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

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 355

where 0 < b < a, x

g(x

) > 0, x

∈ R, x

6= 0, and g(0) = 0. To examine

the passivity of (5.109)–(5.111) we consider the storage function

, x

) =

[βa

+ 2βax

+ x

] + α

g(σ)dσ, (5.112)

where α > 0 and β ∈ (0, 1). Note that V

, x

) is positive deﬁnite and

radially unbounded. Now, computing

, x

) yields

, x

) = α[βa

+ βax

+ g(x

)]x

+ α(βax

+ x

)[−g(x

) −ax

+ u]

= −αβax

g(x

) + α(β − 1)ax

+ α(βax

+ x

)u. (5.113)

Setting α = 1 and β = b/a < 1 if follows that

, x

) = uy − bx

g(x

) − (a −b)x

, (5.114)

which shows that (5.109)–(5.111) is strictly passive. △

The following results present the nonlinear versions of the Kalman-

Yakubovich-Popov positive real lemma and the bounded real lemma.

Corollary 5.2. Let G be zero-state observable and completely reach-

able. G is passive if and only if there exist functions V

: R

→ R,

ℓ : R

→ R

, and W : R

→ R

p×m

such that V

(·) is continuously

diﬀerentiable and positive deﬁnite, V

(0) = 0, and, for all x ∈ R

0 = V

′

(x)f(x) + ℓ

(x)ℓ(x), (5.115)

0 =

′

(x)G(x) − h

(x) + ℓ

(x)W(x), (5.116)

0 = J(x) + J

(x) − W

(x)W(x). (5.117)

If, alternatively,

J(x) + J

(x) > 0, x ∈ R

, (5.118)

then G is passive if and only if there exists a continuously diﬀerentiable

function V

: R

→ R such th at V

(·) is positive deﬁnite, V

(0) = 0, and, for

all x ∈ R

0 ≥ V

′

(x)f(x) + [

′

(x)G(x) − h

(x)]

·[J(x) + J

(x)]

−1

[

′

(x)G(x) − h

(x)]

. (5.119)

Proof. The result is a direct consequence of Theorem 5.6 with l = m,

Q = 0, S = I

, and R = 0. Speciﬁcally, with κ(y) = −y it follows that

r(κ(y), y) = −2y

y < 0, y 6= 0, so that all the assump tions of Theorem 5.6

are satisﬁed.

Example 5.5. Consider the nonlinear controlled Lienard system

¨x(t) +

f(x(t)) ˙x(t) + g(x(t)) = u(t), x(0) = x

, ˙x(0) = ˙x

, t ≥ 0,

(5.120)

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356 CHAPTER 5

with output y(t) =

˙x(t) or, equivalently,

˙x

(t) = x

(t) −F (x

(t)), x

(0) = x

, t ≥ 0, (5.121)

˙x

(t) = −g(x

(t)) + u(t), x

(0) = x

, (5.122)

y(t) =

(t), (5.123)

where x

= x, x

= ˙x + F (x), F (x

) =

f(s)ds,

f(0) = g(0) = 0, and

and g are continuously diﬀerentiable. To examine the passivity of (5.121)–

(5.123) we consider the energy function

, x

) =

g(s)ds, (5.124)

where x

g(x

) ≥ 0, x

∈ R, so that V

, x

) ≥ 0, (x

, x

) ∈ R × R,

is a candidate storage function. Now, we use Corollary 5.2 to determine

conditions on g(x

) and F (x

) that guarantee (5.121)–(5.123) is passive.

