Data-Rate Constrained Observers of Nonlinear Systems
Abstract
:1. Introduction
2. Problem Statement
2.1. Observed Dynamical System
- T is the set of time periods, which is either or ;
- is the evolution function that gives the system state at time , provided that the initial state is ;
- is the focus of our interest in the system.
2.2. Architecture of the Observer, Notations, and General Traits of the Communication Channel
- The sampler and quantizer are built at the measurement site and have access to the dynamics of the system, the set , the current state , and the desired exactness of observation .
- The decoder is built at the remote site and has access to the system dynamics , the set , the desired exactness of observation , and the messages transmitted across the channel.
- (c.1)
- The channel correctly transfers any message to the receiving end provided that the message processing time and the size of the message are in balance:Here, is a channel-dependent function that gives the number of bits processable by the channel during any time period of length .
- (c.2)
- As the processing time increases to infinity, the average number of bits transmittable per unit of time stabilizes and converges to a certain value , called the (bit-rate) channel capacity:
- (c.3)
- The channel is closed for the next message until all bits of the current message have been processed, but is open afterwards.
- (c.4)
- On its way to the destination point , any message incurs a transmission delay :
- (c.5)
- The transmission delays are upper-bounded: .
2.3. Observability via Channels with Limited Bit-Rate Capacity
3. Design of the Proposed Observer
- (e.1)
- —The period between consecutive dispatches of messages via the channel;
- (e.2)
- P—A symmetric and positive definite -matrix;
- (e.3)
- —A function of and for which
- (e.4)
- —A finite covering of the compact set with balls (with respect to the norm ) centered in and with a radius of each.
- (o.1)
- (o.2)
- The quantizer finds an element of the covering from (e.4) that contains and sends its index k over the channel:
- (o.3)
- The decoder performs the following operations at time :
- -
- Extracts the index k from the last message received at a time , where (If no message has been received yet, k is assigned an arbitrarily pre-specified value, e.g., 1.).
- -
- By using the centers from (e.4), forms the current state estimate
- In (o.2), we do not address the case due to the reason stated at the onset of the section.
- The proposed design assumes that both the coder and decoder have access to from (e.1) and the covering from (e.4).
- The observer uses a fixed alphabet , which is shared by the coder and the decoder.
- The quantizer sends data about the estimate of not the current , but the forward-time state , which is computed from the measured by using the known transition map .
- The idea behind this relies on the expectation that these data will be received prior to and put in use at due time, . Then, the exactness of estimation will be at this time.
- These data are also used to estimate the state on the subsequent time interval via applying the matching transition map to the just-discussed estimate at time . By Equation (10), this guarantees the exactness of estimation on this interval.
- (i)
- Let the proposed observer correctly operate for a given and . Then, for any trajectory satisfying Equation (1), the desired exactness of observation ϵ is ensured with respect to the norm ;
- (ii)
- Let a communication channel be given. Also, let any small enough be coupled with some so that the proposed observer with these ϵ and operates correctly via the channel at hand. Then, the system is observable on the set via this communication channel.
4. Criteria for Observability of the System
4.1. The Size of Finite Covering
- (i)
- The smallest number of closed balls of radius δ that cover F;
- (ii)
- The smallest number of closed balls of radius δ and centers in F that cover F;
- (iii)
- The smallest number of cubes of side δ that cover F;
- (iv)
- The number of δ-mesh cubes that intersect F;
- (v)
- The smallest number of sets of diameter at most δ that cover F;
- (vi)
- The largest number of disjoint balls of radius δ with centers in F.
4.2. Balance between the Initial and Forthcoming Estimation Exactness, Respectively
4.3. Correct Operation of the Observer and a Criterion for Observability
5. Constructive Estimates and Analytical Bounds
5.1. Lyapunov-Like Function
5.2. Analytical Upper Bound on the System’s Growth Rate and Related Conditions for Observability
5.3. Analytical Bounds on the Upper Box Dimension and Final Conditions for Observability
6. Examples
6.1. The Smoothened Lozi Map
6.2. The Lorenz System
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
: | the set of nonnegative integers; |
: | the set of nonnegative real numbers; |
: | the absolute value of x; |
: | the cardinality of a set F; |
=; | |
: | the identity matrix of dimension ; |
: | the conjugate transpose of the matrix M; |
: | the inverse of the transpose of the matrix M; |
= ; | |
= , with P symmetric and positive definite; | |
: | the ball in of radius centered in x; |
: | the ball in of radius centered in x. |
Appendix A. Proofs of Section 4
Appendix A.1. Proof of Lemma 3
Appendix A.2. Proof of Proposition 2
Appendix B. Proofs of Section 5
Appendix B.1. Proof of Proposition 3
Appendix B.2. Proof of Proposition 4
Appendix B.3. Proof of Proposition 5
Appendix C. Proofs of Section 6
Proof of Theorem 4
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K (1) | ||||
(bits/s) | 1.0924 | 1.1499 | 1.169431 | 1.212770 |
K (1) | ||||
(bits/s) | 19.814 | 30.739 | 34.614 | 43.714 |
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Voortman, Q.; Pogromsky, A.Y.; Matveev, A.S.; Nijmeijer, H. Data-Rate Constrained Observers of Nonlinear Systems. Entropy 2019, 21, 282. https://doi.org/10.3390/e21030282
Voortman Q, Pogromsky AY, Matveev AS, Nijmeijer H. Data-Rate Constrained Observers of Nonlinear Systems. Entropy. 2019; 21(3):282. https://doi.org/10.3390/e21030282
Chicago/Turabian StyleVoortman, Quentin, Alexander Yu. Pogromsky, Alexey S. Matveev, and Henk Nijmeijer. 2019. "Data-Rate Constrained Observers of Nonlinear Systems" Entropy 21, no. 3: 282. https://doi.org/10.3390/e21030282
APA StyleVoortman, Q., Pogromsky, A. Y., Matveev, A. S., & Nijmeijer, H. (2019). Data-Rate Constrained Observers of Nonlinear Systems. Entropy, 21(3), 282. https://doi.org/10.3390/e21030282