Robust parameter estimation for constrained time-delay systems with inexact measurements

Chongyang Liu; Meijia Han; Zhaohua Gong; Kok Lay Teo

doi:10.3934/jimo.2019113

Article Contents

2021, Volume 17, Issue 1: 317-337. Doi: 10.3934/jimo.2019113

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Robust parameter estimation for constrained time-delay systems with inexact measurements

1.
School of Mathematics and Information Science, Shandong Technology and Business University, Yantai 264005, China
2.
School of Computer Science and Technology, Shandong Technology and Business University, Yantai 264005, China
3.
School of Electrical Engineering, Computing, and Mathematical Sciences, Curtin University, Perth 6845, Australia
4.
Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China

^* Corresponding author: Zhaohua Gong
^* Corresponding author: Zhaohua Gong

Received: December 31, 2018

Revised: March 31, 2019

Early access: September 2019

Published: January 2021

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Abstract

In this paper, we consider estimation problems involving constrained nonlinear systems with the unknown time-delays and unknown system parameters. These unknown quantities are to be estimated such that a least-squares error function between the system output and a set of noisy measurements is minimized subject to the characteristic time constraints specifying the restrictions. We first present the classical estimation formulation, where the expectation of the error function is regarded as the cost function. Then, in order to obtain robust estimates against the noises in measurements, we propose a robust estimation formulation, in which the cost function is the variance of the error function and an additional constraint indicates an allowable sacrifice from the optimal expectation value of the classical estimation problem. For these two estimation problems, we derive the gradients of the corresponding cost and constraint functions with respect to time-delays and system parameters by solving some auxiliary time-delay systems backward in time. On this basis, we develop gradient-based optimization algorithms to determine the optimal time-delays and system parameters. Finally, we consider two example problems, including a parameter estimation problem in microbial batch fermentation process, to illustrate the effectiveness and applicability of our proposed algorithms.

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Mathematics Subject Classification: Primary: 49M37; Secondary: 39B72.

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Figure 1. Reference output trajectory (black line) and corresponding perturbed sample data (blue stars) for Example 1

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Figure 2. Output trajectories corresponding to the optimal time-delay parameter estimates in TABLE 1 for Examplek 1

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Figure 3. Means and variances of the least-squares errors for 100,000 output realization according to (48) for Examplek 1

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Figure 4. Error mean variations with respect to $ \epsilon $ for 100,000 output realization according to (49) for Examplek 1

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Figure 5. State trajectories corresponding to the optimal parameters in TABLE 2 for Examplek 2

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Figure 6. Mean and variance of the least-squares errors for 100,000 output realizations generated according to (59) for Examplek 2

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Figure 7. Error mean variation of the least-squares error for 100,000 output realizations generated according to (61) for Examplek 2

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Table 1. Optimal time-delay and parameter estimates for Examplek 1

$ \beta $	$ \alpha_1^* $	$ \alpha_2^* $	$ \zeta_1^* $	$ \zeta_2^* $
0	0.434658	0.713674	3.068431	0.690517
0.001	0.414390	0.764191	2.986242	0.809873
0.004	0.408673	0.790692	2.930212	0.845580
0.007	0.396182	0.804993	2.830212	0.898728
0.01	0.365811	0.819147	2.805867	0.914341
0.04	0.319229	0.839143	2.761121	0.939468
0.07	0.306182	0.896250	2.748475	0.943645
0.1	0.283349	0.942031	2.536738	0.975285
0.17	0.253673	0.951192	2.245683	1.077120
0.178	0.249934	0.954031	2.231923	1.073528

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Table 2. Bounds, optimal time-delay and parameter estimates for Examplek 2

Parameters	Lower bounds	Upper bounds	Optimal values
Parameters	Lower bounds	Upper bounds	$ \beta=0 $	$ \beta=0.001 $	$ \beta=0.01 $	$ \beta=0.3 $
$ \alpha $	0.0010	1.0000	0.0863	0.0743	0.0700	0.0648
$ \Delta_1 $	0.1000	2.0000	1.5346	1.5366	1.6385	1.9040
$ k_1 $	10.000	500.00	287.91	289.79	331.63	442.56
$ m_2 $	1.0000	$ 20.000 $	5.2545	7.5042	10.759	15.629
$ Y_2 $	0.0001	2.0000	0.0083	0.0092	1.0325	0.0124
$ m_3 $	1.0000	20.000	13.598	15.787	16.587	17.001
$ Y_3 $	10.000	200.00	131.91	136.49	138.26	139.31

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