5.1.1 The Recursive Algorithms
After transforming the given state-space system to the block controller-Hessenberg form , the recursive algorithm constructs a block upper-triangular matrix and a block upper-bidiagonal matrix , whose diagonal blocks , , ..., contain the eigenvalues to be assigned, such that
, where and has full rank.
The following steps are performed:
1. Set .
2. Compute the blocks through of recursively as follows:
2.1 Compute .
2.2 Find the RQ decomposition of : .
3. Compute the feedback matrix by solving the linear system:
Option value for the recursive pole assignment method.
Although the recursive algorithm is the most efficient of all the algorithms, it is not necessarily numerically stable. In the single-input case, however, the algorithm has been proven to be numerically reliable (Arnold and Datta (1998)) in the sense that the ill-conditioning of the matrix is an indicator of the breakdown of the algorithm. Both the single-input and multi-input algorithms have worked well in most test cases. The RQ version of the single-input algorithm is numerically stable (Arnold and Datta (1998)).
Make sure the application is loaded.
Load the collection of test examples.
This is a model of an L-1011 aircraft.
Here are the desired poles, arbitrarily chosen.
The multi-input recursive method gives the following feedback gain matrix.
This computes the norm of the feedback gain matrix.
Here are the computed closed-loop poles.
Make sure the function MultipleListPlot is available.
This displays the closed-loop poles of the L-1011 aircraft model on the complex plane.