A. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control, AC-24 (1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by testing the exact line search method for all those examples. As for the randomly generated examples (see Example 2) no convergence problems occured for uncontrollable data and convergence of the step sizes to zero was never observed. Example 3 This is Example 15 in [5] and Example 4 in [21]. The system matrices describe a mathematical model of position and velocity control for a string of N high speed vehicles. We have n = 2N Gamma 1, m = N , and p = N Gamma 1. 6 6 6 6 6 6 6 6 6 6 6 A 11 A 12 0 : 0 0 A 22 A 23 0 : 0 . ....

....of ones, and Q = diag( 1 ; The exact stabilizing solution is given by = 10 QC: We obtained the starting guess as X 0 = X X ) 2 where X is the solution of (1) computed by the MATLAB function care provided by A. Laub which implements the Schur vector method [21] extended to the generalized algebraic Riccati equation (1) as discussed in [2] Although the stabilizing solution X is symmetric, rounding errors in care may cause it to return a nonsymmetric solution. Observe in Figures 7 and 8 that Newton s method increases the initial residual norm by ....

A. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control, AC-24 (1979.

....consists of about 250 user callable routines and benchmark collections in various domains of systems and control. Webcut has been used to provide large parts of this library with web interfaces. For example, Riccati benchmark collections [14] as well as solvers for the algebraic Riccati equation [15] can be tested on line. The SLICOT web computing project can be found under http: wc2.hpc2n.umu.se. Future developments will concentrate on the coupling of Webcut with grid computing environments. The web interfaces will enable users to direct computations to a heterogeneous set of computers. A ....

Laub, A. J.: A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, 24 (1979) 913-921

....make full use of the existing symmetries, see [5] However, for the sake of brevity, we will only consider the case S = 0, R = rI yielding the Hamiltonian eigenvalue problem #I c = #I 6 r BB Q A 7 . 7) For a treatment of the general case, see e.g. 6,24,28] It was observed in [22], that if c has an n dimensional deflating subspace associated with eigenvalues in the left half plane spanned by the columns of a matrix , partitioned analogous to c as = U , then, if U 1 is invertible, the optimal control is a linear feedback of the form u(t) Kx(t) U 2 U ....

A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, AC-24:913--921, 1979.

....= 0: 1) The matrices defining the problem can be entered by the user in specific fields in the web interface. They can also be uploaded from Matlab binary or Fortran data files. The underlying routine SB02MD is an implementation of a Schur vectors method for solving algebraic Riccati equations [5, 8, 7]. The web interface provides full flexibility in specifying data and computational options. Figures 2 and 3 show the user interface for solving algebraic Riccati 2 Figure 2: Introduction to solving algebraic Riccati equations. 3 Figure 3: Solving algebraic Riccati equations. 4 equations. ....

A. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Autom. Contr., AC-24:913--921, 1979. 8

....method cannot be used to solve a large scale DRE since all n eigenvectors are required. The form of Equation 3.7 is very useful when used to solve a small DRE or when used to analyze the properties of the DRE, e.g. to prove Theorem 3.4. Some researchers prefer the Schur method to compute U ([25]) but, the numerical advantage from the Schur method is not so convincing for ill conditioned problems since the Schur method only orthogonalizes U , not its row partitions U 1 and U 2 . It is the conditioning of U 1 and U 2 a#ects the accuracy of the computed solution of DRE P = U 2 U 1 1 . The ....

A. J. Laub. A schur method for solving algebraic riccati equations. IEEE Trans. Automat. Contr., 24(6), 1979.

....in sense of maximizing some quadratic form [60] We now briefly consider the existence of solution of the equation (1.3) A way to approach the theory of solutions is via the eigensystem of an associated generalized eigenvalue problem. Anderson [2] and Potter[70] and more recently 32 Laub [63], Lancaster and Rodman [60, Thm7.1.2] and Van Dooren [85] considered the characterization of solutions in terms of the eigensystem of the matrix For any m x n matrix X and In, we define the graph of X by where ImX = Xy: y C . We also call a subspace iV c C invariant for the matrix A (or ....

