By Andrew D. Lewis

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**Additional resources for A Mathematical Approach to Classical Control**

**Sample text**

It’s not so hard to see what is happening here. We do not have the ability to “get at” the unstable dynamics of the system with our input. Motivated by this, we come up with another condition on the linear system, different from observability. 20 Definition A pair (A, b) ∈ Rn×n × Rn is controllable if the matrix C(A, b) = b Ab · · · An−1 b has full rank. If Σ = (A, b, ct , D), then Σ is controllable if (A, b) is controllable. The matrix C(A, b) is called the controllability matrix for (A, b). Let us state the result that gives the intuitive meaning for our definition for controllability.

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)} . 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.

V } for ZΣ , and extend this to a basis {v 1 , . . , v n } for Rn . 6. Write Ab,f v 1 = b11 v 1 + · · · + b 1 v .. Ab,f v = b1 v 1 + · · · + b v Ab,f v +1 = b1, +1 v 1 + · · · + b , +1 v + b +1, +1 v +1 + · · · + bn, +1 v n .. Ab,f v n = b1n v 1 + · · · + b n v + b +1,n v +1 + · · · + bnn v n . 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.

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

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