3 Stabilization of MIMO Feedback Systems

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1 3 Stabilization of MIMO Feedback Systems 3.1 Notation The sets R and S are as before. We will use the notation M (R) to denote the set of matrices with elements in R. The dimensions are not explicitly specified, and will be clear from context. Similarly, M (S) will denote the set of matrices with elements in S. In this set of notes, U R will denote square matrices with elements in R, whose inverse is also in M (R). Likewise, U S will denote the units in M (S). 3.2 Coprime Factorizations Definition 28 Suppose N S n y n u, D S n u n u. The pair (N, D) is called right coprime over S if there exist matrices U S n u n y, V S n u n u such that UN + V D = I nu. Definition 29 Suppose N S n y n u, D S n y n y. The pair (N, D) is called left coprime over S if there exist matrices U S n u n y, V S n y n y such that NU + DV = I ny. Definition 30 Suppose N S n y n u, D S n u n u are given, and the pair (N, D) is right coprime over S. If det[d( )] 0, then (N, D) is called a right-coprime factorization (over S) of G := ND R n y n u. Definition 31 Suppose N S n y n u, D S n y n y are given, and the pair (N, D) is left coprime over S. If det[d( )] 0, then (N, D) is called a left-coprime factorization (over S) of G := D N R n y n u. Theorem 32 Suppose that the pair (N, D) is a right coprime factorization of G R n y n u, so N S n y n u, and D S n u n u. If U S n u n u is a unit (ie. 43

2 U M (S)), then the pair (NU, DU) is a right coprime factorization of G. Morover, if (N 1, D 1 ) and (N 2, D 2 ) are both right coprime factorizations of G, then there exists a unit U U (S) such that N 1 = N 2 U, D 1 = D 2 U. Theorem 33 There is a result similar to Theorem 32 for left coprime factorizations. Lemma 34 Every G R n y n u has both a right and left coprime factorization over S. Proof: Let A, B, C, D be a stabilizable and detectable realization of G(s). Choose matrices F and L so that A + BF and A + LC have all of their eigenvalues in the open-left-half-plane. Define the following transfer functions D r U l N r V l = I 0 D I + F (C + DF) (si A BF) [ B L ] In shorthand notation, this is written as D r U l N r V l = A + BF B L F I 0 (C + DF) D I Similarly, define V r N l U l D l := A + LC B + LD L F I 0 C D I Certainly, all of the transfer functions are stable, since the matrices A + BF and A + LC have all of their eigenvalues in the open-left-halfplane. It is left as an exercise to show that 1. det [D l ( )] 0 2. det [D r ( )] 0 3. N r D r = Dl N l = G 44

3 4. V r N l U r D l D r U l N r V l = I 0 0 I 3.3 Multivariable Feedback Systems Consider the standard feedback structure shown below, where C, P M (R). u 2 u e 1 1 C e 2 P Assume that det [I + P( )C( )] 0. This insures that (I + PC) U (R), as well as (I + CP) U (R), and is equivalent to having all closed-loop transfer functions in M (R). Note that if P( ) = 0 ny n u, then this is automatically satisfied. Define H eu to be the transfer matrix from the inputs u to the signals labeled e. Writing loop equations gives H eu = ( Iny + PC ) C (I nu + CP) P (I nu + CP) (I nu + CP) Now, since all entries are matrices, we must be careful about the order in which P and C appear. Let (N pr, D pr ) be any right-coprime factorization of P, and (N cl, D cl ) any left-coprime factorization of C. Define X := D cl D pr + N cl N pr S n u n u Note that det ( I ny + PC ) = det(i nu + CP) = det ( I + Dcl N cl N pr Dpr ) = det ( Dcl ) ( det(x) det D pr 45 )

4 Hence, since det(d cl ( )) and det (D pr ( )) are both finite, and nonzero, we have det (I + P( )C( )) 0 det (X( )) 0. or I + PC 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 the coprime factorizations. (I + PC) = I P (I + CP) C = I N pr Dpr ( I + D = I N pr D pr cl N cl N pr Dpr ( D cl [D cl D pr + N cl N pr ] Dpr = I N pr (D cl D pr + N cl N pr ) N cl = I N pr X N cl (I + CP) = ( I + Dcl N cl N pr Dpr ) = ( Dcl [D cl D pr + N cl N pr ]Dpr ) = D pr [D cl D pr + N cl N pr ] D cl = D pr X D cl ) D cl N cl ) D cl N cl P (I + CP) C (I + PC) = N pr D pr D pr [D cl D pr + N cl N pr ] D cl = N pr [D cl D pr + N cl N pr ] D cl = N pr X D cl = (I + CP) C = D pr [D cl D pr + N cl N pr ] D cl D cl N cl = D pr X N cl Hence H eu = I N pr D pr X [N cl D cl ] Theorem 35 H eu M (S) if and only if X U (S). 46

5 Proof: The main idea is that H eu M (S) if and only if N pr D pr X [N cl D cl ] = H eu I M (S) Certainly, if X M (S), then H eu M (S). Conversely, if H eu M (S), then use the Bezout identities to conclude that X M (S). Theorem 36 Given a multivariable plant P and controller C with det(i + P( )C( )) 0. Suppose that (N pr, D pr ) is a right coprime factorization of P. Then, the closed-loop is stable if and only if there exists a left coprime factorization of C, (N cl, D cl ) such that D cl D pr + N cl N pr = I nu Lemma 37 Let the pair (N r, D r ) be a right coprime factorization of G M (S). Let (N l, D l ) be a left coprime factorization of G. Then X Y : X, Y M (S), XN r + Y D r = 0 = QD l QN l : Q M (S) Theorem 38 Let P R n y n u be given, along with a right and left coprime factorization, and associated matrices completing the double Bezout identity. The set of all stabilizing controllers for P is { (Vpr QN pl ) (U pr + QD pl ) : Q M (S), det ( ) } V pr QN pl s= 0 Separately, we can derive the Multivariable Nyquist criterion. For simplicity, suppose that P and C have no poles on the imaginary axis, and assume well-posedness, so det(i + P( )C( )) 0. Let n P,u denote the number of unstable poles of P (as counted by the number of unstable eigenvalues in a stabilizable and detectable realization). Similarly, let n C,u denote the number of unstable poles of C. Then the closed-loop 47

6 system is stable if and only if the Nyquist plot of det(i + P(jω)C(jω)), as ω ranges from to, encircles the origin n P,u +n C,u times, counterclockwise. 48