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Contemporary Mathematics

Theory and Computations of Some Inverse Eigenvalue Problems for the Quadratic Pencil

Biswa N. Datta and Daniil R. Sarkissian

Abstract. The paper reviews, both from theoretical and computational view points, current developments on the partial modal approach for certain inverse eigenvalue problems for the quadratic pencil associated with a linear control system modeled by a system of matrix second-order di?erential equations. The paper concludes with some future research problems.

1. Introduction An inverse eigenvalue problem for a matrix A is the problem of ?nding A given the complete or a part of the spectrum and/or eigenvectors. There are many di?erent forms of inverse eigenvalue problems and they arise in various applications (see the recent expository paper of [MTC]). In this paper, we focus on certain types of inverse eigenvalue problems associated with a quadratic matrix pencil arising in feedback control of a matrix second-order system. To de?ne our problems, let’s consider the dynamical system (1.1) Mx ¨(t) + Dx ˙ (t) + Kx(t) = f (t), where M , D, and K are symmetric matrices; M is positive de?nite (denoted by M > 0), and x ˙ (t) and x ¨(t), respectively, denote the ?rst and second derivatives of the time dependent vector x(t). The system of the type (1.1) arises in a wide range of applications, especially in the design and analysis of vibrating structures, such as bridges, highways, buildings, airplanes, etc. In vibration analysis, the matrices M , K , and D are known, respectively, as the mass, sti?ness and damping matrices. Upon separation of variables, the system gives rise to the quadratic eigenvalue problem for the pencil (1.2) P (λ) = λ2 M + λD + K. The pencil (1.2) has 2n eigenvalues which are the roots of the equation det(P (λ)) = 0

1991 Mathematics Subject Classi?cation. Primary 34A55, 93B55; Secondary 93B52, 70Q05. Based on an invited presentation at the AMS Research Conference on “Structured Matrices in Operator Theory, Numerical Analysis, Control, Signal and Image Processing”, Boulder, Colorado, July 1999. The paper is comprised of joint work of the authors with S. Elhay and Y. Ram.

c 0000 (copyright holder)

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BISWA N. DATTA AND DANIIL R. SARKISSIAN

and 2n corresponding eigenvectors. If (1.1) represents a vibrating system, then the eigenvalues of P (λ) are related to the natural frequencies of the homogeneous system Mx ¨(t) + Dx ˙ (t) + Kx(t) = 0,

and the eigenvectors are referred to as the modes of vibration of the system (see [BNDa], [DJI]). Dangerous oscillations (called resonance) will occur when one or more eigenvalues of the pencil (1.2) became equal or close to the frequency of the external force. To avoid such unwanted oscillations of the vibratory system modeled by (1.1), a control force f = Bu(t), where B is an n × m matrix and u(t) is a time dependent m × 1 vector needs to be applied to (1.1). Let u(t) be chosen as (1.3) u(t) = FTx ˙ (t) + GT x(t),

where F and G are constant matrices, then the system (1.1) becomes (1.4) Mx ¨(t) + (D ? BF T )x ˙ (t) + (K ? BGT )x(t) = 0.

Mathematically, the problem is then to choose the matrices F and G such that the eigenvalues of the associated closed-loop pencil (1.5) Pc (λ) = λ2 M + λ(D ? BF T ) + K ? BGT can be altered as required to combat the e?ects of resonances or to ensure and improve the stability of the system. In a realistic situation, however, only a few eigenvalues are “troublesome”; so it makes more sense to alter only those “troublesome” eigenvalues, while keeping the rest of the spectrum invariant. This leads to the following inverse eigenvalue problem, known as the partial eigenvalue assignment problem for the pencil (1.2). Problem 1.1. Given 1. Real n × n matrices M = M T > 0, D = DT , K = K T . 2. The n × m (m ≤ n) control matrix B . 3. The self-conjugate subset {λ1 , . . . , λp }, p < n of the open-loop spectrum {λ1 , . . . , λp ; λp+1 , . . . , λ2n } and the corresponding eigenvector set {x1 , . . . , xp }. 4. The self-conjugate set {?1 , . . . , ?p } of numbers. Find real feedback matrices F and G such that the spectrum of the closed-loop pencil (1.5) is {?1 , . . . , ?p ; λp+1 , . . . , λ2n }. While Problem 1.1 is important in its own right, it is to be noted that, if the system response needs to be altered by feedback, both eigenvalue assignment as well as eigenvector assignment should be considered. This is because, the eigenvalues determine the rate at which system response decays or grows while the eigenvectors determine the shape of the response. Such a problem is called the eigenstructure assignment problem. Unfortunately, the eigenstructure assignment problem, in general, is not solvable if the matrix B is given apriori (see [IK]). This consideration leads to the following more tractable (but practical) inverse eigenstructure assignment problem for the quadratic pencil (1.2), known as the partial eigenstructure assignment problem for the pencil (1.2). Problem 1.2. Given 1. Real n × n matrices M = M T > 0, D = DT , K = K T .

SOME INVERSE EIGENVALUE PROBLEMS FOR THE QUADRATIC PENCIL

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2. The self-conjugate subset {λ1 , . . . , λp }, p < n of the open-loop spectrum {λ1 , . . . , λp ; λp+1 , . . . , λ2n } and the corresponding eigenvector set {x1 , . . . , xp }. 3. The self-conjugate sets of numbers and vectors {?1 , . . . , ?p } and {y1 , . . . , yp }, such that ?j = ?k implies yj = yk . Find a real control matrix B of order n × m (m < n), and real feedback matrices F and G of order n × m such that the spectrum of the closed-loop pencil (1.5) is {?1 , . . . , ?p ; λp+1 , . . . , λ2n } and the eigenvector set is {y1 , . . . , yp ; xp+1 , . . . , x2n }, where xp+1 , . . . , x2n are the eigenvectors of (1.2) corresponding to λp+1 , . . . , λ2n . An obvious approach for the above problems is to recast the problem in terms of a ?rst-order reformulation and then apply one of the many well-established techniques for full-order eigenvalue assignment of a ?rst-order system (see e.g. [BNDb]) or, more appropriately, the partial pole placement technique of [YS]. There are some computational di?culties with this approach. If the standard ?rst order transformation z ˙ (t) = 0 ? M ?1 K I ? M ?1 D z (t) + 0 M ?1 B u(t), where z (t) = x(t) x ˙ (t)

is used, then the matrix M has to be inverted, and, if it is ill-conditioned, the state matrix will not be computed accurately. Furthermore, all the exploitable properties such as de?niteness, sparsity, bandness, etc. of the coe?cient matrices M , D, and K , usually o?ered by a practical problem, will be completely destroyed. The use of a nonstandard ?rst-order transformation, such as M 0 0 M z ˙ (t) = 0 ?K M ?D z (t) + 0 B u(t)

