Mesrado Inegrado em Engenharia Elecroécnica e de Compuadores Conrolo em Espaço de Esados Problemas das Aulas Práicas J. Miranda Lemos Fevereiro de 3 Translaed o English by José Gaspar, 6 J. M. Lemos, IST
P. Sae-space model building from fundamenal equaions of physics, chemisry, ec Obain he equaions of a linear sae-space model for he circui of fig.. Use as sae variables he volages of he wo capaciors wih respec o he reference ground node, use as inpu variable he curren source and use as oupu variable he volage a he erminals of he resisor on he righ. R u C C R Fig. problem P. Obain anoher any one sae-space model ha corresponds o he same ransfer funcion. P. Sae-space model building from fundamenal equaions of physics, chemisry, ec Consider he mechanical sysem of fig., where u represens a force era o he weigh of he mass m and such ha a z= he spring has an elongaion ha compensaes he weigh. K z u m Fig. Problema P. a Use Newon's law o obain he equaions of a linear sae-space model describing he sysem. Consider he force u is he inpu and he posiion z is he oupu. b Apply he Laplace ransform, considering null iniial condiions, o deermine he ransfer funcion. Assume K/m=. Consider he case β= J. M. Lemos, IST
and he case β. For each of he cases deermine he posiion of he poles and mark hem in he comple plane. Discuss he posiion of he poles given your inuiion abou he working of he sysem. c Consider now ha u auonomous sysem. Mark one arrow for each of he poins of he sae plane shown in fig.3 given he signals of he derivaives of he sae variables. The arrow indicaes he direcion ha he sae will follow saring a each poin. Compare wih your inuiion abou he working of he sysem. H A B G F E C D Fig. 3 Problema P3. P3. Convering models Obain he ransfer funcion of he sysem described by he sae-space equaions: 5 3 y u P4. Convering models Obain a sae descripion wih minimum order of a sysem wih ransfer funcion: G s s s Modify he realizaion jus done o obain one sae-space model for he ransfer funcion: J. M. Lemos, IST 3
J. M. Lemos, IST 4 3 s s s s G P5. Soluion of he sae-space equaion of an auonomous sysem Consider once more he mass-spring-dumper sysem of problem P. Deermine he eigenvalues of he dynamics mari of he sae space model. Then solve he sae space equaions using a modal decomposiion. Compare he soluion wih your inuiion abou he way he sysem works. P6. Calculaion of he ransiion mari Consider he homogeneous sae space models whose dynamic marices are given by: 4 A 3 A A Deermine he respecive ransiion marices, A e, using he following mehods: a Similariy ransformaion diagonalizaion; b Laplace ransform; P7. Conrollabiliy and observabiliy Consider he sysem described by he sae-space model: u y Indicae for which values of he parameers, e his sae-space realizaion is: a Conrollable b Observable Give one inerpreaion of he resuls based in he ransfer funcion.
P8. Pole Placemen / Full Sae Feedback - FSF / "Realimenação Linear de Variáveis de Esado - RLVE" Consider he sysem wih ransfer funcion s a G s a consan for each case s s a Obain a sae space represenaion for he second order sysem G s. Name he sae variables and. b Considering he sae variables and, indicae he values for he parameer a which allow arbirarily placing he poles of he closed loop sysem, by using linear feedback of all sae variables FSF / RLVE. Do no compue eplicily he characerisic polynomial of he closed loop sysem. c Le a. Find he FSF RLVE gains such ha he closed loop poles are placed in 4 j. Assume in his quesion ha you have access o he direc measuremen of and. d Assume now ha you do no have direc measuremen of and. Indicae values of a such ha he sysem saisfies he condiion of being possible o design an asympoic sae observer such ha he esimaed sae error converges o zero as fas as desired. e Le a. Wrie he equaions of an observer and design is gains such ha he error in he esimaed sae converges o zero, wih eigenvalues he error dynamics. j in P9. Full Sae Feedback In his problem we wan o design a conroller for a permanen-magne moor wih ransfer funcion G s s s where he inpu is he volage applied o he roor and he oupu is he angular posiion of he moor shaf. a Wrie he sae space equaion using as sae he angular posiion and spinning speed of he moor shaf. J. M. Lemos, IST 5
b Compue he gains of a full sae feedback conrol law FSF / RLVE such ha hey place he poles of he closed-loop sysem as he poles of a second order sysem wih n =3, =.5. c Compue he gains of a sae esimaor such ha he characerisic equaion of he sae esimaion error dynamics has n =5, =.5. d Wha is he ransfer funcion of he conroller obained afer solving b and c? P. Full Sae Feedback / RLVE; including an inegral effec for precise acking of references Consider he sysem described by he sae space model d A Bu, A, B, y C C a Le u K Nr. Find a vecor K such ha he closed loop sysem has he eigenvalues placed a -±j. b Compue N such ha if r=r consan hen y=y =r, i.e. he saic posiion error is null. Show ha his propery null saic posiion error is no robus o changes in mari A. c Add an inegraor o he sysem d e y r and choose he gains K and k i, such ha if u=-[k, k i ][, ] hen he eigenvalues of he closed loop are placed a, -±3 / j. Show ha in his case he sysem has null saic posiion error, and ha his propery is robus o changes in mari A, provided ha he closed loop sysem remains sable. 3 J. M. Lemos, IST 6
