hybrid systems course - intranet deibhome.deib.polimi.it/prandini/file/7_input-output.pdf ·...
TRANSCRIPT
![Page 1: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/1.jpg)
INPUT-OUTPUT APPROACH
![Page 2: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/2.jpg)
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 3: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/3.jpg)
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 4: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/4.jpg)
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
impulse response of S
![Page 5: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/5.jpg)
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
impulse response of S
![Page 6: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/6.jpg)
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 7: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/7.jpg)
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
![Page 8: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/8.jpg)
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
The operator H is:
• causal
• biased
• G is the unbiased operator associated with H
induced norm of G
![Page 9: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/9.jpg)
Let
Is H bounded?
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 10: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/10.jpg)
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
![Page 11: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/11.jpg)
Let
Is H bounded?
•
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
free evolution of S
![Page 12: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/12.jpg)
Let
Is H bounded?
•
G bounded H bounded
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
free evolution of S
![Page 13: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/13.jpg)
Let
Is G bounded?
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 14: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/14.jpg)
Let
Is G bounded?
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 15: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/15.jpg)
Let
Is G bounded?
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 16: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/16.jpg)
Let
Is G bounded?
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
impulse response of S
![Page 17: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/17.jpg)
Let
Is G bounded?
G is bounded
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
impulse response of S
![Page 18: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/18.jpg)
Let
H è limitato?
•
• G is bounded
H is bounded
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
free evolution of S
![Page 19: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/19.jpg)
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
![Page 20: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/20.jpg)
Let
•
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 21: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/21.jpg)
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
![Page 22: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/22.jpg)
Let
Let and define
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 23: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/23.jpg)
Let
Let and define
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 24: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/24.jpg)
Let
Let and define
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
𝑔 𝑡 − 𝜏 = 0, 𝜏 > 𝑡 𝑢𝑚 𝜏 = 0, 𝜏 > 𝑚
![Page 25: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/25.jpg)
Let
Let and define
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
𝑔 𝑡 − 𝜏 = 0, 𝜏 > 𝑡 𝑢𝑚 𝜏 = 0, 𝜏 > 𝑚
![Page 26: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/26.jpg)
Let
H (bounded) is weakly bounded because
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 27: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/27.jpg)
Let
H (bounded) is weakly bounded because
We shall show that
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 28: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/28.jpg)
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
![Page 29: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/29.jpg)
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
![Page 30: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/30.jpg)
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)
![Page 31: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/31.jpg)
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
&
![Page 32: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/32.jpg)
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:
![Page 33: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/33.jpg)
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:
![Page 34: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/34.jpg)
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:
![Page 35: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/35.jpg)
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:
![Page 36: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/36.jpg)
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)
![Page 37: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/37.jpg)
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
![Page 38: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/38.jpg)
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
![Page 39: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/39.jpg)
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:
![Page 40: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/40.jpg)
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:
![Page 41: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/41.jpg)
Let
H (bounded) is weakly bounded because
since H is affine
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 42: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/42.jpg)
Let
Is H bounded?
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 43: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/43.jpg)
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
![Page 44: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/44.jpg)
Let
Is G bounded?
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
Fourier transform of the impulse response:
![Page 45: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/45.jpg)
Let
Is G bounded?
By Parseval theorem:
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 46: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/46.jpg)
Let
Is G bounded?
By Parseval theorem:
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
since it is a continuous function
that tends to zero as
![Page 47: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/47.jpg)
Let
Is G bounded?
from which we get
G is bounded
One can show that
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
![Page 48: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/48.jpg)
Let
In conclusion:
H is bounded and weakly bounded with gain
EXAMPLE: LINEAR ASYMPTOTICALLY STABLE
DYNAMICAL SYSTEM
norm of F(s)
![Page 49: Hybrid Systems course - Intranet DEIBhome.deib.polimi.it/prandini/file/7_input-output.pdf · input-output approach. example: linear asymptotically stable dynamical system . example:](https://reader031.vdocuments.site/reader031/viewer/2022022805/5ca2641688c99318568cb28c/html5/thumbnails/49.jpg)
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