5.1.1 The Recursive Algorithms
After transforming the given statespace system to the block controllerHessenberg form , the recursive algorithm constructs a block uppertriangular matrix and a block upperbidiagonal 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 singleinput case, however, the algorithm has been proven to be numerically reliable (Arnold and Datta (1998)) in the sense that the illconditioning of the matrix is an indicator of the breakdown of the algorithm. Both the singleinput and multiinput algorithms have worked well in most test cases. The RQ version of the singleinput algorithm is numerically stable (Arnold and Datta (1998)).
Make sure the application is loaded.
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Load the collection of test examples.
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This is a model of an L1011 aircraft.
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Here are the desired poles, arbitrarily chosen.
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The multiinput recursive method gives the following feedback gain matrix.
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This computes the norm of the feedback gain matrix.
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Here are the computed closedloop poles.
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Make sure the function MultipleListPlot is available.
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This displays the closedloop poles of the L1011 aircraft model on the complex plane.
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