Chapter 4 Transfer Function...

Chapter 4

Transfer Function Models

This chapter introduces models of linear time invariant (LTI) systems defined by theirtransfer functions (or, in general, transfer matrices). The subject is expected to be familiarto the reader, especially in the case of rational transfer functions, which are suitablefor describing systems defined in terms of ordinary differential equations and differenceequations. The main objective of the chapter is to build a mathematical frameworksuitable for handling the non-rational transfer functions resulting from partial differentialequation models which are stabilizable by finite order LTI controllers.

4.1 Fourier Transforms and the Parseval Identity

Fourier transforms play a major role in defining and analyzing systems in terms of non-rational transfer functions. This section requires some background in the theory of inte-gration of functions of a real argument (measureability, Lebesque integrabilty, complete-ness of L2 spaces, etc.), and presents some minimal technical information about Fouriertransforms for ”finite energy” functions on Z and R.

4.1.1 Background: Spaces of Square Integrable Functions

This chapter deals extensively with complex vector-valued square integrable functionsdefined on the sets R, jR (the ”imaginary axis” s ∈ C : Res = 0), T (the unit circlez ∈ C : |z| = 1), and Z (all integers). The sets of all such functions (traditionallycalled ”spaces”) will be denoted by Lm

2 (X), where X ∈ R, jR,T,Z.While there are some technical distinctions caused by the differences between the

four domains X of interest (R, jR, T, and Z), the sets Lm2 (X) share important abstract

35

36 6.245 (Fall 2011) Transfer Functions

features, which allow one to treat them in a way similar to how ordinary vector spacesC

n are treated. Specifically, all these sets are complex vector spaces, i.e. operationsof addition and scaling by a complex scalar are defined on Lm

2 (X), and satisfy the usualcommutative and distributive laws. Moreover the spaces have an inner product operation,mapping every ordered pair (f, g) of functions f, g ∈ Lm

2 (X) to a complex number cdenoted by f ′g. The inner prodiuct satisfies the conditions of Hermitean symmetry (g′fis the complex conjugate of f ′g), linearity with respect to the second argument (f ′(cg) =c(f ′g) for c ∈ C, and f ′(g1+g2) = f ′g1+f ′g2), and positive semidefiniteness (f ′f ≥ 0). Thefunction |f | =

√f ′f is used as a measure of length of a function, and satisfies the triangle

inequality |f+g| ≤ |f |+|g| (or, equivalently, the Cauchy inequality |f ′g| ≤ |f |·|g|), so thatone can talk about convergence in Lm

2 (X): a sequence fk∞k=0 of functions fk ∈ Lm2 (X)

is said to converge to f ∈ Lm2 (X) when fk − f | → 0 as k → ∞. With respect to

this definition of convergence, the sets Lm2 (X) have the completeness property: for every

sequence fk∞k=0 of functions fk ∈ Lm2 (X) satisfying the Cauchy property (|fk − fi| → 0

as k, i → ∞) there exists a function f ∈ Lm2 (X) such that |fk − f | → 0 as k → ∞.

For example, Lm2 (Z) will denote the set of all functions u : Z → C

m (essentially,double-sided sequences of vectors from C

m) such that

∑

t∈Z

|u(t)|2 < ∞,

with the inner product

u′vdef=

∑

t∈Z

u(t)′v(t).

Similarly, Lm2 (R) will be the set of all measureable functions u : R → C

m such that

∫ ∞

−∞

|u(t)|2dt < ∞,

with the inner product

u′vdef=

∫ ∞

−∞

u(t)′v(t)dt.

Note that a function f ∈ Lm2 (R) such that |f | = 0 can take non-zero values, but in this

case the Lebesque measure of the set t : f(t) = 0 is zero.A function F : T → C

k is called integrable when the function F : (−π, π) → Cm

defined by F (Ω) = F (exp(jΩ)) is Lebesque integrable, in which case we use the shortcutnotation

∫

T

F (z)dm(z)def=

1

2π

∫ π

−π

F (exp(jΩ))dΩ.

6.245 (Fall 2011) Transfer Functions 37

The set Lm2 (T) contains integrable functions U : T → C

m such that∫

T

|U(z)|2dm(z) < ∞.

The inner product on Lm2 (T) is defined by

U ′V =

∫

T

U(z)′V (z)dm(z).

Finally, Lm2 (jR) contains functions U : jR → C

m such that the function g : R → Cm

defined by g(ω) = U(jω) is square integrable. The inner product on Lm2 (jR) is defined by

U ′V =1

2π

∫ ∞

−∞

U(jω)′V (jω)dω.

4.1.2 Fourier Transform on Z

For a function (sequence) u : Z → Cm, one expects the Fourier transform to be a function

U : T → Cm given by

U(z) =∑

t∈Z

u(t)z−t. (4.1)

This definition is straightforward for absolutely summable sequences, i.e. those satisfyingthe condition

∑

t∈Z

|u(t)| < ∞,

which guarantees that the sum in (4.1) converges uniformly over z ∈ T. Moreover, thesimple observation that

∫

Z

zizkdm(z) = δikdef=

1, i = k,

0, i 6= k,(4.2)

for all i, k ∈ Z yields the ever helpful Parseval identity

∑

t∈Z

|u(t)|2 =∫

T

|U(z)|2dm(z). (4.3)

The problem is, the class of summable seqences u is too narrow to be useful in manyapplications. The objective of this subsection is to introduce Fourier transform for squaresummable sequences, i.e. elements of Lm

2 (Z).


