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State estimation in mechanical systems of fractional-order based on a family of proportional \({\varvec{\rho }}\)-integral observers

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Abstract

In control theory, there are many proposals to solve the problem of observer design. This paper studies the Mittag-Leffler stability of a class of dynamic observers for nonlinear fractional-order systems, given in the observable canonical form and defined by the Caputo fractional derivative. We prove that the Riemann–Liouville integral could be employed to provide robustness against noisy measurements during the estimation problem. Based on this advantage, the main result of this paper consists of the design of a family of high-gain proportional \(\rho \)-integral observers employed to estimate unmeasured state variables of nonlinear fractional systems of commensurate order. Three illustrative numerical examples of mechanical systems are provided, which corroborate the effectiveness of the proposed algorithms.

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Acknowledgements

Oscar Martínez-Fuentes acknowledges the support provided by the Mexico’s National Council of Science and Technology (CONACyT) under the Postdoctoral Fellowship Program 2022. Guillermo Fernández-Anaya acknowledges the support provided by División de Investigación y Posgrado de la Universidad Iberoamericana.

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Authors and Affiliations

  1. Department of Electronics, INAOE, Luis Enrique Erro No. 1, Santa María Tonantzintla, 72840, San Andrés Cholula, Puebla, Mexico

    Oscar Martínez-Fuentes & Esteban Tlelo-Cuautle

  2. Department of Multidisciplinary Engineering, Higher Education Center at McAllen, Texas A &M University, 6200 Tres Lagos Blvd, McAllen, TX, 78504, USA

    Aldo Jonathan Muñoz-Vázquez

  3. Departamento de Física y Matemáticas, Universidad Iberoamericana Ciudad de México, Prol. Paseo de la Reforma 880, Lomas de Santa Fe, 01219, Mexico City, Álvaro Obregón, Mexico

    Guillermo Fernández-Anaya

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  1. Oscar Martínez-Fuentes
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Appendix

Appendix

As extra information to help the reader, the structures associated with the observers used for each model in the case studies presented in this article are presented below.

1.1 Simple pendulum

The observer structure (33) for the simple pendulum (46), varying the parameter \(1\le \rho \le 5\) takes the following forms:

  • (i) \(\rho =1\)

    $$\begin{aligned} \varSigma _{\hat{{\text {P}}}}=\left\{ \begin{aligned} {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_1&=\hat{x}_2-k_{I_1}z_1-k_{P_{1}}\left( \hat{x}_1-y_{{\text {P}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_2&=-\frac{g}{L} \sin \hat{x}_1-k_{I_2}z_1\\&\qquad -k_{P_{2}}\left( \hat{x}_1-y_{{\text {P}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_1&=\hat{x}_1-y_{{\text {P}}}-k_{z_1} z_1 \end{aligned} \right. \end{aligned}$$
    (54)

  • (ii) \(\rho =2\)

    $$\begin{aligned} \varSigma _{\hat{{\text {P}}}}=\left\{ \begin{aligned} {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_1&=\hat{x}_2-k_{I_1}z_1-k_{P_{1}}\left( \hat{x}_1-y_{{\text {P}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_2&=-\frac{g}{L} \sin \hat{x}_1-k_{I_2}z_1\\&\qquad -k_{P_{2}}\left( \hat{x}_1-y_{{\text {P}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_1&=z_2-k_{z_1} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_2&=\hat{x}_1-y_{{\text {P}}}-k_{z_2} z_1 \end{aligned} \right. \end{aligned}$$
    (55)

  • (iii) \(\rho =3\)

    $$\begin{aligned} \varSigma _{\hat{{\text {P}}}}=\left\{ \begin{aligned} {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_1&=\hat{x}_2-k_{I_1}z_1-k_{P_{1}}\left( \hat{x}_1-y_{{\text {P}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_2&=-\frac{g}{L} \sin \hat{x}_1-k_{I_2}z_1\\&\qquad -k_{P_{2}}\left( \hat{x}_1-y_{{\text {P}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_1&=z_2-k_{z_1} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_2&=z_3-k_{z_2} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_3&=\hat{x}_1-y_{{\text {P}}}-k_{z_3} z_1 \end{aligned} \right. \end{aligned}$$
    (56)

