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

By Andrew D. Lewis

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Let us complete this section by giving a description of the subspace ZΣ . 31 Theorem Let (A, b, ct , 01 ) be a SISO system and let Z be the set of all subspaces V of Rn with the properties (i) V ⊂ ker(ct ) and (ii) A(V ) ⊂ V + span {b}. 28 is the largest subspace in Z . 28 it is clear that ZΣ ∈ Z . We then need only show that ZΣ is the largest subspace in Z . Let V ∈ Z and let x ∈ V . 28 we always have Z1 = ker(ct )). We also have Ax ∈ V + span {b} ⊂ Z1 + span {b} . Therefore x ∈ Z2 . Proceeding in this way we see that x ∈ Zi for i = 1, .

Thus, we can assume that α can be exactly specified by the designer. (b) Based on this information find a value of α in the interval (0, 1) and, for that value of α, the minimal required number of amplifier stages, Nmin , so that the final amplifier design meets the specification noted above. 2. How do the equations change if viscous dissipation is added between each mass and the ground? ) The following two exercises will recur as exercises in succeeding chapters. Since the computations can be a bit involved—certainly they ought to be done with a symbolic manipulation package—it is advisable to do the computations in an organised manner so that they may be used for future calculations.

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 . 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 .