P. Benner, A. J. Laub, and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations II: Discrete-time case, Tech. Rep. SPC 95-23, Fakultat fur Mathematik, TU Chemnitz-Zwickau, Chemnitz, Germany, 1995. Available from http://www.tuchemnitz. de/sfb393/spc95pr.html.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....sti#ness matrices given by D = d ij ) where d ij = # # # # # # # i 1 # i # i , i = j 0 , otherwise and K = k ij ) where k ij = # # # # # # # i 1 # i , i = j 0 , otherwise with # 0 = # n = # 0 = # n = 0, are both symmetric tridiagonal. Using the data provided in [2], the eigenvalues of the uncontrolled system are plotted in Figure 1. It is clear that the decay rate of the vibrations of the axle is governed by its most unstable eigenvalue # 1 = 1.3734 10 6 , which is marked in Figure 1 by sign, whereas the other eigenvalues have much better ....

P. Benner, A.J. Laub and V. Mehrmann, "A collection of benchmark examples for the numerical solutions of algebraic Riccati equations I: Continuoustime case" (1995), available in electronic form from www.tu-chemnitz.de/~pester/sfb/spc95pr.html

....symplectic QR factorization of an m n matrix. the complete symplectic URV factorization of a 2048 2048 (n = 1024) matrix was computed. The accuracy of the block algorithms has been tested for various random matrices as well as Hamiltonian matrices obtained from the Riccati benchmark collection [4]. We measured orthogonality of the factors Q, U and the relative residuals #QR A# 1 #A# 1 , #URV A# 1 #A# 1 . The results for the standard algorithms and the new block algorithms are qualitatively the same. 7 Final Remark While the symplectic QR factorization basically covers the ....

P. Benner, A.J. Laub, and V. Mehrmann. A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case. Technical Report SPC 95 22, Fakultat fur Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995. Available from http://www.tu-chemnitz.de/sfb393/spc95pr.html.

....To measure the performance of the algorithms we employ two classical, well conditioned control problems. The first one arises in the control of a string of high speed vehicles (hereafter, chsv) the second problem is constructed with an a priori known circulant solution matrix (see, among many, [9, 14]) In both examples, E = I n and an standard eigenvalue problem (H std ; I n ) is involved. For our performance evaluation purposes, we work on the equivalent matrix pair (H gen ; K gen ) where A gen : LA, S gen : LAL T , Q gen : Q, and E gen = L a random, well conditioned lower unit ....

P. Benner, A. J. Laub, and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: continuous-time case, Tech. Report SPC 95-22, Faculty of Mathematics, Technische Universitat Chemnitz-Zwickau, 1995.

....algorithm requires 84n 3 flops. 4. Subsequent spectral divisions with the matrix sign function become much cheaper as they operate on smaller matrices. 5. The serial QZ algorithm can still be applied locally when the problem is decoupled into a sufficiently small subproblem. 6. Some applications [4,5,24] do not require all the information provided by the QZ algorithm but only bases for specific deflating subspaces. 3. Accurate Subspace Extraction with the Generalized Newton Iteration The accuracy of the generalized Newton iteration depends, among other factors, on the condition number of A and ....

....of (H; K) along the imaginary axis is thus required. The first problem arises in the control of a string of high speed vehicles (hereafter, chsv) the second problem is constructed with an a priori known circulant solution matrix (hereafter, circ) For a description of these examples see, e.g. [4]. In both examples, E = I n=2 and a standard eigenvalue problem (H std ; In) is involved. For our performance evaluation purposes, we work on the equivalent matrix pair (Hgen ; Kgen ) same block structure as in (4.1) where Egen = L is a random, well conditioned lower unit triangular matrix, ....

P. Benner, A.J. Laub, and V. Mehrmann. A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case. Technical Report SPC 95 22, Fakultat fur Mathematik, TU Chemnitz, 09107 Chemnitz, Germany, 1995. http://www.tu-chemnitz.de/sfb393/spc95pr.html.

