CDS 101: Lecture 5.1 Reachability and State Space Feedback
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1 CDS, Lctur 5. CDS : Lctur 5. Rachability ad Stat Spac Fdback Richard M. Murray ad Hido Mabuchi 5 Octobr 4 Goals: Di rachability o a cotrol systm Giv tsts or rachability o liar systms ad apply to ampls Dscrib th dsig o stat dback cotrollrs or liar systms Radig: Åström ad Murray, Aalysis ad Dsig o Fdback Systms, Ch 5 Lctur Rviw 4.: rom Liar Last Systms Wk u = A + Bu ( = y t ( = At A( t τ ( + ( τ τ + ( τ = yt C C Bu d t Proprtis o liar systms Liarity with rspct to iitial coditio ad iputs Stability charactrizd by igvalus May applicatios ad tools availabl Provid local dscriptio or oliar systms 5 Oct 4 H.Mabuchi, Caltch CDS 5 Octobr 4
2 CDS, Lctur 5. Cotrol Dsig Cocpts Systm dscriptio: sigl iput, sigl output systm (MIMO also OK = (, u R, ( giv y = h(, u u, y Stability: stabiliz th systm aroud a quilibrium poit Giv quilibrium poit R, id cotrol law u=α( such that lim t ( = or all ( Rachability: str th systm btw two poits Giv, R, id a iput u(t such that = u t t = T = (, ( taks ( ( Trackig: track a giv output trajctory Giv y d (t, id u=α(,t such that ( lim yt ( y( t = or all ( d y(t y d (t t 5 Oct 4 H.Mabuchi, Caltch CDS 3 Rachability o Iput/Output Systms = (, u y = h(, u D A iput/output systm is rachabl i or ay, R ad ay tim T > thr ists a iput u:[,t] R such that th solutio o th dyamics startig rom (= ad applyig iput u(t givs (T=. Rmarks I th diitio, ad do ot hav to b quilibrium poits w do t cssarily stay at atr tim T. Rachability is did i trms o stats dos t dpd o output For liar systms, ca charactriz rachability by lookig at th gral solutio: T = A + Bu AT A( T τ ( T = + Bu( τ dτ τ = I itgral is surjctiv (as a liar oprator, th w ca id a iput to achiv ay dsird ial stat. 5 Oct 4 H.Mabuchi, Caltch CDS 4 5 Octobr 4
3 CDS, Lctur 5. = A + Bu Tsts or Rachability T & AT A( T τ = + τ = ( T Bu( τ dτ Thm A liar systm is rachabl i ad oly i th rachability matri is ull rak. Rmarks Vry simpl tst to apply. I MATLAB, us ctrb(a,b ad chck rak w/ dt( I this tst is satisid, w say th pair (A,B is rachabl Som isight ito th proo ca b s by padig th matri potial 5 Oct 4 H.Mabuchi, Caltch CDS 5 Eampl #: Liarizd pdulum o a cart θ m Qustio: ca w locally cotrol th positio o th cart by propr choic o iput? F M Approach: look at th liarizatio aroud th upright positio (good approimatio to th ull dyamics i θ rmais small 5 Oct 4 H.Mabuchi, Caltch CDS 6 5 Octobr 4 3
4 CDS, Lctur 5. Eampl #, co t: Liarizd pdulum o a cart m θ u M Rachability matri B AB A B A 3 B Simpliy by sttig b = Full rak as log as costats ar such that colums ad 3 ar ot multipls o ach othr rachabl as log as g(m+m ca str liarizatio btw poits by propr choic o iput 5 Oct 4 H.Mabuchi, Caltch CDS 7 Cotrol Dsig Cocpts Systm dscriptio: sigl iput, sigl output systm (MIMO also OK = (, u y = h(, u Stability: stabiliz th systm aroud a quilibrium poit Giv quilibrium poit R, id cotrol law u=α( such that lim t ( = or all ( Rachability: str th systm btw two poits Giv, R, id a iput u(t such that = (, u( t taks ( t = ( t = Trackig: track a giv output trajctory Giv y d (t, id u=α(,t such that ( lim yt ( y( t = or all ( d y(t y d (t t 5 Oct 4 H.Mabuchi, Caltch CDS 8 5 Octobr 4 4
