, note

that it follows from Theorem 13.21 and (13.179)–(13.181) that the linearized

system G with storage function V

(x) = x

P x is dissipative with respect to

the quadratic su pply rate r(u, y). Now, the positive deﬁniteness of P follows

from Theorem 13.18.

The following corollaries are immediate from Theorem 13.24 and pro-

vide linearization results for passive and nonexpansive systems, respectively.

Corollary 13.14. Suppose the nonlinear dynamical system G given by

(13.135) and (13.136) is completely reachable and passive. Then there exist

matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P nonnegative deﬁn ite,

such that

P = A

P A + L

L, (13.189)

0 = A

P B − C

+ L

W, (13.190)

0 = D + D

− B

P B − W

W, (13.191)

where A, B, C, and D are given by (13.182). If, in addition, (A, C) is

NonlinearBook10pt November 20, 2007

816 CHAPTER 13

observable, then P > 0.

Corollary 13.15. Suppose the nonlinear dynamical system G given by

(13.135) and (13.136) is completely reachable and nonexpansive. Then there

exist matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P nonnegative

deﬁnite, such that

P = A

P A + C

C + L

L, (13.192)

0 = A

P B + C

D + L

W, (13.193)

0 = γ

− D

D − B

P B − W

W, (13.194)

where A, B, C, and D are given by (13.182) and γ > 0. If, in addition,

(A, C) is observable, then P > 0.

Next, we present a key linearization theorem for geometrically dissi-

pative sys tems.

Theorem 13.25. Let Q ∈ S

, S ∈ R

l×m

, R ∈ S

, and suppose G

given by (13.135) and (13.136) is completely reachable and geometrically

dissipative with respect to the quadratic supply rate r(u, y) = y

Qy +

Su + u

Ru. Then, there exist matrices P ∈ R

n×n

, L ∈ R

p×n

, and

W ∈ R

p×m

, with P nonnegative deﬁn ite, and a scalar ρ > 1, such that

P = A

P A − C

QC + L

L, (13.195)

0 = A

P B − C

(QD + S) + L

W, (13.196)

0 = R + S

D + D

S + D

QD − B

P B − W

W, (13.197)

where A, B, C, and D are given by (13.182). If, in addition, (A, C) is

observable, then P > 0.

Proof. The proof is analogous to the proof of Theorem 13.24.

Linearization results for geometrically passive s ys tems and geometri-

cally nonexpansive systems follow immediately fr om Theorem 13.25.

13.12 Positive Real and Bounded Real Discrete-Time

Dynamical Systems

In this section, we specialize the results of Section 13.10 to the case of linear

discrete-time systems and provide connections to the frequency domain

versions of passivity, geometric p assivity, nonexpansivity, and geometric

nonexpansivity. Speciﬁcally, we consider linear systems

G = G(z) ∼



A B

C D



NonlinearBook10pt November 20, 2007

DISCRETE-TIME THEORY 817

with a state space representation

x(k + 1) = Ax(k) + Bu(k), x(0) = 0, k ∈ Z

, (13.198)

y(k) = Cx(k) + Du(k), (13.199)

where x ∈ R

, u ∈ R

, y ∈ R

, A ∈ R

n×n

, B ∈ R

n×m

, C ∈ R

l×n

, and

D ∈ R

l×m

. To present the main results of this section we ﬁrst give several

key deﬁnitions.

Deﬁnition 13.16. A square transfer function G(z) is positive real if i)

all the entries of G(z) are analytic in |z| > 1 and ii) He G(z) ≥ 0, |z| > 1.

A square transfer function G(z) is strictly positive real if there exists ρ > 1

such that G(z/ρ) is positive real.

Deﬁnition 13. 17. A transfer function G(z) is bounded real if i) all the

entries of G(z) are analytic in |z| > 1 and ii) γ

−G

∗

(z)G(z) ≥ 0, |z| > 1,

where γ > 0. A transfer function G(z) is strictly bounded real if there exists

ρ > 1 such that G(z/ρ) is bounded real.

