TIME VARYING SYSTEM IDENTIFICATION

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1 TIME VARYING SYSTEM IDENTIFICATION FOR GUIDANCE, NAVIGATION AND CONTROL APPLICATIONS 1 Manoranjan Majji Research Associate, Texas A&M University, College Station, TX I warmly acnowledge my colleagues: John Junins, Jer- Nan Juang, Richard Longman, Minh Phan, Fred Hadeagh, Wode Gawronsi, Mar Balas and Todd Griffith for their areciation and motivation. Many thans to: Texas A&M University, National Cheng Kung University, TiiMS, CASS, LASR laboratory, NASA - JPL, NASA - JSC, NRO, Sandia National Labs.

2 LINEAR SYSTEM IDENTIFICATION - ERA Consider a Linear time invariant lant model (Discrete Time). x Ax Bu 1 y Cx Du u y Available To Calculate Aˆ, Bˆ, Cˆ, Dˆ Yi The Inut Outut Reresentation : Zero State Resonse i1 0 0 CA CA B i D Yi i D i1 i1 i 1 y x u u u u : CA B Marov Parameters / Invariants of the Linear System Determination of Marov Parameters is tantamount to Identification. Frequency Resonse Functions (In frequency domain), and OKID in time domain used to determine the Marov Parameters. Given Y determination of the system matrices, is, D Aˆ, Bˆ, Cˆ, Dˆ accomlished by Eigensystem Realization Algorithm (ERA) 2

DETERMINATION OF MARKOV PARAMETERS The inut outut descrition of the lant model in matrix form, Y MU Y y y y y M 0 1 2 l1 l2 mrl D CB CAB CA B R u u u ul u u

1) l mrl ml Variables Conditions Insolvable in the MIMO case in general This leads such to that the CAsystem B 0, of equations.

3 DETERMINATION OF MARKOV PARAMETERS The inut outut descrition of the lant model in matrix form, Y MU Y y y y y M l1 l2 mrl D CB CAB CA B R u u u ul u u ul U u ul u Y MU Y y y y y l ml M D CB CAB CA B u0 u1 u2 u u u u U u u u 0 R R ml rll l2 mr ( 1) u u u u l1 l2 l3 l1 r( 1) l mrl ml Variables Conditions Insolvable in the MIMO case in general This leads such to that the CAsystem B 0, of equations. As a consequence of stability of the origin ˆ M YU Least Squares Solution for Marov Parameters Other Otions: Use Frequency Resonse Functions 3 Use OKID (Based on the so called ARX model)

4 EIGENSYSTEM REALIZATION ALGORITHM Given Marov Parameter sequence, form the Hanel Matrix, H where 1 Y Y Y Y Y Y : Y Y Y P A 1 Q 2 1 Q B AB A B A B Now, mae the observation that H R S P Q P R Q S H 1 T 1/ 2 1/ 2 0, P AQ leading to Aˆ P H Q R H S 1/ 2 T 1/ H 0 : Y1 Y2 Y Y2 Y3 Y 1 PQ Y Y 1 Y C CA 2 P CA T 1 CA Bˆ S E, Cˆ E R 1/ 2 T T 1/ 2 r m 4

5 EXAMPLE 1 True Parameters A Identified Parameters true 0 B 1 ˆ A ˆ 1 D j j C D 1 B ˆ C ˆ ID j j 5

6 True and Estimated Oututs y true y ID 0.4 utest time sin 0.5t 6

7 Outut Error 1.4 x time 7

8 System Resonse 3 2 EXAMPLE Comarison of True and Identified System Resonse - Noise 1e-4 High Fidelity FE Model ERA Identified Hub Aendage Problem time No noise Case Errors of O(1e-9) Feel the ower! Noise (sd 1e-4) in inuts and Oututs Errors of O(1e-4) Mar Balas Beam Parameters 8

Motivation IDENTIFICATION OF TIME VARYING DISCRETE TIME SYSTEMS Most Nonlinear Systems have dearture motions that are linear time varying (continuous or discrete time) models.

