Flight Dynamics & Control Eigenvalue-Eigenvector Assignment

Transcription

1 Flght Dynamcs & Control Egenvalue-Egenvector Assgnment Harry G. Kwatny Department of Mechancal Engneerng & Mechancs Drexel Unversty

2 Outlne State Feedback Pole placement revsted Egenvector assgnment Example: F-16 Example: L1011

3 State Space Desgn Consder the usual lnear plant x& = Ax+ Bu y = Cx+ Du r = 0 The feedback stablzaton problem: fnd an output feedback control that shapes the transent response of the closed loop to meet prescrbed objectves. Ths wll be done n three steps: 1) Desgn a state feedback control, u = Kx 2) Desgn a state estmator, that generates state estmates xt ˆ ( τ) ( τ) τ u() t = xˆ () t from avalable nformaton,.e., u, y [0, t) 3) Implement the composte controller, ( ) x& = Ax + Bu y = Cx+ Du

4 Pole Assgnment Problem Pole assgnment ( ) x& = Ax+ Bu, A, B completely controllable, rank B= m problem: {,, } and vectors { v K v } 1 n 1 Gven a self conjugate set of scalars,, fnd a real m n matrx K such n that the egenvalues and egenvectors of ( A+ BK) are precsely the gven sets. Theorem (Wonham, 1967): The system s controllable f and only f { 1 n} matrx such that ( + { } for every self-conjugate set of scalars,, there exsts a m n K A BK 1 n real ) has,, as ts egenvalues.

5 Some Defntons Defne the matrces S : = [ I - A B ] N n k R : = {columns form a bass for ker[ S]} =, N R, M R M Note: controllablty dm ker = n rank B= m rank N = m N = N [ S ] m k

6 Man Result on Pole Assgnment ( ) { v = K n} { = K n} Theorem (Moore 1976): Let, 1,, be a set of selfconjugate scalars. There exsts a real m n matrx K such that A+ BK v = v, = 1, K, n f and only f 1), 1,, are lnearly ndependent 2) v = v, when = 3) v Im N j Also, f K exsts and rank B= m then K s unque. j

7 Proof: Necessty A+ BK v = v ( ) I A v = BKv ( ) v = Kv [ ] I A B 0 v v S 0 ImR v ImN Kv = Kv

8 Proof: Suffcency, 1 ( ) ( ) { v = K n} assume the set, 1,, satsfes 1), 2), 3) k 3) there exsts z C such that v = N z By defnton S R = 0 I A N + BM = Suppose, K can be chosen such that M z = Kv Then, t would follow that I A+ BK v = 0. Thus, real K s to be chosen so that 0 I A N z + BM z = [ L n ] = 1 0 ( ) K v1 v M z1 L M n z n

9 Proof: Suffcency, 2 Assumpton 1) mples that ths s always possble. If the 's are real, we smply compute K = M [ ] z 1 1 L M z n n v1 L vn If some 's are complex, proceed as follows for each complex conjugate par. Suppose = so that from 2) v = v z = z For smplcty suppose all other egenvalues are real. Defne w : = M z, and use the noton, for any complex quantty a= a + ja R I. Then

10 Proof: Suffcency 3 [ L ] K v1 vn = M z 1 1 L M z n n [ L ] K v + jv v jv v v = w + jw w jw M z L M z 3 n post multply by the nonsngular matrx 1R 1I 1R 1I 3 n 1R 1I 1R 1I 3 n 1/2 j1/2 0 1/2 j1/2 0 I to obtan [ L ] K v v v v = w w M z L M z 3 n 1R 1I 3 n 1R 1I 3 n K = w w 1R 1I 3 n 1R 1I 3 n [ L v ] M z L M z v v v 3 n Fnally, snce a fxed egenstructure unquely defnes be proved that K s unque when rank B= m. ( A+ BK) 1, t can

11 Geometry A subset S of the lnear space (over feld F) X s a lnear subspace of X f x, x S and c, c F, c x + c x S { r s r, s } 1 2 { K } If x X ( = 1, K, k), then span x,, x s a subspace of X. R,S X then R + S = + R S { x S } R S = xx R & x Two subspaces R,S are ndependent f R S = 0 1 k

12 Geometry 2 If R, = 1, K, k are ndependent subspaces, then the sum R = R + L+ R 1 s called an ndrect sum and may be wrtten R = R L R 1 k k The symbol presuposes ndependence. Let X = R S. For each x X, there are unque r R, s S so that x= r+ s. Ths mples a unque functon xa r called the projecton on R along S.

13 Geometry 3 The projecton s a lnear map Q : X X, such that Im Q= R and ker Q= S, and ( ) ( I Q) X = QX I Q X Note that s the projecton on S along R. Thus, 2 ( ) 0 Q I Q = Q = Q Conversly, for any map : such that = 2 Q X X Q Q X = ImQ kerq.e., Q s the projecton on Im Q along ker Q.

