To Design of Nonlinear Control Systems Based on Quasilinear Models

Solution of the design problem of nonlinear control systems is accomplished usually using some transformations of mathematical models. In this case, it is convenient to use the mathematical identities of the algebra of polynomials, vectors, and matrices with numerical and functional coefficients, that are proven in this paper. These identities can be used for the transformations of the mathematical models of both the linear systems with constant parameters and studying the nonlinear control systems represented by quasilinear models. These polynomial-matrix identities also have independent significance, as they can be applied to the algebraic transformations of both some vector-matrix expressions and polynomial-matrix expressions with complex arguments. Applying these identities to the state-dependent coefficients models of control systems is problematic, since these models very often describe nonlinear plants and systems approximately. The polynomial-matrix identities presented below are proved by the equivalent transformations of the operator equations in the state variables of the nonlinear feedback control systems represented by the quasilinear models. These models can accurately represent plants and systems defined by nonlinear differential equations in Cauchy form and output equations, it is only important that the nonlinearities of these equations are differentiable with respect to all their arguments. Using some of the proven polynomialmatrix equalities, the following were obtained: the solution of the eigenvalue placement problem for the system matrix of quasilinear models of closed-loop systems; the controllability criterion of the nonlinear plants output; and the controllability criterion of nonlinear closed-loop systems by reference signals. Two examples of nonlinear plants with uncontrollable output are given, as well as numerical examples demonstrating the correctness of the obtained polynomial-matrix identities.