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Filtered low-power multi-high-gain observer design for a class of nonlinear systems

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Abstract

In this paper, a high-gain observer (HGO) approach is used to design a nonlinear observer for nonlinear systems in canonical observer form. The design uses multiple low-power (LP) observers with low-pass filters that provide a state estimation approach as a weighted sum of the individual observer state variables. As for the typical HGO design, it is shown that the estimation error dynamics of the proposed observer are input-to-state stable (ISS) with regard to measurement noise with a tunable convergence rate. It is also demonstrated that the multiple LP observer yields a smaller bound on the estimation error than the HGO and LP. Finally, the steady-state behaviour of the observer is characterized. For the linear case, the observer improves the sensitivity of the state estimates to measurement noise. The effectiveness of the suggested approach is demonstrated for the identification of a robot arm for which the observer is used as a numerical differentiator.

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  1. Seyed Mohammadmoein Mousavi and Martin Guay contributed equally to this work.

Authors and Affiliations

  1. Department of Chemical Engineering, Queen’s University, 99 University Ave, Kingston, Ontario, K7L 3N6, Canada

    Seyed Mohammadmoein Mousavi & Martin Guay

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  1. Seyed Mohammadmoein Mousavi
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Correspondence to Martin Guay.

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Appendix A Regressor dynamics

Appendix A Regressor dynamics

The regressor used in (43) has the following dynamics. Note that the remainder of the arrays not listed below are zero. Also, \(\varvec{c}_i,\varvec{s}_i, \varvec{c}_{ij},\varvec{s}_{ij}\) indicate the abbreviated form of \(\cos (\Theta _i)\), \(\sin (\Theta _i)\), \(\cos (\Theta _i+\Theta _j)\), \(\sin (\Theta _i+\Theta _j)\), respectively.

