CDS 101: Lecture 5.1 Controllability and State Space Feedback
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1 CDS, Lecture 5. CDS : Lecture 5. Cotrollability ad State Space Feedback Richard M. Murray 8 October Goals: Deie cotrollability o a cotrol system Give tests or cotrollability o liear systems ad apply to eamples Describe the desig o state eedback cotrollers or liear systems Readig: Packard, Poola ad Horowitz, Dyamic Systems ad Feedback, Sectio 5 Optioal: Friedlad, Sectios (cotrollability oly Optioal: A. D. Lewis, A Mathematical Itroductio to Feedback Cotrol, Chapter (available o web page Lecture Review 4.: rom Liear Last Systems Week u = A + Bu y = C+ ( = y t ( = At A( t τ ( + ( τ τ + ( τ = yt Ce Ce Bu d t Properties o liear systems Liearity with respect to iitial coditio ad iputs Stability characterized by eigevalues May applicatios ad tools available Provide local descriptio or oliear systems 8 Oct CDS /8/
2 CDS, Lecture 5. Cotrol Desig Cocepts System descriptio: sigle iput, sigle output oliear system (MIMO also OK = (, u, ( give y = h(, u Stability: stabilize the system aroud a equilibrium poit Give equilibrium poit e, id cotrol law u=α( such that lim t ( = or all ( t e Cotrollability: steer the system betwee two poits Give,, id a iput u(t such that = u t t = T = (, ( takes ( ( Trackig: track a give output trajectory Give y d (t, id u=α(,t such that ( lim yt ( y( t = or all ( t d y(t y d (t t 8 Oct CDS 3 Cotrollability o Liear Systems = A + Bu y = C+, ( give De A liear system is cotrollable i or ay, ad ay time T > there eists a iput u:[,t] such that the solutio o the dyamics startig rom (= ad applyig iput u(t gives (T=. Remarks I the deiitio, ad do ot have to be equilibrium poits we do t ecessarily stay at ater time T. Cotrollability is deied i terms o states does t deped o output Ca characterize cotrollability by lookig at the geeral solutio o a liear system: T AT A( T τ ( T = e + e Bu( τ dτ τ = I itegral is ull rak, the we ca id a iput to achieve ay desired ial state. 8 Oct CDS 4 /8/
3 CDS, Lecture 5. Tests or Cotrollability Thm A liear system is cotrollable i ad oly i the cotrollability matri is ull rak. = A + Bu y = C+, ( give B AB A B A B T ( = AT A( T τ + ( τ τ = T e e Bu dτ Remarks Very simple test to apply. I MATLAB, use ctrb(a,b I this test is satisied, we say the pair (A,B is cotrollable Some isight ito the proo ca be see by epadig the matri epoetial = + ( + ( + + ( + B ABT ( τ ABT ( τ A BT ( τ AT ( τ e B I A T τ A T τ A T τ B = Oct CDS 5 Eample #: Liearized pedulum o a cart θ m Questio: ca we locally cotrol the positio o the cart by proper choice o iput? Approach: look at the liearizatio aroud the upright positio (good approimatio to the ull dyamics i θ remais small M 8 Oct CDS 6 /8/ 3
4 CDS, Lecture 5. Eample #, co t: Liearized pedulum o a cart θ m M Cotrollability matri B AB A B A 3 B Full rak as log as costats are such that colums ad 3 are ot multiples o each other cotrollable as log as g(m+m ca steer liearizatio betwee poits by proper choice o iput 8 Oct CDS 7 Costructive Cotrollability Give that system is cotrollable, how do we id iput to traser betwee states? Simple case: chai o itegrators 3 = = = = u Fid curve (t such that ( ( = ( ( = ( T ( T ( ( T Choose iput as ut ( = ( t ( Cotrollable caoical orm I cotrollable, ca show there eists a liear chage o coordiates such that z = z+ u a a a z z z = ( z ut ( = z ( t + ( az ( + + az 8 Oct CDS 8 /8/ 4