....solutiom of (1.3) has the form X = ZY 1 for some set of Jordan chains Vl, v2, v for T such that Y is nonsingular. For solving the Riccati equations some modified Newton methods were sug gested by some authors [6] 7] 41] A Schur method is also applicable to solve the Riccati equation [63] and for large sparse Riccati equation the conjugate gra dient method has been considered by Ghavimi, Kenny and Laub [34] We apply all these methods for solving the Riccati equation to our equation (1.1) 1.5.3 Matrix Polynomial P(X) Ao Xm d A1X m 1 d . d Am = 0, 1.22) where A0, ....

Alan J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control., AC-24:913 921, 1979.

....flops. Step 1 2 3 total flops 397 1 3 n 3 (real) 40n 3 (complex) 18 2 3 n 3 (complex) # 158n 3 (complex) TABLE 4. 1 Flop counts for Algorithm 1 These numbers compare with 205n 3 complex flops for the computation of the same invariant subspace via the Schur method as suggested in [12]. If only eigenvalues are required, then only Step 1 of Algorithm 1 is performed without accumulating the similarity transformations. This requires about 320 3 n 3 real flops for the computation of the skew Hamiltonian ETNA Kent State University etna mcs.kent.edu 122 Complex Hamiltonian and ....

A. J. LAUB, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control, AC-24 (1979), pp. 913--921.

....In LYAPACK, only the solution of large optimal control problems by solving Riccati equations is considered [6] However, Riccati equation free solution techniques for optimal control problems surely exist. Standard techniques for small, possibly dense Riccati equations are the Schur method [30], standard) Newton method and modi cations [28, 34, 29, 4] and the sign function method, e.g. 42, 10, 17, 27] Numerically reliable and versatile codes for dense problems of moderate size are can be found in the freeware subroutine library SLICOT (Subroutine Library in Control Theory) 7] ....

A. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control, 24 (1979), pp. 913-921.

....service. The matrices defining the problem can be entered by the user in specific fields in the web interface. They can also be uploaded from Matlab binary or Fortran data files. The underlying routine SB02MD is an implementation of a Schur vectors method for solving algebraic Riccati equations [5, 8, 7]. The web interface provides full flexibility in specifying data and computational options. Figures 2 and 3 show the user interface for solving algebraic Riccati equations. Here, the matrices have already been entered and the user may select a number of different options, e.g. the type of ....

A. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Autom. Contr., AC-24:913--921, 1979.

....A number of numericalalgorith5 for obtainingth solution P to (1)h ve been proposed in th literature. However, th computation of th solution P to (1) is of some di#culty especiallywha th dimension ish: 5 One of th numericalalgorith] is a Sch ur typemeth d,wh5 h is described in detail in [15] and [1] Thk meth d is very reliable and used in MATRIX X , CTRL C, and MATLAB. But th number of operations required for th solution of (1) inth] meth d is estimated by more the 75n 3 [15] 75n 3 multiplications take considerable time whe n ishk] even thk]# we h ve powerful computational ....

....th dimension ish: 5 One of th numericalalgorith] is a Sch ur typemeth d,wh5 h is described in detail in [15] and [1] Thk meth d is very reliable and used in MATRIX X , CTRL C, and MATLAB. But th number of operations required for th solution of (1) inth] meth d is estimated by more the 75n 3 [15]. 75n 3 multiplications take considerable time whe n ishk] even thk]# we h ve powerful computational environments. Th initial estimate close to th actual solution reduces considerablyth computation time to be used in th numerical algorith7 Thrith7M it is important to obtain an accurate estimate ....

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A J Laub, "A Schur method for solving for algebraic Riccati equations," IEEE Trans Autom Control, vol 24, p 913--921, 1979

....3 XE E 3 XA 0 (B 3 XE) 3 R 01 (B 3 XE) 0; 1) while the corresponding generalized discrete time Riccati equation takes the form 0E 3 XE A 3 XA C 3 QC 0 (B 3 XA) 3 (R B 3 XB) 01 (B 3 XA) 0; 2) where 3 denotes the conjugate transpose. It is well known, e.g. [16, 17, 18, 19], that the solutions of the algebraic Riccati equations (1) and (2) can be used to obtain solutions to linear quadratic optimal control problems and optimal filter problems. See also the forthcoming book [15] In the continuous time case this is the problem to minimize the cost functional 1 2 ....

