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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