CDS 11: Lecture 5-1 Reachability an State Sace Feeback Richar M. Murray 3 October 6 Goals:! Define reachability of a control syste! Give tests for reachability of linear systes an aly to exales! Describe the esign of state feeback controllers for linear systes Reaing:! Åströ an Murray, Feeback Systes, Ch 6 Review fro Last Week 3 Oct 6 R. M. Murray, Caltech CDS
Control Design Concets Syste escrition: single inut, single outut syste MIMO also OK) Stability: stabilize the syste aroun an equilibriu oint! Given equilibriu oint x e R n, fin control law u=!x) such that Reachability: steer the syste between two oints! Given x o, x f R n, fin an inut ut) such that ẋ = fx, ut)) takes xt ) = x xt ) = x f Tracking: track a given outut trajectory! Given y t), fin u=!x,t) such that x y x f y t 3 Oct 6 R. M. Murray, Caltech CDS 3 Reachability of Inut/Outut Systes Defn An inut/outut syste is reachable if for any x o, x f R n an any tie T > there exists an inut u [,T ] R such that the solution of the ynaics starting fro x )=x an alying inut ut) gives xt)=x f. Rearks! In the efinition, x an x f o not have to be equilibriu oints " we on t necessarily stay at x f after tie T.! Reachability is efine in ters of states " oesn t een on outut! For linear systes, can characterize reachability by looking at the general solution:! If integral is surjective as a linear oerator), then we can fin an inut to achieve any esire final state. 3 Oct 6 R. M. Murray, Caltech CDS 4
Tests for Reachability Th A linear syste is reachable if an only if the n! n reachability atrix is full rank. Rearks! Very sile test to aly. In MATLAB, use ctrba,b) an check rank w/ et)! If this test is satisfie, we say the air A,B) is reachable! Soe insight into the roof can be seen by exaning the atrix exonential 3 Oct 6 R. M. Murray, Caltech CDS 5 Exale #1: Linearize enulu on a cart # Question: can we locally control the osition of the cart by roer choice of inut? Aroach: look at the linearization aroun the uright osition goo aroxiation to the full ynaics if # reains sall) F M t 1 1 ṗ = l g c M t l M t l M t gl M t l y = [ 1 ] x, cl M t l γ l M t l γm t M t l ṗ + M t l l M t l u 3 Oct 6 R. M. Murray, Caltech CDS 6
u Exale #1, con t: Linearize enulu on a cart M Reachability atrix W r = µ l µ # t gl 3 3 µ gl +M) µ µ gl 3 3 µ l µ gl +M) µ B AB A B A 3 B 1 1 ṗ = l g c γ l µ µ µ ṗ + M t gl cl γm t µ µ µ µ = M t l. µ l µ u Silify by setting c,! = Full rank as long as constants are such that coluns 1 an 3 are not ultiles of each other " reachable as long as etw r ) = g l 4 4 µ 4 " can steer linearization between oints by roer choice of inut 3 Oct 6 R. M. Murray, Caltech CDS 7 Control Design Concets Syste escrition: single inut, single outut syste MIMO also OK) Stability: stabilize the syste aroun an equilibriu oint! Given equilibriu oint x e R n, fin control law u=!x) such that " Reachability: steer the syste between two oints! Given x, x f R n, fin an inut ut) such that x x f Tracking: track a given outut trajectory! Given y t), fin u=!x,t) such that y y t 3 Oct 6 R. M. Murray, Caltech CDS 8
State sace controller esign for linear systes Goal: fin a linear control law u = -K x such that the close loo syste is stable at x e =. Rearks ẋ = Ax BKx = A BK)x! Stability base on eigenvalues " use K to ake eigenvalues of A+BK) stable! Can also link eigenvalues to erforance eg, initial conition resonse)! Question: when can we lace the eigenvalues anylace that we want? Theore The eigenvalues of A - BK) can be set to arbitrary values if an only if the air A, B) is reachable. MATLAB: K = lacea, B, eigs) 3 Oct 6 R. M. Murray, Caltech CDS 9 Exale #: Preator rey Natural ynaics H t = r hh 1 H ) ahl H L t = r ll 1 L ) L Controlle ynaics: oulate foo suly H t = r h + u)h 1 H ) ahl L t = r ll 1 L ) Stable Q1: can we ove fro soe initial oulation of foxes an rabbits to a secifie one in tie T by oulation of the foo suly? Q: can we stabilize the oulation aroun the esire equilibriu oint Unstable Aroach: try to answer this question locally, aroun the natural equilibriu oint 3 Oct 6 R. M. Murray, Caltech CDS
Exale #: Proble setu Equilibriu oint calculation H t = r h + u)h 1 H ) ahl L t = r ll 1 L )! x e = 6.5, 1.3), ue =, ye = 6.5 Linearization! Coute linearization aroun equil. oint, x e : % Coute the equil oint % rerey. contains ynaics f = inline'rerey,x)'); xeq = fsolvef, [1, ]); % Coute linearization A = [ rh - *H*rH)/K - a*l)......, rl - *L*rL)/H*k) ]; B = [H*1 - H/K); ];! Reefine local variables: z=x-x e, v=u-u e t [ z1 z ] [ al = ah T h +1) H r h L r l H k K + r h ah ah T h +1 r l L r l H k ] [z1 z ] + [ )] H 1 H K v 3 Oct 6 R. M. Murray, Caltech CDS 11 Exale #: Stabilization via eigenvalue assignent t [ z1 z Control esign: v = Kz + k r r ] [ al = ah T h +1) H r h L r l H k u = u e + Kx x e ) + k r r y e ) K + r h ah ah T h +1 r l L r l H k ] [z1 z ] + [ )] H 1 H K v Place oles at stable values! Choose $=-1, -! K = lacea, B, [-1; -]); Stable Moify NL ynaics to inclue control H t = r h + u)h 1 H ) ahl L t = r ll 1 L ) 3 Oct 6 R. M. Murray, Caltech CDS 1
Ileentation Details Eigenvalues eterine erforance! For each eigenvalue $ i =% i + j& i, get contribution of the for! Reeate eigenvalues can give aitional ters of the for t k e % + j& Iag Axis 1-1 Pole-zero a & - -15-1 -5 5 Real Axis Alitue To: Y1)..15.1.5 Ste Resonse Fro: U1) T' /& 5 1 15 5 3 Tie sec.) Use estiator to eterine the current state if you can t easure it u y! Estiator looks at inuts an oututs of lant an estiates the current state! Can show that if a syste is observable then you can construct an estiator! Use the estiate state as the feeback Estiator! Kalan filter is an exale of an estiator 3 Oct 6 R. M. Murray, Caltech CDS 13 Suary: Reachability an State Sace Feeback x x f u = Kx + k r r Key concets! Reachability: fin u s.t. x ) x f! Reachability rank test for linear systes! State feeback to assign eigenvalues 3 Oct 6 R. M. Murray, Caltech CDS 14