.... Ax(t) Bu(t) x(t 0 ) x 0 ; y(t) Cx(t) 4) In the discrete time case one considers the problem of minimizing the cost functional 1 2 1 k=0 [y 3 k Qy k u 3 k Ru k ] 5) subject to the dynamics Ex k 1 = Ax k Bu k ; x 0 = x 0 ; y k = Cx k : 6) It is also well known, e.g. [16, 17, 18, 19], that the solutions of the algebraic Riccati equations can be obtained via the computation of deflating subspaces of the following pencils. In the continous time case the pencil is of the form E c 0 H c : E 0 0 E 3 0 A BR 01 B 3 C 3 QC 0A 3 = E c 0 0 E 3 c 0 F c G c H c ....

A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, AC-24:913--921, 1979. (see also Proc. 1978 CDC (Jan. 1979), pp. 60-65).

....Approach to Teletraffic Problems 7 which spans the same invariant subspace that T 1 spans and write X = U 12 U Gamma1 11 . Since the major tool is the computation of an invariant subspace, all these methods are referred to invariant subspace methods. These methods include Schur methods [28, 29], iterative refinement techniques [21] and the matrix sign function algorithms [8, 10, 20, 24] It is the invariant subspace approach that has proved to be the most suitable approach for solving the ARE among the three basic approaches mentioned above [29] due to its numerical efficiency and ....

A. J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Auto. Contr., 25:913--921, 1979.

....example, 10, 14] It is easy to see that the matrix X is a solution of (1) if and only if the columns of I n X span an n dimensional invariant subspace of the Hamiltonian matrix H = A R G A T # R 2n,2n (2) where I n is the n n identity matrix. Moreover, it is well known [15, 20] that the unique positive semidefinite solution of (1) when it exists, can be obtained from the stable invariant subspace of H ; i.e. from the invariant subspace corresponding to the eigenvalues of H with negative real parts. More precisely, we have the following well known result (see [20] ....

.... why Newton s method is generally most useful in the iterative refinement of solutions obtained from other methods; see [4] The second class consists of the methods that are based on the computation of the stable invariant subspace of H in (2) These methods include the Schur vector method of Laub [15], the Hamiltonian QR algorithm of Byers [8] the SR algorithm of Bunse Gerstner and Mehrmann [5] the HHDR algorithm of Bunse Gerstner and Mehrmann [6] and the matrix sign function method [9] Unlike earlier attempts based on the computation of eigenvectors of the Hamiltonian matrix H , Laub s ....

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A.J. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automatic Control AC--24 (1979), pp. 912--921.

....In LYAPACK, only the solution of large optimal control problems by solving Riccati equations is considered [6] However, Riccati equation free solution techniques for optimal control problems 41 surely exist. Standard techniques for small, possibly dense Riccati equations are the Schur method [30], standard) Newton method and modi cations [28, 34, 29, 4] and the sign function method, e.g. 42, 10, 17, 27] Numerically reliable and versatile codes for dense problems of moderate size are can be found in the freeware subroutine library SLICOT (Subroutine Library in Control Theory) 7] ....

A. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control, 24 (1979), pp. 913921.

....Q Gamma1 (A i Q B u;i U) 0: 7) This problem is easily transformed into an LMI problem. This formulation allows for a complete solution of the LQG regulator problem addressed in [13] where a non standard Riccati equation was obtained. For solving it, the usual Hamiltonian methods (see e.g. [10]) cannot be applied directly. This Riccati equation is usually solved using homotopy methods, with no guarantee of global convergence [3, 4, 12] Moreover, the LMI formulation can be exploited to solve problems which have no (known) analytical solution , as follows. We note that the constraint ....

A. J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans., AC-24 (1979) 913-- 921.

....than direct iterations on the Riccati equation and are more reliable than solutions based on eigenvalue eigenvector decompositions of the state costate evolution equation. Our survey of these methods draws heavily on Anderson (1978) Gardiner and Laub (1986) Golub, Nash and Van Loan (1979) Laub (1979,1991) and Pappas, Laub and Sandell (1980) This paper is organized as follows. Section 2 decomposes the optimal linear regulator into sub problems that are more efficient to solve and describes classes of economic problems that give rise to such problems. Sections 3, 4, 5, and 6 describe recent ....

....states n. However, merely requiring M to be symplectic permits there to be eigenvalues with absolute values equal to one, and so we will need an additional argument to show that there are exactly n stable eigenvalues. To locate the stable invariant subspace of the symplectic matrix M , we follow Laub (1979) and (block) triangularize M : V Gamma1 MV = W W = W 11 W 12 0 W 22 ; 3:17) where V is a nonsingular matrix. By construction, the matrices M and W are similar. The matrix partitions in (3:17) are built to coincide with the number of stable and unstable eigenvalues. In particular, the ....