....on symplectic pencil (can be implemented efficiently using part of the structure) together with Newton for iterative refinement. Note: a comparison done in [6] shows that Newton s method with line search outperforms the QZ method even without iterative refinement for all benchmark examples of [11]. 2.7 Computation of Transfer Matrix Norms To evaluate the model reduction approximation errors, different norms of TFMs are necessary to be computed. The following routines are provided in the RASP package for this purpose: SHANRM computes the Hankel norm and the Hankel singular values of the ....

P. Benner, A. J. Laub, and V. Mehrmann. A collection of benchmark examples for the numerical solution of algebraic Riccati equations II: Discrete-time case. Technical Report SPC 95 23, Fak. f. Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995. Available from http://www.tu-chemnitz.de/sfb393/spc95pr.html.

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P. Benner, A. Laub, and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case, Tech. Report SPC 95 22, Fak. f. Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995.

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P. Benner, A.J. Laub, and V. Mehrmann. A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case. Technical Report SPC 95 22, Fakultat fur Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995. Available from http://www.tu-chemnitz.de/sfb393/spc95pr.html.

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P. Benner, A.J. Laub, and V. Mehrmann. A collection of benchmark examples for the numerical solution of algebraic Riccati equations II: Discrete-time case. Technical Report SPC 95 23, Fakultat fur Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995. Available from http://www.tu-chemnitz.de/sfb393/spc95pr.html.

....are not aware of a proof for this conjecture. 4 Numerical examples Newton s Method (Algorithms 1) and Exact Line Search (Algorithm 2) were implemented as MATLAB [23] functions. We compared the algorithms on the examples of the benchmark collection for continuous time algebraic Riccati equations [5], several randomly generated examples, and some examples contrived to make Algorithm 1 perform poorly. Although theoretical results ensure the existence of a minimizing t j 2 [ 0; 2 ] as well as kR(X j t j N j )k F kR(X j )k F in each step, rounding errors may cause loss of these properties. ....

....converged to the stabilizing solution for all these examples. This suggests that the convergence theory also holds if controllability is weakened to stabilizability. Convergence of the step sizes t j to zero was never observed for the randomly generated examples. The next examples are taken from [5]. Here, we report only the most intriguing results obtained 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. ....

[Article contains additional citation context not shown here]

P. Benner, A. Laub, and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case, Tech. Report SPC 95 22, Fak. f. Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995.

....of the model reduction algorithms. An upper bound for this difference is given by kG Gamma G r k1 2 n X i=r 1 oe i = ffi; where oe k , k = 1; n, are the Hankel singular values of the system and r is the order of the reducedorder model [10] Example 1 This is Example 18 from [2]. The system matrices come from a linear quadratic regulator problem of one dimensional heat flow. The system is single input single output (m = p = 1) and the order n is scalable. Increasing n results in a finer grid for the underlying finite element (FE) discretization of the space variable. ....

P. Benner, A.J. Laub, and V. Mehrmann. A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case. Technical Report SPC 95 22, Fak. f. Mathematik, TU Chemnitz-- Zwickau, FRG, 1995.

....arithmetic. The root is chosen such that the real part is nonnegative. This subroutine was adapted from the EISPACK [36] subroutine CSROOT. 4 Numerical Examples We tested our subroutines for all Hamiltonian matrices from the benchmark collection for continuous time algebraic Riccati equations [6], the examples given in [39] and some randomly generated examples. Here, we present some selected experiments. The numerical tests reported here used IEEE double precision arithmetic with machine precision 2:2204 Theta 10 Gamma16 . The programs were compiled and run on a Sun Microsystems ....

....time invariant system is described by x(t) Ax(t) Bu(t) for t 0; x(0) x 0 ; y(t) Cx(t) for t 0; where the state vector x contains the state of the jet engine, the actuators, and the sensors. The system matrices A 2 IR 30 Theta30 , B 2 IR 30 Theta3 , and C 2 IR 5 Theta30 appear in [6, 17]. The Hamiltonian matrix is H = A BB T C T C GammaA T # : We know the exact values of only four of the eigenvalues of H: 33:3 and a triple eigenvalue at 20:0. DGEEVX computed these four eigenvalues without error. It returned 33:3 correctly rounded to the precision of the arithmetic. ....

P. Benner, A. Laub, and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case, Tech. Rep. SPC 95 22, Fakultat fur Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995. Available from http://www.tu-chemnitz.de/sfb393/spc95pr.html.

