# MAE 143B - Homework 9

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1 MAE 43B - Homework a) G(s) = (s+)(s+) Polar plot; red for negative ; no encirclements of, a.s. under unit feedback h) G(s) = s+ s(s+) Polar plot; red for negative ; no encirclements of, a.s. under unit feedback; semicircle is traversed

2 a) G(s) = 2 (s+)(s+2) Polar plot; red for negative f) G(s) = s s Polar plot; semicircle traversed at radius ρ ; same contour for positive/negative. 2

3 7.7 a) G(s) = s + s + d) G(s) = s + s 2 h) G(s) = s + (s + ) As seen in the polar plot below, no point on the negative real axis is encircled. Given that the open-loop transfer function has no poles in the right-half plane, this guarantees asymptotic stability for any K >. We achieve both asymptotic tracking of a constant reference and rejection of a constant torque disturbance because our controller has a pole at the origin. Gm = Inf db (at Inf rad/s), Pm = 88.9 deg (at.33 rad/s) Polar plot; red for negative ; semicircle traversed Bode and polar plots for 7. 3

4 7. Note to graders: These solutions explore all possible combinations of positive and negative gains. It is sufficient to give one combination of K p and K i that yields asymptotic stability. As we have seen in a previous assignment, we can design PI controllers in two steps by first choosing the zero of our controller by fixing the ratio between the two gains K p and K i and then choosing K p using the methods from class. This follows from writing our controller transfer function as K(s) = K p(s + K i /K p ). s First note that by the zero at the origin, tracking and disturbance rejection are as in 7. provided K i. calling the polar plot in 7., notice that we have no chance of stabilizing the system with a zero in the right-half plane and positive K p, which would cause the circle traversed at radius ρ to cover the entire left-half plane and thus result in an encirclement of any given point on the negative real axis. Given that our open-loop transfer function does not have poles in the right-half plane, this means that we cannot stabilize the system with a controller zero in the right-half plane and K p >, eliminating all combinations of K p > with K i <. For an example with the zero placed at s = K i /K p =, see the following plots. 4 Gm = Inf, Pm = 34 deg (at 6. rad/s) Polar plot; red for negative ; semicircle traversed Bode and polar plots for 7., zero at s = K i /K p =. We can further see from the above plots how K p < does indeed allow for stabilization with the controller zero in the right-half plane, provided K p is sufficiently small. This becomes apparent by noting that negative values of K p amount to rotating the entire polar plot contour by π around the origin. K p sufficiently small in this case is required to avoid encirclement of the point /K p by the small circle next to the origin. That is, we can stabilize the system by K i > and K p < provided K p is sufficiently small. This leaves us with placing the zero in the left-half plane. However, again using the insights from our polar plot to 7., we can place this left-half plane zero anywhere and choose any positive gain K p to asymptotically stabilize the system. Given that the zero specifies the ratio of our control gains, this implies that any combination of K i > and K p > yields asymptotic stability (from 7. we know that K p = works, too). For an example with the zero placed at s = K i /K p =, see the following plots. 4

5 4 Gm = Inf, Pm = 9 deg (at 6. rad/s) Polar plot; red for negative ; semicircle traversed Bode and polar plots for 7., zero at s = K i /K p =. By the same argument as above (rotation of the polar plot contour by π around the origin), no combination of K i < and K p < can be used to asymptotically stabilize the system.

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