Reduced order observers
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Reduced order observers
- 1. Reduced Order Observers for Linear Systems SOLO HERMELIN Updated: 08.12.08
- 2. Table of Content SOLO Reduced Order Observers for Linear Systems
- 3. SOLO Reduced Order Observers for Linear Systems ∈∈∈+= ∈∈∈∈+= pxmpxnpx nxmnxnmxnx RDRCRyuDxCy RBRARuRxuBxAx 1 11 Plant: We want to construct a Observer such that it’s output will asymptotically converge to .x xˆ
- 4. SOLO Reduced Order Observers for Linear Systems ∈∈∈+= ∈∈∈∈+= pxmpxnpx nxmnxnmxnx RDRCRyuDxCy RBRARuRxuBxAx 1 11 Plant: Assume: npCrank pxn ≤= Find: ( )xnpn RC − ⊥ ∈ such that: nxn C C ⊥ is nonsingular. Solution: Find the Singular Value Decomposition (SVD) of C ( )[ ] H CpnpxCCpxn nxnpxppxp VUC −Σ= 0 where H means Transpose of a matrix and complex conjugate of it’s elements, and: nC H C H CCpC H C H CC IVVVVIUUUU ==== ; ( ) ( ) ( )nxnnpxpp ppC diagIdiagI diagpxp 1,,1,1,1,,1,1 0,,,, 2121 == >≥≥≥=Σ σσσσσσ ( ) ( ) ( ) ( ) ( ) ( ) [ ] H CCxppnCxnpn nxnpnxpnpnxpn VUC −−⊥−−⊥ Σ= −−⊥ 0Then: UC is any orthogonal matrix and ΣC is any non-zero diagonal matrix.
- 5. SOLO Reduced Order Observers for Linear Systems Define: We have : x C C p uDy = − ⊥ ( ) 1 : xpn RpxCp − ⊥ ∈= ( ) − = − = ⊥ − ⊥ p uDy CC p uDy C C x †† 1 or : ( ) pCuDyCx †† ⊥+−= where: ( ) nxpTT RCCCC ∈= −1† is the Right Pseudo-Inverse of C or pICC =† ( ) ( )pnnxTT RCCCC −− ⊥⊥⊥⊥ ∈= 1† is the Right Pseudo-Inverse of C or pnICC −⊥⊥ =† Then: ( ) ( ) ( ) ( ) = = −− − ⊥⊥⊥ ⊥ ⊥ ⊥ pnxppn pnpxp I I CCCC CCCC CC C C 0 0 †† †† †† ( ) nICCCC C C CC =+= ⊥⊥ ⊥ ⊥ ††††
- 6. SOLO Reduced Order Observers for Linear Systems We have: += += uDxCy uBxAx ( ) pCuDyCx †† ⊥+−=and:xCp ⊥= ( ) ( )[ ]{ }uBpCuDyCACuBxACxCp ++−=+== ⊥⊥⊥⊥ †† or: ( ) uBCuDyCACpCACp ⊥⊥⊥⊥ +−+= †† We want to obtain an estimation of . If we add we can see that:ppˆ ( )uBxCyL −− ( )[ ] ( ) ( ) 0ˆ ˆˆ 0 †† †† =−−−−= −+−−=−− − ⊥ ⊥ uDpCCuDyCCy uDpCuDyCCyuDxCy pnpxpI Apparently does not contain any information on , but let compute .py y ( ) ( )[ ]{ } uDuBpCuDyCACuDuBxACuDxCy +++−=++=+= ⊥ ˆ††
- 7. SOLO Reduced Order Observers for Linear Systems We have: Therefore contains the information on .py ( ) ( )[ ]{ } uDuBpCuDyCACuDuBxACuDxCy +++−=++=+= ⊥ ˆ†† ( ) uDuBCpCACuDyCACy +++−= ⊥ ˆ†† Let estimate by using:p ( ) ( )[ ]uDuBCpCACuDyCACyL uBCuDyCACpCACp −−−−−+ +−+= ⊥ ⊥⊥⊥⊥ ˆ ˆˆ †† †† or: ( )[ ] ( ) ( )[ ]uBuDyCApCACLCuDyLp td d +−+−=−− ⊥⊥ †† ˆˆ ( ) pCuDyCx †† ⊥+−=
