Duality of nonlinear dynamical systems and synthesis of observers (Q2704010)

The authors extend their previous results on the application of \(K( x) \)-duality to the synthesis of observers for single-output dynamical systems [\textit{A. P. Krishchenko} and \textit{S. B. Tkachev}, Autom. Remote Control 56, No. 2, Pt. 1, 167-178 (1995; Zbl 0932.93012); translation from Avtom. Telemekh. 1995, No. 2, 21-35 (1995)] to the case of multi-output dynamical systems. NEWLINENEWLINENEWLINEThey show that if a smooth dynamical system with \(m\) outputs is observable then, for any smooth \(m\times m\) matrix, \(K( x) \), there exists a unique affine control system with the same number of inputs, that is \(K( x) \)-dual to the dynamical system. This control system is controllable and can be presented in a canonical form by using a suitable coordinate system. Conversely, for each affine control system that can be presented in canonical form there exists a unique dynamical system with outputs that is its \(K( x) \)-dual. This system is observable and can be reduced to a canonical form. NEWLINENEWLINENEWLINEUsing these results they prove that, for dynamical systems whose canonical forms are of certain types, the system with \(m\) outputs admits certain observers if and only if a certain Lie algebra generated by the fields of its \(K( x) \)-dual control system is commutative. A similar result is obtained for single-input, single-output affine control systems. NEWLINENEWLINENEWLINEThe article is quite self-contained.