Estimation of initial functions for systems with delays from discrete measurements

Krzysztof Fujarewicz

doi:10.3934/mbe.2017011

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2017, Volume 14, Issue 1: 165-178. Doi: 10.3934/mbe.2017011

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Estimation of initial functions for systems with delays from discrete measurements

Krzysztof Fujarewicz

Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland

Received: September 10, 2015

Accepted: July 11, 2016

Published: September 2017

The author is supported by the Polish National Science Centre under grant DEC-2013/11/B/ST7/01713.

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Abstract

The work presents a gradient-based approach to estimation of initial functions of time delay elements appearing in models of dynamical systems. It is shown how to generate the gradient of the estimation objective function in the initial function space using adjoint sensitivity analysis. It is assumed that the system is continuous-time and described by ordinary differential equations with delays but the estimation is done based on discrete-time measurements of the signals appearing in the system. Results of gradient-based estimation of initial functions for exemplary models are presented and discussed.

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Mathematics Subject Classification: Primary: 65M99, 65K05, 93E12; Secondary: 93C57.

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Figure 1. Model of the dynamical system with one isolated discrete delay element

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Figure 2. Alternative structural representation of the delay element with additional input signal and zero initial condition

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Figure 3. The sensitivity model (a) and the adjoint model (b) for one delay element presented in Fig. 2

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Figure 4. The extended model

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Figure 5. The system adjoint to the extended model presented in Fig. 4

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Figure 6. Block diagram of the model A, used in Examples 1-4

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Figure 8. Results of the numerical example 1; (a) － true and estimated initial function $\psi(t)$, (b) － output signal $y(t)$ of the model and the plant, note they are nearly indistinguishable due to very small prediction error, (c) － objective function value, (d) － prediction error i.e. difference between output signals of the plant and the model

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Figure 9. Results of the numerical example 2

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Figure 10. Results of the numerical example 3

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Figure 11. Results of the numerical example 4

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Figure 12. Block diagram of the model B, used in Example 5

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Figure 13. Results of the numerical example 5

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Figure 14. Block diagram of the model C, used in Example 6

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Figure 15. Results of the numerical example 6

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Figure 7. The adjoint system for the model A generating two signals: $\beta(t)$ which is a reversed in time gradient $\nabla_{\psi(t)}J$ and $\gamma(t)$ which integrated over time interval $(0,t_f)$ is equal to the gradient $\nabla_{\tau}J$

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Table 1. Comparison of six numerical examples

Example	Model	Number of delays	Sampling time	Initial function(s)	Delay time(s)	Results
1	A (Fig. 6)	1	0⁺	Estimated	Known	Fig. 8
2	A (Fig. 6)	1	0⁺	Estimated	Estimated	Fig. 9
3	A (Fig. 6)	1	0.1	Estimated	Estimated	Fig. 10
4	A (Fig. 6)	1	0.1	Fixed (=0)	Estimated	Fig. 11
5	B (Fig. 12)	2	0₊	Estimated	Known	Fig. 13
6	C (Fig. 14)	2	0⁺	Estimated	Known	Fig. 15

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