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Algorithms for Robust Inversion of Dynamical Systems

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Abstract

A new methodology for solving inverse dynamics problem is developed. The methodology is based on using a mathematical model of a dynamical system and robust stabilization methods for a system under uncertainty.

Most exhaustively the theory is described for linear finite-dimensional time-invariant scalar systems and multiple-input multiple-output systems.

The study shows that with this approach, the zero dynamics of the original system is of crucial significance. This dynamics, if exists, is assumed to be exponentially stable.

It is established that zero-dynamics, relative order, and the corresponding equations of motion cannot be defined correctly in multiple-input multiple-output systems. For correct inverse transformation of the solution of the problem, additional assumptions have to be introduced, which generally limits the inverse system category.

Special attention is given to the synthesis of elementary (minimal) inverters, i.e., least-order dynamical systems that solve the transformation problem.

It is also established that the inversion methods sustain the efficiency with finite parameter variations in the initial system as well as with uncontrolled exogenous impulses having no impact on the system’s internal dynamics.

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Notes

  1. The relative order of system (1.1) is an integer \(r \) such that the following conditions are met: \(cb=0 \), \(cAb=0,\ldots ,cA^{r-2}b=0 \), \(cA^{r-1}b\ne 0\). In this case, all derivatives of the output \(w=cz\) by virtue of the system up to order \((r-1)\) do not depend on \(\xi \) explicitly, while the \(r \)th derivative depends on \(\xi \) explicitly; i.e., \(w^{(r)}= cA^{r}z + cA^{r-1}b\xi \). It can be shown that \(r=n-m \).

  2. From now on, any function \(f \) with subscript \(T \), i.e., \(f_T \), denotes the filtering operation as in (1.14).

  3. The matrix norm is consistent with the vector norm.

  4. Indeed, if the transfer function of the original system has the form \(W(s)=\frac {\beta (s)}{\alpha ( s)}\), where \(\beta =\mathrm {const} \ne 0\), then with the specified choice of spectrum \(A_l \) one has the equality \(cA_l^{-1}b=\frac {(\beta )^{n+1}}{(\lambda _1\cdot \ldots \cdot \lambda _n)}\ne 0 \).

  5. This can always be achieved by normalizing the input or output.

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Funding

This work was carried out with financial support from the Ministry of Science and Higher Education of the Russian Federation within the framework of the program of the Moscow Center for Fundamental and Applied Mathematics under agreement no. 075-15-2022-284.

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Authors and Affiliations

  1. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang, 310005, China

    A. V. Il’in & V. V. Fomichev

  2. Lomonosov Moscow State University, Moscow, 119991, Russia

    E. I. Atamas’, A. V. Il’in, S. K. Korovin & V. V. Fomichev

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  1. E. I. Atamas’
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  2. A. V. Il’in
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  3. S. K. Korovin
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  4. V. V. Fomichev
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Correspondence to E. I. Atamas’, A. V. Il’in or V. V. Fomichev.

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Translated by V. Potapchouck

CONFLICT OF INTEREST. The authors of this work declare that they have no conflicts of interest.

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Atamas’, E.I., Il’in, A.V., Korovin, S.K. et al. Algorithms for Robust Inversion of Dynamical Systems. Diff Equat 59 (Suppl 2), 73–246 (2023). https://doi.org/10.1134/S001226612314001X

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  • DOI: https://doi.org/10.1134/S001226612314001X

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