Note that (5.121)–(5.123) can be written in the state space form (5.75) and

(5.76) with x = [x

, x

]

, f(x) = [x

− F (x

), −g(x

)]

, G(x) = [0, 1]

h(x) =

, and J(x) = 0. Now, u s ing (5.115) it follows that

0 =



g(x

) x





− F (x

)

−g(x

)



+ ℓ

(x)ℓ(x), (5.125)

which requires that g(x

)F (x

) ≥ 0. If

f(x

) ≥ 0, x

∈ R, then x

F (x

) ≥ 0,

∈ R. Furthermore, since x

and g(x

) have the same sign it follows that

g(x

)F (x

) ≥ 0, x

∈ R. Next, note that (5.116) is automatically satisﬁed

since

0 =



g(x

) x







−

. (5.126)

Hence, if

f(x

) ≥ 0, x

∈ R, then the controlled Lienard s ys tem (5.121)–

(5.123) is passive. △

Corollary 5.3. Let G be zero-state observable and completely reach-

able. G is nonexpansive if and only if there exist functions V

: R

→ R,

ℓ : R

→ R

, and W : R

→ R

p×m

such that V

(·) is continuously

diﬀerentiable and positive deﬁnite, V

(0) = 0, and, for all x ∈ R

0 = V

′

(x)f(x) + h

(x)h(x) + ℓ

(x)ℓ(x), (5.127)

0 =

′

(x)G(x) + h

(x)J(x) + ℓ

(x)W(x), (5.128)

0 = γ

− J

(x)J(x) −W

(x)W(x), (5.129)

where γ > 0. If, alternatively,

− J

(x)J(x) > 0, x ∈ R

, (5.130)

then G is nonexpansive if and only if there exists a continuously diﬀerentiable

function V

: R

→ R such that V

(·) is positive deﬁnite, V

(0) = 0, and, for

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 357

all x ∈ R

0 ≥ V

′

(x)f(x) + h

(x)h(x) + [

′

(x)G(x) + h

(x)J(x)]

·[γ

− J

(x)J(x)]

−1

[

′

(x)G(x) + h

(x)J(x)]

. (5.131)

Proof. The result is a direct consequence of Theorem 5.6 with Q =

−I

, S = 0, and R = γ

. Speciﬁcally, with κ(y) = −

2γ

y it follows that

r(κ(y), y) = −

y < 0, y 6= 0, so that all the assumptions of Theorem 5.6

are satisﬁed.

Example 5.6. Consider the nonlinear controlled dynamical system

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (5.132)

˙x

(t) = −a sin x

(t) −bx

(t) + u(t), x

(0) = x

, (5.133)

y(t) = x

(t), (5.134)

where a, b > 0. Note that (5.132)–(5.134) can be written in the s tate sp ace

form (5.75) and (5.76) with x = [x

, x

]

, f (x) = [x

, −a sin x

− bx

]

G(x) = [0, 1]

, h(x) = x

, and J(x) = 0. To examine the nonexpansivity of

(5.132)–(5.134) we consider the storage function V

(x) = a(1 −cos x

) +

satisfying V

(x) ≥ 0, x ∈ R

. Now, using Corollary 5.3 it follows from

(5.131) that

0 ≥



a sin x





−a sin x

−bx



+ h

(x)

4γ



a sin x









0 1





a sin x



, (5.135)

or, equivalently,

0 ≥ (1 −b)h

(x) +

4γ

(x). (5.136)

Hence, (5.136) is satisﬁed if γ ≥

√

b−1

. △

Finally, the following results present th e nonlinear versions of the

Kalman-Yakubovich-Popov strict positive real lemma and strict bounded real

lemma for exponentially passive and exponentially nonexpansive systems,

respectively.

Corollary 5.4. Let G be zero-state observable and completely reach-

able. G is exponentially passive if and only if there exist functions V

: R

→

R, ℓ : R

→ R

, and W : R

→ R

p×m

and a scalar ε > 0 such that V

(·)

is continuously diﬀerentiable and positive deﬁnite, V

(0) = 0, and, for all

x ∈ R

0 = V

′

(x)f(x) + εV

(x) + ℓ

(x)ℓ(x), (5.137)

0 =

′

(x)G(x) − h

(x) + ℓ

(x)W(x), (5.138)

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358 CHAPTER 5

0 = J(x) + J

(x) − W

(x)W(x). (5.139)

