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Stabilization of linear time invariant systems, Factorization Approach

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Solo Hermelin

This document discusses the stabilization of linear time-invariant (LTI) systems using a factorization approach. It elaborates on concepts such as the coprime factorization of transfer matrices, the well-posedness, stability conditions, and the parameterization of stabilizing compensators. Additionally, it references foundational mathematical principles like Bézout's identity and provides a framework for optimizing the performance of LTI systems.

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Stabilization of Linear Time-Invariant Systems Factorization Approach SOLO HERMELIN Updated: 17.12.10 1
Table of Content SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Introduction Well-Posedness Internal Stability Right and Left Coprime Factorization of a Transfer Matrix 2 Stabilization of Linear Time-Invariant Systems Stabilization of Linear Time-Invariant Systems - State-Space Approach Eigenvalues (modes) of the System Transfer Function of a L.T.I. System State Space Realization of the System Stability of the System Transmission Zeros of the L.T.I. System Controllability, Observability Stabilizability, Detectability Transfer Function of a L.T.I. System:
Table of Content (continue) SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach 3 State-Space Realization of All Coprime Matrices Operations on Linear Systems Change of Variables Cascade of Two Linear Systems G1(s)G2(s) Para-Hermitian Pseodoinverse of G (s) for rank Dpxm=min (p,m) is G(s) † The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator The Eigenvalues of Closed-Loop System The Transfer Function of the Compensator K (s) with O.B.C. Realization Realization of Q (s) Given G (s) and K (s) Realization of Heu (s) Given G (s) and K (s) References
• Assume a Linear Time-Invariant Plant, not necessary stable with m inputs and p outputs. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach • Assume that the plant can be represented by a set of n linear ordinary differential equations with constant coefficients (to assure time-invariance) or by the corresponding transfer matrix G (s)pxm. • Assume also that G (s) is proper: ( ) ∞< ∞→ sG s lim ( G(s) is analytic as s →∞ ) Therefore G (s) is in the Ring of Real-rational proper (Rp) matrices: ( ) pxm pRsG ∈ Using the fact that under those assumptions all G (s) can be factorized in two Real-rational-proper and stable (RH∞) matrices, we will obtain a parameterization of all compensators K (s)mxp that stabilize the given plant (and conversely all plants that can be stabilized by a given compensator). This parameterization will allow to define methods of optimizing the desired performances of the design over all possible stabilizing compensators. 4
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Given the Feedback System, we have: ( ) ( )       =              − 2 1 2 1 u u e e IsK sGI mmxp pxmp from which: ( ) ( ) ( ) ( ) ( ) ( ) ( )               ++ +−+ =       =              − =      −− −− − 2 1 11 11 2 1 2 1 1 2 1 u u GKIKGIK GKIGKGI u u sH u u IsK sGI e e mp mp eu mmxp pxmp 5
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Well-Posedness Definition: The System in Figure is well-posed if the transfer-matrix from u to e; i.e. Heu exists and is proper. ( ) ( ) ( )       =              − =      − 2 1 2 1 1 2 1 u u sH u u IsK sGI e e eu mmxp pxmp Well-Posed ( ) ( ) 1−         − mmxp pxmp IsK sGI exists and is proper ( ) ( ) 1−       ∞− ∞ m p IK GI is invertible ( ) ( )[ ]∞∞+ KGIp is invertible ( ) ( )[ ]∞∞+ GKIm is invertible 6
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Internal Stability Internally Stable ( ) ( ) ( )mpmp eu HsH +×+ ∞∈ ( ) ( ) ( ) ( ) ( ) ( ) ( )               ++ +−+ =      =              − =      −− −−− 2 1 11 11 2 1 2 1 1 2 1 u u GKIKGIK GKIGKGI u u sH u u IsK sGI e e mp mp eu mmxp pxmp Definition: The System in Figure internally stable if it is well-posed and Heu (s)is analytic for all Real (s) ≥ 0 ( )+∈Cs If in addition Heu (s) is real-rational and proper we will write:( ) ( ) ( ) ( )mpmp peu RsH +×+ ∈ Internally Stable ( ) ( ) ( )mpmp eu RHsH +×+ ∞∈ and Internally Stability of a Real-Rational and Proper System ( ) ( ) ( ) ( ) ( ) ( ) ∞−− −− ∞ − ∈         ++ +−+ ⇔∈         − RH GKIKGIK GKIGKGI RH IsK sGI mp mp mmxp pxmp 11 111 7
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Right and Left Coprime Factorization of a Transfer Matrix Right Coprime Factorization (r.c.f.) Left Coprime Factorization (l.c.f.) is a r.c.f. of G(s) pxm if( ) ( )( )mmgmpg sDsN ×× , is a l.c.f. of G(s) pxm if( ) ( )( )ppgmpg sDsN ×× ~ , ~ ( ) ( ) ( ) mmgmpgmp sDsNsG × − ×× = 1 ( ) ( ) ( ) mpgppgmp sNsDsG ×× − × = ~~ 1 ( ) ( ) mm g mp g RHsDRHsN × ∞ × ∞ ∈∈ & ( ) ( ) pp g mp g RHsDRHsN × ∞ × ∞ ∈∈ ~ & ~ ( ) ( ) pm g mm g RHsYRHsX × ∞ × ∞ ∈∈∃ ~ & ~ ( ) ( ) mp g pp g RHsYRHsX × ∞ × ∞ ∈∈∃ & 1 1 2 2 3 3 s.t.: s.t.: ( ) ( )[ ] ( ) ( ) m g g gg I sN sD sYsX =        ~~ ( ) ( )[ ] ( ) ( ) p g g gg I sY sX sNsD =        ~~ Bézout-Diophantine Identities 8
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach In number theory, Bézout's identity or Bézout's lemma is a linear diophantine equation. It states that if a and b are nonzero integers with greatest common divisor d, then there exist integers x and y (called Bézout numbers or Bézout coefficients) such that Additionally, d is the least positive integer for which there are integer solutions x and y for the preceding equation. In mathematics, a Diophantine equation is an indeterminated polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface or more general object, and ask about the lattice points on it. Diophantus of Alexandria (cca 200 – 280) Diophantus, often known as the 'father of algebra', is best known for his Arithmetica, a work on the solution of algebraic equations and on the theory of numbers. However, essentially nothing is known of his life and there has been much debate regarding the date at which he lived. Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac. Étienne Bézout 1730 - 1783 dybxa =+ 9
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Right and Left Coprime Factorization of a Transfer Matrix (continue – 1) ( ) ( )[ ] ( ) ( ) m g g gg I sN sD sYsX =        ~~ ( ) ( )[ ] ( ) ( ) p g g gg I sY sX sNsD =        ~~ Bézout-Diophantine Identities For a real-rational and proper transfer-matrix right and left coprime factorizations always exist (proof in Vidyasagar M., “Control System Synthesis: A Factorization Approach”, MIT Press, 1985) ( ) mp pRsG × ∈ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )         − =         −         − p ggggm gg gg gg gg I sYsXsXsYI sXsN sYsD sDsN sYsX 0 ~~ ~~ ~~ or Define: ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )sDsYsXsXsYsYsY sNsYsXsXsYsXsX ggggggg ggggggg ~~~~ : ~ ~~~~ : ~ 0 0 −+= −+= Pre-multiply by and redefine:( ) ( ) ( ) ( )( )         −− p ggggm I sYsXsXsYI 0 ~~ 4 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       =         −         − p m gg gg gg gg I I sXsN sYsD sDsN sYsX 0 0 ~~ ~~ 0 0 0 0 ( ) ( ) ( ) ( )sYsY sXsX gg gg = = : : 0 0 to obtain: Generalized Bézout Identity 10
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Right and Left Coprime Factorization of a Transfer Matrix (continue – 2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       =         −         − p m gg gg gg gg I I sXsN sYsD sDsN sYsX 0 0 ~~ ~~ 0 0 0 0 we obtained: Generalized Bézout Identity We can see that: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )mpmp gg gg gg gg RH sXsN sYsD sDsN sYsX +×+ ∞ − ∈         − =         − 0 0 1 0 0 ~~ ~~ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )mpmp gg gg gg gg RH sDsN sYsX sXsN sYsD +×+ ∞ − ∈         − =         − 0 0 1 0 0 ~~ ~~ Definition: A square transfer-matrix s.t. is called unimodular, or a unit in the ring of stable real rational and proper transfer matrices. ( ) qqqq RHsU × ∞ × ∈ ( ) qq RHsU × ∞ − ∈1 11
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Right and Left Coprime Factorization of a Transfer Matrix (continue – 3) Proof Proof ( ) ( ) ( ) 1 111 − −−− = == UDUN DUUNDNsG gg gggg Post-multiply by U:[ ] m g g gg I N D YX =        ~~ [ ] U UN UD YX g g gg =        ~~ Pre-multiply this by U-1 : [ ] m g g gg I UN UD YUXU =        −− ~~ 11 ( ) ( )( ) 1 111 ~~~~ ~~~~~~ − −−− = == gg gggg DUNU NUUDNDsG [ ] U UN UD YX g g gg =        ~~ [ ] p g g gg I UY UX NUDU =         − − 1 1 ~ ~ ~~~~ Pre-multiply by :[ ] p g g gg I Y X ND =        ~~ U ~ Post-multiply this by :1~− U Right Coprime Factorization (r.c.f.) Left Coprime Factorization (l.c.f.) 12
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Right and Left Coprime Factorization of a Transfer Matrix (continue – 4) Theorem 1a: If is a r.c.f. of then is also a r.c.f. of G (s) for every U(s) mxm unimodular. ( )gg ND , ( )UNUD gg , ( ) mp pRsG × ∈ Theorem 1b: If is a l.c.f. of then is also a l.c.f. of G (s) for every unimodular. ( )gg ND ~ , ~ ( )gg NUDU ~~ , ~~ ( ) mp pRsG × ∈ Proof (continue – 1) Proof (continue – 1) [ ] m g g gg I UN UD YUXU =        −− ~~ 11 [ ] p g g gg I UY UX NUDU =         − − 1 1 ~ ~ ~~~~ ( ) ppsU × ~ Because ∞ − ∈ RHU 1 ∞ − ∞ − ∈= ∈= RHYUY RHXUX gg gg ~ : ~ ~ : ~ 1 1 1 1 and Because ∞ − ∈ RHU 1~ ∞ − ∞ − ∈= ∈= RHUYY RHUXX gg gg 1 1 1 1 ~ : ~ : and [ ] m g g gg I UN UD YX =         11 ~~ [ ] p g g gg I Y X NUDU =         1 1~~~~ Hence is an r.c.f of G (s).( )UNUD gg , Hence is an l.c.f of G (s). q.e.d. q.e.d. Right Coprime Factorization (r.c.f.) Left Coprime Factorization (l.c.f.) 13 ( )gg NUDU ~~ , ~~
14 SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems We can use the coprime-factorization to find necessary and sufficient conditions s.t. the Linear Time-Invariant System G(s), K(s) is internally stable. The definition of an internal stable system is: ( ) ( ) ( ) ( ) ( ) ∞−− −− ∈         ++ +−+ = RH GKIKGIK GKIGKGI sH mp mp eu 11 11 : Suppose we have any coprime-factorization of G(s) and K(s): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )mxpkmxmkpxpkmxpkmxp pxmgpxpgmxmgpxmgpxm sNsDsDsNsK sNsDsDsNsG ~~ ~~ 11 11 −− −− == == The corresponding generalized Bezout identities are: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       =         −        − =         −         − p m gg gg gg gg gg gg gg gg I I sDsN sYsX sXsN sYsD sXsN sYsD sDsN sYsX 0 0 ~~ ~~ ~~ ~~ 00 0 0 0 000 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       =      −         − =         −       − p m kk kk kk kk kk kk kk kk I I sXsN sYsD sDsN sYsX sDsN sYsX sXsN sYsD 0 0 ~~ ~~ ~~ ~~ 0 00000 0 0
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems ( ) ( ) ( ) ( ) ( ) ∞−− −− ∈         ++ +−+ = RH GKIKGIK GKIGKGI sH mp mp eu 11 11 : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )mxpkmxmkpxpkmxpkmxp pxmgpxpgmxmgpxmgpxm sNsDsDsNsK sNsDsDsNsG ~~ ~~ 11 11 −− −− == == Use Matrix Inverse Lemma: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) gkgkgkgkkgkkg gkkggkkggggkkggp kgkgkgpkkggpp DNNDDDDDNNDDD DDNNDDNNDDDDNNDI NNNDDNINDDNIKGI ~~~~~~ ~~~~~~~~~ ~~~~~ 1111 111111111 11111 −−−− −−−−−−−−− −−−−− +=+= +=+=+= +−=+=+ ( ) ( ) mmmnnmmmmnnnnmmmmmmnnnnmmm CBDCBADCCBADC ×× − ××××××× − × − ×× − × +−=+ 1111 ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( )[ ] ( ) kgkgkgkgkgkgkk kgkgkgkmkkgkgkgpkk gkgkgkgkgkgkkkp NNNDDDNNNDDDDD NNNDDNNIDNNNDDNIND DNNDDNDNNDDDDNKGIK ~~~~~~~~ ~~~~~~~~~~ ~~~~~~ 111 1111 1111 −−− −−−− −−−− +=+= +−=+−= +=+=+ 15
