5.1 Pole Assignment Methods
The problem of finding a feedback matrix , such that the matrix has a desired set of eigenvalues, is called the pole assignment problem. The pole assignment problem routinely is solved to alter certain system behaviors in a desirable way. The eigenvalues of matrix are called open-loop poles and those of the closed-loop matrix are called closed-loop poles.
The following methods for the pole assignment problem have been implemented in Advanced Numerical Methods: the recursive single-input algorithm (Datta (1987)) and a modified version of the recursive multi-input algorithm (Arnold and Datta (1990)); the RQ implementation of the recursive single-input algorithm (Arnold and Datta (1998)); the explicit QR algorithm (Minimis and Paige (1988)); the implicit single-input RQ algorithm (Patel and Misra (1984)); and the Schur method (Varga (1981)). The multi-input recursive algorithm might give a complex feedback matrix in certain cases. Another modification of the algorithm guaranteeing real feedback matrix has been recently proposed in Carvalho and Datta (2001).
These pole assignment methods, except for the Schur method, use the following template:
1. The pair is first transformed to the controller-Hessenberg pair (see Section 4.1).
2. A feedback matrix is computed such that the matrix has the desired set of eigenvalues.
3. A feedback matrix for the original problem is retrieved from the matrix of the Hessenberg problem in step 2: , where is the orthogonal transforming matrix that transforms the pair to the pair .
The various algorithms differ in the way step 2 is implemented.
The default methods are the recursive algorithm for the single-input problem, the multi-input method (Kautsky, Nichols, and Van Dooren method) from Control System Professional for the multi-input problem, and the Schur method for partial eigenvalue assignment.