Analytic Solution of the Problem of Modal Control by Output by means of Bringing to Modal Observation with Fewer Inputs

We propose an effective analytic method for solving the problem of modal control by output for a wide class of linear stationary 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 state vector. The method is based on bringing 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 connecting equation between the matrix of observer 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 block transposing and rank decomposition of matrices. The method significantly expands the class of systems for which an analytic solution exists, compared to the previously proposed approaches, since it is not rigidly tied to the control system dimension and also does not require mandatory zeroing of the column and obtaining a system with a single input. Based on the proposed method, a strict algorithm for analytic solution of problems from the considering class is formed. A simple and convenient necessary condition for the reducibility of modal control by output to modal observation with fewer inputs is also obtained, which allows evaluating the possibility of its analytic solution only by the form of original task. Examples of various order tasks of modal control by output in which the sum of inputs and outputs is less than or equal to the dimension of state vector are considered in symbolic form. A detailed analytic solution of the proposed examples demonstrates the effectiveness of the proposed approach practical application.