According to a ”proper” definition, the Fourier transform U of u ∈ Lm2 (Z) is any

U ∈ Lm2 (T) such that |UT − U | → 0 as T → ∞, where

UT (z) =T∑

t=−T

u(t)z−t. (4.4)

According to this definition, a given sequence u ∈ Lm2 (Z) can have multiple Fourier

transforms, but this non-uniqueness is not really essential, as any two Fourier transformsU, V of the same square summable sequence u must satisfy

|U − V | = |(U − UT )− (V − UT )| ≤ |U − UT |+ |V − UT | → 0 as T → ∞,

which means that |U − V | = 0, i.e. U(z) − V (z) = 0 for almost all z, so that, for allpractical purposes, U and V can be considered equal.

The main statement about Fourier transforms of square summable sequences is theirexistence, coupled with the Parseval identity.

Theorem 4.1. Every u ∈ Lm2 (Z) has a Fourier transform U ∈ Lm

2 (T). Moreover, everyFourier transform U ∈ Lm

2 (T) of u ∈ Lm2 (Z) satisfies (4.3).

Proof. For arbitrary positive integers τ < T the Parseval identity yields

‖UT − Uτ‖22 =T−1∑

t=τ

|u(t)|2 ≤∞∑

t=τ

|u(t)|2 → 0 as τ → ∞,

where UT are defined in (4.4). Hence UT is a Cauchy sequence, and, by completeness ofLm2 (T), has a limit U . By definition, U is a Fourier transform of u.

To prove the Parseval identity, note that

|U | − |UT | = |(U − UT ) + UT | − |UT | ≤ |U − UT |+ |UT | − |UT | → 0 as T → ∞,

|UT | − |U | = |UT − U + U | − |U | ≤ |UT − U |+ |U | − |U | → 0 as T → ∞.

Hence

|U |2 = limT→∞

|UT |2 = limT→∞

T∑

t=−T

|u(t)|2 = |u|2.


4.1.3 Approximations by Trigonometric Polynomials in Lm2 (T)

A trigonometric polynomial (of a single argument) is a continuous function P : T → Cm

which can be expressed in the form

P (z) =T∑

t=−T

p(t)z−t,

where p(t) ∈ Cm are some vectors of coefficients. (The name is likely due to the fact

that zn is a polynomial with respect to cos(Ω) and sin(Ω) when z = exp(jΩ).) Forthe development of Fourier transforms, we need the classical theorem establishing thattrigonometric polynomials can approximate continuous functions V : T → C

m arbitrarilywell.

Theorem 4.2. For every continuous function V : T → Cm and every ǫ > 0 there exists

a trigonometric polynomial P : T → Cm such that

maxz∈T

|P (z)− V (z)| < ǫ.

Proof. For every positive integer n define function Pn : T → Cm by

Pn(z) =

∫

T

hn(z, w)V (w)dm(w), (4.5)

where

hn(z, w) =1

n

∣

∣

∣

∣

1 +w

z+

w2

z2+ · · ·+ wn−1

zn−1

∣

∣

∣

∣

2

.

Since for every fixed n and w the function z → hn(z, w) is a trigonometric polynomial, the resultPn(z) of integration in (4.5) is a trigonometric polynomial, too.

The ”kernel” function hn(z, w) has several useful properties:

(a) hn(z, w) ≥ 0;

(b) limn→∞ supz: |z−w|>δ hn(z, w) = 0 for all δ ∈ (0, π);

(c)∫

Thn(z, w)dm(w) = 1 for all n.

These properties guarantee that

limn→∞

maxz∈T

|Pn(z)− V (z)| = 0. (4.6)

Indeed, since V is a continuous function, for every ǫ > 0 there exists δ ∈ (0, π) such that

|V (w)− V (z)| < 0.5ǫ whenever |w − z| < δ.


For this δ > 0, property (b) of the kernel guarantees existence of n such that

hn(z, w) <0.5ǫ

1 + 2γfor |z − w| > δ, where γ = max

z∈T|V (z)|.

Hence for every z ∈ T

|Pn(z)− V (z)| =

∣

∣

∣

∣

∫

T

hn(z, w)(V (z)− V (w))dm(w)

∣

∣

∣

∣

≤∫

|z−w|<δ

hn(z, w)|V (z)− V (w)|dm(w) +

∫

|z−w|>δ

hn(z, w)|V (z)− V (w)|dm(w)

≤∫

T

0.5ǫdm(w) +

∫

T

0.5ǫ

1 + 2γ2γdm(w)

= ǫ.

4.1.4 Fourier Coefficients

For every U ∈ Lm2 (T) and t ∈ Z the ”Fourier coefficients” integral

u(t) =

∫

T

U(z)ztdm(z) (4.7)

defines a function (sequence) u : Z → Cm.

The following statement, which relies on the result of the previous subsection, showsthat Fourier coefficients deefine a function U ∈ Lm

2 (T) ”almost completely”.

Theorem 4.3. If all Fourier coefficients (4.7) of a function U ∈ Lm2 (T) are zero, then

|U | = 0 (i.e. U(z) = 0 for almost all z ∈ T).