Laub, A.J. (1979). `A Schur Method for Solving Algebraic Riccati Equations'.

....or rational matrix equations can be solved by computing invariant subspaces of matrices and deflating subspaces of matrix pencils. Examples include Schur methods for matrix m th roots, sector functions, algebraic Riccati equations, Sylvester equations, Lyapunov equations and their generalizations [3, 5, 12, 14, 15, 16, 18, 21, 26, 31, 33, 35, 39]. In this paper we consider the computation of deflating subspaces of a generalized matrix pencil of the form ffA Gamma fiBC with complex n Theta n matrices A, B and C. There exist similar methods for real matrices A, B and C that use only real arithmetic. However, for ease of presentation, we ....

A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, AC-24:913--921, 1979.

....to the two CAREs in a (stochastic) linear quadratic Gaussian (LQG) optimal control problem. One common approach to solve (1) is to compute the stable invariant subspace of the Hamiltonian matrix H, i.e. the subspace corresponding to the eigenvalues of H in the open left half plane (e.g. [13, 34, 35, 37, 49]) If this subspace is spanned by U 1 U 2 and U 1 is invertible, then X = U 2 U Gamma1 1 is the stabilizing solution of (1) The examples are grouped in four sections. The first section contains parameter free examples of fixed dimension, the second parameter dependent problems of ....

....of H. On the other hand, a large condition number may also be due to a large norm of H rather than to ill conditioning with respect to inversion as in Example 4.4. If no analytical solution is available, we computed approximations by the multishift algorithm [1] and the Schur vector method [34]. If possible, these approximations were refined by Newton s method [32] possibly combined with an exact line search [8, 9] to achieve the highest possible accuracy. We then chose the approximate solution with smallest residual norm to compute the properties of the example. Note that this is not ....

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A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, AC-24:913--921, 1979.

....as expensive as one iteration of Newton s method [46] Nevertheless, such solutions often lie far from X and a high number of iterations of Newton s method may be required to converge. A different approach consists of finding an initial solution by means of a CARE solver (the Schur vector method [38], the matrix sign function [42] etc. but this requires solving the CARE itself. Newton s method is used most frequently for iterative refinement of approximate solutions computed by some other method. Using the exact line search proposed in the last section often brings the cost of Newton s ....

....steps are performed once the criterion is satisfied. The figures report the time required by Newton s 19 method to refine the initial solution X 0 to maximum accuracy. Example 7 This example describes a mathematical model of position and velocity control of a string of high speed vehicles [38]. The condition number of the example only grows very slowly with n. The left hand plot in Figure 1 reports the execution time of the CARE solvers with X 0 = X and n=249, 499, 749, and 999. Newton s method requires only three iterations to converge in these cases (kR (X 0 ) k F 1:0 Theta ....

A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, AC-24:913--921, 1979.

.... structure: for any S 2 Sp(2n) H 2 H(2n) S Gamma1 HS 2 H(2n) and W 2 W(2n) S Gamma1 WS 2 W(2n) 3 Structural Constraints A variety of methods for computing the eigenvalues and invariant subspaces of Hamiltonian matrices have been described in the literature (for example see [2, 3, 8, 9, 10, 11, 36, 45, 49, 54]) In this section we examine some of the basic structural issues involved in designing Jacobi algorithms that are completely structure preserving. The first issue to consider is the appropriate set of similarities to use. Since orthogonal matrices are perfectly conditioned, and symplectic ....

A. J. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Auto. Control, AC--24 (1979), pp. 913--921. 40

....is to use practical error bounds, similar to the case of solving linear systems of equations [1] and matrix Sylvester equations [13] Such error bounds are implemented in the Fortran 77 solvers ricc and ricm used in the comparison done in this report. The solver ricc implements the Schur method [16, 17] with block scaling [21] which enhances the numerical stability while ricm implements the matrix sign function method [22, 5, 15, 9, 23] Both solvers make use of LAPACK [1] Release 3.0, and are implemented as mex files which makes them easily accessible from MATLAB. Note that the Riccati solvers ....

A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, AC-24:913--921, 1979.

