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

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NonlinearBook10pt November 20, 2007

INPUT-OUTPUT STABILITY AND DISSIPATIVITY 485

7.7 Induced Convolution Operator Norms of Linear

Dynamical Systems

In the remainder of this chapter, we consider the dyn amical system

˙x(t) = Ax(t) + Bu(t), x(0) = 0, t ≥ 0, (7.52)

y(t) = Cx(t), (7.53)

where x(t) ∈ R

, u(t) ∈ R

, y(t) ∈ R

, t ∈ [0, ∞), A ∈ R

n×n

asymptotically stable, B ∈ R

n×m

, and C ∈ R

l×n

. Here, u(·) is an input

signal belonging to the class L

of input signals and y(·) is an output signal

belonging to the class L

of output s ignals, where the notation L

and

denote the set of functions in L

and L

. In applications, (7.52) and

(7.53) may denote a control system in closed-loop conﬁguration where the

objective is to determine the “size” of the output y(·) for a disturbance u(·).

In the remainder of this chapter, we develop explicit formulas for convolution

operator norms and their bounds induced by various norms on several classes

of input-output signal pairs. These results generalize established induced

convolution operator norms for linear dynamical systems.

If the input-output signals are constrained to be ﬁnite-energy signals so

that u, y ∈ L

then the equi-induced (that is, th e domain and range spaces of

the convolution operator are assigned the same temporal and spatial norm s )

signal norm is the H

∞

system norm [121, 478] given by

|||G|||

(2,2),(2,2)

△

= sup

u(·)∈L

|||y|||

2,2

|||u|||

2,2

= sup

ω ∈R

max

[H(ω)], (7.54)

where the notation ||| · |||

p,q

denotes a signal norm with p temporal norm

and q spatial norm, σ

max

(·) denotes maximum singular value, G denotes the

convolution operator of (7.52) and (7.53), and H(s) = C(sI −A)

−1

B is the

corresponding transfer function. Hence, the H

∞

system norm captures the

supremum system energy gain.

Alternatively, if the input-output signals are constrained to be boun-

ded amplitude signals so th at u, y ∈ L

∞

, then the equi-induced signal norm

|||G|||

(∞,∞),(∞,∞)

△

= sup

u(·)∈L

∞

|||y|||

∞,∞

|||u|||

∞,∞

, (7.55)

is the L

system norm

[101,444]. Thus, the L

system norm captures the

worst-case ampliﬁcation from inpu t distur bance signals to output signals,

where the signal size is taken to be the supremum over time of the signal’s

peak value pointwise in time [101, 444].

In the single-input/single-output case, it is well known that the induced norm (7.55)

corresponds to the L

norm of the impulse response matrix fun ction [101,444].

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486 CHAPTER 7

Mixed input-output signals have also been considered. For example,

if u ∈ L

and y ∈ L

∞

then the resulting induced operator norm is ( [461])

|||G|||

(∞,2),(2,2)

△

= sup

u(·)∈L

|||y|||

∞,2

|||u|||

2,2

= λ

max

(CQC

), (7.56)

where λ

max

(·) denotes the m aximum eigenvalue and Q is the unique n × n

nonnegative-deﬁnite solution to the Lyapunov equation

0 = AQ + QA

+ BB

. (7.57)

Hence, |||G|||

(∞,2),(2,2)

provides a worst-case measure of amplitude errors du e

to ﬁnite energy input signals. Alternatively, if the input and output signal

norms are chosen as ||| · |||

2,2

and ||| · |||

∞,∞

, respectively, then the resulting

induced operator norm is ( [461])

|||G|||

(∞,∞),(2,2)

△

= sup

u(·)∈L

|||y|||

∞,∞

|||u|||

2,2

= d

max

(CQC

), (7.58)

where d

max

(·) d en otes the maximum diagonal entry. Hence, |||G|||

(∞,∞),(2,2)

provides a worst-case peak excursion response due to ﬁnite energy distur-

bances.

