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A non-ideally excited pendulum controlled by SDRE technique

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Abstract

In this work, the perturbation theory is applied to the analysis of an electromechanical pendulum system. The frequency response behavior of the system is studied, and the existence of unstable poles is detected using the Routh–Hurwitz criterion. Numerical simulations show the existence of nonlinear behaviors such as hysteresis and the Sommerfeld effect in the resonance region. To damp the electromechanical system oscillations due to the nonlinear characteristics of the system the State Dependent Riccati Equation (SDRE) technique is used. The SDRE control strategy is applied considering two control signals, a feedback control that force the state trajectory of the system to a previously defined periodic orbit, and a nonlinear feedforward control that keeps the system motion synchronized to the periodic orbit. Additionally, the robustness of the control technique is tested for parametric uncertainties.

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Notes

  1. Input signals, forces, or any forcing terms in a differential equation.

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Acknowledgments

The authors would like to acknowledge São Paulo Research Foundation—FAPESP (Grant: 2013/04101-6), and Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPQ (Grant: 447539/2014-0 and 303903/2014-7) for the financial support.

Author information

Authors and Affiliations

  1. Departamento Acadêmico de Matemática - DAMAT, Universidade Tecnológica Federal do Paraná - UTFPR, Ponta Grossa, PR, Brazil

    Angelo Marcelo Tusset

  2. Instituto de Ciência e Tecnologia de Sorocaba - ICTS, Universidade Estadual Paulista - UNESP, Sorocaba, SP, Brazil

    Átila Madureira Bueno

  3. Academia da Força Aérea (AFA), Pirassununga, SP, Brazil

    João Paulo Martins dos Santos

  4. Departamento de Ciência da Computação e Estatística - DCCE, Universidade Estadual Paulista - UNESP, São José do Rio Preto, SP, Brazil

    Masayoshi Tsuchida

  5. Divisão de Engenharia Aeronáutica, Instituto Tecnológico de Aeronática - ITA, São José dos Campos, SP, Brazil

    José Manoel Balthazar

Authors
  1. Angelo Marcelo Tusset
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  2. Átila Madureira Bueno
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  3. João Paulo Martins dos Santos
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  4. Masayoshi Tsuchida
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  5. José Manoel Balthazar
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Corresponding author

Correspondence to Átila Madureira Bueno.

Additional information

Technical Editor: Aline Souza de Paula.

