On the Parametric Solution to the Second-Order Sylvester Matrix Equation EVF 2 − AVF − CV = BW

This paper considers the solution to a class of the second-order Sylvester matrix equation EVF2−AVF −CV = BW . Under the controllability of the matrix triple (E,A,B), a complete, general, and explicit parametric solution to the second-order Sylvester matrix equation, with the matrix F in a diagonal form, is proposed. The results provide great convenience to the analysis of the solution to the second-order Sylvester matrix equation, and can perform important functions in many analysis and design problems in control systems theory. As a demonstration, an illustrative example is given to show the effectiveness of the proposed solution.


Introduction
In vibration and structure analysis, second-order linear systems capture the dynamic behaviour of many natural phenomena and have found wide applications, whose state equation can be given as follows: where q ∈ R n is the generalized coordinate vector, u ∈ R r is the input vector, and E, A, B, C are matrices of appropriate dimensions.The matrices in (1.1) satisfy the following assumption.
Assumption 1.1.The matrix triple (E,A,B) is R-controllable, that is, Assumption 1.2.The matrix B is of full rank, that is, rank(B) = r.
Associated with the second-order linear system (1.1), a generalized second-order Sylvester matrix equation can be written in the form of EV F 2 − AV F − CV = BW, (1.3) where E,A,C ∈ R n×n , B ∈ R n×r and F ∈ R p×p are known matrices; matrices W ∈ R r×p and V ∈ R n×p are to be determined.When E = 0 n , A = −I n , the generalized second-order Sylvester matrix equation (1.3) becomes the following generalized Sylvester matrix equation (1.4) When E = 0 n , A = −B = −I n , W = −Q, the generalized second-order Sylvester matrix equation (1.3) becomes the following normal Sylvester matrix equation Further, if we let C = −F T in (1.5), the above normal Sylvester matrix equation becomes the following well-known Lyapunov matrix equation Thus we can find that the second-order Sylvester equation (1.3) is more general than the first-order Sylvester equations (1.4) and (1.5).As we all know, the Sylvester equation is directly concerned with some control problems for linear systems such as eigenvalue assignment [1,2], observer design [3], eigenstructure assignment design [4][5][6], constrained control [7], and so forth, and has been studied by many authors (see [3][4][5][6], and the references therein).
A solution to the normal Sylvester matrix equation (1.5) has been studied by several researchers.Jameson [8] and Souza and Bhattacharyya [9] gave solutions to this matrix equation in terms of the controllability and observability matrices of some matrix pairs, and Hartwig [10] gave a solution to this equation in terms of the inverse of the related Sylvester resultant, while Jones and Lew [11] presented a solution to this equation in terms of the principal idempotents and nilpotents of the coefficient matrices.Hearon [12] considered the case of Q as being a rank one matrix, and presented some conditions for the matrix V to be nonsingular.For the Lyapunov matrix equation (1.6) with the matrix C in companion form, many authors have considered the solution.Particularly, Sreeram and Agathoklis [13] presented an iterative method based on the Routh's table.Zhou and Duan [14] presented an explicit solution to the generalized Sylvester matrix equation (1.4), and a simple and effective approach for parametric pole assignment is proposed as a demonstration.Duan and Wang [15] proposed two analytical general solutions of (1.3), and by utilizing these results, they investigated many control problems such as the robust control problem [16], the eigenstructure assignment problem [17,18], the model reference control problem [19], the reconfiguring control problem [20], and so on.
Different from the condition in [15], this paper provides another type of general parametric solutions to the second-order Sylvester matrix equation (1.3).In the next section, some preliminaries to be used in this paper are given.In Sections 3 and 4, the general solutions to the generalized Sylvester matrix equation (1.3) are presented in two cases: the undetermined diagonal matrix F and the determined diagonal matrix F. As a demonstration of the proposed solutions to the second-order Sylvester matrix equation, a numerical example is illustrated by utilizing the proposed two algorithms, and general complete parametric forms for the solution to the second-order Sylvester matrix equation are established in Section 5.In Section 6, concluding remarks are drawn.