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems ( ) ( ) ( ) ( ) ( ) ∞−− −− ∈         ++ +−+ = RH GKIKGIK GKIGKGI sH mp mp eu 11 11 : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )mxpkmxmkpxpkmxpkmxp pxmgpxpgmxmgpxmgpxm sNsDsDsNsK sNsDsDsNsG ~~ ~~ 11 11 −− −− == == Use Matrix Inverse Lemma: ( ) ( ) mmmnnmmmmnnnnmmmmmmnnnnmmm CBDCBADCCBADC ×× − ××××××× − × − ×× − × +−=+ 1111 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) kgkgkgkggkggk kggkkggkkkkggkkm gkgkgkmggkkmp DNNDDDDDNNDDD DDNNDDNNDDDDNNDI NNNDDNINDDNIGKI ~~~~~~ ~~~~~~~~~ ~~~~~ 1111 111111111 11111 −−−− −−−−−−−−− −−−−− +=+= +=+=+= +−=+=+ ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( )[ ] ( ) gkgkgkgkgkgkgg gkgkgkgmggkgkgkmgg kgkgkgkgkgkgggm NNNDDDNNNDDDDD NNNDDNNIDNNNDDNIND DNNDDNDNNDDDDNGKIG ~~~~~~~~ ~~~~~~~~~~ ~~~~~~ 111 1111 1111 −−− −−−− −−−− +=+= +−=+−= +=+=+ 16
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems We found for: ( ) gkkgpp DDNNIKGI ~~ 1 2 1 1 1 −−− ∆=∆−=+ ( ) gkkgp DNNDKGIK ~~ 1 2 1 1 1 −−− ∆=∆=+ ( ) gkmkgp NNIDDGKI ~~ 1 1 1 1 1 −−− ∆−=∆=+ ( ) gkkgm NDDNGKIG ~~ 1 2 1 1 1 −−− ∆=∆=+ [ ] [ ] ∞ ∞ ∈      =+=∆ ∈         =+=∆ RH N D NDNNDD RH N D NDNNDD g k kgkgkg g g kkgkgk ~~~~ : ~~~~ : 2 1       =         −        ∆ ∆         − − − p m kg kg kg kg I I DN ND DN ND 0 0 ~~ ~~ 0 0 1 2 1 1 Rearrange those Equations in Matrix Form       ∆ ∆ =         −         − 2 1 0 0 ~~ ~~ kg kg kg kg DN ND DN ND ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )sNsDsDsNsK sNsDsDsNsG kkkk gggg ~~ ~~ 11 11 −− −− == == Also we have: 17
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems We found for: ( ) gkkgpp DDNNIKGI ~~ 1 2 1 1 1 −−− ∆=∆−=+ ( ) gkkgp DNNDKGIK ~~ 1 2 1 1 1 −−− ∆=∆=+ ( ) gkmkgp NNIDDGKI ~~ 1 1 1 1 1 −−− ∆−=∆=+ ( ) gkkgm NDDNGKIG ~~ 1 2 1 1 1 −−− ∆=∆=+ [ ] [ ] ∞ ∞ ∈      =+=∆ ∈         =+=∆ RH N D NDNNDD RH N D NDNNDD g k kgkgkg g g kkgkgk ~~~~ : ~~~~ : 2 1 From those Equations we can write: ( ) ( ) ( ) ( ) ( ) [ ] [ ]gg k k mgkmgk gkgk kk g gp kgkg kgkgp mp mp eu ND N D INNIDN NDDD DN D NI DDND DNNNI GKIKGIK GKIGKGI sH ~~ 0 00 ~~ ~~ ~~ 00 0 ~~ ~~ : 1 21 2 1 2 1 2 1 2 1 11 1 1 1 1 1 1 1 11 11 −∆      +      =         ∆−∆ ∆−∆ = ∆        − +      =         ∆∆ ∆−∆− =         ++ +−+ = − −− −− − −− −− −− −− 18
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems We found for: [ ] [ ] ∞ ∞ ∈      =+=∆ ∈         =+=∆ RH N D NDNNDD RH N D NDNNDD g k kgkgkg g g kkgkgk ~~~~ : ~~~~ : 2 1 ( ) [ ] [ ]gg k k m kk g gp eu ND N D I DN D NI sH ~~ 0 00~~ 00 0 1 2 1 1 −∆      +      =∆        − +      = −− Theorem 2 The Necessary and Sufficient Conditions that Heu(s) is Stable, i.e. are( ) ∞∈ RHsHeu ∞ −− ∞ ∈∆∆∈∆∆⇔∆⇔∆ RHRH 1 2 1 12121 ,&,UnimodularUnimodular Proof Theorem 2 (1) If Δ1(s) (or Δ2(s) ) is Unimodular; i.e. Δ1(s)ϵRH∞ and Δ1(s)-1 ϵRH∞ , then from the Equation above we can se that Heu(s) ϵRH∞ 19
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems We found for: Theorem 2 The Necessary and Sufficient Conditions that Heu(s) is Stable, i.e. are ( ) [ ] [ ]gg k k m kk g gp eu ND N D I DN D NI sH ~~ 0 00~~ 00 0 1 2 1 1 −∆      +      =∆        − +      = −− ( ) ∞∈ RHsHeu ∞ −− ∞ ∈∆∆∈∆∆⇔∆⇔∆ RHRH 1 2 1 12121 ,&,UnimodularUnimodular Proof Theorem 2 (continue) (2) Use and pre-multiply Heu(s) ϵRH∞ by and post-multiply it by ( ) ( )[ ] ( ) ( ) m g g gg I sD sN sXsY =        − − 00 ~~ ( ) ( )[ ] ( ) ( ) m k k kk I sX sY sDsN =      0 0~~ ( ) ( )[ ]sXsY gg 00 ~~ − ( ) ( )      sX sY k k 0 0 [ ] ( ) ( ) ∞∞ − ∈⇔∈         −            −=∆ RHsHRH Y X I sHYX eu g g m eukk 0 0 00 1 2 0 00~~ [ ] ( ) ( ) ∞∞ − ∈⇔∈                    −−=∆ RHsHRH X YI sHXY eu k kp eugg 0 0 00 1 1 00 0~~ Use and pre-multiply Heu(s) ϵRH∞ by and post-multiply it by [ ] p k k kk I N D YX =      00 ~~ [ ] p g g gg I Y X ND =         − − 0 0~~ [ ]00 ~~ kk YX         − 0 0 g g Y X q.e.d. 20
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Corollary ( ) Unimodular~~ ~~ Unimodular         − ⇔         − ⇔∈ ∞ kg kg kg kg eu DN ND DN ND RHsH Proof       =         −        ∆ ∆         − − − p m kg kg kg kg I I DN ND DN ND 0 0 ~~ ~~ 0 0 1 2 1 1         −        ∆ ∆ =         − − −− kg kg kg kg DN ND DN ND ~~ ~~ 0 0 1 2 1 1 1         ∆ ∆         − =         − − − − 1 2 1 1 1 0 0 ~~ ~~ kg kg kg kg DN ND DN ND We found : Therefore: ( ) ( ) ( ) ( ) ∞ − ∞ − ∞ − ∞ − ∞ − ∞ − ∈         − ⇔∈∆∈∆ ∈         − ⇔∈∆∈∆ RH DN ND RHsRHs RH DN ND RHsRHs kg kg kg kg 1 1 1 1 1 1 1 1 1 1 ~~ ~~ & & q.e.d. 21
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Theorem 3 The set of all Proper Linear Compensators K (s)mxp achieving Internal Stability is given by: ( ) ( )( ) ( ) ( ) ( ) ( ) 1 0 1 0 1 0 1 00 0 1 0 1 00 ~ ~~~~ −−−− −− +−= −+=+−= gggpggg gggggggg XQNXIQXXY DQYNQXQNXQDYsK where Q (s) ϵ RH∞ is a free parameter and 0 ~~ det&0det 00 ≠+≠+ gggg NQXQNX Proof Theorem 3 is given in three parts (1) Define ggkggk ggkggk DQYNQDYN NQXDQNXD ~~ : ~ : ~~ : ~ : 00 00 −=−= +=+= We want to prove that are coprime, and that they internally stabilize the System. ( ) ( )kkkk NDandND ~ , ~ , From the definition of K (s) above is clear that: ( ) ( ) ( ) ( ) ( )sNsDsDsNsK kkkk ~~ 11 −− == and: ( ) ( ) ( ) ( ) ∞∈ RHsNsDsNsD kkkk ~ , ~ ,, 22
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 1) ggkggk ggkggg DQYNQDYN NQXDQNXD ~~ : ~ : ~~ : ~ : 00 00 −=−= +=+= (1) To prove that defined previously satisfy the Bezout identity start with ( ) ( )kkkk NDandND ~ , ~ ,       =         −         − p m gg gg gg gg I I XN YD DN YX 0 0 ~~ ~~ 0 0 0 00 Pre-multiply by and post-multiply by       − p m I QI 0       p m I QI 0 to obtain: ( ) ( )       =         + −−         − −+ p m ggg ggg gg gggg I I QNXN QDYD DN DQYNQX 0 0 ~~ ~~~~ 0 000 Which gives by above definition:       =         −         − p m kg kg gg kk I I DN ND DN ND 0 0 ~~ ~~ 0 Bezout Identity 23
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 2) (1) To prove that K (s) stabilizes the System start from the Bezout Identity       =         −         − p m kg kg gg kk I I DN ND DN ND 0 0 ~~ ~~ 0 ∞ − ∈         − =         − RH DN ND DN ND kg kg gg kk 1 0 ~~ ~~ ∞ − ∈         − =         − RH DN ND DN ND gg kk kg kg 0 1 ~~ ~~ According to the Corllary ( ) ∞∞ − ∞ − ∈⇒∈         − ∈         − RHsHRH DN ND RH DN ND eu gg kk kg kg 1 0 1 ~~ ~~ & i.e., K (s) stabilizes the System. 24
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 3) (2) We want to prove that if stabilizes the System then a Q (s) in RH∞ can be found s.t. K (s) will have the form kkkk NDDNK ~~ 11 −− == ( ) ( )( ) ( ) ( ) ( ) ( ) 1 0 1 0 1 0 1 00 0 1 0 1 00 ~ ~~~~ −−−− −− +−= −+=+−= gggpggg gggggggg XQNXIQXXY DQYNQXQNXQDYsK If K (s) stabilizes the System then Δ2 is Unimodular ∞ − ∞ ∈∆∈+=∆ RHandRHNNDD kgkg 1 22 ~~ : Let calculate ( ) ( ) ( ) ( ) ( ) 1 2 1 2 11 22 1 1 2 11 200 1 1 20 1 0 1 20 1 00 ~~~~ ~~~~~~ ~~ −−−−− −−−− −−−− ∆=∆=∆−∆= ∆−=∆−+= ∆−+=∆−+=+ kkggkgg kgmgkgggggg kggggkgggggg DDDDNND NNIDNNYNXDD NYNDXNYDNXQNX ( ) ( )( ) ( )( ) 1 00 11 2 1 2 −−−− +−=∆∆= QNXQDYDNsK ggggkk Therefore Let define Q (s) using the equation ⇒∆=− −1 20 kgg NQDY ( )1 20 1 : −− ∆−= kgg NYDQ 25
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 4) (2) To complete the proof we must show that Q (s) that has been chosen is in RH∞. Pre-multiply by and use the fact that to obtain:[ ] m g g gg I N D YX =         00 [ ]00 gg YX [ ] ∞− − ∈         ∆+− ∆− = RH DX NY YXQ kg kg gg 1 20 1 20 00     ∆=+ ∆=− − − 1 20 1 20 kgg kgg DQNX NQDY The chosen Q (s) satisfies         ∆+− ∆− =         − − 1 20 1 20 kg kg g g DX NY Q N D In the same way we can define Q1 (s) using the equation: gkgk kgg NNDD RHNDQY ~~ : ~~~ 1 1 110 +=∆ ∈∆=− ∞ − ⇒ ( ) 11 101 ~~~ : −− ∆−= gkg DNYQ 26 ⇒
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 5) (2) We defined Q1 (s) using the equation: gkgk kgg NNDD RHNDQY ~~ : ~~~ 1 1 110 +=∆ ∈∆=− ∞ − ( ) 11 101 ~~~ : −− ∆−= gkg DNYQ ( ) ( ) ( ) ( ) ( ) ∞ −−−−− −−−− −−−− ∈∆=∆=−∆∆= ∆−=∆−+= ∆−+=∆−+=+ RHDDDDDNN DNNIDNNNYDX DNNYXNDNYXNQX kggkggk ggkmggkgggg ggkggggkgggg ~~~ ~~~~ ~~~~~~~~~~ 1 1 11 1 1 1 1 1 11 1 11 100 11 100 11 10010 Therefore for the chosen Q1 (s) we obtain: (2) ( ) ( ) ( )kkkk NDNDsK ~~~~ 1 1 11 1 1 −−−− ∆∆== ( ) ( )gggg DQYNQX ~~~~ 10 1 10 −+= − We still have to prove that ( ) ( ) ∞∈= RHsQsQ1 ( ) ( ) ∞∈= RHsQsQ1 ( ) kgggggkgggg NDYDDDNYDDQD ~~~~~~ : ~ 1 10 11 101 −−− ∆−=∆−= Use the fact that gggg DYYD ~~ 00 = Bezout Id. ( ) gggkggkgggg DQDDNYDNDYDQD ~~~~~ 1 20 1 201 =∆−=∆−= −− gkkg DNND ~~ 1 2 1 1 −− ∆=∆and 27
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 6) (3) The last part of the proof is to show that K (s) has the form ( ) ( ) ( ) 1 0 11 0 1 0 1 00 ~ −−−−− +−= gggpggg XQNXIQXXYsK Use the Inversion Matrix Lemma ( ) ( )( ) ( ) ( )[ ]1 0 11 0 1 0 1 00 1 00 −−−−−− +−−=+−= gggmggggggggg XQNXQINXXQDYQNXQDYsK ( ) ( ) ( ) ( ) ( ) 1 0 11 0 1 0 1 0 11 0 1 00 1 0 1 00 11 0 11 0 11 0 −     +− + −−−− + −−−−− −− −−−− +++−−= g QNXII QNXIpQ ggmgggg QNXIQ ggmggggggg XQNXQINXQDXQNXQINXYXQDXY ggpp ggpggp         ( ) ( )[ ] 1 0 11 0 1 0 11 0 1 00 1 0 1 00 −−−−−−−−− +−++−−= gggmpggggpggggggg XQNXIIQDXQNXIQNXYXQDXY ( ) ( )[ ] 1 0 11 0 1 0 11 0 1 00 1 0 1 00 −−−−−−−−− +−++−−= gggppggggmggggggg XQNXIIQDXQNXIQNXYXQDXY ( ) ( ) ( ) 1 0 11 0 1 00 1 00 −−−−− ++−= gggpgggggg XQNXIQNXYDXYsK 28 ( ) ( ) mmmnnmmmmnnnnmmmmmmnnnnmmm CBDCBADCCBADC ×× − ××××××× − × − ×× − × +−=+ 1111
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 7) (3) We want to prove is to show that K (s) has the form ( ) ( ) ( ) 1 0 11 0 1 0 1 00 ~ −−−−− +−= gggpggg XQNXIQXXYsK ( ) ( ) ( ) 1 0 11 0 1 00 1 00 −−−−− ++−= gggpgggggg XQNXIQNXYDXYsK We found 1 000 1 0 1 000 ~~~~~ −−− +=+=⇒=+ ggggggggmgggg XYDNYXDXINYDX Therefore: ( ) ( ) ( ) 1 0 11 0 1 0 1 00 ~ −−−−− +−= gggpggg XQNXIQXXYsK Since 0 1 0 1 00 ~~ gggg YXXY −− = ( ) ( ) ( ) 1 0 11 0 1 00 1 0 ~~~ −−−−− +−= gggpggg XQNXIQXYXsK Uze Bezout Identity: 0 1 0 1 00 ~~ gggg YXXY −− = q.e.d. 29
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 8) (3) We have shown that ( ) ( ) ( ) ( ) ( ) 1 0 11 0 1 00 1 0 1 0 11 0 1 0 1 00 ~~~~ −−−−−−−−−− +−=+−= gggpggggggpggg XQNXIQXYXXQNXIQXXYsK This is equivalent to ff f ggg ggg f yQe e e NXX XXY y y =                  −− =      −− −− 1 1 0 1 0 1 0 1 00 1 ~    To prove this, let develop ( ) 1 1 0 11 0 1 01 1 0 eXQNXIyyQNXeXy gggffgggf −−−−− +−=⇒−−= ( )[ ] ( ) 11 1 0 11 0 1 0 1 00 1 01 1 001 ~~ esKeXQNXIQXXYeXeXYy ggggggfggg =+−=+= −−−−−−− ( ) 1 1 0 11 0 eXQNXIQyQe gggff −−− +−==⇒ q.e.d.30
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Theorem 4 (Dual of Theorem 3) The set of all Proper Linear Time-Invariant Systems G (s) pxm stabilized by the Controller is given by: ( ) ( )( ) ( ) ( ) ( ) ( ) 1 0 11 0 1 0 1 00 0 1 0 1 00 ~ ~~~~ −−−−− −− +−= −+=+−= kkkpkkk kkkkkkkk XSNXISXXY DSYNSXSNXSDYsG where S (s) ϵ RH∞ is a free parameter and 0 ~~ det&0det 00 ≠+≠+ kkkk NSXSNX Proof Theorem 4 The duality of Theorem 4 to Theorem 3 is evident because by replacing in Theorem 3 g to k and Q (s) to S (s), K (s) to G (s) we obtain Theorem 4. Therefore the proof is similar, by performing the above mentioned replacement and interchanging between G (s) and K (s). 31 ( ) ( ) ( ) ( ) ( )sNsDsDsNsK kkkk ~~ 11 −− ==
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 32 From we obtain ( ) ( )( ) ( ) ( )gggggggg DQYNQXQNXQDYsK ~~~~ 0 1 0 1 00 −+=+−= −− ( ) ( ) ( ) ( )KNDKXYQDQYKNQKX XKYNKDQQDYQNKXK gggggggg gggggggg +−=⇒−=+ −+=⇒−=+ − ~~~~~~~ 0000 00 1 00 ( ) ( ) ( ) ( )KNDKXYXKYNKDQ gggggggg +−=−+= − ~~~ 0000 1 From we obtain ( ) ( )( ) ( ) ( )kkkkkkkk DSYNSXSNXSDYsG ~~~~ 0 1 0 1 00 −+=+−= −− ( ) ( ) ( ) ( )GNDGXYSDSYGNSGX XGYNGDSSDYSNGXG kkkkkkkk kkkkkkkk +−=⇒−=+ −+=⇒−=+ − ~~~~~~~ 0000 00 1 00 ( ) ( ) ( ) ( )GNDGXYXGYNGDS kkkkkkkk +−=−+= − ~~~ 0000 1 Go back to Q (s) Realization
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 33 Let calculate Δ1 (s) and Δ2 (s) for ( ) ( )( ) ( ) ( )   kkkk N gg D gg D gg N gg DQYNQXQNXQDYsK ~ 0 1 ~ 0 1 00 ~~~~ −+=+−= −− ( ) ( ) ( ) ( ) ( ) mgggg I gggg gggggggkgk INDDNQNYDX NDQYDNQXNNDDs m =−++= −++=+=∆      0 00 001 ~~~~ ~~~~~~ ( ) ( ) ( ) ( ) ( ) pgggg I gggg ggggggkgkg IQDNNDYNXD QDYNQNXDNNDDs p =−++= −++=+=∆      0 00 002 ~~~~ ~~~~ We can see that Δ1 (s) and Δ2 (s) are Unimodular and according to Theorem 2 Heu(s) is Stable for this choise of K (s).