? (t), and the will give rise to a descriptor system of the form E z ˙ (t) = Az (t) + Bu eigenvalue assignment methods for the descriptor systems, especially, when the matrix E is ill-conditioned, are not well developed. A second approach, popularly known in the engineering literature as the independent modal space control (IMSC) approach, also su?ers from some serious computational di?culties and is almost impossible to implement in practice. The basic idea here is to decouple the problem into a set of n independent problems, solve each of these independent problems separately, and then piece the individual solutions together to obtain a solution of the given problem. The implementation of this idea requires knowledge of the complete spectrum and associated eigenvectors of the pencil P (λ). Unfortunately, numerical methods for the quadratic eigenvalue problem are not well developed, especially for large and sparse problems. The stateof-the-art computational techniques are capable of computing only a few selected extremal eigenvalues and eigenvectors (see [PC] and [SBFV]). Furthermore, for decoupling of the right hand sides of the associated modal equations, some stringent conditions on the control vector need to be imposed (see [DJI]). Speci?cally, if the matrices M , D and K are simultaneously diagonalized by the matrix S , then for a decoupling of the right hand side of the associated modal equations, the following commutativity relations must be satis?ed BF M ?1 D = DM ?1 BF and BGM ?1 D = KM ?1 BG; assuming that BF and BG are both symmetric (see [DJI]).

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BISWA N. DATTA AND DANIIL R. SARKISSIAN

In view of these statements, it is natural to wonder if solutions of the above problems can be obtained using only a partial knowledge of eigenvalues and eigenvectors, and without resorting to a ?rst-order reformulation. A solution technique of this type will be called a direct partial modal approach. It is “direct”, because the solution is obtained directly in the second-order setting without any types of reformulations. It is “partial modal”, because only a part of the spectral data is needed for the solution. Such a direct partial modal approach for the Problem 1.1 and Problem 1.2 have been recently proposed in [DERb], [DS] and [DERS]. The solutions are obtained using only those small number of eigenvalues and the corresponding eigenvectors that are to be reassigned and directly in terms of the coe?cient matrices M , D and K . Variation of Problem 1.1 have also been solved this way in [DR], [DERa] and [CD]. The partial modal solutions for Problem 1.1 (both for the single-input and multiple-input cases) and for Problem 1.2 are described in Section 3 and 4, respectively. Indeed, a uni?ed treatment of solutions to both these problems is given, in the sense, that the results of existences and uniqueness (Theorem 3.1, Theorem 3.3 and Theorem 4.1) are derived and the solutions are expressed in each case using a single matrix Z1 given by (1.6) where (1.7) and (1.8) X1 = (x1 , . . . , xp ), Y1 = (y1 , . . . , yp ). Furthermore, in this paper the above Theorems are derived using a weaker condition than originally used in [DERb] to solve the single-input case of Problem 1.1. The constructive proofs of Theorems 3.1, 3.3 and 4.1, lead, respectively, to Algorithms 3.2, 3.5 and 4.2. Algorithms 3.5 and 4.2 are illustrated with a numerical example in Section 5. In Section 2, three important orthogonality relations between the eigenvectors of a quadratic pencil are stated and proved. One of these relations plays a key role in our derivation of the direct modal approach for Problems 1.1 and 1.2. However, these relations are of independent interest. Based on our discussions and observations in this paper, a few future research problems are stated in the concluding Section 6. Discussions pertaining justi?cation of each of the problems are given and in one case (case (iii) in Section 6) our idea on a possible approach for its solution is stated. The numerical example in Section 5 supports our idea. Some more de?nitive work, however, should be done. 2. Orthogonality Relations of the Eigenvectors of Quadratic Matrix Pencil In this section, we derive three orthogonality relations (due to [DERb]) between the eigenvectors of a symmetric de?nite quadratic pencil. One of these results plays a key role in our later developments. These results generalize the well-known results on orthogonality between the eigenvectors of a symmetric matrix and these of a symmetric de?nite linear pencil (see [BNDa]) of the form K ? λM . Theorem 2.1. (Orthogonality of the Eigenvectors of Quadratic Pencil). Let P (λ) = λ2 M + λD + K, where M = M T > 0, D = DT , and K = K T . Let Λ1 = diag(λ1 , . . . , λp ), Λ′ 1 = diag(?1 , . . . , ?p ) Z1

T T = Λ′ 1 Y1 M X1 Λ1 ? Y1 KX1 ,

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X and Λ = diag(λ1 , . . . , λ2n ) be, respectively, the eigenvector and the eigenvalue matrix of the pencil (1.2). Assume that the eigenvalues λ1 , . . . , λn are all distinct and di?erent from zero. Then there exist diagonal matrices D1 , D2 , and D3 such that (2.1) (2.2) (2.3) Furthermore (2.4) (2.5) (2.6) D1 D2 D2 = = = D3 Λ ? D1 Λ ? D3 Λ 2 . ΛX T M X Λ ? X T KX = D1 ΛX T DX Λ + ΛX T KX + X T KX Λ = D2 ΛX T M X + X T M X Λ + X T DX = D3

Proof. By de?nition, the pair (X, Λ) must satisfy the n × 2n system of equations (called the eigendecomposition of the pencil P (λ) = λ2 M + λD + K ): (2.7) M X Λ2 + DX Λ + KX = 0. Isolating the term in D, we have from above ?DX Λ = M X Λ2 + KX. Multiplying this on the left by ΛX T gives ?ΛX T DX Λ = ΛX T M X Λ2 + ΛX T KX Taking the transpose gives ?ΛX T DX Λ = Λ2 X T M X Λ + X T KX Λ Now, subtracting the latter from the former gives, on rearrangement, ΛX T M X Λ2 ? X T KX Λ = Λ2 X T M X Λ ? ΛX T KX or (2.8) (ΛX T M X Λ ? X T KX )Λ = Λ(ΛX T M X Λ ? X T KX ). Thus, the matrix ΛX T M X Λ ? X T KX which we denote by D1 , must be diagonal since it commutes with a diagonal matrix, the diagonal entries of which are distinct. We thus have the ?rst orthogonality relation (2.1): ΛX T M X Λ ? X T KX = D1 Similarly, isolating the term in M of the eigendecomposition equation, we get ?M X Λ2 = DX Λ + KX, and multiplying this on the left by Λ2 X T gives ?Λ2 X T M X Λ2 = Λ2 X T DX Λ + Λ2 X T KX. Taking the transpose, we have ?Λ2 X T M X Λ2 = ΛX T DX Λ2 + X T KX Λ2. Subtracting the last equation from the previous one and adding ΛX T KX Λ to both sides gives, after some rearrangement, Λ(ΛX T DX Λ + ΛX T KX + X T KX Λ) = (ΛX T DX Λ + ΛX T KX + X T KX Λ)Λ

