Control Systems

Transcription

1 6.5 Cntrl Sytem Tday we are ging t ver part f Chapter 6 and part f Chapter 8 Cntrllability and Obervability State Feedba and State Etimatr Lat Time : Cntrllability Obervability Cannial dempitin Cntrllable/unntrllable Obervable/unbervable Cntrllability: Definitin Cnider the ytem x x Bu x R n ; u R Cntrllability i a relatinhip between tate and input. Definitin: The ytem r the pair (B) i aid t be ntrllable if fr any initial tate x()=x and any final tate x d there exit a finite time T > and an input u(t) t[t] uh that T (T-τ) x(t) e x e Bu( τ)dτ x T p. d ()

2 t τ 'τ (tτ) '(tτ) W (t) e BB'e dτ e BB'e d Equivalent nditin fr ntrllability: ) W (t) i nningular fr any t >. ) The matrix G = [B B B n- B] ha full rw ran i.e. (G ) = n. ) The matrix G n-p+ = [B B B n-p B] ha full rw ran (p=ran(b)) ) M() = [-I B] ha full rw ran at every eigenvalue f. t Obervability: dual nept Cnider an n-dimeninal p-input q-utput ytem: x x Bu; u Sytem y Cx Du ume that we nw the input and an meaure the utput but have n ae t the tate. Definitin: The ytem i aid t be bervable if fr any unnwn initial tate x() there exit a finite t > uh that x() an be exatly evaluated ver [t ] frm the input u and the utput y. Otherwie the ytem i aid t be unbervable. y

3 Equivalent nditin fr bervability: ) W (t) i nningular fr me t >. ) The bervability matrix C C G C C n-p r ( G ) np n C C ha full lumn ran (G n-p+) = n. (p=ran(c)) ) The matrix M ( λ) λi C ha full lumn ran at every eigenvalue f. 5 Cntrllability dempitin Therem: Suppe that (G ) = n < n. Let Q be a nningular matrix whe firt n lumn are LI lumn f G. Let P=Q - and z=px. Then nn np PP B PB R B R C C C B Mrever the pair ( B ) i ntrllab le and C ( I ) B D C(I ) B D z z z y C z C z z B u z z B u y C z i ntrllable and ha the ame tranfer funtin a the riginal ytem. The tate unntrllable z z z i The ntrl ha n effet n it 6

4 Obervability dempitin (fllw frm duality) Therem: Suppe that (G ) = n < n. Let P be a nningular matrix whe firt n rw are LI rw f G. Then PP C nn qn C C R B B PB B R Mrever the pair ( C ) i bervable and C ( I ) B D C(I ) B D B R np Diuin: fter tate tranfrmatin the equivalent ytem i z z B u z may be affeted by z z z z Bu but ha n effet n y r z y C z Du z nt bervable z z Bu y C z Du ha the ame tranfer funtin a the riginal ytem and i bervable. 7 Tday: Cntrllability and bervability ntinued Cntrllability/bervability dempitin Minimal realizatin Cnditin fr Jrdan frm nditin Parallel reult fr direte-time ytem Cntrllability after ampling State feedba deign Ple aignment Uing ntrllable annial frm By lving matrix equatin 8

5 Cntrllability-Obervability dempitin Therem: ll LTI ytem an be tranfrmed via equivalent tranfrmatin int the fllwing frm: z z z z y z B z B u z z C C z Du B where i ntrllab le B ( B ) i ntrllab le and ( C ) i bervable. ( C ) i bervable. Mrever z C ( I ) B D C(I ) B D z B u y C z Du i a ntrllable and bervable realizatin It ha the ame tranfer funtin a the riginal ytem 9 Minimal realizatin f a tranfer matrix Obervatin: Let G() be a prper ratinal tranfer matrix. We learned earlier that there exit (BCD) uh that G()=C(I-) - B+D The realizatin i nt unique. Definitin: realizatin (BCD) f G whih ha the minimal dimenin f tate pae i alled a minimal realizatin f G. Quetin: Whih ne i a minimal realizatin? Hw t btain a minimal realizatin? Therem: (BCD) i a minimal realizatin iff (B) i ntrllable and (C) i bervable. 5

6 Predure t btain a minimal realizatin: n earlier reult: Fr a tritly prper and ratinal matrix G() Let d()= r + a r- + a r- +.+ a r- +a r be the leat mmn denminatr f all entrie Then G() an be expreed a (aume G i qp) G() d() r r qp N N N N N r r i R realizatin f G() i given a: aip a Ip a ri p Ip Ip Ip C N N N r Nr a ri p Ip B Ue equivalene tranfrmatin z = Px uh that PP B B PB B C CP C C ( B) i ntrllable and ( C) i bervable. Then ( C B (I C ) B C(I ) B G() and ) i a minimal realizatin f G(). If G() i nt tritly prper we an firt dempe it a G() = G p () + D where G p () i tritly prper. 6