Appendix

Appendix

$$\begin{aligned}J_{1} &= \frac{{a^{\prime}_{1} \omega_{1}}}{2\omega_{3}}\left\{\cos[ {(\varOmega - \omega_{3} )\tau + (\beta_{1} - \beta_{3} )}]\right. \\& \left. \quad -\, \cos[(\varOmega - \omega_{3})\tau+ (\beta_{1} + \beta_{3})] \right\}\end{aligned}$$
(35)
$$\begin{aligned} J_{2} &= \frac{{a_{1} \beta^{\prime}_{1} \omega_{1} }}{{2\omega_{3} }}\left\{ \sin [ (\varOmega + \omega_{3} )\tau + (\beta_{1} + \beta_{3} )] \right.\\ & \quad \left. - \sin [ (\varOmega - \omega_{3} )\tau + (\beta_{1} - \beta_{3} ) ] \right\} \end{aligned}$$
(36)
$$J_{3} = - \frac{{a_{3} \mu_{3} }}{2}\left[ {1 - \cos \left( {2\omega_{3} \tau + 2\beta_{3} } \right)} \right]$$
(37)
$$\begin{aligned} J_{4} &= - \varepsilon \frac{{\delta a_{1} \omega_{1} \mu_{3} }} {{2\omega_{3} }} \left \{ \cos [(\varOmega - \omega_{3} )\tau + (\beta_{1} - \beta_{3} )] \right. \\ & \left. \quad - \cos [(\varOmega + \omega_{3} )\tau + (\beta_{1} + \beta_{3} )] \right\}\end{aligned}$$
(38)
$$\begin{aligned} J_{5} &= - \frac{{\delta a^{\prime}_{1} \omega_{1} }}{2} \left\{ \cos [ (\varOmega - \omega_{3} )\tau + (\beta_{1} - \beta_{3} ) ] \right. \\ & \left. \quad - \cos [ (\varOmega + \omega_{3} )\tau + (\beta_{1} + \beta_{3} ) ] \right\} \end{aligned}$$
(39)
$$\begin{aligned} J_{6} &= - \frac{{\delta a_{1} \omega_{1} \beta^{\prime}_{1} }}{2} \left\{ \sin [ (\varOmega + \omega_{3} ) \tau + (\beta_{1} + \beta_{3})] \right. \\ & \left. \quad - \sin [ (\varOmega - \omega_{3} ) \tau + (\beta_{1} - \beta_{3}) ] \right\} \end{aligned}$$
(40)
$$\begin{aligned} J_{7} &= - \frac{{\delta a_{1} }}{2}\left( {\omega_{1} \varOmega - \varOmega_{3}^{2} } \right) \left\{ \sin [ (\varOmega + \omega_{3} )\tau + (\beta_{1} + \beta_{3} )] \right. \\ & \left. \quad - \sin [ (\varOmega - \omega_{3} )\tau + (\beta_{1} - \beta_{3} )] \right\} \end{aligned}$$
(41)
$$J^{\prime}_{1} = \frac{{a^{\prime}_{1} \omega_{1} }}{{2a_{3} \omega_{3} }}\left\{ {\sin \left[ {(\varOmega + \omega_{3} )\tau + (\beta_{1} + \beta_{3} )} \right] - \cos \left[ {(\omega_{3} - \varOmega )\tau + (\beta_{3} - \beta_{1} )} \right]} \right\}$$
(42)
$$J^{\prime}_{2} = \frac{{a_{1} \beta^{\prime}_{1} \omega_{1} }}{{2a_{3} \omega_{3} }}\left\{ {\cos \left[ {(\varOmega + \omega_{3} )\tau + (\beta_{1} + \beta_{3} )} \right] + \cos \left[ {(\varOmega - \omega_{3} )\tau + (\beta_{1} - \beta_{3} )} \right]} \right\}$$
(43)
$$J^{\prime}_{3} = \frac{{a_{1} \omega_{1} \varOmega }}{{2a_{3} \omega_{3} }}\left\{ {\cos \left[ {(\varOmega + \omega_{3} )\tau + (\beta_{1} + \beta_{3} )} \right] + \cos \left[ {(\varOmega - \omega_{3} )\tau + (\beta_{1} - \beta_{3} )} \right]} \right\}$$
(44)
$$J^{\prime}_{4} = - \varepsilon \frac{{\mu_{3} }}{2}\sin \left[ {2(\omega_{3} \tau + \beta_{3} )} \right]$$
(45)