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 34 Let calculate Heu(s) for ( ) ( )( ) ( ) ( )   kkkk N gg D gg D gg N gg DQYNQXQNXQDYsK ~ 0 1 ~ 0 1 00 ~~~~ −+=+−= −− ( ) [ ] [ ]gg k k m kk g gp eu ND N D I DN D NI sH ~~ 0 00~~ 00 0 1 2 1 1 −∆      +      =∆        − +      = −− we obtained ( ) ( )[ ] ( ) ( ) [ ]gg gg gg m gggg g gp ND QDY QNX I NQXDQY D NI ~~ 0 00~~~~ 00 0 0 0 00 −         − + +      =+−        − +      = [ ] [ ] [ ] [ ] ∞∈−        − −−         +      = −        − −        − +      = RHNDQ D N ND Y X I NDQ D N XY D NI gg g g gg g g m gg g g gg g gp ~~~~ 0 00 ~~~~ 00 0 0 0 00
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Transfer Function of a L.T.I. System: ( ) ( ) pxmnxmnxnnpxnpxm DBAIsCsG +−= −1 State Space Realization of the System: ( ) 00 xx u x DC BA y x pxmpxn nxmnxn =            =       Eigenvalues (modes) of the System: The Eigenvalues (modes) of the System are the collection of the n complex numbers λ such that: ( ) nAIsrank nxnn <− Stability of the System: The System is stable if ( ) ii ∀λReal 35
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Transmission Zeros of the L.T.I. System: ( ) ( ) ( ) 0,000exp 0 >∀=≠≠= ttyhavewegandxsomeforandtzgtuFor The following definitions are equivalent: (1) The Transmission Zeros of the System are the collection of the complex numbers z s.t.: (2) The Transmission Zeros of the System are the collection of the complex numbers z s.t.: ( )pmn DC BAIz rank pxmpxn nxmnxnn ,min+<      −− 36
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Controllability: The System, or the pair (A,B) is Controllable if, for each time tf > 0 and given state xf, there exists a continuous input u (t) for t ϵ [t0,tf] s.t. x(tf) = xf. Observability: The System, or the pair (A,C) is Observable if, for each time tf > 0 the function y(t), t ϵ [t0,tf] uniquely determines the initial state x0. (1) (A,B) is Controllable (1) (C,A) is Observable (2) The matrix [B AB … An-1 B] has independent rows               −1n AC AC C  (2) The matrix has independent columns (3) The matrix [A – λI B ] has independent rows (This is so called P.B.H. test - Popov-Belevitch-Hautus test) C∈∀λ (3) The matrix has independent columns (This is so called P.B.H. test)       − C IA λ C∈∀λ (4) The eigenvalues of (A+BF) can be freely assigned by suitable choice of F (state feedback) (4) The eigenvalues of (A+HC) can be freely assigned by suitable choice of H (output injection feedback) (5) [BT ,AT ] is Observable (5) [AT ,CT ] is Controllable 37
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Un-Controllability: The modes λ of A for which [A – λI B ] loses rank are called uncontrollable modes. All other modes are controllable Un-Observability: (1) (A,B) is Stabilizable (1) (C,A) is Detectable (2) Exists F s.t. (A+BF) is stable (2) Exists H s.t. (A+HC) is stable (3) The matrix [A - λI B] has independent rows +∈∀ Cλ       − C IA λ The modes λ of A for which loses rank are called uncontrollable modes. All other modes are controllable (3) The matrix has independent columns +∈∀ Cλ       − C IA λ Stabilizability: Detectability: The Poles of the System are all the modes that are both Controllable and Observable. 38
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Transfer Function of a L.T.I. System: ( ) ( ) pxmnxmnxnnpxnpxm DBAIsCsG +−= −1 (A,B,C,D) is called a Realization of the L.T.I. System. The Realization is Minimal if n is the minimal possible degree. That happens if and only if (A,B) is Controllable and (C,A) is Observable. Doyle and Chu introduced the following notation: ( ) ( )           −−−−−−−=+−= − pxmpxn nxmnxn pxmnxmnxnnpxnpxm DC BA DBAIsCsG | | 1 39
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Operations on Linear Systems rnonsingula~ rnonsingula~ rnonsingula~ NzNz RyRy TxTx mxm pxp nxn = = = Suppose we make the following change of variables: ( ) N NDRTCR NBTTAT R DC BA sG pxm           −−−−−−−−−=           −−−= −− −− − 11 11 1 | | | | Change of Variables: Then:               =                           =                  =            =      −− −− − − u x NDTTCR NBTTAT u x N T DC BA R T u x DC BA R T y x R T y x ~ ~ ~ ~ 0 0 0 0 0 0 0 0 ~ ~ 11 11 1 1 40
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Operations on Linear Systems Usefull Notation: † Suppose we have a partition: ( ) ( ) ( ) INIR I TI T I TI T rn r rn r rnrx ==      − =⇒         = − − − − ,, 00 111 ( ) ( ) ( ) ( ) ( ) ( ) ( )               −−−−−−−−−−−−−− = − −−−− − pxmDCC BAA BAA sG rnpxpxdr xmrnrnxrnxrrn rxmrnrxrxr | | | 21 22221 11211 Then: Suppose we want to Change Variables according to:       = −rn r n x x x ( )             +− −−−−−−−−−−−−−−−−−−−−−−−−−−− − +−−++ =           −−−−−−−=           −−−= − − DCTCC BTAAA BTBTATTAATAATA DTC BTTAT DC BA sG pxm | | | | | | | 2111 21212221 21112111112211221111 1 1 ( ) ( ) 221 1211 1 1 ,, 0 columncolumnTcolumn rowrowTrow rn r INIR I TI T rnrx →+•− →•+ −  →⇔         ==         = − 41
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Operations on Linear Systems We have: Cascade of Two Linear Systems G1(s)G2(s): Then: uuyyyu u x DC BA y x u x DC BA y x ===             =                  =      2121 2 2 22 22 2 2 1 1 11 11 1 1                      =           1 2 1 21211 22 21211 1 2 1 0 u x x DDCDC BA DBCBA y x x                       =           1 1 2 21121 21121 22 1 1 2 0 u x x DDCCD DBACB BA y x x   or: From which: ( ) ( )             −−−−−−−−− =             −−−−−−−−− =           −−−−           −−−−= 21121 21121 22 21211 22 21211 22 22 11 11 21 | | |0 | |0 | | | | | DDCCD DBACB BA DDCDC BA DBCBA DC BA DC BA sGsG to ( ) ( ) 1− sDsN gg 42
43 SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Operations on Linear Systems Para-Hermitian: † ( )           −−−−−−−−−− −− = †† †† † | | | DCD BDCBDA sG mxp where is the pseudoinverse (not uniquely defined if p≠m) of D. † D ( ) ( ) mpxmmxp mpxmmxp IsGsG IDDmpif = =⇒> † † & ( ) ( ) ppxmmxp pmxppxm IsGsG IDDmpif = =⇒< † † & Pseodoinverse of G (s) for rank Dpxm=min (p,m) is G(s) is: † ( ) ( ) ( )           − −−−−−−− − =+−−=−= − TT TT TTT n T mxp T mxp H DB CA DCAIsBsGsG | | : 1 ( ) ( ) ppxppxp ppxppxp pxp IsGsG IDD DDmpif = =⇒ == − 1- 1- 1† & ( ) 1− sDgto
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Operations on Linear Systems Proof for p ≥ m ( )           −−−−−−−−−− −− = †† †† † | | DCD BDCBDA sG mxp Pseodoinverse of G (s) for rank Dpxm=min (p,m) is G(s) is: † ( ) ( )               −−−−−−−−−−−−− −−− =           −−−           −−−−−−−−−− −− = m pxmmxp ICDCD BA BCBDCBDA DC BA DCD BDCBDA sGsG | |0 | | | | | †† †† †† †† † Change of Variables INIR I II T I II T n nn n nn ==      − =⇒      = − ,, 00 1 ( ) ( ) ( ) ( ) mmnn m pxmmxp IIBAsICBDAsICD ICD BA CBDA sGsG =+−++−=               −−−−−−−−− − = −−      0 1 0 1†† † † † 00 |0 |0 0|0 Then: q.e.d. 44
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Operations on Linear Systems since is not uniquely defined if p≠m, is not uniquely defined. Moreover it is easy to check that ( ) ( ) ( )           −− −−−−−−−−−−−−−−−−−−− −−+− = † 1 †† † 1 †† † | | | DFDDICD BDFDDIBCBDA sG m m mxp † D Pseodoinverse of G(s) for rank Dpxm=min (p,m) is G(s) is: † ( )sG† ( ) ( ) ( )           −−−−−−−−−−−−−−−−−−−−−−−−−−−− −+−−+− = †† † 1 †† 1 † † | | | DCD DDIHBDCDDIHCBDA sG pp mxp and Are pseudoinverses of G (s), where and are any matrices of the given dimensions. mxn F1 nxp H1 45
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach State-Space Realization of All Coprime Matrices † We have any coprime-factorization of G(s): ( ) ( ) ( ) ( ) ( )sNsDsDsNsG gggg ~~ 11 −− == The corresponding generalized Bezout identities are: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       =         −        − =         −         − p m gg gg gg gg gg gg gg gg I I sDsN sYsX sXsN sYsD sXsN sYsD sDsN sYsX 0 0 ~~ ~~ ~~ ~~ 0 0 0 0 0 0 0 0 Nett, Jacobson and Balas gave the following State-Space Realizations of those Matrices: ( ) ( ) ( ) ( ) ( ) ( )               + −−−−−−−−−−−−− −+ =         − − − 1 1 0 0 | 0| | | WDZDFC ZF HWBZBFA sXsN sYsD gg gg ( ) ( ) ( ) ( ) ( ) ( )               − −−−−−−−−−−−−−− +−+ =         − −− WWDWC ZFZ HHDBHCA sDsN sYsX gg gg | 0| | | ~~ ~~ 11 0 0 ( ) ( ) ( ) ( ) ( )sNsDsDsNsG gggg ~~ 11 −− == where: Fmxn is any Matrix s.t. (A+BF) is stable for (A,B) Stabilizable Hnxp is any Matrix s.t. (A+HC) is stable for (C,A) Detectable s.t. Z and W are any Nonsingular Matrices ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∞∈RHsYsXsNsDsYsXsNsD gggggggg ~ , ~ , ~ , ~ ,,,, 46
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach State-Space Realization of All Coprime Matrices † ( ) ( ) ( ) ( ) ( )sNsDsDsNsG ggggpxm ~~ 11 −− == Nett, Jacobson and Balas gave the following State-Space Realizations of those Matrices: ( ) ( ) ( ) ( ) ( ) ( )               + −−−−−−−−−−−−− −+ =         − − − 1 1 0 0 | 0| | | WDZDFC ZF HWBZBFA sXsN sYsD gg gg ( ) ( ) ( ) ( ) ( ) ( )               − −−−−−−−−−−−−−− +−+ =         − −− WWDWC ZFZ HHDBHCA sDsN sYsX gg gg | 0| | | ~~ ~~ 11 0 0 Proof of (the same way for all others) ( ) ( ) ( ) 1− = sDsNsG ggpxm ( ) ( ) ( )           −−−−−− − =           −−−−−−−−−−−−−−−−− −−+ =           −−−−−−−− + = −−−− −−− − 1111 111 1 | | | | | | | | | ZFZ BA ZFZ ZBZFZBZBFA ZF BZBFA sDg To Pseudoinverse ( ) ∞∈           −−−−−−−− + = RH ZF BZBFA sDg | | | 47
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach State-Space Realization of All Coprime Matrices † ( ) ( ) ( ) ( ) ( )sNsDsDsNsG ggggpxm ~~ 11 −− == Proof of (continue - 1) ( ) ( ) ( ) 1− = sDsNsG ggpxm ( ) ( )             + −−−−−−−−−−− − + =           −−−−−− −           + −−−−−−−− + = −− − DDFDFC BA BBFBFA ZFZ BA DZDFC BZBFA sDsN gg | | |0 | | | | | | | 11 1 q.e.d. to ( ) ( )sGsG 21 ( )sG DC BA DFC BFA DC BA DCDFC BA BFA VariablesChange columncolumncolumn rowrowrow =           −−=           + −−−−− + +           −−=             −+ −−−−−−−−−−− − +  → →+− →+ | | | 0| | 0| | | | | | |0 0|0 0 221 121       48
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach State-Space Realization of All Coprime Matrices † ( ) ( ) ( ) ( ) ( )sNsDsDsNsG ggggpxm ~~ 11 −− == Nett, Jacobson and Balas gave the following State-Space Realizations of those Matrices: ( ) ( ) ( ) ( ) ( ) ( )               + −−−−−−−−−−−−− −+ =         − − − 1 1 0 0 | 0| | | WDZDFC ZF HWBZBFA sXsN sYsD gg gg ( ) ( ) ( ) ( ) ( ) ( )               − −−−−−−−−−−−−−− +−+ =         − −− WWDWC ZFZ HHDBHCA sDsN sYsX gg gg | 0| | | ~~ ~~ 11 0 0 ( ) ( ) ( )           −−−−−− − =           −−−−−−−−−−−−−−−−−− −−+ =           −−−−−−−− + = − − −− −−− − 1 1 11 111 1 | | | | | | | | | ~ WC WHA WCWW WHCWWHHCA WWC HHCA sDg ( ) ∞∈           −−−−−−−− + = RH WWC HHCA sDg | | | ~ Proof of (the same way for all others) ( ) ( ) ( )sNsDsG ggpxm ~~ 1− = 49
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach State-Space Realization of All Coprime Matrices † ( ) ( ) ( ) ( ) ( )sNsDsDsNsG ggggpxm ~~ 11 −− == ( ) ( ) ( )             −−−−−−−−−−−− ++ −− =           −−−−−−−−−− ++           −−−−−− − = − − − DCC HDBHCA DHHCA WDWC HDBHCA WC WHA sNsD see GG gg | | |0 | | | | | | | ~~ 21 1 1 1 Proof of (continue - 1) ( ) ( ) ( )sNsDsG ggpxm ~~ 1− = ( )sG DC BAHDBHCA DC BA DC HDBHCA BA VariablesChange columncolumncolumn rowrowrow =           −−=           −−−−−−−− ++ +           −−=             −−−−−−−−−−−− ++  → →+− →+ | | | 0|0 | | | | | |0 | |0 |0 0 221 121       q.e.d. 50
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Coprime Factorization Example † ( ) ( )0,0.,. 1 21 21 2 >>∈ ++ = ∞ aaeiRH asas sG The Observability Canonical State-Space Realization of G (s) (see Kailath, Linear Systems, pg. 41) is:                         −− =             u x x aa y x x         2 1 122 1 001 1 010 We choose [ ] 1, 0 ,0 ==      − =−= WZ h HfF ( ) ( ) ( ) ( )                 − −−−−−−−−−−−− −−− =         − 10|01 01|0 | 1| 00|10 12 0 0 f hafa sXsN sYsD gg gg ( ) ( ) ( ) ( )                 − −−−−−−−−−−−− −−−−− =         − 10|01 01|0 | 1| 00|10 ~~ ~~ 12 0 0 f haha sDsN sYsX gg gg 51
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Coprime Factorization Example (continue) † ( ) ( )0,0.,. 1 21 21 2 >>∈ ++ = ∞ aaeiRH asas sG By developing the previous Matrix representations we obtain ( ) ( ) ( ) ( ) fasas hf sY fasas hfasas sX fasas sN fasas asas sD g g g g +++ = +++ ++++ = +++ = +++ ++ = 21 20 21 2 21 2 0 21 2 21 2 21 2 1 ( ) ( ) ( ) ( ) hasas hf sY hasas hfasas sX hasas sN hasas asas sD g g g g +++ = +++ ++++ = +++ = +++ ++ = 21 20 21 2 21 2 0 21 2 21 2 21 2 ~ ~ 1~ ~ We can see that are in RH∞ for every f >-a2, and that are in RH∞ for every h >-a2. ( ) ( ) ( ) ( ),,,, sYsXsNsD gggg ( ) ( ) ( ) ( )sYsXsNsD gggg ~ , ~ , ~ , ~ 52
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator We have shown that ( ) ( ) ( ) ( ) ( ) 1 0 11 0 1 00 1 0 1 0 11 0 1 0 1 00 ~~~~ 00 −−−−−−−−−− +−=+−= gggpg K gggggpg K gg XQNXIQXYXXQNXIQXXYsK  This is equivalent to ( ) ff f sJ ggg ggg f yQe e e NXX XXY y y =                  −− =      −− −− 1 1 0 1 0 1 0 1 00 1 ~       53 We want to find the State-Space Realization of K0 (s) and J (s) ( )           + −−−−−−−−−−−−−−−−− −+++           −−−−−−− + =           + −−−−−−−−− −+           −−−−−−− + == − − pp gg IFDC HFDHCHFBA F HFBA IFDC HFBA F HFBA XYsK | | 0| | | | 0| | 1 1 000
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator 54 We want to find the State-Space Realization of K0 (s) ( )           + −−−−−−−−−−−−−−−−− +++           −−−−−−− + =           + −−−−−−−−− −+           −−−−−−− + == − − pp gg IFDC HFDHCHFBA F HFBA IFDC HFBA F HFBA XYsK | | 0| | | | 0| | 1 1 000 ( )           −−−−−−−−−−−−−−−− +++ +           −−−−−−−− + =             −−−−−−−−−−−−−−−−−−−−−− +++ + =  →             −−−−−−−−−−−−−−−−−−−−−− +++ ++ = →+ →− 0| | 0| 0| 0| |0 0|0 0|0 |0 | 0 221 121 F HFDHCHFBA F FBA FF HFDHCHFBA FBA F HFDHCHFBA HFDCHFBA VariablesChange columncolumncolumn rowrowrow      
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator 55 We want to find the State-Space Realization of K0 (s) ( )           −−−−−−−−−−−−−−−− +++ == − 0| | 1 000 F HFDHCHFBA XYsK gg K0 (s) given above is the transfer Matrix of the Estimator-Regulator Compensator (E.R.C.) , well known from the LQG Design Method, given in Figure bellow:
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator 56 We want to find the State-Space Realization of           + −−−−−−−−− −+           −+ −−−−−−−−−−−−−−−−− −+++ =− − pp gg IFDC HFBA IFDC HFDHCHFBA NX | | | | 1 0 ( )             −− = −− −− ggg ggg NXX XXY sJ 1 0 1 0 1 0 1 00 ~    ( ) ( ) ( ) ( )    VariablesChange columncolumncolumn rowrowrow pIFDCFDC BFBA DHFDCHFDCHFBA 221 121 | |0 | →+ →−  →               −+−+ −−−−−−−−−−−−−−−−−−−−−−−− + −+−+++ = ( ) ( )           −+− −−−−−−−−−−−−−−−−−− ++++ == DFDC DHBFDCHFBA | | 
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator 57 We want to find the State-Space Realization of ( )             −− = −− −− ggg ggg NXX XXY sJ 1 0 1 0 1 0 1 00 ~    ( ) ( ) ( )           −−−−−−−−−−−−−−−−−− ++++ =           −−−−−−−−−−−−− +−+ = − − mm g IF DHBFDHBCHA IF DHBCHA X | | | | ~ 1 1 0 Finally
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator 58 We want to find the State-Space Realization of ( )             −− = −− −− ggg ggg NXX XXY sJ 1 0 1 0 1 0 1 00 ~    ( ) ( )           −−−−−−−−−−−−−−−−−− ++++ = − m g IF DHBFDHBCHA X | | ~ 1 0 We found ( ) ( )           −+− −−−−−−−−−−−−−−−−−− ++++ =− − DFDC DHBFDCHFBA NX gg | | 1 0          −+ −−−−−−−−−−−−−−−−− −+++ =− − p g IFDC HFDHCHFBA X | | 1 0           −−−−−−−−−−−−−−−− +++ = − 0| | 1 00 F HFDHCHFBA XY gg Therefore ( ) ( )               −−+− −−−−−−−−−−−−−−−−−−−−−−− ++++ =             −− = −− −− DIFDC IF DHBHFDHCHFBA NXX XXY sJ p m ggg ggg | 0| |~ 1 0 1 0 1 0 1 00   
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator 59 We found ( ) ( )               −−+− −−−−−−−−−−−−−−−−−−−−−−− ++++ =             −− = −− −− DIFDC IF DHBHFDHCHFBA NXX XXY sJ p m ggg ggg | 0| |~ 1 0 1 0 1 0 1 00    ( )                     −−+− ++++ =           11 0 y e x DIFDC IF DHBHFDHCHFBA u u x e p m e  We can see that the realization of K (s) consists of Estimator Regulator Compensator plus the feedback Q (s).