Proof. The theorem can be proven ”component-wise”, i.e. it is sufficient to consider the casem = 1. For 0 < θ < ν < 2π consider the function Fθ,ν : T → C defined by

Fθ,ν(exp(jΩ)) =

1, 0 ≤ Ω < θ,

1− Ω−θν−θ

, θ ≤ Ω < ν,

0, ν ≤ Ω < 2π.

Since Fθ,ν is continuous, there exists a sequence of trigonometric polynomials Pn such that

limn→∞

maxz∈T

|Fθ,ν(z)− Pn(z)| = 0.


Hence

limn→∞

∫

T

(Fθ,ν(z)− Pn(z))V (z)dm(z) = 0.

Since by assumption∫

T

P (z)V (z)dm(z) = 0

for every trigonometric polynomial P , we conclude that

∫

T

Fθ,νV (z)dm(z) = limn→∞

∫

T

Pn(z)V (z)dm(z) = 0.

For 0 < θ < 2π let

Fθ(exp(jΩ)) =

1, 0 ≤ Ω < θ,

0, θ ≤ Ω < 2π..

Since

|Fθ(z)− Fθ,ν(z)| ≤ 1 and limν→θ

Fθ,ν(z) = Fθ(z),

we conclude that∫

T

Fθ(z)V (z)dm(z) = limν→θ

∫

T

Fθ(z)V (z)dm(z) = 0,

i.e.∫ θ

0

V (exp(jΩ))dΩ = 0 for all θ ∈ (0, 2π).

According to the basic properties of Lebesque integration, this means that V (exp(jΩ)) = 0 or

almost all Ω ∈ (0, 2π).

4.1.5 Inverse Fourier Transform on T

As stated by the following theorem, mapping (4.7) of a function U ∈ Lm2 (T) to the

sequence of its Fourier coefficients serves as the inverse of the Fourier transform (4.1).

Theorem 4.4. For every U ∈ Lm2 (T) the sequence u : Z → C

m defined by (4.7) hasfinite energy (i.e. u ∈ ℓm2 ). Moreover, U is a Fourier transform of u.

Proof. First prove that u is square summable. For positive integers T > 0 define functionsUT : T → C

m as in (4.4). Multiplying equality (4.7) by u(t)′ on the left and summing fromt = −T to t = T yields

T∑

t=−T

|u(t)|2 =∫

T

UT (z)′U(z)dm(z).


Applying the Cauchy inequality |U ′TU | ≤ |UT | · |U | to the inner product on the left side yields

|UT |2 ≤ |U | · |UT |, hence |UT |2 ≤ |U |2 for all T > 0, and therefore u ∈ Lm2 (Z).

Let V denote the Fourier transform of u. Since |UT − V | → 0 as T → ∞, and∫

Z

ztUT (z)dm(z) = u(t) for |t| ≤ T,

U and V have same Fourier coefficients. Hence |U − V | = 0 (Theorem 4.3), which means that

U is a Fourier transform of u as well.

4.1.6 Fourier Transform on R

The derivation of Fourier transform on Lm2 (R) follows a path similar to that of Lm

2 (Z). Akey result is the following limited version of the Parseval identity.

Theorem 4.5. The identity

∫ T

−T

|u(t)|2dt = 1

2π

∫ ∞

−∞

|UT (jω)|2dω, (4.8)

where

UT (jω) =

∫ T

−T

e−jωtu(t)dt,

holds for every T > 0 and every function u ∈ Lm2 (R).

Proof (a sketch). It is actually easier to prove the slightly more general identity

∫ T

−T

u(t)′v(t)dt =1

2π

∫ ∞

−∞UT (jω)

′VT (jω)dω, (4.9)

where

UT (jω) =

∫ T

−T

e−jωtu(t)dt, VT (jω) =

∫ T

−T

e−jωtv(t)dt,

and u, v ∈ Lm2 (R) (proving the identity involves showing that UT , VT ∈ Lm

2 (jR)). A goodstarting point is to consider functions u, v of the form

u(t) = exp(jπa/T ), v(t) = exp(jπb/T ), (4.10)

where a, b are arbitrary integer numbers. After some manipulations with integrals, this boilsdown to computing the integrals

∫ ∞

−∞

sin2(ω)

ω(ω + dπ)dω =

π, d = 0,

0, d ∈ Z, d 6= 0.


(Would be easy to do this using the Parseval identity, but this would make for a ”circular”argument: Parseval identity is exactly what is being proven right here!)

Once (4.9) is established for individual exponents (4.10), it extends easily to finite linear

combinations of such exponents. According to the results on inverse discrete Fourier transform

established in the previous subsection, every square integrable function on the interval [−T, T ]

can be approximated arbitrarily well (in terms of L2 convergence) by linear combinations of

functions (4.10), which allows one to prove the general case of (4.9).

Theorem 4.5 shows that, for a given function u ∈ Lm2 (R) the sequence of ”incomplete”

Fourier transforms UT is a Cauchy sequence in Lm2 (jR). Hence, there exists U ∈ Lm

2 (jR)such that U − UT | → 0 as T → ∞ Any such function U ∈ Lm

2 (jR) is called a Fouriertransform of u. Naturally, two different Fourier transforms of the same function u ∈ Lm

2 (R)must be equal almost everywhere on jR.

