Stochastic optimal control of MDOF nonlinear systems under combined harmonic and wide-band noise excitations

R. C. Hu; W. Q. Zhu

doi:10.1007/s11071-014-1727-0

Nonlinear Dynamics

January 2015, Volume 79, Issue 2, pp 1115–1129 | Cite as

Stochastic optimal control of MDOF nonlinear systems under combined harmonic and wide-band noise excitations

Authors
Authors and affiliations

R. C. Hu
W. Q. ZhuEmail author

Original Paper

First Online: 17 October 2014

Received: 24 July 2013

Accepted: 25 September 2014

199 Downloads
2 Citations

Abstract

A strategy for stochastic optimal control of multi-degrees-of-freedom (MDOF) strongly nonlinear systems under combined harmonic and wide-band noise excitations is proposed. First, a stochastic averaging procedure is developed to obtain the partially averaged Itô stochastic differential equations for weakly controlled strongly nonlinear systems under combined harmonic and wide-band noise excitations. Then, the dynamical programming equation for stochastic optimal control problem is derived and solved to yield the optimal control law. Finally, the responses of optimally controlled MDOF nonlinear systems are predicted by solving the Fokker–Planck–Kolmogorov equation associated with the fully averaged Itô stochastic differential equations. As an example, the stochastic optimal control of two coupled Duffing oscillators under combined harmonic and wide-band noise excitations is worked out in detail to illustrate the strategy. The effectiveness of the proposed control strategy is verified using the results from Monte Carlo simulation.

Keywords

Multi-degrees-of-freedom nonlinear systems Harmonic and wind-band noise excitations Stochastic optimal control Stochastic averaging Dynamical programming

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Notes

Acknowledgments

The research reported in the present paper was supported by the National Natural Science Foundation of China under Grant Nos. 10932009, 11072212, 11272279, 11321202, 11372271, 51175474.

Appendix 1: The averaged drift and diffusion coefficients

The coefficients in Eq. (52) are as follows:

$$\begin{aligned} \overline{{m}}_i&= \overline{{H}}_i (\mathbf{A})+F_i (\mathbf{A}) \quad (i=1,2),\end{aligned}$$

(62)

$$\begin{aligned} \overline{{m}}_3&= \overline{{H}}_3 (\mathbf{A})+\Omega _{11} -b_{01} ({A}_1 )\nonumber \\&\quad +\frac{E_{11} \cos \Gamma [2b_{01} (A_1 )-b_{21} (A_1 )]}{[4(\alpha _1 A_1 ^{2}+\omega _1^2 )]} \end{aligned}$$

(63)

$$\begin{aligned} F_1 (A_1 ,\Gamma )&= \frac{E_1 \sin \Gamma \left[ 2b_{01} (A_1 )\!-\!b_{21} (A_1 )\right] }{[4(\alpha _1 A_1 ^{2}\!+\!\omega _{10}^2 )]} \nonumber \\&-\,\frac{A_1 \left[ \beta _{10} (16\omega _1^2 \!+\!10\alpha _1 A_1^2 )\!+\!A_1^2 \beta _{11} (4\omega _1^2 \!+\!3\alpha A_1^2 )\!+\!A_2^2 \beta _{12} (16\omega _1^2 \!+\!10\alpha A_1^2 )\right] }{[32(\omega _1^2 \!+\!\alpha _1 A_1^2 )]} \\ F_2 (A_2 )&= -\frac{A_2 \left[ \beta _{20} (16\omega _2^2 +10\alpha _2 A_2^2 )+A_1^2 \beta _{21} (16\omega _2^2 +10\alpha _2 A_2^2 )+A_2^2 \beta _{22} (4\omega _2^2 +3\alpha _2 A_2^2 )\right] }{[32(\omega _2^2 +\alpha _2 A_2^2 )]} \end{aligned}$$