It is clear from the above discussion that operator norms induced by

classes of input-output signal pairs can be used to capture distu rbance

rejection performance objectives for controlled dynamical systems. In

particular, H

∞

control theory [121, 478] has been developed to addr ess

the problem of disturbance rejection for s ystems with bound ed energy L

signal norms on the disturbance and performance variables. Since the

induced H

∞

transfer function norm (7.54) corresponds to the worst-case

disturbance attenuation, for sys tems with L

disturbances which possess

signiﬁcant power within arbitrarily small bandwidths, H

∞

theory is clearly

appropriate. Alternatively, to address pointwise in time the worst-case peak

amplitude response due to bounded amplitude persistent L

∞

disturbances,

theory is appropriate [101, 444]. The problem of ﬁnding a stabilizing

controller such th at the closed-loop system gain from |||·|||

2,2

to |||·|||

∞,q

, where

q = 2 or ∞, is below a speciﬁed level is solved in [367, 463]. In addition to

the disturbance rejection problem, another application of in duced operator

norms is the problem of actuator amplitude and rate saturation [90, 105].

In particular, since the convolution operator norm |||G|||

(∞,∞),(2,2)

given

by (7.58) captures the worst-case peak amplitude response due to ﬁnite

energy disturbances, deﬁning the output (perf ormance) variables y to

correspond to th e actuator amplitude and actuator rate signals, it follows

that |||G|||

(∞,∞),(2,2)

bounds actuator amplitude and actuator rate excursion.

Furth ermore, since uncertain signals can also be used to model uncertainty

in a sys tem, the treatment of certain classes of uncertain disturbances also

enable the development of controllers that are robust with respect to input-

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INPUT-OUTPUT STABILITY AND DISSIPATIVITY 487

output uncertainty blocks [101,104].

In the papers [461, 462], Wilson developed explicit formulas for

convolution operator norms induced by several classes of input-output signal

pairs. In the remainder of this chapter we extend the results of [461,462] to

a larger class of inp ut-output signal pairs and provide explicit formulas for

induced convolution operator n orms and operator norm bounds for linear

dynamical systems. These results generalize several well-known induced

convolution operator norm results in the literature including results on

∞

equi-induced norms (L

operator norms) and L

equi-induced norms

(resource norms). In cases where the induced convolution operator norm

expressions are not ﬁnitely computable, we p rovide ﬁnitely computable norm

bounds.

To develop induced convolution operator norms for linear systems, we

ﬁrst introduce some notation, deﬁnitions, and several key lemmas. Let k·k

′

and k·k

′′

denote vector norm s on R

and R

, respectively, where m, n ≥ 1.

Then k ·k : R

m×n

→ R deﬁned by

kAk

△

= max

kxk

′

kAxk

′′

is the matrix norm induced by k·k

′

and k·k

′′

. If k·k

′

= k·k

and k·k

′′

= k·k

where p, q ∈ [1, ∞], then the matrix norm on R

m×n

induced by k · k

and

k·k

is denoted by k ·k

q,p

. Let k ·k den ote a vector norm on R

. Then the

dual norm k ·k

of k · k is d eﬁ ned by

kyk

△

= max

kxk=1

x|,

where y ∈ R

[417]. Note that k · k

= k · k [417]. Furthermore, if

p, q ∈ [1, ∞] satisfy 1/p + 1/q = 1, then k · k

= k ·k

[417]. For p ∈ [1, ∞]

we denote the conjugate variable q ∈ [1, ∞] satisfying 1/p + 1/q = 1 by

¯p = p/(p − 1).

Let k·k denote a vector norm on R

. Then k·k is absolute if kxk = k|x|k

for all x ∈ R

. Furth ermore, k · k is monotone if kxk ≤ kyk for all x, y ∈ R

such that |x| ≤≤ |y|. Note th at k · k is absolute if and on ly if k · k is

monotone [201, p. 285].

