MEM633 Lectures 7&8 Chapter 4 Descrptons of MIMO Systems 4- Drect ealzatons y() s s su() s y () s u () s ( s)( s) s y() s u (), s y() s u() s s s y() s u(), s y() s u() s ( s)( s) s () ( s ) y ( s) u ( s) x x u y x () ( s ) y( s) u( s) x x u y x () ( s 3s) y ( s) u ( s) x 3 3 x3 u x 4 0 x 4 0 3 y 0 x x 4
(v) ( s ) y( s) u( s) x x u y 5 5 x 5 Combne (), (), (), and (v), we have x x 0 x x 0 u x 3 3 x3 0 u x 4 0 x 4 0 0 x x 0 5 5 x x 0 0 0 y x 3 0 0 0 x4 x 5
Block Controller Form 3 us ( ) Hs ( ) y( s) Ns Hs ( ) ( ) ds ( ) Ns ( ) s a polynomal matrx n s and ds ( ) s the least common multple of the denomnator polynomals n Hs. ( ) ewrte Hs ( ) as Hs ( ) Ns ( ) D( s ) where D ( s) d( s ). m Now, Defne then ys Ns D s us ( ) ( ) ( ) ( ) () s D () s u() s y() s N() s () s D () s () s u() s
Example: 4 u _ s ds ( ) s asa Ns ( ) Ns N a a u y N N -a m -a m. _ s N N y x am amx m u x m 0 x 0 y N N x x The dmenson of the state space s r m where r s the degree of ds ( ).
Block Observer Form 5 us ( ) Hs ( ) y( s) H() s Ns () ds () Ns ( ) s a polynomal matrx n s and ds ( ) s the least common multple of the denomnator polynomals n Hs. ( ) ewrte Hs ( ) as H s D s N s () L () () where D () s d() s. Now, Defne then L p ys D s Nsus () L() ()() () s D () s y() s L ys D s s () L() () () s Nsus ()()
Example: 6 Defne then ds ( ) s asa Ns ( ) NsN () s DL () sys () ( s as a) ys () y s as y a s y ( sn N ) u u N -a p _ s N -a p _ s y x a p px N u x a p 0 x N x y p 0 x The dmenson of the state space s r p where r s the degree of ds ( ).
Glbert Dagonal ealzatons 7 H() s Ns () ds () Suppose ds ( ) has dstnct roots,.e., r, j ds () ( s ) Then Hs ( ) can be wrtten as where Let H() s r s lm( s ) H( s) s rank and decompose as CB, C : p, B : m Then t s easy to verfy that A = block dag {,,,..., r } T T T T B B B.. B r, C C C C.. r s a realzaton of H () s. The dmenson of the state r space s.
Example: 8 s s H() s s s ( s)( s) s = = A = B = C = Dmenson of A = + = 3.
4- Observablty and Controllablty 9 Def: observablty The lnear tme-nvarant system x () t Ax() t Bu() t yt () Cxt () A: nxn, B: nxm, C: pxn s sad to be observable f the ntal state x(0 - ) can be unquely determned from measurements of the output y(t) over a fnte nterval of tme, 0 - t t f. Theorem: The n-dmensonal lnear tme-nvarant system x () t Ax() t Bu() t yt () Cxt () s observable f and only f the row vectors of the npxn observablty matrx O C CA : n CA span the n-dmensonal space,.e., rank O = n.
Def: controllablty 0 The lnear tme-nvarant system x () t Ax() t Bu() t s sad to be controllable f the state of the system can be transferred from the ntal zero state x(0 - ) = 0 to any fnal state x(t f ) n a fnte tme t f 0, by some control u(t). Theorem: The lnear tme-nvarant system x () t Ax() t Bu() t s controllable f and only f t can be transferred from any ntal state x(0 - ) to any fnal state x(t f ) n a fnte tme t f 0, by some control u(t). Theorem: The n-dmensonal lnear tme-nvarant system x () t Ax() t Bu() t s controllable f and only f the column vectors of the nxnm controllablty matrx C n B AB.. A B span the n-dmensonal space,.e., rank C = n.
The Controllablty Index and the Observablty Index Suppose that the controllablty matrx has rank n. Let C n B AB.. A B q C q B AB.. A B, q n Then the smallest q, say, such that C q has rank n s called the controllablty ndex of {A,B}. Suppose that the observablty matrx O C CA : n CA has rank n. Let O q C CA : q CA Then the smallest q, say, such that O q has rank n s called the observablty ndex of {C,A}.
Smlarty Transformaton Same as scalar case. Theorem: The transfer functon, characterstc polynomal, and ranks of the observablty and controllablty matrces are preserved under smlarty transformaton.
Noncontrollable Canoncal Form 3 Theorem: Let { ABC,, } be such that rank C ( A, B) = r < n Then there exsts a nonsngular T such that the realzaton has the form { A T AT, B T B, C CT } A A 0 A A, B B 0 C C C and the rxr subsystem A, B, C s controllable. Proof: Hnt: Choose T as T T T wth T f f f... r where f, f,..., f r are lnearly ndependent and span the nm columns of C ( AB, ). The columns of T together wth the columns of T span the n- dmensonal space.
Nonobservable Canoncal Form 4 Theorem: Let { ABC,, } be such that ranko ( C, A) = r < n Then there exsts a nonsngular T such that the realzaton { A T AT, B T B, C CT } has the form A 0 A A A, B B B C C 0 and the rxr subsystem A, B, C s observable. Proof: Hnt: Choose T such that U T U where the rows U of are lnearly ndependent and span the np rows of O ( CA, ). The rows of U together wth those of U span the n-dmensonal space.
