ECE602 Exam 1 April 5, You must show ALL of your work for full credit.

Transcription

1 ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b usd. Plas lav fractions as fractions, tc. I do not want th dcimal quivalnts. Cll phons and othr lctronic communication dvics must b turnd off and stowd undr your dsk. Plas do not writ on th backs of th xam or additional pags. Th instructor will grad only on sid of ach pag. Extra papr is availabl from th instructor. Plas writ your nam on vry pag that you would lik gradd

2 ECE62 Exam April 5, ( points) Controllability Dfinitions (a) Provid th dfinition of controllability for a linar tim-invariant (LTI) discrt-tim (DT) systm (A, B). An LTI discrt-tim systm is controllabl if for any initial stat x x, and any x, thr xists a finit lngth input squnc uk that taks th systm stat from x to x. (b) Provid th dfinition of controllability for a linar tim-varying (LTV) continuous-tim (CT) systm (A(t), B(t)). An LTV continuous-tim (CT) systm is controllabl at tim t if for any initial stat x(t ) x and any x, thr xists a finit tim t and an input u(t) dfind on th tim intrval from t to t that taks th systm from x to x.

3 ECE62 Exam April 5, (5 points) Controllability of Continuous-Tim Systms (a) For th systm blow, comput W c (). ẋ(t) x(t) + u(t) First, lt s comput Aτ B. Bcaus th A matrix is in Jordan normal form, w know Aτ without prforming any calculations. Thus w hav Now Aτ B τ τ τ τ τ. W c () τ 2τ 2 Aτ BB T AT τ dτ τ dτ 2 2. (b) Basd on your answr, is th systm controllabl? Th systm is not controllabl bcaus W c (t) has rank on for any valu of t. (c) Would it mak a diffrnc if w chos t 2 instad of t? No. Controllability of LTI systms is not tim dpndnt. LTI systms ar ithr controllabl or uncontrollabl.

4 ECE62 Exam April 5, 27 4 (d) Us on of th othr quivalnt conditions for controllability to chck your answr. Sinc B slcts th first lmnt of ach row of th A matrix, th simplst critrion to chck is th controllability matrix, C BAB Th fact that th controllability matrix dos not hav full rank confirms our conclusion that th systm is uncontrollabl..

5 ECE62 Exam April 5, ( points) Intrnal Stability of a Continuous-Tim LTI Systm (a) Stat th dfinition of marginal stability of a continuous-tim LTI systm. A systm is marginally stabl if any finit initial condition producs a boundd output. (b) What is th additional rquirmnt for asymptotic stability? Th output must go to zro as tim gos to infinity. (c) What is th tim-domain condition for marginal stability? All ignvalus of A must b in th opn lft half plan (hav nonpositiv ral part) and thos ignvalus having zro ral part must b simpl roots of th minimal polynomial.

6 ECE62 Exam April 5, (5 points) Controllability of LTV Continuous-Tim (CT) Systms (a) Stat on of th conditions for controllability of an LTV CT systm. B sur to dfin all of your variabls. Th pair (A(t), B(t)) and th systm it rprsnts is controllabl at tim t if and only if thr xists a finit tim t > t such that t W c (t, t ) Φ(t, τ)b(τ)b T (τ)φ(t, τ)dτ t is nonsingular, whr Φ(t, τ) is th stat transition matrix of ẋ A(t)x. (b) Dtrmin whthr th following systm is controllabl at t. ẋ(t) x(t) + t 2t u(t) Rathr than computing Φ(t, τ) and thn W c (t, t ), it will b asir to chck th sufficint condition. Sinc A(t) and B(t) ar both (n ) tims continuously diffrntiabl, w can chck th rank of th matrix whr M (t) : B(t), and M(t ) : M (t ) M (t )... M n (t ), M m+ (t) A(t)M m (t) + d dt M m(t). W hav M (t) d dt M (t) A(t)M m (t) t 2t t 2 2t t t 2t 2t 2t

7 ECE62 Exam April 5, 27 7 and thus, sinc n 2, M(t ) : M (t ) M (t ) t t 2t 2 2t 2t. This matrix has dtrminant 3t + 2 3t + 2 4t 3t + 2 4t, which is nonzro for all t, and in particular for vry t > t, so th systm is controllabl at any t.

