Necessary and sufficient conditions are given for the existence of state and output transformations, that bring single-input single-output nonlinear state equations into the observer form. The conditions are formulated in terms of differential one-forms, associated with an input-output equation of the system. An algorithm for transformation of the state equations into the observer form is presented and illustrated by an example.
We show how we can transform the H∞ and H2 control problems of descriptor systems with invariant zeros on the extended imaginary into problems with state-space systems without such zeros. Then we present necessary and sufficient conditions for existence of solutions of the original problems. Numerical algorithm for H∞ control is given, based on the Nevanlinna-Pick theorem. Also, we present an explicit formula for the optimal H2 controller.
The paper describes the general form of an ordinary differential equation of an order $n+1$ $(n\ge 1)$ which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form \[ f\biggl (s, w_{00}v_0, \ldots , \sum _{j=0}^n w_{n j}v_j\biggr )=\sum _{j=0}^n w_{n+1 j}v_j + w_{n+1 n+1}f(x,v, v_1, \ldots , v_n), \] where $w_{n+1 0}=h(s, x, x_1, u, u_1, \ldots , u_n)$, $ w_{n+1 1}=g(s, x, x_1, \ldots , x_n, u, u_1, \ldots , u_n)$ and $w_{i j}=a_{i j}(x_1, \ldots , x_{i-j+1}, u, u_1, \ldots , u_{i-j})$ for the given functions $a_{i j}$ is solved on $\mathbb R$, $ u\ne 0.$.