....Approach 8 addition (of size mf Theta mf ) There are variants of matrix sign function iteration algorithms that have a convergence rate of arbitrary order (e.g. cubic convergence rate) but at the expense of requiring more CPU time at each iteration. We will also present the Schur method [32] that is known to be efficient and numerically very reliable, but is slower than the matrix sign function approach and should be considered as an alternative for very ill conditioned problems. We note that the matrices of interest at each iteration of the matrix sign function algorithm are of ....

....the tradeoff is the larger number of flops per iteration versus the higher convergence rates. 2. 2 Schur Approach Another approach to compute the left invariant subspace of a matrix is the so called Schur approach whose use for the solution of algebraic Riccati equations dates back to 1979 [32]. This approach is based on the fact that given a matrix M 2 R m Thetam , then there exists an orthogonal similarity transformation U such that U T MU is quasi upper triangular [22] that is, the reduced form has 2 Theta 2 blocks on the (block) diagonal corresponding to complex conjugate ....

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A. J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Auto. Contr., 25:913--921, 1979.

....1 Gamma U 2 ) T : 53) 3.2 Implementation We focus on two particular methods to compute a basis for each of the left and rightinvariant subspaces of a real matrix: i) matrix sign function method, and (ii) Schur decomposition method. While the latter method is known of its numerical stability [28], 19] the former one is promising especially for large scale problems as it is amenable to parallel implementation [35] Both methods have also been successfully used in the past to solve important equations in control theory, such as matrix Riccati and Lyapunov equations [37] 13] 17] 28] ....

.... [28] 19] the former one is promising especially for large scale problems as it is amenable to parallel implementation [35] Both methods have also been successfully used in the past to solve important equations in control theory, such as matrix Riccati and Lyapunov equations [37] 13] 17] [28]. Here, given the matrix E T m , we aim at simultaneous computation of the matrix geometric factors R 1 and R 2 through finding either the matrix sign or the Schur decomposition of E T m . The two methods are introduced next in that spirit. Akar, Oguz, and Sohraby, A Novel Computational Method ....

A. J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Auto. Contr., 25:913--921, 1979. Akar, Oguz, and Sohraby, A Novel Computational Method for Finite QBD Processes 33

....eigenvalue problem, Hamiltonian pencil (matrix) symplectic pencil (matrix) skew Hamiltonian matrix AMS subject classification. 65F15 1 Introduction The eigenproblem for Hamiltonian and symplectic matrices has received a lot of attention in the last 25 years, since the landmark papers of Laub [13] and Paige Van Loan [20] The reason for this is the importance of this problem in many applications in control theory and signal processing, 17, 12] and also due to the fact that the construction of a completely satisfactory method is still an open problem. Such a method should be numerically ....

A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, AC-24:913--921, 1979. (see also Proc. 1978 CDC (Jan. 1979), pp. 60-65).

.... importance of the Lagrangian invariant subspaces is that most numerical solution methods (with the exception of Newton s method) proceed via the computation of the Lagrangian invariant subspaces to determine the solution of the Riccati equation or directly the solution of the control problems, see [2, 3, 4, 5, 6, 15, 16, 19, 21]. Most modern methods employ transformations with symplectic matrices to compute the desired Lagrangian invariant subspaces and to solve the algebraic Riccati equation. Definition 3 A 2n Theta 2n complex matrix is called symplectic if S H J n S = J n . For a given Hamiltonian matrix H the ....

A.J. Laub. A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control, AC-24:913--921, 1979.

....to Jordan form, and taking the stable eigenvectors as the required basis. However, such a computation is fraught with numerical difficulties and is best avoided. A more appropriate technique is to use a Schur reduction instead. Schur methods to solve Riccati equations have been popularized by Laub [15, 16] and Van Dooren [17] For the DPRE, it goes as follows. Find a unitary matrix T k to put S (k) in Schur form with a particular ordering: 2 4 T 11k T 12k T 21k T 22k 3 5 S (k) 2 4 T 11k T 12k T 21k T 22k 3 5 = 2 4 S 11k S 12k 0 S 22k 3 5 ; 16) where the partitioning conforms to ....

A. J. Laub, "A Schur method for solving algebraic Riccati equations," IEEE Trans. Automat. Control, vol. AC-24, pp. 913--921, December 1979.

