Pole And Optimal Pole Assignment In Linear Systems

The pole assignment problem is one of the most researched problems in linearcontrol system theory. Many systems were controlled via pole assignment and relatedtechniques. Since Wonham showed that this problem has a solution if and only if thesystem {A,B} is controllable, there has been much research considering this problem.This paper is concerned with robust pole assignment and optimal pole assignmentproblem and provides three new methods to solve it.For the first approach an objective function which measures stability property ispresented by virtue of eigenvalue disturbance analysis of the closed-loop system viastate feedback. With the help of the parameterizations of the closed-loop eigenvectormatrix and feedback gain, the gradient of the objective function is explicitly given.Then the objective function is minimized via standard gradient-based optimizationalgorithm. Numerical examples show that the resulting feedback gain is superior tothat given by other approaches.The second method is concerned with minimal norm feedback gain design. Inthis method,the Leverrier algorithm is firstly used to parameterize the feedback gainmatrix. A particular advantage of this method is that no freedom in the design is lost.The gradient of the F-norm of the feedback matrix is also obtained recursively viathe Leverriver algorithm. Finally the F-norm of the feedback matrix is minimized viastandard gradient-based optimization algorithm.The third method considers pole assignment problem ,and simultaneously findsthe feedback gain matrix such that a quadratic objective function is minimized. Theproblem is transformed into the solution of Sylvester equation with two inequalitiesconstraint by using the property of the algebraic Riccati equation and its correspond-ing Hamiltonian matrix. Numerical examples show the effectiveness of the proposedapproach。...