If, alternatively, (5.118) holds, then G is exponentially passive if and only if

there exist a continuously diﬀerentiable function V

: R

→ R and a scalar

ε > 0 such that V

(·) is positive deﬁnite, V

(0) = 0, and, for all x ∈ R

0 ≥ V

′

(x)f(x) + εV

(x) + [

′

(x)G(x) − h

(x)]

·[J(x) + J

(x)]

−1

[

′

(x)G(x) − h

(x)]

. (5.140)

Proof. The result is a direct consequence of Theorem 5.7 with l = m,

Q = 0, S = I

, and R = 0. Speciﬁcally, with κ(y) = −y it follows that

r(κ(y), y) = −2y

y < 0, y 6= 0, so that all the assumptions of Theorem 5.7

are satisﬁed.

Corollary 5.5. Let G be zero-state observable and completely reach-

able. G is exponentially n on exp ansive if and only if there exist functions

: R

→ R, ℓ : R

→ R

, and W : R

→ R

p×m

and a scalar ε > 0 such

that V

(·) is continuously diﬀerentiable and positive d eﬁ nite, V

(0) = 0, and,

for all x ∈ R

0 = V

′

(x)f(x) + εV

(x) + h

(x)h(x) + ℓ

(x)ℓ(x), (5.141)

0 =

′

(x)G(x) + h

(x)J(x) + ℓ

(x)W(x), (5.142)

0 = γ

− J

(x)J(x) −W

(x)W(x), (5.143)

where γ > 0. If, alternatively, (5.130) holds, then G is exponentially

nonexpansive if and only if th ere exist a continuously diﬀerentiable function

: R

→ R and a scalar ε > 0 such that V

(·) is positive deﬁnite, V

(0) = 0,

and, for all x ∈ R

0 ≥ V

′

(x)f(x) + εV

(x) + h

(x)h(x) + [

′

(x)G(x) + h

(x)J(x)]

·[γ

− J

(x)J(x)]

−1

[

′

(x)G(x) + h

(x)J(x)]

. (5.144)

Proof. The result is a direct consequence of Theorem 5.7 with Q =

−I

, S = 0, and R = γ

. Speciﬁcally, with κ(y) = −

2γ

y it follows that

r(κ(y), y) = −

y < 0, y 6= 0, so that all the assumptions of Theorem 5.7

are satisﬁed.

5.5 Linearization of Dissipative Dynamical Systems

In this section, we present several key results on linearization of dissipative,

exponentially dissipative, passive, exponentially passive, nonexpansive, and

exponentially nonexpansive dynamical systems. For these results, we assume

that there exists a fu nction κ : R

→ R

such that κ(0) = 0 and r(κ(y), y) <

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 359

0, y 6= 0, and the available storage function (respectively, exponentially

storage function) V

(x), x ∈ R

, is a smooth function.

Theorem 5.9. Let Q ∈ S

, S ∈ R

l×m

, R ∈ S

, and s uppose G given

by (5.75) and (5.76) is completely reachable and dissipative with respect to

the quadratic supply rate r(u, y) = y

Qy + 2y

Su + u

Ru. Then, there

exist matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P nonnegative

deﬁnite, such that

0 = A

P + P A − C

QC + L

L, (5.145)

0 = P B − C

(QD + S) + L

W, (5.146)

0 = R + S

D + D

S + D

QD − W

W, (5.147)

where

A =

∂f

∂x



x=0

, B = G(0), C =

∂h

∂x



x=0

, D = J(0). (5.148)

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

Proof. First note that since G is completely reachable and dissipative

with respect to a quadratic supply rate there exist f unctions V

: R

→ R,

ℓ : R

→ R

, and W : R

→ R

p×m

, with V

(·) nonnegative deﬁnite, such

that (5.77)–(5.79) are satisﬁed. Now, expanding V

(·) via a Taylor series

expansion about x = 0 and using the fact that V

(·) is nonnegative deﬁnite

and V

(0) = 0 it follows that there exists a nonnegative-deﬁnite matrix

P ∈ R

n×n

such that

(x) = x

P x + V

(x),

where V

: R

→ R contains the higher-order terms of V

(x).

