3 Stabilization of MIMO Feedback Systems3.1 NotationThe sets R and S ar e as before. We wil l use the notation M (R) to denotethe set of matrices with elements in R. The di mensions are not explicitlyspecified, and will be clear from context. Similarly, M (S) will denote theset of matrices with elements in S. In this set o f notes, URwill denote squarematrices with elements in R, whose inverse is also in M (R). Likewise, USwill denote the units in M (S).3.2 Coprime FactorizationsDefinition 28 Suppose N ∈ Sny×nu, D ∈ Snu×nu. The pair (N, D) is calledright coprime over S if there exist matrices U ∈ Snu×ny, V ∈ Snu×nusuchthatUN + V D = Inu.Definition 29 Suppose N ∈ Sny×nu, D ∈ Sny×ny. The pair (N, D) is calledleft coprime over S if there exist matrices U ∈ Snu×ny, V ∈ Sny×nysuchthatNU + DV = Iny.Definition 30 Suppose N ∈ Sny×nu, D ∈ Snu×nuare given, and the pair(N, D) is right coprime over S. If det [D(∞)] 6= 0, then (N, D) is called aright-coprime factorization (over S) of G := ND−1∈ Rny×nu.Definition 31 Suppose N ∈ Sny×nu, D ∈ Sny×nyare given, and the pair(N, D) is left coprime over S. If det [D(∞)] 6= 0, then (N, D) is called aleft-coprime factorization (over S) of G := D−1N ∈ Rny×nu.Theorem 32 Suppose that the pair (N, D) is a right coprime factorizationof G ∈ Rny×nu, so N ∈ Sny×nu, and D ∈ Snu×nu. If U ∈ Snu×nuis a unit (ie.43U−1∈ M (S)), then the pair (NU, DU) is a right coprime factorization ofG. Morover, if (N1, D1) and (N2, D2) are both right coprime factorizationsof G, then there exists a unit U ∈ U (S) such that N1= N2U, D1= D2U.Theorem 33 There is a result similar to Theorem 32 for left coprime fac-torizations.Lemma 34 Every G ∈ Rny×nuhas both a right and left coprime factoriza-tion over S.Proof: Let A, B, C, D be a stabilizable and detectable realizat ion of G(s).Choose matrices F and L so that A + BF and A + LC have all of theireigenvalues in the open-left-half-plane. Define the following transferfunctionsDrUl−NrVl=I 0−D I+F−(C + DF )(sI − A − BF )−1B LIn shorthand notation, this is written a sDrUl−NrVl=A + BFB LF I 0−(C + DF ) −D ISimilarly, defineVr−UlNlDl:=A + LCB + LD L−F I 0CD ICertainly, all of the transfer functions are stable, since the ma tricesA + BF a nd A + LC have all of their eigenvalues in the open-left-half-plane. It is left as an exercise to show that1. det [Dl(∞)] 6= 02. det [Dr(∞)] 6= 03. NrD−1r= D−1lNl= G444.Vr−UrNlDlDrUl−NrVl=I 00 I3.3 Multivariable Feedback SystemsConsider the sta ndard feedback struct ure shown below, wher e C, P ∈ M (R).C P-g- -g?- -6u1e1e2u2−Assume that det [I + P (∞)C(∞)] 6= 0. This insures that (I + P C) ∈ U (R),as well as (I + CP ) ∈ U (R), and is equivalent to having all closed-looptransfer functions in M (R). Note t hat if P (∞) = 0ny×nu, then this isautomatically satisfied.Define Heuto be the transfer matrix from the inputs u to the signals labelede. Writing lo op equations gi vesHeu=Iny+ P C−1−P (Inu+ CP )−1C (Inu+ CP )−1(Inu+ CP )−1Now, since all entries are matr ices, we must be careful about the order inwhich P and C appear.Let (Npr, Dpr) be any right-coprime factorization of P , and (Ncl, Dcl) anyleft-coprime factorization of C. DefineX := DclDpr+ NclNpr∈ Snu×nuNote thatdetIny+ P C= det (Inu+ CP )= detI + D−1clNclNprD−1pr= detD−1cldet(X) detD−1pr45Hence, since det (Dcl(∞)) and det (Dpr(∞)) are both finite, and nonzero, wehavedet (I + P (∞ )C(∞)) 6= 0 ⇔ det (X(∞)) 6= 0.orI + P C ∈ U (R) ⇔ U (R) .Under this condition, all of the closed-loop transfer functions are in M (R),and can be calculated in terms of t he coprime factorizations.(I + P C)−1= I − P (I + CP )−1C= I − NprD−1prI + D−1clNclNprD−1pr−1D−1clNcl= I − NprD−1prD−1cl[DclDpr+ NclNpr] D−1pr−1D−1clNcl= I − Npr(DclDpr+ NclNpr)−1Ncl= I − NprX−1Ncl(I + CP )−1=I + D−1clNclNprD−1pr−1=D−1cl[DclDpr+ NclNpr] D−1pr−1= Dpr[DclDpr+ NclNpr]−1Dcl= DprX−1DclP (I + CP )−1= NprD−1prDpr[DclDpr+ NclNpr]−1Dcl= Npr[DclDpr+ NclNpr]−1Dcl= NprX−1DclC (I + P C)−1= (I + CP )−1C= Dpr[DclDpr+ NclNpr]−1DclD−1clNcl= DprX−1NclHenceHeu=I 00 0−−NprDprX−1[NclDcl]Theorem 35 Heu∈ M (S) if and only if X ∈ U (S).46Proof: The main idea is that Heu∈ M (S) if and only if−NprDprX−1[NclDcl]= Heu−I 00 0∈ M (S)Certainly, if X−1∈ M (S), then Heu∈ M (S). Conversely, if Heu∈M (S), then use the Bezout identities to conclude that X−1∈ M (S).Theorem 36 Given a multivariable plant P and controller C withdet (I + P (∞)C(∞)) 6= 0.Suppose that (Npr, Dpr) is a right coprime factorization of P . Then, theclosed-loop is stable if and only if there exists a left coprime factorization ofC, (Ncl, Dcl) such thatDclDpr+ NclNpr= InuLemma 37 Let the pair (Nr, Dr) be a right coprime factorization of G ∈M (S). Let (Nl, Dl) be a left coprime factorization of G. ThenXY: X, Y ∈ M (S) , XNr+ Y Dr= 0=QDl−QNl: Q ∈ M (S)Theorem 38 Let P ∈ Rny×nube given, along with a right and left coprimefactorization, and associated matrices completing the double Bezout identity.The set of all stabilizing controllers for P isn(Vpr− QNpl)−1(Upr+ QDpl) : Q ∈ M (S) , detVpr− QNpl|s=∞6= 0oSeparately, we can derive the Multivariable Nyquist criterion. For sim-plicity, suppose that P and C have no pol es on the imaginary axis, andassume well-posedness, sodet (I + P (∞)C(∞)) 6= 0.Let nP,udenote the number of unstable poles of P (as counted by the numberof unstable eigenvalues in a stabilizable and detectable realization). Simi-larly, let nC,udenote the number of unstable poles of C. Then the closed-loop47system is stable if and only if the Nyquist plot of det (I + P (jω)C(jω)), asω ranges from −∞ to ∞, encircles the orig in nP,u+nC,utimes,