The derivation of inverse Fourier transform on Lm2 (jR) relies on minor modification of

identity (4.8):1

2π

∫ Ω

−Ω

|U(jω)|2dω =

∫ ∞

−∞

|uΩ(t)|2dt, (4.11)

where

uΩ(t) =1

2π

∫ Ω

−Ω

ejωtU(jω)dω.

Once again, the inverse Fourier transform u ∈ Lm2 (R) of U ∈ Lm

2 (jR) is a limit of theCauchy sequence uΩ as Ω → ∞.

Theorem 4.6. Function U ∈ Lm2 (jR) is a Fourier transform of u ∈ Lm

2 (R) if and only ifu is an inverse Fourier transform of U , in which case

∫ ∞

−∞

|u(t)|2dt = 1

2π

∫ ∞

−∞

|U(jω)|2dω. (4.12)

4.2 Hardy Classes: H2 and H∞

Hardy classes H2 are ”one-sided” Fourier transforms of square integrable signals. Theyare introduced as subsets of Lm

2 (T) (in a ”discrete time” framework) or Lm2 (jR) (in a

”continuous time” framework). However, the functions from H2 are most interestingbecause of the possibility of ”extending” them into the outside of the unit circle (forLm2 (T)) or into the right half plane (for Lm

2 (jR)), where they become analytical functionsof complex argument.

The main objective of this section is to introduce functions from class H∞, as a minormodification of class H2, as they match exactly transfer functions of L2 gain stable causal


LTI systems, and also serve as building blocks for transfer functions of stabilizable causalLTI systems.

4.2.1 Analytical Functions

A function F : Ω → C defined on an open subset Ω of C is called analytical if the limit

F (s) = limδ→0

F (s+ δ)− F (s)

δ

exists at all points s ∈ Ω. As a rule, ”elementary” functions, such as polynomials orexponents, or logarithms (within limited domains), or their compositions, turn out to beanalytical. For example, function F (s) = 1/s is analytical on Ω = s ∈ C : s 6= 0, andfunction

F (s) = log(s)def= log |s|+ arg(s), arg(s) ∈ (−π, π)

is analytical in the right half plane

C+ = s : Res > 0.

In contrast, functions which require separating real and imaginary parts of s, such asF (s) = Res or F (s) = |s| are not analytical. For example, the expression

Res+ δ − Resδ

converges to 1 when δ approaches zero along the real axis, but converges to zero when δapproaches zero along the imaginary axis.

The following classical theorem gives an alternative description of analytical functions.

Theorem 4.7. For some positive numbers R0, R1 such that 0 < R0 < R1 let

Ω = s ∈ C : R0 < |s| < R1.

A function F : Ω → Cm is analytical if and only if there exists a function f : Z → C

m

such that∑

r−t|f(t)| < ∞ for R0 < r < R1,

and

F (s) =∞∑

k=−∞

s−tf(t) ∀ s ∈ Ω.


4.2.2 Class Hm2 (T)

Formally speaking, Hm2 (T) (or simply H2(T) when m = 1) is the subset of Lm

2 (T) con-sisting of those functions U for which the Fourier coefficients u(t) equal zero for t < 0. Inother words, Hm

2 (T) is the set of Fourier transforms of signals of finite energy.

Most significantly, for an arbitrary U ∈ Hm2 (T) the infinite sum

U(w) =∑

t=Z

u(t)w−t =∞∑

t=0

u(t)w−t

converges absolutely on the open set

D+ = w ∈ C : |w| > 1.

Allowing some abuse of notation, we will denote the resulting analytical function on D+

by the same letter, i.e. as U : D+ → C.

The following theorem shows that, combined with analiticity, boundedness of theenergies |Ur|2 for Ur(z) = U(rz) over r > 1 yields a complete description of ”extensions”of functions from class Hm

2 (T) to D+.

Theorem 4.8. Let U : D+ → C

m be a function. The following conditions are equivalent:

(a) U is an extension of a function U ∈ Hm2 (T);

(b) U is analytical on D+, and

supr>1

∫

T

|U(rz)|2dm(z) < ∞. (4.13)

Moreover, when conditions (a),(b) are satisfied, the functions Ur : T → Cm defined for

r > 1 by Ur(z) = U(rz), converge to U , in the sense that |U − Ur| → 0 as r → 1.

One useful corollary of Theorem 4.8 is a criterion for a function U from class Hm2 to

have zero Fourier coefficients u(t) = 0 for all t < T , where T > 0. Since this is equivalentto V (s) = sTU(s) being an extension of a function from class H2, the necessary andsufficient condition is

supr>1

∫

T

r2T |U(rz)|2dm(z) < ∞. (4.14)


Example 4.1. The function defined for |w| > 1 by U(w) = log(1 − 1/w) (where thevalues of the logarithm with imaginary part from (−π, π) are used) is an extension of anH2(T) class function. To establish this, note that that U is analytical on D

+, and use theinequality

∣

∣

∣

∣

1− 1

z

∣

∣

∣

∣

≤ 2

∣

∣

∣

∣

1− 1

rz

∣

∣

∣

∣

∀ z ∈ T, r ≥ 1.

