Control Systems (Lecture note #7)

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1 6.5 Conrol Sysms (Lcur no #7) Ls Tm: Gnrlz gnvcors Jorn form Polynoml funcons of squr mrx bg pcur: on brnch of h cours Vcor spcs mrcs lgbrc quons Egnvlus Egnvcors Dgonl form Cnoncl form Soluons o : x x u; y Cx Du Mrx funcons such s Th lnr lgbr ools wll lso b usful for ohr objcvs. Rvw: gonl form n Jorn form ll gnvlus of r snc gonlzbl Thr r rp gnvlus.g wh mulplcy. If (- I)= n - (- I)= hr xs LI soluons o (- I)v= n hy r ll gnvcors. If hs s h cs for ll rp gnvlus gonlzbl If (- I)=n - (- I) < hr xs gnrlz gnvcors no gonlzbl hr xs Jorn blocs

2 Dfnon. vcor v s gnrlz gnvcor of gr ssoc wh f I v bu I v Dno v v v Iv I v I v I v v I v I v v Iv I v v v Wh s h nw rprsnon w.r.. {v v. v }?.. [v v v ] = [v v v ]Ā v v v v v v v v v :.. Jorn bloc :.... Polynoml funcons of squr mrx L f()= = b polynoml funcon of. If =QĀQ - hn f() = Qf(Ā)Q -. L () b h chrcrsc polynoml of. Cyly-Hmlon Thorm: () = ny polynoml cn b xprss s polynoml of gr n-

3 Thorm. Gvn C nn n polynoml f) L h snc gnvlus of b =...m ch wh mulplcy n (n +n + +n m = n). L Thn f()=g() ff whr f g( ) β l n β βn f (l) ( ) = g (l) ( ) l =..n - =.. m l f l f () ( ) f( ) Unr h bov conon h coffcns s cn b rmn 5 Gnrl funcons of squr mrx: Dfnon: Gvn C nn. L h snc gnvlus of b =...m ch wh mulplcy n (n +n + +n m = n). L f() b gnrl funcon wh {f (l) ( )} wll fn. Suppos h g() s polynoml ssfyng f (l) ( ) = g (l) ( ) l =..n - =.. m Thn f() g(). Gnrlly g s polynoml of gr n-. 6

4 Toy: Som proprs of ; Soluon o connuous-m sysm x x u; y Cx Du Soluon o h scr-m sysm x[ ] [] u[]; y[] Cx[] Du[] Equvln s quons 7 Som proprs for From h fnon I!! Th followng cn b vrf I; ( ) ; ; Cuon: + usully os no qul o. W only hv + = whn = 8

5 Mor proprs:? I!!!! L -! 9 Mor proprs: τ τ τ? τ! τ! I I! τ τ τ ( ~ ssumng h - xss! = I) I I Ths wll b us o compu h oupu rspons unr consn npus.! 5

6 Exmpl. Lplc Trnsform of L L! ~ ssumng < si L s s s s I s s s L or L si si L How o compu (si - ) -? si ~ ssumng s s suffcnly lrg Exmpl. f() = (s - ) -. Compu f() = (si - ) -. () = ( - ) wh of mulplcy f () ( ) = (s - ) - f () ( ) = (s - ) - f () ( ) = (s - ) - g() = + ( - ) + ( - ) g () ( ) = = (s - ) - g () ( ) = = (s - ) - g () ( ) = = (s - ) - g() = (s - ) - + (s - ) - ( - ) + (s - ) - ( - ) g() = (s - ) - I + (s - ) - ( - ) + (s - ) - ( - ) 6

7 7 s s s s s s g() ) (si I) ( I ) (s ) (s ) (s g() / L ) L( W wll compu ; Som of s proprs; Soluon o connuous-m sysm Du Cx y u; x x Toy: Du[] Cx[] y[] u[]; [] ] x[ Soluon o h scr-m sysm Equvln s quons

8 S-Spc Soluons n Rlzons Soluons of Dynmc Equons x: n u: p y: q Consr lnr sysm: x x u; y Cx Du : nn rl mrx; : np rl mrx C: qn rl mrx; D: qp rl mrx Gvn x( ) = x n u() unqu soluon x() y() Wh s h soluon? 5 Rcll h rlr w rv h soluon for h npu/oupu scrpon bs on suprposon: y() G( τ)u(τ)τ Qusons: g g G( τ) g q ( - τ) ( - τ) ( - τ) g g g q ( - τ) ( - τ) ( - τ) Gvn sysm mrcs CD wh s G()? Wh s h rspons u o nl s? nohr pproch s by usng Lplc rnsform: ŷ (s) C(sI ) x() [C(sI ) D]û(s) g g g p p qp ( - τ) ( - τ) ( - τ) owns: h Lplc rnsform of u() my b no vlbl you my n o pproxm. 6 8