$$\begin{aligned} \overline{{H}}_i (A_i )&= \overline{{m}}_{i1} +\overline{{m}}_{i2} +\overline{{m}}_{i3} +\overline{{m}}_{i4} \quad (i=1,2),\\ \overline{{H}}_3 (A_1 )&= \overline{{m}}_{31} +\overline{{m}}_{32} +\overline{{m}}_{33} +\overline{{m}}_{34} ,\\ \overline{{m}}_{i1}&= \overline{{m}}_{i11} S_{i1} (\omega _i (A_i))+\overline{{m}}_{i13} S_{i1} (3\omega _i (A_i))\\&+\, \overline{{m}}_{i15} S_{i1} (5\omega _i (A_i))+\overline{{m}}_{i17} S_{i1} (7\omega _i (A_i )),\\ \overline{{m}}_{i3}&= \overline{{m}}_{i31} S_{i1} (\omega _i (A_i ))+\overline{{m}}_{i33} S_{i1} (3\omega _i (A_i ))\\&+\,\overline{{m}}_{i35} S_{i1} (5\omega _i (A_i ))+\overline{{m}}_{i37} S_{i1} (7\omega _i (A_i )),\\ \overline{{m}}_{i2}&= \overline{{m}}_{i22} S_{i2} (2\omega _i (A_i ))+\overline{{m}}_{i24} S_{i2} (4\omega _i (A_i ))\\&+\,\overline{{m}}_{i26} S_{i2} (6\omega _i (A_i ))+\overline{{m}}_{i28} S_{i2} (8\omega _i (A_i )),\\ \overline{{m}}_{i4}&= \overline{{m}}_{i42} S_{i2} (2\omega _i (A_i ))+\overline{{m}}_{i44} S_{i2} (4\omega _i (A_i ))\\&+\,\overline{{m}}_{i46} S_{i2} (6\omega _i (A_i ))+\overline{{m}}_{i48} S_{i2} (8\omega _i (A_i )),\\ \overline{{m}}_{31}&= \overline{{m}}_{311} I_1 (\omega _i (A_i ))+\overline{{m}}_{313} I_1 (3\omega _i (A_i ))\\&+\,\overline{{m}}_{315} I_1 (5\omega _i (A_i ))+\overline{{m}}_{317} I_1 (7\omega _i (A_i )),\\ \overline{{m}}_{33}&= \overline{{m}}_{331} I_1 (\omega _i (A_i ))+\overline{{m}}_{333} I_1 (3\omega _i (A_i ))\\&+\,\overline{{m}}_{335} I_1 (5\omega _i (A_i ))+\overline{{m}}_{337} I_1 (7\omega _i (A_i )),\\ \end{aligned}$$

$$\begin{aligned} \overline{{m}}_{32}&= \overline{{m}}_{322} I_2 (2\omega _i (A_i ))+\overline{{m}}_{324} I_2 (4\omega _i (A_i ))\\&+\,\overline{{m}}_{326} I_2 (6\omega _i (A_i ))+\overline{{m}}_{328} I_2 (8\omega _i (A_i )),\\ \overline{{m}}_{34}&= \overline{{m}}_{342} I_2 (2\omega _i (A_i ))+\overline{{m}}_{344} I_2 (4\omega _i (A_i ))\\&+\,\overline{{m}}_{346} I_2 (6\omega _i (A_i ))+\overline{{m}}_{348} I_2 (8\omega _i (A_i )),\\ I_i (\omega )&= \frac{D_{1i} }{\omega _{1i} }\frac{\omega }{\omega ^{2}+\omega _{1i}^2 }\quad { (i=1,2)} \end{aligned}$$

$$\begin{aligned} \overline{{m}}_{i11}&= \pi \left[ b_{2i} (A_i )-2b_{0i} (A_i )\right] \frac{\left\{ 2\alpha A_i \left[ 2b_{0i} (A_i )-b_{2i} (A_i )\right] +(A_i^2 \alpha _i +\omega _i^2 )\left[ db_{2i} (A_i )/dA_i -2(db_{0i} (A_i )/dA_i )\right] \right\} }{\left[ 8(A_i^2 \alpha _i +\omega _i^2 )^{3}\right] }\\ \overline{{m}}_{i13}&= \pi [b_{2i} (A_i )-b_{4i} (A_i )] \frac{\left\{ 2\alpha A_i [b_{4i} (A_i )-b_{2i} (A_i )]+(A_i^2 \alpha _i +\omega _i^2 )\left[ db_{2i} (A_i )/dA_i -2(db_{4i} (A_i )/dA_i )\right] \right\} }{[8(A_i^2 \alpha _i +\omega _i^2 )^{3}]} \\ \overline{{m}}_{i15}&= \pi \left[ b_{4i} (A_i )-2b_{6i} (A_i )\right] \frac{\left\{ 2\alpha A_i \left[ b_{6i} (A_i )-b_{4i} (A_i )\right] +(A_i^2 \alpha _i +\omega _i^2 )\left[ db_{4i} (A_i )/dA_i -2(db_{6i} (A_i )/dA_i )\right] \right\} }{\left[ 8(A_i^2 \alpha _i +\omega _i^2 )^{3}\right] } \\ \overline{{m}}_{i17}&= \frac{\pi b_{6i} (A_i )\left\{ -2\alpha A_i b_{6i} (A_i )+(A_i^2 \alpha _i +\omega _i^2 )(db_{6i} (A_i )/dA_i )\right\} }{\left[ 8(A_i^2 \alpha _i +\omega _i^2 )^{3}\right] }\\ \overline{{m}}_{i22}&= \pi A_i \left[ 2b_{0i} (A_i )-b_{4i} (A_i )\right] \frac{\left\{ \left[ 2b_{0i} (A_i )-b_{4i} (A_i )\right] (A_i^2 \alpha _i -\omega _i^2 )-A_i (A_i^2 \alpha _i -\omega _i^2 )\left[ 2db_{0i} (A_i )/dA_i -(db_{4i} (A_i )/dA_i )\right] \right\} }{\left[ 32(A_i^2 \alpha _i +\omega _i^2 )^{3}\right] } \\ \overline{{m}}_{i24}&= \pi A_i \left[ b_{2i} (A_i )-b_{6i} (A_i )\right] \frac{\left\{ \left[ b_{6i} (A_i )-b_{2i} (A_i )\right] (A_i^2 \alpha _i -\omega _i^2 )-A_i (A_i^2 \alpha _i -\omega _i^2 )\left[ db_{2i} (A_i )/dA_i -(db_{6i} (A_i )/dA_i )\right] \right\} }{\left[ 32(A_i^2 \alpha _i +\omega _i^2 )^{3}\right] } \\ \overline{{m}}_{i26}&= \frac{\pi A_i b_{4i} (A_i )\left\{ b_{4i} (A_i )(A_i^2 \alpha _i -\omega _i^2 )-A_i (A_i^2 \alpha _i -\omega _i^2 )(db_{4i} (A_i )/dA_i )\right\} }{\left[ 32(A_i^2 \alpha _i +\omega _i^2 )^{3}\right] }\\ \overline{{m}}_{i28}&= \frac{\pi A_i b_{6i} (A_i )\left\{ b_{6i} (A_i )(A_i^2 \alpha _i -\omega _i^2 )-A_i (A_i^2 \alpha _i -\omega _i^2 )(db_{6i} (A_i )/dA_i )\right\} }{\left[ 32(A_i^2 \alpha _i +\omega _i^2 )^{3}\right] } \end{aligned}$$