- 8. SOLO Reduced Order Observers for Linear Systems We have: ( )[ ] ( ) ( )[ ]uBuDyCApCACLCuDyLp td d +−+−=−− ⊥⊥ †† ˆˆ ( ) pCuDyCx ˆˆ †† ⊥+−=
- 9. SOLO Reduced Order Observers for Linear Systems We also have: ( )[ ] ( ) [ ]uBxACLCuDyLp td d +−=−− ⊥ ˆˆ ( ) pCuDyCx ˆˆ †† ⊥+−=
- 10. SOLO Reduced Order Observers for Linear Systems One other form: ( )[ ] ( ) ( ) ( ) ( ) uBCLCuDyCACLC pCACLCuDyLp td d −+−−+ −=−− ⊥⊥ ⊥⊥ † † ˆˆ ( ) pCuDyCx ˆˆ †† ⊥+−=
- 11. SOLO Reduced Order Observers for Linear Systems And another form: ( )[ ] ( ) ( )[ ] ( ) ( )( ) ( ) uBCLCuDyLCCACLC uDyLpCACLCuDyLp td d −+−+−+ −−−=−− ⊥⊥⊥ ⊥⊥ †† † ˆˆ ( )[ ] ( )( )uDyLCCuDyLpCx −++−−= ⊥⊥ ††† ˆˆ
- 12. SOLO Reduced Order Observers for Linear Systems We have: ( ) ( )[ ]uDuBCpCACuDyCACyL uBCuDyCACpCACp −−−−−+ +−+= ⊥ ⊥⊥⊥⊥ ˆ ˆˆ †† †† Subtract those equations: Define the estimation error: ( ) ( )[ ]uDuBCpCACuDyCACyL uBCuDyCACpCACp −−−−−+ +−+= ⊥ ⊥⊥⊥⊥ †† †† ( ) ( )ppCACLppCACpp ˆˆˆ †† −−−=− ⊥⊥⊥ ppp ˆ:~ −= ( ) pCACLCp ~~ † ⊥⊥ −= p~We can see that ( the estimation error) is uncontrollable and is stable iff. ( )[ ] iCACLCi ∀<− ⊥⊥ 0Real † λ ppp →→ ˆ&0~
- 13. SOLO Reduced Order Observers for Linear Systems Note: Define: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) H C pnxpnCxppn pnpxC pnxpnCxppn pnpxC xnpn pxn nxn pxppxp V U U C C Σ Σ = −−− − −−− − −⊥ ⊥⊥ 0 0 0 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) − = − = −−⊥−⊥−−− − − pxnxppnxnpn pxn xnpn pxn pnxpnxppn pnpx xnpn pxn CLC C C C IL I T C pxp 0 : Since: ( ) [ ] ( ) ( ) = − −− ⊥ ⊥− pn p pn p pn p I I IL I CC C C IL I 0 000 †† Define: [ ] [ ] ( ) [ ]††††† 0 : ⊥⊥ − ⊥ += = CLCC IL I CCMH pn p
- 14. SOLO Reduced Order Observers for Linear Systems Note (continue – 1): Define: CLCT −= ⊥: Then: ( )[ ] ( ) ( )[ ] ( ) ( )( ) ( ) uBCLCuDyLCCACLC uDyLpCACLCuDyLp td d −+−+−+ −−−=−− ⊥⊥⊥ ⊥⊥ †† † ˆˆ ( )[ ] ( )( )uDyLCCuDyLpCx −++−−= ⊥⊥ ††† ˆˆ [ ] [ ]††† : ⊥⊥+= CLCCMH ( )uDyLpz −−= ˆ: ( ) ( ) −+= +−+= uDyHzMx uBTuDyHATzMAT td zd KF ˆ ( ) ( ) ( ) ( ) AT C T HMAT C T HATMAT I T C MH I I MH T C KF n pn p = = = = − 0 0 Those are the well known Reduced Order Observer Equations
- 15. SOLO Reduced Order Observers for Linear Systems Note (continue – 2): Then: ( ) ( ) −+= +−+= uDyHzMx uBTuDyHATzMAT td zd GF ˆ ( ) ( ) ( ) ( ) AT C T HMAT C T HATMAT I T C MH I I MH T C GF n pn p = = = = − 0 0 ( ) CGCHATTMIAT TMATATTFAT DHSuSyHzMx DGBTJuJyGzF td zd n ==−= −=− −=++= −=++= :ˆ : =+ =+ −= =− − 0DHS ITMCH DGBTJ CGTFAT valueseigenstablehasF n nxpnxmnxq qxpqxmqxq nq HSM GJF xz xx yHuSzMx yGuJzFz RRR RRR RR ∈∈∈ ∈∈∈ ∈∈ → ++= ++= ,, ,, ,ˆ, ˆ ˆ
