Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Information Theory for Control Systems: Causality
and Feedback
Charalambos D. Charalambous
Department of Electrical and Computer Engineering
University of Cyprus
E-mail: [email protected]
Workshop on Communication Networks and Complexity
Athens, Greece, 2006
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Outline
1
Introduction
2
Information Theory for Causal Systems
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
3
Control of Dynamic Systems over Finite Capacity Channels
Asymptotic Stability and Observability
4
Example: Necessary and Sufficient Conditions
Asymptotic Stability and Observability
5
Conclusions
6
References
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Outline
1
Introduction
2
Information Theory for Causal Systems
3
Control of Dynamic Systems over Finite Capacity Channels
4
Example: Necessary and Sufficient Conditions
5
Conclusions
6
References
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Introduction I
Yt
Control/Communication
System
Information Theory for
Causal Control Systems
Channel Capacity with
Feedback (Causal)
Rate Distortion (Causal)
Plant
(Information
Source)
Encoder
Zt
Discrete Time
Channel
Ut
~
Zt
Feedback Control
Observability
Stability
Controller
Decoder
~
Yt
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Introduction II
Objective
Design Encoders, Decoders, Controllers to Achieve Control and
Communication Objectives
Trade Offs
What are the Trade Offs Between Control and Communication
Design Objectives?
Separation Principle
Does Separation Hold Between Communication and Control
System Design?
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Introduction III
Structure of Presentation
Information Theory for Causal Systems
Control of Stochastic Systems over Limited Capacity Channels
Partially Observed Linear Stochastic Control Systems
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Outline
1
Introduction
2
Information Theory for Causal Systems
3
Control of Dynamic Systems over Finite Capacity Channels
4
Example: Necessary and Sufficient Conditions
5
Conclusions
6
References
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Definition of Subsystems I
Information Source
The information source is identified by the joint probability
4
distribution P(dY T ), Y T = (Y0 , . . . , Y T −1 )
Communication Channel
The communication channel is modeled by a feedback channel
with memory via the family of stochastic kernels
t
t
{P(d Z̃t ; z t , z̃ t−1 )}T
t=0 , t, T ∈ N+ , where Z = z is the specific
realization of the channel input, and Z̃ t−1 = z̃ t−1 is a specific
realization of the previous channel outputs.
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Definition of Subsystems II
C
Causality Y T → Z̃ N .
Given any two sequences Y T and Z̃ N , T , N ∈ N+ we shall say that
the stochastic kernel connecting Y T to Z̃ N is causal if and only if
P(d Z̃t ; y n , z̃ t−1 ) = P(d Z̃t ; y t , z̃ t−1 ), ∀n > t, n, t ∈ N+ .
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Definition of Subsystems III
Causal Channel, Feedback, Compression
C
Z T → Z̃ T is equivalent to
P(d Z̃t ; z n , z̃ t−1 ) = P(d Z̃t ; z t , z̃ t−1 ), ∀n > t, where
t, n ∈ N+ , which means the communication channel is
non-anticipative or causal.
C
Z T ← Z̃ T is equivalent to P(dZt ; z̃ n , z t−1 ) = P(Zt ; z̃ t , z t−1 ),
∀n > t, where t, n ∈ N+ , which means that the
communication channel is used with non-anticipative or causal
feedback.
C
Y T → Ỹ T is equivalent to
P(d Ỹt ; ỹ t−1 , y n ) = P(d Ỹt ; ỹ t−1 , y t ), ∀n > t, where
t, n ∈ N+ , which means that the source reproduction
stochastic kernel is non-anticipative or causal.
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Restricted Mutual Information for Causal Systems I
Shannon’s Self-Mutual Information
Given a communication channel with input Z N−1 and output
Z̃ N−1 , the self-mutual information is defined by
4
i(Z N−1 ; Z̃ N−1 ) = log
= log
P(d Z̃ N−1 ; Z N−1 )
P(d Z̃ N−1 )
P(d Z̃ N−1 , dZ N−1 )
,
P(d Z̃ N−1 ) × P(dZ N−1 )
P(d Z̃ N−1 ;Z N−1 )
is the RND between the stochastic
P(d Z̃ N−1 )
P(d Z̃ N−1 ; Z N−1 ) and distribution P(d Z̃ n−1 ).
