Analytical Solution of the Problem of Modal Control by Output via Reducing to Modal Observation with Fewer Inputs

Abstract

An effective analytical method is proposed to solve the problem of modal control by output for a wide class of linear time-invariant systems in which the sum of inputs and outputs can be not only greater than or equal to but also less than the dimension of a state vector. The method is based on reducing the modal control by output to modal observation with fewer inputs. At the same time, it is not necessary to additionally ensure the solvability of the equation connecting the matrix of observer matrix and the desired matrix of controller by output. The reduction is performed by constructing a generalized dual canonical form of control using the operations of the block transpose and the rank decomposition of matrices. The method significantly expands the class of systems for which an analytical solution exists compared to the previously proposed approaches, since it is not strictly tied to the control system’s dimension and also does not require mandatory zeroing of the column and obtaining a system with a scalar input. Based on the proposed method, a strict algorithm for the analytical solution of problems from the considered class is formed. A simple and convenient necessary condition of reducibility of modal control by output to modal observation with fewer inputs is also obtained, which allows evaluating the possibility of analytical solution of the original problem basing only on its formulation. Examples of various problems of modal control by output in which the sum of inputs and outputs is less than or equal to the dimension of a state vector are considered in symbolic form. A detailed analytical solution of the considered examples demonstrates the efficiency of the proposed approach practical application.