Since the imaginary part of log(1− 1/w) for w ∈ D+ is bounded (lies within the (−π, π)

interval), and the real part cannot be larger than log(2), we have∣

∣

∣

∣

log(1− 1

rz)

∣

∣

∣

∣

≤ π + log(2) +

∣

∣

∣

∣

log

∣

∣

∣

∣

1− 1

z

∣

∣

∣

∣

∣

∣

∣

∣

,

which allows one to bound the integrals∫

T

|log(1− 1/(rz))|2 dm(z) ≤∫

T

|π + log(2) + |log(1− 1/z)||2 dm(z) < ∞

(the last inequality holds because log(|Ω|) is square integrable over Ω ∈ (−π, π)).Naturally, the function U(z) = log(1−1/z) (defined arbitrarily for z = 1) provides the

values of U to the unit circle. Since we have already established an integrable majorantfor |U(z)− U(rz)|, the convergence

limr→1

∫

T

|U(z)− Ur(z)|2dm(z) = 0

follows from the point-wise convergence Ur(z) → U(z) as r → 1 for z 6= 1.

Proof of Theorem 4.8, (a)⇒(b). Let u ∈ Lm2 (Z) be the inverse Fourier transform of U .

By assumption u(t) = 0 for t < 0. Hence the expansion

U(s) =∑

t∈Z

u(t)s−t

satisfies the absolute convergence condition∑

t∈Z

|u(t)|r−t < ∞,

which makes U analytical on D+. Since Ur(z) = U(rz) is the Fourier transform of ur(t) =

r−tu(t), the Parseval identity yields

∫

T

r2T |U(rz)|2dm(z) =

∞∑

t=0

r−2t|u(t)|2 ≤ |U |2.


Moreover, a similar use of the Parseval identity yields

∫

T

r2T |U(rz)− U(z)|2dm(z) =∞∑

t=0

(1− r−t)2|u(t)|2 → 0

as r → 1.

Proof of Theorem 4.8, (b)⇒(a). By Theorem 4.7 there exists an expansion

U(s) =∑

t∈Z

u(t)s−t

which converges absolutely for |s| > 1. Hence the sequence ur(t) = r−tu(t) is square summablefor all r > 1, and, by the Parseval identity,

∫

T

r2T |U(rz)|2dm(z) =∞∑

t=0

r−2t|u(t)|2

for all r > 1. Since the sum on the right converges to +∞ as r → ∞, unless u(t) = 0 for all

t < 0, we conclude that u(t) = 0 for t < 0. Subject to this, the sum on the right converges to

|u|2 as r → 1, which proves that U ∈ Hm2 (T).

4.2.3 Class Hm,k∞ (T)

Functions from class Hm,k∞ (T) will serve as transfer matrices of causal L2 gain stable LTI

models (with m inputs and k outputs) in discrete time. Each element G ∈ Hm,k∞ (T) can

be viewed as a k-by-m matrix with entries from H∞(T) = H1,1∞ (T). For the purpose of

this section, it is sufficient to study the properties of the scalar functions U ∈ H∞(T).Formally, the class H∞(T) is the subset of those functions G ∈ H2(T) = H1

2 (T) whichare uniformly bounded on T, i.e. satisfy the condition

supz∈T

|U(z)| < ∞.

The following statement describes the extensions of functions from H∞(T) to D+.

Theorem 4.9. Let U : D+ → C be a function. The following conditions are equivalent:

(a) U is an extension of a function U ∈ H∞(T);

(b) U is analytical, andγ = sup

z∈D+

|U(z)| < ∞. (4.15)


Moreover, if conditions (a),(b) are satisfied then γ equals the essential supremum of |U(z)|over T, i.e. the minimal upper bound of those h ≥ 0 for which the set of z ∈ T satisfying|U(z)| ≥ h has positive Lebesque measure.

Example 4.2. Out of the two functions

G1(z) = exp

(

1− z

1 + z

)

, G2(z) = exp

(

z − 1

z + 1

)

,

the first belongs to class H∞(T) because it has an extension which is analytical and bounded

in the domain |z| > 1. In contrast, function G2 ∈ L2(T) does not belong to any of the classes

H2(T) or H∞(T), because it grows very large as z approaches −1 from the outside of the unit

circle.

Proof of Theorem 4.9, (a)⇒(b). For every s ∈ D+ consider the continuous function

Rs : T → R+ defined by

Rs(z) = 1 +∞∑

k=1

(z

s+

z

s

)

=|s|2 − 1

|s− z|2 .

By construction,

U(s) =

∫

T

Rs(z)U(z)dm(z) ∀ s ∈ D+. (4.16)

Since Rs ≥ 0 and∫

T

Rs(z)dm(z) = 1,

identity (4.17) implies

|U(s)| =∣

∣

∣

∣

∫

T

Rs(z)U(z)dm(z)

∣

∣

∣

∣

≤∫

T

Rs(z)|U(z)|dm(z) ≤ γ

for all s ∈ D+.

Proof of Theorem 4.9, (a)⇒(b). For every s ∈ D+ consider the continuous function

Rs : T → R+ defined by

Rs(z) = 1 +∞∑

k=1

(z

s+

z

s

)

=|s|2 − 1

|s− z|2 .

