**Author details**

For the estimation of time-invariant parameters, *g*1ðÞ¼ *t g*2ðÞ¼ *t* 0, so take *l*<sup>31</sup> ¼ *l*<sup>41</sup> ¼ 0. Take design parameters in simulation analysis *k*<sup>1</sup> ¼ *k*<sup>2</sup> ¼ 0, *kd*<sup>1</sup> ¼ *kd*<sup>1</sup> ¼ 0, *ω*<sup>11</sup> ¼ *ω*<sup>12</sup> ¼ 0*:*35, *γ*<sup>11</sup> ¼ *γ*<sup>12</sup> ¼ 10, *k*<sup>11</sup> ¼ *k*<sup>21</sup> ¼ 10, *k*<sup>12</sup> ¼ *k*<sup>22</sup> ¼ 5, *k*<sup>31</sup> ¼ *k*<sup>32</sup> ¼ 15, *k*<sup>41</sup> ¼ *k*<sup>42</sup> ¼ 10, *b*<sup>11</sup> ¼ *b*<sup>21</sup> ¼ 1, *b*<sup>12</sup> ¼ *b*<sup>22</sup> ¼ 25, *α*<sup>1</sup> ¼ *α*<sup>2</sup> ¼ 0*:*1, *c*<sup>1</sup> ¼ *c*<sup>2</sup> ¼ 6, *ω*<sup>1</sup> ¼ *ω*<sup>2</sup> ¼ 0*:*35, *γ*<sup>1</sup> ¼ 10, *γ*<sup>2</sup> ¼ 15, *μ*1ð Þ¼ 0 *mu*2ð Þ¼ 0 0. Suppose the initial state of the state observer is 0, 0 ð Þ. The state and parameter estimation, and estimation error obtained by simulation are shown in **Figure 9**. The simulation result in **Figure 9** shows that the above parameter method is still applicable to the estimation of time-invariant parameters. We can see from the simulation results that the estima-

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

*<sup>θ</sup>*<sup>1</sup> � ^*θ*1). If we tune the design parameters properly, we can get a more accurate

When the observation of the system (29) contains noise, for example, there is noise that obeys uniformly distribute in ½ � �0*:*001, 0*:*001 , that is, *y t*ðÞ¼ *x t*ðÞþ Eð Þ*t* , where EðÞ� *t U*½ � �0*:*001, 0*:*001 . In this case, the design parameters are the same as above, and the estimated errors of the states and parameters are shown in **Figure 10**, where the estimated error of the states are controlled within 5 � <sup>10</sup>�<sup>3</sup>

The parameter estimation error is larger than the parameter estimation error without noise, but the parameter estimation error can still be controlled within 5 � <sup>10</sup>�<sup>2</sup>

In summary, this section analyzes the estimation problem of multiple timevarying parameters in nonlinear systems based on the parameter estimation method combined the observer with the new stripping principle. Simulation research shows that the parameter estimation method proposed this chapter can estimate multiple time-varying parameters (this section only considers the estimation of two parameters), and the time-invariant and time-varying conditions of the parameters in the analysis both illustrate the applicability of the parameter estimation method. In addition, the simulation research on whether there is observation noise in the observations verifies the robustness and feasibility of the parameter estimation

This chapter studies the state observer method of nonlinear system parameter estimation. When the unknown parameters have explicit expressions, we can use the nonlinear tracking-differentiator-based method to estimate the parameters. The unknown parameters which is relatively non-linear system in nonlinear form or is not easy to express by explicit are main considered in this chapter. According to the different characteristics of the parameters contained in the dynamic process, based on the research of the existing literatures, this chapter proposes a new parameter estimation method based on the state observer and NSP. The parameter estimation method based on the combination of state observer with new stripping principle for dynamic systems containing multiple time-varying parameters. This chapter not only proves the feasibility of the method in theory, but also do the simulations. The simulation results show that the design method can approximate the true value of the parameter within a certain error range. The simulations also consider the presence or absence of observation noise. The simulation results not only show that the parameter estimation method introduced in this chapter is robust to noise, but also show the adaptability of the design parameters. Because it is found in the design parameter adjustment that: adjusting the design parameters within a certain range has little effect on the accuracy of parameter estimation, so in the adjustment of

, and the estimation error of the

.

.

, or even better (see **Figure 9**

tion error of the state is controlled within 5 � <sup>10</sup>�<sup>3</sup>

parameter estimation is controlled within 2 � <sup>10</sup>�<sup>2</sup>

method proposed in this section.

**4. Conclusions**

**128**

estimate.

Jinping Feng<sup>1</sup> \*† and Wei Wang2†

1 School of Mathematics and Statistics, Henan University, KaiFeng, China

2 School of Mathematics, Renmin University of China, Beijing, China

\*Address all correspondence to: fjinping@henu.edu.cn

† These authors contributed equally.

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