....expert driver routine DGEEVX [3] applied to the Hamiltonian matrix and using quadruple precision. Moreover, we tested the effects of balancing when solving algebraic Riccati equations with the structure preserving multishift method presented in [1] for the examples from the benchmark collection [6]. We only present some of the most intriguing results. Example 6.1. 6, Example 6] The system data come from an optimal control problem for a J 100 jet engine as a special case of a multivariable servomechanism problem. The resulting Hamiltonian matrix H 2 R 60 Theta60 has 8 isolated ....

P. Benner, A. Laub, and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case, Tech. Rep. SPC 95 22, Fakultat fur Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995. Available from http://www.tu-chemnitz.de/sfb393/spc95pr.html.

....equations. The collections consist of the FORTRAN77 subroutines CAREX (for the Continous algebraic Riccati equation) and DAREX (for the Discrete algebraic Riccati equation) and analogous MATLAB functions carex.m and darex.m. Currently the codes that create these collections and technical reports [1, 2] that describe the collections and how to use them, are available via anonymous ftp at TU Chemnitz Zwickau. Fakultat fur Mathematik, Technische Universitat Chemnitz Zwickau, 09107 Chemnitz, FRG. e mail: benner mathematik.tu chemnitz.de, mehrmann mathematik.tu chemnitz.de. These authors have been ....

P. Benner, A. Laub, and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations II: Discrete-time case, Tech. Report SPC 95 23, Fak. f. Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995. 2

....equations. The collections consist of the FORTRAN77 subroutines CAREX (for the Continous algebraic Riccati equation) and DAREX (for the Discrete algebraic Riccati equation) and analogous MATLAB functions carex.m and darex.m. Currently the codes that create these collections and technical reports [1, 2] that describe the collections and how to use them, are available via anonymous ftp at TU Chemnitz Zwickau. Fakultat fur Mathematik, Technische Universitat Chemnitz Zwickau, 09107 Chemnitz, FRG. e mail: benner mathematik.tu chemnitz.de, mehrmann mathematik.tu chemnitz.de. These authors have been ....

P. Benner, A. Laub, and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case, Tech. Report SPC 95 22, Fak. f. Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995.

.... Hamiltonian matrix corresponding to the CARE; H) 2 norm condition number of the Hamiltonian matrix corresponding to the stabilizing solution of the CARE; KU upper bound for the CARE condition number as given in (6) The given benchmark collection is based on the collection described in [10, 11]. The FORTRAN 77 subroutine has been modified, a couple of bugs have been corrected, and in particular a new example (Example 2.9) has been added. 1 Parameter free problems of fixed size Example 1.1 [34, Example 1] n m p parameter j re min j jjH jj (H) jjX jj (X ) KU 2 1 2 1.0 2.4 5.8 3.0 ....

P. Benner, A.J. Laub, and V. Mehrmann. A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case. Technical Report SPC 95 22, Fakultat fur Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995. Available from http://www.tu-chemnitz.de/sfb393/spc95pr.html.

....until the resulting symplectic matrix pencil has no more zero and infinity eigenvalues and e A is nonsingular. Note that neither the rank of A nor the number of zero eigenvalues of A determine the number of zero and infinity eigenvalues of the symplectic pencil. For instance, in Example 10 of [12], n = 6, A HYBRID METHOD FOR THE NUMERICAL SOLUTION OF DAREs 9 rank(A) 5, A has three zero eigenvalues and L Gamma M as in (1.2) has two zero and infinite eigenvalues each. All the computation in this algorithm can be carried out in a numerically reliable way. The solution of the linear ....

....is a trademark of The MathWorks, Inc. A HYBRID METHOD FOR THE NUMERICAL SOLUTION OF DAREs 11 reported. Only in Example 14, line search reduced the number of required Newton iterations by one in both methods. The two algorithms are compared for the examples from the DARE benchmark collection [12, 13]. We do not give results for Examples 3 and 4 as there, R is singular and hence the approach via the symplectic pencil is not possible. All computations were done using Matlab Version 5.3 [24] under Linux on a Tangent workstation with a 200 MHz PentiumPro CPU using IEEE double precision ....