7 Cnditin fr Jrdan frm equatin Equivalene tranfrmatin d nt hange ntrllability and bervability Thee prpertie are eay t ee frm Jrdan frm. Therem: ume that ha m ditint eigenvalue m and ha a Jrdan frm arranged by the eigenvalue with bl J diag[jj... JJ J mj m...] m Let the rw f B rrepnding t the lat rw f J ij be b ij. Let the lumn f C rrepnding t the firt lumn f J ij be ij Then the ytem i ntrllable iff fr eah i the rw {b i b i } are LI. The ytem i bervable iff fr eah i the lumn { i i } are LI. Example: λ λ λ λ λ λ λ b * b B b * b * C * * *. (B) i ntrllable iff {b b b } i LI and b (C) i bervable iff { } i LI and The lumn f C and the rw f B mared by * have n effet n ntrllability r bervability. 7

8 8 5 Example: λ λ λ λ λ λ λ * * * B * * * C Cae :. Cae : =. b b b b Example: C b b b b B λ λ λ λ Cae :. Therem: Fr a ingle input ytem It i ntrllable iff fr eah ditint eigenvalue there i nly ne Jrdan bl and eah element f B rrepnding t the lat rw f a Jrdan bl i nnzer; It i bervable iff fr eah ditint eigenvalue there i nly ne Jrdan bl and eah element f C rrepnding t the firt lumn f a Jrdan bl i nnzer. (B) i ntrllable iff b and b (C) bervable iff and Cae : = then what?

9 Direte-Time Sytem The ytem deribed by differene equatin: x[+] = x[]+bu[] y[]= Cx[]+Du[] Reult n ntrllability and bervability are quite imilar t the fr ntinuu-time ytem. 7 Definitin Cnider the differene equatin x[ ] x[] Bu[] y[] Cx[] D[] where xr n ur p. Definitin : The ytem r the pair (B) i aid t be ntrllable if fr any initial tate x()=x and any final tate x d there exit an integer > and a equene f input u[] [ ] uh that m --m x[ ] x u[m] x Definitin : The ytem r the pair (C) i aid t be bervable if fr any unnwn initial tate x() there exit a finite > uh that x() an be exatly evaluated ver [ ] frm the input u and the utput y. Otherwie the ytem i aid t be unbervable. d () 8 9

10 Equivalent nditin fr ntrllability: The fllwing are equivalent nditin fr the pair (B) t be ntrllable: ) The matrix G = [B B B n- B] ha full rw ran i.e. (G ) = n. ) The matrix M () = [I B] ha full rw ran at every eigenvalue f. ) The fllwing nn matrix i nningular W d [n ] n m m BB'( Nte: There may exit an integer n < n uh that W d (n -) i nningular. m )' 9 Equivalent nditin fr bervability: ) The bervability matrix C G C n C ha full lumn ran. ) The matrix M ( λ) C λi ha full lumn ran at every eigenvalue f. ) The fllwing nn matrix i nningular W d [n ] n m ( m )' C' C m

11 Cntrllability after ampling ntinuu-time ytem x x Bu; Let the ampling perid be T. During the ampling perid u(t) = u(t) fr all t [T (+)T) = Define u[]:= u[t]; x[]=x[t]. The relatin between u[] and x[] i gverned by the differene equatin: where x[ ] x[] B u[] d e T d B d d [ d - I]B Quetin: I ntrllability retained after diretizatin? Summary f reult frm 6.7 If the pair (B) i unntrllable then ( d B d ) i al unntrllable fr any ampling time T. If all the eigenvalue f i real then (B) ntrllable implie that ( d B d ) i ntrllable. If ha mplex eigenvalue ntrllability maybe lt fr me peial ampling perid T. We ue Re[x] and Im[x] t dente the real part and the imaginary part f a mplex number x. Suppe (B) i ntrllable. uffiient nditin fr ( d B d ) t be ntrllable i that Im[ i j ] m/t fr m= whenever Re[ i - j ]=. The nditin i t enure that the number f Jrdan bl will nt inreae fr a partiular eigenvalue. Nte that if i i an eigenvalue f then e it i an eigenvalue f d. If i and j have ame real part e it and e jt may be the ame.