$$J^{\prime}_{5} = - \varepsilon \frac{{\mu_{3} \delta a^{\prime}_{1} \omega_{1} }}{{2a_{3} \omega_{3} }}\left\{ {\sin \left[ {(\varOmega + \omega_{3} )\tau + (\beta_{1} + \beta_{3} )} \right] - \sin \left[ {(\omega_{3} - \varOmega )\tau + (\beta_{3} - \beta_{1} )} \right]} \right\}$$
(46)
$$J^{\prime}_{6} = - \frac{{\delta a^{\prime}_{1} \omega_{1} }}{{2a_{3} \omega_{3} }}\left\{ {\sin \left[ {(\varOmega + \omega_{3} )\tau + (\beta_{1} + \beta_{3} )} \right] - \sin \left[ {(\omega_{3} - \varOmega )\tau + (\beta_{1} - \beta_{3} )} \right]} \right\}$$
(47)
$$J^{\prime}_{7} = - \frac{{\delta a_{1} \omega_{1} \beta^{\prime}_{1} }}{{2a_{3} \omega_{3} }}\left\{ {\cos \left[ {(\varOmega + \omega_{3} )\tau + (\beta_{1} + \beta_{3} )} \right] + \sin \left[ {(\varOmega - \omega_{3} )\tau + (\beta_{1} - \beta_{3} )} \right]} \right\}$$
(48)
$$J^{\prime}_{8} = - \frac{{\delta a_{1} }}{{2a_{3} \omega_{3} }}\left( {\omega_{1} \varOmega - \omega_{3}^{2} } \right)\left\{ {\cos \left[ {(\varOmega + \omega_{3} )\tau + (\beta_{1} + \beta_{3} )} \right] + \cos \left[ {(\varOmega - \omega_{3} )\tau + (\beta_{1} - \beta_{3} )} \right]} \right\}$$
(49)
$$G_{1} = \left( {\frac{{a_{3}^{2} \omega_{3}^{2} }}{2} + \frac{{a_{1}^{2} \delta^{2} a_{3} }}{4}} \right)\left\{ {\sin \left[ {(2\varOmega + 2\omega_{3} )\tau + (\beta_{1} + 3\beta_{3} )} \right] - \sin \left[ {(\omega_{3} - \varOmega )\tau + (\beta_{3} - \beta_{1} )} \right]} \right\}$$
(50)
$$G_{2} = \left( { - \frac{{a_{3}^{2} \omega_{3}^{2} }}{2}} \right)\left\{ {\sin \left[ {(\varOmega + 3\omega_{3} )\tau + (\beta_{1} + 3\beta_{3} )} \right] - \sin \left[ {(3\omega_{3} - \varOmega )\tau + (3\beta_{3} - \beta_{1} )} \right]} \right\}$$
(51)
$$G_{3} = \left( { - \frac{{a_{3}^{2} \omega_{3} a_{1} \delta }}{4}} \right)\left\{ {\sin \left[ {(2\varOmega + 2\omega_{3} )\tau + (2\beta_{1} + 2\beta_{3} )} \right] - \sin (2\omega_{3} \tau + 2\beta_{3} )} \right\}$$
(52)
$$G_{4} = \left( { - \frac{{a_{1}^{2} \delta^{3} }}{8}} \right)\left\{ {\sin \left[ {(3\varOmega + \omega_{3} )\tau + (3\beta_{1} + \beta_{3} )} \right] - \sin \left[ {(\varOmega + \omega_{3} )\tau + (\beta_{3} + \beta_{1} )} \right]} \right\}$$
(53)
$$G_{5} = \left( { - \frac{{a_{1}^{2} \delta^{2} a_{3} }}{8}} \right)\left\{ {\sin \left[ {(3\varOmega - \omega_{3} )\tau + (3\beta_{1} - \beta_{3} )} \right] - \sin \left[ {(\varOmega - \omega_{3} )\tau + (\beta_{1} - \beta_{3} )} \right]} \right\}$$
(54)
$$G_{6} = \left( {\frac{{a_{3}^{2} \omega_{3}^{2} a_{1} \delta }}{4} + \frac{{a_{3}^{2} \delta^{2} }}{2} - \frac{{a_{1}^{3} \delta^{3} }}{8}} \right)\left[ {\sin (2\varOmega \tau + 2\beta_{1} )} \right]$$
(55)
$$G_{7} = \left( { - \frac{{a_{3}^{2} \omega_{3}^{2} \delta a_{1} }}{8}} \right)\left\{ {\sin \left[ {(2\varOmega + 2\omega_{3} )\tau + (2\beta_{1} + 2\beta_{3} )} \right] - \sin (2\omega_{3} \tau + 2\beta_{3} )} \right\}$$
(56)