As in the continuous case, note that ii) in Deﬁnition 13.17 implies that

G(z) is analytic in |z| ≥ 1, and hence, a bounded real transfer function is

asymptotically stable. To see this, note that γ

−G

∗

(z)G(z) ≥ 0, |z| > 1,

implies that

[γ

− G

∗

(z)G(z)]

(i,i)

= γ

−

j=1

(j,i)

(z)|

≥ 0, |z| > 1, (13.200)

and hence, |G

(i,j)

(z)| is bounded by γ

at every point in |z| > 1. Hence,

(i,j)

(z) cannot possess a pole in |z| = 1 since in this case, |G

(i,j)

(z)|

would take on arbitrary large values in |z| > 1 in the vicinity of this

pole. Hence, G(e

θ

) = lim

σ→1,σ>1

G(σe

θ

) exists for all θ ∈ [0, 2π] and

− G

∗

θ

)G(e

θ

) ≥ 0, θ ∈ [0, 2π]. Now, since G

∗

θ

)G(e

θ

) ≤ γ

θ ∈ [0, 2π], is equivalent to sup

θ∈[0,2π]

max

[G(e

θ

)] ≤ γ, it f ollows that G(z)

is bounded real if and only if G(z) is asymptotically stable and |||G(z)|||

∞

≤ γ.

Similarly, it can be shown that strict bounded realness is equivalent to

G(z) asymptotically stable and G

∗

θ

)G(e

θ

) < γ

, θ ∈ [0, 2π], or,

equivalently, |||G(z)|||

∞

< γ.

The following theorem gives a frequency domain test for positive

realness.

Theorem 13.26. Let G(z) be a square, real rational transfer function.

G(z) is positive real if and only if the following conditions hold:

i) No entry of G(z) has a pole in |z| > 1.

NonlinearBook10pt November 20, 2007

818 CHAPTER 13

ii) He G(e

θ

) ≥ 0 for all θ ∈ [0, 2π], with e

θ

not a pole of any entry of

G(z).

iii) If e



is a pole of any entry of G(z) it is at most a simple pole, and the

residue matrix G

△

= lim

z→e



(z − e



)G(z) is nonnegative deﬁnite.

Proof. The proof follows from the m aximum modulus theorem of

complex variable theory by f orming a Nyquist-type closed contour Γ in |z| >

1 and analyzing the function f (z) = x

∗

G(z

−1

)x, x ∈ C

, on Γ. For details

see [197].

Now, we p resent the key results of this section for characterizing

positive realness, strict positive realness, bounded realness, and strict

bounded realness of a linear discrete-time dynamical system in terms of

the system matrices A, B, C, and D.

Theorem 13.27 (Positive Real Lemma). Consider the dynamical

system

G(z)

min

∼



A B

C D



with input u(·) ∈ U and output y(·) ∈ Y. Then the following statements are

equivalent:

i) G(z) is positive real.

ii)

K−1

k=0

(k)y(k) ≥ 0, K − 1 ≥ 0.

iii) There exist matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P

positive deﬁnite, such that

P = A

P A + L

L, (13.201)

0 = A

P B − C

+ L

W, (13.202)

0 = D + D

− B

P B − W

W. (13.203)

If, alternatively, D + D

−B

P B > 0, then G(z) is positive real if and only

if there exists an n × n positive-deﬁnite matrix P such that

P ≥ A

P A + (B

P A −C)

(D + D

− B

P B)

−1

P A −C). (13.204)

Proof. First, we show that i) implies ii). Suppose G(z) is positive

real. Th en it follows from Parseval’s theorem (see Problem 13.24) that, for

all K − 1 ≥ 0, and the truncated input function

(k) =



u(k), k = {0, 1, . . . , K − 1},

0, otherwise,

(13.205)

NonlinearBook10pt November 20, 2007

DISCRETE-TIME THEORY 819

K−1

k=0

(k)u(k) =

∞

k=−∞

(k)u

(k)

2π

−π

∗

θ

)dθ

4π

−π

∗

θ

)[G(e

θ

) + G

∗

θ

)]u

θ

)dθ

≥ 0,

which implies that G(z) is passive.

Next, we show that ii) implies iii). If G(z) is passive, then it follows

from Corollary 13.14 with f(x) = Ax, G(x) = B, h(x) = Cx, J(x) = D,

that th ere exist matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P

positive deﬁnite, such that (13.201)–(13.203) are satisﬁed.

