7.2.Balanced Realizations 103 4.Then G(s)=_ AU UB is a controllable realization. CU D A dual procedure can also be applied to eliminate non-observable subsystems. Now assume that A>0 is diagonal and is partitioned as A1=diag(A1,A12)such that Ama(A12)Amin(A1),then it is tempting to conclude that one can also discard those states corresponding to A12 without causing much error.However,this is not necessarily true as shown in the following example. Example 7.,Consider a stable transfer function G(s)= 3s+18 s2+3s+18 Then G(s)has a state space realization given by ,1 .4/a1 G 4a .2 2a .12/a0 where a is any nonzero number.It is easy to check that the controllability Gramian of the realization is given by 0.5 P= a2· Since the last diagonal term of P can be made arbitrarily small by making a small, the controllability of the corresponding state can be made arbitrarily weak.If the state corresponding to the last diagonal term of P is removed,we get a transfer function G- 11 which is not close to the original transfer function in any sense.The problem may be easily detected if one checks the observability Gramian Q,which is Q=0.5 1/a2· Since 1/a2is very large if a is small,this shows that the state corresponding to the last diagonal term is strongly observable. This example shows that controllability (or observability)Gramian alone can not give an accurate indication of the dominance of the system states in the input/output
Balanced Realizations Then G s U AU U B CU D is a controllable realization A dual procedure can also be applied to eliminate nonobservable subsystems Now assume that is diagonal and is partitioned as diag such that max min then it is tempting to conclude that one can also discard those states corresponding to without causing much error However this is not necessarily true as shown in the following example Example Consider a stable transfer function G s s s s Then G s has a state space realization given by G s where is any nonzero number It is easy to check that the controllability Gramian of the realization is given by P Since the last diagonal term of P can be made arbitrarily small by making small the controllability of the corresponding state can be made arbitrarily weak If the state corresponding to the last diagonal term of P is removed we get a transfer function G s which is not close to the original transfer function in any sense The problem may be easily detected if one checks the observability Gramian Q which is Q Since is very large if is small this shows that the state corresponding to the last diagonal term is strongly observable This example shows that controllability or observability Gramian alone can not give an accurate indication of the dominance of the system states in the inputoutput
104 BALANCED MODEL REDUCTION be havior.This motivates the int roduction of a balanced realization which gives balanced G ramians for controllabitity and observability. A Suppose G= is stable,i.e.,Ais stable.Let P and Q denote the control- lability Gramian and observability G ramian,respectively.Then by Lemma 7.1,P and Q satisfy the following Lyapunov equations AP+PA*+ EB:o (7.2) AQ+QA+C*C=0, (7.3) andP≥Q.Q≥Fthermo,the pair(A.B。>Qand(GNis observable iff Q>0. Suppose the state is transformed by a nonsingular T toi=Tz to yield the realiza- tion TAT-LTB CT D Then the Gramians are transformed to P=TPT*and Q=(T-D*QT-t Note that PQ TPOT-and therefore the eigenvalues of the product of the Gramians are invariant under state transformation. Consider the similarity transformation T which gives the eigenvector decomposition PQ=T-AAT A=diag(A Is1,...,ANIsN ) Then the columns of T are eigenvectors of PQ corresponding to the eigenvalues fAig. Later,it will be shown that PQ has a real diagonal Jordan form and that A>0,which are consequences of P≥0andQ≥0. Although the eigenvectors are not unique,in the case of a minimal realization they can always be chosen such that P=TPT=∑， Q=(T-0QT-=, where diag(o Is,Is2,...,NI)and >(=A.This new realization with con- trollability and observablity Gramians P=Q-will be referred to as a balanced realization (also called internally balanced realization).The decreasingly order num- bers,a >o>...>N >0,are called the Hankel singular values of the system. More generally,if a realization of a stable sy stem is not minimal,then there is a trans- formation such that the controllability and observability Gramians for the transformed realization are diagonal and the controllable and observable subsystem is balanced.This is a consequence of the following matrix fact
