Download e-book for kindle: A Mathematical Approach to Classical Control by Andrew D. Lewis

By Andrew D. Lewis

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5. We choose the basis {v 1 = (1, 1), v 2 = (1, −1)} for R2 , noting that v 1 is a basis for ZΣ . 6. We compute Ab,f (v 1 ) = (1, 1) = 1v 1 + 0v 2 Ab,f (v 2 ) = (−1, 1) = 0v 1 − 1v 2 . 3 Input/output response versus state behaviour 47 7. The matrix N Σ is 1 × 1 and is given by NΣ = 1 . 8. The zero dynamics are then w˙ 1 = 1 w1 which is a scalar system. Okay, so how is our bad behaviour reflected here? Well, note that the zero dynamics are unstable! This, it turns out, is the problem. 30 Remarks 1.

Things are a bit more subtle with this example. The problem is that the large input is not being transmitted to the output. Describing the general scenario here is not altogether easy, but we work through it so that you may know what is going on. 28 Algorithm for determining zero dynamics We start with a SISO linear system Σ = (A, b, ct , D) with A ∈ Rn×n and b, c ∈ Rn . We do not assume that (A, b) is controllable or that (A, c) is observable. 1. Define Z0 = Rn . 2. Inductively define a sequence of subspaces of Rn by Zk+1 = ker(ct ) ∩ {x ∈ Rn | Ax ∈ Zk + span {b}} .

17 are readily made for MIMO systems. Also, for a MIMO system Σ = (A, B, C, D) one can certainly define   C  CA    O(A, C) =  , ..   . 17 hold in this case. 2 Bad behaviour due to lack of controllability Okay, so we believe that a lack of observability may be the cause of problems in the state, regardless of the good behaviour of the input/output map. Are there other ways in which things can go awry? Well, yes there are. Let us look at a system that is observable, but that does not behave nicely.

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A Mathematical Approach to Classical Control by Andrew D. Lewis


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