Robust observed-state feedback design. for discrete-time systems rational in the uncertainties

Transcription

1 Robust observed-state feedback desgn for dscrete-tme systems ratonal n the uncertantes Dmtr Peaucelle Yosho Ebhara & Yohe Hosoe Semnar at Kolloquum Technsche Kybernetk, May 10, 016 Unversty of Stuttgart Extends results from two papers presented at 19th IFAC World Congress (Cape Town). Results submtted to Automatca: hal.archves-ouvertes.fr/hal v1

2 Motvaton n Lterature full of robust state-feedback desgn results, few for robust observer desgn n Output or state estmaton flter 6= Observer (no open loop stablty assumpton) n Observers of states n gven state-space + assumng MIMO systems.e. not restrcted to SISO systems n canoncal form (ntegrators n seres) ẋ = x (f(x, )+g(x, )u) 7 5 D. Peaucelle 1 May 10, 016

3 Motvaton n Dscrete-tme lnear system wth uncertantes x k+1 = A r ( )x k + B r ( )u k, y k = Cx k n Luenberger-lke observer ˆx k+1 = A oˆx k + B o u k + L(C ˆx k y k ) n Observed-state feedback u k = K ˆx k l Our goals: s Buld a separaton-lke heurstc wth frst, K desgn, then, A o,b o,ldesgn s Use up-to-date SV-LMI tools s For systems ratonal n the uncertantes D. Peaucelle May 10, 016

4 Motvaton n Closed-loop dynamcs (state x and error e = x 0 x k+1 e k+1 A = ˆx) drven by the state matrx 4 A r( )+B r ( )K B r ( )K A( )+ B ( )K A o + LC B( )K where A( ) =A r ( ) A o and B( ) =B r ( ) B o. 0 x k e k 1 A l Separaton obtaned when A( ) =0and B( ) =0 0 x k+1 A = 4 A r( )+B r ( )K B r ( )K e k+1 0 A o + LC 0 x k e k 1 A s Impossble when are uncertantes (Noton of observed-state not qute defned for uncertan systems) D. Peaucelle May 10, 016

5 Motvaton n Closed-loop dynamcs (state x and error e = x 0 x k+1 e k+1 A = ˆx) drven by the state matrx 4 A r( )+B r ( )K B r ( )K A( )+ B ( )K A o + LC B( )K where A( ) =A r ( ) A o and B( ) =B r ( ) B o. 0 x k e k 1 A s Choces from the lterature: A o = A r ( nom ), but why? s Possble choce mn Ao max ka r ( ) A o k, but what propertes? l Our choce: optmze the nput/output performances of e k+1 =(A o + LC B( )K)e k +( A( )+ B ( )K)x k, k = Ke k where x k s treated as the nput and k s the output. D. Peaucelle 4 May 10, 016

6 Outlne Descrptor mult-affne modelng of ratonal systems LMI results for robust desgn and robust analyss Observed-state feedback desgn heurstc π Example D. Peaucelle 5 May 10, 016

7 Descrptor mult-affne modelng of ratonal systems n p ndependent uncertan vectors p R m p ndexed by p =1 p = {( 1,..., p ) 1... p }. n o n Each p n a polytope wth v p vertces V p = p [1],..., [ v p] p p = Co(V p )= ( p = v p X v=1 p,v [v] p : p,v 0, v p X v=1 p,v =1 l Example: scalar uncertanty n an nterval: p [ [1] p, [] p ]. l Example: D vector n convex hull of ponts ssued from dentfcaton process ). D. Peaucelle 6 May 10, 016

8 Descrptor mult-affne modelng of ratonal systems n Mult-affne matrces: affne n each p l Example for two scalar uncertantes 1 [ [1] 1, [] 1 ], [ [1], [] ] = 1,1,1 (1 + [1] 1 + [1] 1 [1] ) + 1,1, (1 + [1] 1 + [1] 1 [] ) + 1,,1 (1 + [] 1 + [] 1 [1] ) + 1,, (1 + [] 1 + [] 1 [] ). s Not the same as the convex hull of all possble vertces l Example: h 1 1 wth 1 [1, ] and [1, ]. 1 h h 4 = h 5 6= h 9 4 D. Peaucelle 7 May 10, 016

9 Descrptor mult-affne modelng of ratonal systems n Any matrx ratonal n admts a descrptor mult-affne representaton (DMAR) R( ) =M 1 ( )M 1 ( )M ( ) where M 1 ( ), M ( ), M ( ) are mult-affne n. l Alternatve to lnear-fractonal representatons l Usually of smaller sze, and easer to buld l Example: = D. Peaucelle 8 May 10, 016