9 S-Spc Soluons Th sysm: x x u; y Cx Du Gvn x() n u() for. Th soluon for x n y s x() y() C x() x() ( τ) C u ( ) ; ( τ) u ( ) Du() Clrly wo prs: zro-npu rsp. + zro-s rsp. Lnry lso obvous. W now how o compu. Th ngron cn b on numrclly hrough scrzon. ( τ) u ( ) G(-)=C (-) ( )Δ u( ) 7 W frs consr h s x: x () x() u(); Rcll h Th y pr x x ( )x x x (**) Plug (*) no (**) x x u x u x() u() Ingr from o ; x( ) u( ) x( ) x() u( ) (*) Prmulplyng o boh ss nong - = I x( ) x() (- ) u( ) 8 9

10 W vrfy h h soluon x( ) x() u( ) ssfs x () x() u(); (-τ) x() x() u( τ)τ (-τ) (-τ) x() u( τ)τ u( τ) x() x() u() (-τ) u( τ)τ u( ) lso s clr h h nl conon s ssf. Fnlly y( ) Cx() Du() C x() C (- ) (- ) τ u( ) Du() 9 Dffrn wys o compu : From Dfnon : Form () n fn { } n ( ) (l) Consruc n (n - ) h orr polynoml such h g (l) ( ) = ( ) (l) for ll n l = g() From Dfnon :! Us Jorn form =QĀQ - = Q Ā Q - subl for compur Us h nvrs Lplc rnsform of (si-) -. =L - (si-) -

11 Exmpl: n LTI sysm: x () x() u(); y [ ]x Gvn x()=; u()= for. Compu y(). Sp : Compu. Egnvlus of r L g() = +b; f()=. From g(-)=-+b= - ; g(-)=-+b= -. = ; b= - - -; bi ( ) ( ) Sp : (- ) From y( ) C x() C u( ) y() - [ ] τ ( τ) [ u( τ)τ ] τ ( ( τ) ( τ) ) τ Som proprs bou h zro-npu rspons x() x Consr Jorn bloc!!! For gnrl h rms of r lnr combnons of n m R( ) < for ll. hn s ll rms convrgs o x() lwys convrgs o. Sbl sysm. R( ) > for som. hn s som rms vrg. Thr xs x such h x() grows unboun. Unsbl R( for ll. ll gnvlus wh rl prs r smpl s boun for ll bu no convrg o. crcl cs R( ) for ll. som gnvlus wh rl prs r rp unboun; x() unboun for som x. unsbl

12 Toy: W wll compu ; Som of s proprs; Soluon o connuous-m sysm x x u; y Cx Du Soluon o h scr-m sysm x[ ] [] u[]; Equvln s quons y[] Cx[] Du[] Dscrzon connuous-m sysm W us scrzon for x x u; y Cx Du Dgl smulon wh compur; Implmnon hrough gl conrollr pproch : Suppos w now x(t). If T s smll nough x(t T) x(t) x(t)t (x(t) u(t))t x(( )T) x(t) Tx(T) Tu(T) (I T)x(T) Tu(T) y(t) Cx(T) Dy(T) x[]: x(t); u[]: u(t) Smpl bu no ccur. x() x(t) x(t) x(t) x[ ] (I T)x[] Tu() y[] Cx[] Du[]

13 pproch : Rl suon: conrol u mplmn by compur n gl-nlog convrr. Durng holng pro u() = u(t) for ll [T (+)T) = Soluon T n (+)T x[ ] T T ( T (T-τ) x[] : x( T) x() u( τ)τ T ( )T )T (( x() )T-τ) u( τ)τ ( )T T (T-τ) T x() u( τ) τ T TT T (T-τ) x() u( τ)τ T T (T-τ) x[] u[]τ (()T-τ) T T (T-τ) x[] τ u[] : x[] u[] u( τ)τ 5 Th scrz sysm: whr x[ ] x[] u[] y[] C x[] D u[] T T (T-τ) τ C C Ths xcly scrbs h npu-s npu-oupu rlonshp nsns T T T For noc h T T T (T-τ) T τ T τ τ ( T T τ T T T T I [ I] τ ) τ [ D I] D 6