Next, to show that iii) implies i), note that if there exist matrices

P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P positive deﬁnite, such that

(13.201)–(13.203) are satisﬁed, then, for all |z| > 1,

G(z) + G

∗

(z)

= C(zI

− A)

−1

B + D + B

∗

− A)

−T

+ D

= W

W + B

P B + (B

P A + W

L)(zI

− A)

−1

∗

− A)

−T

P B + L

W )

= W

W + W

L(zI

− A)

−1

B + B

∗

− A)

−T

∗

− A)

−T

[(z

∗

− A)

P (zI

− A) + (z

∗

− A)

P A

P (zI

− A)](zI

− A)

−1

= W

W + W

L(zI

− A)

−1

B + B

∗

− A)

−T

∗

− A)

−T

L + (|z| − 1)P ](zI

− A)

−1

≥ [W + L(zI

− A)

−1

∗

[W + L(zI

− A)

−1

≥ 0.

To show analyticity of the entries of G(z) in |z| > 1 note that an entry of

G(z) will have a pole at z = λ only if λ ∈ spec(A). Now, it follows from

(13.201) and the fact th at P > 0 that all eigenvalues of A have magnitude

less than or equal to unity. Hence, G(z) is analytic in |z| > 1, which implies

that G(z) is positive real. Finally, (13.204) follows from (13.163) with th e

linearization given above.

Theorem 13.28 (Strict Positive Real Lemma). Consider the dynam-

ical system

G(z)

min

∼



A B

C D



NonlinearBook10pt November 20, 2007

820 CHAPTER 13

with input u(·) ∈ U and output y(·) ∈ Y. Then the following statements are

equivalent:

i) G(z) is s trictly positive real.

ii)

K−1

k=0

k+1

(k)y(k) ≥ 0, K − 1 ≥ 0.

iii) There exist matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P

positive deﬁnite, and a scalar ρ > 1 such that

P = A

P A + L

L, (13.206)

0 = A

P B − C

+ L

W, (13.207)

0 = D + D

− B

P B − W

W. (13.208)

If, alternatively, D + D

− B

P B > 0, then G(z) is strictly positive real if

and only if there exist n ×n positive-deﬁnite matrices P and R such that

P = A

P A+(B

P A−C)

(D+D

−B

P B)

−1

P A−C)+R. (13.209)

Proof. The equivalence of i) and iii) is a direct consequence of

Theorem 13.27 by noting that G(z) is strictly positive real if and only if

there exists ρ > 1 such that

G(z/

√

ρ)

min

∼



√

ρA

√

ρB

C D



is positive real. The fact that iii) implies ii) follows from Corollary 13.12

with f(x) = Ax, G(x) = B, h(x) = Cx, J(x) = D, V

(x) = x

P x, ℓ(x) =

Lx, an d W(x) = W . To show that ii) imp lies iii), note that if G(z) is

geometrically passive, then it follows from Theorem 13.25 with f(x) = Ax,

G(x) = B, h(x) = Cx, J(x) = D, Q = 0, S = I

, and R = 0, that

there exist matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P positive

deﬁnite, and a scalar ρ > 1 such that (13.206)–(13.208) are satisﬁed.

Finally, with the linearization given above, it follows from (13.173)

that if D + D

− B

P B > 0, then G(z) is strictly positive real if an d only

if there exist a scalar ρ > 1 and a positive-deﬁnite matrix P ∈ R

n×n

such

that

P ≥ A

P A + (B

P A −C)

(D + D

−B

P B)

−1

P A −C). (13.210)

Now, if there exist a s calar ρ > 1 and a positive-deﬁnite matrix P ∈ R

n×n

such that (13.210) is satisﬁed, then there exists an n × n positive-deﬁnite

matrix R such that (13.209) is satisﬁed. Conversely, if D +D

−B

P B > 0

and there exists an n × n positive-deﬁnite matrix R such that (13.209) is

satisﬁed, then, with ρ = σ

min

(R)/σ

max

(P ), (13.209) implies (13.210). Hence,

NonlinearBook10pt November 20, 2007

DISCRETE-TIME THEORY 821

if D + D

− B

P B > 0, then G(z) is strictly positive real if and only if

there exist n × n positive-deﬁnite matrices P and R s uch that (13.209) is

satisﬁed.

Next, we present analogous results for bounded real systems.