  • (iv) \(\rho =4\)

    $$\begin{aligned} \varSigma _{\hat{{\text {P}}}}=\left\{ \begin{aligned} {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_1&=\hat{x}_2-k_{I_1}z_1-k_{P_{1}}\left( \hat{x}_1-y_{{\text {P}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_2&=-\frac{g}{L} \sin \hat{x}_1-k_{I_2}z_1\\&\qquad -k_{P_{2}}\left( \hat{x}_1-y_{{\text {P}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_1&=z_2-k_{z_1} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_2&=z_3-k_{z_2} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_3&=z_4-k_{z_3} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_4&=\hat{x}_1-y_{{\text {P}}}-k_{z_4} z_1 \end{aligned} \right. \end{aligned}$$
    (57)

  • (v) \(\rho =5\)

    $$\begin{aligned} \varSigma _{\hat{{\text {P}}}}=\left\{ \begin{aligned} {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_1&=\hat{x}_2-k_{I_1}z_1-k_{P_{1}}\left( \hat{x}_1-y_{{\text {P}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_2&=-\frac{g}{L} \sin \hat{x}_1-k_{I_2}z_1\\&\qquad -k_{P_{2}}\left( \hat{x}_1-y_{{\text {P}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_1&=z_2-k_{z_1} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_2&=z_3-k_{z_2} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_3&=z_4-k_{z_3} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_4&=z_5-k_{z_4} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_5&=\hat{x}_1-y_{{\text {P}}}-k_{z_5} z_1 \end{aligned} \right. \end{aligned}$$
    (58)

1.2 The flexible joint mechanism

On the other hand, doing similar calculations the observer structure (33) for the robot with flexible joint (49), varying the parameter \(1\le \rho \le 5\) takes the following forms:

  • (i) \(\rho =1\)

    $$\begin{aligned} \varSigma _{\hat{{\text {P}}}}=\left\{ \begin{aligned} {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_1&=\hat{x}_2-k_{I_1}z_1-k_{P_{1}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_2&=\hat{x}_3-9.81\sin \hat{x}_1-0.1\hat{x}_2-\hat{x}_1\\&\qquad -k_{I_2}z_1-k_{P_{2}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_3&=\hat{x}_4-k_{I_3}z_1-k_{P_{3}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_4&=5\hat{x}_1-5\hat{x}_3-\hat{x}_4+5u\\&\qquad -k_{I_4}z_1-k_{P_{4}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_1&=\hat{x}_1-y_{{\text {R}}}-k_{z_1} z_1 \end{aligned} \right. \end{aligned}$$
    (59)

  • (ii) \(\rho =2\)

    $$\begin{aligned} \varSigma _{\hat{{\text {P}}}}=\left\{ \begin{aligned} {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_1&=\hat{x}_2-k_{I_1}z_1-k_{P_{1}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_2&=\hat{x}_3-9.81\sin \hat{x}_1-0.1\hat{x}_2-\hat{x}_1\\&\qquad -k_{I_2}z_1-k_{P_{2}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_3&=\hat{x}_4-k_{I_3}z_1-k_{P_{3}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_4&=5\hat{x}_1-5\hat{x}_3-\hat{x}_4+5u\\&\qquad -k_{I_4}z_1-k_{P_{4}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_1&=z_2-k_{z_1} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_2&=\hat{x}_1-y_{{\text {R}}}-k_{z_2} z_1 \end{aligned} \right. \end{aligned}$$
    (60)

  • (iii) \(\rho =3\)

    $$\begin{aligned} \varSigma _{\hat{{\text {P}}}}=\left\{ \begin{aligned} {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_1&=\hat{x}_2-k_{I_1}z_1-k_{P_{1}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_2&=\hat{x}_3-9.81\sin \hat{x}_1-0.1\hat{x}_2-\hat{x}_1\\&\qquad -k_{I_2}z_1-k_{P_{2}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_3&=\hat{x}_4-k_{I_3}z_1-k_{P_{3}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_4&=5\hat{x}_1-5\hat{x}_3-\hat{x}_4+5u\\&\qquad -k_{I_4}z_1-k_{P_{4}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_1&=z_2-k_{z_1} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_2&=z_3-k_{z_2} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_3&=\hat{x}_1-y_{{\text {R}}}-k_{z_3} z_1 \end{aligned} \right. \end{aligned}$$
    (61)