$$G_{8} = \left( { - \frac{{a_{3}^{2} \omega_{3}^{2} \delta a_{1} }}{8}} \right)\left\{ {\sin (2\omega_{3} \tau + 2\beta_{3} ) - \sin \left[ {(2\omega_{3} - 2\varOmega )\tau + (2\beta_{3} - 2\beta_{1} )} \right]} \right\}$$
(57)
$$G_{9} = \left( {\frac{{a_{1}^{2} \delta^{2} a_{3} \omega_{3} }}{4}} \right)\left\{ {\sin \left[ {(3\varOmega - \omega_{3} )\tau + (3\beta_{1} - \beta_{3} )} \right] - \sin \left[ {(\varOmega - \omega_{3} )\tau + (\beta_{1} - \beta_{3} )} \right]} \right\}$$
(58)
$$G_{10} = \left( { - \frac{{a_{1}^{2} \delta a_{3} \omega_{3} }}{4}} \right)\left\{ {\sin \left[ {(3\varOmega + \omega_{3} )\tau + (3\beta_{1} + \beta_{3} )} \right] - \sin \left[ {(\varOmega + \omega_{3} )\tau + (\beta_{1} + \beta_{3} )} \right]} \right\}$$
(59)
$$G_{11} = \left( {\frac{{a_{1} \delta a_{3} \omega_{3} }}{8}} \right)\left\{ {\sin (4\varOmega \tau + 4\beta_{1} ) - \sin (2\varOmega \tau + 2\beta_{1} )} \right\}$$
(60)
$$G_{12} = \left( {\frac{{a^{\prime}_{3} \omega_{3} }}{2}} \right)\left\{ {\cos \left[ {(\omega_{3} - \varOmega + )\tau + (\beta_{3} - \beta_{1} )} \right] - \sin \left[ {(\omega_{3} - \varOmega )\tau + (\beta_{3} - \beta_{1} )} \right]} \right\}$$
(61)
$$G_{13} = \left( {\frac{{a_{3} \omega_{3} \beta^{\prime}_{3} }}{2}} \right)\left\{ {\sin \left[ {(\omega_{3} + \varOmega )\tau + (\beta_{3} + \beta_{1} )} \right] - \sin \left[ {(\omega_{3} - \varOmega )\tau + (\beta_{3} - \beta_{1} )} \right]} \right\}$$
(62)
$$G_{14} = \left( {\frac{{a_{3} \omega_{3}^{2} }}{2}} \right)\left\{ {\sin \left[ {(\omega_{3} + \varOmega )\tau + (\beta_{3} + \beta_{1} )} \right] - \sin \left[ {(\omega_{3} - \varOmega )\tau + (\beta_{3} - \beta_{1} )} \right]} \right\}$$
(63)
$$G^{\prime}_{1} = \left( {\frac{{a_{3}^{3} \omega_{3}^{2} }}{4} - \frac{{a_{3}^{3} \omega_{3}^{3} }}{8} + \frac{{a_{1}^{2} \delta^{2} \omega_{1}^{2} a_{3} }}{4}} \right)\left\{ {\cos \left[ {(\varOmega + \omega_{3} )\tau + (\beta_{1} + \beta_{3} )} \right]} \right\}$$
(64)
$$G^{\prime}_{2} = \left( { - \frac{{a_{3}^{3} \omega_{3}^{2} }}{4}} \right)\left\{ {\cos \left[ {(3\omega_{3} + \varOmega )\tau + (3\beta_{3} + \beta_{1} )} \right] + \cos \left[ {(3\omega_{3} - \varOmega )\tau + (3\beta_{3} - \beta_{1} )} \right]} \right\}$$
(65)
$$G^{\prime}_{3} = \left( {\frac{{a_{3}^{2} \omega_{3} a_{1} \delta }}{4}} \right)\left\{ \begin{array}{l} \left[ {\cos (2\omega_{3} \tau ) + \cos (2\omega_{3} - 2\varOmega )\tau + (2\beta_{3} - 2\beta_{1} )} \right] \hfill \\ - \left[ {\cos \left[ {(2\omega_{3} + 2\varOmega )\tau + (3\beta_{3} + 2\beta_{1} )} \right] + \cos (2\omega_{3} \tau + 2\beta_{3} )} \right] \hfill \\ \end{array} \right\}$$
(66)
$$G^{\prime}_{4} = \left( { - \frac{{a_{1}^{2} \omega_{1}^{3} a_{3} \delta^{2} }}{8}} \right)\left\{ \begin{array}{l} \left[ {\cos \left( {(3\varOmega + \omega_{3} )\tau + (3\beta_{1} + \beta_{3} )} \right) + \cos \left( {(\varOmega + \omega_{3} )\tau } \right)} \right] \hfill \\ + \left[ {\cos \left[ {(2\varOmega - \omega_{3} )\tau + (3\beta_{1} - \beta_{3} )} \right] + \cos \left[ {(\varOmega - \omega_{3} )\tau + (\beta_{1} - \beta_{3} )} \right]} \right] \hfill \\ \end{array} \right\}$$
