Numerical Linear Algebra Techniques for Systems and Control

The author discusses a number of numerical linear algebra techniques for large scale problems in systems and control. Attention is focused on special matrix-problems, i.e. matrices which are either sparse, patterned or structured. The topics considered all point to the significant role linear algebra problems play in the control, optimization and model reduction of multivariable linear systems. The individual topics discussed all pertain to the common problem of designing low-order robust controllers for large scale plants, and the techniques are closely related

[1]  G. Stewart Introduction to matrix computations , 1973 .

[2]  Petros G. Voulgaris,et al.  On optimal ℓ∞ to ℓ∞ filtering , 1995, Autom..

[3]  M. Malek-Zavarei,et al.  Book reviews - ORACLS - A design system for linear multivariable control , 1980, IEEE Control Systems Magazine.

[4]  Alan J. Laub,et al.  Frequency response computation via rational interpolation , 1993, IEEE Trans. Autom. Control..

[5]  K. Glover,et al.  State-space formulae for all stabilizing controllers that satisfy and H ∞ norm bound and relations to risk sensitivity , 1988 .

[6]  John M. Mulvey,et al.  On Reporting Computational Experiments with Mathematical Software , 1979, TOMS.

[7]  P. Misra Hessenberg-triangular reduction and transfer function matrices of singular systems , 1989 .

[8]  K.-W.E. Chu A controllability condensed form and a state feedback pole assignment algorithm for descriptor systems , 1988 .

[9]  A. Laub Schur techniques in invariant imbedding methods for solving two-point boundary value problems , 1982, 1982 21st IEEE Conference on Decision and Control.

[10]  George Cybenko,et al.  The Numerical Stability of the Levinson-Durbin Algorithm for Toeplitz Systems of Equations , 1980 .

[11]  S. Nash,et al.  Approaches to robust pole assignment , 1989 .

[12]  Alan J. Laub,et al.  A Newton-squaring algorithm for computing the negative invariant subspace of a matrix , 1993, IEEE Trans. Autom. Control..

[13]  J. H. Wilkinson,et al.  AN ESTIMATE FOR THE CONDITION NUMBER OF A MATRIX , 1979 .

[14]  P. Dooren,et al.  The eigenstructure of an arbitrary polynomial matrix : Computational aspects , 1983 .

[15]  G. Zames Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses , 1981 .

[16]  M. A. Jenkins,et al.  A Three-Stage Algorithm for Real Polynomials Using Quadratic Iteration , 1970 .

[17]  James R. Bunch,et al.  Stability of Methods for Solving Toeplitz Systems of Equations , 1985 .

[18]  Dianne P. O'Leary,et al.  Constrained Matrix Sylvester Equations , 1992, SIAM J. Matrix Anal. Appl..

[19]  C. Loan,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix , 1978 .

[20]  David J. Kuck,et al.  On Stable Parallel Linear System Solvers , 1978, JACM.

[21]  C. Paige Properties of numerical algorithms related to computing controllability , 1981 .

[22]  G. Hewer,et al.  The sensitivity of the algebraic and differential riccati equations , 1990 .

[23]  A. Laub,et al.  Computation of supremal (A,B)-invariant and controllability subspaces , 1977, 1977 IEEE Conference on Decision and Control including the 16th Symposium on Adaptive Processes and A Special Symposium on Fuzzy Set Theory and Applications.

[24]  A. Laub Efficient multivariable frequency response computations , 1980, 1980 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[25]  A. Laub,et al.  Feedback properties of multivariable systems: The role and use of the return difference matrix , 1981 .

[26]  J. Rice A Theory of Condition , 1966 .

[27]  Alan J. Laub,et al.  Error bounds for Newton refinement of solutions to algebraic riccati equations , 1990, Math. Control. Signals Syst..

[28]  Rajni V. Patel,et al.  Numerical algorithms for eigenvalue assignment by constant and dynamic output feedback , 1989 .

[29]  Vladimír Kucera,et al.  Discrete linear control: The polynomial equation approach , 1985, IEEE Transactions on Systems, Man, and Cybernetics.

[30]  A. Laub,et al.  Parallel algorithms for algebraic Riccati equations , 1991 .

[31]  B MolerCleve,et al.  Solution of the Sylvester matrix equation AXBT + CXDT = E , 1992 .

[32]  Charles Kenney,et al.  Polar Decomposition and Matrix Sign Function Condition Estimates , 1991, SIAM J. Sci. Comput..

[33]  A. Laub,et al.  Condition Estimates for Matrix Functions , 1989 .

[34]  R. V. Patel,et al.  Computation of minimal-order state-space realizations and observability indices using orthogonal transformations , 1981 .

[35]  E. Feron,et al.  Computing Bounds for the Structured Singular Value via an Interior Point Algorithm , 1992, 1992 American Control Conference.

[36]  P. Dooren The Computation of Kronecker's Canonical Form of a Singular Pencil , 1979 .

[37]  P. Henrici Fast Fourier Methods in Computational Complex Analysis , 1979 .

[38]  Charles L. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[39]  Alan J. Laub,et al.  Algorithm 640: Efficient calculation of frequency response matrices from state space models , 1986, TOMS.

[40]  George Miminis,et al.  A direct algorithm for pole assignment of time-invariant multi-input linear systems using state feedback , 1988, Autom..

[41]  B. Francis,et al.  A Course in H Control Theory , 1987 .

[42]  K. Glover All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[43]  V. Mehrmann The Autonomous Linear Quadratic Control Problem , 1991 .

[44]  Stephen P. Boyd,et al.  A bisection method for computing the H∞ norm of a transfer matrix and related problems , 1989, Math. Control. Signals Syst..