9 Motivation IDENTIFICATION OF TIME VARYING DISCRETE TIME SYSTEMS Most Nonlinear Systems have dearture motions that are linear time varying (continuous or discrete time) models. In such roblems, * * A Coule of Observations x fx x fu u x 1 F x GKaboom! uu 1. Linear time No Matrix invariant Exonential identification Genie Use schemes to => erform High order this => conversion exm Large Comutational (discretization) exense. in the Time 2. A time varying Varying identification Case scheme oens new doors Most for imortantly, model comression all based nonlinear estimation models and control. may NOT be VanLoan correct in ractice! 3. Time varying system theory, being well ( Not even Lagrange/Euler develoed, one has the hoe to derive concrete methods Thanfully And We you are are in going to see them business for this very reason) Eg: Rotor craft models are LPV, Unsteady Aerodynamic Models, Tyical Guidance Problems the list goes on 9

10 LINEAR TIME VARYING DYNAMICAL SYSTEMS Sniets from Linear System Theory we all now and love! Plant Model x A x B u 1 y C x D u Inut Outut Reresentation y 1 h u, i i i0 Solution Notice: In the shift invariant case h h Y, j, j j 1,0, 1 x x i B u 0 i i i0 State Transition Matrix A..., 1A 2 Ai i, i I, i undefined i Generalized Marov Parameters h i, C, i 1 Bi, i 1 0, i1 Seminal Wors: PhD Dissertations S. Shooohi, L. Silverman 2D ulse resonse sequences Note that the eigenvalues of the system matrix are not an index of stability of the origin. 10

11 DIGRESSION TIME VARYING COORDINATES IN LINEAR SYSTEMS Tyical Thought : What? He s Crazy! I dint read this in standard linear system theory text boos (Kailath, Chen, Juang ) My Resonse: Well that s why I (the crazy art IS true) got a PhD For Time Invariant Systems, x Ax Bu 1 y Cx Du 1 A T AT x 1 B T B, C CT Tz z Az Bu 1 y Cz Du T - Time Invariant Transformation Matrix AA, - Similar Matrices Consequences: Modal Coordinates and all the good stuff (IMSC) 11

TIME VARYING COORDINATES IN LINEAR SYSTEMS For Time Varying Linear System

Kinematically Similar or A, B, C, D A, B, C, D Toologically Equivalent

Only defining characteristic is that the elements of the sequence need to be

12 TIME VARYING COORDINATES IN LINEAR SYSTEMS For Time Varying Linear System Models (Discrete Time) x A x B u 1 y C x D u x T z z A z B u 1 y C z D u Kinematically Similar or A, B, C, D A, B, C, D Toologically Equivalent Realizations A T A T 1 1 B T B C 1 1 C T Lyaunov Transformation Sequence T Only defining characteristic is that the elements of the sequence need to be nonsingular! Nothing else 12 Consequence: A, A NOT Similar Research resentation at the University at

13 TIME VARYING COORDINATES IN LINEAR SYSTEMS Fundamental Matrices (Continuous Time Setting) t t t t0 : 0 1 t0 2 t0 t ξ ξ ξ n 0 where t: ξ1t ξ2t ξn t ξi t tξi t and i t 0 Initial Conditions (Matrix) ξ i1,2,..., n State Transition Matrices (Continuous Time Setting) t t, t 0 t t, t0 1 t, t 0 t t0 Sans state sace Initial Condition (Unit Matrix) t, t 0 In First evidence of Coordinate Transformations change of basis 13 is evident!

14 TIME VARYING COORDINATES IN LINEAR SYSTEMS state sace at time ste TRANSFORMATION T state sace at time ste t t 1 e 3 t, t e1 e2 e 3 t, t ξ1 t ξ2 t ξ3 t ξ t 1 ξ ξ t 2 3 INVERSE t e 2 e 1 1 T t 1, t 1 t1 2 t1 3 t1 t 1, t 1 t 1 2 t 1 3 t ξ ξ ξ 1 ξ 3 t 1 2 t 1 ξ 1 t 1 ξ3t 1 1 t 1 2 t 1 14

15 x A x B u 1 y C x D u TIME VARYING COORDINATES IN LINEAR SYSTEMS x T z z F z G u 1 y H z D u O C C C C 1A C 1A C 1A : O : T O T C A 1... A C A 1... A C A 1... A R B A B 1... R B A B T 1 B A B 1... T 1R h, 1 h, 2 C H h h C A B A B O R O R 15 1, 1 1,

O T 1 T T 1 O O T O O T Consider T T Consequence: 1 1 1 F : O O F T O

16 x A x B u 1 y C x D u TIME VARYING COORDINATES IN LINEAR SYSTEMS x T z z F z G u 1 y H z D u In general we have, leads to O T O T and T O O T 1 T T 1 O O T O O T Consider T T Consequence: F : O O F T O O T T A T A, F Similar T O O T 1 1 A T 1 with A T A : O O A 1 16