Symmetry: i(Z N−1 ; Z̃ N−1 ) = i(Z̃ N−1 ; Z N−1 )
kernel
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Restricted Mutual Information for Causal Systems II
Mutual Information
I (Z N−1 ; Z̃ N−1 ) = EP(d Z̃ N−1 ,dZ N−1 ) i(Z N−1 ; Z̃ N−1 )
= I (Z̃ N−1 ; Z N−1 ).
The Shannon information capacity for the time horizon N is
4
CN =
sup
N−1
P(dZ N−1 )∈DCI
I (Z N−1 ; Z̃ N−1 )
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Restricted Mutual Information for Causal Systems III
Restricted Self-Mutual Information
Given a channel with input Z N−1 and output Z̃ N−1 , the restricted
self-mutual information is defined by
4
iR (Z N−1 ; Z̃ N−1 ) = log
= log
P(d Z̃ N−1 ; Z N−1 )
|R
P(d Z̃ N−1 )
P(d Z̃ N−1 , dZ N−1 )
|R
P(d Z̃ N−1 ) × P(dZ N−1 )
which denotes the logarithm of the restricted RND associated with
C
Z N → Z̃ N .
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Restricted Mutual Information for Causal Systems IV
The Restricted Mutual Information
4
I (Z N−1 ; Z̃ N−1 )|R = EP(d Z̃ N−1 ,dZ N−1 ) iR (Z N−1 ; Z̃ N−1 )
=
N−1
X
i=0
I (Z i ; Z̃i |Z̃ i−1 ) =
N−1
X
i=0
log
P(d Z̃i ; Z i , Z̃ i−1 )
× P(dZ i , d Z̃ i )
i−1
P(d Z̃i ; Z̃ )
Symmetry Fails: I (Z N−1 ; Z̃ N−1 )|R 6= I (Z̃ N−1 ; Z N−1 )|R
Tighter Bound: I (Z N−1 ; Z̃ N−1 )|R ≤ I (Z N−1 ; Z̃ N−1 )
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Restricted Mutual Information for Causal Systems V
Relation to Previous Work: Directed Information
H. Marko, 1973 (IEEE Communication Systems)
J. Massey, 1990 (ISIT)
G. Kramer, 1998 (Ph.D. Thesis)
S. Tatikonda, 2000 (Ph.D. Thesis)
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Causal Rate Distortion I
Shannon’s Rate Distortion
Let Y T −1 and Ỹ T −1 denote the source and reproduction outputs,
respectively, and denote the distortion measure
o
n
T 4
DDC
= P(d Ỹ T −1 ; y T −1 ); E ρT (Y T −1 , Ỹ T −1 ) ≤ Dv
The information rate distortion function for the time horizon T is
defined by
4
RTD (Dv ) =
inf
T
P(d Ỹ T −1 ;y T −1 )∈DDC
I (Y T −1 ; Ỹ T −1 )
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Causal Rate Distortion II
The Minimizing Kernel is Non-Causal
The infimizing reproduction
can be factored
Q −1stochastic Kernel
t−1 , y T −1 ), which is a
as P(d Ỹ T −1 ; y T −1 ) = T
P(d
Ỹ
;
ỹ
t
i=0
non-causal operation on the source data.
