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The Explicit and Implicit QR AlgorithmsPartial Pole Assignment

5.1.3 The Schur Method

The Schur method is based on the ordered real Schur decomposition of the matrix given by , where the matrix has all the "bad" eigenvalues of and the "good" eigenvalues of are contained in . The matrix is constructed such that has the desired eigenvalues , where . Then the feedback matrix is such that has the eigenvalues . That is, only the "bad" eigenvalues of the open-loop system are modified, while the remaining eigenvalues are unchanged. The Schur method works in steps. First, it modifies the open-loop eigenvalue(s) contained in the uppermost diagonal block and the Schur form is then reordered to bring the next open-loop eigenvalue(s) to be reassigned in the next step on the top of the Schur form, and so on.

The Schur method is the most expensive of the direct methods if it is used to assign all the poles. From the preceding discussion, however, it is the best-suited method to solve the partial pole assignment problem (see Section 5.2).

Option value for the Schur method of pole assignment.

This is a linearized model of the vertical-plane dynamics of a flight control system.

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Here are the desired poles, chosen arbitrarily.

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This computes the feedback gain matrix using the Schur method to assign all the eigenvalues.

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This computes the norm of the feedback gain matrix.

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Here are the computed closed-loop poles.

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This displays the closed-loop poles of an aircraft model on the complex plane.

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This computes the feedback gain matrix using the recursive algorithm.

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This computes the norm of the feedback gain matrix. The feedback gains are larger than those obtained earlier, but the closed-loop poles are the same.

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The Explicit and Implicit QR AlgorithmsPartial Pole Assignment



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