1.3 Multivariate Linear Matrix Equations
1.3.2 First-Order Generalized Sylvester Matrix Equations
In controller design of linear systems, the following Sylvester matrix equation is often encountered:
AX+BY =XF, (1.20)
whereA∈Rn×n,B∈Rn×r, and F∈Rp×pare known matrices, andX ∈Rn×pand Y ∈Rr×pare the matrices to be determined. This matrix equation plays a very vital role in eigenstructure assignment [130, 169], pole assignment [214], and so on. Its dual form is the following so-called Sylvester-observer matrix equation:
XA+YC =FX, (1.21)
whereA∈Rn×n,C∈Rm×n, and F∈Rp×pare known matrices, andX∈Rp×nand Y ∈Rp×mare the matrices to be determined. It is well-known that the existence of a Luenberge observer for linear systems can be characterized based on this equation [60]. A more general form of (1.20) is
AX+BY =EXF. (1.22) This generalized Sylvester matrix equation appears in the field of descriptor linear systems [81], and can be used to solve the problems of eigenstructure assignment [64] and output regulation [211] for descriptor linear systems. An important variation of (1.22), called generalized Sylvester-observer equation
XA+YC=FXE, (1.23)
where X andY need to be determined, arises in observer design [32, 254], fault detection [127] of descriptor linear systems. A more general form of the matrix equation (1.22) is the following equation
AX−EXF =B0Y+B1YF, (1.24)
which was studied in [82] for solving eigenstructure assignment in a type of general- ized descriptor linear systems. Besides the Sylvester matrix equations (1.20)–(1.23), in [305] the following matrix equation was investigated:
MXF−X=TY, (1.25)
where M ∈ Rn×n,F ∈ Rp×p, and T ∈ Rn×r are known matrices. Obviously, the matrix equations (1.20)–(1.25) are homogeneous. In fact, their nonhomogeneous counterparts have also been investigated. For example, the so-called regulator equa- tion
AX+BY =XF+R
was considered in [300]. The nonhomogeneous generalized Sylvester matrix equa- tion
AX+BY =EXF+R was studied in [78, 278]. The nonhomogeneous equation
AX−EXF=B0Y+B1YF+R
was studied in [86]. In the aforementioned equations, the largest degree of square matrices is 1. Due to this reason, all these equations are called first-order generalized Sylvester matrix equations.
1.3.2.1 Solution Approaches
There have been many numerical algorithms for solving these matrix equations.
In [60], an orthogonal-transformation-based algorithm was proposed to solve the Sylvester-observer matrix equation (1.21). In this algorithm, the matrix pair(A,C)
is first transformed via a unitary state-space transformation into staircase form. With such a transformation, the solution of the Sylvester-observer matrix equation can be obtained by a reduced dimensional matrix equation with Schur form. The advantage of this approach is that one can use more degrees of freedom in the equation to find a solution matrix with some desired robustness properties such as the minimum norm. In [45], a computational method for solving the matrix equation (1.21) was proposed whenAis large and sparse. This method uses Amoldi’s reduction in the initial process, and allows an arbitrary choice of distinct eigenvalues of the matrix F. The numerical aspects of the method in [45] was discussed in [30], and a strategy was presented for choosing the eigenvalues of F. In [22], in view of the design requirement the generalized Sylvester matrix equation (1.20) was first changed into a normal Sylvester matrix equation by choosing F in a block upper Hessenberg matrix and fixingY to a special matrix, and then a parallel algorithm was given by reducing the matrixAto lower Hessenberg form. In [44], an algorithm was proposed to construct an orthogonal solution of the Sylvester-observer matrix equation (1.21) by generalizing the classical Arnoldi method. In [31], a block algorithm was proposed to compute a full rank solution of (1.21). This algorithm does not require the reduction of the matrixA.
On the numerical solutions of (1.22) and (1.23), only a few results were reported in literature. In [32], a singular value decomposition (SVD) based block algorithm was proposed for solving the generalized Sylvester-observer matrix equation (1.23).
In this algorithm, the matrixFneeds to be chosen in a special block form, and the matricesE,A,andCare not necessarily reduced to any canonical forms. In [33], a new algorithm was proposed to numerically solve the generalized Sylvester-observer matrix equation (1.23). This algorithm can be viewed as a natural generalization of the well-known observer-Hessenberg algorithm in [22]. In this algorithm, the matrices E andAshould be respectively transformed into an upper triangular matrix and a block upper Hessenberg matrix by orthogonal transformation. The algorithm in [33]
was improved in [34], and was applied to state and velocity estimation in vibrating systems.
