Algebraic design method of low order control systems

Control system design with low order controllers still remains a challenge even for linear SISO systems. PI and PID controllers with two or three parameters still predominate in the industrial and technical practice. This is due to, on the one hand, well established tuning methods for each type of objects, on the other hand, to the lack of convenient design methods for systems, which controllers provide more than 3 free parameters. The authors propose an optimization approach to the modal design of low order controllers. As a rule, one can formulate the requirements for a transient quality in the form of conditions on the system pole location in the left semiplane or certain domain within. The farther an objective domain moves to the left, the better is the location of poles contained therein. We suggest fixing the shape of the objective domain only and considering abscissa of its right border as the objective function - R-graduation. However, numerical minimization of R-graduation is associated with serious difficulties: nonconvexity, multiextremality, nondifferentiality et al. The article is devoted to finding optimal and suboptimal pole location with algebraic means. The leftmost position of the objective domain as a rule is characterized by location at its right border of the largest possible number of poles. Such locations are called in the article critical root diagrams. Their number depends on the parameter dimension as Fibonacci law. Each diagram is associated with the root polynomial. Residue of division of the characteristic polynomial by the root one forms algebraic equation system, which connects the control parameters and root coordinates, hence the first become algebraic functions of the latter. It allows finding the critical pole locations and optimal control parameters.