14 From CT sys. o DT sys. x x u y Cx Du T x[ ] x[] u[] y[] C x[] D u[] L h smplng pro b T. Thn Exmpl: [ I] C C D T. D Us mlb: =xpm(*t); =nv()*(-y())*; Soluon of Dscr-m Equons Th DT sysm: x[ ] x[] u[] y[] Cx[] Du[] Th soluon s rv n srghforwr wy: x[]=x[]+u[] x[]=x[]+u[]=(x[]+u[])+u[] = x[]+u[]+u[] x[]=x[]+u[]= x[]+ u[]+u[]+u[] x[] x[] y[] C x[] m m m C u[m] m u[m] Du[] 8

15 Som proprs bou h zro-npu rspons x[] x Consr Jorn bloc For gnrl h rms of r lnr combnons of ( ) m ( ) ( )( ) ( )! < for ll. hn s ll rms convrgs o x[] lwys convrgs o. Sbl sysm. > for som. hn s som rms vrg. Thr xs x such h x[] grows unboun. Unsbl for ll. ll gnvlus wh un mgnu r smpl s boun for ll bu no convrg o. Crcl cs for ll. som gnvlus wh un mgnu r rp unboun; x[] unboun for som x Unsbl!! 9 n Erlr Exmpl: Inrs n morzon How o scrb pyng bc cr lon ovr four yrs wh nl b D nrs r n monhly pymn p? L x[] b h moun you ow h bgnnng of h h monh. Thn x[+] = ( + r) x[] p Inl n rmnl conons: x[] = D n fnl conon x[8] = How o fn p? y solvng h sysm x[8]= D+ p p 5

16 Th sysm: x[+] = ( + r) x[] + () p Soluon: u x[] x[] ( r) ( r) x[] D m m m m ( r) ( r) u[m] m m ( )p p ( r) Gvn D=; r=.; x[8]=; (.) (.). 8 8 p ( r) D r p Your monhly pymn p= Toy: W wll compu ; Som of s proprs; Soluon o connuous-m sysm x x u; y Cx Du Soluon o h scr-m sysm x[ ] [] u[]; Equvln s quons y[] Cx[] Du[] 6

17 Equvln s quons Gvn s-spc scrpon: x x u; y Cx Du L P b nonsngulr mrx. - Dfn x Px hn x P x x Px Px Pu PP x Pu y Cx Du CP x Du Dno PP x x u; - P y Cx Du C CP D D No: For DT sysms h quvln rnsformon s h sm. - (*) (**) (*) n (**) r s o b quvln o ch ohr n h procur from (*) o (**) s cll n quvln rnsformon Rcll: PP - n r smlr o ch ohr Thy hv sm gnvlus. Sm sbly prf. Wh o w xpc from h wo rnsfr funcons: n G(s) C(sI ) D G(s) C(sI ) D To vrfy G(s) G(s) G(s) C(sI ) D CP (spp PP ) P - - CP - (P(sI - )P ) P D - CP P(sI ) P P D D (XYZ) Z C(sI ) D Y X 7

18 8 5 u x x x Q Dfn z xs). nvrs (h L Q - u z z such h n Compu Exmpl: Gvn s quon Q Q Q - - Soluon:. L Q Immly. How o g? Q ; Q ; Q ; Q Q Q Q Q Q Q Q Q 6 hs o ssfy (*) L =[ ] (*) cn b wrn s (**) From Cyly-Hmlon s horm: ()=. ) )(s (s si Δ(s) s s s s s Δ() I = = = = =[ ] ssfs For

19 Nx Tm: How o l wh complx gnvlus Rlzon of rnsfr funcon Smulon of sysms by usng Smuln Cours projc n mor from lnr lgbr Qurc funcons n posv-fnnss 7 Problm S #7. Th sysm: x x x() Compu x() for.. For h LTI sysm - x() x() - u(); y [ ]x ) Gvn x()=[ ] compu h zro-npu rspons y(); b) Gvn u()= for compu h zro-s rspons y(); c) L h smplng pro b T=.. Us mlb o compu h scrz sysm mrcs. 8 9