12 Example: x x Bu β G B B detg β β β B The CT ytem i ntrllable if. Nw uppe. Let the ampling perid be T. x[ ] d x[] B u[] d d e T e T βt in T in T βt B d βt βin βt β β T β in βt What happen when T=? S far we have tudied ntrllability and bervability Main Prblem f the Cure nalyi: Slutin t LTI ytem tability et. Cntrllability and bervability; Feedba deign and ntrutin f berver Optimal ntrl Next we will tart t addre deign prblem

13 Stabilizatin prblem Given a LTI ytem x x Bu. typial ntrl prblem i t bring the tate x frm any initial nditin t the rigin and eep it there. If i table we nly need t et u= and x(t) will nverge t the rigin aympttially. nther prblem i t bring x t a deirable pint x d a fat a pible and eep it there. Bth f thee prblem are abut tabilizatin at an equilibrium pint. The end prblem an be tranfrmed int the firt ne. 5 Fr example given an LTI ytem: z z Bv; y Cz Dv Suppe that i nningular and v = u + u e. (u e a given ntant). We have z z Bu e Bu; Let z e = Bu e and define x = z z e. Then x z z Bue Bu (z - ze) Bu x Bu; x x Bu. Suppe that z e i a deirable pint where we wuld lie t eep z there. If i table then by etting u = x(t) will nverge t frm any initial x and will tay there. z(t) = x(t) + z e nverge t z e and tay there. Quetin: What if i nt table? What if i table but the nvergene rate i t lw? 6

14 Fr the equatin x x Bu. Reall that if (B) i ntrllable then the fllwing ntrl u(t) B'e t) t W (t )[e x x ' (t an bring x frm any initial nditin x t any final detinatin x d. The time duratin [t ] an be arbitrarily mall. nd the ntrl i f minimal energy. Hwever thi ntrl trategy i nt ued in pratie. Rean: Very enitive t parameter hange and implementatin errr; Even if the tate i at the rigin diturbane eep driving it away frm the rigin. Nt eay t mpute. In ummary: nt reliable mpliated and frutrating. d ] (*) 7 pratial and effetive lutin: tate feedba Fr the ytem x x Bu y Cx Du If we let u = r K x. Then x ( BK)x Br. If i untable but (B) i ntrllable we an mae BK table by hing K prperly; If i table but the nvergene rate i t lw we an imprve the nvergene prperty by deigning K prperly. The feedba law u = r Kx i imple fr implementatin but very effetive. We hall find ut hw t deign a tate feedba law. 8

15 n additinal tl: State etimatin What if the tate annt be btained thrugh meaurement? ume that all the infrmatin that an be meaured i y = Cx+Du. If the ytem i bervable we hall ue a tate-etimatr alled an berver t etimate the tate frm the meaurement y and the input u. The berver i al an LTI ytem with input a u and y and it utput i the etimate f the tate x: u xˆ(t) - an etimate f x(t) Oberver y We will learn hw t deign an berver. We Start with State Feedba Deign 9 State feedba deign: ingle input ae ingle input ingle utput ytem x x bu y x (aume D= fr impliity) where R nn br n ha nly ne lumn and R n ha ne rw. p=q=. Let R n be a rw vetr. Then x R. With tate feedba u= r x we have x ( b)x br y x Therem: The pair (-b b) i ntrllable iff (b) i ntrllable. (ee page fr prf.) Cmment: tate-feedba de nt hange ntrllability prperty. Hwever the bervability f (-b) might be different frm that f (). 5

16 What an be gained frm uing tate feedba? The riginal ytem: x x bu y x With tate feedba we have: x ( b)x br y x reult t be hwn later: if (b) i ntrllable then the eigenvalue f -b an be plaed anywhere by hing prperly. Example: b b Eigenvalue f : = = untable. Charateriti plynmial fr -b i ()= +( -)+( - -8)= +a +a The tw effiient a and a an tae any value. Cntrllable Cannial Frm Fr impliity we nider a th-rder ytem. The reult fr the general ae an be eaily extended frm the pattern. Therem: Suppe that (b) i ntrllable and det(i ) - Let Q : P b b b b With the tate tranfrmatin z = Px we have - PP b Pb P β β β β Cntrllable Cannial frm Furthermre β (I) b β β β 6

17 7 Prf: We an brea the tranfrmatin int tw tep: x P x P P x where Q P b b b b Q P - With the firt tranfrmatin we btain B Q B Q Q With the end tranfrmatin we btain B Q B Q Q Here we an verify that B B Q Q Q Exat ple aignment Therem: Suppe that (b) i ntrllable. Then the eigenvalue f -b an be arbitrarily aigned prvided that mplex njugate eigenvalue are aigned in pair. Prf: Let z = Px be the tate tranfrmatin that tranfrm the equatin int ntrllable annial frm: Pb b PP - b - have we With ) ( ) ( ) ( b) det(i