....finite or infinite. See, among others Kucera (1989) Lancaster and Rodman (1980) Willems (1971) This paper is restricted to real symmetric solutions of the ARE. Both the closed loop system eigenvalues, and the solution R to the ARE can be determined from the state co state system matrix ( 7) Laub (1979), Di Ruscio and Balchen (1990) The state co state system matrix (the Hamiltonian matrix) is derived from optimal control theory by augmenting the co state equation p = GammaQx Gamma A T p to the state space model ( 1) with the optimal control input vector u = GammaP Gamma1 B T p. The ....

Laub, A. J. (1979). A Schur method for Solving Algebraic Riccati Equations. IEEE Trans. on Automatic control, vol. AC-24, pp. 913-921.

....the tractor. 4 Taken from [14] 5 Matlab is a registered trademark of The MathWorks, Inc. The controller then switched between various gains based on the velocity V . Employing the SLICOT subroutine SB02MD 6 which provides an implementation of the Schur vector method for solving (3) see [9]) it is now possible to compute the control gains on line. The DARE is solved in roughly 5 of the 200msec sampling time which gives plenty of time for doing the other necessary calculations. Having the ability to solve the DARE in real time now means that one can use information from an on line ....

A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, AC-24:913--921, 1979.

....in the preceding section. Conceivably, this could be done by direct calculation of eigenvectors and, possibly, generalized eigenvectors. However, this strategy presents difficult problems of numerical stability and is generally avoided. A class of more stable algorithms originates with Laub [11]. They are known as Schur methods because they involve reduction to triangular (or quasitriangular) form by orthogonal similarity following a technique of Schur from 1909. The use of orthogonal similarity is the basis of more reliable numerical performance. Thus, a real orthogonal matrix U = U ....

Laub, A.J., A Schur method for solving algebraic Riccati equations, IEEE Trans. on Automatic Control, AC-24, 1979, 913-921.

....U 11 X = GammaU 21 with the QR (or LR ) decomposition (with pivoting) see e.g. 15] Step 3:Use defect correction (Algorithm 3.5 Step 2) to refine X and compute an error estimate P . The costs are O(n 3 ) operations. This way of computing the solution of the ARE has been introduced in [18]. Nowadays it is the most frequently used numerical method to compute the stabilizing solution of (1.1) and it is implemented in software packages like NAG or the MATLAB control toolbox. Even though the QR algorithm is a numerically stable method to solve the eigenvalue problem, in this context ....

A.J. Laub, A Schur Method for Solving Algebraic Riccati Equations, IEEE Trans.Autom.Control AC-24 (1979), 913-- 921.

....submatrices of R, solving a linear algebraic system in a total least squares sense [10] The covariance matrices are computed using the residuals of a least squares problem. The Kalman gain is obtained by solving a discrete time algebraic matrix Riccati equation using the Schur vectors approach [11] for the dual of an optimal control problem. To give further details, let us partition the triangular factor of H as R = U p U f Y p Y f ] where the subscripts p and f stand for past and future data, respectively, and the four blocks have ms, ms, s, and s columns, respectively. Define W ....

A. J. Laub, "A Schur method for solving algebraic Riccati equations," IEEE Trans. Automat. Control, vol. AC--24, no. 6, pp. 913--921, 1979.

....known and thoroughly studied nonlinear matrix equations in engineering. It arises in different aspects of optimal control and estimation problems, see e.g. 1, 3, 23, 24] etc. A lot of research effort has been devoted to finding stable and fast numerical algorithms for solving the ARE, see e.g. [5, 25, 26, 30]. In this paper we discuss how computer algebra can be used to solve the ARE symbolically. By solving we mean triangulating the system of polynomial equations that is rendered by considering each matrix entry of the matrix equation. Thus variables are eliminated successively so that we eventually ....

A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Aut. Contr., AC-24(6):913--921, December 1979.

....[10, 14] It is easy to see that the matrix X is a solution of (1) if and only if the columns of I n X span an n dimensional invariant subspace of the Hamiltonian matrix H = A R G GammaA T 2 R 2n;2n (2) where I n is the n Theta n identity matrix. Moreover, it is well known [15, 20] that the unique positive semidefinite solution of (1) when it exists, can be obtained from the stable invariant subspace of H; i.e. from the invariant subspace corresponding to the eigenvalues of H with negative real parts. More precisely, we have the following well known result (see [20] ....