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 60 The Eigenvalues of Closed-Loop System
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 61 The Eigenvalues of Closed-Loop System The State-Space Equation of the System are ( )sQ uDxCy uBxAx Estimator uDxCy uHuBxAx System uDxCy uBxAx q qq ee ee     += +=    += −+=    += += 11 1 1    ( ) ( ) ( ) ( ) rDxCxCDFxCDyxFu rxCxC ruDxCuDxCyyru qeeeqqe e ee −+−+=+= −−= −+−+=−−−= 1 1 ( ) ( )                             −−+ −− −−+++− −−+ =               r x x x DDCDCDFDCDDI BACBCB DBHCBCDBCHFBACDBCH DBCBCDBFBCDBA y x x x q e qqqqp qqqq qqqq qqqq q e    with
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 62 The Eigenvalues of Closed-Loop System ( ) ( )    VariablesChange columncolumncolumn rowrowrow qqqqp qqqq qqqq qqqq DDCDCDFDCDDI BACBCB DBHCBCDBCHFBACDBCH DBCBCDBFBCDBA 221 121 | | | | →+ →−  →                   −−+ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −− −−+++− −−+ ( ) ( ) ( ) ( ) ( ) ( )                     −++ −−−−−−−−−−−−−−−−−−−−−−−−− − −++− −+ = qqqp qqq qqq cl DDCDFDCCDDI BACB DBHCBFBACDBH HCHA sG | |0 | | |00     We can see that the Eigenvalues of the Closed Loop System are the Eigenvalues of (A+B F), (A+H C) and Aq. We can see the Separation between the Eigenvalues of the Regulator, Estimator and Q (s). The separation of Eigenvalues of Q (s) is due to the fact that the Input to Q (s) is from the Estimator Error, also Input to Estimator.
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 63 The Transfer Function of the Compensator K (s) with O.B.C. Realization We want to compute the Transfer Matrix of K (s) using the State-Space Representation euDxCu uDxCxFu uBxAx uHuBxAx e qqqe qq ee −−−= ++= += −+= 1 1 1 1   From the last two equations ( )[ ] ( )[ ] qpkkqqkeqk qmkqqqeqk DDIReRxCRDxCDFRDCu DDIReDxCxCDFRu +=−−−+−= +=−+−= −−− − : : 2211 11 111 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               −− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−+− −+−+++ = −−− −−− −−− 111 111 111 211 212 211 | | | kqqkqk kqqkqqkq kqqkqk RDCRCDFR RBCRDBAFDCRB RDBHCRDHBCDFRDHBCHA sK Therefore we obtain:
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 64 The Transfer Function of the Compensator K (s) with O.B.C. Realization ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               −− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−+− −+−+++ = −−− −−− −−− 111 111 111 211 212 211 | | | kqqkqk kqqkqqkq kqqkqk RDCRCDFR RBCRDBAFDCRB RDBHCRDHBCDFRDHBCHA sK We obtained: It is easy to prove that: ( ) ( ) ( ) ( ) ( ) ( )FDCRDBHFBA CDFRDHBCHA kq qk +−++= −+++ − − 1 1 2 1 Therefore a certain duality exists between (A+B F) and (A+H C), (B+H D) and (C+D F), (F-DqC) and (H-B Dq), respectively. The Realization of K (s) has n + nq States, where nq is number of states of Q (s). But this realization may not be minimal.
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 65 Realization of Q (s) Given G (s) and K (s) Suppose we have a Minimal State-Realization of a Compensator ( )           −−−−−−−= kk kk DC BA sK | | | We found that [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )             +++ −−−−−−−−−−−−−−−−−−− + + =             + −−−−−−−−−−−−−− + + +           −−−−−−−− + =           + −−−−−−− +           −−−−−−−+           −−−−−−−− + =+ DDIFDCDFC BBFA DBFDCBA DDFDCDC BBFA DBFDCBA IF BZBFA DDFC BBFA DC BA IF BZBFA NKD kmkk kkk kkk kkk m kk kk m gg | | |0 | | | |0 | | | | | | | | | | | | | [ ]    += +=               + −−−−−−−−−−−−−−−−−−−−−−− −− − =+ −−− −− −− − kp km kk kk kkkkk gg DDIR DDIR RCDRFCR BCDRBACRB DBCRBCRDBA NKD : : | | | | 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 ( ) ( )00 1 gggg XKYNKDQ −+= −
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 66 Realization of Q (s) Given G (s) and K (s) Suppose we have a Minimal State-Realization of a Compensator ( )           −−−−−−−= kk kk DC BA sK | | | We found that ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )             −++ −−−−−−−−−−−−−−−− + −+ =             −+ −−−−−−−−−−−−− + −+ −           −−−−−−− + =           + −−−−−−− +           −−−−−−−−           −−−−−−− + =− kkk kkk kkk kkk pkk kk gg DFDCDFC HFBA BFDCBA DFDCDC HFBA BFDCBA F HBFA IDFC HBFA DC BA F HBFA XKY | | |0 | | | |0 | 0| | | | | | | | | 0| | | 00 ( ) ( )00 1 gggg XKYNKDQ −+= −
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 67 Realization of Q (s) Given G (s) and K (s) Suppose we have a Minimal State-Realization of a Compensator ( )           −−−−−−−= kk kk DC BA sK | | | We found that ( ) ( ) ( ) ( )             −++ −−−−−−−−−−−−−−−− + −+               + −−−−−−−−−−−−−−−−−−−−−−− −− − =−+= −−− −− −− − kkk kkk kk kk kkkkk gggg DFDCDFC HFBA BFDCBA RCDRFCR BCDRBACRB DBCRBCRDBA XKYNKDQ | | |0 | | | | | 1 1 1 1 1 1 1 1 1 1 1 2 1 1 00 1 ( ) ( )00 1 gggg XKYNKDQ −+= − - Perform the Multiplication - Change Variables according to    VariablesChange columncolumncolumn rowrowrow 331 131 →+− →+  →    VariablesChange columncolumncolumn rowrowrow 442 242 →+− →+  →- Change Variables according to - Delete Unobservable Modes, to obtain ( ) ( )               + −−−−−−−−−−−−−−−−−−−−−−−−−−− +−− −− =−+= −−− −−− −−− − kkk kkk kkkkk gggg DRCDRFCR HDRBCDRBACRB RBCRBCRDBA XKYNKDQ 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 1 00 1 | | | |
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 68 Example Suppose we have the System G (s) and the Compensator K (s) ( )           −−−−−−== k ksK |0 | 0|0 ( ) ( )0,0.,. 0|01 1| 0|10 1 21 12 21 2 >>∈             −−−−−−−−− −− = ++ = ∞ aaeiRH aa asas sG ( ) ( ) k kasas hkfk kfk hkaka DRCDRFCR HDRBCDRBACRB RBCRBCRDBA Q kkk kkk kkkkk − +++ −− =             −− −−−−−−−−−−−−− −−−− =               + −−−−−−−−−−−−−−−−−−−−−−−−−−− +−− −− = −−− −−− −−− 21 2 12 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 1 |0 | 0|10 | | | | Note 1: The Poles of Q (s) are the Poles of the Closed-Loop System Note 2: The order of the System is n = 2 and the order of Q is nq = 2. A O.B.E. State-Space Realization of the Compensator K will have nk = n + nq = 4. But the minimum realization of K (s) is nk = 0. This shows that the realization of K (s) by O.B.E. is not minimal. Using the previous result we obtain:
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 69 Realization of Heu (s) Given G (s) and K (s) Suppose we have a Minimal State-Realization of a Compensator and of the System ( )           −−−−−−−= kk kk DC BA sK | | | ( )           − −−−−−− −  →           −−−−−−= =−= = DC BA DC BA sG mp n INIR IT | | | | | | We want to find a Realization of ( ) 1−       − = m p eu IK GI sH ( )                   −− −− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−− −− =                 −−−−−−−−−− − = −−−− −−−− −−−− −−−−− 1 1 1 2 1 1 1 1 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 1 1 | | | | | |0 0|0 | 0|0 0|0 RRDCDRCR RDRCRCDR RBHDRBCDRBACRB DRBRBCRBCRDBA IDC IC BA BA sH kkk k kkk kkkkkk mkk p kk eu         Note 1: We can see that Heu (s) and Q (s) have the same eigenvalues, therefore { } ( ){ } ( ){ }StablesQStablesHStableInternallySystem eu ⇔⇔    += += kp km DDIR DDIR : : 2 1
SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 70 Realization of Heu (s) Given G (s) and K (s) ( ) ( ) ( ) ( ) ( )                   −− −− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−− −− =         ++ +−+ = −−−− −−−− −−−− −−−− −− −− 1 1 1 2 1 1 1 1 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 1 11 11 | | | | | RRDCDRCR RDRCRCDR RBHDRBCDRBACRB DRBRBCRBCRDBA GKIKGIK GKIGKGI sH kkk k kkk kkkkkk mp mp eu     Note 2: If G (s) is stable, and therefore, if we choose for F and H the particular values F = 0 and H = 0, we obtain: ( ) ( ) ( )         − =      ∞− ∞ = −− −−− ∞→ 1 1 1 2 1 1 1 2 1 lim RRD RDR IK GI sH km p eu s   ( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 1 | | | | − −−− −−− −−− +−=               −−−−−−−−−−−−−−−−−−−−−−−−− −− −− = KGIK DRCDRCR DRBCDRBACRB RBCRBCRDBA Q p kkk kkk kkkkk That means that for a stable G (s) is enough to check the stability of K (Ip+G K)-1 or any other entry of Heu (s). Note 3:    += += kp km DDIR DDIR : : 2 1 ( ) ( ) ( )           += += ⇔                 ∞− ∞ ⇔       − Invertible DDIR DDIR Invertible IK GI PosedWell sH kp km m p eu : : 2 1
References SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach S. Hermelin, “Robustness and Sensitivity Design of Linear Time-Invariant Systems”, PhD Thesis, Stanford University, 1986 M. Vidyasagar, “Control System Synthesis: A Factorization Approach MIT Press, 1985”, 71 K. Zhou, J.C. Doyle, K. Glover, “Robust and Optimal Control”, Lecture Notes , 1993 B.A. Francis, “A Course in H∞ Control Theory”, Lecture Notes in Control and Information Sciences, vol .88, Springer-Verlag, 1987 K. Zhou, “Essential of Robust Control”, Pdf Slides on Homepage at Louisiana University, 2000 J.C. Doyle, B.A. Francis, A.R. Tannenbaum, “Feedback Control Theory”, Macmillan Publishing Company, 1992 A. Weinmann, “Uncertain Models and Robust Control”, Springer-Verlag, 1991 D.C.McFarlane, K. Glover, “Robust Controller Design Using Normalized Coprime Factor Plant Descriptions”, Lecture Notes in Control and Information Sciences, vol .138, Springer-Verlag, 1990
72 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
Jhon C. Doyle California Institute of Technology Kemin Zhou Louisiana State University Bruce A. Francis University of Toronto Mathukumalli Vidyasagar University of Texas Keith Glover University of Cambridge Pramod Khargonekar University of Florida 73

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Stabilization of linear time invariant systems, Factorization Approach

  • 1. Stabilization of Linear Time-Invariant Systems Factorization Approach SOLO HERMELIN Updated: 17.12.10 1
  • 2. Table of Content SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Introduction Well-Posedness Internal Stability Right and Left Coprime Factorization of a Transfer Matrix 2 Stabilization of Linear Time-Invariant Systems Stabilization of Linear Time-Invariant Systems - State-Space Approach Eigenvalues (modes) of the System Transfer Function of a L.T.I. System State Space Realization of the System Stability of the System Transmission Zeros of the L.T.I. System Controllability, Observability Stabilizability, Detectability Transfer Function of a L.T.I. System:
  • 3. Table of Content (continue) SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach 3 State-Space Realization of All Coprime Matrices Operations on Linear Systems Change of Variables Cascade of Two Linear Systems G1(s)G2(s) Para-Hermitian Pseodoinverse of G (s) for rank Dpxm=min (p,m) is G(s) † The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator The Eigenvalues of Closed-Loop System The Transfer Function of the Compensator K (s) with O.B.C. Realization Realization of Q (s) Given G (s) and K (s) Realization of Heu (s) Given G (s) and K (s) References
