5.1.2 The Explicit and Implicit QR Algorithms
The explicit and implicit QR algorithms are based on the same idea as the QR algorithm for the eigenvalue problem, but they are direct rather than iterative in nature, since the eigenvalues to be assigned are used as shifts (for a description of the QR iteration method, see Datta (1995)). In the single-input case, there exist RQ implementations of both the explicit and implicit QR algorithms (see Arnold and Datta (1998)). The option ControlInput, described in Section 9.1 of Control System Professional, can be used with the single-input methods described in this chapter.
The setting MethodQRDecomposition invokes the explicit QR algorithm for both single-input and multi-input pole assignment problems. In the multi-input case, the explicit QR algorithm, as implemented, may return a complex feedback matrix for a real state-space system.
The multi-input explicit QR algorithm requires the solution of a single-input problem, for which the implicit RQ method has been used. For the single-input problem in exact arithmetic, both the implicit and explicit RQ algorithms generate the same data at each deflation step, up to a sign scaling. These algorithms are numerically stable.
Option value for the explicit QR pole assignment method.
Consider the aircraft problem from Section 5.1.1. The explicit QR algorithm gives the following feedback gain matrix.
This computes the norm of the feedback gain matrix.
Although the feedback gain matrix is different from the one obtained by recursive method, the computed poles of the closed-loop system are essentially the same.
In the single-input case, the option values RecursiveRQDecomposition and ImplicitRQDecomposition invoke the RQ algorithms. These methods are available for only single-input problems.
Option values for the single-input pole assignment methods.
The same multi-input aircraft problem is now solved with only one input using the single-input RQ version of the recursive algorithm.
The norm of the feedback gain matrix is bigger than the one obtained by the explicit QR algorithm.
Here are the computed closed-loop poles.
The problem solved with the same control input using the implicit RQ algorithm obtains the same feedback as in the explicit RQ algorithm.
This verifies that the single-input pole assignment problem has a unique solution, regardless of the algorithm used.