Lemma 7.1. Let p ∈ [1, ∞] and let A ∈ R

m×n

. Then

kAk

2,2

= σ

max

(A), (7.59)

kAk

p,1

= max

i=1,...,n

kcol

(A)k

, (7.60)

and

kAk

∞,p

= max

i=1,...,m

krow

(A)k

¯p

. (7.61)

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488 CHAPTER 7

Proof. Expression (7.59) is standard; see [417] for a proof. To sh ow

(7.60), note that, for all x ∈ R

kAxk



i=1

col

(A)



≤

i=1

|kcol

(A)k

≤ max

i=1,...,n

kcol

(A)k

kxk

and hence, kAk

p,1

≤ max

i=1,...,n

kcol

(A)k

. Next, let j ∈ {1, . . . , n} be such

that kcol

(A)k

= m ax

i=1,...,n

kcol

(A)k

. Now, since ke

= 1, it follows

that kAe

= kcol

(A)k

, which implies kAk

p,1

≥ max

i=1,...,n

kcol

(A)k

and h en ce, (7.60) holds.

Finally, to show (7.61) n ote that, for all x ∈ R

, it follows from

H¨older’s inequality that

kAxk

∞

= max

i=1,...,m

|row

(A)x| ≤ max

i=1,...,m

krow

(A)k

¯p

kxk

which implies that kAk

∞,p

≤ max

i=1,...,m

krow

(A)k

¯p

. Next, let j ∈ {1, . . . ,

n} be such that krow

(A)k

¯p

= max

i=1,...,m

krow

(A)k

¯p

and let x be such that

|row

(A)x| = krow

(A)k

¯p

kxk

. Hence, kAxk

∞

= max

i=1,...,m

|row

(A)x| ≥

|row

(A)x| = krow

(A)k

¯p

kxk

, which implies th at kAk

∞,p

≥ max

i=1,...,m

krow

(A)k

¯p

, and hence, (7.61) holds.

Note that (7.60) and (7.61) generalize the well-known expressions

kAk

1,1

= max

i=1,...,n

kcol

(A)k

[201], kAk

∞,∞

= max

i=1,...,m

krow

(A)k

[201], and kAk

∞,1

= kAk

∞

[225]. Furthermore, since max

i=1,...,n

kcol

(A)k

= d

1/2

max

A) and

max

i=1,...,m

krow

(A)k

= d

1/2

max

(AA

it follows fr om (7.60) with p = 2 that kAk

2,1

= d

1/2

max

A) and from (7.61)

with p = 2 that kAk

∞,2

= d

1/2

max

(AA

Lemma 7.2. Let k ·k

′

and k · k

′′

denote absolute vector norms on R

and R

, respectively, and let k·k : R

m×n

→ R be the matrix norm induced

by k ·k

′

and k · k

′′

. Then the following statements hold:

i) L et A ∈ R

m×n

be such that A ≥≥ 0. Then there exists x ∈ R

such

that x ≥≥ 0, kxk

′

= 1, and kAk = kAxk

′′

ii) Let A, B ∈ R

m×n

be s uch that 0 ≤≤ A ≤≤ B. Th en kAk ≤ kBk.

Proof. To p rove i) let y ∈ R

be such that kyk

′

= 1 and kAk = kAyk

′′

Now, since k·k

′

and k·k

′′

are absolute (and hence monotone) vector norms

it follows that

kAk = kAyk

′′

≤ kA|y|k

′′

≤ kAkk|y|k

′

= kAkkyk

′

= kAk,

NonlinearBook10pt November 20, 2007

INPUT-OUTPUT STABILITY AND DISSIPATIVITY 489

which implies that kAk = kA|y|k

′′

, and hence, i) follows with x = |y|.

Next, to prove ii) let x ∈ R

be such that x ≥≥ 0, kxk

′

= 1, and

kAk = kAxk

′′

(existence of such an x follows from i)). Hence, since k · k

′′

an absolute vector norm and Ax ≤≤ Bx it follows that

kAk = kAxk

′′

≤ kBxk

′′

≤ kBk,

which implies ii).

The following result generalizes H¨older’s inequality to mixed-signal

norms.

Lemma 7.3. Let p, r ∈ [1, ∞], and let f ∈ L

and g ∈ L

¯p

. Then

hf, gi ≤ |||f|||

p,r

|||g|||

¯p,¯r

. (7.62)

Finally, the following two r esults are needed for the results given in

Section 7.8.

