Please feel free to refer to the French texts as we couldn't translate formulas in english.

Problem

They consider the enslavement of position of the face below (see face).

R (t) represents a baggage room to be followed;
P1 (t) represents a disturbance of type echelon, of level 0.1;
P2 (t) represents a sinusoidal noise of measure of expression: 0.01sin10t.

a) The schema of compensation corresponds in what type of regulator? Why? (1 pt)

b) By assuming the noise of measure of constant amplitude whatever is the measured point, suggest, by justifying it, an alternative in the establishment of the compensator which could show better performances.

c) Show, by itemizing the counting of schemata corresponding blocks that:

d) Specify the stocks of k1 and k2 positive for which the system will be stable. (0.5 point)

e) Which precision is allowed by this system in monitoring (R (t) unit echelon; no disturbance)? Justify your answer. (1 point)

f) If the error of regulation in the presence of disturbance P1(t) must be in most than 0.01, determine a corresponding pressure on parameter k1 of the compensator. (1 point)

g) Let us imagine k1 fixed currently. Draw the place of the poles of the system in a continuous loop closed when A (k2 - 1) vary 0 in the infinity, by specifying:
1) existence on the axle of reals;
2) double root (according to k1).

Note: In the variable of the place of roots is. It will be necessary to put characteristic equation under form 1+ A G (s)= 0 before

h) They want to assure a percentage of overtaking %OS consisted of between 5 and 10 %. On the place of roots in g, point out regions corresponding to the stocks of k2 allowed, and specify the corresponding mailmen of amortizement.

i) From the diagram in g (k1 fixes), discuss with justification of the effect of an increase of k2 on:

(I) The mailman of amortizement of poles;

(Ii) The time of answer to 5 % of the system.

j) For a mailman of given amortizement, specify with justification from diagram in g, the action to be taken to ameliorate the time of answer to 5 % of the system.

k) In all previous pressures (%OS, precision in comparison with P1), they want to add pressure that in permanent regime, the effect on the exit O (t) of the error of measure P2 (t) should not be superior to 0.01 of amplitude.

By considering that the function of transfer O (t) / P2 (t) (please see french text)
can be approached by: (please see french text) and by assuming inequality: (please see french text)

(I) Draw the asymptotes of the diagram of Bode amplitude linked according to k1, k2.
(II) (*) by treating asymptotes as though it was about the diagram of precise Bode, to deduct a pressure on parameter k2 so that pressure linked to P2 is satisfied.

For question 1.L; 1.m and 1.n we consider that k1 >= 10 ( K1 is greater or equal to 10 ).

l) For %OS 5 %, determine the stocks of k1 and k2 which obey all pressures of precision, while minimizing the time of answer to 5 %. Specify the value of this last.

m) Repeat question for %OS 10 %.

n) Among the choices of parameters in l or m, favorisezvous one particularly, and why?