INPUT-OUTPUT APPROACH

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

impulse response of S

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

impulse response of S

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

The operator H is:

• causal

• biased

• G is the unbiased operator associated with H

G is a linear operator

induced norm of G

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

The operator H is:

• causal

• biased

• G is the unbiased operator associated with H

induced norm of G

Let

Is H bounded?

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

WEAKLY BOUNDED/BOUNDED OPERATOR

H u y

causal

Definition (weakly bounded operator):

A causal operator is weakly bounded (or with finite gain)

if

Definition (bounded operator):

A causal operator is bounded if

•

• is bounded

Remark: H bounded H weakly bounded

Let

Is H bounded?

•

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

free evolution of S

Let

Is H bounded?

•

G bounded H bounded

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

free evolution of S

Let

Is G bounded?

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

Let

Is G bounded?

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

Let

Is G bounded?

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

Let

Is G bounded?

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

impulse response of S

Let

Is G bounded?

G is bounded

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

impulse response of S

Let

H è limitato?

•

• G is bounded

H is bounded

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

free evolution of S

Let

H è limitato?

•

• G is bounded

H is bounded

Let us compute the zero bias gain of G

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

free evolution of S

Let

•

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

Let

•

• If we find a sequence of inputs um, with such that

G bounded operator in with zero bias gain

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

Let

Let and define

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

Let

Let and define

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

Let

Let and define

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

𝑔 𝑡 − 𝜏 = 0, 𝜏 > 𝑡 𝑢𝑚 𝜏 = 0, 𝜏 > 𝑚

Let

Let and define

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

𝑔 𝑡 − 𝜏 = 0, 𝜏 > 𝑡 𝑢𝑚 𝜏 = 0, 𝜏 > 𝑚

Let

H (bounded) is weakly bounded because

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

Let

H (bounded) is weakly bounded because

We shall show that

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

AFFINE CAUSAL OPERATOR

H u y

causal operator

Definition (affine operator):

The causal operator is affine if the associated unbiased

operator G

Is linear.

G

y0

u y

AFFINE CAUSAL OPERATOR

H u y

affine causal operator

Theorem

A weakly bounded affine causal operator

• is bounded

• its gain is equal to its zero bias gain

AFFINE CAUSAL OPERATOR

H u y

affine causal operator

Theorem

A weakly bounded affine causal operator

• is bounded

• its gain is equal to its zero bias gain

Proof:

H weakly bounded causal such that

(first condition for H to be bounded)

AFFINE CAUSAL OPERATOR

H u y

affine causal operator

Theorem

A weakly bounded affine causal operator

• is bounded

• its gain is equal to its zero bias gain

Proof:

For any

&

AFFINE CAUSAL OPERATOR

H u y

affine causal operator

Theorem

A weakly bounded affine causal operator

• is bounded

• its gain is equal to its zero bias gain

Proof:

AFFINE CAUSAL OPERATOR

H u y

affine causal operator

Theorem

A weakly bounded affine causal operator

• is bounded

• its gain is equal to its zero bias gain

Proof:

AFFINE CAUSAL OPERATOR

H u y

affine causal operator

Theorem

A weakly bounded affine causal operator

• is bounded

• its gain is equal to its zero bias gain

Proof:

AFFINE CAUSAL OPERATOR

H u y

affine causal operator

Theorem

A weakly bounded affine causal operator

• is bounded

• its gain is equal to its zero bias gain

Proof:

AFFINE CAUSAL OPERATOR

H u y

affine causal operator

Theorem

A weakly bounded affine causal operator

• is bounded

• its gain is equal to its zero bias gain

Proof:

G bounded (second condition for H to be bounded)

AFFINE CAUSAL OPERATOR

H u y

affine causal operator

Theorem

A weakly bounded affine causal operator

• is bounded

• its gain is equal to its zero bias gain

Proof:

G bounded (second condition for H to be bounded) H bounded

AFFINE CAUSAL OPERATOR

H u y

affine causal operator

Theorem

A weakly bounded affine causal operator

• is bounded

• its gain is equal to its zero bias gain

Proof:

G bounded ( H bounded) with

AFFINE CAUSAL OPERATOR

H u y

affine causal operator

Theorem

A weakly bounded affine causal operator

• is bounded

• its gain is equal to its zero bias gain

Proof:

AFFINE CAUSAL OPERATOR

H u y

affine causal operator

Theorem

A weakly bounded affine causal operator

• is bounded

• its gain is equal to its zero bias gain

Proof:

Let

H (bounded) is weakly bounded because

since H is affine

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

Let

Is H bounded?

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

Let

Is H bounded?

•

If G bounded

H bounded (and, hence, weakly bounded) and, since H is affine,

its gain is equal to the zero bias gain, i.e.,

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

free evolution of S

Let

Is G bounded?

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

Fourier transform of the impulse response:

Let

Is G bounded?

By Parseval theorem:

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

Let

Is G bounded?

By Parseval theorem:

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

since it is a continuous function

that tends to zero as

Let

Is G bounded?

from which we get

G is bounded

One can show that

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

Let

In conclusion:

H is bounded and weakly bounded with gain

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

norm of F(s)

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

One can show that the H operator is bounded (and, hence, weakly

bounded, with gain equal to the zero bias gain) in Lpe, for any p