Lemma 7.4 ([461]). Let p ∈ [1, ∞) and r ∈ [1, ∞], and let f ∈ L

Then

|||f |||

p,r

= sup

g ∈G

hf, gi, (7.63)

where G

△

= {g ∈ L

¯p

: |||g|||

¯p,¯r

≤ 1}.

Lemma 7.5 ([461]). Let p ∈ [1, ∞), r ∈ [1, ∞], and f : [0, ∞) ×

[0, ∞) → R

be such that f(t, ·) is integrable for almost all t ∈ [0, ∞),

f(·, τ) ∈ L

for almost all τ ∈ [0, ∞), and g ∈ L

, where g(τ )

△

= [

∞

kf(t, τ)k

dt]

1/p

. Then

|||y|||

p,r

≤

∞

g(τ)dτ, (7.64)

where

y(t) =

∞

f(t, τ )dτ, t ≥ 0. (7.65)

7.8 Induced Convolution Operator Norms of Linear Dynamical

Systems and L

Stability

In this section, we develop induced convolution operator norms. For the

system (7.52) and (7.53), let G : R → R

l×m

denote the impulse response

function

G(t)

△



0, t < 0,

B, t ≥ 0.

(7.66)

NonlinearBook10pt November 20, 2007

490 CHAPTER 7

Next, let G : L

→ L

denote the convolution operator

y(t) = G[u](t) = (G ∗ u)(t)

△

∞

G(t −τ)u(τ)dτ, (7.67)

and d eﬁ ne the induced norm |||G|||

(q,s),(p,r)

|||G|||

(q,s),(p,r)

△

= sup

|||u|||

p,r

|||G ∗ u|||

q,s

. (7.68)

First, note that if the induced norm |||G|||

(p,s),(p,r)

is bounded for p, r, s ∈

[1, ∞], then G is L

-stable with ﬁnite gain. The following result presents a

suﬃcient condition for L

stability of G.

Proposition 7.2. Consider the linear dynamical system given by (7.52)

and (7.53) where A is Hurwitz. Then, for every p ∈ [1, ∞], G is L

-stable

with ﬁnite gain.

Proof. Since A is Hurwitz it follows that there exist positive d eﬁ nite

matrices P ∈ R

n×n

and R ∈ R

n×n

such that

0 = A

P + P A + R.

Now, the result is a direct consequence of Theorem 7.6. Speciﬁcally, with

F (x, u) = Ax + Bu, H(x, u) = Cx, V (x) = x

P x, α = σ

min

(P ), β =

max

(P ), γ

= σ

min

(R), γ

= 2σ

max

(P ), L = σ

max

(B), η

= σ

max

(C), and

= 0, all the conditions of Theorem 7.6 are satisﬁed.

Note that if (A, B) is controllable and (A, C) is observab le, then the

converse of Proposition 7.2 is also true (see, for example [445]). In the

remainder of the chapter we provide several explicit expressions for induced

norms of G.

The following lemma provides an explicit expression for |||G|||

(∞,∞),(r,r)

for the case in wh ich G is a single-input/single-output operator.

Lemma 7.6. Let r ∈ [1, ∞] and let l = m = 1. Then G : L

→ L

∞

there exists u ∈ L

such that lim

t→∞

(G ∗u)(t) = |||G|||

¯r,¯r

|||u|||

r,r

, and

|||G|||

(∞,∞),(r,r)

= |||G|||

¯r,¯r

. (7.69)

Proof. For r = 1 and r = ∞, (7.69) is standard; see [461] and [104,

pp. 23-24], respectively. Next, let r ∈ (1, ∞) and note for all t ≥ 0, it follows

from Lemma 7.3 with p = r that

|y(t)| =



∞

G(t −τ)u(τ)dτ



NonlinearBook10pt November 20, 2007

INPUT-OUTPUT STABILITY AND DISSIPATIVITY 491

≤



∞

|G(t − τ)|

¯r

dτ



1/¯r

|||u|||

r,r



|G(τ)|

¯r

dτ



1/¯r

|||u|||

r,r

≤ |||G|||

¯r,¯r

|||u|||

r,r

which implies

|||G|||

(∞,∞),(r,r)

