An On-Line Tracker for a Stochastic Chaotic System Using Observer/Kalman Filter Identification Combined with Digital Redesign Method
Abstract
:1. Introduction
2. OKID Formulation
2.1. Basic Observer Equation
2.2. Computation of Markov Parameters
2.2.1. System Markov Parameters
2.2.2. Observer Gain Markov Parameters
3. Digital Redesign of Full-Order Observer
4. Effective Design Procedures for a Stochastic Chaotic System
- Step 1.
- Perform the off-line system identification scheme to obtain both system and observer-gain Markov parameters of the OKID model.
- (i)
- Compute the observer Markov parameters. Choose a value of that determines the number of observer Markov parameters from the given set of input-output data, and then compute the least-squares solution of the Markov parameter matrix in Equation (10).
- (ii)
- Identify both system and observer-gain Markov parameters. Use the Markov parameters identified in (i) above, and Equations (14) and (18) to determine the combined system and observer-gain Markov parameters. Moreover, set up the Hankel matrix and as shown in Equation (19).
- (iii)
- Realize a state-space model of the system and the corresponding observer gain from the identified sequence of the system and observer-gain Markov parameters by using the ERA method to obtain the desired discrete system realization , and in Equations (21)–(23), where the non-causal term is assumed to be zero.
- Step 2.
- Set the full-order observer-based sampled-data system for the OKID combined with a digital redesign method.
- (i)
- Find the optimal observer gain . Select appropriate weighting matrices in Equation (34).
- (ii)
- By using the new digitally redesigned observer form, obtain the desired discrete system realization , and in Equation (38).
5. An Illustrative Example
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Method | The Error Range of the Outputs Values | ||
---|---|---|---|
OKID | ±0.3 | ±0.5 | ±0.02 |
OKID combined digital redesign | ±0.03 | ±0.05 | ±0.002 |
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Chien, T.-H.; Chen, Y.-C. An On-Line Tracker for a Stochastic Chaotic System Using Observer/Kalman Filter Identification Combined with Digital Redesign Method. Algorithms 2017, 10, 25. https://doi.org/10.3390/a10010025
Chien T-H, Chen Y-C. An On-Line Tracker for a Stochastic Chaotic System Using Observer/Kalman Filter Identification Combined with Digital Redesign Method. Algorithms. 2017; 10(1):25. https://doi.org/10.3390/a10010025
Chicago/Turabian StyleChien, Tseng-Hsu, and Yeong-Chin Chen. 2017. "An On-Line Tracker for a Stochastic Chaotic System Using Observer/Kalman Filter Identification Combined with Digital Redesign Method" Algorithms 10, no. 1: 25. https://doi.org/10.3390/a10010025