Advanced Control Systems Problem Sheet for Part B: Multivariable Systems

Transcription

1 Advanced Conrol Ssems Problem Shee for Par B: Mlivariable Ssems Qesion B 998 Given a lan o be conrolled, which is described b a sae-sace model A B C Oline he rocess b which o wold design a discree LQG reglaor incororaing a Kalman sae esimaor. Draw a simlaion diagram of he closed-loo ssem ha dislas he srcre of he reglaor. Wrie down he eqaions describing he Kalman esimaor, and describe how he wold be imlemened in a conrol comer. Qesion B 998 Under wha circmsances wold i be aroriae o se a fll-order esimaor, even hogh some of he lan saes are available for direc measremen? Derive he eqaions describing a redced-order esimaor. Show ha for an ideal redced-order esimaor erfec lan model, no rocess or measremen disrbances, he errors in he esimaed saes will aroach zero asmoicall. Qesion B3 998 Show how he reference ins ma be inrodced ino an LQG-reglaed ssem so ha i will have zero sead-sae errors in resonse o se-oin changes. Show how inegral conrol ma be inrodced ino an LQG-reglaed ssem so ha i will have zero sead-sae errors in resonse o se-oin changes and consan lan load disrbances. Qesion B4 998 Consider a lan whose ransfer fncion is /s and which has a ZOH on is in. If he o is samled wih a samling eriod s, he discree sae eqaions are.5 [ ] Wha is he discree ransfer fncion from o? Wrie down he smmeric roo locs eqaion for his ssem. Wha relaion does his have o i oimal conrol of he lan, and ii oimal esimaion of he saes? he SRL is drawn in figre. If oles are chosen for a sae-feedbac conroller which have roeries eqivalen o s-lane oles wih ω n.π rad/s and ζ.7, wha is he weigh laced on o ecrsions relaive o conrol effor in he erformance inde? Problem_A.doc

2 d e f Design a digial sae-feedbac conrol law ha will ield hese closed-loo oles. Design a fll-order esimaor wih dnamics five imes faser han he conroller. If he esimaor oles are chosen from he SRL, wha do he chosen oles iml abo he relaive levels of he assmed rocess and measremen noise? Draw a simlaion diagram of he closed-loo ssem, srcred so ha i will resond o se in commands wih zero sead-sae error..5 Smmeric roo locs.5 Im ag A is Qesion A Real Ais Figre Given a lan o be conrolled, which is described b a sae-sace model A B C Oline he rocess b which o wold design a discree LQ reglaor incororaing a crren sae esimaor. Draw a simlaion diagram of he closed-loo ssem ha dislas he srcre of he reglaor. Wrie down he eqaions describing he crren esimaor, and describe how he wold be imlemened in a conrol comer. Problem_A.doc

3 Qesion A Describe he rocess b which disrbances of nown form, b nnown magnide, can be esimaed and comensaed for in a sae-sace conrol ssem design. A endlm wih a orqe moor a is hinge is described b [ ] b w where [ ] [ ] θ θ, θ being he angle of he endlm from verical, is a conrol orqe, w is a orqe bias, and b is a measremen bias. he o is samled wih a samle eriod. s, and he oal orqe w is alied hrogh a zero-order hold. Wih his discreisaion, he ssem and in marices become ξ ξ.998.5, d e e B A A i Wih no measremen bias b, agmen he discree ssem model so ha he orqe bias is an elemen of he sae vecor. Is his ssem observable? ii Wih no orqe bias w, agmen he discree ssem model so ha he measremen bias is an elemen of he sae vecor. Is his ssem observable? Problem_A.doc 3

4 Qesion A3 Consider a doble-inegraor lan: s s U s Y Sech a smmeric roo locs SRL for his lan. Elain how his SRL cold be sed in he design of a feedbac reglaor. Wha is he heoreical basis for sch ses? A conroller canonical realisaion of he lan is [ ] Show ha he oimal conrol law which will minimise he erformance inde [ ]d J ρ is ρ ρ Hin: he erformance inde [ ]d J R Q is minimised b he conrol law K, where P B R K and P is a smmeric, osiive-definie solion of he coninos algebraic Riccai eqaion Q B B PBR PA P A For his simle lan i is sraighforward o manall solve he mari Riccai eqaion for P. he mari 3 P is osiive-definie if and onl if and >. 3 > Problem_A.doc 4