Next, let f(x) = Ax+f

(x), ℓ(x) = Lx+ℓ

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

(x),

where f

(x), ℓ

(x), and h

(x) contain the nonlinear terms of f(x), ℓ(x),

and h(x), respectively, and let G(x) = B + G

(x), J(x) = D + J

(x), and

W(x) = W + W

(x), where W

△

= W(0) and G

(x), J

(x), and W

(x) contain

the nonconstant terms of G(x), J(x), and W(x), respectively. Using the

above expressions (5.77) and (5.78) can be written as

0 = x

P + P A − C

QC + L

L)x + γ(x), (5.149)

0 = x

(P B − C

(QD + S) + L

W ) + Γ(x), (5.150)

where

γ(x)

△

= V

′

(x)f(x) + ℓ

(x)ℓ

(x) + 2x

(P f

(x) − C

(x) + L

ℓ

(x))

−h

(x)Qh

(x),

Γ(x)

△

′

(x)G(x) − h

(x)(QJ(x) + S) −x

(x) + x

P G

(x)

(x) + ℓ

(x)W.

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360 CHAPTER 5

Now, viewing (5.149) and (5.150) as the Taylor series expansion of (5.77)

and (5.78), respectively, about x = 0, and noting that

lim

kxk→0

|γ(x)|

kxk

= 0, lim

kxk→0

kΓ(x)k

kxk

= 0,

where k · k denotes the Euclidean vector norm, it follows th at P satisﬁes

(5.145) and (5.146). Now, (5.147) follows from (5.79) by setting x = 0.

Next, it follows from Theorem 5.6 and (5.145)–(5.147) th at the linear

system

˙x(t) = Ax(t) + Bu(t), x(0) = x

, t ≥ 0,

y(t) = Cx(t) + Du(t),

with a storage function V

(x) = x

P x is dissipative with respect to the

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

from Theorem 5.3.

The following corollaries are immediate from Theorem 5.9 and provide

linearization results for passive and nonexpansive dynamical systems,

respectively.

Corollary 5. 6. Suppose the nonlinear d ynamical system G given by

(5.75) and (5.76) is completely reachable and passive. Th en th ere exist

matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P nonnegative deﬁnite,

such that

0 = A

P + P A + L

L, (5.151)

0 = P B − C

+ L

W, (5.152)

0 = D + D

− W

W, (5.153)

where A, B, C, and D are given by (5.148). If, in addition, (A, C) is

observable, then P > 0.

Corollary 5. 7. Suppose the nonlinear d ynamical system G given by

(5.75) and (5.76) 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

0 = A

P + P A + C

C + L

L, (5.154)

0 = P B + C

D + L

W, (5.155)

0 = γ

− D

D − W

W, (5.156)

where A, B, C, and D are given by (5.148) and γ > 0. If, in addition, (A, C)

is observable, then P > 0.

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 361

Next, we present a linearization theorem for exponentially dissipative

systems.

Theorem 5.10. Let Q ∈ S

, S ∈ R

l×m

, R ∈ S

, and suppose G given

by (5.75) and (5.76) is completely reachable and exponentially dissipative

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

Qy + 2y

Su + u

Ru.

Then, there exist matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P

nonnegative deﬁnite, and a scalar ε > 0 su ch that

0 = A

P + P A + εP − C

QC + L

L, (5.157)

0 = P B − C

(QD + S) + L

W, (5.158)

0 = R + S

D + D

S + D

QD − W

W, (5.159)

where A, B, C, and D are given by (5.148). If, in addition, (A, C) is

observable, then P > 0.

Proof. The proof is analogous to the proof of Theorem 5.9.

Linearization results for exponentially passive and exponentially non-

expansive dynamical systems follow immediately from Theorem 5.10.

5.6 Positive Real and Bounded Real Dynamical Systems

In this section, we specialize the results of Section 5.4 to the case of linear

systems and provide connections to the fr equ en cy domain versions of passiv-

ity, exponential passivity, nonexpansivity, and exponential nonexpansivity.