- 16. SOLO Observers Generic Observer for a Linear Time Invariant (LTI) System pxmpxnnxmnxn pmn DCBA yux uDxCy uBxAx RRRR RRR ∈∈∈∈ ∈∈∈ += += ,,, ,, Observer nxpnxmnxq qxpqxmqxq nq RSM GJF xz xx yRuSzMx yGuJzFz RRR RRR RR ∈∈∈ ∈∈∈ ∈∈ → ++= ++= ,, ,, ,ˆ, ˆ ˆ A Necessary Condition for obtaining an Observer is that (A,C) is Observable. The Observer will achieve if and only if: xx →ˆ =+ =+ −= =− − 0DRS ITGCR DGBTJ CGTFAT valueseigenstablehasF n L.T.I. System [ ] [ ]††† : ⊥⊥+= CLCCMH CLCT −= ⊥: HATGMATF == :&:
- 17. SOLO Reduced Order Observers for Linear Systems Let use a constant feedback from the Reduced Order Observer to control the plant: xK ˆ The control is xKrpCKxKru ˆ~† −=+−= ⊥ ( )[ ] ( ) ( )[ ]ppCpCuDyCKr pCuDyCKrxKru ˆ ˆˆ ††† †† −−+−−= +−−=−= ⊥⊥ ⊥ The augmented system is ( ) [ ] [ ] + −=+= −+=+−= + − = ⊥ ⊥⊥ ⊥⊥ rD p x CKKDCuDxCy p x CKKrpCKxKru u B p x CACLC A p x † †† † ~ 00 0 ~
- 18. SOLO Reduced Order Observers for Linear Systems The augmented system is ( ) [ ] [ ] + −= + − + − = ⊥ ⊥ ⊥⊥ rD p x CKKDCy r B p x CKK B CACLC A p x † † † 000 0 ~ or The poles of the closed loop system are given by: ( ) [ ] + −= + − − = ⊥ ⊥⊥ ⊥ rD p x CKKDCy r B p x CACLC CKBKBA p x † † † 00~ ( ) ( ) [ ] ( ) ( )[ ] ObservertheofPoles pn ControllertheofPoles n pn n CACLCIsKBAIs CACLCIs CKBKBAIs † † † detdet 0 det ⊥⊥− ⊥⊥− ⊥ −−⋅+−= −− −+− Hence the Reduced Order Controller has the “Separation Property” of the Controller and Observer.
- 19. SOLO Reduced Order Observers for Linear Systems Compensator Transfer Function By tacking the Laplace Transform of the compensator dynamics we obtain: ( )[ ] ( ) [ ]uBxACLCuDyLp td d +−=−− ⊥ ˆˆ ( ) pCuDyCx ˆˆ †† ⊥+−= xKu ˆ−= ( ) ppCCuDyCCxC I ˆˆˆ † 0 † =+−= ⊥⊥⊥⊥ ( )[ ] ( ) ( ) xKBACLCuDyLxCs ˆˆ −−=−− ⊥⊥ ( ) ( ) ( )[ ]( ) xKBACLCKDLCsyLs xnpn ˆ −⊥⊥ −−−−= ( ) ( ) ( )[ ] ( ) ( ) 1 †ˆ mxxmpnpnnx yLsKBACLCKDLCsx −−⊥⊥ −−−−= where Therefore ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ] ( ) ( )pnpnnxxnpn IKBACLCKDLCsKBACLCKDLCs −−⊥⊥−⊥⊥ =−−−−−−−− † ( ) ( ) ( )[ ] ( ) ( ) 1 † mxxmpnpnnx yLsKBACLCKDLCsKu −−⊥⊥ −−−−−=