The aforementioned numerical approaches for solving the four first-order gen- eralized Sylvester matrix equations (1.20), (1.21), (1.22), and (1.23) can only give one solution each time. However, for several applications it is important to obtain general solutions of these equations. For example, in robust pole assignment prob- lem one encounters optimization problems in which the criterion function can be expressed in terms of the solutions to a Sylvester matrix equation [164]. WhenFis in a Jordan form, an attractive analytical and restriction-free solution was presented in [231] for the matrix equation (1.21). Reference [66] proposes two solutions to the Sylvester matrix equation (1.20) for the case where the matrixFis in a Jordan form.
One is in a finite iterative form, and the other is in an explicit form. To obtain the explicit solution given in [66], one needs to carry out a right coprime factorization of(sI−A)−1B(when the eigenvalues of the Jordan matrixFare undetermined) or a series of singular value decompositions (when the eigenvalues ofF are known).
When the matrixFis in a companion form, an explicit solution expressed by a Hankel matrix, a symmetric operator and a controllability matrix was established in [301].
In many applications, for example, model reference control [96, 100], Luenberger observer design [93], the Sylvester matrix equation in the form of (1.20) with F being an arbitrary matrix is often encountered. Therefore, it is useful and interesting to give complete and explicit solutions by using the general matrixFitself directly.
For such a case, a finite series solution to the Sylvester matrix equation (1.20) was proposed in [300]. Some equivalent forms of such a solution were also provided in that paper.
On the generalized Sylvester matrix equation (1.22), an explicit solution was provided in [64] when F is in a Jordan form. This solution is given by a finite iteration. In addition, a direct explicit solution for the matrix equation (1.22) was established in [69] with the help of the right coprime factorization (sE−A)−1B.
The results in [64, 69] can be viewed as a generalization of those in [66]. The case whereFis in a general form was firstly investigated in [303], and a complete parametric solution was presented by using the coefficient matrices of a right coprime factorization of(sE−A)−1B. In [247], an explicit solution expressed by generalized R-controllability matrix and generalized symmetric operator matrix was also given.
In order to obtain this solution, one needs to solve a standard unilateral matrix equation. In [262], an explicit solution was also given for the generalized Sylvester matrix equation (1.22) in terms of R-controllability matrix, generalized symmetric operator matrix and observability matrix by using Leverrier algorithm for descriptor linear systems. In [202], the generalized Sylvester matrix equation (1.22) was solved by transforming it into a linear vector equation with the help of Kronecker products.
Now, we give the results in [69, 303] on the generalized Sylvester matrix equation (1.22). Due to the block diagonal structure of Jordan forms, when the result is stated for the case of Jordan forms, the matrixFis chosen to be the following matrix
F=
⎡
⎢⎢
⎢⎢
⎣ θ 1
θ ...
...1 θ
⎤
⎥⎥
⎥⎥
⎦∈Cp×p. (1.26)
Theorem 1.10 ([69])Given matrices E,A∈Rn×nand B∈Rn×rsatisfying rank
sE−A B
=n, (1.27)
for all s ∈ C, let F be in the form of (1.26). Further, let N(s) ∈ Rn×r[s]and D(s)∈Rr×r[s]be a pair of right coprime polynomial matrices satisfying
(A−sE)N(s)+BD(s)=0. (1.28)
Then, all the solutions to the generalized Sylvester matrix equation (1.22) are given by
xk
yk
= N(θ)
D(θ)
fk+ d ds
N(θ) D(θ)
fk−1+ ã ã ã + 1 (k−1)!
dk−1 dsk−1
N(θ) D(θ)
f1, k ∈ I[1,p],
where fi∈Cr, i∈I[1,p], are a group of arbitrary vectors. In the preceding expres- sion, xkand yk, are the k-th column of the matrices X and Y , respectively.
Theorem 1.11 ([303]) Given matrices E,A ∈ Rn×n,B ∈ Rn×r, and F ∈ Rp×p satisfying (1.27) for all s∈C, let
N(s)=
ω−1
i=0
Nisi∈Rn×r[s],and D(s)= ω
i=0
Nisi∈Rr×r[s]
be a pair of right coprime polynomial matrices satisfying (1.28). Then, all the solu- tions to the generalized Sylvester matrix equation (1.22) are given by
X =N0Z+N1ZF+ ã ã ã +Nω−1ZFω−1 Y =D0Z+D1ZF+ ã ã ã +DωZFω , where Z ∈Rr×pis an arbitrary matrix.