  • 4. • Assume a Linear Time-Invariant Plant, not necessary stable with m inputs and p outputs. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach • Assume that the plant can be represented by a set of n linear ordinary differential equations with constant coefficients (to assure time-invariance) or by the corresponding transfer matrix G (s)pxm. • Assume also that G (s) is proper: ( ) ∞< ∞→ sG s lim ( G(s) is analytic as s →∞ ) Therefore G (s) is in the Ring of Real-rational proper (Rp) matrices: ( ) pxm pRsG ∈ Using the fact that under those assumptions all G (s) can be factorized in two Real-rational-proper and stable (RH∞) matrices, we will obtain a parameterization of all compensators K (s)mxp that stabilize the given plant (and conversely all plants that can be stabilized by a given compensator). This parameterization will allow to define methods of optimizing the desired performances of the design over all possible stabilizing compensators. 4
  • 5. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Given the Feedback System, we have: ( ) ( )       =              − 2 1 2 1 u u e e IsK sGI mmxp pxmp from which: ( ) ( ) ( ) ( ) ( ) ( ) ( )               ++ +−+ =       =              − =      −− −− − 2 1 11 11 2 1 2 1 1 2 1 u u GKIKGIK GKIGKGI u u sH u u IsK sGI e e mp mp eu mmxp pxmp 5
  • 6. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Well-Posedness Definition: The System in Figure is well-posed if the transfer-matrix from u to e; i.e. Heu exists and is proper. ( ) ( ) ( )       =              − =      − 2 1 2 1 1 2 1 u u sH u u IsK sGI e e eu mmxp pxmp Well-Posed ( ) ( ) 1−         − mmxp pxmp IsK sGI exists and is proper ( ) ( ) 1−       ∞− ∞ m p IK GI is invertible ( ) ( )[ ]∞∞+ KGIp is invertible ( ) ( )[ ]∞∞+ GKIm is invertible 6
  • 7. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Internal Stability Internally Stable ( ) ( ) ( )mpmp eu HsH +×+ ∞∈ ( ) ( ) ( ) ( ) ( ) ( ) ( )               ++ +−+ =      =              − =      −− −−− 2 1 11 11 2 1 2 1 1 2 1 u u GKIKGIK GKIGKGI u u sH u u IsK sGI e e mp mp eu mmxp pxmp Definition: The System in Figure internally stable if it is well-posed and Heu (s)is analytic for all Real (s) ≥ 0 ( )+∈Cs If in addition Heu (s) is real-rational and proper we will write:( ) ( ) ( ) ( )mpmp peu RsH +×+ ∈ Internally Stable ( ) ( ) ( )mpmp eu RHsH +×+ ∞∈ and Internally Stability of a Real-Rational and Proper System ( ) ( ) ( ) ( ) ( ) ( ) ∞−− −− ∞ − ∈         ++ +−+ ⇔∈         − RH GKIKGIK GKIGKGI RH IsK sGI mp mp mmxp pxmp 11 111 7
  • 8. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Right and Left Coprime Factorization of a Transfer Matrix Right Coprime Factorization (r.c.f.) Left Coprime Factorization (l.c.f.) is a r.c.f. of G(s) pxm if( ) ( )( )mmgmpg sDsN ×× , is a l.c.f. of G(s) pxm if( ) ( )( )ppgmpg sDsN ×× ~ , ~ ( ) ( ) ( ) mmgmpgmp sDsNsG × − ×× = 1 ( ) ( ) ( ) mpgppgmp sNsDsG ×× − × = ~~ 1 ( ) ( ) mm g mp g RHsDRHsN × ∞ × ∞ ∈∈ & ( ) ( ) pp g mp g RHsDRHsN × ∞ × ∞ ∈∈ ~ & ~ ( ) ( ) pm g mm g RHsYRHsX × ∞ × ∞ ∈∈∃ ~ & ~ ( ) ( ) mp g pp g RHsYRHsX × ∞ × ∞ ∈∈∃ & 1 1 2 2 3 3 s.t.: s.t.: ( ) ( )[ ] ( ) ( ) m g g gg I sN sD sYsX =        ~~ ( ) ( )[ ] ( ) ( ) p g g gg I sY sX sNsD =        ~~ Bézout-Diophantine Identities 8
  • 9. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach In number theory, Bézout's identity or Bézout's lemma is a linear diophantine equation. It states that if a and b are nonzero integers with greatest common divisor d, then there exist integers x and y (called Bézout numbers or Bézout coefficients) such that Additionally, d is the least positive integer for which there are integer solutions x and y for the preceding equation. In mathematics, a Diophantine equation is an indeterminated polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface or more general object, and ask about the lattice points on it. Diophantus of Alexandria (cca 200 – 280) Diophantus, often known as the 'father of algebra', is best known for his Arithmetica, a work on the solution of algebraic equations and on the theory of numbers. However, essentially nothing is known of his life and there has been much debate regarding the date at which he lived. Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac. Étienne Bézout 1730 - 1783 dybxa =+ 9
  • 10. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Right and Left Coprime Factorization of a Transfer Matrix (continue – 1) ( ) ( )[ ] ( ) ( ) m g g gg I sN sD sYsX =        ~~ ( ) ( )[ ] ( ) ( ) p g g gg I sY sX sNsD =        ~~ Bézout-Diophantine Identities For a real-rational and proper transfer-matrix right and left coprime factorizations always exist (proof in Vidyasagar M., “Control System Synthesis: A Factorization Approach”, MIT Press, 1985) ( ) mp pRsG × ∈ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )         − =         −         − p ggggm gg gg gg gg I sYsXsXsYI sXsN sYsD sDsN sYsX 0 ~~ ~~ ~~ or Define: ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )sDsYsXsXsYsYsY sNsYsXsXsYsXsX ggggggg ggggggg ~~~~ : ~ ~~~~ : ~ 0 0 −+= −+= Pre-multiply by and redefine:( ) ( ) ( ) ( )( )         −− p ggggm I sYsXsXsYI 0 ~~ 4 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       =         −         − p m gg gg gg gg I I sXsN sYsD sDsN sYsX 0 0 ~~ ~~ 0 0 0 0 ( ) ( ) ( ) ( )sYsY sXsX gg gg = = : : 0 0 to obtain: Generalized Bézout Identity 10
  • 11. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Right and Left Coprime Factorization of a Transfer Matrix (continue – 2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       =         −         − p m gg gg gg gg I I sXsN sYsD sDsN sYsX 0 0 ~~ ~~ 0 0 0 0 we obtained: Generalized Bézout Identity We can see that: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )mpmp gg gg gg gg RH sXsN sYsD sDsN sYsX +×+ ∞ − ∈         − =         − 0 0 1 0 0 ~~ ~~ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )mpmp gg gg gg gg RH sDsN sYsX sXsN sYsD +×+ ∞ − ∈         − =         − 0 0 1 0 0 ~~ ~~ Definition: A square transfer-matrix s.t. is called unimodular, or a unit in the ring of stable real rational and proper transfer matrices. ( ) qqqq RHsU × ∞ × ∈ ( ) qq RHsU × ∞ − ∈1 11
  • 12. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Right and Left Coprime Factorization of a Transfer Matrix (continue – 3) Proof Proof ( ) ( ) ( ) 1 111 − −−− = == UDUN DUUNDNsG gg gggg Post-multiply by U:[ ] m g g gg I N D YX =        ~~ [ ] U UN UD YX g g gg =        ~~ Pre-multiply this by U-1 : [ ] m g g gg I UN UD YUXU =        −− ~~ 11 ( ) ( )( ) 1 111 ~~~~ ~~~~~~ − −−− = == gg gggg DUNU NUUDNDsG [ ] U UN UD YX g g gg =        ~~ [ ] p g g gg I UY UX NUDU =         − − 1 1 ~ ~ ~~~~ Pre-multiply by :[ ] p g g gg I Y X ND =        ~~ U ~ Post-multiply this by :1~− U Right Coprime Factorization (r.c.f.) Left Coprime Factorization (l.c.f.) 12
  • 13. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Right and Left Coprime Factorization of a Transfer Matrix (continue – 4) Theorem 1a: If is a r.c.f. of then is also a r.c.f. of G (s) for every U(s) mxm unimodular. ( )gg ND , ( )UNUD gg , ( ) mp pRsG × ∈ Theorem 1b: If is a l.c.f. of then is also a l.c.f. of G (s) for every unimodular. ( )gg ND ~ , ~ ( )gg NUDU ~~ , ~~ ( ) mp pRsG × ∈ Proof (continue – 1) Proof (continue – 1) [ ] m g g gg I UN UD YUXU =        −− ~~ 11 [ ] p g g gg I UY UX NUDU =         − − 1 1 ~ ~ ~~~~ ( ) ppsU × ~ Because ∞ − ∈ RHU 1 ∞ − ∞ − ∈= ∈= RHYUY RHXUX gg gg ~ : ~ ~ : ~ 1 1 1 1 and Because ∞ − ∈ RHU 1~ ∞ − ∞ − ∈= ∈= RHUYY RHUXX gg gg 1 1 1 1 ~ : ~ : and [ ] m g g gg I UN UD YX =         11 ~~ [ ] p g g gg I Y X NUDU =         1 1~~~~ Hence is an r.c.f of G (s).( )UNUD gg , Hence is an l.c.f of G (s). q.e.d. q.e.d. Right Coprime Factorization (r.c.f.) Left Coprime Factorization (l.c.f.) 13 ( )gg NUDU ~~ , ~~
  • 14. 14 SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems We can use the coprime-factorization to find necessary and sufficient conditions s.t. the Linear Time-Invariant System G(s), K(s) is internally stable. The definition of an internal stable system is: ( ) ( ) ( ) ( ) ( ) ∞−− −− ∈         ++ +−+ = RH GKIKGIK GKIGKGI sH mp mp eu 11 11 : Suppose we have any coprime-factorization of G(s) and K(s): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )mxpkmxmkpxpkmxpkmxp pxmgpxpgmxmgpxmgpxm sNsDsDsNsK sNsDsDsNsG ~~ ~~ 11 11 −− −− == == The corresponding generalized Bezout identities are: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       =         −        − =         −         − p m gg gg gg gg gg gg gg gg I I sDsN sYsX sXsN sYsD sXsN sYsD sDsN sYsX 0 0 ~~ ~~ ~~ ~~ 00 0 0 0 000 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       =      −         − =         −       − p m kk kk kk kk kk kk kk kk I I sXsN sYsD sDsN sYsX sDsN sYsX sXsN sYsD 0 0 ~~ ~~ ~~ ~~ 0 00000 0 0
  • 15. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems ( ) ( ) ( ) ( ) ( ) ∞−− −− ∈         ++ +−+ = RH GKIKGIK GKIGKGI sH mp mp eu 11 11 : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )mxpkmxmkpxpkmxpkmxp pxmgpxpgmxmgpxmgpxm sNsDsDsNsK sNsDsDsNsG ~~ ~~ 11 11 −− −− == == Use Matrix Inverse Lemma: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) gkgkgkgkkgkkg gkkggkkggggkkggp kgkgkgpkkggpp DNNDDDDDNNDDD DDNNDDNNDDDDNNDI NNNDDNINDDNIKGI ~~~~~~ ~~~~~~~~~ ~~~~~ 1111 111111111 11111 −−−− −−−−−−−−− −−−−− +=+= +=+=+= +−=+=+ ( ) ( ) mmmnnmmmmnnnnmmmmmmnnnnmmm CBDCBADCCBADC ×× − ××××××× − × − ×× − × +−=+ 1111 ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( )[ ] ( ) kgkgkgkgkgkgkk kgkgkgkmkkgkgkgpkk gkgkgkgkgkgkkkp NNNDDDNNNDDDDD NNNDDNNIDNNNDDNIND DNNDDNDNNDDDDNKGIK ~~~~~~~~ ~~~~~~~~~~ ~~~~~~ 111 1111 1111 −−− −−−− −−−− +=+= +−=+−= +=+=+ 15
  • 16. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems ( ) ( ) ( ) ( ) ( ) ∞−− −− ∈         ++ +−+ = RH GKIKGIK GKIGKGI sH mp mp eu 11 11 : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )mxpkmxmkpxpkmxpkmxp pxmgpxpgmxmgpxmgpxm sNsDsDsNsK sNsDsDsNsG ~~ ~~ 11 11 −− −− == == Use Matrix Inverse Lemma: ( ) ( ) mmmnnmmmmnnnnmmmmmmnnnnmmm CBDCBADCCBADC ×× − ××××××× − × − ×× − × +−=+ 1111 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) kgkgkgkggkggk kggkkggkkkkggkkm gkgkgkmggkkmp DNNDDDDDNNDDD DDNNDDNNDDDDNNDI NNNDDNINDDNIGKI ~~~~~~ ~~~~~~~~~ ~~~~~ 1111 111111111 11111 −−−− −−−−−−−−− −−−−− +=+= +=+=+= +−=+=+ ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( )[ ] ( ) gkgkgkgkgkgkgg gkgkgkgmggkgkgkmgg kgkgkgkgkgkgggm NNNDDDNNNDDDDD NNNDDNNIDNNNDDNIND DNNDDNDNNDDDDNGKIG ~~~~~~~~ ~~~~~~~~~~ ~~~~~~ 111 1111 1111 −−− −−−− −−−− +=+= +−=+−= +=+=+ 16
  • 17. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems We found for: ( ) gkkgpp DDNNIKGI ~~ 1 2 1 1 1 −−− ∆=∆−=+ ( ) gkkgp DNNDKGIK ~~ 1 2 1 1 1 −−− ∆=∆=+ ( ) gkmkgp NNIDDGKI ~~ 1 1 1 1 1 −−− ∆−=∆=+ ( ) gkkgm NDDNGKIG ~~ 1 2 1 1 1 −−− ∆=∆=+ [ ] [ ] ∞ ∞ ∈      =+=∆ ∈         =+=∆ RH N D NDNNDD RH N D NDNNDD g k kgkgkg g g kkgkgk ~~~~ : ~~~~ : 2 1       =         −        ∆ ∆         − − − p m kg kg kg kg I I DN ND DN ND 0 0 ~~ ~~ 0 0 1 2 1 1 Rearrange those Equations in Matrix Form       ∆ ∆ =         −         − 2 1 0 0 ~~ ~~ kg kg kg kg DN ND DN ND ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )sNsDsDsNsK sNsDsDsNsG kkkk gggg ~~ ~~ 11 11 −− −− == == Also we have: 17