P. Benner, A.J. Laub, and V. Mehrmann. A collection of benchmark examples for the numerical solution of algebraic Riccati equations II: Discrete-time case. Technical Report SPC 95 23, Fakultat fur Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995. Available from http://www.tu-chemnitz.de/sfb393/spc95pr.html.

....Besides a computational cost of O(p 2 n 3 ) for forming all the products, the accumulation of roundoff errors in Pi k will result in inaccurate DPRE solutions X k . Furthermore, it requires the nonsingularity of the A k which often does not hold, even in case p = 1 (see the examples in [5]) All these problems do not arise using the periodic QR or cyclic QZ algorithm [10, 16, 17] which requires O(pn 3 ) flops. Using this method, the Pi k are not computed (neither explicitly nor implicitly) and the assumption of the A k and E k being nonsingular can be removed. The methods ....

P. Benner, A.J. Laub, and V. Mehrmann. A collection of benchmark examples for the numerical solution of algebraic Riccati equations II: Discrete-time case. Technical Report SPC 95 23, Fakultat fur Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995. Available from http://www.tu-chemnitz.de/sfb393/spc95pr.html.

....tried as this is nothing but setting t j = 2. In that sense, Newton s method with exact line search includes the algorithm proposed in [26] All the above considerations were employed in [8] for testing Newton s method with exact line search for all examples of the CARE benchmark collection [11]. All examples were solved successfully, employing the standard stabilization procedure from [43] for selecting X 0 and also using X 0 = 0 in case A is stable. Restarts were only encountered for two examples in both cases due to the stagnation detected by (11) In many applications in control ....

....(5 7 4 log p) 2 iter) Table 2 Communication cost of a single iteration of Newton s method for the CARE. 5 Numerical Experiments In this section we compare the performance of our CARE solvers for several benchmark examples. These problems are generated with the Fortran routine carex.f [11]. For comparisons of Newton s method with and without line search see [8,7,9] The experiments were performed using Fortran 77 and ieee double precision arithmetic ( 2:2 Theta 10 Gamma16 ) on an ibm sp2 platform. We made use of the vendor supplied BLAS (essl) and the LAPACK library [2] ....

[Article contains additional citation context not shown here]

P. Benner, A.J. Laub, and V. Mehrmann. A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case. Technical Report SPC 95 22, Fakultat fur Mathematik, TU Chemnitz-- Zwickau, 09107 Chemnitz, FRG, 1995.

....( Gamma) have the same magnitude. If this is true, then the bound (53) is as good as the standard perturbation bound. 6 Numerical Examples Algorithm 2 was implemented in Fortran 77 and was tested for all examples given in the benchmark collections for continuous time algebraic Riccati equations [5], the examples given in [25] and some randomly generated examples. Here, we present the most interesting results obtained by these experiments. The numerical tests were performed using IEEE double precision arithmetic with machine precision 2:2204 Theta 10 Gamma16 on a HP Model 712 60 ....

P. Benner, A. Laub, and V. Mehrmann. A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case. Technical Report SPC 95 22, Fak. f. Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995.

....9th order discrete state space model of a tubular ammonia reactor. It should be noted that the underlying model includes a disturbance term which is neglected in this context. The continuous state space model of this problem was presented as Example 5 in the first part of the benchmark collection [5]. Sampling every 30 seconds yields the following system matrices for the discrete model: A = 10 Gamma2 Theta 2 6 6 6 6 6 6 6 6 6 6 6 6 4 87:01 13:50 1:159 0:05014 Gamma3:722 0:03484 0 0:4242 0:7249 7:655 89:74 1:272 0:05504 Gamma4:016 0:03743 0 0:4530 0:7499 Gamma12:72 35:75 81:70 0:1455 ....

P. Benner, A. Laub, and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case, Tech. Report SPC 95 22, Fak. f. Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995.

....we did not use the hybrid algorithm proposed in Remark 16 in order to demonstrate the behaviour of Exact Line Search and to monitor the possible convergence of the t j s to zero. We compared the algorithms on the examples of the benchmark collection for continuous time algebraic Riccati equations [5], several randomly generated examples, and some examples contrived to make Algorithm 1 perform poorly. All computations were done under MATLAB Version 4.2a [18] on Hewlett Packard Apollo series 700 computers under IEEE double precision and machine precision 2:2204 Delta 10 Gamma16 at the ....