17 TIME VARYING EIGENSYSTEM REALIZATION ALGORITHM Generalized Hanel Matrix h h h, 1, 2, h, 1 h, 2 h 1, 1 h 1, 2 h 1, q, 1, 1 1, 2 H h h H h q, 1 h q, 2 h q, q, - Parameters chosen aroriately Assuming the generalized Marov arameters are available, 1 1, q qt T 2 2 T H O R 1 U V Generalized Hanel Matrix sequences can always be formed! 17

18 TIME VARYING EIGENSYSTEM REALIZATION ALGORITHM Ushifted Hanel Matrix h 1, 1 h 1, 2 h 1, q, h 2, 1 h 2, 2 h 2, H : h q1, 1 h q1, 2 h q1, Decomosition 1 1, q qt T 2 2 T H O R 1 U V qt O O T O 1A T where O,,, 1 A B R such that 1 1 O O T qt in true unnown coordinates can be written 18

19 TIME VARYING EIGENSYSTEM REALIZATION ALGORITHM Therefore qt qt 1 1 O O T A T 1 1 Accordingly, ˆ 1 qt1 qt A T A T O O 1 1 R T 1 ˆ 1 T 1 :,1: 1 R B T B R r ˆ qt 1:,: C O m C T T 1 That s it! we can all go home (well not really, there is more than what meets the eye here.. I will let you go through our recent aers!) 19

20 EXAMPLE (NO DAMPING, UNSTABLE, OSCILLATOR) A ex Ac t B D , C, A c K t 0 K t I t t sin 10, : cos 10 Time Varying synthetic roblem Stiffness is time varying Thin dearture motion dynamics of a Duffing oscillator about a nominal reference. Thin CW equations (in resence of atmosheric drag)! 20

21 EXAMPLE Hanel Matrix Sequence Singular Values 21

22 EXAMPLE Outut Comarison (True Vs. Identified - Forced Resonse) Test Inuts: 0.5sin 12, cos 7 u t t u t t

23 EXAMPLE Outut Comarison (True Vs. Identified - Forced Resonse) 23

24 EXAMPLE Coefficients of the Characteristic Equation : True and Identified (in Time 24 Varying Coodinate Systems)

25 EXAMPLE Coefficients of the Characteristic Equation : True and Identified (in Reference 25 Coordinate System)

26 TIME VARYING OKID Why? Recall I/O relationshi,0 Calculation of Generalized Marov Parameters??? 1 y C x h u D u 0, j j j0 Comlex roblem to solve in general! y # of unnowns : 1 mr Thin a model sequence at 10 Hz for 10 seconds u 1 C A 1... A A D C B 1 C A 1... A B u x u 0 In a tyical Least Squares tye solution # of exeriments required grows raidly Therefore, although rigorous, historical TV Identification methods are difficult to imlement in ractice We changed this! 26

27 TIME VARYING OKID Philosohy : When in doubt use (time varying) feedbac! x A x B u G y G y 1 A G C x B G D u G y A x B G D G y A x B u υ I/O relationshi becomes, y C A... A x D u h υ 1, j j 0 0 j1 0 1 ARX model => OKID theory is based on this reresentation Lets not worry about coordinates for clarity Notice the structural similarity to ARX models y a y... a y 1 1 n1 n b u... b u 0 n1 n 27

28 TIME VARYING OKID h C A... A B, j 1 j1 j C A... A B G C C A... A G h 1 j1 j j j 1 j1 j h 1 2, j, j Generalized Observer Marov Parameters => [Gen. System Marov Parameters, -Gen. Observer Gain Marov Parameters] y C A... A x D u h υ 1, 1 1 j j j1 D u, j1 j1 h min υ j1 28 Each time ste has a different ARX model Hence GTV-ARX model Minimum # - naturally leads to a time varying dead beat observer Let us construct the observer and then come bac to discuss deadbeat

29 TIME VARYING OKID y u υ 1 D h h h υ υ, 1, 2, 2 m equations m r * r m 1 2 N 1 2 N Y y y y 1 2 N N D h, 1 h, 2 h, υ 2 υ 2 υ 2 MV u u u υ υ υ υ υ υ 1 2 N No growth in the number of arameters ˆ Y V M unnowns OK we can solve this 29 But what we need are System Marov Parameters