20 Mrm Rvw (Lcur #-Lcur #6) Molng of LTI sysms Lnr lgbr Vcor spcs: LI LD bss nnr prouc orhogonl Lnr lgbrc quon: rng spc null spc conons for h xsnc of soluon ll soluons Egnvlus gnvcors gonl form Gnrlz gnvcors Jorn from Polynoml funcons of mrx 9 Mol of crcu: S vrbls? n v S n oupu quons? u R v L v L R L L R vl v R L v v v C C C C R L v C x R L C L L u L v x R L L u() C v - - R y - L L v v L y R R v x x u y Cx Du u

21 Ingrors + mplfrs + u() - x x x x u x /s /s x y = x x x y() Wh r h s vrbls? Slc oupu of ngrors s SVs Wh r h s n oupu quons? x x x u x y x x u C D Exmpl u + + /s u /s + /s + y + +

22 Lnr lgbr scs Elmnry oprons h prsrv rmnn Elmnry oprons h prsrv rn Us lmnry opron o rnsform mrx no uppr or lowr rngulr form Lnr Inpnnc s of vcors {x x.. x m } n R n s LD f {.. m } n R no ll zro s.. x + x m x m = (*) If h only s of { } = o m s.. h bov hols s = =.. = m = hn {x } = o m s s o b LI Gvn {x x.. x m } form m x x... x If = hs unqu soluon LI; If = hs nonunqu soluon LD. If rn()=m h soluon s unqu LI If rn()<m h soluon s no unqu LD.

23 5 Exmpls: r h followng ss of vcors LI or LD? c b f c b c b LI f h rn of h mrx quls h numbr of columns 6 ss n Rprsnons s of LI vcors {.. n } of R n s s o b bss of R n f vry vcor n R n cn b xprss s unqu lnr combnon of hm For ny x R n hr xs unqu {.. n }s.. n n n.. x n n :... x n... x : Rprsnon of x wh rspc o h bss Thorm: In n n-mnsonl vcor spc (or subspc) ny s of n LI vcors qulfs s bss

24 Chng of bss: Gvn bss... ; n L h nw bss b: Thn Q n n n Q For x such h x... n β n W hv x -... Q β n 7 Lnr lgbrc quon x = y If () ([ : y]) (.. y R()) hn h quons r nconssn n hr s no soluon If () = ([ : y]) hn ls on soluon If () = ([ : y]) < n (.. () > ) hn hr r nfn numbr of soluons If () = ([ : y]) = n (.. () = ) hn hr s unqu soluon For n nn mrx x = y hs unqu soluon y R m ff - xss or 8

25 Ky concps: ssum R mn. Rng spc R(): {yr m : xss xr n s.. y=x} subspc of R m mnson = () rn of bss: form by h mxml numbr of LI columns of Null spc N(): {xr n : x=} subspc of R n mnson ()=n-() bss: form by () LI soluons o x=. 9 Exmpl: Th rng spc R() s spnn by { } Wh s h rlonshp mong h vcors? Wh r () ()? Th mnson of R()? Th bss of R()? Th null spcs? 5 5

26 Prmrzon of ll soluons Thorm: Gvn mn mrx n m vcor y. L x p b soluon o x = y. L ()=. Suppos > n h null spc s spnn by {n n n } Th s of ll soluons s gvn by {x = x p + n + n + + n : R} 5 Egnvlus gnvcors n gonl form sclr s cll n gnvlu of C nn f nonzro x C n such h x = x n x s h gnvcor ssoc wh. Cs : ll gnvlus r snc Thorm: h ss of gnvcors {v v.v n } s LI. L Q=[v v v n ] hn - Q Q : : : n 5 6

27 Dfnon. vcor v s gnrlz gnvcor of gr ssoc wh f I v bu I v Dno v v v Iv I v I v I v v I v I v v Iv I v v v Wh s h nw rprsnon w.r.. {v v. v }?.. [v v v ] = [v v v ]Ā... v v v v v v v v : : Jorn bloc v Polynoml funcons of squr mrx. Compuon of. Toy s mrl wll no b nclu n h mrm. Qusons? 5 7

28 Mrm xm: 6.-9: pm Oc (Thursy) Opn boo opn nos No clculor No Lpop Goo luc! 55 8

The Mathematics of Harmonic Oscillators

The Mathematics of Harmonic Oscillators Th Mhcs of Hronc Oscllors Spl Hronc Moon In h cs of on-nsonl spl hronc oon (SHM nvolvng sprng wh sprng consn n wh no frcon, you rv h quon of oon usng Nwon's scon lw: con wh gvs: 0 Ths s sos wrn usng h