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BISWA N. DATTA AND DANIIL R. SARKISSIAN

Again, this commutativity property implies, since Λ has distinct diagonal entries, that ΛX T DX Λ + ΛX T KX + X T KX Λ = D2 is a diagonal matrix. This is the second orthogonality relation (2.2). The ?rst and second orthogonality relations together easily imply the third orthogonality relation (2.3): ΛX T M X + X T M X Λ + X T DX = D3 To prove (2.4) we multiply the last equation on the right by Λ giving ΛX T M X Λ + X T M X Λ2 + X T DX Λ = D3 Λ, which, using the eigendecomposition equation, becomes ΛX T M X Λ + X T (?KX ) = D3 Λ. So, from the ?rst orthogonality relation (2.1) we see that D1 = D3 Λ Next, using the eigendecomposition equation (2.7), we rewrite the second orthogonality relation (2.2) as D2 = ΛX T (DX Λ + KX ) + X T KX Λ = ΛX T (?M X Λ2 ) + X T KX Λ = (?ΛX T M X Λ+ X T KX )Λ By the ?rst orthogonality relation we then have D2 = ? D1 Λ Finally, from D1 = D3 Λ and D2 = ?D1 Λ we have D2 = ? D3 Λ 2 . We remind the reader that matrix and vector transposition here does not mean conjugation for complex quantities. Remark 2.2. If the condition “the eigenvalues λ1 , . . . , λ2n are all distinct and di?erent from zero” in Theorem 2.1 is replaced by the weaker condition “the sets {λ1 , . . . , λp } and {λp+1 , . . . , λ2n } are disjoint” then the weaker version of the ?rst orthogonality relation (2.1) holds: (2.9) (2.10) where (2.11) and (2.12) X1 = (x1 , . . . , xp ), X2 = (xp+1 , . . . , x2n ); N Λ = ΛN, where N = (nij ) = ΛX T M X Λ ? X T KX, then nij λi = λj nij implies nij = 0 if 1 ≤ i ≤ p < j ≤ 2n or 1 ≤ j ≤ p < i ≤ 2n. Therefore, (2.9) and (2.10) follow from (2.8). Indeed, if (2.8) is written as Λ1 = diag(λ1 , . . . , λp ), Λ2 = diag(λp+1 , . . . , λ2n )

T T Λ 1 X1 M X2 Λ 2 ? X1 KX2 T Λ 2 X2 M X1 Λ 1

= 0, = 0,

?

T X2 KX1

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3. Solution to Problem 1.1 In this section, we present a direct partial modal approach for the solution of Problem 1.1. We consider the single-input and multi-input cases separately. 3.1. Case 1. Single-input Case. In the single-input case, Problem 1.1 reduces to the following problem: Given n × n matrices M = M T > 0, D = DT , K = K T , the n × 1 (m = 1) control vector b; a part of the spectrum {λ1 , . . . , λp } of the open-loop pencil P (λ), and the set {?1 , . . . , ?p }, both closed under complex conjugation, ?nd real feedback vectors f and g such that the spectrum of Pc (λ) = λ2 M + λ(D ? bf T ) + K ? bg T is precisely the set {?1 , ?2 , . . . , ?p , λp+1 , . . . , λ2n }, where λp+1 , . . . , λ2n are the remaining eigenvalues of the pencil P (λ). As before, let (X, Λ) be the eigenvector-eigenvalue matrix pair of the quadratic pencil P (λ) = λ2 M + λD + K. Partition X and Λ in the form X = (X1 , X2 ), Λ = diag(Λ1 , Λ2 ), where Λ1 , Λ2 and X1 , X2 are de?ned by (2.11) and (2.12), respectively. De?ne the vectors f and g by: f = M X1 Λ1 β and g = ?KX1 β, where β is an arbitrary p × 1 vector. We ?rst show that with this choice of f and g , the (2n ? p) eigenvalues λp+1 , · · · , λ2n of the closed-loop pencil Pc (λ) = λ2 M + λ(D ? bf T ) + (K ? bg T ) remain unchanged by feedbacks; that is, they are the same as those of the open-loop pencil: P (λ) = λ2 M + λD + K. In terms of the eigenvalue and eigenvector matrices, this amounts to proving that T T M X2 Λ 2 2 + (D ? bf )X2 Λ2 + (K ? bg )X2 = 0. To prove this, we consider the eigendecomposition equation again: (3.1) From this, we obtain

T T M X2 Λ 2 2 + (D ? bf )X2 Λ2 + (K ? bg )X2 T T = M X2 Λ2 + DX2 Λ2 + KX2 ? bβ T (Λ1 X1 M X2 Λ 2 ? X1 KX2 ) T T = ?bβ T (Λ1 X1 M X2 Λ 2 ? X1 KX2 ).

M X Λ2 + DX Λ + KX = 0.

Indeed, (3.1) implies M X2 Λ2 2 + DX2 Λ2 + KX2 = 0 and, furthermore, if we assume that {λ1 , . . . , λp } ∩ {λp+1 , . . . , λ2n } = ?, then by (2.9) in Remark 2.2, we have

T T Λ 1 X1 M X2 Λ 2 ? X1 KX2 = 0. T T Thus M X2 Λ2 2 + (D ? bf )X2 Λ2 + (K ? bg )X2 = 0. Choosing β for Partial Assignment of Eigenvalues. In order to solve Problem 1.1 completely, we still need to choose β which will move eigenvalues {λj }p j =1 of p the pencil P (λ) to {?j }j =1 in Pc (λ), if that is possible. If there is such a vector β , then there exists an eigenvector matrix Y1 of ordfer n × p:

Y1 = (y1 , y2 , . . . , yp ), yj = 0, j = 1, 2, ..., m, such that

2 T ′ T M Y1 (Λ′ 1 ) + (D ? bf )Y1 Λ1 + (K ? bg )Y1 = 0,

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BISWA N. DATTA AND DANIIL R. SARKISSIAN

where Λ′ 1 = diag(?1 , ?2 , . . . , ?p ). Substituting for f, g and rearranging, we have

2 ′ M Y1 (Λ′ 1 ) + DY1 Λ1 + KY1

= =

T T bβ T (Λ1 X1 M Y1 Λ′ 1 ? X1 KY1 ) T bβ T Z1 = bcT ,

where Z1 is given by (1.6) and c = Z1 β is a vector that will depend on the scaling chosen for the eigenvectors in Y1 . If we assume that the open-loop pencil (1.2) is partially controllable with respect to ?1 , . . . , ?p then we can solve for each of the eigenvectors yi using the equations (3.2) (?2 j M + ?j D + K )yj = b, j = 1, 2, ..., p to obtain Y1 . This corresponds to choosing the vector c = (1, 1, ..., 1)T , so, having computed the eigenvectors we could solve the p × p linear system (3.3) Z1 β = (1, 1, ..., 1)T

for β , and hence determine the vectors f and g . We now show that the vectors f and g obtained this way are real vectors. Since the set {λ1 , . . . , λp } is self-conjugate and the coe?cient matrices M , D and K of the open-loop pencil P (λ) are real, we know that λj = λk implies that xj = xk (conjugate eigenvectors correspond to conjugate eigenvalues). Therefore, there exists a nonsingular permutation matrix T such that X1 = X1 T and X1 Λ1 = X1 Λ1 T. Similarly, there is a permutation matrix T ′ such that

′ ′ Y1 = Y1 T ′ and Y1 Λ′ 1 = Y1 Λ1 T .

