We therefore develop a framework for calculating eigenvalues and eigenvectors for the Rosenbrock system's matrix S(λ). More precisely, we describe the construction of families of Fiedler pencils of the Rosenbrock system matrix S(λ) and show that these Fiedler pencils are linearizations of S(λ).
Preliminaries
Polynomial Eigenvalue Problem(PEP)
Denote Lm(Cn×n) is the set of n × n matrix polynomials of degree less than even tom of the form (1.8). The right and left zero spaces of a singular matrix polynomial P(λ), denoted by Nr(P) and Nl(P), respectively, are given by.
Fiedler linearizations for matrix polynomial
Then t is said to be in standard columnar form if d+t is in standard columnar form. 1}, for some d ≥ 1, then q satisfies the SIP if and only if q is equivalent to a (unique) tuple in columnar standard form.
Rational matrix function
13] For the regular rational matrix function G(λ), one can study the polar and zero polynomials without computing the Smith-Mcmillan form, since this is the case according to Theorem 1.3.1. Unfortunately, if the pole and zero polynomials have common roots, the corresponding zeros and poles do not appear in detG(λ).
State space theory
Minimal realizations
A realization of a rational matrix is not unique, that is, the realization of a rational function can be represented in different forms. Using Theorem 1.4.3 and Theorem 1.4.4 it is easy to see that this realization is controllable and observable and therefore minimal.
Sensitivity analysis of eigenvalues
Therefore, we calculate the eigenvalues by solving a generalized eigenvalue problem for Fiedler pencil of the system matrix. More precisely, in this chapter we describe the construction of Fiedler pencils of the Rosenbrock system.
Zeros and poles
A complex numberλ is said to be an invariant zero of the LTI system Σ associated with S(λ) ifrank(S(λ)) Note that σ(i) indicates the position of the factor Mi in the product Mσ; i.e. σ(i) =j means that Mi is the jth factor in the product Mσ. If σ has a sequence at m−2, we can write using the commutativity relations of the matrices Mi. The Fiedler pencils of the system matrix can be considered, under appropriate assumptions, as linearizations of the transfer function G(λ). We have already seen that Fiedler pencils of the Rosenbrock system matrix S(λ) are linearizations for S(λ). We now give a direct proof of this fact without any recourse to Fiedler pencils of matrix polynomials. Since the product of unimodular matrices is again unimodular and Ri and Qi have the desired block diagonal form, the final product together with premultiplication by diag[−I(m−1)n, In, Ir], we have (2.32). The main goal of this chapter is to analyze recovery of eigenvector from Fiedler pencils of system matrix. We also describe recovery of eigenvector of transfer function from that of Fiedler pencil of system matrix. Indeed, we establish isomorphisms between null spaces of transfer functions, system matrix and Fiedler pencils. We want to recover eigenvectors of S(λ) and G(λ) from the eigenvectors of Lσ(λ). For the special case of companion pencil C1(λ) given by. It is also easy to recover the left eigenvector of G(λ) from that of C1(λ). We actually have. The vector y ∈ Cn is a left eigenvector of G(λ) corresponding to an eigenvalue λ if and only if. We now show how we can derive the right and left eigenvectors of G(λ) and S(λ) from those of Fiedler pencils Lσ(λ). The right and left null spaces of a singular G(λ) ∈ C(λ)n×n, denoted by Nr(G) and Nl(G), respectively, are given by. Now we define the left inverse of a matrix that will play an important role in recovering the eigenvector of S(λ) from its linearizations. Then E is called invertible if there exists E` :Cm →Cn such that E`E =In. In that case, E` is called a left inverse of E. We refer to F(P) as the recovery of the right eigenvector map and K(P) as the left eigenvector recovery map. The inverse of the bijection plays an important role in showing that there is an isomorphism between the left null space of S(λ) and the left null space of the pencil Lσ(λ). Let Lσ(λ) be the Fiedler pencil for S(λ) associated with the bijectionσ. We have shown that there is an isomorphism between the null spaces G(λ), S(λ) and Lσ(λ). Next, we consider the singular system matrix and the singular transfer function, i.e. S(λ) and G(λ) are singular, and then we analyze the isomorphism between N(S(λ)) and N(Lσ(λ)), where Lσ(λ) is a linearization of S(λ). If Sσ(λ) is a Fiedler pencil for S(λ) associated with a bijection σ, then the state space system Σ2 associated with Sσ(λ) is given by. We present a generalized Fiedler (GF) pencil of the system matrix and show that GF pencils are also linearizations of the system matrix. We now analyze pencils that are strictly equivalent to Fiedler pencils and pencil block entries. Generalized Fiedler pencils are further classified into two groups, namely proper generalized Fiedler pencils (PGF pencils) and non-proper generalized Fiedler pencils (NPGF pencils). It is well known [6] that for a matrix polynomialP(λ) there are more than one Fiedler pencils that are strictly equivalent to a given GF pencil. Like matrix polynomial, there is more than one Fiedler pencil of S(λ) that is strictly equivalent to a given GF. The following result describes PGF pencils that are equivalent to Fiedler pencils that consistently preserve at 0. The following result analyzes the existence of self-contiguous linearizations of self-contiguous system matrix. We also mention that for the matrix polynomial