A Mathematical Approach to Classical Control by Andrew D. Lewis

By Andrew D. Lewis

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T n−1 (n−1) (n−1) cA ct An−1 z1 (0) z2 (0) However, since (A, c) is observable, this gives O(A, c)x1 (0) = O(A, c)x2 (0) =⇒ x1 (0) = x2 (0), which is, as we have seen, equivalent to the assertion that u1 = u2 and y1 = y2 implies that x1 = x2 . Now suppose that rank(O(A, c)) = n. Then there exists a nonzero vector x0 ∈ Rn so that O(A, c)x0 = 0. By the Cayley-Hamilton Theorem it follows that ct Ak x1 (t) = 0, k ≥ 1, if x1 (t) is the solution to the initial value problem ˙ x(t) = Ax(t), x(0) = x0 .

In particular, if we choose x1 = 0 we see that if x2 ∈ V then x2 ∈ span {b}. Similarly, if x2 = 0 we see that if x1 ∈ V then Ax1 ∈ span {b}. ). It now remains to show that T (A, b) = span {b} ∩ A−1 (span {b}). We consider the two cases where (1) b is an eigenvector for A and (2) b is not an eigenvector for A. In the first case, b ∈ A−1 (span {b}) so we clearly have span {b} ∩ A−1 (span {b}) = span {b} . In the second case, b ∈ A−1 (span {b}) so that span {b} ∩ A−1 (span {b}) = {0}. But this is our result.

1 The impulse response for causal systems . . . . . . . . . . . . 2 The impulse response for anticausal systems . . . . . . . . . . . 53 Canonical forms for SISO systems . . . . . . . . . . . . . . . . 5 Controller canonical form . . . . . . . . . . . . . . . . 2 Observer canonical form . . . . . . . . . . . . . . . . 3 Canonical forms for uncontrollable and/or unobservable systems . . . . . 59 Summary . . . . . . .

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