.... why Newton s method is generally most useful in the iterative refinement of solutions obtained from other methods; see [4] The second class consists of the methods that are based on the computation of the stable invariant subspace of H in (2) These methods include the Schur vector method of Laub [15], the Hamiltonian QR algorithm of Byers [8] the SR algorithm of Bunse Gerstner and Mehrmann [5] the HHDR algorithm of Bunse Gerstner and Mehrmann [6] and the matrix sign function method [9] Unlike earlier attempts based on the computation of eigenvectors of the Hamiltonian matrix H, Laub s ....

[Article contains additional citation context not shown here]

A.J. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automatic Control AC--24 (1979), pp. 912--921.

....E in (3) may be swapped. In all other cases, the QZ algorithm has to be employed as initial stage when solving these equations via the most frequently used HessenbergSchur and Bartels Stewart methods; see [11, 21] and the references therein. Solving (4) by the (generalized) Schur vector method [19, 3], again the QR (QZ) algorithm is applied to the corresponding Hamiltonian matrix (Hamiltonian skew Hamiltonian matrix pencil) When iteratively refining an approximate solution or even solving (4) directly using Newton s method [3, 17, 18, 20] in each iteration step a (generalized) Lyapunov ....

A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, AC24: 913--921, 1979.

....optimization problems. In [16] Freund and Jarre use the simplified QMR from BCG Algorithm 5.1 to solve the symmetric indefinite linear systems that arise in each iteration of a primal dual interior point method for linear programs. Our last example is a Hamiltonian eigenvalue problem, taken from [33]. This is a family of Hamiltonian matrices A of size N = 2(2v Gamma 1) v = 1; 2; that arises in the context of position and velocity control for a string of high speed vehicles. Moreover, the submatrix F in the partitioning (28) is tridiagonal, and the submatrices F and H in (28) are ....

A.J. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control 24 (1979) 913--921.

....algebraic Riccati equation 0 = F T X XF H Gamma XGX; 1) where F; G; H are the blocks in H and X is a real n Theta n symmetric matrix. It is well known, that if X is symmetric and the columns of the matrix I n GammaX # span an invariant subspace of H then X solves (1) e.g. [19, 24, 20, 23, 18]. Paige Van Loan [24] showed that if H 2 H 2n , then it has a Hamiltonian Schur form, i.e. there exist a matrix Q 2 US 2n such that Q T HQ = T N 0 GammaT T # ; 2) where T is quasi upper triangular and N = N T . The first n columns of Q then span the desired Lagrangian subspace. ....

....flop counts for the four steps are given in Table 1. Step 1 2 3 4 total flops 103 n 3 9 n 3 9 n 3 42 n 3 163 n 3 Table 1: Flop counts for Algorithm 1 These numbers compare with 203n 3 flops for the computation of the same invariant subspace via the standard QR algorithm as suggested in [19]. The storage requirement for this algorithm is about 9n 2 , a little more than for the standard QR algorithm. Remark 3.4 Up to now we have discussed only the computation of the stable invariant subspace of the Hamiltonian matrix and not the solution of algebraic Riccati equation (1) since the ....

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A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, AC-24:913--921, 1979. (See also Proc. 1978 CDC (Jan. 1979), pp. 60-65).

....Riccati equation 0 = F T X XF H Gamma XGX; 1) where F; G; H are the blocks in H and X is a real n Theta n symmetric matrix. It is well known, that if X is symmetric and the columns of the matrix I n GammaX # span a Lagrangian invariant subspace of H then X solves (1) e.g. [23, 29, 24, 28, 22]. An invariant subspace is called Lagrangian if it is a maximal isotropic subspace. Paige Van Loan [29] showed that if H 2 H 2n , then it has a Hamiltonian Schur form, i.e. there exist a matrix Q 2 US 2n such that Q T HQ = T N 0 GammaT T # ; 2) where T is quasi upper triangular ....

....flop counts for the four steps are given in Table 1. Step 1 2 3 4 total flops 103 n 3 9 n 3 9 n 3 42 n 3 163 n 3 Table 1: Flop counts for Algorithm 1 These numbers compare with 203n 3 flops for the computation of the same invariant subspace via the standard QR algorithm as suggested in [23]. The storage requirement for this algorithm is about 9n 2 , a little more than the 8n 2 required for the Schur vector method [23] based on an implementation of the standard QR algorithm [4] Remark 3.4 Up to now we have discussed only the computation of the stable invariant subspace of the ....

[Article contains additional citation context not shown here]

A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, AC-24:913--921, 1979. (See also Proc. 1978 CDC (Jan. 1979), pp. 60-65).