30 TIME VARYING OKID h C B, C B 1 G 1D 1 G 1 h, 1 h, 1 Some adhoc oerations (We will never now why I tried them) h h D C B 1 2, 1, h, 1 h h D C A B C A G D 1 2, 2, C A B G C A G D C A B 1 2 C A G C B C A B h C B 1 2, h h h, 2, 1 1, 2 Loos romising Really? 30

31 TIME VARYING OKID Induction Ste: h h D C A A... A B G D C A A... A G D 1 2,, And the term C A A... A B C A A... A A G C B C A A... A A B C A A... A G C B C A A... A A B h h , 1 1, C A A... A A B C A... A A G C A B C A... A A B C A... G C A B C A... A A B h , 2 2, C A... A B h h h h... h h 1 1, 1 1,, 2 2,, 2 2, 2 2 h h h h h... h 2,, 1 1,, 2 2,, 2 2, h h

32 TIME VARYING OKID Therefore the general term becomes h h D h h h h h... h h,,,, 1 1,, 2 2,, 1 1, 1 2 h h h,, 1 1, j1 Leading to:... h h h D 1 2, 1, 1, h h h h h D, 2, 1 1, 2, 2, h h h... h h h h D,, 1 1,, 1 1,,, More clearly, defining, r : h h D 1 2 i, j i, j i, j j 32

33 TIME VARYING OKID i h h h D h h First time stes, i, i, i i, j j, i j1 2 h h h, i, j j, i j Im h, 1 h, 2 h, 1 h, 1 h, 2 h, I 0 m h h 1, 2 1, 2 1, 2 h h 1, I 0 0 h m 1, BLOCK TRIANGULAR EQUATIONS FOR SYSTEM MARKOV PARAMETERS r r r 0 r r All revious time stes, 1, 2, 0 0 1, 2 1, r 1, 33

34 TIME VARYING OKID Now the observer gain calculations! Observer Gain Marov Parameters are Defined: C A A... A G, i i1 i o, i 1 h C G, i 1 0, i1 Exact same miracles haen here as well: 2 h C G h o, 1 1, 1 2 h C A G, o 2 C A G C G h h h o , 2, 1 1, 2 and so on including the induction ste. 34

35 TIME VARYING OKID j1 2 o 2 o, j, j, i i, j i1 h h h h o 2 o, j, i i, j i1 h h h First time stes All revious time stes o o o Im h, 1 h, 2 h, 1 h, 1 h, 2 h, 2 2 o o 0 I 0 m h h 1, 2 1, 2 1, 2 h h 1, o I 0 0 h m 1, h h h 0 h h 0 0 h, 1, 2, 2 2 1, 2 1, 2 1, 35

36 TIME VARYING OKID Gain Calculations: h C G C o 1, 1 1 o h 2, C 2A 1G C2A 1 m 1: 1 P G O G o h m, C m A m1... A 1G Cm A m1... A 1 Estimate for the gains: Aˆ T A T 1 1 Bˆ T B Cˆ 1 1 C T 1 1 mt Gˆ T G O P Automatically in the right coordinates Aˆ Gˆ Cˆ T A G C T

37 TIME VARYING OKID Error in Marov Parameters Comutations 37

38 TIME VARYING DEADBEAT OBSERVER Closed Loo Vs. Oen Loo Resonse for a test situation showing exonential decay (=4) 38

39 APPLICATION Model for controller design: x x, x x x u, x u Nominal Solution arameters 39

40 APPLICATION: PRESENCE OF MODEL ERRORS In a new medium, 40

41 APPLICATION: IDENTIFICATION OF FIRST ORDER ERROR MODEL Perturbed Nominal Motions Deviations from Nominal 41

42 APPLICATION: RESULTS Identification Results: Singular Values of Hanel Matrix Sequence Perturbation model order automatically determined! 42

43 APPLICATION: VALIDATION OF IDENTIFIED MODELS Identification Results: System Resonse (Identified vs True Nonlinear System) 43

44 APPLICATION: VALIDATION OF IDENTIFIED MODEL Error in System Resonse : Identified Resonse True system Resonse 44

45 APPLICATION: PERTURBATION GUIDANCE USING DEVIATION MODEL Set discrete time reference trajectory as, Such that : 45

46 APPLICATION: PERTURBATION GUIDANCE USING DEVIATION MODEL Guidance with identified model: comarison with uncomensated motions 46