  • 18. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems We found for: ( ) gkkgpp DDNNIKGI ~~ 1 2 1 1 1 −−− ∆=∆−=+ ( ) gkkgp DNNDKGIK ~~ 1 2 1 1 1 −−− ∆=∆=+ ( ) gkmkgp NNIDDGKI ~~ 1 1 1 1 1 −−− ∆−=∆=+ ( ) gkkgm NDDNGKIG ~~ 1 2 1 1 1 −−− ∆=∆=+ [ ] [ ] ∞ ∞ ∈      =+=∆ ∈         =+=∆ RH N D NDNNDD RH N D NDNNDD g k kgkgkg g g kkgkgk ~~~~ : ~~~~ : 2 1 From those Equations we can write: ( ) ( ) ( ) ( ) ( ) [ ] [ ]gg k k mgkmgk gkgk kk g gp kgkg kgkgp mp mp eu ND N D INNIDN NDDD DN D NI DDND DNNNI GKIKGIK GKIGKGI sH ~~ 0 00 ~~ ~~ ~~ 00 0 ~~ ~~ : 1 21 2 1 2 1 2 1 2 1 11 1 1 1 1 1 1 1 11 11 −∆      +      =         ∆−∆ ∆−∆ = ∆        − +      =         ∆∆ ∆−∆− =         ++ +−+ = − −− −− − −− −− −− −− 18
  • 19. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems We found for: [ ] [ ] ∞ ∞ ∈      =+=∆ ∈         =+=∆ RH N D NDNNDD RH N D NDNNDD g k kgkgkg g g kkgkgk ~~~~ : ~~~~ : 2 1 ( ) [ ] [ ]gg k k m kk g gp eu ND N D I DN D NI sH ~~ 0 00~~ 00 0 1 2 1 1 −∆      +      =∆        − +      = −− Theorem 2 The Necessary and Sufficient Conditions that Heu(s) is Stable, i.e. are( ) ∞∈ RHsHeu ∞ −− ∞ ∈∆∆∈∆∆⇔∆⇔∆ RHRH 1 2 1 12121 ,&,UnimodularUnimodular Proof Theorem 2 (1) If Δ1(s) (or Δ2(s) ) is Unimodular; i.e. Δ1(s)ϵRH∞ and Δ1(s)-1 ϵRH∞ , then from the Equation above we can se that Heu(s) ϵRH∞ 19
  • 20. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems We found for: Theorem 2 The Necessary and Sufficient Conditions that Heu(s) is Stable, i.e. are ( ) [ ] [ ]gg k k m kk g gp eu ND N D I DN D NI sH ~~ 0 00~~ 00 0 1 2 1 1 −∆      +      =∆        − +      = −− ( ) ∞∈ RHsHeu ∞ −− ∞ ∈∆∆∈∆∆⇔∆⇔∆ RHRH 1 2 1 12121 ,&,UnimodularUnimodular Proof Theorem 2 (continue) (2) Use and pre-multiply Heu(s) ϵRH∞ by and post-multiply it by ( ) ( )[ ] ( ) ( ) m g g gg I sD sN sXsY =        − − 00 ~~ ( ) ( )[ ] ( ) ( ) m k k kk I sX sY sDsN =      0 0~~ ( ) ( )[ ]sXsY gg 00 ~~ − ( ) ( )      sX sY k k 0 0 [ ] ( ) ( ) ∞∞ − ∈⇔∈         −            −=∆ RHsHRH Y X I sHYX eu g g m eukk 0 0 00 1 2 0 00~~ [ ] ( ) ( ) ∞∞ − ∈⇔∈                    −−=∆ RHsHRH X YI sHXY eu k kp eugg 0 0 00 1 1 00 0~~ Use and pre-multiply Heu(s) ϵRH∞ by and post-multiply it by [ ] p k k kk I N D YX =      00 ~~ [ ] p g g gg I Y X ND =         − − 0 0~~ [ ]00 ~~ kk YX         − 0 0 g g Y X q.e.d. 20
  • 21. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Corollary ( ) Unimodular~~ ~~ Unimodular         − ⇔         − ⇔∈ ∞ kg kg kg kg eu DN ND DN ND RHsH Proof       =         −        ∆ ∆         − − − p m kg kg kg kg I I DN ND DN ND 0 0 ~~ ~~ 0 0 1 2 1 1         −        ∆ ∆ =         − − −− kg kg kg kg DN ND DN ND ~~ ~~ 0 0 1 2 1 1 1         ∆ ∆         − =         − − − − 1 2 1 1 1 0 0 ~~ ~~ kg kg kg kg DN ND DN ND We found : Therefore: ( ) ( ) ( ) ( ) ∞ − ∞ − ∞ − ∞ − ∞ − ∞ − ∈         − ⇔∈∆∈∆ ∈         − ⇔∈∆∈∆ RH DN ND RHsRHs RH DN ND RHsRHs kg kg kg kg 1 1 1 1 1 1 1 1 1 1 ~~ ~~ & & q.e.d. 21
  • 22. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Theorem 3 The set of all Proper Linear Compensators K (s)mxp achieving Internal Stability is given by: ( ) ( )( ) ( ) ( ) ( ) ( ) 1 0 1 0 1 0 1 00 0 1 0 1 00 ~ ~~~~ −−−− −− +−= −+=+−= gggpggg gggggggg XQNXIQXXY DQYNQXQNXQDYsK where Q (s) ϵ RH∞ is a free parameter and 0 ~~ det&0det 00 ≠+≠+ gggg NQXQNX Proof Theorem 3 is given in three parts (1) Define ggkggk ggkggk DQYNQDYN NQXDQNXD ~~ : ~ : ~~ : ~ : 00 00 −=−= +=+= We want to prove that are coprime, and that they internally stabilize the System. ( ) ( )kkkk NDandND ~ , ~ , From the definition of K (s) above is clear that: ( ) ( ) ( ) ( ) ( )sNsDsDsNsK kkkk ~~ 11 −− == and: ( ) ( ) ( ) ( ) ∞∈ RHsNsDsNsD kkkk ~ , ~ ,, 22
  • 23. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 1) ggkggk ggkggg DQYNQDYN NQXDQNXD ~~ : ~ : ~~ : ~ : 00 00 −=−= +=+= (1) To prove that defined previously satisfy the Bezout identity start with ( ) ( )kkkk NDandND ~ , ~ ,       =         −         − p m gg gg gg gg I I XN YD DN YX 0 0 ~~ ~~ 0 0 0 00 Pre-multiply by and post-multiply by       − p m I QI 0       p m I QI 0 to obtain: ( ) ( )       =         + −−         − −+ p m ggg ggg gg gggg I I QNXN QDYD DN DQYNQX 0 0 ~~ ~~~~ 0 000 Which gives by above definition:       =         −         − p m kg kg gg kk I I DN ND DN ND 0 0 ~~ ~~ 0 Bezout Identity 23
  • 24. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 2) (1) To prove that K (s) stabilizes the System start from the Bezout Identity       =         −         − p m kg kg gg kk I I DN ND DN ND 0 0 ~~ ~~ 0 ∞ − ∈         − =         − RH DN ND DN ND kg kg gg kk 1 0 ~~ ~~ ∞ − ∈         − =         − RH DN ND DN ND gg kk kg kg 0 1 ~~ ~~ According to the Corllary ( ) ∞∞ − ∞ − ∈⇒∈         − ∈         − RHsHRH DN ND RH DN ND eu gg kk kg kg 1 0 1 ~~ ~~ & i.e., K (s) stabilizes the System. 24
  • 25. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 3) (2) We want to prove that if stabilizes the System then a Q (s) in RH∞ can be found s.t. K (s) will have the form kkkk NDDNK ~~ 11 −− == ( ) ( )( ) ( ) ( ) ( ) ( ) 1 0 1 0 1 0 1 00 0 1 0 1 00 ~ ~~~~ −−−− −− +−= −+=+−= gggpggg gggggggg XQNXIQXXY DQYNQXQNXQDYsK If K (s) stabilizes the System then Δ2 is Unimodular ∞ − ∞ ∈∆∈+=∆ RHandRHNNDD kgkg 1 22 ~~ : Let calculate ( ) ( ) ( ) ( ) ( ) 1 2 1 2 11 22 1 1 2 11 200 1 1 20 1 0 1 20 1 00 ~~~~ ~~~~~~ ~~ −−−−− −−−− −−−− ∆=∆=∆−∆= ∆−=∆−+= ∆−+=∆−+=+ kkggkgg kgmgkgggggg kggggkgggggg DDDDNND NNIDNNYNXDD NYNDXNYDNXQNX ( ) ( )( ) ( )( ) 1 00 11 2 1 2 −−−− +−=∆∆= QNXQDYDNsK ggggkk Therefore Let define Q (s) using the equation ⇒∆=− −1 20 kgg NQDY ( )1 20 1 : −− ∆−= kgg NYDQ 25
  • 26. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 4) (2) To complete the proof we must show that Q (s) that has been chosen is in RH∞. Pre-multiply by and use the fact that to obtain:[ ] m g g gg I N D YX =         00 [ ]00 gg YX [ ] ∞− − ∈         ∆+− ∆− = RH DX NY YXQ kg kg gg 1 20 1 20 00     ∆=+ ∆=− − − 1 20 1 20 kgg kgg DQNX NQDY The chosen Q (s) satisfies         ∆+− ∆− =         − − 1 20 1 20 kg kg g g DX NY Q N D In the same way we can define Q1 (s) using the equation: gkgk kgg NNDD RHNDQY ~~ : ~~~ 1 1 110 +=∆ ∈∆=− ∞ − ⇒ ( ) 11 101 ~~~ : −− ∆−= gkg DNYQ 26 ⇒
  • 27. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 5) (2) We defined Q1 (s) using the equation: gkgk kgg NNDD RHNDQY ~~ : ~~~ 1 1 110 +=∆ ∈∆=− ∞ − ( ) 11 101 ~~~ : −− ∆−= gkg DNYQ ( ) ( ) ( ) ( ) ( ) ∞ −−−−− −−−− −−−− ∈∆=∆=−∆∆= ∆−=∆−+= ∆−+=∆−+=+ RHDDDDDNN DNNIDNNNYDX DNNYXNDNYXNQX kggkggk ggkmggkgggg ggkggggkgggg ~~~ ~~~~ ~~~~~~~~~~ 1 1 11 1 1 1 1 1 11 1 11 100 11 100 11 10010 Therefore for the chosen Q1 (s) we obtain: (2) ( ) ( ) ( )kkkk NDNDsK ~~~~ 1 1 11 1 1 −−−− ∆∆== ( ) ( )gggg DQYNQX ~~~~ 10 1 10 −+= − We still have to prove that ( ) ( ) ∞∈= RHsQsQ1 ( ) ( ) ∞∈= RHsQsQ1 ( ) kgggggkgggg NDYDDDNYDDQD ~~~~~~ : ~ 1 10 11 101 −−− ∆−=∆−= Use the fact that gggg DYYD ~~ 00 = Bezout Id. ( ) gggkggkgggg DQDDNYDNDYDQD ~~~~~ 1 20 1 201 =∆−=∆−= −− gkkg DNND ~~ 1 2 1 1 −− ∆=∆and 27
  • 28. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 6) (3) The last part of the proof is to show that K (s) has the form ( ) ( ) ( ) 1 0 11 0 1 0 1 00 ~ −−−−− +−= gggpggg XQNXIQXXYsK Use the Inversion Matrix Lemma ( ) ( )( ) ( ) ( )[ ]1 0 11 0 1 0 1 00 1 00 −−−−−− +−−=+−= gggmggggggggg XQNXQINXXQDYQNXQDYsK ( ) ( ) ( ) ( ) ( ) 1 0 11 0 1 0 1 0 11 0 1 00 1 0 1 00 11 0 11 0 11 0 −     +− + −−−− + −−−−− −− −−−− +++−−= g QNXII QNXIpQ ggmgggg QNXIQ ggmggggggg XQNXQINXQDXQNXQINXYXQDXY ggpp ggpggp         ( ) ( )[ ] 1 0 11 0 1 0 11 0 1 00 1 0 1 00 −−−−−−−−− +−++−−= gggmpggggpggggggg XQNXIIQDXQNXIQNXYXQDXY ( ) ( )[ ] 1 0 11 0 1 0 11 0 1 00 1 0 1 00 −−−−−−−−− +−++−−= gggppggggmggggggg XQNXIIQDXQNXIQNXYXQDXY ( ) ( ) ( ) 1 0 11 0 1 00 1 00 −−−−− ++−= gggpgggggg XQNXIQNXYDXYsK 28 ( ) ( ) mmmnnmmmmnnnnmmmmmmnnnnmmm CBDCBADCCBADC ×× − ××××××× − × − ×× − × +−=+ 1111
  • 29. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 7) (3) We want to prove is to show that K (s) has the form ( ) ( ) ( ) 1 0 11 0 1 0 1 00 ~ −−−−− +−= gggpggg XQNXIQXXYsK ( ) ( ) ( ) 1 0 11 0 1 00 1 00 −−−−− ++−= gggpgggggg XQNXIQNXYDXYsK We found 1 000 1 0 1 000 ~~~~~ −−− +=+=⇒=+ ggggggggmgggg XYDNYXDXINYDX Therefore: ( ) ( ) ( ) 1 0 11 0 1 0 1 00 ~ −−−−− +−= gggpggg XQNXIQXXYsK Since 0 1 0 1 00 ~~ gggg YXXY −− = ( ) ( ) ( ) 1 0 11 0 1 00 1 0 ~~~ −−−−− +−= gggpggg XQNXIQXYXsK Uze Bezout Identity: 0 1 0 1 00 ~~ gggg YXXY −− = q.e.d. 29
  • 30. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Proof Theorem 3 (continue - 8) (3) We have shown that ( ) ( ) ( ) ( ) ( ) 1 0 11 0 1 00 1 0 1 0 11 0 1 0 1 00 ~~~~ −−−−−−−−−− +−=+−= gggpggggggpggg XQNXIQXYXXQNXIQXXYsK This is equivalent to ff f ggg ggg f yQe e e NXX XXY y y =                  −− =      −− −− 1 1 0 1 0 1 0 1 00 1 ~    To prove this, let develop ( ) 1 1 0 11 0 1 01 1 0 eXQNXIyyQNXeXy gggffgggf −−−−− +−=⇒−−= ( )[ ] ( ) 11 1 0 11 0 1 0 1 00 1 01 1 001 ~~ esKeXQNXIQXXYeXeXYy ggggggfggg =+−=+= −−−−−−− ( ) 1 1 0 11 0 eXQNXIQyQe gggff −−− +−==⇒ q.e.d.30
  • 31. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems Theorem 4 (Dual of Theorem 3) The set of all Proper Linear Time-Invariant Systems G (s) pxm stabilized by the Controller is given by: ( ) ( )( ) ( ) ( ) ( ) ( ) 1 0 11 0 1 0 1 00 0 1 0 1 00 ~ ~~~~ −−−−− −− +−= −+=+−= kkkpkkk kkkkkkkk XSNXISXXY DSYNSXSNXSDYsG where S (s) ϵ RH∞ is a free parameter and 0 ~~ det&0det 00 ≠+≠+ kkkk NSXSNX Proof Theorem 4 The duality of Theorem 4 to Theorem 3 is evident because by replacing in Theorem 3 g to k and Q (s) to S (s), K (s) to G (s) we obtain Theorem 4. Therefore the proof is similar, by performing the above mentioned replacement and interchanging between G (s) and K (s). 31 ( ) ( ) ( ) ( ) ( )sNsDsDsNsK kkkk ~~ 11 −− ==
  • 32. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 32 From we obtain ( ) ( )( ) ( ) ( )gggggggg DQYNQXQNXQDYsK ~~~~ 0 1 0 1 00 −+=+−= −− ( ) ( ) ( ) ( )KNDKXYQDQYKNQKX XKYNKDQQDYQNKXK gggggggg gggggggg +−=⇒−=+ −+=⇒−=+ − ~~~~~~~ 0000 00 1 00 ( ) ( ) ( ) ( )KNDKXYXKYNKDQ gggggggg +−=−+= − ~~~ 0000 1 From we obtain ( ) ( )( ) ( ) ( )kkkkkkkk DSYNSXSNXSDYsG ~~~~ 0 1 0 1 00 −+=+−= −− ( ) ( ) ( ) ( )GNDGXYSDSYGNSGX XGYNGDSSDYSNGXG kkkkkkkk kkkkkkkk +−=⇒−=+ −+=⇒−=+ − ~~~~~~~ 0000 00 1 00 ( ) ( ) ( ) ( )GNDGXYXGYNGDS kkkkkkkk +−=−+= − ~~~ 0000 1 Go back to Q (s) Realization
  • 33. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 33 Let calculate Δ1 (s) and Δ2 (s) for ( ) ( )( ) ( ) ( )   kkkk N gg D gg D gg N gg DQYNQXQNXQDYsK ~ 0 1 ~ 0 1 00 ~~~~ −+=+−= −− ( ) ( ) ( ) ( ) ( ) mgggg I gggg gggggggkgk INDDNQNYDX NDQYDNQXNNDDs m =−++= −++=+=∆      0 00 001 ~~~~ ~~~~~~ ( ) ( ) ( ) ( ) ( ) pgggg I gggg ggggggkgkg IQDNNDYNXD QDYNQNXDNNDDs p =−++= −++=+=∆      0 00 002 ~~~~ ~~~~ We can see that Δ1 (s) and Δ2 (s) are Unimodular and according to Theorem 2 Heu(s) is Stable for this choise of K (s).
