Static state feedback linearization of nonlinear control systems on homogeneous time scales

Abstract

The paper addresses the problem of static state feedback linearization for nonlinear control systems defined on homogeneous time scales. Necessary and sufficient conditions for generic local linearizability of the considered systems by static state feedback and state transformation are presented in terms of a sequence of subspaces of differential one-forms related to the system.

Constructing a method for solving the riccati equations to describe objects parameters in an analytical form

“…Another feature of the proposed method is the use of dualism in the way of solving some of the equations from the list under consideration. In this case, it is an equation 5with the possibility of its analytical solution in the form y = tg(x) and numerical one in form (6). The application of a special nondimensionalization procedure makes it possible to bring a discrete analog (4) of any equation from the list under consideration (in this case, (3) to the form (6) for equation (5).…”

“…When applying it to a certain type of equation (in the considered case, (3)), a supporting (basic) solution to the same type of equation (in the considered case, (5)) is used. Its feature is the presence of a simple analytical solution, as well as the representation of the solution on the basis of a discrete analog (6). Next, the analytical representation of the solution to all equations of the type in question is recorded based on a basic solution.…”

“…Another, more widely used, method is the linearization via feedback [5]. This method, similarly to logarithmization, applies the introduction of a new frame of reference, namely a new time dimension [6]. More to the point, its calculation, which makes it possible to represent the original nonlinear model in a linear form.…”

“…Denote Δx i , x i , y i , y i+1 are the values from a discrete ana log (12), Δx i , x i , y i , y i+1 are the values of the discrete analog (6), which serves as a nondimensionalized version of analog (12). They are related according to (7), where, from (9), the following values are the normalizing ones:…”

“…Assuming that, after nondimensionalization, the values derived from (6) and compare the values obtained with the corresponding x i (Fig. 2).…”