Multivariable Control Lecture 05 Multivariable Poles and Zeros John T. Wen September 4, 2006
SISO poles/zeros SISO transfer function: G(s) = n(s) d(s) (no common factors between n(s) and d(s)). Poles: values of s such that d(s) = 0. Zeros: values of s such that n(s) = 0. Watch out for pole/zero cancellation in cascaded systems! (Find the state space realization.) September 4, 2006Copyrighted by John T. Wen Page
MIMO Case For MIMO: G(s) = D+C(sI A) B (rational matrix: matrix of rational functions). We will write G(s) as G(s) = D L (s)n L(s) }{{} left MFD where D L, N L, N R, D R are polynomial matrices. = N R (s)d R (s) }{{} right MFD September 4, 2006Copyrighted by John T. Wen Page 2
Some Definitions Definition R p m [s] = p m polynomial matrix of s with coefficients in R R p m (s) = p m rational matrix of s with coefficients in R R p (s) R p,o (s) = proper rational functions = strictly proper rational functions. Definition The normal rank of G R p m [s] is the maximum rank of G over all s C. Definition (Generalization of the concept of scalar constants in R[s]) G R k k [s] is unimodular if G R k k [s]. Fact: G is unimodular iff G R[s] iff detg = constant. s+ is unimodular, but 2 is not. 0 2 0 s+ September 4, 2006Copyrighted by John T. Wen Page 3
Elementary Operations row/column interchange scaling scaling and addition row: pre-multiplication column: post-multiplication All elementary operations are unimodular. September 4, 2006Copyrighted by John T. Wen Page 4
Smith Form Let M R p m [s] with rank(m) min(p,m). Then M(s) can be transformed using elementary operations to U(s)M(s)V(s) = S(s) = diag{ε (s),...,ε r (s)} 0 0 0 where ε i (s) is monic and ε i ε i+ (ε i divides ε i+ ). {ε i } are called the invariant factors of M(s). September 4, 2006Copyrighted by John T. Wen Page 5
Smith-McMillan Form Let G R p m (s). G can be written as G(s) = N(s) d(s) where N Rp m [s] and d R[s] (l.c.m. of all denominators of G i j (s)). Reduce N(s) to its Smith Form, then Reduce σ i (s)/d(s) to lowest terms: U(s)G(s)V(s) = U(s)N(s)V(s)/d(s) = S(s)/d(s). σ i (s) d(s) = ε i(s) ψ i (s). Then U(s)G(s)V(s) = ε (s) ψ (s)... 0 ε r (s) ψ r (s) 0 0. September 4, 2006Copyrighted by John T. Wen Page 6
MIMO poles/zeros zero polynomial: z(s) = ε (s) ε r (s). Roots are called transmission zeros of G(s) pole polynomial: p(s) = ψ (s) ψ r (s). Roots are called poles of G(s). p (order of p(s)) is called the McMillan degree of G(s) (i.e., number of poles). If G is square: transmission zeros are also the roots of detg(s) = 0 (but must be careful with pole/zero cancellation). if, in addition, G( ) is invertible, poles of G (s) are the transmission zeros of G(s) (again watch out for pole/zero cancellation). September 4, 2006Copyrighted by John T. Wen Page 7
Example Consider G(s) = s+ s+3 s s+4 0 (s+)(s+3) Smith-McMillan Form: G SM (s) = (s+)(s+3)(s+4) 0 0 (s+)(s+4) (s+3) Clearly, there is no pole/zero cancellation for the and 4 modes since these poles and zeros occur in different channels. September 4, 2006Copyrighted by John T. Wen Page 8
MFD Write the Smith-McMillan Form of G(s) as Then G(s) can be written as U(s)G(s)V(s) = E(s)Ψ r (s) = Ψ l (s)e(s). G(s) = (U (s)e(s))(v(s)ψ r (s)) = (Ψ l (s)u(s)) (E(s)V (s)). }{{}}{{}}{{}}{{} N r (s) D r (s) D l (s) N l (s) September 4, 2006Copyrighted by John T. Wen Page 9
Coprime MFD MIMO transfer matrix: G(s) = N r (s)d r (s) = D l (s)n l (s) where N r, D r, N l, D l are polynomial matrices (matrix fraction description, MFD). D r is m m, D l is p p, N r is p m, N l is m p. Coprime MFD: (N r,d r ) are right coprime: If there are polynomial matrices N r, D r, R such that (so that N r D r N r = N r R, D r = D r R = N r D r ), then R must be unimodular, i.e., R and R are both polynomial matrices (like nonzero constants for SISO systems). (D l,n l ) are left coprime: If there are polynomial matrices D l, N l, L such that D l = LD l, N l = LN l (so that D l N l = D l N l ), then L must be unimodular. September 4, 2006Copyrighted by John T. Wen Page 0
