LetG(λ)∈C(λ)n×n be nonproper and be given by G(λ) :=P(λ) +Gsp(λ), whereP(λ) is a matrix polynomial of degree m and Gsp(λ) is a strictly proper rational matrix. Let Gsp(λ) =C(λE−A)−1B be a minimal realization ofGsp(λ), whereλE −A is an r×r pencil with E being nonsingular, C ∈Cn×r and B ∈Cr×n.Then
G(λ) = P(λ) +C(λE−A)−1B (5.5) is a minimal realization of G(λ). Consequently, the (n+r)×(n+r) Rosenbrock system matrix
S(λ) :=
P(λ) C B A−λE
(5.6)
associated with G(λ) is irreducible. Recall that S(λ) is irreducible if and only if rank h
B A−λE i
=r = rank
C (A−λE)
for all λ∈C.
Assumption: For the rest of this chapter, we assume that P(λ) := Pm
i=0λiAi with Am 6= 0 and the realization G(λ) = P(λ) +C(λE −A)−1B of G(λ) given by (5.5) is minimal. The system matrix S(λ) associated with G(λ) is given by (5.6).
We now define Rosenbrock strong linearizationof the G(λ).
Definition 5.2.1 (Biproper rational matrix, [59]). An n × n proper rational matrix F(λ) is said to be biproper if F(∞) is nonsingular.
Definition 5.2.2 (Rosenbrock strong linearization). Let G(λ) and S(λ) be as in (5.5) and (5.6), respectively. Also let L(λ)be an(mn+r)×(mn+r)irreducible system matrix associated with the transfer function G(λ) given by
L(λ) :=
X −λY C B H−λK
and G(λ) :=X −λY+C(λK−H)−1B,
where H−λK is an r×r pencil withK being nonsingular, C ∈Cmn×r, B ∈Cr×mn and X −λY is anmn×mnpencil. ThenL(λ)is said to be a Rosenbrock strong linearization of G(λ) provided that the following conditions hold.
(a) There exist mn×mn unimodular matrix polynomials U(λ) andV(λ), and r×r nonsingular matrices U0 and V0 such that
U(λ) 0 0 U0
L(λ)
V(λ) 0 0 V0
=
I(m−1)n 0
0 S(λ)
for all λ∈C.
(b) There exist mn×mn biproper rational matrices O`(λ) and Or(λ) such that O`(λ)λ−1G(λ)Or(λ) =
I(m−1)n 0
0 λ−mG(λ)
for all λ∈C. We also refer to L(λ) as a Rosenbrock strong linearization of S(λ).
We mention that Rosenbrock strong linearization can be defined for rectangular rational matrices by appropriately modifying Definition 5.2.2. We also mention that by [4, Theorem 3.5], the Rosenbrock strong linearization is equivalent to the strong linearization of rational matrices introduced in [5].
The condition (a) in Definition 5.2.2 ensures that the finite eigenstructure of L(λ) (resp., H−λK) is the same as the finite zero (resp., pole) structure ofG(λ), see [2, 4].
On the other hand, the condition (b) ensures that the pole-zero structure of G(λ) at infinity can be easily recovered from the pole-zero structure ofL(λ) at infinity. The finite and infinite pole-zero structure of L(λ) can be computed from the Kronecker canonical form of L(λ),see [58, 57].
Twon×n rational matrices G1(λ) and G2(λ) are said to beequivalent at infinity if there exist biproper rational matrices FL(λ) and FR(λ) such that FL(λ)G1(λ)FR(λ) = G2(λ), see [59]. We write G1(λ) ∼ei G2(λ) when G1(λ) and G2(λ) are equivalent at infinity. Note that if G1(λ) ∼ei G2(λ) then G1(λ) and G2(λ) have the same pole-zero structure at infinity. The pole-zero structure at infinity of a rational matrix is given by the Smith-McMillan form at infinity.
Theorem 5.2.3 ([59], Chapter 3). Let H(λ)∈C(λ)n×n with normal rank `.Then there exist biproper rational matrices FL(λ) and FR(λ) such that
FL(λ)H(λ)FR(λ) = diag(λp1, . . . , λpk, λ−gk+1, . . . , λ−g`,0, . . . ,0) =:SM∞(H(λ)), where p1 ≥p2 ≥ · · · ≥pk ≥0>−gk+1 ≥ · · · ≥ −g` withpi (resp.,gj) being nonnegative (resp., positive) integers for i = 1 : k (resp., j = k + 1 : `). Further, the pole-zero index at infinity given by Ind∞(H(λ)) := (p1, . . . , pk,−gk+1, . . . ,−g`)∈Z` is a complete invariant under biproper equivalence. Furthermore, if H(λ) is nonproper and m is the degree of the polynomial part of H(λ) then p1 =m.
The diagonal matrixSM∞(H(λ)) is the Smith-McMillan form ofH(λ) at infinity. If pi >0 then H(λ) has a pole of orderpi at infinity and ifgj >0 then H(λ) has a zero of order gj at infinity. IfH(λ) is a regular polynomial thenp1+· · ·+pk ≥gk+1+· · ·+g`, see [59, Corollary 3.87].