Thus, conjugating (1.6), we obtain

T ′ T T Z1 = (T ′ )T Λ′ 1 Y1 M X1 Λ1 T ? (T ) Y1 KX1 T

= (T ′ )T Z1 T,

and conjugation of (3.3) gives Z1 β = ((T ′ )T Z1 T )β = (T ′ )T (1, 1, . . . , 1)T , implying that β = T T β . Therefore, f = M (X1 Λ1 T )(T T β ) = f and g = ?K (X1 T )(T T β ) = g which shows that f and g are real vectors. Theorem 3.1. (Solution to Single-input Partial Eigenvalue Assignment Problem for a Quadratic Pencil). If {λ1 , . . . , λp } ∩ {λp+1 , . . . , λ2n } = ? then (i) For any arbitrary vector β , the feedback vectors f and g de?ned by (3.4) (3.5) f and g = M X1 Λ 1 β = ?KX1 β.

are such that 2n ? p eigenvalues λp+1 , . . . , λ2n of the closed-loop pencil Pc (λ) = λ2 M + λ(D ? bf T ) + K ? bg T are the same as these of the open-loop pencil P (λ) = λ2 M + λD + K .

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(ii) Let y1 , . . . , yp be the set of p vectors such that for each k = 1, 2, . . . , p, yk 1 ∈ null(?2 k M + ?k D + K, ?b).

(Equivalently, the pencil P (λ) is partially controllable with respect to ?1 , . . . , ?p ). De?ne Z1 =

T T Λ′ 1 Y1 M X1 Λ1 ? Y1 KX1

as in (1.6). Then Problem 1.1 has a solution in the form (3.4)-(3.5) if and only if the system of equations Z1 β has a solution. Based on Theorem 3.1 we can state the following algorithm. Algorithm 3.2. (An Algorithm For the Single-input Partial Eigenvalue Assignment Problem for the Quadratic Pencil). Inputs: 1. The n × n matrices M, K , and D; M = M T > 0, D = DT and K = KT . 2. The n × 1 control (input) vector b. 3. The set {?1 , · · · , ?p }, closed under complex conjugation. 4. The self-conjugate subset {λ1 , . . . , λp } of the open-loop spectrum {λ1 , . . . , λp ; λp+1 , . . . , λ2n } and the associated eigenvector set {x1 , . . . , xp }. Outputs: The feedback vectors f and g such that the spectrum of the closed-loop pencil Pc (λ) = λ2 M + λ(D ? bf T ) + (K ? bg T ) is {?1 , . . . , ?p , λp+1 , . . . , λ2n }. Assumptions: 1. The quadratic pencil is (partially) controllable with respect to the “eigenvalues to be assigned” ?1 , . . . , ?p . 2. {λ1 , . . . , λp } ∩ {λp+1 , . . . , λ2n } = ?. Step 1. Form Λ1 = diag(λ1 , . . . , λp ) and X1 = (x1 , . . . , xp ). Step 2. Solve for y1 , . . . , yp : (?2 j M + ?j D + K )yj = b, j = 1, . . . , p. Step 3. Form

T T Z1 = Λ′ 1 Y1 M X1 Λ1 ? Y1 KX1 ,

=

(1, 1, . . . , 1)T .

where Y1 = (y1 , . . . , yp ) and Λ′ 1 = diag(?1 , . . . , ?p ). If Z1 is ill-conditioned, then warn the user that the problem is ill-posed. Step 4. Solve for β : Z1 β = (1, 1, · · · , 1)T . Step 5. Form f g = M X1 Λ 1 β = ?KX1 β.

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3.2. Case 2. Multi-input case. In the multi-input case, we obtain the following generalization of Theorem 3.1. Theorem 3.3. (Solution to Multi-input Partial Eigenvalue Assignment Problem for a Quadratic Pencil). If {λ1 , . . . , λp } ∩ {λp+1 , . . . , λ2n } = ?. then (i) For any arbitrary matrix Φ, the feedback matrices F and G de?ned by (3.6) (3.7) F and G = M X 1 Λ 1 ΦT = ?KX1 ΦT .

are such that 2n ? p eigenvalues λp+1 , . . . , λ2n of the closed-loop pencil Pc (λ) = λ2 M + λ(D ? BF T ) + K ? BGT are the same as those of the open-loop pencil P (λ) = λ2 M + λD + K . (ii) Let {y1 , . . . , yp } and {γ1 , γ2 , . . . , γp } be the two sets of vectors chosen in such a way that ?j = ?k implies γj = γk and for each k = 1, 2, . . . , p, (3.8) yk γk ∈ null(?2 k M + ?k D + K, ?B )

(equivalently, the pair (P (λ), B ) is partially controllable with respect to the modes ?1 , . . . , ?p ). De?ne Z1 and Y1 as in Theorem 3.1. Then Problem 1.1 (in the multiinput case) has a solution with F and G given by (3.6), and (3.7), respectively, provided that Φ satis?es the linear system of equations: (3.9)

T ΦZ 1

= Γ, where Γ = (γ1 , . . . , γp ).

Proof. Using the ?rst orthogonality relation (2.1), it is easy to verify that

T T M X2 Λ 2 2 + (D ? BF )X2 Λ2 + (K ? BG )X2 T T = (M X2 Λ2 2 + DX2 Λ2 + KX2 ) ? B Φ(Λ1 X1 M X2 Λ2 ? X1 KX2 ) = 0,

which proves Part (i). To prove Part (ii), we note using (3.6) and (3.7), that ? ? ? ?

p

p

?D ? B Pc (?k )yk = ??2 k M + ?k = Bγk ? B ? ?

p

j =1

? + ?K + B φj λj xT j M ? ?

j =1

?? yk φj xT j K

p

?? ?

j =1

where Φ = (φ1 , φ2 , . . . , φp ) and zkj ’s are the elements of the matrix Z1 . Then Pc (?k )yk = 0 for k = 1, 2, . . . , p can be written in the form of the single matrix equation

T ΦZ 1

? ?γk ? φj xT j (?k λj M ? K ) yk = B

j =1

φj zkj ? ,

= Γ,

which proves (3.9). We now show that the matrices F and G obtained this way are real matrices. Since, if γ1 , . . . , γp are chosen in such a way that ?j = ?k implies γj = γk , then this also implies yj = yk and, then, as in the proof of Theorem 3.1, there exist permutation matrices T and T ′ such that

′ ′ X1 = X1 T, X1 Λ1 = X1 Λ1 T, Γ = ΓT ′ , Y1 = Y1 T ′ and Y1 Λ′ 1 = Y1 Λ1 T .