8 ECE62 Exam April 5, (2 points) Controllability and Stability for Discrt Tim (a) For th systm blow, comput W dc n. xk + xk + uk Th formula for W dc n is W dc n n m For this systm n 2, so w hav A m BB T (A T ) m. W dc A BB T (A T ) + A BB T (A T ). Th matrix B slcts th scond column of any matrix it postmultiplis, so w hav W dc +. 2 (b) Basd on your answr, is th systm controllabl? Ys, W dc has full rank so th systm is controllabl. (c) Is th systm stabl? Plas b spcific. Th matrix A has a rpatd ignvalu with magnitud on. Thus w must dtrmin whthr it is a simpl root of th minimal polynomial. Th charactristic polynomial is (s ) 2 and A I so on is not a simpl root of th minimal polynomial and th systm is nithr marginally nor asymptotically stabl. W ar not givn th C and D matrics, so w do not know ĝ(s); howvr, w know that th pols of th transfr function ar a

9 ECE62 Exam April 5, 27 9 subst of th ignvalus of A. Thus, unlss thr ar no pols, w can conclud that th systm is not BIBO stabl. If, for xampl w assum that C I and D, thn w find that thr ar pols, so th systm is not BIBO stabl. (d) Find an input uk that taks initial stat x to x, or show that such an input dos not xist. W hav x Ax + Bu + + u so u will achiv th dsird x. u

10 ECE62 Exam April 5, (2 points) BIBO Stability of a Continuous-Tim Systm (a) Dfin BIBO Stability. A systm is boundd-input boundd-output stabl if vry boundd input producs a boundd output. (b) Dtrmin whthr th following systm is BIBO stabl. g(t) t + A sufficint condition for th systm to b BIBO stabl is that th transfr function g(t) b absolutly intgrabl. For nonngativ t, g(t) is always nonngativ, so th absolut valu of g(t) is th sam as g(t). g(t) dt dt lim t + ln (t + b ) b lim ln (t + ) ln. b This limit dos not xist so g(t) is not absolutly intgrabl and thus th systm is not BIBO stabl. (c) Dfin absolut intgrability of a function f(t). A function f(t) is absolutly intgrabl if thr xists a finit M such that f(t) dt < M <. Of cours w alrady said that M was finit, so in som sns th < is suprfluous; howvr it is hlpful to hav that information includd in th inquality. (d) Dtrmin whthr th following systm is BIBO stabl. g(t) (t + ) 2

11 ECE62 Exam April 5, 27 As bfor, a sufficint condition for th systm to b BIBO stabl is that th transfr function g(t) b absolutly intgrabl. For nonngativ t, g(t) is always nonngativ, so th absolut valu of g(t) is th sam as g(t). g(t) dt dt (t + ) 2 ( ), t + so th transfr function is absolutly intgrabl and this systm is BIBO stabl.

12 ECE62 Exam April 5, ( points) Find a valid Q matrix for dtrmining th Kalman Controllability form of th following systm, and indicat th dimnsions of th x c vctor. ẋ(t) x(t) + u(t) For xtra crdit, you may calculat th Kalman Dcomposition. To find th Kalman controllability dcomposition, w must first find th controllability matrix, which is C BAB This matrix having rank n, which is lss than full rank, w slct a maximal st of linarly indpndnt columns for th first n columns of th Q matrix. W thn add additional columns to mak Q invrtibl. Thus w hav only a singl choic for th first column of Q, but th scond column is arbitrary so long as it is not a multipl of th first column. Th simplst choic for Q is thus th idntity matrix. Bcaus n is on, th x c vctor is of lngth on. Th Kalman dcomposition is thus A c, A 2, A c, B c, and B c,.