Fall 2001 16.3110-10 Summary of Zeros: Great feature of solving for zeros using the generalized eigenvalue matrix condition is that it can be used to find mimo zeros of a system with multiple inputs/outputs B det D Need to be very careful when we find MIMo zeros that have the same fre- quency as the poles of the system, because it is not obvious that a pole/zero cancellation will occur(for MIMO systems The zeros have a directionality associated with them, and that must agree"as well, or else you do not get cancellation More on this topic later e Relationship to transfer function matrix If z is a zero with(right)direction s, u,then Ⅰ-A BlS 0 D -If z not an eigenvalue of A, then S=(zl-A)-Bu, which gives C(aI-A)-B+D]=G(2)i=0 Which implies that G(s)loses rank ats=z -If G(s) is square, can test det G(s=0 If any of the resulting roots are also eigenvalues of A, need to re-check the generalized eigenvalue matrix conditionFall 2001 16.31 10–10 • Summary of Zeros: Great feature of solving for zeros using the generalized eigenvalue matrix condition is that it can be used to find MIMO zeros of a system with multiple inputs/outputs. det s0I − A − B C D = 0 • Need to be very careful when we find MIMO zeros that have the same fre￾quency as the poles of the system, because it is not obvious that a pole/zero cancellation will occur (for MIMO systems). – The zeros have a directionality associated with them, and that must “agree” as well, or else you do not get cancellation – More on this topic later. • Relationship to transfer function matrix: – If z is a zero with (right) direction [ζT , u˜T ] T , then zI − A − B C D ζ u˜ = 0 – If z not an eigenvalue of A, then ζ = (zI − A)−1Bu˜, which gives  C(zI − A) −1 B + D u˜ = G(z)˜u = 0 – Which implies that G(s) loses rank at s = z – If G(s) is square, can test: det G(s) = 0 – If any of the resulting roots are also eigenvalues of A, need to re-check the generalized eigenvalue matrix condition