$$\begin{aligned} \overline{{m}}_{i31}&= \frac{-\pi [b_{2i}^2 (A_i )-4b_{0i}^2 (A_i )]}{[8A_i (A_i^2 \alpha _i +\omega _i^2 )^{2}]},\\ \overline{{m}}_{i33}&= \frac{-3\pi [b_{4i}^2 (A_i )-b_{2i}^2 (A_i )]}{[8A_i (A_i^2 \alpha _i +\omega _i^2 )^{2}]},\\ \overline{{m}}_{i35}&= \frac{-5\pi [b_{6i}^2 (A_i )-b_{4i}^2 (A_i )]}{[8A_i (A_i^2 \alpha _i +\omega _i^2 )^{2}]},\\ \overline{{m}}_{i37}&= \frac{7\pi b_{6i}^2 (A_i )}{[8A_i (A_i^2 \alpha _i +\omega _i^2 )^{2}]}, \end{aligned}$$

$$\begin{aligned} \overline{{m}}_{i42}&= \frac{\pi A_i [2b_{0i} (A_i )-b_{4i} (A_i )][2b_{0i} (A_i )+2b_{2i} (A_i )+b_{4i} (A_i )]}{[16(A_i^2 \alpha _i +\omega _i^2 )^{2}]},\\ \overline{{m}}_{i44}&= \frac{\pi A_i [b_{2i} (A_i )-b_{6i} (A_i )][b_{2i} (A_i )+2b_{4i} (A_i )+b_{6i} (A_i )]}{[8(A_i^2 \alpha _i +\omega _i^2 )^{2}]}, \end{aligned}$$

$$\begin{aligned} \overline{{m}}_{i46}&= \frac{3\pi A_i b_{4i} (A_i )[b_{4i} (A_i )+2b_{6i} (A_i )]}{[16(A_i^2 \alpha _i +\omega _i^2 )^{2}]},\\ \overline{{m}}_{i48}&= \frac{\pi A_i b_{6i}^2 (A_i )}{[4(A_i^2 \alpha _i +\omega _i^2 )^{2}]},\\ \overline{{m}}_{311}&= \frac{[2b_{01} (A_1 )+b_{21} (A_1 )]^{2}}{[8A_1^2 (A_1^2 \alpha _1 +\omega _1^2 )^{2}]},\\ \end{aligned}$$