....to the stabilizing solution for all of these examples. This suggests that the convergence theory also holds if controllability is weakened to stabilizability. Convergence of the step sizes t j to zero was never observed for the randomly generated examples. Example 3 This is Example 14 from [5] and Example 2 from [2] Here, A depends upon a parameter ffi. If ffi 0, the system approaches one which is unstabilizable and a conjugate complex pair of the closed loop eigenvalues approaches the imaginary axis. The system matrices are given by A = 2 6 6 4 Gammaffi 1 0 0 Gamma1 Gammaffi ....

P. Benner, A. Laub, and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case, Tech. Report SPC 95 22, Fak. f. Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995.

....such testing and validation environments for the area of numerical methods in control and to accompany SLICOT with benchmark collections for each of the major problem areas. Currently, SLICOT contains two such benchmark collections in the areas of linear quadratic control (Riccati equations) [5, 6, 7]. Other collections are currently being developed. See the WGS homepage for details. 5 Contents of SLICOT 5.1 Current contents of the library A detailed presentation of the previous SLICOT Releases 1 and 2, including chapter by chapter summaries, and individual user callable routine documents, ....

....benchmarks for control methods and their implementation [18] These benchmarks should be practically oriented. Carefully chosen benchmarks give insight in the state of the art with respect to the performance of methods in the language of the control system analyst or control engineer; see, e.g. [5, 6, 7] and Section 4.5. The need for this kind of insight is rapidly increasing due to today s widespread availability of a wealth of methods and implementations. Therefore, a SLICOT benchmark library and accompanying standards will be set up and made available through the WGS ftp site. To assess the ....

P. Benner, A. Laub, and V. Mehrmann. A collection of benchmark examples for the numerical solution of algebraic Riccati equations II: Discrete-time case. Technical Report SPC 95 23, Fak. f. Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995. Available from http://www.tu-chemnitz.de/sfb393/spc95pr.html. xxx

....such testing and validation environments for the area of numerical methods in control and to accompany SLICOT with benchmark collections for each of the major problem areas. Currently, SLICOT contains two such benchmark collections in the areas of linear quadratic control (Riccati equations) [5, 6, 7]. Other collections are currently being developed. See the WGS homepage for details. 5 Contents of SLICOT 5.1 Current contents of the library A detailed presentation of the previous SLICOT Releases 1 and 2, including chapter by chapter summaries, and individual user callable routine documents, ....

....benchmarks for control methods and their implementation [18] These benchmarks should be practically oriented. Carefully chosen benchmarks give insight in the state of the art with respect to the performance of methods in the language of the control system analyst or control engineer; see, e.g. [5, 6, 7] and Section 4.5. The need for this kind of insight is rapidly increasing due to today s widespread availability of a wealth of methods and implementations. Therefore, a SLICOT benchmark library and accompanying standards will be set up and made available through the WGS ftp site. To assess the ....

P. Benner, A. Laub, and V. Mehrmann. A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case. Technical Report SPC 95 22, Fak. f. Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995. Available from http://www.tu-chemnitz.de/sfb393/spc95pr.html.

....attempts, the restart attempts are stopped and an explicit restart with a new random starting vector is initiated. We tested the restarted symplectic Lanczos method for the Hamiltonian matrices corresponding to the continuous time algebraic Riccati equations given in the benchmark collection [4]. Restarts were only encountered in very few cases and we never had to perform an explicit restart when choosing a random starting vector. To demonstrate the restart process we report the two most intriguing of those examples. Due to a special starting vector the implicit restart fails for the A ....

P. Benner, A. Laub, and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case, Tech. Report SPC 95 22, Fak. f. Mathematik, TU Chemnitz--Zwickau, 09107 Chemnitz, FRG, 1995.

No context found.

P. Benner, A. J. Laub, and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations II: Discrete-time case, Tech. Rep. SPC 95-23, Fakultat fur Mathematik, TU Chemnitz-Zwickau, Chemnitz, Germany, 1995. Available from http://www.tuchemnitz. de/sfb393/spc95pr.html.

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