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Causal Rate Distortion III
Rate Distortion for Causal Systems
Let Y T −1 and Ỹ T −1 denote the source output and the
reproduction of the source output, respectively, and
T 4
DDC
=
T
−1
n
o
X
T −1 T −1
P(d Ỹ
;y
);
E ρt (Y t , Ỹ t ) ≤ Dv
t=0
The restricted information rate distortion function is defined by
4
RTD (Dv )|R =
inf
T
P(d Ỹ T −1 ;y T −1 )∈DDC
I (Y T −1 ; Ỹ T −1 )|R
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Causal Rate Distortion IV
The Minimizing Kernel is Causal
The infimizing reproduction stochastic Kernel is given by
∗
t
P (d Ỹt ; y , ỹ
t−1
)= R
e sρt (y
Ỹt
t ,ỹ t−1 ,Ỹ )
t
P ∗ (d Ỹt ; ỹ t−1 )
e sρt (y t ,ỹ t−1 ,Ỹt ) P ∗ (d Ỹt ; ỹ t−1 )
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Data Processing Inequalities I
Data Processing Inequalities for Causal Channels
Suppose Y T −1 → Z N−1 → Z̃ N−1 → Ỹ T −1 → U T −1 form a
Markov chain.
C
C
For Z n → Z̃ n , Y t → Ỹ t , t, n ∈ N+ then
I (Z n ; Z̃ n ) ≥ I (Z n ; Z̃ n )|R ≥ I (Y t ; Z̃ n ) ≥ I (Y t ; Ỹ t ) ≥ I (Y t ; Ỹ t )|R
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Definition of Control/Communication Subsystems
Mutual Information for Causal Systems
Causal Rate Distortion
Data Processing Inequalities
Information Transmission Theorem
Information Transmission Theorem I
Information Transmission Theorem
Suppose Y T −1 → Z N−1 → Z̃ N−1 → Ỹ T −1 → U T −1 form a
Markov chain.
C
C
For Z n → Z̃ n , Y t → Ỹ t , t, n ∈ N+
Then
CN |R ≥ RTD (Dv )|R
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Asymptotic Stability and Observability
Outline
1
Introduction
2
Information Theory for Causal Systems
3
Control of Dynamic Systems over Finite Capacity Channels
4
Example: Necessary and Sufficient Conditions
5
Conclusions
6
References
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Asymptotic Stability and Observability
Asymptotic Stability and Observability I
Asymptotic Observability
Consider the general control/communication system.
The system is asymptotically observable in r -mean if there exist a
control sequence
and an encoder and a decoder such that
1 PT −1
limT →∞ T t=0 E ρ(Yt , Ỹt ) ≤ Dv , where ρ(Y , Ỹ ) = ||Y − Ỹ ||,
r > 0 and Dv ≥ 0 is finite.
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Asymptotic Stability and Observability
Asymptotic Stability and Observability II
Asymptotic Stabilizability
Consider the general control/communication system in which,
Yt = Ht + Υt , where Υt , t ∈ N+ is a function of the measurement
noise.
This system is asymptotically stabilizable in r -mean if there exist a
controller, encoder,
and decoder such that
1 PT −1
limT →∞ T t=0 E ρ(Ht , 0) ≤ Dv , where ρ(H, 0) = ||H − 0||r ,
r > 0 and Dv ≥ 0 is finite.
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Asymptotic Stability and Observability
Asymptotic Stability and Observability III
Necessary Conditions for Asymptotic Observability and
Stabilizability
Suppose Y T −1 → Z N−1 → Z̃ N−1 → Ỹ T −1 → U T −1 form a
Markov chain.
C
C
For Z n → Z̃ n , Y t → Ỹ t , t, n ∈ N+ the
Necessary Conditions are
d
C|R ≥ R D (Dv )|R ≥ HS (Y) − log e r + log(
d d
r
(
)r )
d
rD
dVd Γ( r )
v
where HS (Y) is Shannon Entropy Rate of Output Process Y T .
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Asymptotic Stability and Observability
Outline
1
Introduction
2
Information Theory for Causal Systems
3
Control of Dynamic Systems over Finite Capacity Channels
4
Example: Necessary and Sufficient Conditions
5
Conclusions
6
References
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Asymptotic Stability and Observability
Asymptotic Stability and Observability I
Stochastic Control System
Xt+1 = AXt + NUt + BWt , X0 = X ,
Yt = Ht + DVt , Ht = CXt
AGN Communication Channel
Z̃t = Zt + W̃t , 0 ≤ t ≤ T
P −1
2
where with power constraint T1 T
i=0 E ||Zt || ≤ P.
