A Natural Gradient Algorithm for Stochastic Distribution Systems
Abstract
:1. Introduction
2. Model Description
- (1)
- The inverse function of y = f(u, ω) with respect to ω exists and is denoted by ω = f−1(y, u), which is at least C2 with respect to all variables (y, u).
- (2)
- The output PDF p(y; u) is at least C2 with respect to all variables (y, u).
Definition 1
Definition 2 ([5,6,24])
3. Natural Gradient Algorithm
Lemma 1 ([15])
Proposition 1
Proof
Theorem 1
Proof
- (1)
- Initialize u0.
- (2)
- At the sample time k − 1, formulate ∇J(uk−1) and use Equation (1) to give the inverse of the Fisher metric Gk−1.
- (3)
- (4)
- If J(uk) < δ, where δ is a positive constant, which is determined by the precision needed, escape. Additionally, at the sample time k, the output PDF p(y; uk) is the final one. If not, turn to Step 5.
- (5)
- Increase k by one and go back to Step 2.
4. Convergence of the Algorithm
Lemma 2
Proof
Lemma 3
Proof
Theorem 2
Proof
5. Simulations
6. Conclusions
- (1)
- By the statistical characterizations of the stochastic distribution control systems, we formulate the controller design in the frame of information geometry. By virtue of the natural gradient algorithm, a steepest descent algorithm is proposed.
- (2)
- The convergence of the obtained algorithm is proven.
- (3)
- An example is discussed in detail to demonstrate our algorithm.
Acknowledgments
Author Contributions
Conflicts of Interest
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Zhang, Z.; Sun, H.; Peng, L.; Jiu, L. A Natural Gradient Algorithm for Stochastic Distribution Systems. Entropy 2014, 16, 4338-4352. https://doi.org/10.3390/e16084338
Zhang Z, Sun H, Peng L, Jiu L. A Natural Gradient Algorithm for Stochastic Distribution Systems. Entropy. 2014; 16(8):4338-4352. https://doi.org/10.3390/e16084338
Chicago/Turabian StyleZhang, Zhenning, Huafei Sun, Linyu Peng, and Lin Jiu. 2014. "A Natural Gradient Algorithm for Stochastic Distribution Systems" Entropy 16, no. 8: 4338-4352. https://doi.org/10.3390/e16084338
APA StyleZhang, Z., Sun, H., Peng, L., & Jiu, L. (2014). A Natural Gradient Algorithm for Stochastic Distribution Systems. Entropy, 16(8), 4338-4352. https://doi.org/10.3390/e16084338