18 det(i b) Thi mean that the eigenvalue f aigned. Hw abut - b? ( ) ( ) ( ) b an be arbitrarily - If we let Q P - then - b QQ QbQ Q( b)q The eigenvalue f - b are the ame a the f b Frm the prf a predure t deign the feedba gain an be derived. 5 Predure fr aigning the eigenvalue f -b. Step. Che the deired eigenvalue et { i i= n} whih ntain njugate mplex pair e.g. i = -+j and i+ = j and btain the effiient f n n n () (λ )(λ ) (λ ) d Step. Cmpute the harateriti plynmial f () det(i) n n n and the tranfrmatin matrix e.g. fr n = Q : P - b b b b n n n 6 8

19 9 7 Pb b PP Then - b - Step : i i i Che b Then - Step : P. Cmpute...n} i {λ the deired eigenvalue ha b)q Q( - b then i 8 Example: le. ntrllab (B) nningular B B B G untable - eigenvalue : B Step : The deired eigenvalue - -+j --j 8 5 j) j)( )( ( ) ( d Step : Charateriti plynmial f ) I det( Q P G Q

20 9 Step : 8 8; ; ; Step : 9 8 P Step 5: Verify: b j - - j b : f Eigenvalue ) ) )(( ( ) )( ( 8 9 ) det(i Tranfer funtin f the feedba ytem: β β β β b The riginal ytem x y bu x x The ytem with tate feedba x y br ( - b)x x Tranfer funtin frm u t y: β β β β b ) (I -b Tranfer funtin frm r t y: ) ( ) ( ) ( β β β β b b) (I

21 Cmpare: (I) b β β β β (I b) β β β β b ( ) ( ) ( ) Cnluin: State feedba de nt hange the zer f the ytem. If (b) i ntrllable the ple an be arbitrarily aigned. The feedba gain that aign the eigenvalue i unique. (Nt unique if the ytem ha multiple input). If a new ple i the ame a ne f the zer the rder f the led-lp ytem an be redued. mut be unbervable. (ine the ntrllability i the ame). Deirable eigenvalue regin t the firt tep f the predure we need t he the deirable eigenvalue. Hw t d thi? There are me general rule depending n the perfrmane pe. Suh a the verht rie time ettling time (nvergene rate). Generally Large real part f eigenvalue fat nvergene hrt ettling time Large imaginary part f eigenvalue big illatin and big verht. If the rati between the imag part and the real part i apprpriate we may have mall verht and fat rie time typial regin fr deired eigenvalue Im Re

22 State feedba deign: multiple input ae Cnider a ytem x x Bu; y Cx Du where R nn BR np CR qn. We an al tranfrm the ytem int a ntrllable annial frm. The idea i extended frm the ingle-input ae; The annial frm al reveal the truture t ee hw the ple are mved; Hwever the predure an be very mpliated. (ee 8.6.) Here we will tudy a quite different apprah. It al applie t ingle input ytem. State feedba deign: By lving matrix equatin In thi apprah we dn t tranfrm a ytem int a ntrllable annial frm Hw de it wr? The main idea i a fllw. The prblem: Find K.t. - BK ha a et f deired eigenvalue ay the eigenvalue f F. Thi i the ae if -BK and F are imilar i.e. there exit a nningular matrix T.t. - BK = TFT - ~ Similar matrie have ame eigenvalue Key: Find bth K and T

23 The new prblem: Given B and F find K and nningular T uh that - BK = TFT - Multiply bth ide frm right with T we btain T -BKT = TF Sine T i nningular there i a ne t ne rrepndene between KT and K. If we let K = KT then K=K T -. Nw T BK = TF T T F = BK The predure: he K R pn. Slve T T F = BK fr T. If T i nningular let K=K T -. Then -BK and F are imilar. Then -BK ha the deired eigenvalue. Main nern: Hw t lve the matrix equatin T-TF=BK? Under what nditin i the lutin T nningular? 5 Main nern: Hw t lve the matrix equatin T T F = BK? Under what nditin i the lutin T nningular? Summary f the main pint: The matrix equatin an be tranfrmed int a regular linear algebrai equatin with nn unnwn. It ha a unique lutin iff and F have n mmn eigenvalue. If (B) i ntrllable then the lutin i generally nningular with K arbitrarily hen. If K i generated by rand(pn) r randn(pn) then the prbability that T i nningular i. When p = the reulting K=K T - i unique. When p > the reulting K=K T - i nt unique. Baed n thee reult ptimizatin algrithm an be develped fr imprving ther perfrmane while the eigenvalue are at the deired latin. 6