5 CDS, Lctur 5. Stat spac cotrollr dsig or liar systms = A + Bu T ( = AT A( T τ + ( τ τ = T Bu dτ Goal: id a liar cotrol law u=k such that th closd loop systm = A + BK = ( A + BK is stabl at =. Rmarks Stability basd o igvalus us K to mak igvalus o (A+BK stabl Ca also lik igvalus to prormac (g, iitial coditio rspos Qustio: wh ca w plac th igvalus ayplac that w wat? Thorm Th igvalus o (A+BK ca b st to arbitrary valus i ad oly i th pair (A,B is rachabl. MATLAB: K = plac(a, B, igs 5 Oct 4 H.Mabuchi, Caltch CDS 9 Natural dyamics = br a = a d Eampl #: Prdator pry Cotrolld dyamics: modulat ood supply = br ( + u a = a d Q: ca w mov rom som iitial populatio o os ad rabbits to a spciid o i tim T by modulatio i th ood supply? Q: ca w stabiliz th populatio aroud th dsird quilibrum poit Approach: try to aswr this qustio locally, aroud th atural quilibrium poit Oct 4 H.Mabuchi, Caltch CDS ustabl stabl 5 Octobr 4 5
6 CDS, Lctur 5. Eampl #: Problm stup Equilibrium poit calculatio = br ( + u a = a d = (5, 35 Liarizatio Comput liarizatio aroud quil. poit, : A= B = u (, u (, u % Comput th quil poit % prdpry.m cotais dyamics = ili('prdpry(,'; q = solv(, [5,5]; % Comput liarizatio A = [ br - a*q( - a*q(; a*q(, -d + a*q( ]; B = [br*q(; ]; Rdi local variabls: z=-, v=u-u d z br a, a, z b r, v dt z = a, d a, z + + Rachabl? YES, i b r,a (chck [B AB] ca locally str to ay poit 5 Oct 4 H.Mabuchi, Caltch CDS Eampl #: Stabilizatio via igvalu assigmt d z br a, a, z b r, v dt z = + a, d + a, z Cotrol dsig: v = Kz = K( u = u + v = u K( Plac pols at stabl valus Choos λ=-, - K = plac(a, B, [-; -]; Modiy dyamics to iclud cotrol = b ( K( a = a d r stabl ustabl Oct 4 H.Mabuchi, Caltch CDS 5 Octobr 4 6
7 CDS, Lctur 5. Implmtatio Dtails Eigvalus dtrmi prormac For ach igvalu λ i =σ i + jω i, gt cotributio o th orm ( ω ω σt y ( t = asi( t + bcos( t i Imag Ais - Pol-zro map ω p Amplitud To: Y( Stp Rspos From: U( T π/ω p Rpatd igvalus ca giv additioal trms o th orm t k σ +jω Ral Ais Tim (sc. Us stimator to dtrmi th currt stat i you ca t masur it u = A+ Bu y Estimator looks at iputs ad outputs o plat ad stimats th currt stat Ca show that i a systm is obsrvabl th you ca costruct ad stimator Us th stimatd stat as th dback Estimator ˆ u = Kˆ Kalma iltr is a ampl o a stimator 5 Oct 4 H.Mabuchi, Caltch CDS 3 Summary: Rachability ad Stat Spac Fdback = A + Bu u = u K( Ky cocpts Rachability: id u s.t. Rachability rak tst or liar systms Stat dback to assig igvalus Oct 4 H.Mabuchi, Caltch CDS 4 5 Octobr 4 7
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