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OverviewThe Explicit and Implicit QR Algorithms

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.


OverviewThe Explicit and Implicit QR Algorithms