10 Descrptor mult-affne modelng of ratonal systems n Dscrete-tme lnear system, wth performance I/O, ratonal n the uncertantes x k+1 z k = A r ( )x k + B rw ( )w k = C rz ( )x k + D rzw ( )w k l The DMAR 4 A r( ) B rw ( ) C rz ( ) D rzw ( ) 5 = 4 E x( ) E z ( ) 5 E 1 ( ) h A( ) B w ( ) n gves the followng descrptor mult-affne representaton of the system 4 I 0 E x ( ) I E z ( ) E ( ) A( ) B w ( ) 0 5 x k+1 s k : exogenous vector = E 1 ( )(A( )x k + B w ( )w k ) z k k x k w k 1 C A = E( ) k =0 D. Peaucelle 9 May 10, 016

11 LMI results for robust desgn and robust analyss n If there exsts P [v] = P [v]t 0, S and µ such that for all vertces [v] V h dag P [v] I 0 P [v] µ I (SE( [v] )) + (SE( [v] )) T then the system s robustly stable (.e. 8 ) wth robust H 1 performance µ. l Proof - step 1 - By convexty the condton holds for all : h dag P ( ) I 0 P ( ) µ I (SE( )) + (SE( )) T wth mult-affne Lyapunov matrx P ( ) 0. l Proof - step - Snce E( ) k =0one gets T k dag h P ( ) I 0 P ( ) µ I k = x T k+1 P ( )x k+1 + z T k z k x T k P ( )x k µ w T k w k < 0 D. Peaucelle 10 May 10, 016

12 LMI results for robust desgn and robust analyss n If there exsts P [v] = P [v]t 0, S and µ such that for all vertces [v] V h dag P [v] I 0 P [v] µ I (SE( [v] )) + (SE( [v] )) T then the system s robustly stable (.e. 8 ) wth robust H 1 performance µ. l S-varable result l Extended n present work to mult-affne representatons l Exst tools to reduce numercal burden (sometmes lossless) s Example: no S f plant s mult-affne n & common P = P ( ) l Extensons to mxed constant/tme-varyng uncertantes D. Peaucelle 11 May 10, 016

13 LMI results for robust desgn and robust analyss n SV-LMI for robust state-feedback desgn If there exst P [v] d 0, S dx, S dy, S d such that LMIs L sf ( [v] ) hold for all [v] V then K = S dy T (S dx T ) 1 s a robustly stablzng state-feedback gan s.t. x k+1 z k = A r ( )x k + B r ( )u k + B rw ( )w k, u k = Kx k = C rz ( )x k + D rzu ( )u k + D rzw ( )w k has an H 1 performance smaller than µ d whatever. l Lnearzng change of varables on S-varables l Proof uses equvalence wth dual system x d,k+1 = A T r ( )x d,k +... l Result s new because for ratonal systems l Easy extensons for regonal pole locaton, H performance, etc. D. Peaucelle 1 May 10, 016

14 LMI results for robust desgn and robust analyss n SV-LMI for analyss of state trajectores under fxed state-feedback K = K If there exst P [v] 0, Q and S such that LMIs L sf,a ( [v] ) hold for all [v] V, then x k+1 = A r ( )x k + B r ( )u k, u k = Kx k + k s robustly stable and x k s bounded for bounded control errors k : kw xk applek k where W = Q 1/. l Allow to estmate the state trajectores n case of corrupted state-feedback (nevtable when feedback s wth observed-state) D. Peaucelle 1 May 10, 016

15 LMI results for robust desgn and robust analyss n SV-LMI for robust observer desgn under fxed K = K and expected state trajectores W = W If there exst P 1 [v] 0, P p [v] K T K, S x, S a, S b, S l, S, S p such that LMIs L ob ( [v] ) hold for all [v] V, then A o = S 1 x S a, B o = S 1 x S b, L = S 1 x S l defne an observer that guarantees: k k apple kwxk, k k p apple p kwxk where k = Ke k. The propertes hold whatever bounded x and whatever. l Norm-to-norm perf: asymptotc couplng of observaton error on system dynamcs l Norm-to-peak perf: avod waterbed effects of transent peaks l Small gan theorem: f < 1 observed-state feedback robustly stablzes D. Peaucelle 14 May 10, 016