14 Geometry 4 X Im B ( ) 1 Q= B CB C 1 ( ) ( 1 ( ) ) ( ) ( ) ( ) Q = B CB CB CB C = B CB C = Q ImQ= Im B ker Q= ker C X = Im B ker C B CB C s the projecton on Im B along ker C kerc I B CB C s the projecton on ker C along Im B

15 Example: F-16 landng approach u& u 0 & α α = + δ E q& q & θ θ 0 u α y = [ ] q θ phugod: = ± j h= ± j short perod: = , h = ,

16 Example: F-16 state feedback Desred poles - short perod: = 1.25 ± j ,4 1,2 phugod: = 0.01± j Re R , Im R = 1 = Re R , Im R = 3 = K = [ ]

17 Example: F-16 Rynask robust observer "place observer poles at LHP plant zeros, remander are placed arbtrarly" = 0, , , R = 0, R = , R L T [ ] = = , R =

18 Example: F-16 G G p c ( s) ( s) = ( )( s ) s s 2 ( s )( s )( s s ) ( s )( s ± j ) ( )( )( ) = s s s s db MAGNITUDE rad êsec

19 Example F sd

20 F-16 CCV The frst YF-16 ( ) was rebult n December 1975 to become the USAF Flght Dynamcs Laboratory's Control Confgured Vehcle (CCV). CCV arcraft have ndependent or "decoupled" flght control surfaces, whch make t possble to maneuver n one plane wthout movement n another--for example, turnng wthout havng to bank. The CCV YF-16 was ftted wth twn vertcal canards added underneath the ar ntake, and flght controls were modfed to permt use of wng tralng edge flaperons actng n combnaton wth the all movng stablator. The YF-16/CCV flew for the frst tme on March 16, 1976, ploted by Davd J. Thgpen. On June 24, 1976, t was serously damaged n a crash landng after ts engne faled durng a landng approach. The arcraft was repared and ts flght test program was resumed. The last flght of the YF-16/CCV was on June 31, 1977, after 87 sortes and 125 ar hours had been logged.

21 F-16 AFTI The Flght Dynamcs Laboratory of the Ar Force Systems Command sponsored an Advanced Fghter Technology Integraton (AFTI) program. In 1979, General Dynamcs was awarded a contract to convert the ffth FSD F-16A ( ) nto an AFTI arcraft. It captalzed on the experence ganed wth the CCV (Control Confgured Vehcle) F-16 ( ). The AFTI F-16 was ftted wth twn canard surfaces mounted below the ar ntake, these surfaces havng been taken from the CCV/F-16. It had a fullauthorty trplex Dgtal Flght Control System (DFCS) and an Automated Maneuverng Attack System (AMAS). Ths system provdes sx ndependent degrees of freedom. It was desgned to be fault tolerant, so that no sngle falure should affect correct operaton. In the event of a second fault, the system reverts to a standby condton whch wll permt safe flght to contnue. The system ncorporates an analog backup flght-control system. The AFTI frst took to the ar July 10. Phase I testng nvolved the demonstraton of drect translatonal maneuverng capablty. Phase II testng ( ) nvolved F-16C-standard avoncs wth AMAS. The AMAS enabled the AFTI/F- 16 to translate n all three axes at a constant angle of attack and to be ponted up to sx degrees off the flght vector. In recent years, the AFTI/F-16 became assocated wth close ar support (CAS) studes, some of them conducted by NASA. These studes began n 1991.

22 Multmode, Hgh Maneuverablty Flght Control Sobel & Shapro, 1985 Longtudnal Ptch pontng/ vertcal translaton - command the ptch angle wthout a change n flght path angle Drect lft command normal acceleraton (or flght path angle rate) wthout affectng angle of attack Lateral Yaw pontng/ lateral translaton decouple the lateral drectonal response from roll (bank) angle and rate and yaw rate Drect sdeforce command lateral acceleraton wthout a change n sdeslp angle

23 Example: F-16 CCV

24 F-16 CCV ptch pontng Objectves: command the ptch angle whle mantanng the flght path angle Stablze short perod mode, ζ=0.8, ω=7 rad/s Measured varables: ptch rate, normal acceleraton (at plot staton), flght path angle, surface deflectons

25 F-16 CCV ptch pontng Replace θ by γ+α, so that θ equaton s replaced by θ equaton. Choose egenvectors n an attempt to decouple ptch rate and flght path angle.

26 F-16 CCV ptch ponng

27 Example L-1011, Shapro & Chung, 1983 d dt β β.02 0 p p δ r = + r r δ a φ φ 0 0 Egenvalue Desred Egenvalue Dutch Roll ± ± 1.5 Roll Subsdence ± 1.5 Spral x x 1 Dutch roll: roll: 1x x