5 CDS, Lecture 5. Cotrol Desig Cocepts System descriptio: sigle iput, sigle output oliear system (MIMO also OK = (, u, ( give y = h(, u Stability: stabilize the system aroud a equilibrium poit Give equilibrium poit e, id cotrol law u=α( such that lim t ( = e or all ( t Cotrollability: steer the system betwee two poits Give,, id a iput u(t such that = (, u( t takes ( t = ( t = Trackig: track a give output trajectory Give y r (t, id u=α(,t such that ( lim yt ( y( t = or all ( t d 8 Oct CDS 9 y(t y d (t t State space cotroller desig or liear systems = A + Bu y = C+, ( give T ( = AT A( T τ + ( τ τ = T e e Bu dτ Goal: id a liear cotrol law u=k such that the closed loop system = A + BK = ( A + BK is stable at e =. Remarks Stability determied by eigevalues use K to make eigevalues o (A+BK stable Ca also lik eigevalues to perormace (eg, iitial coditio respose Questio: whe ca we place the eigevalues ayplace that we wat? Thm The eigevalues o (A+BK ca be set to arbitrary values i ad oly i the pair (A,B is cotrollable. MATLAB: K = place(a, B, eigs 8 Oct CDS /8/ 5
6 CDS, Lecture 5. Natural dyamics = b a b r = a d b Eample #: Predator prey Cotrolled dyamics: modulate ood supply = b ( + u a b r = a d b 8 6 Q: ca we move rom some iitial populatio o oes ad rabbits to a speciied oe i time T by modulatio i the ood supply? Q: ca we stabilize the populatio aroud the curret equilibrum poit Approach: try to aswer this questio locally, aroud the equilibrium poit Oct CDS Eample #: Problem setup Equilibrium poit calculatio = br ( + u a b = a d b e = ( Liearizatio Compute liearizatio aroud equil. poit, e : A= B = u ( e, ue ( e, ue Redeie local variables: z=- e, v=u-u e % Compute the equil poit % predprey.m cotais dyamics = ilie('predprey(,'; eq = solve(, [5,5]; % Compute liearizatio A = [ br - a*eq( - *b*eq(, -a*eq(; a*eq(, -d + a*eq( - *b*eq( ]; B = [br*eq(; ]; d z br a, e b, e a, e z b r, e v dt z = a, e d a, e b, e z + + Cotrollable? YES, i b r,a (check [B AB] ca locally steer to ay poit 8 Oct CDS /8/ 6
7 CDS, Lecture 5. Eample #: Stabilizatio via eigevalue assigmet d z br a, e b, e a, e z b r, e v dt z = a, e d a, e b +, e z + Cotrol desig: v = Kz = K( e u = u + v Place poles at stable values Choose λ=-, - K = place(a, B, [-; -]; e Modiy dyamics to iclude cotrol = b ( K( a b a d b r e = Oct CDS 3 How to Assig Eigevalues Eigevalue locatio determies time respose For each eigevalue λ i =σ i + jω i, get cotributio o the orm ( ω ω σt y ( t = e asi( t + bcos( t i Imag Ais - Pole-zero map ω p Amplitude To: Y( Step Respose From: U( T π/ω p Repeated eigevalues ca give additioal terms o the orm t k e σ +jω Real Ais Time (sec. Eigevalue assigemet chages the dyamics o the system Illustrates oe o the mai priciples o eedback Ca be used to make ustable poits stable, icrease the speed o the respose, etc Cautio: eigevalue assigmet aects magitude o iput required The amout o actuator eort required to chage the dyamics rom the atural (ope loop dyamics to the desired (closed loop dyamics ca be large We will lear more sophisticated ways to deal with this tradeo later i the course 8 Oct CDS 4 /8/ 7
8 CDS, Lecture 5. State Space Feedback Desig Tools Eigevalue assigmet (also called pole placemet Choose eigevalue locatios to correspod to desired dyamics Optimal cotrol (CDS b Choose the eedback that miimizes a cost uctio: ( T T J = Q+ urudt u = K K = lqr(a, B, Q, R Key advatage is the eplicit tradeo betwee state error ad iput magitude Ote very heard to relate the weights (Q, R to the desired system behavior Noliear systems (CDS I liearizatio is cotrollable, ca show that liear eedback provides local stability More geerally, ca look at oliear state eedback, u = α( Optimal cotrol problem ca t be solved i closed orm Alterative: Lyapuov based desig ( itegrator backsteppig, Lypuov redesig, etc 8 Oct CDS 5 Summary: Cotrollability ad State Space Feedback = A + Bu y = C+ B AB A B A B u = u + K( e e Key cocepts Cotrollability: id u s.t. Cotrollability rak test or liear systems State eedback to assig eigevalues Oct CDS 6 /8/ 8
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