Remark 5.2.4. Let G(λ), S(λ), L(λ) and G(λ) be as in Definition 5.2.2. Then it is easily seen that S(λ) ∼ei diag(λIr, G(λ)) and L(λ) ∼ei diag(λIr,G(λ)) showing that G(λ) and S(λ) have the same zero structure at infinity and, G(λ) and L(λ) have the same zero structure at infinity. Indeed, we have
In C(λE−A)−1
0 Ir
S(λ)
In 0
(λE−A)−1B Ir
=
G(λ) 0
0 A−λE
which shows that S(λ) ∼ei diag(λIr, G(λ)). Similarly, L(λ) ∼ei diag(λIr,G(λ)). Now by Definition 5.2.2(b), we have G(λ)∼ei diag(λI(m−1)n, λ1−mG(λ)) and hence L(λ)∼ei diag( λI(m−1)n+r, λ1−mG(λ)). This shows that we can deduce pole-zero structure of L(λ) from those of G(λ) and vice-versa.
Theorem 5.2.5. Let G(λ),G(λ) and L(λ) be as in Definition 5.2.2. Suppose that q is the normal rank of G(λ) and s is the rank of Y. Then we have
SM∞(L(λ)) =λIr+s⊕diag(λ−g1, . . . , λ−g`)⊕0n−q
where 0≤g1 ≤ · · · ≤g` are integers and s+`= (m−1)n+q with ` < q. Furthermore, SM∞(G(λ)) = λIs⊕diag(λ−g1, . . . , λ−g`)⊕0n−q,
SM∞(G(λ)) = λmIq−`⊕diag λm−(g1+1), . . . , λm−(g`+1)
⊕0n−q.
Equivalently, we have Ind∞(L(λ)) = (1, . . . ,1,−g1, . . . ,−g`),where 1appearsr+stimes and 0≤g1 ≤ · · · ≤g` are integers, and
Ind∞(G(λ)) = m, . . . , m, m−(g1+ 1), . . . , m−(g`+ 1) , where m appears q−` times.
Proof. By [58, Theorem 2], L(λ) has exactly r+s poles at infinity of order 1 and by [60, Corollary 1], L(λ) has no zero at infinity whenr+s= nrank(L).By Remark 5.2.4, we have L(λ) ∼ei diag( λI(m−1)n+r, λ1−mG(λ)) which shows that L(λ) has at least (m−1)n+r poles at infinity. Consequently, we have s ≥(m−1)n. Since G(λ) has at least one pole at infinity and nrank(L) = (m−1)n+r+q, there exist a nonnegative integer
` < qsuch thats+` = (m−1)n+qandSM∞(L(λ)) = λIr+s⊕diag(λ−g1, . . . , λ−g`)⊕0n−q. Next, SM∞(G(λ)) follows from the fact that L(λ)∼ei diag(λIr, G(λ)).
Finally, by Theorem 5.2.3, the highest order of a pole of G(λ) at infinity is m. Since SM∞(λ1−mG(λ)) =λ1−mSM∞(G(λ)), by Remark 5.2.4, we have
L(λ) ∼ei diag λI(m−1)n+r, λ1−mG(λ)
∼ei diag λI(m−1)n+r, λ1−mSM∞(G(λ))
= λIr+s⊕diag(λ−g1, . . . , λ−g`)⊕0n−q =SM∞(L(λ)).
Now equating the last n diagonal entries of SM∞(L(λ)) to λ1−mSM∞(G(λ), we ob- tain the desired SM∞(G(λ)).The pole-zero indices Ind∞(L(λ)) and Ind∞(G(λ)) follow immediately.
We mention that Theorem 5.2.5 also holds when G(λ) is proper. In such a case, in Theorem 5.2.5, we have L(λ) =S(λ), s= 0 and `≤q.
The next result shows that the Smith-McMillan form ofG(λ) at infinity can be used to deduce Smith-McMillan forms of the system matrix as well as Rosenbrock strong linearizations of G(λ).
Theorem 5.2.6. Let G(λ),S(λ) and L(λ) be as in Definition 5.2.2. Suppose that SM∞(G(λ)) = diag(λp1, . . . , λpk, It, λ−g1, . . . , λ−g`)⊕0n−q, where q is the normal rank of G(λ), k+t+` = q, p1 ≥ · · · ≥ pk >0 and 0 < g1 ≤ · · · ≤ g` are natural numbers.
Then we have the following
SM∞(S(λ)) = diag(λp1, . . . , λpk, λIr, It, λ−g1, . . . , λ−g`)⊕0n−q,
SM∞(L(λ)) = λIu⊕diag(λpˆ2, . . . , λpˆk, λ−(m−1)It, λ−ˆg1, . . . , λ−ˆg`)⊕0n−q, where u := (m−1)n+r+ 1, pˆj :=pj + 1−m ≤1 for j = 2 : k and ˆgj :=gj+m−1 for j = 1 :`.
Proof. Note that by Theorem 5.2.3, we have p1 = m. Now by Remark 5.2.4, we have S(λ) ∼ei diag(λIr, G(λ)) and L(λ) ∼ei diag λI(m−1)n+r, λ1−mG(λ)
which yield the desired results.