Column-wise relative degree and it’s properties

Andrey Vladimirovich Kraev, A. I. Rogovskiy

Article ID: 675
Vol 1, Issue 4, 2018, Article identifier:

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Many questions of control theory are well studied for systems which satisfy to the relative degree definition. If this definition is fulfilled then there exists linear state-space transform reducing system to a very convenient canonical form where zero dynamics is a part of system’s equations. Algorithms of such reduction are well-known. However, there exist systems which don’t satisfy this definition. Such systems are the subject of investigation in the presented paper. To investigate their properties here we suggest to consider an analogue of the classical relative degree definition – the so-called column-wise relative degree. It turned out that this definition is satisfied in some cases when classical relative degree doesn’t exist. We introduce this notion here, investigate it properties and suggest algorithm for reducing systems to the column-wise relative degree compliant form if possible. It is possible to show that systems with column-wise relative degree also can be reduced to a convenient canonical form by a linear state-space transformation. Some problems arise from the fact that some systems which do not have relative degree can be reduced to a form with it using linear inputs or outputs transform. Here we show that this is an interesting mathematical problem, which can be solved with the help of properties of relative degree, formulated and proved in this paper.


Control theory; linear dynamical systems; relative degree; zero dynamics; canonical forms

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Isidori A. Nonlinear Control Systems // London: Springer Verlag, 1995.

Il’in, A.V., Korovin, S.K., and Fomichev, V.V., Metody robastnogo obrashcheniya dinamicheskikh sistem (Methods of Robust Inversion of Dynamical Systems), Moscow, 2009

Kraev A. V. On the reduction of third-order vector linear system to a form with Isidori relative order // Differential Equations. — 2012. — Vol. 48, no. 11. — P. 1537–1539.

Kraev A. V. On the analogue of the Isidori relative degree definition for linear dynamical MIMO systems // Doklady Mathematics. — 2014. — Vol. 89, no. 1. — P. 128–133



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