Mnmal ealzatons 5 Def: mnmal realzaton A mnmal realzaton of a gven transfer functon Hs ( ) s one that has the smallest-sze A matrx for all trples {A,B,C} satsfyng Theorem: Cs ( A) BHs. ( ) A realzaton {A,B,C} s mnmal f and only f t s controllable and observable. Proof: "Necessty": If {A,B,C} s ether noncontrollable or nonobservable, then by the theorems n p.3 and 4, we could obtan another realzaton A, B, C wth the same transfer functon but a smaller number of states. Therefore {A,B,C} s not mnmal f t s noncontrollable or nonobservable. "Suffcency": To prove that a controllable and observable realzaton { ABC,, } s mnmal. Suppose t s not mnmal, and let { ABC,, } be another controllable and observable realzaton wth a lower state dmenson, n<n. Snce CA B CA B all we have
OC OC n n 6 From Sylvester s nequalty, we have ( O ) ( C )- n ( OC ) mn{ ( O), ( C )} Snce O and C have rank n, hence ( OC ) n But { ABC,, } s also controllable and observable, t must follow smlarly that ( OC ) n n n Thus, n n. It s a contradcton. Therefore a controllable and observable realzaton {A,B,C} must be mnmal. Theorem: Suppose that A, B, C,, are two mnmal realzatons of a transfer functon, then there exsts a unque nonsngular matrx T such that and A T AT, B T B, C CT T CC ( CC ) T T T T T ( OO) O O
Theorem: 7 The Glbert realzaton s mnmal. Proof: A block dag,,..., r B B B B r T T T T.. C C C C.. r The controllablty matrx can be wrtten as C = n B AB.. A B = B. V It s easy to see that ( V ) rm f j.
By Sylvester s nequalty 8 ( B ) ( V )- rm ( C ) mn{ ( B ), ( V )} we have ( C ) ( B ). By the Glbert constructon, r ( B ) ( B ) n r so that the realzaton s controllable. The observablty can be smlarly proved.
PBH Tests for Controllablty and Observablty 9 Theorem: PBH Egenvector Tests. { A, B } s noncontrollable f and only f there exsts a nonzero row vector q such that qa = q, and qb = 0.. { C, A } s nonobservable f and only f there exsts a nonzero column vector p such that Ap = p, and Cp = 0. Theorem: PBH ank Tests. { A, B } s controllable f and only f rank s A B n for all s. { C, A } s observable f and only f sa rank n C for all s Here, n s the sze of A.
4-3 Matrx -Fracton Descrptons 0 us ( ) Hs ( ) y( s) Ns Hs ( ) ( ) ds ( ), deg ds ( ) r Wrte Hs ( ) as a matrx fracton, Hs ( ) N ( s) D( s) Wth D( s) d( s) m, N( s) N( s) and defne the degree of the denomnator matrx as deg D ( s) deg det D ( s) rm then the order of the block controller realzaton would be equal to r m, the degree of D ( s ). Smlarly, f we wrte Hs D s N s ( ) L( ) L( ) Wth DL( s) d( s) p, NL( s) N( s) then the degree of the denomnator matrx s deg D ( s) deg det D ( s) rp L and the order of the block observer realzaton s equal to r p, the degree of DL ( s ). L
Clam A: Gven any rght MFD of Hs, ( ) Hs ( ) N ( s) D( s) we can always obtan a controllable state-space realzaton {A,B,C} of order Clam B: n = deg det D ( s ) := the degree of the MFD Gven any left MFD of Hs, ( ) Hs D s N s ( ) L( ) L( ) we can always obtan an observable state-space realzaton {A,B,C} of order n = deg det DL ( s ) := the degree of the MFD An MFD wth mnmal degree s sad to be a mnmal MFD (rreducble MFD). A realzaton of a mnmal MFD s a mnmal realzaton.
Example: s s Hs ( ) ( s)( s) s =
ght Dvsors and Irreducble MFDs 3 Gven an MFD Ds of ( ) Ns ( ) ( ) Hs. If we defne N s s, N ( s) ( ) ( ) D( s) D( s) ( s) where s ( ) s any nonsngular polynomal matrx, then Hs ( ) Ns ( ) Ds ( ) N( s) D( s).e., N s D s s also an MFD of ( ) ( ) ( ) Hs. Note that Ns ( ) N( ss ) ( ) Ds ( ) D( ss ) ( ), s ( ) s called a rght dvsor of Ns ( ) and Ds ( ). Snce and det Ds ( ) det D( s) det s ( ) degdet Ds ( ) deg det D( s) degdet s ( ) we have deg det Ds ( ) deg det D( s ) It s easy to see that the degree of the MFD can be reduced by removng rght common dvsors of the numerator and denomnator matrces. Therefore, we can get a mnmum-degree MFD by extractng a greatest common rght dvsor (gcrd) of Ns ( ) and Ds ( ).
The MFD and only f NsDs ( ) ( ) s mnmal (rreducble) f 4 deg det Ds ( ) = deg det D( s ) for all nonsngular rght dvsors s ( ) of Ns ( ) and Ds ( ). That s, det s ( ) = a nonzero constant ndependent of s A polynomal matrx whch satsfes the above equalty s called unmodular. Two polynomal matrces Ns ( ) and Ds ( ) wth the same number of columns s relatvely rght prme (rght coprme) f they only have unmodular common rght dvsors. And the MFD NsDs ( ) ( ) s rreducble (mnmal) f Ns ( ) and Ds ( ) are rght coprme. Irreducble MFDs are not unque.