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By Andrew D. Lewis

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We then have Ax = Ab A2 b · · · An b y. The result will follow if we can show that each of the vectors Ab, . . , An b is in the columnspace of C(A, b). It is clear that Ab, . . , An−1 b are in the columnspace of C(A, b). By the Cayley-Hamilton Theorem we have An b = −pn−1 An−1 b − · · · − p1 Ab − p0 b, which shows that An b is also in the columnspace of C(A, b). Now we show that if V is an A-invariant subspace with b ∈ V then V contains the columnspace of C(A, b). If V is such a subspace then b ∈ V .

1. We take V0 = R2 as directed. 2. As per the instructions, we need to compute ker(ct ) and we easily see that ker(ct ) = span {(1, 1)} . Now we compute x ∈ R2 Ax ∈ Z0 + span {b} = R2 since Z0 = R2 . Therefore Z1 = ker(ct ). To compute Z2 we compute x ∈ R2 Ax ∈ Z1 + span {b} = R2 since ker(ct ) and span {b} are complementary subspaces. Therefore Z2 = ker(ct ) and so our sequence terminates at Z1 . 3. We have ZΣ = ker(ct ) = span {(1, 1)} . 4. Let f = (f1 , f2 ). We compute Ab,f = 0 1 0 + −2 −3 1 f1 f2 = 0 1 .

32 Proposition For any u ∈ U the output of the system ˙ x(t) = Ax(t) + bu(t) y(t) = ct x(t) with the initial condition x = x0 is t y(t) = ct eAt x0 + hΣ (t − τ )u(τ ) dτ. 0 That is to say, from the impulse response one can construct the solution associated with any input by performing a convolution of the input with the impulse response. This despite the fact that no input in U will ever produce the impulse response itself! We compute the impulse response for the mass-spring-damper system. 33 Examples For this example we have A= 0 1 , k − m − md b= 0 .

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A Mathematical Approach to Classical Control by Andrew D. Lewis

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