# A Mathematical Approach to Classical Control by Andrew D. Lewis PDF

By Andrew D. Lewis

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Example text

3. 27. The problem here is that all the input energy can be soaked by the unstable modes of the zero dynamics, provided the input is of the right type. It is important to note that if we have any of the badness of the type listed above, there ain’t nothing we can do about it. It is a limitation of the physical system, and so one has to be aware of it, and cope as best one can. We shall see these ideas arise in various ways when we discuss transfer functions in Chapter 3. As we say, the connection here is a little deep, so if you really want to see what is going on here, be prepared to invest some effort—it is really a very neat story, however.

47 The impulse response . . . . . . . . . . . . . . . . . . . 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 .

If the resulting state vector is x(t), we then compute T AT −AT x(T ) = e e eA(T −τ ) bu(τ ) dτ. 8), we have xt0 x(T ) = xt0 x0 . Therefore, it is not possible to find a control for which x(T ) = 0. This test of controllability for linear systems was obtained by Kalman, Ho, and Narendra [1963]. The idea is quite simple to comprehend: controllability reflects that we can reach any state from any other state. We can easily see how this comes up in an example. 19 to be 0 0 C(A, b) = 1 −1 which has rank 1 < 2 and so the system is not controllable.