62 CHAPTER 5.STABILIZATION yielding T2=(1+q192)m-92n. So we should take x=、更 1+92 T2 T2 that is, x(A)=-30λ+19，y()=5λ+1. Next is a procedure for doing a coprime factorization of G.The main idea is to transform variables,s→λ，so that polynomials inλyield funct ions in S. Procedure B Input:G Step 1 If G is stable,set N=G,M=1,X=0,Y 1,and stop;else,continue. Step 2 Transform G(s)to G(A)under the mapping s =(1-A)/A.Write G as a ratio of coprime poly nomials: G(A)=zQ) m() Step 3 Using Euclid's algorithm,find polynomials r(A),y(A)such that nx my =1. Step 4 Transform n(),m(),x(),y()to N(s),M(s),X(s),Y(s)under the mapping λ=1/(s+1). The mapping used in this procedure is not unique;the only requirement is that polynomialsn,and so on,map to N,and so on,in S. Example 4 For 1 G(8)=8-08-2 the algorit hm gives A2 G()= 6λ2-5λ+1’ n(A)=2, m(A)=6λ2-5λ+1， x()=-30λ+19， y()=5+1 (from Example 3), 1 N(s)= (8+1)2
CHAPTER STABILIZATION yielding r qq m qn So we should take x q r y qq r that is x y Next is a procedure for doing a coprime factorization of G The main idea is to transform variables s so that polynomials in yield functions in S Procedure B Input G Step If G is stable set N G M X Y and stop else continue Step Transform Gs to G under the mapping s Write G as a ratio of coprime polynomials G n m Step Using Euclids algorithm nd polynomials x y such that nx my Step Transform n m x y to Ns Ms Xs Y s under the mapping s The mapping used in this procedure is not unique the only requirement is that polynomials n and so on map to N and so on in S Example For Gs s s the algorithm gives G n m x y from Example Ns s
,.X COPRIME FACTORIZATION BY STATE-SPACE METHODS (OPTIONAL) 63 M(s)= (8-1)(8-22 (8+1)2 X(s)= 19s-11 8+1 Y()= s+6 s+1 513 Coprime Factorization by State4space Methods (Optional) This optional section presents a state space procedure for computing a coprime factorizat ion over S of a proper G.This procedure is more efficient than the poly nomial method in the preceding sect ion. We start with a new data structure.Suppose that A,B,C,D are real matrices of dimensions n×n2n×121×n21×1. The transfer funct ion going along with this quartet is D+C(sI-A)B. Note that the const ant D equals the value of the transfer funct ion at s=oo;if the transfer funct ion is strictly proper,then D=0.It is convenient to write AB instead of D+C(sI-A)B. Beginning with a realizat ion of G, G(s)= A B LGD the goal is to get state space realizations for four functions N,M,X,Y,all in S,such that G-- N M2 NX+MY =1 First,we look at how to get N and M.If the input and output of G are denoted u and y, respectively,then the state model of G is 元=Ax+Bu2 (.2) y=Cx+Du. (.3) Choose a real matrix F,1x n,such that A+BF is stable (i.e.,all eigenvalues in Res <0).Now define the signal v:=u-Fr.Then from (..2)and (..3)we get 元=(A+BF)x+B2 u=Fx+v2 =(C+DF)x+Dv
COPRIME FACTORIZATION BY STATESPACE METHODS OPTIONAL Ms s s s Xs s s Y s s s Coprime Factorization by StateSpace Methods Optional This optional section presents a statespace procedure for computing a coprime factorization over S of a proper G This procedure is more ecient than the polynomial method in the preceding section We start with a new data structure Suppose that A B C D are real matrices of dimensions n n n n The transfer function going along with this quartet is D CsI A B Note that the constant D equals the value of the transfer function at s if the transfer function is strictly proper then D It is convenient to write A B C D instead of D CsI A B Beginning with a realization of G Gs A B C D the goal is to get statespace realizations for four functions N M X Y all in S such that G N M NX M Y First we look at how to get N and M If the input and output of G are denoted u and y respectively then the state model of G is x Ax Bu y Cx Du Choose a real matrix F n such that A BF is stable ie all eigenvalues in Res Now dene the signal v u F x Then from and we get x A BF x Bv u F x v y C DF x Dv
64 CHAPTER 5.STABILIZATION Evidently from these equations,the transfer function from v to u is (5.4) and that from v to y is N(s):= A+BFB C+DF D (5.5) Therefore, u=Mv,y=Nv, so that y=NM-u,that is,G=N/M.Clearly,N and M are proper,and they are stable because A+BF is.Thus N,ME S.Suggestion:Test the formulas above for the simplest case,G(s)=1/s (A=0B=1,C=1,D=0). The theory behind the formulas for X and Y is beyond the scope of this book.The procedure is to choose a real matrix H,n x 1,so that A+HC is stable,and then set X(s):= 「A+HC|H] (5.6) Y(s):= A+HC-B-HD F 1 (5.7) In summary,the procedure to do a coprime factorization of G is this: Step 1 Get a realization (A,B,C,D)of G. Step 2 Compute matrices F and H so that A+BF and A+HC are stable. Step 3 Using formulas(5.4)to (5.7),compute the four functions N,M,X,Y. 5.4 Controller Parametrization:General Plant The transfer function P is no longer assumed to be stable.Let P=N/M be a coprime factorization over S and let X,Y be two functions in S sat isfying the equation NX+MY =1. (5.8) Theorem 2 The set of all Cs for which the feedback system is internally stable equals ∫X+MQ Y-NQ It is useful to note that Theorem 2 reduces to Theorem 1 when P ES.To see this,recall from Section 5.2(Step 1 of Procedure B)that we can take N=P,M=1,X=0,Y=1 when P∈S.Then X+MQ Q Y-NQ=1-PQ The proof of Theorem 2 requires a preliminary result
CHAPTER STABILIZATION Evidently from these equations the transfer function from v to u is Ms A BF B F and that from v to y is Ns A BF B C DF D Therefore u Mv y Nv so that y NMu that is G NM Clearly N and M are proper and they are stable because A BF is Thus NM S Suggestion Test the formulas above for the simplest case Gs s A B C D The theory behind the formulas for X and Y is beyond the scope of this book The procedure is to choose a real matrix H n so that A HC is stable and then set Xs A HC H F Y s A HC B HD F In summary the procedure to do a coprime factorization of G is this Step Get a realization A B C D of G Step Compute matrices F and H so that A BF and A HC are stable Step Using formulas to compute the four functions N M X Y Controller Parametrization General Plant The transfer function P is no longer assumed to be stable Let P NM be a coprime factorization over S and let X Y be two functions in S satisfying the equation NX M Y Theorem The set of al l Cs for which the feedback system is internal ly stable equals X MQ Y NQ Q S It is useful to note that Theorem reduces to Theorem when P S To see this recall from Section Step of Procedure B that we can take N P M X Y when P S Then X MQ Y NQ Q P Q The proof of Theorem requires a preliminary result