- 20. SOLO Reduced Order Observers for Linear Systems How to Find K & L For a Stable Closed Loop This can be done by solving the following Asymptotic Minimum Variance Control Problem: xKu ˆ−= ( ) ( ){ } ( ) ( ) ( ){ } += −===++= uDxCy tWwtwEtwExwuBxAx T τδτ 0,0,00 ( ) ( ) ( ) ( ){ } )(0lim definitepositiveRtuRtutxQtxEJ TT t >+= ∞→ System with no output noise to allow us to use a Reduced Order Observer. The solution to this problem is: where: PBRK T1− = and P is the solution of the Algebraic Riccati Equation: 01 =−++ − PBRBPQAPPA TT or: HT T ARic AQ BRBA RicP = −− − = −1 Minimize: A stabilizing solution (and unique) exists iff: 1 (A,B) is stabilizable 2 AH has no jω axis eigenvalues If Q ≥ 0 then P ≥ 0
- 21. SOLO Reduced Order Observers for Linear Systems How to Find K & L For a Stable Closed Loop (continue – 1) ( ) wCuBCuDyCACpCACp inputknown ⊥⊥⊥⊥⊥ ++−+= 1 †† ( ) pCuDyCx †† ⊥+−=and:xCp ⊥= and ( )wuBxACxCp ++== ⊥⊥ ( ) ( )[ ]{ } ( ) wCuDuBCuDyCACpCAC uDwuBpCuDyCACuDwuBxACuDxCy inputknown +++−+= ++++−=+++=+= ⊥ ⊥ 2 †† †† ˆ The measurements are given by (instead of )y y Let define: †*†* **** :,: :,:,:,: ⊥⊥⊥ ⊥ == ==== CACCCACA wCvwCwyypx
- 22. SOLO Reduced Order Observers for Linear Systems How to Find K & L For a Stable Closed Loop (continue – 2) ( ) wCuBCuDyCACpCACp inputknown ⊥⊥⊥⊥⊥ ++−+= 1 †† ( ) wCuDuBCuDyCACpCACy inputknown +++−+= ⊥ 2 †† Define: †*†* **** :,: :,:,:,: ⊥⊥⊥ ⊥ == ==== CACCCACA wCvwCwyypx The Estimation Problem becomes: ( ) ∗∗∗∗ ++= winputknownxAx 1 ( ) ∗∗∗∗ ++= vinputknownxCy 2 [ ] = = ⊥ ⊥⊥⊥ ** ** 00 00** * * : RS SP CWCCWC CWCCWC vw v w E TTT TT TT
- 23. SOLO Reduced Order Observers for Linear Systems How to Find K & L For a Stable Closed Loop (continue – 3) The Estimation Problem: ( ) ∗∗∗∗ ++= winputknownxAx 1 ( ) ∗∗∗∗ ++= vinputknownxCy 2 [ ] = = ⊥ ⊥⊥⊥ ** ** 00 00** * * : RS SP CWCCWC CWCCWC vw v w E TTT TT TT The Solution to the Estimation Problem is: ( )( ) ( )[ ]( ) 1 0 † 0 1 − ⊥⊥ −∗∗∗ +=+= TTTT CWCCACYCWCRCYSL or ( )[ ] ( ) 1 0 † 0 − ⊥⊥ += TTT CWCCCAYWCL where ( )[ ] ( ) ( )( ) ( )[ ] −−−− −− = ∗−∗∗∗∗−∗∗∗ ∗−∗∗∗−∗∗∗ CRSASRSP CRCCRSA RicY T TT 11 11
- 24. SOLO Reduced Order Observers for Linear Systems How to Find K & L For a Stable Closed Loop (continue – 4) In explicit form (the Algebraic Riccati Equation) is: But ( )[ ] ( )[ ] ( ) ( )( ) 0 1111 =−+−−+− ∗−∗∗∗∗−∗∗∗−∗∗∗∗−∗∗∗ TTT SRSPYCRCYCRSAYYCRSA ( ) ( ) ( ) ( )[ ] ( )pnnx TT nxnpn TT CACCWCCWIC CACCWCCWCCACCRSA −⊥ − −⊥ ⊥ − ⊥⊥⊥ ∗−∗∗∗ −= −=− †1 00 †1 00 †1 ( ) ( ) ( ) ( ) ( ) †1 0 † †1 0 †1 ⊥ − ⊥ ⊥ − ⊥ ∗−∗∗ = = CACCWCCAC CACCWCCACCRC TTTT TT ( ) ( )( ) ( ) ( )[ ] TTT n TTTTT CWCCWCCWIC CWCCWCCWCCWCSRSP ⊥ − ⊥ − ⊥⊥⊥ ∗−∗∗∗ −= −=− 0 1 00 0 1 000 1
