INPUT-OUTPUT APPROACH

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(2)

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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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

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EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

The operator H is: • causal

• biased

• G is the unbiased operator associated with H

(9)

Let

Is H bounded?

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

(10)

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

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Let

Is H bounded? •

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

(12)

Let

Is H bounded? •

G bounded  H bounded

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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Let

Is G bounded?

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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Let

Is G bounded?

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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Let

Is G bounded?

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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Let

Is G bounded?

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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Let

Is G bounded?

 G is bounded

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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Let

H è limitato? •

• G is bounded

 H is bounded

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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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

(20)

Let •

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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Let •

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

 G bounded operator in with zero bias gain

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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Let

Let and define

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

� 0 𝑡𝑡 𝑔𝑔 𝑡𝑡 − 𝜏𝜏 𝑢𝑢_𝑚𝑚 𝜏𝜏 𝑑𝑑𝜏𝜏 = � 0 min{𝑡𝑡,𝑚𝑚} 𝑔𝑔 𝑡𝑡 − 𝜏𝜏 𝑢𝑢_𝑚𝑚 𝜏𝜏 𝑑𝑑𝜏𝜏 Let

Let and define

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Let

Let and define

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

� 0 min{𝑡𝑡,𝑚𝑚} 𝑔𝑔 𝑡𝑡 − 𝜏𝜏 𝑢𝑢_𝑚𝑚 𝜏𝜏 𝑑𝑑𝜏𝜏 ≤ � 0 𝑡𝑡 𝑔𝑔 𝑡𝑡 − 𝜏𝜏 𝑑𝑑𝜏𝜏,𝑡𝑡 < 𝑚𝑚 = � 0 𝑚𝑚 |𝑔𝑔 𝑡𝑡 − 𝜏𝜏 |𝑑𝑑𝜏𝜏, 𝑡𝑡 ≥ 𝑚𝑚 � 0 𝑡𝑡 𝑔𝑔 𝑡𝑡 − 𝜏𝜏 𝑢𝑢_𝑚𝑚 𝜏𝜏 𝑑𝑑𝜏𝜏 =

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Let

Let and define

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

= � 0

𝑚𝑚

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Let

Let and define

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

= � 0

𝑚𝑚

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Let

H (bounded) is weakly bounded because

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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Let

We shall show that

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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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

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AFFINE CAUSAL OPERATOR

H

u y

affine causal operator

Theorem

A weakly bounded affine causal operator • is bounded

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AFFINE CAUSAL OPERATOR

H

u y

Theorem

• its gain is equal to its zero bias gain Proof:

H weakly bounded causal  such that

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AFFINE CAUSAL OPERATOR

H

u y

Theorem

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AFFINE CAUSAL OPERATOR

H

u y

Theorem

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AFFINE CAUSAL OPERATOR

H

u y

Theorem

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AFFINE CAUSAL OPERATOR

H

u y

Theorem

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AFFINE CAUSAL OPERATOR

H

u y

Theorem

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AFFINE CAUSAL OPERATOR

H

u y

Theorem

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AFFINE CAUSAL OPERATOR

H

u y

Theorem

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AFFINE CAUSAL OPERATOR

H

u y

Theorem

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AFFINE CAUSAL OPERATOR

H

u y

Theorem

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AFFINE CAUSAL OPERATOR

H

u y

Theorem

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Let

since H is affine

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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Let

Is H bounded?

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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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

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Let

Is G bounded?

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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Let

Is G bounded?

By Parseval theorem:

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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Let

Is G bounded?

By Parseval theorem:

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

since it is a continuous function that tends to zero as ω  ∞

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Let

Is G bounded?

from which we get

 G is bounded One can show that

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

(49)

Let

In conclusion:

H is bounded and weakly bounded with gain

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE

DYNAMICAL SYSTEM

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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 L_pe, for any p