P(λ), the GF pencil Tω(λ) = λO−1 −M0EMm−1 is self-adjoint/ symmetric when P(λ) is self-adjoint/ symmetric, where O = M1M3. Note that ξ2 has m−c0−2 sequences at c0+ 1, so by the structure of Fiedler matrices we have. Note that since Tω(λ) is a PGF and has ω0 c0 sequences at 0, then m /∈σ1 and contains indices greater than c0 + 1. With the commutativity given in (2.11) and the fact that ω0 has c0 sequences at 0 allow us to write Mω. This chapter is devoted to a new family of linearizations, referred to as generalized Fiedler pencil with repetition (GFPR), for system matrix. Given a system matrix of an LTI system, the main objective of this chapter is to investigate a new family of linearizations referred to as generalized Fiedler pencil with repetition (GFPR) for the system matrix and for the associated transfer function. An important property of the companion form is that there is a simple relationship between its eigenvectors and those of the system matrix S(λ) that it linearizes. We have already seen the recovery of eigenvectors of the system matrix from the Fiedler pencils and from the GF pencils. In this chapter we also discuss the eigenvector recovery of the system matrix and of the transfer function from GFPR. We have seen that a self-adjoint system matrix may not have a self-adjoint GF pencil. Note that this relation is an equivalence relation, and if Mq2 can be obtained from Mq1 by applying the commutativity relations given in (5.6), then q1 is equivalent to q2. Now we analyze the product of Fiedler matrices where the entries in the product block consist of 0, In, Ai, A, B, C, E. So the question is: if we let iterate the Fiedler matrices, will that product still be a product without operation. The following theorem provides a canonical form of an action-free product, which helps us construct structure while preserving linearizations of the system matrix. The assumption that 0 is a simple index cannot be relaxed in Theorem 5.2.5. is not company-free because M0:1M0:0 is not company-free. We develop a general framework for sensitivity analysis of the invariant zeros of the LTI system and the eigenvalues of rational matrix functions. We derive explicit computable expressions for the conditional numbers of invariant zeros of the LTI system and thus for the eigenvalues of rational matrix functions. We analyze the influence of linearization on the conditioning of the invariant zeros of the LTI system and the eigenvalues of the rational matrix function. Also recall that Sp(S) is the set of invariant zeros of Σ, and Sp(G) is the set of transmission zeros of Σ. We analyze the sensitivity of invariant zeros of Σ with respect to small perturbations in the coefficient matrices of the LTI system Σ. Since invariant zeros of Σ are the eigenvalues of the system matrix S(λ), for sensitivity analysis we must choose perturbations ∆S so that S(λ) + ∆S(λ) is again a system matrix. Note that if λ is not a pole of the transfer function G(λ) =P(λ) +C(λE−A)−1B, condS(λ,S) measures the sensitivity of λ to small perturbations in the coefficient matrices. The condition number condS(λ,S) with respect to the standard in (6.3) is given by. b) The condition number cond(λ, G) with respect to the subspace norm in (6.3) is given by. We then compare the unstructured condition number of the simple eigenvalue of the system matrix with the condition number of Fiedler linearizations. The following result compares the condition number of the quadratic eigenvalue problem with that of the companion form given in (6.14). 15] Let λ be a simple, finite and nonzero eigenvalue of the square matrix polynomialQ(λ) given in (6.11) and L be the corresponding accompanying linearization given in (6.12). Therefore, we have developed a framework for computing eigenvalues and eigenvectors of the Rosenbrock system matrix S(λ). To this end, we have introduced three families of linearizations—which we referred to as Fiedler pencils, Generalized Fiedler (GF) pencils, and Generalized Fiedler pencils with repetition (GFPR)—of the Rosenbrock system matrix S(λ). Thus, the invariant zeros of the LTI system Σ could be computed by solving generalized eigenvalue problems for Fiedler pencils. We described the construction of the Fiedler pencils of the matrix of the Rosenbrock system S(λ) and showed that these Fiedler pencils are linearizations for S(λ). We further showed that Fiedler pencils for S(λ) are linearizations of G(λ) when LTI. We also described the recovery of the eigenvector G(λ) from the Fiedler pencils of the system matrix S(λ). Furthermore, we have shown that the linearized systems obtained in this way are strictly system equivalent to the higher order systems and therefore retain system characteristics of the original systems. We also developed a framework for sensitivity analysis of eigenvalues of the system matrixS(λ) and the transfer functionG(λ). We defined state numbers for simple eigenvalues of S(λ) and G(λ), and obtained explicit computable expressions. tions for the condition numbers. We also analyzed the effect of linearization on the conditioning of the eigenvalues of the system matrix S(λ).Fiedler pencils for Rosenbrock system matrix
Fielder pencils are linearizations
Eigenvector recovery for system matrix
Linearized state-space system
Higher order state space system
Eigenvector recovery from GF pencils
Operation-free products of Fiedler matrices
Generalized Fiedler pencil with repetition
Eigenvector formula for GFPR
Condition number of Fiedler pencils
Comparison of Condition Numbers for QEP