P. Nonlinear sysems: relaionship beween nonlinear dynamics and dynamics linearized a equilibrium poins There were recenly discovered wo new species of herbivorous, nicely named Necs and Plaks, living in he Melanesia island. A number of biological sudies have shown ha he wo species compee for he same food and he mean numbers of he populaions can be modeled by a sysem of nonlinear differenial equaions: dn N N P dp 3 P P N 4 4 where N is he number of Necs and P is he number of Plaks. These numbers are normalized. More in deail, one has o muliply he numbers by o obain he real numbers of he populaions. Given he model jus inroduced, deermine if he wo populaions can coeis in he long erm. Suggesion: Sar by showing ha N, P is an equilibrium poin of he nonlinear sysem and sudy wha happens o he populaions if his equilibrium is slighly disurbed. P. Consider he auonomous sysem i.e. a sysem wihou inpus, of second order, described by he sysem of nonlinear equaions: d d a Show ha he origin is an equilibrium poin of he sysem. Obain he equaions of he sysem linearized around he origin. Classify he origin in J. M. Lemos, IST 7
erms of he eigenvalues of he linearized sysem. Say wha you can conclude abou he behavior of he nonlinear sysem around he origin. b Using he Lyapunov's nd mehod, and considering he Lyapunov candidae funcion V, nonlinear sysem?, wha can you ell abou he origin of he P3. Consider he nd order nonlinear sysem, wihou inpus, described by he sae space equaions: d d 3 4 4 a Considering Lyapunov's nd mehod and using he Lyapunov candidae funcion V, 3 show ha he origin, is an equilibrium poin, which is sable, a leas in he sense of Lyapunov. Say if you can guaranee ha he poin is asympoically sable. b Using he Invarian Se Theorem show ha he origin is effecively asympoically sable. P4. Consider he sysem of fig.4 where he ank inpu flow, "caudal" in Poruguese, u is conrolled in order o regulae he level h o a reference level r consan and known. Assume ha all horizonal secions of he ank have he same consan value A, which is known. Assume A. The area of he oupu opening a he ank base is described by a, and has an unknown value. The dynamic of he level of he ank is described by J. M. Lemos, IST 8
dh A where is a parameer o esimae. u h u h A a Fig. 4 Problem P4. a Assuming a perfec knowledge of, deermine a saic feedback of he sysem oupu such ha he sysem ank + feedback behaves like an inegraor. b Sill assuming perfec knowledge of, apply a linear conrol law o he resuling inegraor such ha he following error e h r of he conrolled sysem converges o zero wih a ime consan of seconds. c Using he Lyapunov's nd Mehod, obain a conrol law adjusing he parameer which guaranees he complee sysem is sable. Say, in a jusified manner, wheher or no can be guaraneed he following error e h r ends o zero as ends o infiniy. P5. The company Confeiaria Rainha Regional, well known since 89, makes he delicious and well know flour Farinha Inegral 33, essenial for he many nuriious breakfass of he greaes engineering schools, is sudying he opimal invesmen policy in one of is producion lines. Afer horoughly sudies by he company managers, has been concluded ha he producion, P, has a relaionship wih he invesmen I ime varying given by he model: dp.p.5i P J. M. Lemos, IST 9
where he ime uni is year. The producion line is epeced o operae 5 years, and afer ha will be sold by a price proporional o he producion a ha ime. The oal value of he producion line is herefore: 5 J P 5 P I The invesmen is posiive and canno eceed he maimal value I ma, i.e.: I I Using he Maimum Principle, deermine he opimal invesmen policy I, ha maimizes J for 5. ma Helping formulae: d f, u J u T d ' ' f ' T L, u, u L, u T H,, u ' f, u L, u T Some more help: The soluion of he differenial equaion: a b where a and b are consans, is given by b a ce where c is a consan ha depends on he iniial condiions. a J. M. Lemos, IST
P6. Consider he sysem represened in figure in which one wans o sop a ball rolling in a rack, "calha", by applying a volage o a DC moor. u Moor Bola Calha y Fig. Problem P. Equilibrium of a ball on a rack. Using simplifying hypohesis, he sysem can be approimaed by a model based in a ransfer funcion wih wo poles a he origin: G s s Using Chang-Leov's heorem, deermine he posiion of he poles of he closed loop sysem ha opimizes: J y u 6 J. M. Lemos, IST