Preliminaries
If the matrix F ∈ R p×p in the matrix equation (1.3) is not diagonal, there must exist a nonsingular matrix M ∈ R p×p such that where λ 1 ,λ 2 ,...,λ p are the eigenvalues of F. Thus (1.3) can be changed into the following form: Then the above equation is changed into where F = diag(λ 1 ,λ 2 ,...,λ p ) is a diagonal Jordan form.Therefore, without loss of generality, we assume that the matrix F in (1.3) is in a diagonal Jordan form.Let Γ = {s i ,s i ∈ C, re(s i ) < 0, i = 1,2,..., p}, which is symmetric about the real axis and whose elements are distinct, be the set of eigenvalues of F. Thus the diagonal Jordan matrix F can be written in the following form: then (1.3) can be further changed into The matrices V , V , and W can be written by their columns as then (2.8) can be decomposed into or In the next two sections, we will consider the solution to the second-order Sylvester matrix equation (1.3) in two cases: one is the case of the undetermined diagonal matrix F, and the other is the case of the determined diagonal matrix F.

Case of the undetermined diagonal matrix F
If the matrix triple (E,A,B) is R-controllable, there exists a pair of unimodular matrices Partition Q(s) as follows: where Then we have the following lemma, which gives the solutions to (2.8).
Lemma 3.1.Assume that the matrix triple (E,A,B) is R-controllable, then all the solutions V and W in (2.8) can be given by their column vectors as Wang Guo-Sheng et al. 5 or where matrices .., p, are a group of free parametric vectors.
Proof.Substitute (3.2) into (3.1),we can obtain It is clear to see that (3.5) is equivalent with the following equation By utilizing (3.3a), we can obtain thus (3.3) are the solutions to (2.8).Now let us show that the vectors v i and w i satisfying (2.10a) can be expressed in the form of (3.3a).Premultiplying by P(s i ) both sides of (2.10b) and using (3.1) yield which produces Substituting (3.11) into (3.9),we can get (3.3a).Finally, noticing that the equivalence between (3.3a) and (3.3b) is obvious, the proof is completed.
For control applications, the second-order linear system (1.1) is usually transformed into the following first-order linear system where Wang Guo-Sheng et al. 7 The above matrix can easily be transformed, by elementary transformations, into the following form: Partition L(s) as follows: where Equation (2.7) can be equivalently written in the following vector form: which is clearly in the form of (3.20).Applying Lemmas 3.2 and 3.3 to (3.29) gets ) where (2) Solve the pair of unimodular matrices H(s) and L(s) satisfying (3.18) by applying elementary transformations to the matrix [ Q12(s)P(s)C+sI −Q11(s) ] and partition matrix L(s) as in (3.19).
(3) Establish the general parametric forms for the matrices V and W according to (3.30) and (3.32), respectively.
Remark 3.5.The advantage of the proposed solutions to the second-order Sylvester equation is that it can provide the general explicit parametric solution.In the case of the undetermined eigenvalues of the matrix F, the free parameters in the solutions are composed of two parts: one is clearly the group of parameter vectors g i ∈ C r , i = 1,2,..., p, while the other is the group of eigenvalues s i , i = 1,2,..., p of the matrix F. Both the two parts of degrees of freedom may be selected to design some control problems such as the eigenstructure assignment problem, the model tracking problem, the observer design problem, and so forth.