By construction,

U(s) =

∫

T

Rs(z)U(z)dm(z) ∀ s ∈ D+. (4.17)

Since Rs ≥ 0 and∫

T

Rs(z)dm(z) = 1,


identity (4.17) implies

|U(s)| =∣

∣

∣

∣

∫

T

Rs(z)U(z)dm(z)

∣

∣

∣

∣

≤∫

T

Rs(z)|U(z)|dm(z) ≤ γ

for all s ∈ D+.

Proof of Theorem 4.9, (a)⇒(b). By Theorem 4.8, U is an extension of an H2(T) functionV , and |Ur − V | → 0 as r → 1, where Ur(z) = U(rz). It is sufficient to show that the essentialsupremum of V is not larger than γ. If this is not true, i.e. for some γ1 > γ the inequality|V (z)| ≥ γ1 holds on a subset Z ⊂ T of measure δ > 0, then

|Ur − V |2 =∫

T

|Ur(z)− V (z)|2dm(z) ≥∫

Z

|Ur(z)− V (z)|2dm(z) ≥ δ(γ1 − γ)2 > 0,

which contradicts the convergence |Ur − V | → 0 as r → 1.

4.2.4 Class Hm2 (jR)

Formally speaking, Hm2 (jR) (or simply H2(jR) when m = 1) is the subset of Lm

2 (jR)consisting of those functions U for which an inverse Fourier transform u(t) equals zerofor t < 0 (which is equivalent to every inverse Fourier transform u of U satisfying thecondition u(t) = 0 for almost all t < 0). In other words, Hm

2 (jR) is the set of Fouriertransforms of CT signals of finite energy.

Most significantly, for an arbitrary U ∈ Hm2 (jR) the ”Laplace transform” integral

U(s) =

∫ ∞

−∞

u(t)e−stdt =∞∑

t=0

u(t)e−stdt

converges absolutely on the open set

C+ = s ∈ C : Res > 0.

Allowing some abuse of notation, we will denote the resulting analytical function on C+

by the same letter, i.e. as U : C+ → C

m.The following theorem shows that, combined with analiticity, boundedness of the

energies |Uσ|2 for Uσ(jω) = U(jω + σ) over σ > 0 yields a complete description of”extensions” of functions from class Hm

2 (jR) to C+.

Theorem 4.10. Let U : C+ → C

m be a function. The following conditions are equivalent:

(a) U is an extension of a function U ∈ Hm2 (jR);


(b) U is analytical, and

supσ>0

∫ ∞

−∞

|U(jω + σ)|2dω < ∞. (4.18)

Moreover, when conditions (a),(b) are satisfied, the functions Uσ : jR → Cm defined for

σ > 0 by Uσ(jω) = U(σ + jω), converge to U , in the sense that |U − Uσ| → 0 as σ → 0.

One useful corollary of Theorem 4.10 is a criterion for a function U from class Hm2 (jR)

to be a Fourier transform of u ∈ Lm2 (R) such that u(t) = 0 for all t < T , where T > 0.

Since this is equivalent to V (s) = exp(Ts)U(s) being an extension of a function from classH2, the necessary and sufficient condition is

supσ>0

e2σT∫ ∞

−∞

|U(σ + jω)|2dω < ∞. (4.19)

Example 4.3. The function defined for Res > 0 by U(s) = log(s)/(s + 1) (where thevalues of the logarithm with imaginary part from (−π, π) are used) is an extension of anH2(jR) class function.

4.2.5 Class Hm,k∞ (jR)

Functions from class Hm,k∞ (jR) will serve as transfer matrices of causal L2 gain stable LTI

models (with m inputs and k outputs) in continuous time. Each element G ∈ Hm,k∞ (jR)

can be viewed as a k-by-m matrix with entries from H∞(jR) = H1,1∞ (jR). For the purpose

of this section, it is sufficient to study the properties of the scalar functions U ∈ H∞(T).Formally, the classH∞(jR) can be defined as the subset of those functionsG : jR → C

which are uniformly bounded on jR, and can be represented in the form

G(jω) = (1 + jω)G(jω) (4.20)

for some G ∈ H2(jR).Extending H2 class function G from (4.20) into the right half plane C

+ defines anatural extension for G ∈ H∞(jR), via G(s) = (1 + s)G(s). The following statementdescribes the extensions of functions from H∞(jR) to C

+.

Theorem 4.11. Let U : C+ → C be a function. The following conditions are equivalent:

(a) U is an extension of a function U ∈ H∞(jR);

(b) U is analytical, andγ = sup

s∈C+

|U(s)| < ∞. (4.21)


Moreover, if conditions (a),(b) are satisfied then γ equals the essential supremum of |U(s)|over s ∈ jR, i.e. the minimal upper bound of those h ≥ 0 for which the set of ω ∈ R

satisfying |U(jω)| ≥ h has positive Lebesque measure.

4.3 L2 Gain Stable Transfer Matrix Models

Defining LTI system models in terms of their transfer functions is supposed to be straight-forward: apply Fourier transform to the input, multiply the result by the transfer function,and then apply inverse Fourier transform to the product. There are several importantquestions to resolve, though:

(a) What are the boundary conditions for the resulting system?

(b) What to do when the signals involved are not square integrable

(c) Which conditions guarantee that the resulting system is well defined (e.g. the inverseFourier transform involved does exist)?