- 25. SOLO Reduced Order Observers for Linear Systems How to Find K & L For a Stable Closed Loop (continue – 5) Therefore Y(n-p)x(n-p) is given by the following (n-p) Algebraic Riccati Equation: ( )[ ] ( )[ ]{ }T TT n TT n CACCWCCWICYYCACCWCCWIC †1 00 †1 00 ⊥ − ⊥⊥ − ⊥ −+− ( ) ( ) YCACCWCCACY TTTT †1 0 † ⊥ − ⊥− ( )[ ] 00 1 00 =−+ ⊥ − ⊥ TTT n CWCCWCCWIC Note: 1 ( )[ ] ( ) ( ) ( ) ( ) PCCWCCWI CCWCCCWCWCCWCCWCCWICCWCCWIP TT n T I TTTTT n TT n =−= +−=−= − −−−− 1 00 1 00 1 00 1 00 21 00 2 2: This is a Projection, since P2 = P, but oblique because P is not symmetrical. 2 For W0 = In we get: ( ) ( ) ⊥⊥ −− =−=−=− CCCCICCCCICCWCCWI n TT n TT n ††11 00 ( ) ( ) ( ) ††††1 0 † ⊥⊥⊥ − ⊥ = CACCACCACCWCCAC TTTTTT ( )[ ] ††††1 00 ⊥⊥⊥⊥⊥⊥⊥ − ⊥ ==− − CACCACCCCACCWCCWIC pnI TT n ( )[ ] TT I TTT n CCCCCCCWCCWCCWIC ⊥⊥⊥⊥⊥⊥⊥ − ⊥ ==− † 0 1 00
- 26. SOLO Reduced Order Observers for Linear Systems How to Find K & L For a Stable Closed Loop (continue – 6) Hence, for W0=In, Y(n-p)x(n-p) is given by the following (n-p) Algebraic Riccati Equation: Notice (continue – 1): 3 With this L , A*- L C* will have stable eigenvalues, but 2 ( ) ( ) ( ) 0 ††††† † =+−+ ⊥⊥⊥ − ⊥⊥⊥⊥⊥ ⊥⊥ T CCI TTT CCYCACCACYCACYYCAC n ( )[ ] ( ) ( ) †† 0 1† † † CACYCCCCAYCL TTCC C TTT ⊥ = − ⊥⊥ ⊥⊥ =+= ( ) ††† ** ⊥⊥⊥⊥⊥ −=−=− CACLCCACLCACCLA Therefore has stable eigenvalues, and the Reduced Order Estimator is stable ( ) † ⊥⊥ − CACLC
- 27. SOLO Reduced Order Observers for Linear Systems How to Find K & L For a Stable Closed Loop (continue – 7) Notice (continue – 2): 4 Following P.J. Blanvillain and T.L. Johnson (IEEE Tr. AC., Vol. AC-23, No.1, June 1978) this Problem is equivalent to the following ( ) ( ) = += xCy WNxuBxAx 0,0~0 Given Find the Dynamic Compensator Parameters (F, G, H, M) += += yMzHu yGzFz Compensator Which minimizes the Quadratic Performance Index: ( ) ( ) ( ) ( ) ( )[ ] += ∫ ∞ 0 ,,, dttuRtutxQtxEMHGFJ TT
- 28. SOLO Reduced Order Observers for Linear Systems Let append to the Reduced Order Observer the Stable Transfer Matrix ( ) ( ) =+−= − DC BA DBAIsCsQ ˆˆ ˆˆ :ˆˆˆˆ 1 =−−− uBCpCACyCACy ˆ†† The input to the Stable Transfer Function will be the same as for the Reduced Order Observer.
- 29. References SOLO Kwakernaak, H., Sivan, R., “Linear Optimal Control Systems”, Wiley Inter-science, 1972, pg.335 Reduced Order Observers for Linear Systems Gelb A. Ed, “Applied Optimal Estimation”, The Analytic Science Corporation, 1974, pg.320
- 30. August 13, 2015 30 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 –2013 Stanford University 1983 – 1986 PhD AA