SOME INVERSE EIGENVALUE PROBLEMS FOR THE QUADRATIC PENCIL

11

Thus, conjugating (1.6) gives Z1 = (T ′ )T Z1 T and, conjugating (3.9), we get ΦT T Z 1 T ′ which implies that Φ = ΦT . Therefore, F = M (X1 Λ1 T )(T T ΦT ) = F and G = ?K (X1 T )(T T ΦT ) = G showing that F and G are real matrices. Remark 3.4. The results of Theorem 3.3 provide a parametric solution to Problem 1.1 in the multi-input case. The freedom of choosing these parameters can be conveniently exploited to obtain a solution with certain desirable properties such as the one having minimal norm, etc. See Section 6 for further discussions. Based on the Theorem 3.3 we can state the following algorithm. Algorithm 3.5. (An Algorithm For the Multi-input Partial Eigenvalue Assignment Problem for the Quadratic Pencil). Inputs: 1. The n × n matrices M, K , and D; M = M T > 0, K = K T and D = DT . 2. The n × m control (input) matrix B . 3. The set {?1 , · · · , ?p }, closed under complex conjugation. 4. The self-conjugate subset {λ1 , . . . , λp } of the open-loop spectrum {λ1 , . . . , λp ; λp+1 , . . . , λ2n } and the associated eigenvector set {x1 , . . . , xp }. Outputs: The feedback matrices F and G such that the spectrum of the closed-loop pencil Pc (λ) = λ2 M +λ(D?BF T )+(K ?BGT ) is {?1 , . . . , ?p , λp+1 , . . . , λ2n }. Assumptions: 1. The quadratic pencil is (partially) controllable with respect to the “eigenvalues to be assigned” ?1 , . . . , ?p . 2. {λ1 , . . . , λp } ∩ {λp+1 , . . . , λ2n } = ?. Step 1. Form Λ1 = diag(λ1 , . . . , λp ) and X1 = (x1 , . . . , xp ). Step 2. Choose arbitrary vectors γ1 , . . . , γp in such a way that ?j = ?k implies γj = γk and solve for y1 , . . . , yp : (?2 j M + ?j D + K )yj = Bγj , j = 1, . . . , p. Step 3. Form

T T Z1 = Λ′ 1 Y1 M X1 Λ1 ? Y1 KX1 ,

=

ΓT ′ ,

where Y1 = (y1 , . . . , yp ) and Λ′ 1 = diag(?1 , . . . , ?p ). If Z1 is ill-conditioned, then return to Step 2 and select di?erent vectors γ1 , . . . , γp . Step 4. Form Γ = (γ1 , γ2 , . . . , γp ) and solve for Φ:

T ΦZ 1 = Γ.

Step 5. Form F = G = M X 1 Λ 1 ΦT ?KX1 ΦT .

12

BISWA N. DATTA AND DANIIL R. SARKISSIAN

4. Solution to Problem 1.2 The solution process consists of two stages: ?, F ? , and G ? (generally complex) which satisfy Stage I. Determine matrices B (4.1) ?F ? T )Y Λ′ + (K ? B ?G ? T )Y M Y (Λ′ )2 + (D ? B = 0, where Y = (Y1 , X2 ); Y1 = (y1 , y2 , . . . , yp ), X2 = (xp+1 , . . . , x2n ), and Λ′ = (Λ′ 1 , Λ2 ) = diag(?1 , ?2 , . . . , ?p , λp+1 , . . . , λ2n ). ?, F ? , and G ? in Stage I, ?nd real matrices B , F , and G such that Stage II. From B ?F ? T and BGT = B ?G ?T ; BF T = B solving Problem 1.2. Let’s ?rst focus on Stage I. Let Λ′ 1 = diag(?1 , . . . , ?p ). Suppose that the triplet ? ?, G ? ) is a solution. Then (4.1) implies (B, F (4.2)

2 ′ M Y1 (Λ′ 1 ) + DY1 Λ1 + KY1

=

? F ? T Y1 . ? T Y1 Λ′ + G B 1

?, F ? , and G ? constitute a solution to Problem 1.2, then for any invertible Note that if B ? = BW ? , F ? =F ? W ?T , and G ? = GW ? ?T also constitute a solution; because W, B T T T T ?F ? =B ?F ? and B ?G ? =B ?G ? . Denote B ? T Y1 Λ′ + G ? T Y1 . (4.3) W = F

1

? = BW ? ? and G ?. Then, provided that W is invertible, B is admissible for some F Thus we can take ? = M Y1 (Λ′ )2 + DY1 Λ′ + KY1 (4.4) B

1 1

by virtue of (4.2) and (4.3). Relations (4.4) and (4.1) together imply that (4.5) ? T Y1 Λ′ ?T F 1 + G Y1 = I. It was shown in Theorem 3.3 that for any Φ, the matrices ? = M X1 Λ1 ΦT and G ? = ?KX1 ΦT (4.6) F satisfy ? ?T ? ?T M X2 Λ 2 2 + (D ? B F )X2 Λ2 + (K ? B G )X2 Substituting (4.6) into (4.5), we obtain Φ =

T T Λ 1 X1 M Y1 Λ′ 1 ? X1 KY1 ?1

=

0.

? and G ? can be determined. from which F Now, consider Stage II. Since ?j = ?k implies yj = yk , as in the proof of Theorem 3.3, there exist permutation matrices T and T ′ such that

′ ′ ′ 2 ′ ′ 2 X1 = X1 T, X1 Λ1 = X1 Λ1 T, Y1 = Y1 T ′ , Y1 Λ′ 1 = Y1 Λ1 T and Y1 (Λ1 ) = Y1 (Λ1 ) T .