In this section we use functions from Hardy classes H∞ as transfer matrices L2 gain stabletransfer matrix models.

The following shortcut notation (applicable to signals of arbitrary dimension d), whereX = Z, Y = T, and Z = ℓ for discrete time systems, or X = R, Y = jR, and Z = L forCT systems) will be used in this section:

• F : Fourier transform F : Ld2(X) → Ld

2(Y );

• F−1: inverse Fourier transform F−1 : Ld2(Y ) → Ld

2(X);

• PT : ”past projection” PT : Zd ∪ Ld2(X) → Ld

2(X),

PTf = g means g(t) =

f(t), 0 ≤ t ≤ T,

0, otherwise;

• P+: ”future projection” P+ : Ld2(X) → Ld

2(X),

P+f = g means g(t) =

f(t), t ≥ 0,

0, t < 0;


4.3.1 Real Symmetry

So far, we have applied Fourier transforms to general complex-valued functions and se-quences. Since we use real-valued functions to describe actual ”real world” signals, adescription of Fourier transforms of real-valued functions will be useful.

If U ∈ Lm2 (T) is a Fourier transform of u ∈ Lm

2 (Z) then the function Uc ∈ Lm2 (T)

defined by Uc(z) = U(z) is a Fourier transform of uc ∈ Lm2 (Z) defined by uc(t) = u(t).

Since Uc(z) = U(z) for almost all z ∈ T if and only if uc(t) = u(t) for all t ∈ Z, inverseFourier transform of a function U ∈ Lm

2 (T) is real-valued if and only if U(z) = U(z)for almost all z ∈ T. Similarly, inverse Fourier transform of a function U ∈ Lm

2 (jR) isreal-valued if and only if U(−jω) = U(jω) for almost all ω ∈ R.

This observation suggests that transfer matrices of LTI systems should also have thereal symmetry property, which leads us to the definition of class Hk,m

∞|Re(X) (where X = T

or X = jR) as the subset of those functions G from Hk,m∞ (X) (i.e. k-by-m matrices with

entries from the class H∞(X)) which satisfy the real symmetry condition G(z) = G(z).Recall that a rational function satisfies the real symmetry condition if and only if it hasreal coefficients. Naturally, one expects transfer matrices of ”physically implementable”systems to have real symmetry.

4.3.2 Discrete Time Stable Transfer Matrix Models

We will use the following preliminary observation.

Theorem 4.12. For every G ∈ Hk,m

∞|Re(T) and w ∈ ℓm there exists a unique y

def= LGw ∈ ℓk

such thatPTy = PTF−1GFPTw ∀ T ≥ 0. (4.22)

Moreover, if w(t) = 0 for t ≤ τ then y(t) = 0 for t ≤ τ .

The function LG defined in Theorem 4.12 generates the ”zero-state” response of thetransfer matrix model defined by G. Its definition follows the familiar ”Fourier tranform,multiplication, inverse Fourier transform” pattern, with projections PT inserted to ensurelegality of the associated operations.

Also, for G ∈ Hk,m

∞|Re(T), let XG denote the subset of sequences x0 ∈ Lk

2(Z) which can

be approximated arbitrarily well (in the sense of |x0 − y0| → 0 by sequences y0 of theform y0 = P+F−1GFw0 (i.e. those obtained by taking w0 ∈ Lm

2 (Z), applying Fouriertransform, multiplying the result by G, applying inverse Fourier transform, and thenzeroing out the samples with t < 0), where w0 ∈ Lm

2 (Z) is such that w0(t) = 0 for t ≥ 0,and w0(t) ∈ R

m for t < 0. Due to the real symmetry assumption, all sequences in XG


are real. The set XG will be used to represent the ”boundary conditions” or ”zero inputresponse” of the LTI system associated with G.

Finally, the transfer matrix model STMDT (G) defined by G ∈ Hk,m

∞|Re(T) is the discrete

time system STMDT (G) ∈ Sm,k

DT (XG), such that

STMDT (G)(w, x0) = x0 + LGw ∀ w ∈ ℓm, x0 ∈ XG. (4.23)

Example 4.4. Let G : T → C be defined by G(z) = 1/z. Since G is the Fourier transform ofreal-valued signal

g(t) =

1, t = 1,

0, t 6= 1,

we conclude that G ∈ H∞|Re(T). What kind of LTI system is STMDT (G)?

Let us begin by studying time-domain implications of the Fourier transform relation Y (z) =G(z)W (z), where Y,W ∈ L2(T). Since Y (z) = G(z)W (z) means zY (z) = W (z), the corre-sponding inverse Fourier transforms y, w satisfy the difference equation y(t + 1) = w(t) for allt ∈ Z.

The original observation proves that y = LGw means

y(t) =

w(t− 1), t > 0,

0, t = 0.

Similarly, sequences of the form y0 = P+F−1GFw0 are such that

y0(t) =

w0(t− 1), t ≥ 0,

0, t < 0,

where w0(t) = 0 for t ≥ 0, which means that y0(t) = 0 for all t except at t = 0, while y0(0) canbe arbitrary. Therefore the set XG consists of all sequences x0 : Z → R such that x0(t) = 0 forall t 6= 0. While, formally speaking, XG is a set of sequences, its elements can be parametrizedlinearly by a single real parameter!