Condition for Coprimeness Row reduce N r(s) to R(s) (i.e., find an unimodular matrix X such that D r (s) 0 X(s) N r(s) = R(s) ). Then (N r,d r ) is right coprime if and only if R(s) is uni- D r (s) 0 modular. [ Column reduce D l (s) N l (s) [ ] [ that D l (s) N l (s) Y(s) = L(s) is unimodular. ] [ ] to L(s) 0 (i.e., find an unimodular matrix Y such ] L(s) 0 ). Then (D l,n l ) is left coprime if and only if September 4, 2006Copyrighted by John T. Wen Page
To row reduce N r D r Row and Column Reductions, first find its Smith Form U U N r D r N r D r = ΣV = R 0 V = Σ, then. Similiarly for column reduction. MATLAB Tools Use smith command in MAPLE (in MATLAB Symbolic toolbox) to transform [ or D l (s) N l (s) ] to the Smith Form. N r(s) D r (s) September 4, 2006Copyrighted by John T. Wen Page 2
Bezout Identity If (N r,d r ) is right coprime, then there exist polynomial matrices (U r,v r ) such that U r N r +V r D r = I. If (D l,n l ) is left coprime, then there exist polynomial matrices (V l,u l ) such that Generalized Bezout Identity D l V l + N l U l = I. If (N r,d r ) are right coprime and (D l,n l ) are left coprime, then there exist polynomial matrices (U r,v r,u l,v l ) such that U r V r N r V l = I 0. D l N l D r U l 0 I (U r,v r,u l,v l ) may be found usingsmith function in MATLAB symbolic toolbox. September 4, 2006Copyrighted by John T. Wen Page 3
Computation of Coprime MFD Given a right MFD for G(s) = N r (s)d r (s) (e.g., G(s) = N(s) d(s) where d(s) is the least common multiple of all denominators of G(s)). Extract the greatest common right divisor, R, for (N r,d r ). Then N r (s) = N r (s)r (s) and D r (s) = D r (s)r (s) form a right coprime MFD for G(s). Similarly, to find the left coprime MFD, start with any left MFD (D l,n l ). Extract the greatest common left divisor, L. Then D l = L D l and N l = L N l form a left coprime MFD. Coprime MFD can be used in pole/zero characterization, state space realization and polynomial based control design. Example: G (s) = s+ 2 s+2 s+2 s+ and G 2 (s) = s+ s+3 s s+4 0 (s+)(s+3). September 4, 2006Copyrighted by John T. Wen Page 4
MIMO Pole/Zero: Characterization through MFD Given coprime MFD s for G(s): (N r (s),d r (s)) and (D l (s),n l (s)). Poles: values of s such that detd r (s) = detd l (s) = 0. If (A,B,C,D) is a minimal realization of G(s) (state dimension, n = # of poles), then poles are also given by the roots of det(si A) = 0 (characteristic equation), i.e., eigenvalues of A. (Note that G(s) = CAdj(sI A)B/det(sI A)+D.) Transmission Zeros: values of s such that N r (s) (or N l (s)) loses rank. Blocking zeros: values of s such that G(s) = 0. Blocking zeros Transmission zeros Pole/zero cancellation in MIMO system may not be obvious, e.g., consider K(s)G(s) below G(s) = s+ 2 s+2 s+2 s+, K(s) = s+2 s 2 s+ s 2 0. September 4, 2006Copyrighted by John T. Wen Page 5
Interpretation of poles and zeros If s o is a pole of G(s), then there exists an initial condition x o such that the zero input response of G is e s ot Cx o (choose x o to be the eigenvector of A corresponding to s o ). If z o is a transmission zero and p m (i.e., G is square or tall), then there exist u o and x o such that the output under the input u(t) = u o e z ot and initial state x o is zero (certain input direction is blocked). If z o is a transmission zero and p m (i.e., G is square or fat), then there exist v o and x o such that under any input of the form u o e z ot and initial state x o, output y(t) is orthogonal to v o, i.e., v T o y(t) = 0 (certain output direction cannot be achieved with any input). If z o is a blocking zero, then G(s)u o e z ot = 0 for all u o. MATLAB commands: pole,tzero September 4, 2006Copyrighted by John T. Wen Page 6
Example poles: {-,-,-2,-2} zero: -.5 G(s) = s+ 2 s+2 s+2 s+ Evaluate G at the transmission zero: G(.5) = 2 2 2 2 zero input direction: u =. unreachable output direction: v =. September 4, 2006Copyrighted by John T. Wen Page 7