Thus, Z1 = (T ′ )T Z1 T and using (4.4) and (4.6) we obtain

2 ′ ′ ′ ′ ? = M Y1 (Λ′ B 1 ) T + DY1 Λ1 T + KY1 T

= = =

? ′, BT ?F ? T and B ?G ?T , B

?F ? T = BT ? ′ (M X1 Λ1 T (T T Z ?1 T ′ ))T = BM ? X 1 Λ 1 Z ?1 B 1 1

?1 ?G ? T = BT ? ′ (?KX1 T (T T Z ?1 T ′ ))T = ?BKX ? B 1 Z1 1

?F ? T and G ?G ? T are real matrices. which implies that both B

SOME INVERSE EIGENVALUE PROBLEMS FOR THE QUADRATIC PENCIL

13

De?ne now the real n × 2n matrix H ? F ?T , G ?T = B

and let LR = H be a factorization of H ; where L and R are, respectively, of order n × m and m × 2n. Then we can take B to be L, the ?rst n columns of R to be F T , and the last n columns of R to be GT . Either the economy size QR factorization of H or the economy singular value decomposition of H can be used to compute B , F , and G (see [GVL] or [BNDa]). The above discussion leads to the following theorem. Theorem 4.1. (Solution to the Partial Eigenstructure Assignment for a Quadratic Pencil). Let X1 = (x1 , x2 , . . . , xp ), Y1 = (y1 , y2 , . . . , yp ), Λ1 = diag(λ1 , λ2 , . . . , λp ), and Λ′ 1 = diag(?1 , ?2 , . . . , ?p ). Then provided that the matrix Z1 ? B ? F =

T T Λ′ 1 Y1 M X1 Λ1 ? Y1 KX1

? F ?, G ? ) de?ned by is nonsingular, the triplet (B,

2 ′ = M Y1 (Λ′ 1 ) + DY1 Λ1 + KY1 , ?1 = M X1 Λ 1 Z1 , and

? = ?KX1 Z ?1 G 1 constitutes a (possibly complex) solution to the Problem 1.2. ? F ?, G ? ) by taking A solution with real B , F , and G is obtained from the triplet (B, ? F ?T , G ?T . the economy size QR factorization or the SVD of the real matrix H = B If QR factorization is used and if LR = H is the economy QR factorization of H , then B F

T T

= = =

L, (r1 , r2 , . . . , rn ) and (rn+1 , rn+2 , . . . , r2n );

G

where R = (r1 , . . . , r2n ). If SVD H = U ΣV T is used then the above formulae could be used either with L = U, R = ΣV T , or with L = U Σ, R = V T . Based on the Theorem 4.1 we can state the following algorithm. Algorithm 4.2. (An Algorithm For the Partial Eigenstructure Assignment Problem for a Quadratic Pencil). Inputs: 1. The n × n matrices M, K , and D; M = M T > 0, K = K T , D = DT . 2. The set of p numbers {?1 , · · · , ?p } and the set of p vectors {y1 , . . . , yp }, both closed under complex conjugation. 4. The self-conjugate subset {λ1 , . . . , λp } of the open-loop spectrum {λ1 , . . . , λp ; λp+1 , . . . , λ2n } and the associated eigenvector set {x1 , . . . , xp }.

14

BISWA N. DATTA AND DANIIL R. SARKISSIAN

Outputs: The feedback matrices F and G such that the spectrum of the closed-loop pencil Pc (λ) = λ2 M +λ(D?BF T )+(K ?BGT ) is {?1 , . . . , ?p , λp+1 , . . . , λ2n } and the eigenvectors corresponding to ?1 , . . . , ?p are y1 , . . . , yp , respectively. Assumptions: 1. {λ1 , . . . , λp } ∩ {λp+1 , . . . , λ2n } = ?. 2. ?j = ?k implies yj = yk for all 1 ≤ j, k ≤ p. Step 1. Obtain the ?rst p eigenvalues λ1 , . . . , λp that need to be reassigned and the corresponding eigenvectors x1 , . . . , xp . Form Λ1 = diag(λ1 , . . . , λp ), Λ′ 1 = diag(?1 , . . . , ?p ), X1 = (x1 , . . . , xp ) and Y1 = (y1 , . . . , yp ). ? and Z1 Step 2. Form the matrices B ? B Z1 ? ∈ Step 3. Solve for H = =

2 ′ M Y1 (Λ′ 1 ) + DY1 Λ1 + KY1 , T T Λ′ 1 Y1 M X1 Λ1 ? Y1 KX1 .

p×2n T ? T T Z1 H = (ΛT 1 X1 M, ?X1 K )

?H ?. and form H = B Step 4. Compute the economy size QR decomposition of H = BR. m×2n m× n Step 5. Partition R ∈ ? to get F, G ∈ ? R = (F T , GT ). (This step can also be implemented using the SVD of H , as shown in Theorem 4.1). Note 4.3. MATLAB codes for Algorithm 3.5 and 4.2 are available from the authors upon request. 5. Illustrative Numerical Examples Example 5.1. We illustrate Algorithm 3.5 for the quadratic pencil P (λ) = λ2 M + λD + K with random matrices M , D, K and B given by ? ? ? ? 1.3525 1.2695 0.7967 0.8160 1.4685 0.7177 0.4757 0.4311 ?1.2695 1.3274 0.9144 0.7325? ?0.7177 2.6938 1.2660 0.9676? ? ? ? M =? ?0.4757 1.2660 2.7061 1.3918? , D = ?0.7967 0.9144 0.9456 0.8310? 0.8160 0.7325 0.8310 1.1536 0.4311 0.9676 1.3918 2.1876 ? ? ? ? 1.7824 0.0076 ?0.1359 ?0.7290 0.3450 0.4578 ? 0.0076 ?0.0579 0.7630? 1.0287 ?0.0101 ?0.0493? ? ? ? K=? ??0.1359 ?0.0101 2.8360 ?0.2564? and B = ?0.5967 0.9990? ?0.7290 ?0.0493 ?0.2564 1.9130 0.2853 0.3063 The open-loop eigenvalues of P (λ), computed via MATLAB, are ?0.0861 ± 1.6242i, ?0.1022 ± 0.8876i, ?0.1748 ± 1.1922i and ? 0.4480 ± 0.2465i. We will solve Problem 1.1, reassigning only the most unstable pair of the openloop eigenvalues; namely, ?0.0861 ± 1.6242i to the locations ?0.1 ± 1.6242i. That is, we want the closed-loop pencil Pc (λ) = λ2 M + λ(D ? BF T ) + K ? BGT to have the spectrum (5.1) {?0.1 ± 1.6242i, ?0.1022 ± 0.8876i, ?0.1748 ± 1.1922i, ?0.448 ± 0.2465i}.

