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 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. In[1]:= Load the collection of test examples. In[2]:= This is a model of an L-1011 aircraft. In[3]:= Out[3]= Here are the desired poles, arbitrarily chosen. In[4]:= The multi-input recursive method gives the following feedback gain matrix. In[5]:= Out[5]= This computes the norm of the feedback gain matrix. In[6]:= Out[6]= Here are the computed closed-loop poles. In[7]:= Out[7]= Make sure the function MultipleListPlot is available. In[8]:= This displays the closed-loop poles of the L-1011 aircraft model on the complex plane. In[9]:=