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1 POLITONG SHANGHAI BASIC AUTOMATIC CONTROL June Academic Year / Exam grade NAME (pinyin/italian)... MATRICULATION NUMBER... SIGNATURE Ue only thee page (including the bac) for anwer. Do not ue additional heet. Ue of any boo, note, or other didactic material i not allowed. Write clearly and be explicit and concie in your anwer [N] in the text mut be ubtituted with the number of letter of your given name. EXERCISE Conider the continuoutime ytem defined by the following matrice A c [ ] ; b a) I it table? Why? b) Chec if it i poible to determine an algebraic control law u x v uch that the reulting matrix Ab ha arbitrary eigenvalue. c) Chec if it i poible to determine an algebraic control law u x v uch that the reulting ytem i table with a ettling time T time unit. Aume [N] a) The trace of A i >, thu the ytem i untable. α [ N] b) To aign arbitrary eigenvalue, the ytem mut be completely reachable. We thu chac the ran of the reachability matrix R: R 9 thu det( R) and the ran i <. The ytem i not completely reachable. c) Depite not completely reachable, the ytem may till perhap be tabilized. Let chac if thi i true by looing at the eigenvalue of the matrix ABK. June POLITONG /

2 Baic Automatic Control June POLITONG / [ ] BK A To examine the eigenvalue of uch a matrix, we note that it can be divided into two bloc: The firt, with the firt two variable and the econd with the third variable. BK A One of the eigenvalue i thu. The other two eigenvalue can be fixed by electing uitable value of and. To have a ettling time of unit, we need the dominant time contant to be and thu the dominant eigenvalue to be.5. Let aume for intance that we want to et both the other eigenvalue to.5. Thu, we et.5 from which.5 Which mean that we can tabilize the ytem in the required time, ince the third eigenvalue, that cannot be changed by an algebraic control law i already compatible with the pecification.

3 Baic Automatic Control d) EXERCISE Compute the aymptotic output of the following ytem when u ( tep( and u (in(. u u () () y Y Y ' Y" Y ' ( )( ) G ( ) U ( )( ) Y" G U The ytem i table ince it i nd order and all the coefficent of We can thu apply the final value theorem. lim y'( lim Y '( ) lim G ( ) We evaluate the aymptotic repone to u Thu, y ( G ( ) ( ) ( ) ( )( ) ( j) in( t arg( G G( j) j. ( j)) y (.9in( t.9).9 with the frequency reapone theorem : arg( G ( j)) atan() atan().9 the characteritic polynomial >. June POLITONG /

4 Baic Automatic Control EXERCISE Conider the control ytem repreented by the following bloc diagram: u( Σ u y Σ Σ y( _ Σ where Σ i a pure gain µ [ N] A Σ i a tranfer function Σ i a ingle time contant at the denominator with µ, T 5 Σ i a unit integrator / Compute the tranfer function G TOT (). a) Dicu the ytem tability, determining if ocillation are preent during the tranient; b) Determine the main characteritic of the tep repone; c) Draw the Bode plot of the magnitude of the frequency repone. R (ω) db.6 ω G TOT ( ) 5 5 ( ) The firt term i aymptotically table (a poitive time contan and alo the econd term i aymptotically table ince all the coeff. To compute them exactly, ±, that are indeed complex conjugate with negative real part. Thi mean that there will be an ocillating repone. Other characteritic of the tep repone : lim y( lim Y ( ) lim G( ) t of the characteritic polynomial at the denominator are > June POLITONG /

5 Baic Automatic Control lim y& ( lim Y ( ) t lim&& y( lim Y ( ) > t lim y( lim Y ( ) The true tep repone: Amplitude Step Repone Time (ec) The Bode plot of the magnitude tart with a egment of lope croing the db axi in /.. Then the lope become in the firt pole. and finally become for the double pole in ω n.6 The true diagram i preented below jut a a reference: Bode Diagram Magnitude (db) 6 9 Phae (deg) 9 8 Frequency (rad/ec) June POLITONG 5/

6 Baic Automatic Control EXERCISE Anwer the following quetion, uing only the available pace and WRITING CLEARLY. a) Two DISCRETE TIME ytem have the following tate tranition matrice. Which i fater in reaching the equilibrium? Why?.. A A [ ]/. N [.] We can compute the eigenvalue of A uing the definition det(ia ). They turn out to be. and., o both ytem are aynt. tab and will reach the equilibrium, however, ince the dominant eigenvalue of the firt (.) i cloer to the tability boundary than the econd (.), it will tae longer.. b) What i the phae margin of a feedbac control ytem? It i a robutne indicator. It i the angle left to 8 when the Nyquit plot enter the unit circle.. c) What i the meaning of obervability and why it i important? A tate i obervable if the output obtained tarting from it i different from that obtained tarting from, at leat for ome t. If all tate are obervable we can build an aymptotic tate recontructor.. d) Draw the approximate Nyquit plot for the tranfer function L()[N]/() and demontrate if it i table or not when connected in a direct feedbac (a in the figure). The interection with the real axi can be obtained finding the value of ω uch that Im(L(jω))8 and we can verify that the correpondent Re(L(jω))>. L( jω) ( jω) ( jω(ω ) ω ) ω (ω ) ( ω ) etting the imaginary part to, we obtain ω (beide the obviouω ). The correpondent real part i ( 9) /8 >. ( 9) jω( ω ) ω u ( Imaginary Axi 6 L() Nyquit Diagram y( Real Axi June POLITONG 6/

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