....Cholesky factors. Therefore, in these applications it is very important that such rank decisions to be based on results computed with the highest achievable accuracy. The standard method to compute the stabilizing, non negative definite solution X of (1) is the Schur vectors method proposed by Laub (1979). X is computed as X = Z 2 Z Gamma1 1 (5) where Z = Z 1 Z 2 # is a 2n Theta n matrix with orthonormal columns which span the stable eigenspace of the Hamiltonian matrix H = A Q GammaR GammaA T # : 6) Although each computational step to determine X can be performed by ....

....is that, this rank information must be recovered later, after performing some operations with the computed factor. The purpose of this section is to discuss several other possible methods to compute S. Method 1. We discuss first the computation of the Cholesky factor S by using the Schur method (Laub, 1979). Accuracy losses arise here due to the instability of the inversion step when computing X = Z 2 Z Gamma1 1 . Provided the Riccati equation is well scaled, that is k Xk 1 (Kenney et al. 1989) we can avoid the explicit inversion of Z 1 by using the following approach suggested by Singer ....

Laub, A. (1979). A Schur method for solving the algebraic Riccati equation. IEEE Trans.

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A. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control, AC-24 (1979.

....2 2 2 1.05 0.69 5:07 Theta 10 Gamma2 4.97 4. 74 This example illustrates a linear quadratic control problem as defined by (6) 8) The coefficient matrices are A = 0:9512 0 0 0:9048 ; B = 4:877 4:877 Gamma1:1895 3:569 ; R = 1 3 0 0 3 ; Q = 0:005 0 0 0:02 : In [16, 17], solution matrices are given. We omit reproducing them here since they are not derived analytically. Example 3 [31, Example II] n m p parameter (A) j C max j jjXjj (X) KDARE 2 1 1 5.83 0.00 1.00 1.00 1 This example was used in [31] to demonstrate a compression technique for the extended ....

A. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control, AC-24 (1979), pp. 913--921. (see also Proc. 1978 CDC (Jan. 1979), pp. 60-65).

....to the eigenvalues inside the open unit disk, respectively. If this subspace is spanned by h U T 1 U T 2 i T (or h U T 1 U T 2 U T 3 i T for the ESP (13) and U 1 is invertible, then X = U 2 U Gamma1 1 (14) is the stabilizing solution of (1) or (2) respectively, e.g. [16, 38, 39, 43, 60]. All other solutions are obtained via different Lagrange invariant (deflating) subspaces, too. At this point it should be noted that it is possible to transform a continuous time algebraic Riccati equation (CARE) into a DARE (and vice versa) via a (generalized) Cayley transformation, i.e. the ....

....cause eigenvalues with small real part to cross the imaginary axis; see, e.g. 59] For computing the system properties we use the exact stabilizing solution if an analytical solution is available. Otherwise, we computed approximations by the multishift algorithm [2] and the Schur vector method [38]. If possible, these approximations were refined by Newton s method [33] possibly combined with an exact line search [9, 10] to achieve the highest possible accuracy. We then chose the approximate solution with smallest residual norm to compute the properties of the example. Note that this is not ....

[Article contains additional citation context not shown here]

A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, AC-24:913--921, 1979. (see also Proc. 1978 CDC (Jan. 1979), pp. 60-65).

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A. J. Laub, "A Schur Method for Solving Algebraic Riccati Equations," IEEE Trans. Aut. Control, 1979, vol. 24, pp. 913-914.

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A.J. Laub. A Schur Method for Solving Algebraic Riccati Equations, IEEE Trans. Automat. Control, AC-24:913--921, 1979.

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Laub A. A Schur method for solving algebraic Riccati equations. IEEE Transactions on Automatic Control, AC-24, pp. 913-921, 1979.

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A.J. Laub, "A Schur method for solving algebraic Riccati equations", IEEE Trans. Auto. Control, Vol. 24, pg. 913-921, 1979.

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A.J. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automatic Control AC--24 (1979), pp. 912--921.

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A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, AC-24:913--921, 1979.

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A.J. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Autom. Control AC 24 (1976), 213--238.

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A.J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control, AC-24:913--921, 1979. See ftp://wgs.esat.kuleuven.ac.be/pub/WGS/SLICOT/doc/SB02MD.html for more details. 21

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A. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control, AC-24 (1979), pp. 913--921.

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