Finally, combining these descriptions of LG and XG, we conclude that the set of input-output

pairs (w, y) in system STMDT (G) is defined by equation y(t+1) = w(t), where the initial value y(0)

can be an arbitrary real number, and is determined by the boundary condition vector x0 ∈ XG.

No surprises here!

Proof of Theorem 4.12 To simplify notation and terminology, the proof is given for thecase m = k = 1.

For every u ∈ ℓ and T > 0 the Fourier transform UT of uT = PTu is real symmetric, andhas an analytical extension UT : D

+ → C such that

limr→1

supz∈T

|UT (rz)− UT (z)| = 0.


As an element of H∞(T), function G also has an analytical extension G : D+ → C which is

uniformly bounded, and such that

limr→1

∫

T

|G(rz)−G(z)|2dm(z) = 0.

Hence the function YT : D+ → C defined by YT (z) = G(z)UT (z) is an analytical extension of

YT = GUT , and

∫

T

|YT (z)− YT (rz)|2dm(z) =

∫

T

|G(z)UT (z)− G(rz)UT (rz)|2dm(z)

= 2

∫

T

|G(z)(UT (z)− UT (rz))|2 + |(G(z)− G(rz))UT (rz)|2dm(z)

≤ 2maxz∈T

|UT (z)− UT (rz)|2∫

T

|G(z)|2dm(z)

+2maxz∈T

|G(z)|2∫

T

|G(z)− G(rz)|2dm(z) → 0

as r → 1. Hence YT ∈ H2(T) has inverse Fourier transform yT . To complete the proof, it issufficient to show that for every τ > T the equality yT (t) = yτ (t) holds for all t ≤ T . Indeed,since uT (t) = uτ (t) for t ≤ T we have

supr>1

r2T∫

T

|UT (rz)− Uτ (rz)|2dm(z) < ∞.

Since G(z) is uniformly bounded in D+, this implies

supr>1

r2T∫

T

|YT (rz)− Yτ (rz)|2dm(z) = supr>1

r2T∫

T

|G(rz)(UT (rz)− Uτ (rz))|2dm(z) < ∞,

and hence yT (t) = yτ (t) for t ≤ T .

.

4.3.3 Continuous Time Stable Transfer Matrix Models

The development follows the DT case closely. We start by defining zero state response.

Theorem 4.13. For every G ∈ Hk,m

∞|Re(jR) and w ∈ Lm there exists a unique y

def= LGw ∈

Lk such thatPTy = PTF−1GFPTw ∀ T ≥ 0. (4.24)

Moreover, if w(t) = 0 for t ≤ τ then y(t) = 0 for t ≤ τ .


For G ∈ Hk,m

∞|Re(R), let XG denote the subset of sequences x0 ∈ Lk

2(R) which can

be approximated arbitrarily well (in the sense of |x0 − y0| → 0 by sequences y0 of theform y0 = P+F−1GFw0 (i.e. those obtained by taking w0 ∈ Lm

2 (R), applying Fouriertransform, multiplying the result by G, applying inverse Fourier transform, and thenzeroing out the samples with t < 0), where w0 ∈ Lm

2 (Z) is such that w0(t) = 0 for t ≥ 0,and w0(t) ∈ R

m for t < 0. Due to the real symmetry assumption, all sequences in XG arereal.

Finally, the transfer matrix model STMCT (G) defined by G ∈ Hk,m

∞|Re(R) is the continuous

time system STMCT (G) ∈ Sm,k

CT (XG), such that

STMCT (G)(w, x0) = x0 + LGw ∀ w ∈ ℓm, x0 ∈ XG. (4.25)

Example 4.5. Let G : jR → C be defined by G(s) = e−τs, where τ > 0 is a parameter. SinceG is uniformly bounded on jR, and G(s)/(s+ 1), as the Fourier transform of

g(t) =

0, t < τ,

eτ−t, t ≥ τ,

is from class H2(jR), we conclude that G ∈ H∞(jR). Moreover, since G has real symmetry,G ∈ H∞|Re(jR). What kind of LTI system is STM

CT (G)?We begin by studying time-domain implications of the Fourier transform relation Y (jω) =

G(jω)W (jω), where Y,W ∈ L2(jR): y(t) = w(t− τ) for all t ∈ R.The original observation proves that y = LGw means

y(t) =

w(t− τ), t ≥ τ,

0, t < τ.

Similarly, functions of the form y0 = P+F−1GFw0 are such that

y0(t) =

w0(t− τ), t ≥ 0,

0, t < 0,

where w0(t) = 0 for t ≥ 0, which means that y0(t) = 0 for all t ≥ 0, and the restriction of y0 tothe interval (0, τ) can be arbitrary square integrable function. Therefore the set XG consists ofall square integrable functions x0 : R → R such that x0(t) = 0 for all t ≥ τ . Note that the setof ”boundary conditios” does not have a finite basis: this is what we expect when the transferfunction is not rational.

Finally, combining these descriptions of LG and XG, we conclude that the set of input-output

pairs (w, y) in system STMCT (G) is defined by equation y(t+ τ) = w(t), where the value y(t) for

t < τ can be an arbitrary, as long as they are real, and the resulting function is square integrable

over (0, τ).

Chapter 4 Transfer Function...

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