≤ |||G|||

¯r,¯r

. (7.70)

Next, let T > 0 and let u(·) be such that u(t) = sgn(G(T −t))|G(T −

t)|

1/(r−1)

, t ≥ 0, where sgn(·) denotes the signum function. Now, since

|||u|||

r,r



∞

|G(T − t)|

¯r



1/r

, it follows that

|y(T )| =



∞

G(T − τ )u(τ)dτ



∞

|G(T − τ)|

¯r

dτ



∞

|G(T − τ)|

¯r

dτ



1/¯r

|||u|||

r,r



|G(τ)|

¯r

dτ



1/¯r

|||u|||

r,r

Hence,

|||y|||

∞,∞

≥ lim

T →∞

|y(T )| = |||G|||

¯r,¯r

|||u|||

r,r

which, with (7.70), proves the result.

Note that it follows from L emm a 7.6 that there exists u ∈ L

such

that lim

t→∞

(G ∗ u)(t) = |||G|||

¯r,¯r

|||u|||

r,r

Next, deﬁne P ∈ R

m×m

and Q ∈ R

l×l

△

∞

(t)G(t)dt, Q

△

∞

G(t)G

(t)dt. (7.71)

Note that P = B

P B and Q = CQC

, where the observability and

controllability Gramians P and Q, respectively, are the unique n × n

nonnegative-deﬁnite solutions to the Lyapu nov equations

0 = A

P + P A + C

C, 0 = AQ + QA

+ BB

. (7.72)

Furth ermore, let G

[p,q]

denote the l × m matrix whose (i, j)th entry is

|||G

(i,j)

|||

(p,p),(q,q)

Theorem 7.7. The f ollowing statements hold:

NonlinearBook10pt November 20, 2007

492 CHAPTER 7

i) G : L

→ L

, and

|||G|||

(2,2),(2,2)

= sup

ω ∈R

max

(H(ω)). (7.73)

ii) Let r ∈ [1, ∞]. Then G : L

→ L

, and

|||G|||

(2,2),(1,r)

= kP

1/2

2,r

. (7.74)

iii) Let p ∈ [1, ∞]. Then G : L

→ L

∞

, and

|||G|||

(∞,p),(2,2)

= kQ

1/2

2,¯p

. (7.75)

iv) Let p, r ∈ [1, ∞]. T hen G : L

→ L

∞

, and

|||G|||

(∞,p),(1,r)

= sup

t≥0

kG(t)k

p,r

. (7.76)

v) Let r ∈ [1, ∞]. Then G : L

→ L

∞

, and

|||G|||

(∞,∞),(r,r)

= max

i=1,...,l

krow

[¯r,¯r]

¯r

. (7.77)

vi) Let p ∈ [1, ∞]. Then G : L

→ L

, and

|||G|||

(p,p),(1,1)

= max

j=1,...,m

kcol

[p,p]

. (7.78)

Proof. i) It follows fr om Theorem 5.12 that

|||y|||

2,2

∞

(t)y(t)dt

2π

∞

−∞

∗

(ω)y(ω)dω

2π

∞

−∞

∗

(ω)H

∗

(ω)H(ω)u(ω)dω

≤ sup

ω ∈R

max

(H(ω))

2π

∞

−∞

∗

(ω)u(ω)dω

= sup

ω ∈R

max

(H(ω))|||u|||

2,2

which implies that

|||G|||

(2,2),(2,2)

≤ sup

ω ∈R

max

(H(ω)).

Next, let γ

△

= |||G|||

(2,2),(2,2)

and note that since G is causal, for every

u ∈ L

and for every T > 0, |||y

|||

2,2

≤ γ|||u

|||

2,2

or, equivalently,

(t)y(t)dt ≤ γ

(t)u(t)dt, T ≥ 0.