  • 34. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 34 Let calculate Heu(s) for ( ) ( )( ) ( ) ( )   kkkk N gg D gg D gg N gg DQYNQXQNXQDYsK ~ 0 1 ~ 0 1 00 ~~~~ −+=+−= −− ( ) [ ] [ ]gg k k m kk g gp eu ND N D I DN D NI sH ~~ 0 00~~ 00 0 1 2 1 1 −∆      +      =∆        − +      = −− we obtained ( ) ( )[ ] ( ) ( ) [ ]gg gg gg m gggg g gp ND QDY QNX I NQXDQY D NI ~~ 0 00~~~~ 00 0 0 0 00 −         − + +      =+−        − +      = [ ] [ ] [ ] [ ] ∞∈−        − −−         +      = −        − −        − +      = RHNDQ D N ND Y X I NDQ D N XY D NI gg g g gg g g m gg g g gg g gp ~~~~ 0 00 ~~~~ 00 0 0 0 00
  • 35. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Transfer Function of a L.T.I. System: ( ) ( ) pxmnxmnxnnpxnpxm DBAIsCsG +−= −1 State Space Realization of the System: ( ) 00 xx u x DC BA y x pxmpxn nxmnxn =            =       Eigenvalues (modes) of the System: The Eigenvalues (modes) of the System are the collection of the n complex numbers λ such that: ( ) nAIsrank nxnn <− Stability of the System: The System is stable if ( ) ii ∀λReal 35
  • 36. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Transmission Zeros of the L.T.I. System: ( ) ( ) ( ) 0,000exp 0 >∀=≠≠= ttyhavewegandxsomeforandtzgtuFor The following definitions are equivalent: (1) The Transmission Zeros of the System are the collection of the complex numbers z s.t.: (2) The Transmission Zeros of the System are the collection of the complex numbers z s.t.: ( )pmn DC BAIz rank pxmpxn nxmnxnn ,min+<      −− 36
  • 37. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Controllability: The System, or the pair (A,B) is Controllable if, for each time tf > 0 and given state xf, there exists a continuous input u (t) for t ϵ [t0,tf] s.t. x(tf) = xf. Observability: The System, or the pair (A,C) is Observable if, for each time tf > 0 the function y(t), t ϵ [t0,tf] uniquely determines the initial state x0. (1) (A,B) is Controllable (1) (C,A) is Observable (2) The matrix [B AB … An-1 B] has independent rows               −1n AC AC C  (2) The matrix has independent columns (3) The matrix [A – λI B ] has independent rows (This is so called P.B.H. test - Popov-Belevitch-Hautus test) C∈∀λ (3) The matrix has independent columns (This is so called P.B.H. test)       − C IA λ C∈∀λ (4) The eigenvalues of (A+BF) can be freely assigned by suitable choice of F (state feedback) (4) The eigenvalues of (A+HC) can be freely assigned by suitable choice of H (output injection feedback) (5) [BT ,AT ] is Observable (5) [AT ,CT ] is Controllable 37
  • 38. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Un-Controllability: The modes λ of A for which [A – λI B ] loses rank are called uncontrollable modes. All other modes are controllable Un-Observability: (1) (A,B) is Stabilizable (1) (C,A) is Detectable (2) Exists F s.t. (A+BF) is stable (2) Exists H s.t. (A+HC) is stable (3) The matrix [A - λI B] has independent rows +∈∀ Cλ       − C IA λ The modes λ of A for which loses rank are called uncontrollable modes. All other modes are controllable (3) The matrix has independent columns +∈∀ Cλ       − C IA λ Stabilizability: Detectability: The Poles of the System are all the modes that are both Controllable and Observable. 38
  • 39. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Transfer Function of a L.T.I. System: ( ) ( ) pxmnxmnxnnpxnpxm DBAIsCsG +−= −1 (A,B,C,D) is called a Realization of the L.T.I. System. The Realization is Minimal if n is the minimal possible degree. That happens if and only if (A,B) is Controllable and (C,A) is Observable. Doyle and Chu introduced the following notation: ( ) ( )           −−−−−−−=+−= − pxmpxn nxmnxn pxmnxmnxnnpxnpxm DC BA DBAIsCsG | | 1 39
  • 40. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Operations on Linear Systems rnonsingula~ rnonsingula~ rnonsingula~ NzNz RyRy TxTx mxm pxp nxn = = = Suppose we make the following change of variables: ( ) N NDRTCR NBTTAT R DC BA sG pxm           −−−−−−−−−=           −−−= −− −− − 11 11 1 | | | | Change of Variables: Then:               =                           =                  =            =      −− −− − − u x NDTTCR NBTTAT u x N T DC BA R T u x DC BA R T y x R T y x ~ ~ ~ ~ 0 0 0 0 0 0 0 0 ~ ~ 11 11 1 1 40
  • 41. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Operations on Linear Systems Usefull Notation: † Suppose we have a partition: ( ) ( ) ( ) INIR I TI T I TI T rn r rn r rnrx ==      − =⇒         = − − − − ,, 00 111 ( ) ( ) ( ) ( ) ( ) ( ) ( )               −−−−−−−−−−−−−− = − −−−− − pxmDCC BAA BAA sG rnpxpxdr xmrnrnxrnxrrn rxmrnrxrxr | | | 21 22221 11211 Then: Suppose we want to Change Variables according to:       = −rn r n x x x ( )             +− −−−−−−−−−−−−−−−−−−−−−−−−−−− − +−−++ =           −−−−−−−=           −−−= − − DCTCC BTAAA BTBTATTAATAATA DTC BTTAT DC BA sG pxm | | | | | | | 2111 21212221 21112111112211221111 1 1 ( ) ( ) 221 1211 1 1 ,, 0 columncolumnTcolumn rowrowTrow rn r INIR I TI T rnrx →+•− →•+ −  →⇔         ==         = − 41
  • 42. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Operations on Linear Systems We have: Cascade of Two Linear Systems G1(s)G2(s): Then: uuyyyu u x DC BA y x u x DC BA y x ===             =                  =      2121 2 2 22 22 2 2 1 1 11 11 1 1                      =           1 2 1 21211 22 21211 1 2 1 0 u x x DDCDC BA DBCBA y x x                       =           1 1 2 21121 21121 22 1 1 2 0 u x x DDCCD DBACB BA y x x   or: From which: ( ) ( )             −−−−−−−−− =             −−−−−−−−− =           −−−−           −−−−= 21121 21121 22 21211 22 21211 22 22 11 11 21 | | |0 | |0 | | | | | DDCCD DBACB BA DDCDC BA DBCBA DC BA DC BA sGsG to ( ) ( ) 1− sDsN gg 42
  • 43. 43 SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Operations on Linear Systems Para-Hermitian: † ( )           −−−−−−−−−− −− = †† †† † | | | DCD BDCBDA sG mxp where is the pseudoinverse (not uniquely defined if p≠m) of D. † D ( ) ( ) mpxmmxp mpxmmxp IsGsG IDDmpif = =⇒> † † & ( ) ( ) ppxmmxp pmxppxm IsGsG IDDmpif = =⇒< † † & Pseodoinverse of G (s) for rank Dpxm=min (p,m) is G(s) is: † ( ) ( ) ( )           − −−−−−−− − =+−−=−= − TT TT TTT n T mxp T mxp H DB CA DCAIsBsGsG | | : 1 ( ) ( ) ppxppxp ppxppxp pxp IsGsG IDD DDmpif = =⇒ == − 1- 1- 1† & ( ) 1− sDgto
  • 44. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Operations on Linear Systems Proof for p ≥ m ( )           −−−−−−−−−− −− = †† †† † | | DCD BDCBDA sG mxp Pseodoinverse of G (s) for rank Dpxm=min (p,m) is G(s) is: † ( ) ( )               −−−−−−−−−−−−− −−− =           −−−           −−−−−−−−−− −− = m pxmmxp ICDCD BA BCBDCBDA DC BA DCD BDCBDA sGsG | |0 | | | | | †† †† †† †† † Change of Variables INIR I II T I II T n nn n nn ==      − =⇒      = − ,, 00 1 ( ) ( ) ( ) ( ) mmnn m pxmmxp IIBAsICBDAsICD ICD BA CBDA sGsG =+−++−=               −−−−−−−−− − = −−      0 1 0 1†† † † † 00 |0 |0 0|0 Then: q.e.d. 44
  • 45. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Operations on Linear Systems since is not uniquely defined if p≠m, is not uniquely defined. Moreover it is easy to check that ( ) ( ) ( )           −− −−−−−−−−−−−−−−−−−−− −−+− = † 1 †† † 1 †† † | | | DFDDICD BDFDDIBCBDA sG m m mxp † D Pseodoinverse of G(s) for rank Dpxm=min (p,m) is G(s) is: † ( )sG† ( ) ( ) ( )           −−−−−−−−−−−−−−−−−−−−−−−−−−−− −+−−+− = †† † 1 †† 1 † † | | | DCD DDIHBDCDDIHCBDA sG pp mxp and Are pseudoinverses of G (s), where and are any matrices of the given dimensions. mxn F1 nxp H1 45
  • 46. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach State-Space Realization of All Coprime Matrices † We have any coprime-factorization of G(s): ( ) ( ) ( ) ( ) ( )sNsDsDsNsG gggg ~~ 11 −− == The corresponding generalized Bezout identities are: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       =         −        − =         −         − p m gg gg gg gg gg gg gg gg I I sDsN sYsX sXsN sYsD sXsN sYsD sDsN sYsX 0 0 ~~ ~~ ~~ ~~ 0 0 0 0 0 0 0 0 Nett, Jacobson and Balas gave the following State-Space Realizations of those Matrices: ( ) ( ) ( ) ( ) ( ) ( )               + −−−−−−−−−−−−− −+ =         − − − 1 1 0 0 | 0| | | WDZDFC ZF HWBZBFA sXsN sYsD gg gg ( ) ( ) ( ) ( ) ( ) ( )               − −−−−−−−−−−−−−− +−+ =         − −− WWDWC ZFZ HHDBHCA sDsN sYsX gg gg | 0| | | ~~ ~~ 11 0 0 ( ) ( ) ( ) ( ) ( )sNsDsDsNsG gggg ~~ 11 −− == where: Fmxn is any Matrix s.t. (A+BF) is stable for (A,B) Stabilizable Hnxp is any Matrix s.t. (A+HC) is stable for (C,A) Detectable s.t. Z and W are any Nonsingular Matrices ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∞∈RHsYsXsNsDsYsXsNsD gggggggg ~ , ~ , ~ , ~ ,,,, 46
  • 47. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach State-Space Realization of All Coprime Matrices † ( ) ( ) ( ) ( ) ( )sNsDsDsNsG ggggpxm ~~ 11 −− == Nett, Jacobson and Balas gave the following State-Space Realizations of those Matrices: ( ) ( ) ( ) ( ) ( ) ( )               + −−−−−−−−−−−−− −+ =         − − − 1 1 0 0 | 0| | | WDZDFC ZF HWBZBFA sXsN sYsD gg gg ( ) ( ) ( ) ( ) ( ) ( )               − −−−−−−−−−−−−−− +−+ =         − −− WWDWC ZFZ HHDBHCA sDsN sYsX gg gg | 0| | | ~~ ~~ 11 0 0 Proof of (the same way for all others) ( ) ( ) ( ) 1− = sDsNsG ggpxm ( ) ( ) ( )           −−−−−− − =           −−−−−−−−−−−−−−−−− −−+ =           −−−−−−−− + = −−−− −−− − 1111 111 1 | | | | | | | | | ZFZ BA ZFZ ZBZFZBZBFA ZF BZBFA sDg To Pseudoinverse ( ) ∞∈           −−−−−−−− + = RH ZF BZBFA sDg | | | 47
  • 48. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach State-Space Realization of All Coprime Matrices † ( ) ( ) ( ) ( ) ( )sNsDsDsNsG ggggpxm ~~ 11 −− == Proof of (continue - 1) ( ) ( ) ( ) 1− = sDsNsG ggpxm ( ) ( )             + −−−−−−−−−−− − + =           −−−−−− −           + −−−−−−−− + = −− − DDFDFC BA BBFBFA ZFZ BA DZDFC BZBFA sDsN gg | | |0 | | | | | | | 11 1 q.e.d. to ( ) ( )sGsG 21 ( )sG DC BA DFC BFA DC BA DCDFC BA BFA VariablesChange columncolumncolumn rowrowrow =           −−=           + −−−−− + +           −−=             −+ −−−−−−−−−−− − +  → →+− →+ | | | 0| | 0| | | | | | |0 0|0 0 221 121       48
  • 49. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach State-Space Realization of All Coprime Matrices † ( ) ( ) ( ) ( ) ( )sNsDsDsNsG ggggpxm ~~ 11 −− == Nett, Jacobson and Balas gave the following State-Space Realizations of those Matrices: ( ) ( ) ( ) ( ) ( ) ( )               + −−−−−−−−−−−−− −+ =         − − − 1 1 0 0 | 0| | | WDZDFC ZF HWBZBFA sXsN sYsD gg gg ( ) ( ) ( ) ( ) ( ) ( )               − −−−−−−−−−−−−−− +−+ =         − −− WWDWC ZFZ HHDBHCA sDsN sYsX gg gg | 0| | | ~~ ~~ 11 0 0 ( ) ( ) ( )           −−−−−− − =           −−−−−−−−−−−−−−−−−− −−+ =           −−−−−−−− + = − − −− −−− − 1 1 11 111 1 | | | | | | | | | ~ WC WHA WCWW WHCWWHHCA WWC HHCA sDg ( ) ∞∈           −−−−−−−− + = RH WWC HHCA sDg | | | ~ Proof of (the same way for all others) ( ) ( ) ( )sNsDsG ggpxm ~~ 1− = 49
  • 50. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach State-Space Realization of All Coprime Matrices † ( ) ( ) ( ) ( ) ( )sNsDsDsNsG ggggpxm ~~ 11 −− == ( ) ( ) ( )             −−−−−−−−−−−− ++ −− =           −−−−−−−−−− ++           −−−−−− − = − − − DCC HDBHCA DHHCA WDWC HDBHCA WC WHA sNsD see GG gg | | |0 | | | | | | | ~~ 21 1 1 1 Proof of (continue - 1) ( ) ( ) ( )sNsDsG ggpxm ~~ 1− = ( )sG DC BAHDBHCA DC BA DC HDBHCA BA VariablesChange columncolumncolumn rowrowrow =           −−=           −−−−−−−− ++ +           −−=             −−−−−−−−−−−− ++  → →+− →+ | | | 0|0 | | | | | |0 | |0 |0 0 221 121       q.e.d. 50
  • 51. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Coprime Factorization Example † ( ) ( )0,0.,. 1 21 21 2 >>∈ ++ = ∞ aaeiRH asas sG The Observability Canonical State-Space Realization of G (s) (see Kailath, Linear Systems, pg. 41) is:                         −− =             u x x aa y x x         2 1 122 1 001 1 010 We choose [ ] 1, 0 ,0 ==      − =−= WZ h HfF ( ) ( ) ( ) ( )                 − −−−−−−−−−−−− −−− =         − 10|01 01|0 | 1| 00|10 12 0 0 f hafa sXsN sYsD gg gg ( ) ( ) ( ) ( )                 − −−−−−−−−−−−− −−−−− =         − 10|01 01|0 | 1| 00|10 ~~ ~~ 12 0 0 f haha sDsN sYsX gg gg 51
  • 52. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems - State-Space Approach Coprime Factorization Example (continue) † ( ) ( )0,0.,. 1 21 21 2 >>∈ ++ = ∞ aaeiRH asas sG By developing the previous Matrix representations we obtain ( ) ( ) ( ) ( ) fasas hf sY fasas hfasas sX fasas sN fasas asas sD g g g g +++ = +++ ++++ = +++ = +++ ++ = 21 20 21 2 21 2 0 21 2 21 2 21 2 1 ( ) ( ) ( ) ( ) hasas hf sY hasas hfasas sX hasas sN hasas asas sD g g g g +++ = +++ ++++ = +++ = +++ ++ = 21 20 21 2 21 2 0 21 2 21 2 21 2 ~ ~ 1~ ~ We can see that are in RH∞ for every f >-a2, and that are in RH∞ for every h >-a2. ( ) ( ) ( ) ( ),,,, sYsXsNsD gggg ( ) ( ) ( ) ( )sYsXsNsD gggg ~ , ~ , ~ , ~ 52
  • 53. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator We have shown that ( ) ( ) ( ) ( ) ( ) 1 0 11 0 1 00 1 0 1 0 11 0 1 0 1 00 ~~~~ 00 −−−−−−−−−− +−=+−= gggpg K gggggpg K gg XQNXIQXYXXQNXIQXXYsK  This is equivalent to ( ) ff f sJ ggg ggg f yQe e e NXX XXY y y =                  −− =      −− −− 1 1 0 1 0 1 0 1 00 1 ~       53 We want to find the State-Space Realization of K0 (s) and J (s) ( )           + −−−−−−−−−−−−−−−−− −+++           −−−−−−− + =           + −−−−−−−−− −+           −−−−−−− + == − − pp gg IFDC HFDHCHFBA F HFBA IFDC HFBA F HFBA XYsK | | 0| | | | 0| | 1 1 000
  • 54. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator 54 We want to find the State-Space Realization of K0 (s) ( )           + −−−−−−−−−−−−−−−−− +++           −−−−−−− + =           + −−−−−−−−− −+           −−−−−−− + == − − pp gg IFDC HFDHCHFBA F HFBA IFDC HFBA F HFBA XYsK | | 0| | | | 0| | 1 1 000 ( )           −−−−−−−−−−−−−−−− +++ +           −−−−−−−− + =             −−−−−−−−−−−−−−−−−−−−−− +++ + =  →             −−−−−−−−−−−−−−−−−−−−−− +++ ++ = →+ →− 0| | 0| 0| 0| |0 0|0 0|0 |0 | 0 221 121 F HFDHCHFBA F FBA FF HFDHCHFBA FBA F HFDHCHFBA HFDCHFBA VariablesChange columncolumncolumn rowrowrow      
  • 55. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator 55 We want to find the State-Space Realization of K0 (s) ( )           −−−−−−−−−−−−−−−− +++ == − 0| | 1 000 F HFDHCHFBA XYsK gg K0 (s) given above is the transfer Matrix of the Estimator-Regulator Compensator (E.R.C.) , well known from the LQG Design Method, given in Figure bellow:
  • 56. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator 56 We want to find the State-Space Realization of           + −−−−−−−−− −+           −+ −−−−−−−−−−−−−−−−− −+++ =− − pp gg IFDC HFBA IFDC HFDHCHFBA NX | | | | 1 0 ( )             −− = −− −− ggg ggg NXX XXY sJ 1 0 1 0 1 0 1 00 ~    ( ) ( ) ( ) ( )    VariablesChange columncolumncolumn rowrowrow pIFDCFDC BFBA DHFDCHFDCHFBA 221 121 | |0 | →+ →−  →               −+−+ −−−−−−−−−−−−−−−−−−−−−−−− + −+−+++ = ( ) ( )           −+− −−−−−−−−−−−−−−−−−− ++++ == DFDC DHBFDCHFBA | | 
  • 57. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator 57 We want to find the State-Space Realization of ( )             −− = −− −− ggg ggg NXX XXY sJ 1 0 1 0 1 0 1 00 ~    ( ) ( ) ( )           −−−−−−−−−−−−−−−−−− ++++ =           −−−−−−−−−−−−− +−+ = − − mm g IF DHBFDHBCHA IF DHBCHA X | | | | ~ 1 1 0 Finally
  • 58. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator 58 We want to find the State-Space Realization of ( )             −− = −− −− ggg ggg NXX XXY sJ 1 0 1 0 1 0 1 00 ~    ( ) ( )           −−−−−−−−−−−−−−−−−− ++++ = − m g IF DHBFDHBCHA X | | ~ 1 0 We found ( ) ( )           −+− −−−−−−−−−−−−−−−−−− ++++ =− − DFDC DHBFDCHFBA NX gg | | 1 0          −+ −−−−−−−−−−−−−−−−− −+++ =− − p g IFDC HFDHCHFBA X | | 1 0           −−−−−−−−−−−−−−−− +++ = − 0| | 1 00 F HFDHCHFBA XY gg Therefore ( ) ( )               −−+− −−−−−−−−−−−−−−−−−−−−−−− ++++ =             −− = −− −− DIFDC IF DHBHFDHCHFBA NXX XXY sJ p m ggg ggg | 0| |~ 1 0 1 0 1 0 1 00   
  • 59. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems The Equivalence Between Any Stabilizing Compensator and the Observer Based Compensator 59 We found ( ) ( )               −−+− −−−−−−−−−−−−−−−−−−−−−−− ++++ =             −− = −− −− DIFDC IF DHBHFDHCHFBA NXX XXY sJ p m ggg ggg | 0| |~ 1 0 1 0 1 0 1 00    ( )                     −−+− ++++ =           11 0 y e x DIFDC IF DHBHFDHCHFBA u u x e p m e  We can see that the realization of K (s) consists of Estimator Regulator Compensator plus the feedback Q (s).
