Joint Parameter and State Estimation of Fractional-Order Singular Systems Based on Amsgrad and Particle Filter
Abstract
1. Introduction
2. System Description
3. Algorithm Derivation
3.1. Particle Filter (PF)
3.2. Amsgrad Optimization
- Learning rate λ: Initial values should be higher for smooth problems and lower for noisy or nonlinear systems. Time decay () is implemented to balance rapid convergence in the early stage and the precise refinement in the later stage.
- First-order moment factor : For fractional-order systems with long memory effects, dynamic decay () is used to mitigate historical gradient interference. And its initial value is close to 1.
- Second-order moment factor : Maintaining with max operation of Amsgrad can prevent premature learning rate decay while preserving gradient variance information.
- Stability constant ε: Setting ε to a tiny number such as can prevent the denominator of (19) from reporting an error when the second-order moment estimate is 0.
3.3. Joint Estimation Based on Amsgrad-Particle Filter (Ams-PF)
- Initialize the particle number N, the recursion count K, and set parameters and . Small initial variables are defined as , is a relatively large number.
- Collect and , then set . Construct parameter and information vectors , by (7) and (8).
- The system states based on the PF phase begins. Firstly, solutions of N particles are initialized according to (9). Then, update both the particle weights and states using (13), (20) and (21), and implement the resampling strategy. Finally, the particle weights are normalized to recursively obtain the state estimates by (22).
- In the Amsgrad optimization phase, the system parameters are optimized. Firstly, the first moment and second moment are computed using (14) and (16), respectively. The parameter estimates are updated via gradient information, ensuring the system output converges closer to the true observed values. Then, compute the bias-corrected first moment estimate by (15) and the bias-corrected second moment estimate by (17). Besides, perform the maximum value operation based on (18). Finally, calculate the estimated parameter vector by (19).
- Increase k by 1 and return to Step 3. Continue the recursive computation until k reaches the total data length K.
3.4. Convergence and Error Bound Proof of the Ams-PF Algorithm
4. Illustrative Examples
4.1. Three-Order Fractional Singular System
4.2. Four-Order Fractional Singular System
- By horizontally looking at the values in Table 1 and Table 3, it can be observed that the parameters identified by Ams-PF gradually approach the true values as k increases. It can be seen from the last line of each method that the final identification errors of Ams-PF are and , which are significantly smaller than those of the Amsgrad ( and ) and GSA-KF ( and ) methods. Thus, the Ams-PF algorithm can effectively identify fractional singular systems and its identification performance is satisfactory.
- In Figure 2, Figure 3 and Figure 4, Figure 8 and Figure 9, the Ams-PF algorithm exhibits excellent convergence speed and identification accuracy. Moreover, the system parameters identified by Ams-PF all converge near their true values with minimal estimation errors. Thus, it exhibits a stronger ability to escape local optima and has a faster convergence speed than Amsgrad and GSA-KF.
- In Figure 5 and Figure 10, the black line represents the true output, the red stars indicate the estimated outputs of Ams-PF, the blue dots represent the estimated outputs of PF, and the green forks denote the estimated outputs of GSA-KF. The system states in Figure 6 and Figure 11 are the same. As can be seen from these figures, the red stars are closest to the black line, which means that the estimated outputs and states of Ams-PF have a best fit with the true output and state values. These performances further demonstrate that the Ams-PF algorithm achieves satisfactory fitting between estimated and actual states, estimated outputs and actual outputs. The fitting performance is excellent and clearly superior to that of the PF and GSA-KF. Thus, the Ams-PF algorithm can effectively accomplish state estimation for fractional singular systems.
- From Table 2 and Figure 7, it can be seen that as the noise variance increases, the parameter estimation error becomes larger. However, this change is very minor. In Table 4, the final average value of Ams-PF identification error is 1.15654 and the standard deviation is 0.30091. Thus, we can conclude that the Ams-PF method can guarantee a certain robustness under different noise interferences or parameter uncertainty.