$$\begin{aligned} \overline{{m}}_{313}&= \frac{3[b_{21} (A_1 )+b_{41} (A_1 )]^{2}}{[8A_1^2 (A_1^2 \alpha _1 +\omega _1^2 )^{2}]},\\ \overline{{m}}_{315}&= \frac{5[b_{41} (A_1 )+b_{61} (A_1 )]^{2}}{[8A_1^2 (A_1^2 \alpha _1 +\omega _1^2 )^{2}]},\\ \overline{{m}}_{317}&= \frac{7[b_{61} (A_1 )]^{2}}{[8A_1^2 (A_1^2 \alpha _1 +\omega _1^2 )^{2}]},\\ \overline{{m}}_{322}&= \frac{[2b_{01} (A_1 )+2b_{21} (A_1 )+b_{41} (A_1 )]^{2}}{[16(A_1^2 \alpha _1 +\omega _1^2 )^{2}]},\\ \overline{{m}}_{324}&= \frac{[b_{21} (A_1 )+2b_{41} (A_1 )+b_{61} (A_1 )]^{2}}{[8(A_1^2 \alpha _1 +\omega _1^2 )^{2}]},\\ \end{aligned}$$

$$\begin{aligned} \overline{{m}}_{326}&= \frac{3[b_{41} (A_1 )+2b_{61} (A_1 )]^{2}}{[16(A_1^2 \alpha _1 +\omega _1^2 )^{2}]},\\ \overline{{m}}_{328}&= \frac{[b_{61} (A_1 )]^{2}}{[4(A_1^2 \alpha _1 +\omega _1^2 )^{2}]}, \end{aligned}$$

$$\begin{aligned} \overline{{m}}_{331}&= \left[ 2b_{01} (A_1 )-b_{21} (A_1 )\right] \frac{\left\{ -\left[ b_{21} (A_1 )+2b_{01} (A_1 )\right] (3A_1^2 \alpha _1 +\omega _1^2 )+A_1 (\alpha _1 A_1^2 +\omega _1^2 )\left[ 2(db_{01} (A_1 )/dA_1 )+(db_{21} (A_1 )/dA_1 )\right] \right\} }{\left[ 8(A_1^2 \alpha _1 +\omega _1^2 )^{3}\right] }, \\ \overline{{m}}_{333}&= \left[ b_{21} (A_1 )-b_{41} (A_1 )\right] \frac{\left\{ -\left[ b_{21} (A_1 )+b_{41} (A_1 )\right] (3A_1^2 \alpha _1 +\omega _1^2 )+A_1 (\alpha _1 A_1^2 +\omega _1^2 )\left[ (db_{21} (A_1 )/dA_1 )+(db_{41} (A_1 )/dA_1 )\right] \right\} }{\left[ 8(A_1^2 \alpha _1 +\omega _1^2 )^{3}\right] }, \\ \overline{{m}}_{335}&= \left[ b_{41} (A_1 )-b_{61} (A_1 )\right] \frac{\left\{ -\left[ b_{41} (A_1 )+b_{61} (A_1 )\right] (3A_1^2 \alpha _1 +\omega _1^2 )+A_1 (\alpha _1 A_1^2 +\omega _1^2 )\left[ (db_{41} (A_1 )/dA_1 )+(db_{61} (A_1 )/dA_1 )\right] \right\} }{\left[ 8(A_1^2 \alpha _1 +\omega _1^2 )^{3}\right] }, \\ \overline{{m}}_{337}&= \frac{b_{61} (A_1 )\left\{ -b_{61} (A_1 )(3A_1^2 \alpha _1 +\omega _1^2 )+A_1 (\alpha _1 A_1^2 +\omega _1^2 )\left[ (db_{61} (A_1 )/dA_1 )\right] \right\} }{\left[ 8(A_1^2 \alpha _1 +\omega _1^2 )^{3}\right] } \end{aligned}$$

$$\begin{aligned} \overline{{m}}_{342}&= \big \{-2A_1 \alpha _1 \left[ 2b_{01} (A_1 )+2b_{21} (A_1 )+b_{41} (A_1 )\right] \\&+\,(A_1^2 \alpha _1 +\omega _1^2 )\Bigg [\frac{2db_{01} (A_1 )}{dA_1 }+\frac{2db_{21} (A_1 )}{dA_1 }\\&+\frac{2db_{41} (A_1 )}{dA_1 }\Bigg ] \frac{A_1 [(2b_{01} (A_1 )-b_{41} (A_1 )\big \}}{[32(A_1^2 \alpha _1 +\omega _1^2 )^{3}]} \\ \overline{{m}}_{344}&= \big \{-2A_1 \alpha _1 [b_{21} (A_1 )+2b_{41} (A_1 )+b_{61} (A_1 )]\\&+\,(A_1^2 \alpha _1 +\omega _1^2 )\Bigg [\frac{2db_{21} (A_1 )}{dA_1 }+\frac{2db_{41} (A_1 )}{dA_1 }\\&+\,\frac{2db_{61} (A_1 )}{dA_1 }\Bigg ] \frac{A_1 [(2b_{21} (A_1 )-b_{61} (A_1 )\big \}}{[32(A_1^2 \alpha _1 +\omega _1^2 )^{3}]} \end{aligned}$$

