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Estimating and enlarging the region of attraction of multi-equilibrium points system by state-dependent edge impulses

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Abstract

When an isolated equilibrium of a nonlinear system is locally attractive, it might be difficult to estimate and then enlarge its Region of Attraction (RA). To solve this problem, this paper, by means of impulsive control, provides an effective method, i.e., state-dependent edge impulse (STDEI) combining with SOS programming and trajectory reversing with convex hull. Based on SOS programming and trajectory reversing, the initial estimation of RA is established. Then, by applying STDEI, we can design and then obtain the enlarged estimation of RA. Three numerical examples are provided to demonstrate the performance of the proposed approaches.

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Acknowledgements

This work was supported by National Key Research and Development Project (2018AAA0100101) , Fundamental Research Funds for the Central Universities under Grant (XDJK 2019B054) , and in part by National Natural Science Foundation of China (61873213, 61633011).

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Authors and Affiliations

  1. The Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing and also with the College of Electronic and Information Engineering, Southwest University, Chongqing, 400715, People’s Republic of China

    Yi Li, Chuandong Li, Zhilong He & Zixiang Shen

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  1. Yi Li
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  2. Chuandong Li
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  3. Zhilong He
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Correspondence to Chuandong Li.

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Appendices

A Algorithm for STDEI method

figure b

B Results of SOS programming

In this paper, we use a linearization system to get candidate Lyapunov function. For system (18), setting \(Q=diag[1\ 1]\), then P can be obtained through solve \(A^{T}P+PA = Q\), where A is the jacobian matrix of system (18).

$$\begin{aligned} P= \left[ \begin{matrix} 3.4833 &{} 0.5 \\ 0.5 &{} 3.3333 \end{matrix} \right] \end{aligned}$$

Then, applying the SOS programing, obtaining \(\gamma =1.0033,\)

$$\begin{aligned} V= & {} 0.0018944\delta ^{6} + 0.0060405\delta ^{5}\omega \\&+ 0.014242\delta ^{4}\omega ^{2} \\&+ 0.0023657\delta ^{3}\omega ^{3} + 0.020956\delta ^{2}\omega ^{4}\\&- 0.0067449\delta \omega ^{5}\\&+ 0.0062642\omega ^{6}+ 8.4643\times 10^{-9}\delta ^{5} \\&+ 9.9009\times 10^{-9}\delta ^{4}\omega \\&- 7.1832\times 10^{-10}\delta ^3\omega ^{2} \\&- 1.3331\times 10^{-8}\delta ^{2}\omega ^{3} \\&- 1.8875\times 10^{-8}\delta \omega ^{4}\\&- 1.1032\times 10^{-8}\omega ^{5} \\&+ 0.00062251\delta ^{4}+ 0.020627\delta ^{3}\omega \\&+ 0.028212\delta ^{2}\omega ^{2} \\&+ 0.026518\delta \omega ^{3} + 0.0051252\omega ^{4}\\&- 5.2122\times 10^{-8}\delta ^{3} \\&+ 1.0591\times 10^{-8}\delta ^{2}\omega \\&+ 2.064\times 10^{-8}\delta \omega ^{2}\\&+ 6.8358\times 10^{-8}\omega ^{3} \\&+ 0.097865\delta ^{2} + 0.022028\delta \omega + 0.099737\omega ^{2}. \end{aligned}$$

Similar, for system (19), set \(Q=diag[1\ 1]\), by simple calculation, we obtain

$$P= \left[ \begin{matrix} 0.001&{} 0 \\ 0 &{} 0.0009 \end{matrix} \right] . $$

For point (0, 0), \(\gamma =1.0004,\)

$$\begin{aligned} V= & {} 0.00033869x_{1}^{6} - 2.8363\times 10^{-5}x_{1}^{5}x_{2}\\&+ 0.0016375x_{1}^{4}x_{2}^{2} \\&+ 0.00011389x_{1}^{3}x_{2}^{3} + 0.00055266x_{1}^{2}x_{2}^{4}\\&+ 0.001014x_{1}x_{2}^{5}\\&+ 0.00095217x_{2}^{6} - 0.0033133x_{1}^{5}\\&+ 0.00011255x_{1}^{4}x_{2} \\&- 0.0031276x_{1}^{3}x_{2}^{2} \\&- 0.00021796x_{1}^{2}x_{2}^{3} - 0.0010689x_{1}x_{2}^{4} \\&- 0.00021932x_{2}^{5} + 0.0064202x_{1}^{4}\\&+ 0.00040366x_{1}^{3}x_{2} \\&- 0.0040643x_{1}^{2}x_{2}^{2} - 0.00012829x_{1}x_{2}^{3} \\&+ 0.0022644x_{2}^{4} \\&+ 0.03111x_{1}^{3} - 0.0008091x_{1}^{2}x_{2}\\&- 0.0020231x_{1}x_{2}^{2} \\&+ 0.001224x_{2}^{3} + 0.034594x_{1}^{2} - 0.0031531x_{1}x_{2}\\&+ 0.0050539x_{2}^{2}. \\ \end{aligned}$$

For point (6,-6), \(\gamma =1.0014,\) and

$$\begin{aligned} V= & {} 0.00034639x_{1}^{6} + 1.4702\times 10^{-5}x_{1}^{5}x_{2} \\&+ 0.0022435x_{1}^{4}x_{2}^{2} \\&+0.00051484x_{1}^{3}x_{2}^{3} + 0.0013379x_{1}^{2}x_{2}^{4}\\&+ 0.00017818x_{1}x_{2}^{5} \\&+ 0.00094429x_{2}^{6} + 0.0034406x_{1}^{5}\\&+ 0.00016752x_{1}^{4}x_{2} \\&+0.0034072x_{1}^{3}x_{2}^{2} - 0.00059315x_{1}^{2}x_{2}^{3}\\&+ 0.00031562x_{1}x_{2}^{4} \\&+ 0.00045504x_{2}^{5} + 0.0062919x_{1}^{4} \\&+ 0.00049161x_{1}^{3}x_{2} \\&- 0.0054393x_{1}^{2}x_{2}^{2} + 0.0016782x_{1}x_{2}^{3}\\&+ 0.0024157x_{2}^{4} \\&- 0.031493x_{1}^{3} - 0.0010251x_{1}^{2}x_{2}\\&+ 0.0046324x_{1}x_{2}^{2} \\&- 0.0011864x_{2}^{3} + 0.037281x_{1}^{2}\\&- 0.0043478x_{1}x_{2}\\&+ 0.0057891x_{2}^{2}. \end{aligned}$$

Similar, for system (21), by setting \(Q = diag[1\ 1\ 1]\), P can be obtained as follow

$$ \left[ \begin{matrix} 0.15073&{} 0.0187315 &{} -0.0187315 \\ 0.0187315&{} 0.14423 &{} 0 \\ -0.0187315&{} 0 &{} 0.28846 \end{matrix} \right] $$

\(V=0.15073x^{2} + 0.037463xy - 0.037463xz + 0.28846y^{2} + 0.28846z^{2}\), and \(\gamma =0.2890.\)

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Li, Y., Li, C., He, Z. et al. Estimating and enlarging the region of attraction of multi-equilibrium points system by state-dependent edge impulses. Nonlinear Dyn 103, 2421–2436 (2021). https://doi.org/10.1007/s11071-021-06259-9

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