DESIGN OF HYBRID CONTROL SYSTEM FOR NONAFFINE OBJECTS
Abstract
In the theory of automatic control, an urgent problem is the development of design methods by
nonaffine control systems. In such systems, the control affects the input of the plant nonlinearly, so it
affects the state variables non-additively. The purpose of this article is to develop a design method that
ensures the stability of the zero equilibrium position of a closed control system in a certain area.
The object described by a nonlinear system of differential equations with one control is considered.
A restriction is introduced, consisting in the differentiability of the right part of the differential equations
for all state variables. The task of designing control in the form of a function of the reference signal, a
vector of state variables and control values at previous points in time is set. This problem is solved using
a quasilinear model of the control plant. This model of description allows you to preserve all the features
of a nonlinear plant without simplifying them. In the quasilinear model, matrices and vectors are
functions of the variables of the state of the control plant. The control is performed using an algebraic
polynomial matrix method. This method allows you to find control when the control condition of the
plant are met in the form of inequalities. This article presents the expressions for calculating the control
according to the polynomial matrix method. Based on the given coefficients of the desired polynomial,
as a result of solving an algebraic system of equations, coefficients are found that are a function of control
and state variables. At the same time, the fulfillment of the controllability condition guarantees the
existence of a solution of the specified algebraic system. An expression has been found that allows calculating
the control by the coefficients found. The article also finds a condition for the possibility of
providing a non-zero value of the output controlled quantity of a nonlinear Hurwitz system in a steadystate
mode. Under this condition, a zero value of the static error for the setting effect can also be provided.
Further, the transformation of the obtained continuous control into a discrete one is proposed, which
is implemented in a digital computer. The article also provides a numerical example of the control design
of a second-order nonlinear control and the results of modeling a closed nonaffine system.
The given example confirms the theoretical results obtained. Thus, the proposed approach makes it possible
to design stable Hurwitz control systems for nonaffine objects using the algebraic polynomial matrix
method with sufficiently small sampling periods of variables of the control object and small modules of the
roots of the characteristic polynomial of the matrix of a closed system in its quasilinear model.
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