$$\begin{aligned} \overline{{m}}_{346}&= Ab_{41} (A_1 ) \times \frac{\left\{ -2A_1 \alpha _1 [b_{41} (A_1 )+2b_{61} (A_1 )]+(A_1^2 \alpha _1 +\omega _1^2 )[(db_{41} (A_1 )/dA_1 +(2db_{61} (A_1 )/dA_1 )]\right\} }{[32(A_1^2 \alpha _1 +\omega _1^2 )^{3}]}, \\ \overline{{m}}_{348}&= \frac{A_1 b_{61} (A_1 )\left\{ -2A_1 \alpha _1 b_{61} (A_1 )+(A_1^2 \alpha _1 +\omega _1^2 )db_{61} (A_1 )/dA_1 )\right\} }{[32(A_1^2 \alpha _1 +\omega _1^2 )^{3}]}, \end{aligned}$$

$$\begin{aligned} \overline{{b}}_{ii}&= \overline{{b}}_{ii1} +\overline{{b}}_{ii2} , \quad (i=1,2)\end{aligned}$$

(64)

$$\begin{aligned} \overline{{b}}_{33}&= \overline{{b}}_{331} +\overline{{b}}_{332} , \end{aligned}$$

(65)

$$\begin{aligned}&\overline{{b}}_{ij} =0, \quad i\ne j,\\ \overline{{b}}_{ii1}&= \overline{{b}}_{ii11} S_1 (\omega _i (A_i ))+\overline{{b}}_{ii13} S_1 (3\omega _i (A_i))\\&\quad +\,\overline{{b}}_{ii15} S_1 (5\omega _i (A_i ))+\overline{{b}}_{ii17} S_1 (7\omega _i (A_i )),\\ \overline{{b}}_{ii2}&= \overline{{b}}_{ii22} S_2 (\omega _i (A_i ))+\overline{{b}}_{ii24} S_2 (4\omega _i (A_i))\\&\quad +\,\overline{{b}}_{ii26} S_1 (6\omega _i (A_i ))+\overline{{b}}_{ii28} S_1 (8\omega _i (A_i )),\\ \overline{{b}}_{331}&= \overline{{b}}_{3311} S_1 (\omega _1 (A_1 ))+\overline{{b}}_{3313} S_1 (3\omega _1 (A_1))\\&\quad +\,\overline{{b}}_{3315} S_1 (5\omega _1 (A_1 ))+\overline{{b}}_{3317} S_1 (7\omega _1 (A_1 )),\\ \overline{{b}}_{333}&= \overline{{b}}_{3330} S_2 (0)\!+\!\overline{{b}}_{3332} S_2 (2\omega _1 (A_1 ))\!+\!\overline{{b}}_{3334} S_2 (4\omega _1 (A_1))\\&\quad +\,\overline{{b}}_{3336} S_2 (6\omega _1 (A_1 ))+\overline{{b}}_{3338} S_2 (8\omega _1 (A_1 )),\\ \end{aligned}$$

$$\begin{aligned} b_{ii11}&= \frac{\pi [b_{2i} (A_i )-2b_{0i} (A_i )]^{2}}{[4(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{ii13}&= \frac{\pi [b_{2i} (A_i )-2b_{4i} (A_i )]^{2}}{[4(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{ii15}&= \frac{\pi [b_{4i} (A_i )-b_{6i} (A_i )]^{2}}{[4(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{ii17}&= \frac{\pi b_{6i}^2 (A_i )}{[4(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{ii22}&= \frac{\pi A_i^2 [b_{4i} (A_i )-2b_{0i} (A_i )]^{2}}{[16(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{ii24}&= \frac{\pi A_i^2 [b_{2i} (A_i )-b_{6i} (A_i )]^{2}}{[16(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{ii26}&= \frac{\pi A_i^2 b_{4i}^2 (A_i )}{[16(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{ii28}&= \frac{\pi A_i^2 b_{6i}^2 (A_i )}{[16(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{3311}&= \frac{\pi [2b_{02} (A_1 )+b_{22} (A_1 )]^{2}}{[4A_1^2 (\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},\\ b_{3313}&= \frac{\pi [b_{21} (A_1 )+b_{41} (A_1 )]^{2}}{[4A_1^2 (\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},,\\ b_{3315}&= \frac{\pi [b_{41} (A_1 )+b_{61} (A_1 )]^{2}}{[4A_1^2 (\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},\\ b_{3317}&= \frac{\pi b_{61} ^{2}(A_1 )}{[4A_1^2 (\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},\\ b_{3320}&= \frac{\pi [2b_{01} (A_1 )+b_{21} (A_1 )]^{2}}{[8(\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},\\ b_{3322}&= \frac{\pi [2b_{01} (A_1 )+2b_{21} (A_1 )+b_{41} (A_1 )]^{2}}{[16(\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},\\ b_{3324}&= \frac{\pi [b_{21} (A_1 )+2b_{41} (A_1 )+b_{61} (A_1 )]^{2}}{[16(\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},\\ b_{3326}&= \frac{\pi [b_{41} (A_1 )+2b_{61} (A_1 )]^{2}}{[16(\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},\\ b_{3328}&= \frac{\pi b_{61}^2 (A_1 )}{[16(\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]}. \end{aligned}$$