SOME INVERSE EIGENVALUE PROBLEMS FOR THE QUADRATIC PENCIL

15

The em random choices of γ1 and γ2 produce matrix Z1 with the condition ?1 number Cond2 (Z1 ) = ||Z1 ||2 ||Z1 ||2 = 1.64 and the feedback matrices ? ? ? ? 3.3599 ?2.4691 ?4.1868 1.4318 ? ?2.5437 1.2692 ? ? ?0.3352 0.4506 ? ? ? ? F =? ? 10.3080 ?6.8494 ? and G = ? ?6.4921 1.2369 ? ?8.0702 5.6643 7.2495 ?2.2698 with the norms ||F ||2 = 16.6 and ||G||2 = 10.99 such that the spectrum of the closed-loop pencil Pc (λ) is precisely (5.1). The method, essentially similar to the method 2/3 in [KNVD], that uses the freedom in chosing vectors γ1 and γ2 in order to improve the condition number (robust) of Z1 , converges after 3 steps, producing the matrix Z1 with the condition (robust) number Cond2 (Z1 ) = 1.1 and the feedback matrices ? ? ? ? 1.3988 ?0.7501 ?0.6532 0.4781 ? ?0.0075 ?0.0285 ? ? 0.7079 ?0.4183 ? (robust) ? ? =? F (robust) = ? ? 2.5185 ?1.2554 ? ? ?2.2620 1.5349 ? and G ?2.4964 1.3185 1.6636 ?1.1733 with the norms ||F (robust) ||2 = 3.6 and ||G(robust) ||2 = 4.3 such that the spectrum (robust) of the “robust” closed-loop pencil Pc (λ) is precisely (5.1). We call the last closed-loop pencil “robust” because, aside from the mere reduction in the norm of the feedback matrices, our numerical experiments suggest (robust) that the eigenvalues of Pc (λ) are less a?ected by the random perturbations of feedback matrices. This is illustrated Figure 1 that plots the convex hulls of the closed-loop eigenvalues, when the feedback matrices F , G, F (robust) and G(robust) are perturbed, respectively, by ?F , ?G, ?F (robust) and ?G(robust) , such that ||?F ||2 < 0.01||F ||2 , ||?G||2 < 0.01||G||2 and ||?F (robust) ||2 < 0.01||F (robust) ||2 , ||?G(robust) ||2 < 0.01||G(robust) ||2 , with 200 random perturbations. Example 5.2. The same quadratic pencil is now used to illustrate Algorithm 4.2. We will solve Problem 1.2, reassigning again the most unstable pair of the openloop eigenvalues; namely, ?0.0861 ± 1.6242i to the same locations ?0.1 ± 1.6242i. That is, we want the closed-loop pencil to have the spectrum (5.1). Let the matrix of vectors to be assigned be: ? ? 1.0000 1.0000 ?0.0535 + 0.3834i 0.0535 ? 0.3834i? ? Y1 = ? ?0.5297 + 0.0668i 0.5297 ? 0.0668i? . 0.6711 + 0.4175i 0.6711 ? 0.4175i Algorithm 4.2 produces the control matrix ? ? ?0.3814 ?0.5751 ??0.5555 ?0.4821? ? B=? ??0.5191 0.4311 ? ?0.5258 0.5010

16

2

BISWA N. DATTA AND DANIIL R. SARKISSIAN

1.5

1

0.5

0

?0.5

?1

?1.5

?2 ?0.5

?0.4

?0.3

?0.2

?0.1

0

0.1

0.2

Figure 1. The convex hulls of closed-loop eigenvalues under 200 random 1% perturbation of the feedback matrices for quadratic (robust) pencils Pc (λ) (solid lines) and Pc (λ) (dashed lines). with ||B ||2 = 1 and the feedback matrices ? ? ? ? 0.1688 ?1.3245 ?8.1693 ?0.2320 ? ? 2.5326 0.2236 ? ?0.4130? ? ? and G = ? 2.0584 F =? ? ? ??20.6466 ?5.7223 ?3.0184? 0 0.9821 2.4989 18.0019 0.2962 with the norms ||F ||2 = 28.6976 and ||G||2 = 7.07673. The spectrum of the closedloop pencil Pc (λ) is precisely (5.1) and the columns of Y1 are the eigenvectors, corresponding to the eigenvalues ?0.1 ± 1.6242i. Remark 5.3. Examples 5.1 and 5.2 are purely illustrative ones. A real-life example involving an 211 × 211 quadratic pencil with sparse matrices has been solved in [DS], using Algorithm 3.5. 6. Conclusions and Future Research An uniform treatment of solutions, both theoretical and algorithmic, is presented for two important inverse eigenvalue problems for the quadratic pencil (1.2). The two problems are the problems of partial eigenvalue assignment and the partial eigenstructure assignment arising in feedback control of the matrix second-order control systems (1.1). The solutions have the following important practical features: 1. They are “direct” in the sense that they are obtained directly in the matrix second-order settings without resorting to a ?rst-order reformulation, so that the important structures, such as sparsity, de?niteness, symmetry, etc. can be exploited.

SOME INVERSE EIGENVALUE PROBLEMS FOR THE QUADRATIC PENCIL

17

2. They are “partial modal”, meaning that only a part of the spectrum (in fact, only that part that needs to be reassigned) and the corresponding eigenvectors are required. 3. No spill-over occurs; that is, the eigenvalues and eigenvectors that are not required to be altered do not become a?ected by application of feedback. 4. No explicit knowledge of damping matrix is needed in ?nding the feedback matrices. Damping is needed only to compute the small number of eigenvalues and the corresponding eigenvectors that need to be reassigned and in ?nding the matrix B in Problem 1.2. Future Research. We conclude this section by mentioning some future research problems in this area. Our discussions in this paper reveal that the “direct partial modal” approach for feedback problems is quite attractive for practical computations even for very large and sparse systems. Thus, some further studies on this approach for the problems under consideration and related problems are in order. The studies should include: (i) Robust eigenvalues assignment for the quadratic pencil (1.1). (ii) Finding a computational algorithm for minimizing the feedback norms, both for Problem 1.1 and Problem 1.2. (iii) Finding an optimization-based algorithm, which allows one to choose the eigenvalues to be assigned from the speci?ed stability subregion of the complex plane in such a way, that the conditioning of the closed-loop eigenvalues is a small as possible. (iv) Extending the partial modal approach to the feedback problems of distributed parameter systems. To justify the studies (i)-(iii), we ?rst consider the following well-known fact: Even if a feedback matrix is computed accurately by a numerically viable algorithm, there is no guarantee in practice that the eigenvalues of the closed-loop pencil are the same as those prescribed. There are several factors associated with this phenomenon (see [BNDb] for details): (a) (b) (c) (d) Conditioning of the eigenvector matrix of the closed-loop system. Large norm of the feedback matrix. Wide separation of the closed-loop and open-loop eigenvalues. Nearness to uncontrollability: Small perturbations in the data can make the system uncontrollable.

Factor (a) prompts the robust eigenvalue assignment. The robust eigenvalue assignment is concerned with choosing the eigenvector matrix of the closed-loop system in such a way that the condition number of the eigenvector matrix is as small as possible. Recall that the closed-loop eigenvector matrix for Problem 1.1 is the matrix (6.1) Y1 Y1 Λ′ 1 X2 X2 Λ 2 .

Since the matrices X2 and Λ2 are to remain unaltered, the problem then reduces Y1 to choosing the matrix in such a way that the condition number of (6.1) Y1 Λ′ 1 is minimized.