Speciﬁcally, we consider linear systems

G = G(s) ∼



A B

C D



with a state space representation

˙x(t) = Ax(t) + Bu(t), x(0) = 0, t ≥ 0, (5.160)

y(t) = Cx(t) + Du(t), (5.161)

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 5.15. A function p : C → C of the form

p(s) = α

+ α

n−1

+ ··· + α

s + α

, (5.162)

where α

, . . . , α

are real numbers, is called a polynomial. If the leading

co eﬃcient α

is nonzero, then the degree of p(s), denoted by deg p(s), is n,

whereas if p(s) = 0, then deg p(s) = −∞. If α

= 1, then p(s) is monic.

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362 CHAPTER 5

Deﬁnition 5.16. A function f : C → C is analytic at a point z

∈ C if,

for every ε > 0, there exists δ > 0 such that f

′

(z) exists for all z ∈ B

). f

is said to be analytic in C if f is analytic at each point in C. f (s) is a rational

function if there exist polynomials n(s) and d(s) such that f (s) = n(s)/d(s).

f(s) is called strictly proper (respectively, proper) if deg n(s) < deg d(s)

(respectively, deg n(s) ≤ deg d(s)). The relative degree of f (s) den oted by

r, is r

△

= deg d(s) −deg n(s).

Deﬁnition 5.17. An l ×m m atrix transfer function G(s) is a rational

transfer function if every entry of G(s) is a rational function. G(s) is

strictly proper (respectively, proper) if every entry of G(s) is strictly p roper

(respectively, proper).

In this book all transfer functions are assumed to be rational proper

transfer functions.

Deﬁnition 5.18. A square transfer fu nction G(s) is positive real if i) all

the entries of G(s) are analytic in Re[s] > 0 and ii) He G(s) ≥ 0, Re[s] > 0.

A square tran s fer function G(s) is strictly positive real if there exists ε > 0

such that G(s − ε) is positive real. Finally, a square trans fer function G(s)

is strongly positive real if it is strictly positive real and D + D

> 0, where

△

= G(∞).

Deﬁnition 5.19. A transfer function G(s) is bounded real if i) all the

entries of G(s) are analytic in Re[s] > 0 and ii) γ

− G

∗

(s)G(s) ≥ 0,

Re[s] > 0, where γ > 0. A transfer function G(s) is strictly bounded real if

there exists ε > 0 such that G(s − ε) is bounded real. Finally, a transfer

function G(s) is strongly bounded real if it is strictly bounded real and γ

−

D > 0, where D

△

= G(∞).

It is interesting to note that ii) in Deﬁnition 5.19 implies that G(s)

is analytic in Re[s] ≥ 0, and hence, a bounded real transfer function is

asymptotically stable. To see this, note that γ

−G

∗

(s)G(s) ≥ 0, Re[s] >

0, implies that

[γ

− G

∗

(s)G(s)]

(i,i)

= γ

−

j=1

(j,i)

(s)|

≥ 0, Re[s] > 0, (5.163)

and hence, |G

(i,j)

(s)| is bound ed by γ

at every point in Re[s] > 0. Hence,

(i,j)

(s) cannot possess a pole in Re[s] = 0 since in this case |G

(i,j)

(s)|

would take on arbitrary large values in Re[s] > 0 in the vicinity of this

pole. Hence, G(ω) = lim

σ→0,σ>0

G(σ + ω) exists for all ω ∈ R and γ

−

∗

(ω)G(ω) ≥ 0, ω ∈ R. Now, since G

∗

(ω)G(ω) ≤ γ

, ω ∈ R, is

equivalent to sup

ω ∈R

max

[G(ω)] ≤ γ, it follows that G(s) is bounded real if

and only if G(s) is asymp totically stable and |||G(s)|||

∞

≤ γ. Similarly, it can

NonlinearBook10pt November 20, 2007

DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 363

be shown that strict bounded realness is equ ivalent to G(s) asymptotically

stable and G

∗

(ω)G(ω) < γ

, ω ∈ R, or, equivalently, |||G(s)|||

∞

< γ.