  • 60. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 60 The Eigenvalues of Closed-Loop System
  • 61. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 61 The Eigenvalues of Closed-Loop System The State-Space Equation of the System are ( )sQ uDxCy uBxAx Estimator uDxCy uHuBxAx System uDxCy uBxAx q qq ee ee     += +=    += −+=    += += 11 1 1    ( ) ( ) ( ) ( ) rDxCxCDFxCDyxFu rxCxC ruDxCuDxCyyru qeeeqqe e ee −+−+=+= −−= −+−+=−−−= 1 1 ( ) ( )                             −−+ −− −−+++− −−+ =               r x x x DDCDCDFDCDDI BACBCB DBHCBCDBCHFBACDBCH DBCBCDBFBCDBA y x x x q e qqqqp qqqq qqqq qqqq q e    with
  • 62. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 62 The Eigenvalues of Closed-Loop System ( ) ( )    VariablesChange columncolumncolumn rowrowrow qqqqp qqqq qqqq qqqq DDCDCDFDCDDI BACBCB DBHCBCDBCHFBACDBCH DBCBCDBFBCDBA 221 121 | | | | →+ →−  →                   −−+ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −− −−+++− −−+ ( ) ( ) ( ) ( ) ( ) ( )                     −++ −−−−−−−−−−−−−−−−−−−−−−−−− − −++− −+ = qqqp qqq qqq cl DDCDFDCCDDI BACB DBHCBFBACDBH HCHA sG | |0 | | |00     We can see that the Eigenvalues of the Closed Loop System are the Eigenvalues of (A+B F), (A+H C) and Aq. We can see the Separation between the Eigenvalues of the Regulator, Estimator and Q (s). The separation of Eigenvalues of Q (s) is due to the fact that the Input to Q (s) is from the Estimator Error, also Input to Estimator.
  • 63. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 63 The Transfer Function of the Compensator K (s) with O.B.C. Realization We want to compute the Transfer Matrix of K (s) using the State-Space Representation euDxCu uDxCxFu uBxAx uHuBxAx e qqqe qq ee −−−= ++= += −+= 1 1 1 1   From the last two equations ( )[ ] ( )[ ] qpkkqqkeqk qmkqqqeqk DDIReRxCRDxCDFRDCu DDIReDxCxCDFRu +=−−−+−= +=−+−= −−− − : : 2211 11 111 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               −− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−+− −+−+++ = −−− −−− −−− 111 111 111 211 212 211 | | | kqqkqk kqqkqqkq kqqkqk RDCRCDFR RBCRDBAFDCRB RDBHCRDHBCDFRDHBCHA sK Therefore we obtain:
  • 64. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 64 The Transfer Function of the Compensator K (s) with O.B.C. Realization ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               −− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−+− −+−+++ = −−− −−− −−− 111 111 111 211 212 211 | | | kqqkqk kqqkqqkq kqqkqk RDCRCDFR RBCRDBAFDCRB RDBHCRDHBCDFRDHBCHA sK We obtained: It is easy to prove that: ( ) ( ) ( ) ( ) ( ) ( )FDCRDBHFBA CDFRDHBCHA kq qk +−++= −+++ − − 1 1 2 1 Therefore a certain duality exists between (A+B F) and (A+H C), (B+H D) and (C+D F), (F-DqC) and (H-B Dq), respectively. The Realization of K (s) has n + nq States, where nq is number of states of Q (s). But this realization may not be minimal.
  • 65. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 65 Realization of Q (s) Given G (s) and K (s) Suppose we have a Minimal State-Realization of a Compensator ( )           −−−−−−−= kk kk DC BA sK | | | We found that [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )             +++ −−−−−−−−−−−−−−−−−−− + + =             + −−−−−−−−−−−−−− + + +           −−−−−−−− + =           + −−−−−−− +           −−−−−−−+           −−−−−−−− + =+ DDIFDCDFC BBFA DBFDCBA DDFDCDC BBFA DBFDCBA IF BZBFA DDFC BBFA DC BA IF BZBFA NKD kmkk kkk kkk kkk m kk kk m gg | | |0 | | | |0 | | | | | | | | | | | | | [ ]    += +=               + −−−−−−−−−−−−−−−−−−−−−−− −− − =+ −−− −− −− − kp km kk kk kkkkk gg DDIR DDIR RCDRFCR BCDRBACRB DBCRBCRDBA NKD : : | | | | 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 ( ) ( )00 1 gggg XKYNKDQ −+= −
  • 66. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 66 Realization of Q (s) Given G (s) and K (s) Suppose we have a Minimal State-Realization of a Compensator ( )           −−−−−−−= kk kk DC BA sK | | | We found that ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )             −++ −−−−−−−−−−−−−−−− + −+ =             −+ −−−−−−−−−−−−− + −+ −           −−−−−−− + =           + −−−−−−− +           −−−−−−−−           −−−−−−− + =− kkk kkk kkk kkk pkk kk gg DFDCDFC HFBA BFDCBA DFDCDC HFBA BFDCBA F HBFA IDFC HBFA DC BA F HBFA XKY | | |0 | | | |0 | 0| | | | | | | | | 0| | | 00 ( ) ( )00 1 gggg XKYNKDQ −+= −
  • 67. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 67 Realization of Q (s) Given G (s) and K (s) Suppose we have a Minimal State-Realization of a Compensator ( )           −−−−−−−= kk kk DC BA sK | | | We found that ( ) ( ) ( ) ( )             −++ −−−−−−−−−−−−−−−− + −+               + −−−−−−−−−−−−−−−−−−−−−−− −− − =−+= −−− −− −− − kkk kkk kk kk kkkkk gggg DFDCDFC HFBA BFDCBA RCDRFCR BCDRBACRB DBCRBCRDBA XKYNKDQ | | |0 | | | | | 1 1 1 1 1 1 1 1 1 1 1 2 1 1 00 1 ( ) ( )00 1 gggg XKYNKDQ −+= − - Perform the Multiplication - Change Variables according to    VariablesChange columncolumncolumn rowrowrow 331 131 →+− →+  →    VariablesChange columncolumncolumn rowrowrow 442 242 →+− →+  →- Change Variables according to - Delete Unobservable Modes, to obtain ( ) ( )               + −−−−−−−−−−−−−−−−−−−−−−−−−−− +−− −− =−+= −−− −−− −−− − kkk kkk kkkkk gggg DRCDRFCR HDRBCDRBACRB RBCRBCRDBA XKYNKDQ 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 1 00 1 | | | |
  • 68. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 68 Example Suppose we have the System G (s) and the Compensator K (s) ( )           −−−−−−== k ksK |0 | 0|0 ( ) ( )0,0.,. 0|01 1| 0|10 1 21 12 21 2 >>∈             −−−−−−−−− −− = ++ = ∞ aaeiRH aa asas sG ( ) ( ) k kasas hkfk kfk hkaka DRCDRFCR HDRBCDRBACRB RBCRBCRDBA Q kkk kkk kkkkk − +++ −− =             −− −−−−−−−−−−−−− −−−− =               + −−−−−−−−−−−−−−−−−−−−−−−−−−− +−− −− = −−− −−− −−− 21 2 12 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 1 |0 | 0|10 | | | | Note 1: The Poles of Q (s) are the Poles of the Closed-Loop System Note 2: The order of the System is n = 2 and the order of Q is nq = 2. A O.B.E. State-Space Realization of the Compensator K will have nk = n + nq = 4. But the minimum realization of K (s) is nk = 0. This shows that the realization of K (s) by O.B.E. is not minimal. Using the previous result we obtain:
  • 69. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 69 Realization of Heu (s) Given G (s) and K (s) Suppose we have a Minimal State-Realization of a Compensator and of the System ( )           −−−−−−−= kk kk DC BA sK | | | ( )           − −−−−−− −  →           −−−−−−= =−= = DC BA DC BA sG mp n INIR IT | | | | | | We want to find a Realization of ( ) 1−       − = m p eu IK GI sH ( )                   −− −− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−− −− =                 −−−−−−−−−− − = −−−− −−−− −−−− −−−−− 1 1 1 2 1 1 1 1 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 1 1 | | | | | |0 0|0 | 0|0 0|0 RRDCDRCR RDRCRCDR RBHDRBCDRBACRB DRBRBCRBCRDBA IDC IC BA BA sH kkk k kkk kkkkkk mkk p kk eu         Note 1: We can see that Heu (s) and Q (s) have the same eigenvalues, therefore { } ( ){ } ( ){ }StablesQStablesHStableInternallySystem eu ⇔⇔    += += kp km DDIR DDIR : : 2 1
  • 70. SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach Stabilization of Linear Time-Invariant Systems 70 Realization of Heu (s) Given G (s) and K (s) ( ) ( ) ( ) ( ) ( )                   −− −− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−− −− =         ++ +−+ = −−−− −−−− −−−− −−−− −− −− 1 1 1 2 1 1 1 1 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 1 11 11 | | | | | RRDCDRCR RDRCRCDR RBHDRBCDRBACRB DRBRBCRBCRDBA GKIKGIK GKIGKGI sH kkk k kkk kkkkkk mp mp eu     Note 2: If G (s) is stable, and therefore, if we choose for F and H the particular values F = 0 and H = 0, we obtain: ( ) ( ) ( )         − =      ∞− ∞ = −− −−− ∞→ 1 1 1 2 1 1 1 2 1 lim RRD RDR IK GI sH km p eu s   ( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 1 | | | | − −−− −−− −−− +−=               −−−−−−−−−−−−−−−−−−−−−−−−− −− −− = KGIK DRCDRCR DRBCDRBACRB RBCRBCRDBA Q p kkk kkk kkkkk That means that for a stable G (s) is enough to check the stability of K (Ip+G K)-1 or any other entry of Heu (s). Note 3:    += += kp km DDIR DDIR : : 2 1 ( ) ( ) ( )           += += ⇔                 ∞− ∞ ⇔       − Invertible DDIR DDIR Invertible IK GI PosedWell sH kp km m p eu : : 2 1
  • 71. References SOLO Stabilization of Linear Time-Invariant Systems Factorization Approach S. Hermelin, “Robustness and Sensitivity Design of Linear Time-Invariant Systems”, PhD Thesis, Stanford University, 1986 M. Vidyasagar, “Control System Synthesis: A Factorization Approach MIT Press, 1985”, 71 K. Zhou, J.C. Doyle, K. Glover, “Robust and Optimal Control”, Lecture Notes , 1993 B.A. Francis, “A Course in H∞ Control Theory”, Lecture Notes in Control and Information Sciences, vol .88, Springer-Verlag, 1987 K. Zhou, “Essential of Robust Control”, Pdf Slides on Homepage at Louisiana University, 2000 J.C. Doyle, B.A. Francis, A.R. Tannenbaum, “Feedback Control Theory”, Macmillan Publishing Company, 1992 A. Weinmann, “Uncertain Models and Robust Control”, Springer-Verlag, 1991 D.C.McFarlane, K. Glover, “Robust Controller Design Using Normalized Coprime Factor Plant Descriptions”, Lecture Notes in Control and Information Sciences, vol .138, Springer-Verlag, 1990
  • 72. 72 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
  • 73. Jhon C. Doyle California Institute of Technology Kemin Zhou Louisiana State University Bruce A. Francis University of Toronto Mathukumalli Vidyasagar University of Texas Keith Glover University of Cambridge Pramod Khargonekar University of Florida 73

Editor's Notes

  • #10: http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity http://en.wikipedia.org/wiki/Diophantine_equation http://en.wikipedia.org/wiki/Diophantus
  • #38: Kailath, “Linear Syatems”, PBH Tests, pp. 135-136
  • #47: C.N.Nett, C.A. Jacobson, M.J. Balas,”A Connection Between State-Space and Doubly Coprime Fractional Representations”, IEEE Trans. On Automatic Control, Vol. AC-29, pp.831-832, Sep. 1984
  • #48: C.N.Nett, C.A. Jacobson, M.J. Balas,”A Connection Between State-Space and Doubly Coprime Fractional Representations”, IEEE Trans. On Automatic Control, Vol. AC-29, pp.831-832, Sep. 1984
  • #49: C.N.Nett, C.A. Jacobson, M.J. Balas,”A Connection Between State-Space and Doubly Coprime Fractional Representations”, IEEE Trans. On Automatic Control, Vol. AC-29, pp.831-832, Sep. 1984
  • #50: C.N.Nett, C.A. Jacobson, M.J. Balas,”A Connection Between State-Space and Doubly Coprime Fractional Representations”, IEEE Trans. On Automatic Control, Vol. AC-29, pp.831-832, Sep. 1984
  • #51: C.N.Nett, C.A. Jacobson, M.J. Balas,”A Connection Between State-Space and Doubly Coprime Fractional Representations”, IEEE Trans. On Automatic Control, Vol. AC-29, pp.831-832, Sep. 1984
  • #74: http://arnetminer.org/viewperson.do?naid=328332&amp;keyword=John%20C.%20Doyle