  • (iv) \(\rho =4\)

    $$\begin{aligned} \varSigma _{\hat{{\text {P}}}}=\left\{ \begin{aligned} {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_1&=\hat{x}_2-k_{I_1}z_1-k_{P_{1}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_2&=\hat{x}_3-9.81\sin \hat{x}_1-0.1\hat{x}_2-\hat{x}_1\\&\qquad -k_{I_2}z_1-k_{P_{2}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_3&=\hat{x}_4-k_{I_3}z_1-k_{P_{3}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_4&=5\hat{x}_1-5\hat{x}_3-\hat{x}_4+5u\\&\qquad -k_{I_4}z_1-k_{P_{4}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_1&=z_2-k_{z_1} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_2&=z_3-k_{z_2} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_3&=z_4-k_{z_3} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_4&=\hat{x}_1-y_{{\text {R}}}-k_{z_4} z_1 \end{aligned} \right. \end{aligned}$$
    (62)

  • (v) \(\rho =5\)

    $$\begin{aligned} \varSigma _{\hat{{\text {P}}}}=\left\{ \begin{aligned} {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_1&=\hat{x}_2-k_{I_1}z_1-k_{P_{1}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_2&=\hat{x}_3-9.81\sin \hat{x}_1-0.1\hat{x}_2-\hat{x}_1\\&\qquad -k_{I_2}z_1-k_{P_{2}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_3&=\hat{x}_4-k_{I_3}z_1-k_{P_{3}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_4&=5\hat{x}_1-5\hat{x}_3-\hat{x}_4+5u\\&\qquad -k_{I_4}z_1-k_{P_{4}}\left( \hat{x}_1-y_{{\text {R}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_1&=z_2-k_{z_1} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_2&=z_3-k_{z_2} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_3&=z_4-k_{z_3} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_4&=z_5-k_{z_4} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_5&=\hat{x}_1-y_{{\text {R}}}-k_{z_5} z_1 \end{aligned} \right. \end{aligned}$$
    (63)

1.3 The inverted pendulum

The observer structure (33) for the inverted pendulum (50), varying the parameter \(1\le \rho \le 2\), has the following forms:

  • (i) \(\rho =1\)

    $$\begin{aligned} \varSigma _{\hat{{\text {P}}}}=\left\{ \begin{aligned} {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_1&=\hat{x}_2-k_{I_1}z_1-k_{P_{1}}\left( \hat{x}_1-y_{{\text {IP}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_2&=\hat{A} \left( \sin \hat{x}_1+u\cos \hat{x}_1\right) -k_{I_2}z_1\\&\qquad -k_{P_{2}}\left( \hat{x}_1-y_{{\text {IP}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_1&=\hat{x}_1-y_{{\text {IP}}}-k_{z_1} z_1 \end{aligned} \right. \end{aligned}$$
    (64)

  • (ii) \(\rho =2\)

    $$\begin{aligned} \varSigma _{\hat{{\text {P}}}}=\left\{ \begin{aligned} {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_1&=\hat{x}_2-k_{I_1}z_1-k_{P_{1}}\left( \hat{x}_1-y_{{\text {IP}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha \hat{x}_2&=\hat{A}\left( \sin \hat{x}_1+u\cos \hat{x}_1\right) -k_{I_2}z_1\\&\qquad -k_{P_{2}}\left( \hat{x}_1-y_{{\text {IP}}}\right) \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_1&=z_2-k_{z_1} z_1 \\ {_0^{C}}{\mathcal {D}}_t^\alpha z_2&=\hat{x}_1-y_{{\text {IP}}}-k_{z_2} z_1 \end{aligned} \right. \end{aligned}$$
    (65)

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Martínez-Fuentes, O., Muñoz-Vázquez, A.J., Fernández-Anaya, G. et al. State estimation in mechanical systems of fractional-order based on a family of proportional \({\varvec{\rho }}\)-integral observers. Nonlinear Dyn 111, 19879–19899 (2023). https://doi.org/10.1007/s11071-023-08919-4

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