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12.4: Exercises

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    Exercise 12.1

    Use the expression in (12.1) to find the transfer functions of the DT versions of the controller canonical form and the observer canonical form defined in Chapter 8. Verify that the transfer functions are consistent with what you would compute from the input-output difference equation on which the canonical forms are based.

    Exercise 12.2

    Let \(v\) and \(w^{\prime}\) be the right and left eigenvectors associated with some non-repeated eigenvalue \(\lambda\) of a matrix \(A\), with the normalization \(w^{\prime}v = 1\). Suppose \(A\) is perturbed infinitesimally to \(A + dA\), so that \(\lambda\) is perturbed to \(\lambda +d\lambda\), \(v\) to \(v + dv\), and \(w^{\prime}\) to \(w^{\prime} + dw^{\prime}\). Show that \(d\lambda =w^{\prime}(dA)v\).

    This page titled 12.4: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Mohammed Dahleh, Munther A. Dahleh, and George Verghese (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.