Theorem 13.29 (Bounded Real Lemma). Consider the dynamical

system

G(z)

min

∼



A B

C D



with input u(·) ∈ U and output y(·) ∈ Y. Then the following s tatements are

equivalent:

i) G(z) is bounded real.

ii)

K−1

k=0

(k)y(k) ≤ γ

K−1

k=0

(k)u(k), K − 1 ≥ 0, γ > 0.

iii) There exist matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P

positive deﬁnite, such that

P = A

P A + C

C + L

L, (13.211)

0 = A

P B + C

D + L

W, (13.212)

0 = γ

− D

D − B

P B − W

W. (13.213)

If, alternatively, γ

− D

D − B

P B > 0, then G(z) is bounded r eal if

and only if there exists an n × n positive-deﬁnite matrix P such th at

P ≥ A

P A + C

C + (B

P A + D

(γ

− D

D − B

P B)

−1

·(B

P A + D

C). (13.214)

Proof. First, we show that i) implies ii). Suppose G(z) is bound ed

real. Th en it follows from Parseval’s theorem (see Problem 13.24) that, for

all K − 1 ≥ 0, and the truncated input function u

(·) given by (13.205),

K−1

k=0

(k)y(k) =

∞

k=−∞

(k)y(k)

2π

−π

∗

θ

)y(e

θ

)dθ

2π

−π

∗

θ

∗

θ

)G(e

θ

)dθ

≤

2π

−π

∗

θ

)dθ

NonlinearBook10pt November 20, 2007

822 CHAPTER 13

= γ

K−1

k=0

(k)u(k),

which implies that G(z) is n on exp ansive.

Next, we show that ii) implies iii). If G(z) is nonexpansive, then

it follows from Corollary 13.15 with f (x) = Ax, G(x) = B, h(x) = Cx,

J(x) = D, that there exist matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

with P positive deﬁnite, such that (13.211)–(13.213) are satisﬁed.

Now, to show that iii) implies i), note that if there exist matrices

P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P positive deﬁnite, such that

(13.211)–(13.213) are satisﬁed, then, for all |z| ≥ 1,

− G

∗

(z)G(z)

= γ

− [C(zI

− A)

−1

B + D]

∗

[C(zI

−A)

−1

B + D]

= [γ

− D

D] − [C(zI

− A)

−1

∗

[C(zI

− A)

−1

−[C(zI

− A)

−1

∗

D − D

[C(zI

− A)

−1

= W

W + B

P B + [B

P A + W

L](zI

−A)

−1

B + B

(zI

−A)

−∗

P B + L

W ] − B

(zI

− A)

−∗

C(zI

− A)

−1

= W

W + B

P B + W

L(zI

− A)

−1

B + B

(zI

− A)

−∗

−B

(zI

−A)

−∗

C(zI

− A)

−1

(zI

−A)

−∗

[(zI

− A)

∗

P A + A

P (zI

−A)](zI

− A)

−1

= W

W + W

L(zI

− A)

−1

B + B

(zI

− A)

−∗

(zI

−A)

−∗

L(zI

− A)

−1

+(|z| − 1)B

(zI

− A)

−∗

P (zI

− A)

−1

≥ [W + L(zI

− A)

−1

∗

[W + L(zI

− A)

−1

≥ 0.

To show analyticity of the entries of G(z) in |z| ≥ 1 note that it follows

from (13.211) and the fact that P > 0 and (A, C) is observable that all

the eigenvalues of A lie inside the unit disk. Hence, G(z) is analytic in

|z| ≥ 1, which implies that G(z) is bounded real. Finally, (13.214) follows

from (13.168) with the linearization given above.

Theorem 13.30 (S trict Bounded Real Lemma). Consider the dy-

namical system

G(z)

min

∼



A B

C D



with input u(·) ∈ U and output y(·) ∈ Y. Then the following statements are

equivalent:

NonlinearBook10pt November 20, 2007

DISCRETE-TIME THEORY 823

i) G(z) is s trictly bounded real.

ii)

K−1

k=0

k+1

(k)y(k) ≤ γ

K−1

k=0

k+1

(k)u(k), K − 1 ≥ 0, γ >

iii) There exist matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P

positive deﬁnite, and a scalar ρ > 1 such th at

P = A

P A + C

C + L

L, (13.215)

0 = A

P B + C

D + L

W, (13.216)

0 = γ

− D

D − B

P B − W

W. (13.217)

If, alternatively, γ

− D

D − B

P B > 0, then G(z) is s trictly positive

real if and only if there exist n × n positive-deﬁnite matrices P and R such

that

P = A

P A + C

C + (B

P A + D

(γ

− D

D − B

P B)