47 APPLICATION: PERTURBATION GUIDANCE USING DEVIATION MODEL 47

48 APPLICATION: PERTURBATION GUIDANCE USING DEVIATION MODEL 48

49 Time invariant Case: Review 1 0 A 1 2 C 0 1 TIME VARYING OKID TIME VARYING DEADBEAT OBSERVER A 1 1 A GC 1 1 1g g 3g GC 2 3 g2 g1 2 g Closed loo oles are laced at origin 49

50 GTV-ARX MODEL With this observer in the loo, the inut outut relation (at each time ste) becomes, u v1 y D h, 1 h, 2 h, v2 v Using the following definition, The I/O relation becomes D, h, h m r 1 m r 2 m m,0, j, j, j, j y u u y,0, j j, j j j1 Subsequent time ste can be written as y u u y 1 1,0 1 1, j 1 j 1, j 1 j j1 50

51 GTV-ARX MODEL Exanding and (not quite) simlifying y u u y u y 1 1,0 1 1,1 1,1 1, j 1 j 1, j 1 j j2 1 u u y u y 1,0 1 1,1 1,1 1, j1 j 1, j1 j j1 u u u u y 1,0 1 1,1 1,1,0, j j, j j j1 1 j1 u y Motivates a re-grou of the outut at next time ste as, y u u u y 1 1,0 1 1,1 1, j j 1, j j j1 1, j1 j 1, j1 j A SECOND redefinition of variables! WHY? 51

52 GTV-ARX MODEL OK method to madness : this is why! Y B U Δ U Γ Y : s : s : s : s 1: : s 1: y y 1 u u 1 y : s, y 1:, u : s, u Y Y U U 1: ys y us u B : s , ,1 1, ,2 2,1 2, : 0 0 q, q q, q1 q, q2 q,0 0 0 q1, q1 q 1, q q 1, q1 q 1,1 q 1,0 0 s, s s, s1 s, s2 s, sq s, sq1 s,0 Γ Δ : s : s Matrices at last! There is Hoe!,1,2, 1,1 1, q,1 q, s,1 s,,1,2, 1,1 1, q,1 q, s,1 s, 52

53 TV DEADBEAT CONTROL Question: What should be the control future control inuts u, u 1,..., u the future outut sequence is set to zero? y, y,..., y q1 q2 q such that Obviously, the control action is to start at and end at +q, that is tvd tvd tvd uq 1 uq2... u 0 and y 1 y 2... y 0 q q Using these conditions in the giant matrix equation above, we have Y B U Δ U Γ Y q1: s tvd q1: s : s : q q1: s 1: q1: s 1: tvd y y q 1 1 u u 1 tvd yq2 2 tvd 1 2 q1: s, y 1:, u : q, u Y Y U U 1: y u u tvd ys q 53

54 TV DEADBEAT CONTROL with coefficient matrices aroriately defined as B q1: s : s q1, q1 q1, q q1,1 q2, q2 q2, q1 q2,2 s, s s, s1 s, sq Δ q1: s Γ q1: s q1,1 q1,2 q1, q2,1 q2,2 q2, s,1 s,2 s, Necessary conditions for outut stabilization yield Y 0 q1,1 q1,2 q1, q2,1 q2,2 q2, s,1 s,2 s, q1: s Leading to the control inuts for secified time stes as tvd q1: s : q : s q1: s 1: q1: s 1: U B Δ U Γ Y 54

55 APPLICATION TO SPACECRAFT ATTITUDE CONTROL Sin stabilized sacecraft configuration Precession Desired states: q ω 1 rev/hr 22.5 deg rm Initial conditions t t 0 T deg/ sec Inertia arameters J 15.2, J 50.2, J T Sun Consider a Wie Barba solution for regulation 55

56 SPACECRAFT ATTITUDE CONTROL Consider erturbations (eg. Gyro bias and uncomensated state feedbac) ω 2 t rad/sec 56

57 SPACECRAFT ATTITUDE CONTROL Attitude Errors Incurred 57

58 SPACECRAFT ATTITUDE CONTROL Alication of Identified Deadbeat Controller to stabilize attitude motions 58

59 SPACECRAFT ATTITUDE CONTROL Angular velocity deviation rofile comarison (with and without deadbeat control) 59

60 SPACECRAFT ATTITUDE CONTROL At what cost? Comarison of torque requirements Not bad at all rigorous constraints on control can be imosed in DPC easily 60

Observer/Kalman Filter Time Varying System Identification

Observer/Kalman Filter Time Varying System Identification Manoranjan Majji Texas A&M University, College Station, Texas, USA Jer-Nan Juang 2 National Cheng Kung University, Tainan, Taiwan and John L. Junins