18

BISWA N. DATTA AND DANIIL R. SARKISSIAN

This can perhaps be done using the same type of technique as used in the well-known paper [KNVD] for the ?rst-order system. Empirical results suggest that this is indeed a good thing to do and the condition number obtained this way, in each case of our numerical experiments, has turned out to be smaller then that Y1 obtained without applying any speci?c criterion for choosing . Some Y1 Λ′ 1 more de?nitive work needs to be done. See also a related paper by [CD] where two numerical algorithms for robust eigenvalue assignment for a quadratic pencil have been proposed for the full-order eigenvalue assignment. The consideration of factor (b) gives rise to (ii). There exists an algorithm due to [KFB] for minimizing the norm of the feedback matrix for the ?rst-order system. Factor (c) is related to (iii). An optimization-based algorithm has been recently proposed in [CLRa] and [CLRb] for the ?rst-order model. An analogous algorithm for the quadratic pencil (1.2) is to be developed. Finally, regarding (iv), we note that the second-order model (1.2) is just a discretized approximation (say, by the ?nite element method) of a distributed parameter system; thus, in spite of the fact that a second-order model is much used in practice for convenience, it has some severe limitations. For example, suppose that starting with a distributed model, ?rst a second-order model is obtained by discretization and then Problems 1.1 and 1.2 are solved using the direct partial-modal approach of this paper. Even though Theorem 2.1 and Theorem 3.1 guarantee no spill-over of the 2n ? p eigenvalues that are not required to be reassigned, there still remains obvious uncertainity with the remaining in?nite number of eigenvalues of the in?nite-order system. It is, therefore, desirable (though extremely hard) to obtain solutions directly from the distributed model without going through a discretization procedure. Some attempts, however, have been made already in this direction. Generalizing the results of [DERb], a solution to a single-input version of Problem 1.1 for a distributed gyroscopic system entirely in terms of the distributed parameters has been recently obtained in [DRS] (see also [YMR]). Speci?cally, the following problem has been solved: Given the self-adjoint positive de?nite operators M and K , a gyroscopic operator G, and a self-conjugate set ? = {?1 , . . . , ?p }, ?nd feedback functions f (x) and g (x) such that each member of ? is an eigenvalue of the closed-loop operator system M ?ν (t, x) ? 2 ν (t, x) +G + Kν (t, x) 2 ?t ?t = b(x)(f (x), ?ν (t, x) ) + b(x)(g (x), ν (t, x)), ?t

where (·, ·) is a scalar product, and the remaining in?nite number of eigenvalues λp+1 , λp+2 , . . . remain the same as those of the open-loop operator system M ?u(t, x) ? 2 u(t, x) +G + Ku(t, x) = ?t2 ?t 0.

The solution has been obtained in terms of the quantities given and entirely in the distributed parameter setting (that is, without any use of the discretization technique). The results obtained in this paper are the ?rst and only results available for inverse eigenvalue problems for a quadratic operator pencil. Clearly, much remains to be done in this area.

SOME INVERSE EIGENVALUE PROBLEMS FOR THE QUADRATIC PENCIL

19

References

[CLRa] D. Calvetti, B. Lewis and L. Reichel, On the solution of the single input pole placement problem, in Mathematical Theory of Networks and Systems, eds. A. Beghi, L. Finesso and G. Picci, Il Poliografo, Padova (1998), 585–588. [CLRb] , On the selection of the poles in the single input pole placement problem, To appear in Linear Alg. Appl. (special issue dedicated to Hans Schneider) (1999). [CD] E. K. Chu and B. N. Datta, Numerically Robust Pole Assignment for the Second-Order Systems, Int. J. Control 4 (1996), 1113–1127. [MTC] M. T. Chu, Inverse Eigenvalue Problems, SIAM Rev. 40 (1998), no. 1, 1–39. [BNDa] B. N. Datta, Numerical Linear Algebra and Applications, Brook/Cole Publishing Co., Paci?c Grove, California (1998). [BNDb] B. N. Datta, Numerical Methods for Linear Control Systems Design and Analysis, Academic Press: New York (1999) (To appear). [DERa] B. N. Datta, S. Elhay and Y. M. Ram, An algorithm for the partial multi-input pole assignment problem of a second-order control system, Proceedings of the IEEE Conference on Decision and Control (1996), 2025–2029. [DERb] , Orthogonality and Partial Pole Assignment for the Symmetric De?nite Quadratic Pencil, Linear Algebra and its Applications 257 (1997), 29–48. [DERS] B. N. Datta, S. Elhay, Y. M. Ram and D. R. Sarkissian, Partial Eigenstructure Assignment for the quadratic Pencil, Journal of Sound and Vibration, in press (1999). [DRS] B. N. Datta, Y. M. Ram and D. R. Sarkissian, Spectrum modi?cation for gyroscopic systems, to be submitted for publication (1999). [DR] B. N. Datta and F. Rinc? on, Feedback Stablization of the Second-Order Model: A Nonmodal Approach, Lin. Alg. Appl. 188 (1993), 138–161. [DS] B. N. Datta and D. R. Sarkissian, Multi-input Partial Eigenvalue Assignment for the Symmetric Quadratic Pencil, Proceedings of the American Control Conference (1999), 2244–2247. [GVL] G. Golub and C. Van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore (1984) & 3rd edition (1996). [DJI] D. J. Inman, Vibrations: Control, Measurement and Stability, Prentice Hall (1989). [IK] D. J. Inman and A. Kress, Eigenstructure Assignment via inverse eigenvalue methods, AIAA J. Guidance, Control and Dynamics 18 (1995), 625–627. [KNVD] J. Kautsky, N. K. Nichols and P. van Dooren, Robust pole assignment in linear state feedback, Int. J. Contr. 41 (1985), no. 5, 1129–1155. [KFB] L. H. Keel, J. A. Fleming, S. P. Bhattacharyya, Minimum norm pole assignment via Sylvester equation, Contemporary Mathematics 47 (1985), 265–272. [PC] B. N. Parlett and H. C. Chen, Use of inde?nite pencils for computing damped natural modes, Lin. Alg. Appl. 140 (1990), 53–88. [YMR] Y. M. Ram, Pole assignment for the vibrating rod, Quarterly Journal of Mechanics and Applied Mathematics 51 (1998), no. 3, 461–476. [YS] Y. Saad, A projection method for partial pole assignment in linear state feedback, IEEE Trans. Auto. Control 33 (1988), no. 3, 290–297. [SBFV] G. L. G. Sleijpen, A. G. L. Booten, D.R. Fokkema and H. A. van der Vorst, JacobiDavidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT 36 (1996), no. 3, 595–633. Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, 60115 E-mail address : dattab@math.niu.edu Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, 60115 E-mail address : sarkiss@math.niu.edu

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