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Asymptotic Stability and Observability
Asymptotic Stability and Observability II
Channel Does Not Assume Feedback
The encoder is an innovations encoder without channel feedback
Zt = Yt − E [Yt |Y t−1 , U t−1 ]
Necessary and Sufficient Condition for Asymptotic Observability
C|R ≥ R D (Dv )|R =
1
1
log Λ∞ − log Dv
2
2
where Λ∞ is the steady state value of the innovations error
covariance
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Asymptotic Stability and Observability
Asymptotic Stability and Observability III
The Sufficiency is obtained by matching the source to the
channel
HS (Y) ≥
1
2
log(2πeD 2 ) + max{0, log |A|}
This encoder only works for stable control systems (the
decoder cannot stabilize the control system
Next, we present one that works!
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Asymptotic Stability and Observability
Asymptotic Stability and Observability IV
AGN Communication Channel With Feedback
Encoder (uses channel feedback)
Zt = A0 (Z̃ t−1 , U t−1 ) + A1 (Z̃ t−1 , U t−1 )Yt
Decoder (Least-Squares Decoder)
Yt|t−1 = E Yt |Z̃ t−1 , U t−1 = CE Xt |Z̃ t−1 , U t−1
Controller (LQG)
Ut = Kt Yt|t−1
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Outline
1
Introduction
2
Information Theory for Causal Systems
3
Control of Dynamic Systems over Finite Capacity Channels
4
Example: Necessary and Sufficient Conditions
5
Conclusions
6
References
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Conclusions I
Causality of Rate Distortion
Separation Principle Holds for Gaussian Control and
Communication Channels
Uncertain Control Systems and Channels (done some work)
Nonlinear Stochastic Control Systems (future work)
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
Outline
1
Introduction
2
Information Theory for Causal Systems
3
Control of Dynamic Systems over Finite Capacity Channels
4
Example: Necessary and Sufficient Conditions
5
Conclusions
6
References
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
References I
C. E. Shannon, The Mathematical Theory of Communication,
Bell Systems Technical Journal, vol. 27, pp. 379–423,
pp. 623–656, 1948.
R. L. Dubrushin, Information Transmission in Channel with
Feedback, Theory of Probability and its Applications, vol. III,
No. 4, pp. 367–383, 1958.
H. Marko, The Bidirectional Communication Theory-A
Generalization of Information Theory, IEEE Transactions on
Communication Systems, vol. 21, pp. 1345–1351, 1973.
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
References II
J. Massey, “Causality, Feedback and Directed Information”, in
the in Proceedings of the 1990 IEEE International Symposium
on Information Theory and its Applications, pp. 303–305,
Nov.27–30, Hawaii, U.S.A.1990.
S. Tatikonda, Control Under Communication Constraint,
Ph.D. Thesis, Department of Electrical Enginnering and
Computer Science, MIT., September 2000.
G. Kramer, Directed Information for Channels with Feedback,
Ph.D. Thesis, Swiss Federal Institute of Technology, Diss. ETH
No. 12656, 1998.
Introduction
Information Theory for Causal Systems
Control of Dynamic Systems over Finite Capacity Channels
Example: Necessary and Sufficient Conditions
Conclusions
References
References III
S. Tatikonda and S. Mitter, Control Over Noisy Channels,
IEEE Transactions on Automatic Control, vol. 49, No. 7,
pp. 1196–1201, 2004.
S. Tatikonda and S. Mitter, Control Under Communication
Constraints, IEEE Transactions on Automatic Control, vol. 49,
No. 7, pp. 1056–1068, 2004.
S. Yang and A. Kavcic and S. Tatikonda, Feedback Capacity
of Finite-State Machine Channels, IEEE Transactions on
Information Theory, vol. 51, No. 3, pp. 799–810, 2005.