24 Tranfrmatin int a regular algebrai equatin: Example: Slve T - TF = BK F S T TF t t BK t t t T t t t t t t t t t t t t t BK De it have a lutin? Regnizing that we have variable and nditin the abve an be nverted t: t t t t 7 but the lutin t T TF = BK Therem : If and F have n mmn eigenvalue then the equatin ha a unique lutin. (.7) Therem : If and F have n mmn eigenvalue the neeary nditin fr T t be nningular are that { B} i ntrllable and {FK } i bervable. Fr the ingle input ae (p=) T i nningular iff { B} i ntrllable and {FK } i bervable. Therem : Suppe that and F have n mmn eigenvalue and (B) i ntrllable. Then fr almt all K T i nningular. 8

25 lgrithm Selet F having deired led-lp eigenvalue whih are different frm the f Che an arbitrary K uh that {FK } i bervable Slve T-TF=BK t btain the unique T. The matlab mmand t lve the equatin i T=lyap(-F-B*K) If T i nn-ingular let K = K T -. Then -BK ha the deired eigenvalue. If T i ingular whih i rarely the ae he a different K and try again Finally dn t frget t he if -BK ha the deired eigenvalue. Yu might have typed the wrng number. eig(-b*k)=? 9 but the eletin f F: Firt elet the deired eigenvalue with me rule If the deired eigenvalue are all real imply let F=diag{ n } If the deired eigenvalue ha mplex njugate pair ay +j -j +j -j he λ F β β β β 5 5

26 Example: B F 5 Ue T= lyap(-f-b*k) and K=K*inv(T) K: K Oberve that me K have mall element but me may have 5 big element. In implementatin we lie t ue mall valued K. Obervatin: If there are mre than ne K that aign the eigenvalue f -BK t the ame latin then there are infinitely many f them. n intereting and meaningful prblem: Pi ne frm the K whih aign the eigenvalue uh that the petral nrm f K i.e. K i minimized. We may al develp algrithm t he K t ptimize r imprve ther perfrmane ee e.g. T. Hu Z. Lin and J. Lam `` unified gradient apprah t perfrmane ptimizatin under ple aignment ntraint" Jurnal f Optimizatin Thery and ppliatin July T. Hu and J. Lam ``Imprvement f parametri tability margin under ple aignment'' IEEE Tranatin n utmati Cntrl Vl.~ N.~ pp.~

27 Hw t realize tate-feedba in Simulin? u x x Bu y Cx Same ytem an be equivalently realized with u x xbu y x Under tate feedba u=v-kx v u y x x Bu x y x K x nt available in thi bl C x y C y y=cy =Cx The purpe f ding thi i t get x 5 Tday: Cntrllability and bervability ntinued Cntrllability/bervability dempitin Minimal realizatin Cnditin fr Jrdan frm nditin Parallel reult fr direte-time ytem Cntrllability after ampling State feedba deign Uing ntrllable annial frm By lving matrix equatin Next Time: Regulatin and traing Rbut traing and diturbane rejetin Stabilizatin State etimatin 5 7

28 8 55 Prblem et #. I the fllwing tate equatin ntrllable? Obervable? x y u x x 56. Fr the fllwing tate equatin x y u x x ) Find a tate feedba u = r - x t plae the ple at ---. Ue bth methd (via ntrllable annial frm via lving matrix equatin hw all tep) and mpare the reult. ) Find a tate feedba u = r - f x t plae the ple at -+j --j -8 Ue bth methd and mpare the reult. ) Ue imulin t imulate the led-lp ytem reulting frm ) and ) repetively under initial nditin x()=[ - ] and r(t) =unit tep. Plt y(t) fr the tw ae in the ame figure.

29 . Fr the fllwing tate equatin x x u y x ) Find tw different tate feedba u = r K x and u = r- K x t plae the ple at -+j --j -6. Try t find K and K uh that ne ha relatively larger element and the ther ne ha relatively mall element. ) Ue imulin t imulate the led-lp ytem reulting frm Cae : u = r - K x x()=[ ] and r(t) =. Cae : u = r - K x x()=[ ] and r(t) =. Plt y(t) fr the tw ae in the ame figure. Plt u (t) fr the tw ae in the ame figure. Plt u (t) fr the tw ae in the ame figure. u Nte that u u 57 9