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
PF | Particle Filter |
Ams-PF | Amsgrad-Particle Filter |
Adam | Adaptive Moment Estimation |
GL | Grünwald–Letnikov |
GSA-KF | Gravitational Search Algorithm-Kalman Filter |
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Algorithm | Parameter | k = 100 | k = 200 | k = 500 | k = 1000 | k = 2000 | k = 3000 | True Value |
---|---|---|---|---|---|---|---|---|
Ams-PF | −0.61541 | −0.02476 | 1.14192 | 4.72356 | 3.35476 | 1.35720 | 1.45000 | |
−1.06909 | −0.92999 | −0.24511 | 6.48938 | 7.48246 | 7.68150 | 7.62000 | ||
−0.62677 | −0.01994 | 1.18192 | 4.73852 | 3.42233 | 1.23715 | 1.29000 | ||
−1.12186 | −0.97711 | −0.28772 | 6.38395 | 7.35942 | 7.56279 | 7.50000 | ||
−4.49469 | −5.55102 | −6.21625 | −11.97485 | −12.32042 | −12.28608 | −12.23000 | ||
−4.56647 | −5.58235 | −6.20746 | −11.42739 | −11.01381 | −10.81964 | −10.85000 | ||
81.67777 | 75.76372 | 68.40623 | 25.73892 | 14.63015 | 0.77586 | 0.00000 | ||
Amsgrad | −0.00006 | −0.00006 | −0.00007 | −0.00007 | −0.00008 | −0.00009 | 1.45000 | |
−0.00006 | −0.00006 | −0.00007 | −0.00007 | −0.00008 | −0.00009 | 7.62000 | ||
−0.00006 | −0.00006 | −0.00007 | −0.00007 | −0.00008 | −0.00009 | 1.29000 | ||
−0.00006 | −0.00006 | −0.00007 | −0.00007 | −0.00008 | −0.00009 | 7.50000 | ||
−0.32544 | −0.58082 | −2.51044 | −5.70348 | −10.32070 | −13.04685 | −12.23000 | ||
−0.40543 | −0.70195 | −3.22139 | −7.46451 | −11.53878 | −10.95777 | −10.85000 | ||
97.83775 | 96.21691 | 83.81867 | 66.83424 | 56.31184 | 55.51342 | 0.00000 | ||
GSA-KF | 1.60228 | 1.87892 | 2.13652 | 0.55312 | 1.70922 | 0.71478 | 1.45000 | |
1.11366 | 0.46282 | 2.17060 | 1.04038 | 2.37803 | 2.59275 | 7.62000 | ||
0.43591 | −0.22426 | 0.78218 | −0.10266 | 0.24588 | −0.31317 | 1.29000 | ||
0.99366 | 0.74029 | −0.33393 | 0.82055 | −0.26198 | 0.97589 | 7.50000 | ||
−5.61093 | −6.65074 | −11.22424 | −13.54027 | −13.27158 | −12.65576 | −12.23000 | ||
−3.68962 | −4.99514 | −9.76241 | −8.89775 | −10.09497 | −8.62924 | −10.85000 | ||
68.43794 | 56.61571 | 45.42038 | 44.01233 | 40.03237 | 40.03237 | 0.00000 |
Noise | k | |||||||
---|---|---|---|---|---|---|---|---|
100 | −0.61993 | −1.07816 | −0.63087 | −1.12662 | −4.51960 | −4.59531 | 81.58529 | |
200 | −0.02616 | −0.93060 | −0.02162 | −0.97967 | −5.58548 | −5.62083 | 75.62647 | |
500 | 1.12818 | −0.25172 | 1.16804 | −0.29629 | −6.24719 | −6.24261 | 68.32139 | |
1000 | 4.68612 | 6.42875 | 4.69650 | 6.32621 | −11.92001 | −11.38289 | 25.59878 | |
2000 | 3.26999 | 7.44564 | 3.35706 | 7.30810 | −12.31156 | −11.00405 | 14.11934 | |
3000 | 1.56226 | 7.55420 | 1.38692 | 7.44336 | −12.32993 | −10.86013 | 1.01396 | |
100 | −0.62497 | −1.07610 | −0.63717 | −1.13089 | −4.57133 | −4.65109 | 81.37875 | |
200 | −0.03067 | −0.93304 | −0.02977 | −0.98243 | −5.64181 | −5.68185 | 75.41060 | |
500 | 1.10890 | −0.26537 | 1.14559 | −0.31072 | −6.29412 | −6.29484 | 68.21139 | |
1000 | 4.61111 | 6.31260 | 4.61662 | 6.21463 | −11.79260 | −11.27850 | 25.36458 | |
2000 | 3.24096 | 7.30739 | 3.32794 | 7.16698 | −12.27024 | −10.98292 | 14.03265 | |
3000 | 1.63235 | 7.34108 | 1.42873 | 7.24212 | −12.31576 | −10.87492 | 2.30512 | |
True Value | 1.45000 | 7.62000 | 1.29000 | 7.50000 | −12.23000 | −10.85000 | 0.00000 |
Algorithm | Parameter | k = 100 | k = 200 | k = 500 | k = 1000 | k = 2000 | k = 3000 | True Value |
---|---|---|---|---|---|---|---|---|
