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By Andrew D. Lewis

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2 Example Let’s look at a system that can be written in the above form. 10. 1 Properties of finite-dimensional, time-invariant linear control systems 25 where we denote by u(t) the input force. 1) we define x1 = x and x2 = x. ˙ The governing equations are then x˙ 1 = x˙ = x2 k x˙ 2 = x¨ = − m x− d x˙ m + 1 u m k = −m x1 − d x m 2 + 1 u. m We can write this in matrix/vector form as 0 1 x˙ 1 = k x˙ 2 −m − md x1 0 + 1 u. 1). We shall look at three ways in which the output equation may appear in this example.

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 . Since V is A-invariant, Ab ∈ V . Proceeding in this way we ascertain that A2 b, . . , An−1 b ∈ V , and therefore the columnspace of C(A, b) is contained in V . 42 2 State-space representations (the time-domain) 22/10/2004 There is a somewhat subtle thing happening here that should be understood.

46 2 State-space representations (the time-domain) 22/10/2004 7. Define an × matrix by   b11 b12 · · · b1 b21 b22 · · · b2    N Σ =  .. . ..  .  . .  . b 1 b 2 ··· b 8. The linear differential equation ˙ = N Σw w is called the zero dynamics for Σ. This is plainly nontrivial! Let’s illustrate what is going on with our example. 27 cont’d) We shall go through the algorithm step by step. 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)} .

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