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INPUT-OUTPUT STABILITY AND DISSIPATIVITY 493

Hence, it follows from Theorem 5.15 that H(s) is bounded real (with gain

γ), that is,

sup

ω ∈R

max

(H(ω)) ≤ γ = |||G|||

(2,2),(2,2)

which implies that

|||G|||

(2,2),(2,2)

= sup

ω ∈R

max

(H(ω)).

ii) I t follows from Lemma 7.5 that

|||y|||

2,2

≤

∞

|||G(t − τ )u(τ)|||

2,2

dτ

∞



∞

(τ)G

(t −τ )G(t − τ)u(τ)dt



1/2

dτ

∞

1/2

u(τ)k

dτ

≤

∞

1/2

2,r

ku(τ)k

dτ

= kP

1/2

2,r

|||u|||

1,r

which implies that |||G|||

(2,2),(1,r)

≤ kP

1/2

2,r

Next, let u

(·) = ˆuv

(·), k = 1, 2, . . ., where ˆu ∈ R

is such that

kˆuk

= 1, kP

1/2

ˆuk

= kP

1/2

2,r

kˆuk

, and measurable v

: [0, ∞) → R is

such that |||v

|||

1,1

= 1 and, as k → ∞, v

(·) → δ(·), where δ(·) is the Dirac

delta function. Note that |||u

|||

1,r

= 1, k = 1, 2, . . ., and y

(t) → G(t)ˆu,

t ≥ 0, as k → ∞, where y

(t)

△

= (G ∗ u

)(t). Hence,

|||G|||

(2,2),(1,r)

≥ lim

k→∞

|||y

|||

2,2



∞

kG(t)ˆuk



1/2



∞

ˆu

(t)G(t)ˆudt



1/2

= (ˆu

Pˆu)

1/2

= kP

1/2

ˆuk

= kP

1/2

2,r

which implies that |||G|||

(2,2),(1,r)

= kP

1/2

2,r

NonlinearBook10pt November 20, 2007

494 CHAPTER 7

iii) With p = r = 2 it follows from Lemma 7.3 that for all t ≥ 0,

ky(t)k



∞

G(t −τ)u(τ)dτ



= max

{ˆu∈R

: kˆuk

¯p

=1}

∞

ˆu

G(t −τ)u(τ)dτ

≤ max

{ˆu∈R

: kˆuk

¯p

=1}



∞

kG(t − τ)ˆuk

dτ



1/2

|||u|||

2,2

= max

{ˆu∈R

: kˆuk

¯p

=1}



ˆu

G(τ)G

(τ)dτ ˆu



1/2

|||u|||

2,2

≤ max

{ˆu∈R

: kˆuk

¯p

=1}

(ˆu

Qˆu)

1/2

|||u|||

2,2

= kQ

1/2

2,¯p

|||u|||

2,2

which implies that |||y|||

∞,p

≤ kQ

1/2

2,¯p

|||u|||

2,2

for all y ∈ L

∞

and u ∈ L

, and

hence, |||G|||

(∞,p),(2,2)

≤ kQ

1/2

2,¯p

Next, let ˆu ∈ R

be such that kˆuk

¯p

= 1 and kQ

1/2

ˆuk

= kQ

1/2

2,¯p

and let T > 0 and

u(t) =

1/2

2,¯p

(T − t)ˆu,

so that |||u|||

2,2

≤ 1. Now, since k ·k

= k · k

¯p

, it follows that

|||y|||

∞,p

= sup

t≥0

ky(t)k

= sup

t≥0

max

{ˆy∈R

:kˆyk

¯p

=1}

ˆy

y(t)

≥ sup

t≥0

ˆu

y(t)

1/2

2,¯p

sup

t≥0

∞

ˆu

G(T −τ)G

(T − τ)ˆudτ

1/2

2,¯p

sup

t≥0

ˆu

G(τ)G

(τ)ˆudτ,

which implies that, for every T > 0, there exists u ∈ L

such that |||u|||

2,2

≤ 1

and

|||y|||

∞,p

≥

1/2

2,¯p

sup

t≥0

ˆu

G(τ)G

(τ)ˆudτ,

or, equivalently,

|||G|||

(∞,p),(2,2)

≥ sup

T >0

1/2

2,¯p

sup

t≥0

ˆu

G(τ)G

(τ)ˆudτ