Appendix 2: An approximate solution of final dynamical programming equation

For ergodic control of 2-DOF Duffing system, the final dynamical programming equation is of the form:

$$\begin{aligned} \eta _c&= L_1 (\mathbf{H},\Gamma )+\tilde{m}_1 (\mathbf{H},\Gamma )\frac{\partial V}{\partial H_1 }+\tilde{m}_2 (\mathbf{H})\frac{\partial V}{\partial H_2 }\nonumber \\&\quad +\,\tilde{m}_3 (\mathbf{H},\Gamma )\frac{\partial V}{\partial \Gamma }-\frac{m_{11} (\mathbf{H})}{4R_1 }\left( {\frac{\partial V}{\partial H_1 }} \right) ^{2} \nonumber \\&\quad -\,\frac{m_{22} (\mathbf{H})}{4R_2 }\left( {\frac{\partial V}{\partial H_2 }} \right) ^{2}-\frac{m_{33} (\mathbf{H})}{4R_1 }\left( {\frac{\partial V}{\partial \Gamma }} \right) ^{2}\nonumber \\&\quad +\,\frac{b_{11} }{2}\frac{\partial ^{2}V}{\partial H_1^2 }+\frac{b_{22} }{2}\frac{\partial ^{2}V}{\partial H_2^2 }+\frac{b_{33} }{2}\frac{\partial ^{2}V}{\partial \Gamma ^{2}} \end{aligned}$$

(66)

Suppose that $L_1 (\mathbf{H},\Gamma )$ can be expanded into Fourier series with respect to$\Gamma $,

$$\begin{aligned} L_1 (\mathbf{H},\Gamma )&= L_{10} (\mathbf{H})+\sum _{i=1}^\infty (L_{1i}^c (\mathbf{H})\cos i\Gamma \nonumber \\&\quad +\, L_{1i}^s (\mathbf{H})\sin i\Gamma ) \end{aligned}$$

(67)

Then the solution of Eq. (66) can be assumed of the form of the following Fourier series:

$$\begin{aligned} V(\mathbf{H},\Gamma )=V_0 (\mathbf{H})+\sum _{i=1}^\infty {(V_i^c (\mathbf{H})\cos i\Gamma +V_i^s (\mathbf{H})\sin i\Gamma )}\nonumber \\ \end{aligned}$$

(68)

Note that coefficients $\tilde{m}_1 , \tilde{m}_2 $ and $\tilde{m}_3 $ in Eq. (54) are of the form

$$\begin{aligned} \tilde{m}_1 (\mathbf{H},\Gamma )&= m_{10} (\mathbf{H})+m_{11}^s (\mathbf{H})\sin \Gamma \nonumber \\ \tilde{m}_2 (\mathbf{H})&= m_{20} (\mathbf{H})\nonumber \\ \tilde{m}_3 (\mathbf{H},\Gamma )&= m_{30} (\mathbf{H})+m_{31}^c (\mathbf{H})\cos \Gamma \end{aligned}$$

(69)

and $m_{i0} (i=1,2,3),m_{11}^s ,m_{31}^c $ are all independent of $\Gamma $. Substituting Eqs. (67)–(69), the following series of equation can be obtained:

$$\begin{aligned}&\eta _c =L_{10} (\mathbf{H})+m_{10} (\mathbf{H})\frac{\partial V_0 }{\partial H_1 }+m_{20} (\mathbf{H})\frac{\partial V_0 }{\partial H_2 }\nonumber \\&\quad -\,\frac{1}{2}m_{11}^s (\mathbf{H})\frac{\partial V_1^c }{\partial H_1 }+\frac{1}{2}m_{31}^c (\mathbf{H})V_1^s\nonumber \\&\quad -\,\frac{m_{11} (\mathbf{H})}{4R_1 }\!\left\{ {\left( \!{\frac{\partial V_0 }{\partial H_1 }}\!\right) ^{2}\!+\!\frac{1}{2}\sum _{i=1}^\infty {\left[ \!\!{\left( \!{\frac{\partial V_i^c }{\partial H_1 }} \right) ^{2}\!+\! \left( \!{\frac{\partial V_i^s }{\partial H_1 }} \right) ^{2}}\!\right] }}\!\right\} \nonumber \\&\quad -\,\frac{m_{22} (\mathbf{H})}{4R_2 }\!\left\{ {\left( \! {\frac{\partial V_0 }{\partial H_2 }} \!\right) ^{2}\!+\!\frac{1}{2}\sum _{i=1}^\infty {\!\left[ {\!\left( \! {\frac{\partial V_i^c }{\partial H_2 }} \!\right) ^{2}\!+\!\left( \! {\frac{\partial V_i^s }{\partial H_2 }} \!\right) ^{2}} \!\right] } }\!\right\} \nonumber \\&\quad -\,\frac{m_{33} (\mathbf{H})}{8R_1 }\!\left[ {\sum _{i=1}^\infty {i^{2}\!\left( \!{V_i^{c^{2}}\!+\!V_i^{s^{2}} } \right) } }\!\right] \!+\!\frac{b_{11} }{2}\frac{\partial ^{2}V_0 }{\partial H_1^2 }\!+\!\frac{b_{22} }{2}\frac{\partial ^{2}V_0 }{\partial H_2^2 }\nonumber \\\end{aligned}$$