Ams-PF | −4.15262 | −5.26322 | −5.93478 | −5.67812 | −5.17051 | −4.28444 | −4.35000 | |
−3.89106 | −4.56233 | −4.68299 | −4.80976 | −5.13349 | −5.62614 | −5.60000 | ||
−3.37188 | −3.87983 | −4.22355 | −4.13936 | −3.58811 | −3.22194 | −3.22000 | ||
−4.02288 | −5.03670 | −5.81737 | −5.54977 | −5.08730 | −4.22023 | −4.30000 | ||
−3.91768 | −4.78959 | −4.76688 | −4.92696 | −5.06612 | −5.55220 | −5.55000 | ||
−3.42535 | −3.75496 | −4.09824 | −3.97158 | −3.57478 | −3.16901 | −3.17000 | ||
−2.37429 | −1.65605 | 0.06044 | 0.18526 | −0.13507 | −0.31770 | −0.38000 | ||
−1.52845 | 0.18578 | 4.08040 | 5.78942 | 6.28496 | 6.49679 | 6.50000 | ||
−1.30907 | −0.48574 | 1.46846 | 1.76074 | 1.57158 | 1.36158 | 1.33000 | ||
70.32583 | 54.41356 | 29.41611 | 20.40704 | 11.59974 | 0.99473 | 0.00000 | ||
Amsgrad | −2.00252 | −2.08265 | −2.09940 | −2.09940 | −2.09940 | −2.09941 | −4.35000 | |
−2.00252 | −2.08265 | −2.09940 | −2.09940 | −2.09940 | −2.09941 | −5.60000 | ||
−2.00252 | −2.08265 | −2.09940 | −2.09940 | −2.09940 | −2.09941 | −3.22000 | ||
−2.00252 | −2.08265 | −2.09940 | −2.09940 | −2.09940 | −2.09941 | −4.30000 | ||
−2.00252 | −2.08265 | −2.09940 | −2.09940 | −2.09940 | −2.09941 | −5.55000 | ||
−2.00252 | −2.08265 | −2.09940 | −2.09940 | −2.09940 | −2.09941 | −3.17000 | ||
2.40826 | −0.55162 | −0.33212 | −0.45776 | 1.02542 | −1.80296 | −0.38000 | ||
8.37223 | 6.67781 | 6.45143 | 6.78119 | 8.72513 | 5.86281 | 6.50000 | ||
3.13573 | 0.18506 | 0.42633 | 0.29708 | 1.43440 | −0.39791 | 1.33000 | ||
64.55746 | 61.10604 | 61.10887 | 61.24270 | 62.02571 | 61.68410 | 0.00000 | ||
GSA-KF | 0.82003 | 1.00473 | 2.03327 | −2.68556 | −3.48806 | −3.42608 | −4.35000 | |
1.67698 | 1.20865 | 0.93494 | 2.64296 | 1.08677 | −2.40842 | −5.60000 | ||
0.46681 | 0.01001 | −0.43139 | −1.08602 | 1.30994 | −2.98696 | −3.22000 | ||
0.75629 | −0.19207 | −0.36270 | 0.99605 | −0.31427 | −1.31883 | −4.30000 | ||
0.49804 | −0.54719 | −0.10848 | −0.79584 | −0.28896 | −1.59978 | −5.55000 | ||
−0.25889 | −0.73610 | −0.09367 | −0.08238 | 0.51672 | −1.45601 | −3.17000 | ||
0.49905 | −0.41279 | −0.91987 | −0.02331 | −0.96026 | −0.36071 | −0.38000 | ||
4.54039 | 5.34174 | 5.50382 | 6.17914 | 6.01744 | 6.21313 | 6.50000 | ||
0.84229 | 0.78674 | 0.43152 | 2.44888 | 2.52241 | 0.42249 | 1.33000 | ||
95.02548 | 88.95787 | 86.56646 | 85.57912 | 63.56020 | 46.87218 | 0.00000 |
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Sun, T.; Zhao, K.; Wang, Z.; Zong, T. Joint Parameter and State Estimation of Fractional-Order Singular Systems Based on Amsgrad and Particle Filter. Fractal Fract. 2025, 9, 480. https://doi.org/10.3390/fractalfract9080480
Sun T, Zhao K, Wang Z, Zong T. Joint Parameter and State Estimation of Fractional-Order Singular Systems Based on Amsgrad and Particle Filter. Fractal and Fractional. 2025; 9(8):480. https://doi.org/10.3390/fractalfract9080480
Chicago/Turabian StyleSun, Tianhang, Kaiyang Zhao, Zhen Wang, and Tiancheng Zong. 2025. "Joint Parameter and State Estimation of Fractional-Order Singular Systems Based on Amsgrad and Particle Filter" Fractal and Fractional 9, no. 8: 480. https://doi.org/10.3390/fractalfract9080480
APA StyleSun, T., Zhao, K., Wang, Z., & Zong, T. (2025). Joint Parameter and State Estimation of Fractional-Order Singular Systems Based on Amsgrad and Particle Filter. Fractal and Fractional, 9(8), 480. https://doi.org/10.3390/fractalfract9080480