BALANCED MODEL REDUCTION behavior This motivates the introduction of a balanced realization which gives balanced Gramians for controllability and observability Suppose G A B C D is stable ie A is stable Let P and Q denote the control lability Gramian and observability Gramian respectively Then by Lemma P and Q satisfy the following Lyapunov equations AP P A BB AQ QA CC and P Q Furthermore the pair A B is controllable i P and C A is observable i Q Suppose the state is transformed by a nonsingular T to x T x to yield the realiza tion G A B C D T AT T B CT D Then the Gramians are transformed to P TPT and Q T QT Note that P Q T P QT and therefore the eigenvalues of the product of the Gramians are invariant under state transformation Consider the similarity transformation T which gives the eigenvector decomposition P Q T T diag Is N IsN Then the columns of T are eigenvectors of P Q corresponding to the eigenvalues fig Later it will be shown that P Q has a real diagonal Jordan form and that which are consequences of P and Q Although the eigenvectors are not unique in the case of a minimal realization they can always be chosen such that P TPT Q T QT where diag Is Is N IsN and This new realization with con trollability and observability Gramians P Q will be referred to as a balanced realization also called internally balanced realization The decreasingly order num bers N are called the Hankel singular values of the system More generally if a realization of a stable system is not minimal then there is a trans formation such that the controllability and observability Gramians for the transformed realization are diagonal and the controllable and observable subsystem is balanced This is a consequence of the following matrix fact
7a2a Bacice d Re dizctions 105 Thom75 La P and Q be postive semidefinite marices Then there erists a nonsingular matriz T such that TPT 0 respecvetyhdagonal and se definie Ploof.Since P is a positive semidefinite matrix,there exists a transformation Tsuch that Ir-[8] Now let "r"-[8黑8] and there exists a unitary matrix Uv such that -[8], %01 EyV0. Let and then % 0 Qwv (T2"I2"QI"T2"= 0 0 Qww But Q>0implies Qi=0.So now let 0 0 (T2"= 0 0 giving E%0 0 (T92"T2"(I2"QI%"T2"T)2"= 00 0 00 Qw2Qtv2“y」 Next find a unitary matrix Uw such that V(Q. /0
Balanced Realizations Theorem Let P and Q be two positive semidenite matrices Then there exists a nonsingular matrix T such that TPT T QT respectively with diagonal and positive denite Proof Since P is a positive semidenite matrix there exists a transformation T such that TP T I Now let T QT Q Q Q Q and there exists a unitary matrix U such that UQU Let T U I and then T T QT T Q Q Q Q Q But Q implies Q So now let T I I Q I giving T T T QT T T Q Q Q Next nd a unitary matrix U such that U Q Q Q U
102 LANCED MODEL REDUCTION Define m-81 0 0U6 and let T=TTIT5 Then PT= with 6=I. ∠ CThe product of tiveatritov semixde'nite matrix- Prf Let Pand Qbe any positive semi-definite matrices.Then it is easy to see that with the transformation given above 601 TPQT-005 ∠ CGry For any staHle system G= D <there exists a nonsingular T 6升罗 has controllalility Gramian P and observalility Gramian Q given by 1 respectively<with∑：l∑6l∑ff diagonal and positive de'nite- In the special case where B D is a minimal realization,a balanced realization can be obtained through the following simplified procedure: 1.Compute the controllability and observability Gramians P 01Q 0
BALANCED MODEL REDUCTION Dene T I U and let T TTTT Then TPT T QT with I Corollary The product of two positive semidenite matrices is similar to a positive semidenite matrix Proof Let P and Q be any positive semidenite matrices Then it is easy to see that with the transformation given above T P QT Corollary For any stable system G A B C D there exists a nonsingular T such that G T AT T B CT D has control lability Gramian P and observability Gramian Q given by P Q respectively with diagonal and positive denite In the special case where A B C D is a minimal realization a balanced realization can be obtained through the following simplied procedure Compute the controllability and observability Gramians P Q
7.2.Balanced Realizations 07 A Filn mati RslCtato RR 3 DigCQR:Q(tROR UxU- (i[器晋] 1 baaKCr A.hcttte.neate..eicceiacc.ct.t agT6ct8智 Ir vaslefor.15551oN are ( 9g6 caid.cnoc.色 Tf(rhaxCab OCtys品x TwC((CerGa r EGLC input normal realizig wTeP I qth1at1ee色6地 1o3 eccenialan ofe置e ile el taretcceit adbg(c:I.1o6Lo 1555aNLs)0 55,ow≥0.sICtat 少+8A+BB*,0A8+8A+©CO5 Tevctedointce. Theorem 7.1 ≤Ialw≤tt。 where g(。CB 通 战密桂器e格程整 rGbata.CemakEther((f%sIp
Balanced Realizations Find a matrix R such that P RR Diagonalize RQR to get RQR U U Let T RU Then TPT T QT and T AT T B CT D is balanced Assume that the Hankel singular values of the system is decreasingly ordered so that diag Is Is N IsN and N and suppose r r for some r then the balanced realization implies that those states corresponding to the singular values of rN are less controllable and observable than those states corresponding to r Therefore truncating those less controllable and observable states will not lose much information about the system Two other closely related realizations are called input normal realization with P I and Q and output normal realization with P and Q I Both realizations can be obtained easily from the balanced realization by a suitable scaling on the states Next we shall derive some simple and useful bounds for the H norm and the L norm of a stable system Suppose G s A B C RH is a balanced realization ie there exists diag Is Is N IsN with N such that A A BB A A CC Then we have the following theorem Theorem kGk Z kg tk dt X N i i where g t CeAtB Remark It should be clear that the inequalities stated in the theorem do not depend on a particular state space realization of G s However use of the balanced realization does make the proof simple