(70)

$$\begin{aligned}&0=L_{11}^c (\mathbf{H})+m_{10} (\mathbf{H})\frac{\partial V_1^c }{\partial H_1 }+m_{20} (\mathbf{H})\frac{\partial V_1^c }{\partial H_2 }\nonumber \\&\quad +\,m_{30} (\mathbf{H})V_1^s (\mathbf{H})+\frac{1}{2}m_{11}^s (\mathbf{H})\frac{\partial V_2^c }{\partial H_1 }+m_{31}^c (\mathbf{H})V_2^s \nonumber \\&\quad -\,\frac{m_{11} (\mathbf{H})}{4R_1 }\!\left[ {2\frac{\partial V_0 }{\partial H_1 }\frac{\partial V_1^c }{\partial H_1 }\!+\!\!\sum _{i=1}^\infty {\left( {\frac{\partial V_i^c }{\partial H_1 }\frac{\partial V_{i+1}^c }{\partial H_1 }+\!\frac{\partial V_i^s }{\partial H_1 }\frac{\partial V_{i+1}^s }{\partial H_1 }} \right) }}\! \right] \nonumber \\&\quad -\,\frac{m_{22} (\mathbf{H})}{4R_2 }\!\left[ {2\frac{\partial V_0 }{\partial H_2 }\frac{\partial V_1^c }{\partial H_2 }\!+\!\!\sum _{i=1}^\infty {\!\left( {\frac{\partial V_i^c }{\partial H_2 }\frac{\partial V_{i+1}^c }{\partial H_2 }\!+\!\frac{\partial V_i^s }{\partial H_2 }\frac{\partial V_{i+1}^s }{\partial H_2 }} \right) }} \!\right] \nonumber \\&\quad -\,\frac{m_{33} (\mathbf{H})}{4R_1 }\left[ {\sum _{i=1}^\infty {i(i+1)\left( {V_i^c V_{i+1}^c +V_i^s V_{i+1}^s } \right) } } \right] \nonumber \\&\quad +\,\frac{b_{11} }{2}\frac{\partial ^{2}V_1^c }{\partial H_1^2 }+\frac{b_{22} }{2}\frac{\partial ^{2}V_1^c }{\partial H_2^2 }-\frac{b_{33} }{2}V_1^c \end{aligned}$$

(71)

$$\begin{aligned}&0=L_{11}^c (\mathbf{H})+m_{10} (\mathbf{H})\frac{\partial V_1^s }{\partial H_1 }+m_{20} (\mathbf{H})\frac{\partial V_1^s }{\partial H_2 }\nonumber \\&\quad -\, m_{30} (\mathbf{H})V_1^c (\mathbf{H})+m_{11}^s (\mathbf{H})\frac{\partial V_0 }{\partial H_1 }\nonumber \\&+\frac{1}{2}m_{11}^s (\mathbf{H})\frac{\partial V_2^c }{\partial H_1 }-m_{31}^c (\mathbf{H})V_2^c \nonumber \\&\quad -\,\frac{m_{11} (\mathbf{H})}{4R_1 }\!\left[ \!{2\frac{\partial V_0 }{\partial H_1 }\frac{\partial V_1^s }{\partial H_1 }\!+\!\!\sum _{i=1}^\infty {\left( {\frac{\partial V_i^c }{\partial H_1 }\frac{\partial V_{i+1}^s }{\partial H_1 }+\frac{\partial V_i^s }{\partial H_1 }\frac{\partial V_{i+1}^c }{\partial H_1 }} \!\right) } } \!\!\right] \nonumber \\&\quad -\,\frac{m_{22} (\mathbf{H})}{4R_2 }\!\left[ \!{2\frac{\partial V_0 }{\partial H_2 }\frac{\partial V_1^s }{\partial H_2 }\!+\!\!\sum _{i=1}^\infty {\left( {\frac{\partial V_i^c }{\partial H_2 }\frac{\partial V_{i+1}^s }{\partial H_2 }+\!\frac{\partial V_i^s }{\partial H_2 }\frac{\partial V_{i+1}^c }{\partial H_2 }} \right) } }\! \!\right] \nonumber \\&\quad -\,\frac{m_{33} (\mathbf{H})}{4R_1 }\left[ {\sum _{i=1}^\infty {i(i+1)\left( {V_i^s V_{i+1}^c +V_i^c V_{i+1}^s } \right) } } \right] \nonumber \\&\quad +\,\frac{b_{11} }{2}\frac{\partial ^{2}V_1^s }{\partial H_1^2 }+\frac{b_{22} }{2}\frac{\partial ^{2}V_1^s }{\partial H_2^2 }-\frac{b_{33} }{2}V_1^s \end{aligned}$$

