For the PD controlled positioning system, where Ge(s) = K (s+2) and Gp(S) = s(s+1) (a) Plot the root-locus of the system using the Matlab command of "rlocus" for the positive values of gain. (b) Using the root-locus you plotted, write the range of gain for stability of the system. (c) What would be the value of the gain to have a PD Controller Plant closed-loop pole at -2.5? Find it by hand and show it on the root-locus you plotted. (d) Where would be the second pole for the same value of gain? Find it by hand calculations and show it on the root-locus you plotted. R(S) Ge(s) Gp(s) C(s)

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For the PD controlled positioning system, where Ge(s) = K(s + 2) and Gp(S) = s(s+1)
(a) Plot the root-locus of the system using the Matlab command of "rlocus" for the positive values of gain.
(b) Using the root-locus you plotted, write the range of gain for stability of the system.
(c) What would be the value of the gain to have a
PD Controller
closed-loop pole at -2.5? Find it by hand and
show it on the root-locus you plotted.
(d) Where would be the second pole for the same
value of gain? Find it by hand calculations and
show it on the root-locus you plotted.
(e) Suppose that the plant has an additional
unstable pole at s = +4 and hence looks as follows,
R(S)
Gc(s)
Plant
Gp(s)
s+3
G₂ (S) = S(S+1)(S-4)
Plot the root-locus of this new system, Ge(s)G, (s), for the positive values of gain in Matlab.
(f) Write the stability range for gain of this new system by means of the root locus plot and also by hand
calculations.
(g) What would be the frequency of sustained oscillation in Hz?
(h) Plot this response for the first 4-5 periods and show that the response confirms what you found in (g).
C(s)
Transcribed Image Text:s+3 For the PD controlled positioning system, where Ge(s) = K(s + 2) and Gp(S) = s(s+1) (a) Plot the root-locus of the system using the Matlab command of "rlocus" for the positive values of gain. (b) Using the root-locus you plotted, write the range of gain for stability of the system. (c) What would be the value of the gain to have a PD Controller closed-loop pole at -2.5? Find it by hand and show it on the root-locus you plotted. (d) Where would be the second pole for the same value of gain? Find it by hand calculations and show it on the root-locus you plotted. (e) Suppose that the plant has an additional unstable pole at s = +4 and hence looks as follows, R(S) Gc(s) Plant Gp(s) s+3 G₂ (S) = S(S+1)(S-4) Plot the root-locus of this new system, Ge(s)G, (s), for the positive values of gain in Matlab. (f) Write the stability range for gain of this new system by means of the root locus plot and also by hand calculations. (g) What would be the frequency of sustained oscillation in Hz? (h) Plot this response for the first 4-5 periods and show that the response confirms what you found in (g). C(s)
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