For the matrix equation (1.24), based on the concept ofF-coprimeness, degrees of freedom existing in the general solution to this type of equations were first given in [82], and then a general complete parametric solution in explicit closed form was established based on generalized right factorization. On the matrix equation (1.25), a neat explicit parametric solution was given in [305] by using the coefficients of the characteristic polynomial and adjoint polynomial matrices. For the matrix equation AX +BY = EXF+R, an explicit parametric solution was established in [78] by elementary transformation of polynomial matrices whenFis in a Jordan form; while in [278] the solution of this equation was given based on the solution of a standard matrix equation without any structural restriction on the coefficient matrices.
1.3.2.2 Applications in Control Systems Design
The preceding several kinds of Sylvester matrix equations have been extensively applied to control systems design. In eigenstructure assignment problems of linear systems, the Sylvester matrix equation (1.20) plays vital roles. By using a parametric solution to time-varying Sylvester matrix equations, eigenstructure assignment prob- lems were considered in [105] for time-varying linear systems. In [66], a parametric approach was proposed for state feedback eigenstructure assignment in linear sys- tems based on explicit solutions to the Sylvester matrix equation (1.20). In [63], the robust pole assignment problem was solved via output feedback for linear systems by combining parametric solutions of two Sylvester matrix equations in the form of
(1.20) with eigenvalue sensitivity theory. In [73, 224], the output feedback eigen- structure assignment was investigated for linear systems. In [224], the problem was solved by using two coupled Sylvester matrix equations and the concept of(C,A,B) -invariance; while in [73] the problem was handled by using an explicit parametric solution to the Sylvester matrix equation based on singular value decompositions.
In [67], the problem of eigenstructure assignment via decentralized output feedback was solved by the parametric solution proposed in [66] for the Sylvester matrix equation (1.20). In [65], a complete parametric approach for eigenstructure assign- ment via dynamical compensators was proposed based on the explicit solutions of the Sylvester matrix equation (1.20). In [91], the parametric approach in [65] was utilized to deal with the robust control of a basic current-controlled magnetic bearing by an output dynamical compensator.
In [39], disturbance suppressible controllers were designed by using Sylvester equations based on left eigenstructure assignment scheme. In addition, some observer design problems can also be solved in the framework of explicit parametric solutions of the Sylvester matrix equation (1.20). For example, in [99] the design of Luenberger observers with loop transfer recovery was considered; an eigenstructure assignment approach was proposed in [95] to the design of proportional integral observers for continuous-time linear systems. A further application of parametric solutions to the Sylvester matrix equation (1.20) is in fault detection. In [98], the problem of fault detection in linear systems was investigated based on Luenberger observers. In [101], the problem of fault detection based on proportional integral observers was solved by using the parametric solutions given in [66].
In some design problems of descriptor linear systems, the generalized Sylvester matrix equations (1.22) and (1.23) are very important. In [64], the eigenstructure assignment via state feedback was investigated for descriptor linear systems based on the proposed explicit solution to the generalized Sylvester matrix equation (1.22).
The parametric solution of (1.22) proposed in [69] was applied to state feedback eigenstructure assignment and response analysis in [70], output feedback eigen- structure assignment in [68] and eigenstructure assignment via static proportional plus derivative state feedback in [97]. In [291], the obtained iterative solution to the generalized Sylvester matrix equation (1.22) was used to solve the eigenstructure assignment problem for descriptor linear systems. In [94], disturbance decoupling via output feedback in descriptor linear systems was investigated. This problem was tackled by output feedback eigenstructure assignment with the help of parametric solutions to the generalized Sylvester matrix equation (1.22). Also, the parametric solution of (1.22) in [69] was used to design some proportional integral observers for descriptor linear systems in [244–246, 249]. In [251], the parametric solution given in [69] was applied to the design of generalized proportional integral deriva- tive observers for descriptor linear systems. In addition, the result on a parametric solution in [247] to the matrix equation (1.22) has been used to design proportional multi-integral observers for descriptor linear systems in [254] for discrete-time case and in [263] for continuous-time case.