−1

·(B

P A + D

C) + R. (13.218)

Proof. The equivalence of i) and iii) is a direct consequence of

Theorem 13.29 by noting that G(z) is strictly boun ded real if and only

if there exists ρ > 1 such that

G(z/

√

ρ)

min

∼



√

ρA

√

ρB

C D



is bounded real. The fact that iii) implies ii) follows from Corollary 13.13

with f(x) = Ax, G(x) = B, h(x) = Cx, J(x) = D, V

(x) = x

P x, ℓ(x) =

Lx, an d W(x) = W . To sh ow that ii) implies iii), note that if G(z) is

geometrically nonexpansive, then it f ollows from Theorem 13.25 with f (x) =

Ax, G(x) = B, h(x) = Cx, J(x) = D, Q = −I

, S = 0, and R = γ

, that

there exist matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P positive

deﬁnite, such that (13.215)–(13.217) are satisﬁed.

Finally, with the linearization given above, it follows from (13.178)

that if γ

−D

D −B

P B > 0, then G(z) is strictly bounded real if and

only if there exist a scalar ε > 0 and a positive-deﬁnite matrix P ∈ R

n×n

such that

P ≥ ρA

P A+(ρB

P A−C)

(D+D

−ρB

P B)

−1

(ρB

P A−C). (13.219)

Now, if there exist a s calar ρ > 1 and a positive-deﬁnite matrix P ∈ R

n×n

such that (13.219) is satisﬁed, then there exists an n × n positive-deﬁnite

matrix R such that (13.218) is satisﬁed. Conversely, if γ

− D

D −

P B > 0 and there exists an n × n positive-deﬁnite matrix R such that

(13.218) is satisﬁed, then, with ε = σ

min

(R)/σ

max

(P ), (13.218) imp lies

NonlinearBook10pt November 20, 2007

824 CHAPTER 13

(13.219). Hence, if γ

−D

D −B

P B > 0, then G(z) is strictly bounded

real if and only if there exist n × n positive-deﬁnite matrices P and R such

that (13.218) is satisﬁed.

The following result characterizes n ecessary and suﬃcient conditions

for a transfer function to be strictly positive r eal in terms of a frequency

domain test.

Theorem 13.31. Let

G(z) ∼



A B

C D



be an m ×m transfer function and suppose that G(z) is not singular. Then,

G(z) is strictly positive real if and only if the following conditions hold:

i) No entry of G(z) has a pole in |z| ≥ 1.

ii) He G(e

θ

) > 0 for all θ ∈ [0, 2π].

Proof. Suppose i) and ii) hold. Let ρ > 1 and note that

) = C(

I − A)

−1

B + D

= C(zI − A)

−1

(zI − A)(

I − A)

−1

B + D

= C(zI − A)

−1

(

I − A +

ρ−1

zI))(

I − A)

−1

B + D

= G(z) +

ρ−1

(z), (13.220)

where G

(z)

△

= zC(zI − A)

−1

(

I − A)

−1

B. Since A is Schur, zI − A is

nonsingular for all z = e

θ

and there exists ˆρ such th at

I −A is nonsin gular

for all ρ ∈ [1, ˆρ] and for all z = e

θ

. Hence, f(θ, ρ)

△

= max

|λ

[He G

θ

)]|

is ﬁnite f or all ρ ∈ [1, ˆρ] and for all θ ∈ [0, 2π]. Since f (θ, ·) is continuous it

follows that there exists k

> 0 such that f(θ, ρ) < k

for all ρ ∈ [1, ˆρ] and

for all θ ∈ [0, 2π], so that −k

≤ He G

(z) ≤ k

for all ρ ∈ [1, ˆρ] and

for all z = e

θ

. Now, s ince He G(e

θ

) > 0 for all θ ∈ [0, 2π], it follows that

there exists k

> 0 such that He G(θ) ≥ k

> 0, θ ∈ [0, 2π]. Choosing

1 < ρ < min{ˆρ,

−k

}, it follows that

He G(

) = He G(z) +

ρ − 1

He G

(z) ≥ k

−

ρ − 1

> 0, (13.221)

for all z = e

θ

. Hence, G(

) is positive real, and hence, by d eﬁ nition, G(z)

is strictly positive real.

Conversely, suppose G(z) is strictly positive real an d let ρ > 1 be

such that G(

) is positive real. Note that G(z) is asymptotically stable