(72)

The higher harmonic terms in the solution can be neglected. An approximated solution of Eq. (66)

$$\begin{aligned} V(\mathbf{H},\Gamma )\!=\!V_0 (\mathbf{H})\!+\!V_1^c (\mathbf{H})\cos \Gamma \!+\! V_1^s (\mathbf{H})\sin \Gamma \end{aligned}$$

(73)

To evaluate $V_0 (\mathbf{H}), V_1^c (\mathbf{H})$ and $V_1^s (\mathbf{H})$ from the coupled differential equations (70)–(72), the Fourier coefficients are further expanded into Taylor series:

$$\begin{aligned} m_{10}&= m_{101} H_1 +m_{102} H_1^3 +m_{103} H_2^2 H_1 +\cdots \nonumber \\ m_{20}&= m_{201} H_2 +m_{202} H_1^2 H_2 +m_{203} H_2^3 +\cdots \nonumber \\ m_{30}&= m_{300} +m_{302} H_1^2 +\cdots \nonumber \\ m_{11}^s&= m_{111}^s +m_{113}^s H_1^2 +\cdots \nonumber \\ m_{31}^c&= m_{311}^c H_1^{-1} +m_{313}^c H_1 +\cdots \nonumber \\ m_{11}&= m_{110} +m_{112} H_1^2 +\cdots \nonumber \\ m_{22}&= m_{220} +m_{222} H_2^2 +\cdots \nonumber \\ m_{33}&= m_{330} H_1^{-2} +m_{332} +\cdots \nonumber \\ b_{11}&= b_{110} +b_{112} H_1^2 +\cdots \nonumber \\ b_{22}&= b_{220} +b_{222} H_2^2 +\cdots \nonumber \\ b_{33} :&= b_{330} H_1^{-2} +b_{332} +\cdots \end{aligned}$$

(74)

$$\begin{aligned} L_{10}&= L_{100} H_1^4 +L_{101} H_1^3 +(L_{102} H_2^2 +L_{103} )H_1^2\nonumber \\&+\, L_{104} H_1 +L_{105} H_2^4 +L_{106} H_2^2 +L_{107} +\cdots \nonumber \\ L_{11}^c&= L_{200} H_1^4 +(L_{201} H_2^2 +L_{202} )H_1^2 +L_{203} +\cdots \nonumber \\ L_{11}^s&= L_{300} H_1^4 +L_{301} H_1^3 +(L_{302} H_2^2\nonumber \\&+\, L_{303} )H_1^2 +L_{304} H_1 +L_{305} +\cdots \end{aligned}$$

(75)

$$\begin{aligned} V_0&= V_{011} H_1 +V_{012} H_2 +V_{021} H_1^2\nonumber \\&+V_{022} H_2^2 +V_{023} H_1 H_2 +\cdots \nonumber \\ V_1^c&= V_{11}^c H_1 +V_{12}^c H_2 +V_{13}^c H_1^2\nonumber \\&+V_{14}^c H_2^2 +V_{15}^c H_1 H_2 +\cdots \nonumber \\ V_1^s&= V_{11}^s H_1 +V_{12}^s H_2 +V_{13}^s H_1^2\nonumber \\&+V_{14}^s H_2^2 +V_{15}^s H_1 H_2 +\cdots \end{aligned}$$

(76)

Substituting Eqs. (74)–(76) into Eqs. (70)–(72)and letting the coefficients of the same power of $H_1 ,H_2 $ vanish yield the coefficients of polynomials $V_0,V_1^c $ and $V_1^s $.

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R. C. Hu
- 1
W. Q. Zhu
- 1
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1.Department of Mechanics, State Key Lab of Fluid Power Transmission and ControlZhejiang UniversityHangzhouPeople’s Republic of China

Stochastic optimal control of MDOF nonlinear systems under combined harmonic and wide-band noise excitations

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