Case of the determined diagonal matrix F
When the eigenvalues s i , i = 1,2,..., p of the matrix F are prescribed, the solutions to the generalized Sylvester matrix equation (1.3) are actually dependent on the constant matrices which satisfy the following: Therefore, we can, instead of seeking the polynomial matrices P(s) and Q(s) satisfying the relation (3.1), find these constant matrices directly so as to avoid polynomial matrix manipulations.
The constant matrices P i = P(s i ) ∈ C n×n and Q i = Q(s i ) ∈ C (n+r)×(n+r) , i = 1,2,..., p can be easily obtained by singular value decompositions.In fact, by applying singular value decomposition to the matrix [ A−siE B ], we obtain two orthogonal matrices P i ∈ C n×n and Q i ∈ C (n+r)×(n+r) satisfying the following equations: where Σ i , i = 1,2,..., p are diagonal matrices with positive diagonal elements.By rearranging (4.3) in the following form: we can obtain the constant matrices P i , and Q i , i = 1,2,..., p satisfying (4.2) as Partitioning the matrix Q i , i = 1,2,..., p is as follows: where Then we can give a corollary of Lemma 3.1 as follows.
Corollary 4.1.Let the matrix triple (E,A,B) be R-controllable, and let the diagonal matrix F be defined as in (2.6) with s i , i = 1,2,..., p known.Then all the solutions V and W in (2.8) can be given by their columns as or, equivalently, as where , we obtain two orthogonal matrices H i ∈ R n×n and L i ∈ R (n+r)×(n+r) satisfying the following equation: where Ξ i , i = 1,2,..., p, are diagonal matrices with positive diagonal elements.Note that (4.9) can be arranged in the following form:  , and Σ i be given by ( 4.3) and (4.6).Then all the vectors y and z satisfying Wang Guo-Sheng et al. 11 are given as with g ∈ C r being an arbitrary parameter vector.With the help of the above two corollaries, via a similar development as in Section 3, the following theorem can be obtained for solution to Problem SOS in the case that the eigenvalues of the matrix F are given a priori.
Theorem 4.3.Let the matrix triples (E,A,B) and (E ,A ,B ) both be R-controllable, and let the desired closed-loop eigenvalues s i , i = 1,2,..., p be given a priori.Then all the solutions V and W to Problem SOS can be given by their column vectors, respectively: where g i ∈ C r , i = 1,2,..., p, are a group of parameter vectors.Based on the above theorem, an algorithm for solution to Problem SOS in the case of the determined eigenvalues of the matrix F can be given as follows.
(3) Establish the general parametric forms for the matrices V and W according to (4.14) and (4.15), respectively.Remark 4.4.In the case that the eigenvalues of the matrix F are given a priori, the proposed solution to the second-order Sylvester equation also provides the general explicit parametric solution.The free parameters are composed of the group of parameter vectors g i ∈ C r , i = 1,2,..., p, which can also be selected to design some control problems.

An illustrative example
Consider a second-order Sylvester equation in the form of (1.1) with the following parameters: ( It can be easily verified that the matrix triples (E,A,B) and (E ,A ,B ) are both R-controllable.

Case of the undetermined diagonal matrix F.
We assume that the matrix F = diag(s 1 ,s 2 ,s 3 ,s 4 ,s 5 ,s 6 ).In the following, we solve the second-order Sylvester equation using Algorithm SOS1.
(1) By applying some elementary transformations to matrix [ A−sE B ], a pair of unimodular matrices P(s) and Q(s) satisfying (3.1) can be obtained as (2) Note that the matrix (5.7) According to (3.30), the matrix V in (1.3) is given by its column vectors as (5.9) Wang Guo-Sheng et al. 13 Thus from (3.32), the matrix W in (1.3) is given by its column vectors as (5.10) 5.2.Case of the determined diagonal matrix F. We assume that the matrix F = diag(s 1 , s 2 ,s 3 ), where s 1,2 = −1 ± 2 j, s 3 = −3.In the following, we solve the second-order Sylvester equation using Algorithm SOS2.

Conclusions
This paper considers the solutions to a second-order Sylvester matrix equation.Under the controllability of some matrix triples, complete, general, and explicit parametric solutions to the generalized second-order Sylvester matrix equation are proposed in two cases of the undetermined eigenvalues of the matrix F and the determined eigenvalues of the matrix F. The general explicit parametric solutions to the second-order Sylvester matrix equation are presented.These results provide great convenience to the analysis of the solution to the equation, and can perform important functions in many analysis and design problems in control systems theory.As a demonstration, a numerical example is offered to show the effectiveness of the proposed approaches.
2,..., p are a group of parametric vectors.Q 21 s i L 21 s i − Q 22 s i P s i CL 11 s i g i , i = 1,2,..., p. Algorithm [SOS1].(1) Solve the pair of unimodular matrices P(s) and Q(s) satisfying (3.1) by applying elementary transformations to matrix [ A−sE B ] and partition matrix Q(s) as in (3.2).
Theorem 3.4.Let the matrix triples (E,A,B) and (E ,A ,B ) both be R-controllable, then all the solutions V and W to Problem SOS are given by(3.30)and(3.32)intheir column vectors, respectively.