5 Qesion Simlaion diagrams for reglaed ssems incororaing i a discree redicion sae esimaor, and ii a discree crren sae esimaor, are shown in figres 4 and 5 resecivel. Show ha he esimaion error dnamics for he redicion esimaor are [ C] ~ ~ L where he redicion esimae error is defined as ~ ˆ. For a mli-in, mli-o MIMO ssem, discss some sraegies for deermining he esimaor gain L which will lace he esimaor oles a desired locaions in he z-lane. In a ole-lacemen design of a redicion esimaor for a MIMO ssem, he esimaor gain L can be fond wih he MALAB Conrol Ssems oolbo command L lacephi, C, de where he desired esimaor oles are defined in he vecor de. If he esimaion error of he crren esimaor is defined as ~ ˆ c derive an eqaion for he esimaion error dnamics, and hence show how he lace command can be sed o calclae he crren esimaor gain L c which will resl in he same esimaor oles, de. Qesion 3 Describe he fndamenal objecives which nderlie he mahemaical formlaion of an oimal conroller and an oimal esimaor. Describe he rocess of designing an LQG reglaor for a MIMO ssem. m q C m q C Plan Plan K ˆ q L ˆ q C ˆ Feedbac comensaor C ˆ Esimaor K Feedbac comensaor ˆ L c Esimaor Figre 4: Predicion esimaor Figre 5: Crren esimaor Problem_A.doc 5

6 Problem 7.58 [adaed from Franlin e al. ] Consider he endlm roblem wih conrol orqe c and disrbance orqe d : θ θ 4 c d here g/l 4. Assme here is a oeniomeer a he in ha measres he o angle θ, b wih a consan nnown bias b. hs he measremen eqaion is θ b. a b c d e f ae he agmened sae vecor o be θ where w is he in-eqivalen bias. Wrie he ssem eqaions in sae-sace form. Give vales for he marices A, B, and C. Using sae-variable mehods, show ha he characerisic eqaion of he model is ss 4. Show ha w is observable if we assme θ, and wrie he esimaor eqaions for ˆ ˆ θ θ wˆ. Pic esimaor gains [L L L 3 ] o lace all he roos of he esimaor-error characerisic eqaion a. Using fll sae feedbac of he esimaed conrollable sae-variables, derive a conrol law o lace he closed-loo oles a ± j. Draw a simlaion diagram of he comlee closed-loo ssem esimaor, lan, and conroller sing inegraor blocs. Inrodce he esimaed bias ino he conrol so as o ield zero sead-sae error o he o bias b. Demonsrae he erformance of or design b loing he resonse of he ssem o a se change in b; ha is, b changes from o some consan vale. θ w Problem_A.doc 6

7 Problem Problems reqiring MALAB An aroimae linear model for he laeral dnamics of an aircraf, wrien in erms of small errbaions from a rim condiion, is r β φ r β φ δ δ Sose a malfncion revens manilaion of he rdder in δ r. Is i ossible o conrol he aircraf sing onl he aileron in δ a? If no, wha aircraf moions can and canno be conrolled? Is he aircraf conrollable wih js he rdder in δ r? d Verif ha he aircraf is conrollable wih boh ins oerable. If he onl o is a measremen of he roll rae rovided b a rae gro, is he ssem observable? e Is he ssem observable from ban indicaor measremens of φ? f g h Design a discree sae-feedbac conroller, wih a samle eriod of s. s, wih closed-loo oles which have roeries corresonding o s-lane oles a s, 5, 8,. Show how reference ins for aw rae and roll angle can be inrodced sch ha he resonse o se commands will ehibi zero sead-sae errors. Design a discree LQ oimal reglaor, wih a samle eriod of s. s. Sar wih he following rial weighing marices: Q diag.,.,.,., R I. Se he closed loo ssem o rac aw and roll commands. Problem An nsable discree-ime ssem is described b [ ] 3 a r Find he infinie horizon feedbac gain mari for he oimal reglaor roblem, assming all saes are available for se. Also deermine he resling closed-loo eigenvales. Use he following weighs: i Q I, R ii Q I, R iii Q I, R iv Q I, R Problem_A.doc 7

8 he ssem is now assmed o have addiive rocess noise w and measremen noise v. he noise covariance marices are: 9 Q and 6, resecivel 4 R A Kalman filer is o be sed raher han a deerminisic observer. Find he seadsae Kalman gain K and he oles of he filer s dnamics. he above filer is sed wih he LQ conroller gains fond above. How do he resence of he filer and he se of esimaed saes raher han acal saes affec he overall ssem erformance in each case? he searaion heorem garanees ha his aroach is he bes ha can be done. However, i does no garanee ha he resls will be wha he conroller ased for, or ha he ssem will even wor well. d Reea wih he measremen noise covariance redced o R. Problem 3 For a lan described b: [...] design a linear/qadraic conroller ha minimises he cos fncion: q r J d Perform he design for differen vales of q and r. Chec he sabili of he closed-loo ssem wih a linear/qadraic conroller for all design cases. Obain he oen-loo ransfer fncion o calclae he gain and hase margins. Comare he resls for differen vales of q and r. Comare he sensiivi of ole and zero locaions and se resonses of he closedloo ssem, for changing vales of q and r. Commen on he robsness of LQ-designed conrollers Problem_A.doc 8