$$\begin{aligned}&\varvec{{\textbf {Y}}}(1,1)=\ddot{\Theta }_1\\&\varvec{{\textbf {Y}}}(1,2)=\varvec{c}_2^2\ddot{\Theta }_1-2\varvec{s}_2\varvec{c}_2\dot{\Theta }_1\dot{\Theta }_2\\&\varvec{{\textbf {Y}}}(1,3)=\varvec{s}_2^2\ddot{\Theta }_1+2\varvec{s}_2\varvec{c}_2\dot{\Theta }_1\dot{\Theta }_2\\&\varvec{{\textbf {Y}}}(1,4)=\varvec{s}_{23}^2\ddot{\Theta }_1+2\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1\dot{\Theta }_2 +2\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1\dot{\Theta }_3\\&\varvec{{\textbf {Y}}}(1,5)=\varvec{c}_{23}^2\ddot{\Theta }_1-2\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1\dot{\Theta }_2 -2\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1\dot{\Theta }_3\\&\varvec{{\textbf {Y}}}(1,6)=-2\varvec{c}_2\varvec{s}_{23}\ddot{\Theta }_1+2(\varvec{s}_2\varvec{s}_{23}\\&\qquad \qquad \quad -\varvec{c}_2\varvec{c}_{23}) \dot{\Theta }_1\dot{\Theta }_2 -2\varvec{c}_2\varvec{c}_{23}\dot{\Theta }_1\dot{\Theta }_3\\&\varvec{{\textbf {Y}}}(1,7)=\varvec{c}_{23}^2\ddot{\Theta }_1-\varvec{c}_{23}\ddot{\Theta }_4-2\varvec{s}_{23}\varvec{c}_{23} \dot{\Theta }_1\dot{\Theta }_2\\&\qquad \qquad \quad -2\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1\dot{\Theta }_3 + \varvec{s}_{23}(\dot{\Theta }_2\dot{\Theta }_4 +\dot{\Theta }_3 \dot{\Theta }_4)\\&\varvec{{\textbf {Y}}}(1,8)=-2\varvec{c}_2\varvec{s}_{23}\ddot{\Theta }_1+2(\varvec{s}_2\varvec{s}_{23}\\&\qquad \qquad \quad -\varvec{c}_2\varvec{c}_{23}) \dot{\Theta }_1\dot{\Theta }_2 -2\varvec{c}_2\varvec{c}_{23}\dot{\Theta }_1\dot{\Theta }_3\\&\varvec{{\textbf {Y}}}(1,9)=\varvec{s}_{23}^2\ddot{\Theta }_1+2\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1\dot{\Theta }_2 +2\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1\dot{\Theta }_3\\&\varvec{{\textbf {Y}}}(1,10)=-2\varvec{c}_2\varvec{s}_{23}\ddot{\Theta }_1+2(\varvec{s}_2\varvec{s}_{23}\\&\qquad \qquad \quad -\varvec{c}_2\varvec{c}_{23}) \dot{\Theta }_1\dot{\Theta }_2 -2\varvec{c}_2\varvec{c}_{23}\dot{\Theta }_1\dot{\Theta }_3\\&\varvec{{\textbf {Y}}}(1,13)=\text {sign}(\dot{\Theta }_1)\\&\varvec{{\textbf {Y}}}(1,17)=\dot{\Theta }_1\\&\varvec{{\textbf {Y}}}(2,2)=\ddot{\Theta }_2+\varvec{s}_2\varvec{c}_2\dot{\Theta }_1^2\\&\varvec{{\textbf {Y}}}(2,3)=-\varvec{s}_2\varvec{c}_2\dot{\Theta }_1^2\\&\varvec{{\textbf {Y}}}(2,4)=\ddot{\Theta }_2+\ddot{\Theta }_3-\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1^2\\&\varvec{{\textbf {Y}}}(2,5)=\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1^2\\&\varvec{{\textbf {Y}}}(2,6)=-2\varvec{s}_3\ddot{\Theta }_2-(\varvec{s}_2\varvec{s}_{23}\\&\qquad \qquad \quad -\varvec{c}_2\varvec{c}_{23})\dot{\Theta }_1^2 -\varvec{c}_3\dot{\Theta }_3^2 -2\varvec{c}_3\dot{\Theta }_2\dot{\Theta }_3-\varvec{s}_3\ddot{\Theta }_3\\&\varvec{{\textbf {Y}}}(2,7)=\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1^2-\varvec{s}_{23}\dot{\Theta }_1\dot{\Theta }_4\\&\varvec{{\textbf {Y}}}(2,8)=2\ddot{\Theta }_2-\varvec{s}_3\ddot{\Theta }_3-(\varvec{s}_2\varvec{s}_{23}-\varvec{c}_2\varvec{c}_{23}) \dot{\Theta }_1^2-\varvec{c}_3\dot{\Theta }_3^2\\&\varvec{{\textbf {Y}}}(2,9)=\ddot{\Theta }_2-\ddot{\Theta }_3-\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1^2\\&\varvec{{\textbf {Y}}}(2,10)=-2\varvec{s}_3\ddot{\Theta }_2+\varvec{s}_3\ddot{\Theta }_3-(\varvec{s}_2\varvec{s}_{23} -\varvec{c}_2\varvec{c}_{23})\dot{\Theta }_1^2 \\&\qquad \qquad \quad +\varvec{c}_3\dot{\Theta }_3^2-2\varvec{c}_3\dot{\Theta }_2\dot{\Theta }_3\\&\varvec{{\textbf {Y}}}(2,11)=-\varvec{c}_2\\&\varvec{{\textbf {Y}}}(2,12)=\varvec{s}_{23}\\&\varvec{{\textbf {Y}}}(2,14)=\text {sign}(\dot{\Theta }_2)\\&\varvec{{\textbf {Y}}}(2,18)=\dot{\Theta }_2\\&\varvec{{\textbf {Y}}}(3,4)=\ddot{\Theta }_2+\ddot{\Theta }_3-\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1^2\\&\varvec{{\textbf {Y}}}(3,5)=\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1^2\\&\varvec{{\textbf {Y}}}(3,6)=-\varvec{s}_3\ddot{\Theta }_2+\varvec{c}_2\varvec{c}_{23}\dot{\Theta }_1^2+\varvec{c}_3\dot{\Theta }_2^2\\&\varvec{{\textbf {Y}}}(3,7)=\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1^2-\varvec{s}_{23}\dot{\Theta }_1\dot{\Theta }_4\\&\varvec{{\textbf {Y}}}(3,8)=-\varvec{s}_3\ddot{\Theta }_2+\varvec{c}_2\varvec{c}_{23}\dot{\Theta }_1^2\\&\varvec{{\textbf {Y}}}(3,9)=-\ddot{\Theta }_2+\ddot{\Theta }_3-\varvec{s}_{23}\varvec{c}_{23}\dot{\Theta }_1^2\\&\varvec{{\textbf {Y}}}(3,10)=\varvec{s}_3\ddot{\Theta }_2+\varvec{c}_2\varvec{c}_{23}\dot{\Theta }_1^2+\varvec{c}_3\dot{\Theta }_2^2\\&\varvec{{\textbf {Y}}}(3,12)=\varvec{s}_{23}\\&\varvec{{\textbf {Y}}}(3,15)=\text {sign}(\dot{\Theta }_3)\\&\varvec{{\textbf {Y}}}(3,19)=\dot{\Theta }_3\\&\varvec{{\textbf {Y}}}(4,7)=-\varvec{c}_{23}\ddot{\Theta }_1+\ddot{\Theta }_4+\varvec{s}_{23} (\dot{\Theta }_1\dot{\Theta }_2+\dot{\Theta }_1\dot{\Theta }_3)\\&\varvec{{\textbf {Y}}}(4,16)=\text {sign}(\dot{\Theta }_4)\\&\varvec{{\textbf {Y}}}(4,20)=\dot{\Theta }_4 \end{aligned}$$

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Mousavi, S.M., Guay, M. Filtered low-power multi-high-gain observer design for a class of nonlinear systems. Nonlinear Dyn 112, 2745–2762 (2024). https://doi.org/10.1007/s11071-023-09205-z

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