(67)
$$G^{\prime}_{5} = \left( {\frac{{a_{3}^{2} \omega_{3} a_{1} \delta }}{4} + \frac{{a_{3}^{3} \omega_{1}^{2} \delta^{3} }}{4} - \frac{{a_{1}^{3} \delta^{3} \omega_{1}^{2} }}{8}} \right)\left[ {1 - \cos (2\varOmega \tau + 2\beta_{1} )} \right]$$
(68)
$$G^{\prime}_{6} = \left( { - \frac{{a_{3}^{2} \omega_{3}^{2} a_{1} \delta }}{8}} \right)\left\{ \begin{array}{l} \left[ {\cos \left( {(2\varOmega + \omega_{3} )\tau + (2\beta_{1} + 2\beta_{3} )} \right) + \cos \left( {2\omega_{3} \tau + 2\beta_{3} } \right)} \right] \hfill \\ + \left[ {\cos (2\omega_{3} \tau + 2\beta_{3} ) + \cos \left[ {(2\omega_{3} - 2\varOmega )\tau + (2\beta_{3} - 2\beta_{1} )} \right]} \right] \hfill \\ \end{array} \right\}$$
(69)
$$G^{\prime}_{7} = \left( {\frac{{a_{1}^{2} \omega_{3}^{2} a_{3} \delta }}{4}} \right)\left\{ \begin{array}{l} \left[ {\cos \left( {(3\varOmega - \omega_{3} )\tau + (3\beta_{1} - \beta_{3} )} \right) + \cos \left( {(\varOmega - \omega_{3} )\tau + (\beta_{1} - \beta_{3} )} \right)} \right] \hfill \\ - \left[ {\cos \left( {(\omega_{3} + 3\varOmega )\tau + (3\beta_{1} + \beta_{3} )} \right) + \cos \left[ {(\omega_{3} + \varOmega )\tau + (\beta_{3} + \beta_{1} )} \right]} \right] \hfill \\ \end{array} \right\}$$
(70)
$$\begin{aligned} G^{\prime}_{8} &= \left( { - \frac{{a_{3}^{3} \omega_{1}^{2} \delta^{3} }}{8}} \right)\left[ {\cos \left( {4\varOmega \tau + 4\beta_{1} } \right) + \cos \left( {2\varOmega \tau + 2\beta_{1} } \right)} \right] \\ & \quad + \, \left( {\frac{{a^{\prime}_{3} \omega_{3} }}{2}} \right)\left[ {\sin \left( {\left( {\varOmega + \omega_{3} } \right)\tau + (\beta_{1} + \beta_{3} )} \right) - \sin \left( {\left( {\varOmega - \omega_{3} } \right)\tau + (\beta_{1} - \beta_{3} )} \right)} \right] \\ \end{aligned}$$
(71)
$$G^{\prime}_{9} = \left( {\frac{{a_{3} \omega_{3} \beta^{\prime}_{3} }}{2}} \right)\left\{ {\cos \left[ {(\varOmega + \omega_{3} )\tau + (\beta_{3} + \beta_{1} )} \right] + \cos \left[ {(\omega_{3} - \varOmega )\tau + (\beta_{3} - \beta_{1} )} \right]} \right\}$$
(72)
$$G^{\prime}_{10} = \left( {\frac{{a_{3} \omega_{3}^{2} }}{2}} \right)\left\{ {\cos \left[ {(\varOmega + \omega_{3} )\tau + (\beta_{3} + \beta_{1} )} \right] + \cos \left[ {(\omega_{3} - \varOmega )\tau + (\beta_{3} - \beta_{1} )} \right]} \right\}$$
(73)

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Tusset, A.M., Bueno, Á.M., dos Santos, J.P.M. et al. A non-ideally excited pendulum controlled by SDRE technique. J Braz. Soc. Mech. Sci. Eng. 38, 2459–2472 (2016). https://doi.org/10.1007/s40430-016-0517-7

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