16 LMI results for robust desgn and robust analyss n SV-LMI for observed-state feedback analyss under fxed K = K, A o = A o etc. If there exst P [v] c 0, S c such that LMIs L ob,a ( [v] ) hold for all [v] V, then x k+1 z k = A r ( )x k + B r ( )u k + B rw ( )w k = C rz ( )x k + D rzu ( )u k + D rzw ( )w k ˆx k+1 = A oˆx k + B o u k + L(C ˆx k y k ), u k = K ˆx k has an H 1 performance smaller than µ c whatever. l Exsts also SV-LMI condtons L ob,da for the dual system: µ dc upper-bound l No apror relaton between upper-bounds µ d (deal state-feedback), µ c and µ dc D. Peaucelle 15 May 10, 016

17 Observed-state feedback desgn heurstc n 4 steps l 1- Desgn stablzng state-feedback K (for example usng LMI L sf ) l - Get estmate of state trajectores represented by W (usng LMIs L sf,a ) (max mn (W T W ) leads to tght estmates) l - Desgn observer (usng LMIs L ob ) (mn + p p to adjust tradeoff between norm and peak performances) l 4- Analyze observed-state closed-loop (usng LMIs L ob,a or L ob,da ) s No guarantee that next step would be feasble s No guarantee to fnd a robustly stablzng control when exsts s All step are purely LMI wth clear control theory justfcaton s Each step based on new LMI condtons D. Peaucelle 16 May 10, 016

18 π Example n Academc example for llustraton x k+1 = z k = 4 1 / h x k u k + h x k + u k, y k = x k 5 w k l DMAR E x = A = h 5, E z = 0 1, E = 4 0 5, 0 1 5, B = 4 0 5, B w = l 1 [1 1, 1+ 1 ], [1, 1+ ] at lmt of stablty when 1 = =1 D. Peaucelle 17 May 10, 016

19 π Example l Results when µ d = 10 at frst step ( 1, ) (, p) (, p) µ c µ dc (0, 0) (1, 1) (10 4, 10 4 ) (0.1, 0) (1, 1) (1.0747, ) (0, 0.1) (1, 1) (0.947, 0.54) (0.1, 0.1) (10, 1) (1.76, ) (0.1, 0.1) (1, 1) (1.96, ) (0.1, 0.1) (1, 10) (1.9, ) (0., 0.1) (1, 1) (1.006, 1.181) (0.1, 0.) (1, 1) (1.4, 1.155) (0., 0.) (1, 1) (.59,.08) D. Peaucelle 18 May 10, 016

20 π Example l Results when µ d s mnmzed at frst step and (, p) =(1, 1). ( 1, ) µ d µ d (, p) µ c µ dc (0, 0) 1 1 (10 4, 10 4 ) (0.1, 0) (0.49, 0.49) (0, 0.1) (0.061, 0.777) (0.1, 0.1) (0.5611, ) (0., 0.1) (1.190, ) (0.1, 0.) (1.4, ) (0., 0.) (.0445,.5166) 1 1 s µ d computed on SV-LMI wth reduced sze D. Peaucelle 19 May 10, 016

21 π Example l Sze of LMI condtons (applyng non-conservatve sze reducton procedures) ( 1, ) L sf Lsf L sf,a L ob L ob,a L ob,da (0, 0) {5, 0} {5, 0} {, 0} {6, 0} {5, 0} {5, 0} (1, 0) {6, 1} {6, 1} {5, } {7, 1} {7, } {7, } (0, 1) {7, } {6, 1} {5, } {8, } {7, } {10, 5} (1, 1) {7, } {6, 1} {6, } {8, } {8, } {11, 6} s {N r,n c } wth N r rows n each LMI (to be multpled by nb of vertces) s {N r,n c } wth N c columns of S-varables D. Peaucelle 0 May 10, 016

22 π Example n System model x k+1 = 4 1 / x k u k w k l Observer model (desgn for 1 = =0.) A o = , B o = D. Peaucelle 1 May 10, 016

23 Conclusons n Revsted Luenberger observer desgn n case of uncertan systems l Separaton prncple replaced by a mxed norm/peak performance measure l LMI desgn of state-feedback and observer gans one after the other l Heurstc wth no guarantee of success s Surprsngly, K desgn for fxed observer s more complex (prospectve work) s Extensons for contnuous-tme systems: rases ssues about S-varable condtons for desgn (tunng parameters) n Promsng descrptor mult-affne representaton combned to SV-LMIs D. Peaucelle May 10, 016