Alternatively, for a positive real trans fer function it follows from ii) of

Deﬁnition 5.19, using a limiting argument, that He G(ω) ≥ 0 for all ω ∈ R

such th at ω is not a pole of any entry of G(s). However, unlike bounded

real transfer functions, there are positive real transfer functions possessing

poles on the ω axis. A simple example is G(s) = 1/s which is analytic in

Re[s] > 0 and He G(s) = 2Re[s]/|s|

> 0, Re[s] > 0. Hence, He G(ω) ≥ 0,

ω ∈ R, does not provide a frequency domain test for positive realness. In

the case where G(s) is analytic in Re[s] > 0 and He G(ω) ≥ 0 holds for all

ω ∈ R for which ω is not a pole of any entry of G(s), one might surmise that

G(s) is positive real. Once again this is not true. A simple counterexample

is G(s) = −1/s which is analytic in Re[s] > 0 and satisﬁes He G(ω) = 0,

ω ∈ R. However, He G(s) ≥ 0, Re[s] > 0, is not s atisﬁed. The following

theorem gives a frequency domain test for positive realness.

Theorem 5.11. Let G(s) be a square, real rational transfer function.

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

i) No entry of G(s) has a pole in Re[s] > 0.

ii) He G(ω) ≥ 0 for all ω ∈ R, with ω not a pole of any entry of G(s).

iii) If ˆω is a pole of any entry of G(s) it is at most a simple pole, and

the residue matrix G

△

= lim

s→ˆω

(s − ˆω)G(s) is nonnegative-deﬁnite

Hermitian. Alternatively, if the limit G

∞

△

= lim

ω →∞

G(ω)/ω exists,

then G

∞

is nonnegative-deﬁnite Hermitian.

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

complex variable theory by forming a Nyqu ist-type closed contour Γ in

Re[s] > 0 and analyzing the fu nction f(s) = x

∗

G(s)x, x ∈ C

, on Γ.

For details see [11].

Next, we present the key results of this section for characterizing

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

bounded realness of a linear d ynamical system in terms of the system

matrices A, B, C, and D. First, however, we present a key theorem due to

Parseval.

Theorem 5. 12 (Parseval’s Theorem). Let u : [0, ∞) → R

and y :

[0, ∞) → R

be in L

, p ∈ [0, ∞), and let u(s) and y(s) denote their Laplace

NonlinearBook10pt November 20, 2007

364 CHAPTER 5

transforms, respectively. Then

∞

(t)y(t)dt =

2π

∞

−∞

∗

(ω)y(ω)dω. (5.164)

Proof. Since y(·) ∈ L

, p ∈ [0, ∞), it follows that the inverse Fourier

transform of y(t) is given by

y(t) =

2π

∞

−∞

y(ω)e

ωt

dω. (5.165)

Now, for all s ∈ C, Re[s] ≥ 0,

∞

(τ)y(τ )]e

−sτ

dτ =

∞

−sτ

(τ)



2π

∞

−∞

y(ω)e

ωτ

dω



dτ

2π

∞

−∞



∞

(τ)e

−(s−ω)τ

dτ



y(ω)dω

2π

∞

−∞

(s − ω)y(ω)dω. (5.166)

Setting s = 0 yields (5.164).

Theorem 5.13 (Positive Real Lemma). Consider the linear dynamical

system

G(s)

min

∼



A B

C D



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

equivalent:

i) G(s) is positive real.

ii)

(t)y(t)dt ≥ 0, T ≥ 0.

iii) There exist matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P

positive deﬁnite, such that

0 = A

P + P A + L

L, (5.167)

0 = P B − C

+ L

W, (5.168)

0 = D + D

− W

W. (5.169)

If, alternatively, D + D

> 0, then G(s) is positive real if and only if there

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

0 ≥ A

P + P A + (B

P − C)

(D + D

)

−1

P − C). (5.170)