[46]  Alan J. Laub,et al.  Algorithm 705; a FORTRAN-77 software package for solving the Sylvester matrix equation AXBT + CXDT = E , 1992, TOMS.

[47]  G. Hewer,et al.  The sensitivity of the stable Lyapunov equation , 1987, 26th IEEE Conference on Decision and Control.

[48]  Alan S. Willsky,et al.  Digital Signal Processing and Control and Estimation Theory: Points of Tangency, Areas of Intersection, and Parallel Directions , 1979 .

[49]  Alan J. Laub,et al.  Systolic computation of multivariable frequency response , 1988 .

[50]  George Cybenko The numerical stability of the lattice algorithm for least squares linear prediction problems , 1984 .

[51]  Xiaomei Yang Rounding Errors in Algebraic Processes , 1964, Nature.

[52]  S. Nash,et al.  Numerical methods and software , 1990 .

[53]  G. Robel On computing the infinity norm , 1989 .

[54]  A. Laub,et al.  Improving the efficiency of matrix operations in the numerical solution of large implicit systems of linear differential equations , 1987 .

[55]  Alan J. Laub,et al.  Calculation of transmission zeros using QZ techniques , 1977, 1977 IEEE Conference on Decision and Control including the 16th Symposium on Adaptive Processes and A Special Symposium on Fuzzy Set Theory and Applications.

[56]  Alan J. Laub,et al.  A Parallel Algorithm for the Matrix Sign Function , 1990, Int. J. High Speed Comput..

[57]  J. H. Wilkinson,et al.  Similarity reduction of a general matrix to Hessenberg form , 1968 .

[58]  A. Laub,et al.  Generalized eigenproblem algorithms and software for algebraic Riccati equations , 1984, Proceedings of the IEEE.

[59]  G. Forsythe,et al.  Computer solution of linear algebraic systems , 1969 .

[60]  B. Parlett The Symmetric Eigenvalue Problem , 1981 .

[61]  Jagdish J. Modi,et al.  Parallel algorithms and matrix computation , 1988 .

[62]  John R. Rice Matrix Computations and Mathematical Software , 1983 .

[63]  Judith Gardiner,et al.  A generalization of the matrix sign function solution for algebraic Riccati equations , 1985, 1985 24th IEEE Conference on Decision and Control.

[64]  Cleve B. Moler,et al.  Matrix computations with Fortran and paging , 1972, CACM.

[65]  A. Laub,et al.  Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms , 1987 .

[66]  Alan J. Laub,et al.  Controllability and stability radii for companion form systems , 1988, Math. Control. Signals Syst..

[67]  A. Laub,et al.  The singular value decomposition: Its computation and some applications , 1980 .

[68]  B. Kågström Bounds and perturbation bounds for the matrix exponential , 1977 .

[69]  N. Nichols,et al.  Eigenstructure assignment in descriptor systems , 1986 .

[70]  Alan J. Laub,et al.  Optimal control study for the Space Station solar dynamic power module , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[71]  Alan J. Laub Robust stability of linear systems : some computational considerations , 1979 .

[72]  Clifford T. Mullis,et al.  Synthesis of minimum roundoff noise fixed point digital filters , 1976 .

[73]  Huibert Kwakernaak,et al.  Linear Optimal Control Systems , 1972 .

[74]  Alan J. Laub,et al.  Small-Sample Statistical Condition Estimates for General Matrix Functions , 1994, SIAM J. Sci. Comput..

[75]  James Demmel,et al.  Accurate Singular Values of Bidiagonal Matrices , 1990, SIAM J. Sci. Comput..

[76]  Adhemar Bultheel,et al.  Error analysis of incoming and outgoing schemes for the trigonometric moment problem , 1981 .

[77]  P. Dooren,et al.  Condensed forms for efficient time-invariant Kalman filtering , 1988 .

[78]  B. Moore On the flexibility offered by state feedback in multivariable systems beyond closed loop eigenvalue assignment , 1975, 1975 IEEE Conference on Decision and Control including the 14th Symposium on Adaptive Processes.

[79]  Alan J. Laub,et al.  Schur techniques for riccati differential equations , 1982 .

[80]  Michel Verhaegen,et al.  On the use of unitary state-space transformations , 1985 .

[81]  G. Bierman Factorization methods for discrete sequential estimation , 1977 .

[82]  A. J. Laub,et al.  Hypercube implementation of some parallel algorithms in control , 1988 .

[83]  Sun-Yuan Kung,et al.  A new identification and model reduction algorithm via singular value decomposition , 1978 .

[84]  Robert L. Johnston,et al.  Numerical methods: A software approach , 1982 .

[85]  John C. Doyle Analysis of Feedback Systems with Structured Uncertainty , 1982 .

[86]  L. D. Jong,et al.  Towards a formal definition of numerical stability , 1977 .

[87]  Paul Van Dooren,et al.  Computation of zeros of linear multivariable systems , 1980, Autom..

[88]  C. Paige,et al.  An algorithm for pole assignment of time invariant multi-input linear systems , 1982, 1982 21st IEEE Conference on Decision and Control.

[89]  W. Kahan Conserving Confluence Curbs Ill-Condition , 1972 .

[90]  A. Tits,et al.  Characterization and efficient computation of the structured singular value , 1986 .

[91]  Jack J. Dongarra,et al.  Matrix Eigensystem Routines — EISPACK Guide Extension , 1977, Lecture Notes in Computer Science.

[92]  A. Laub A schur method for solving algebraic Riccati equations , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[93]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[94]  Thomas F. Coleman,et al.  Handbook for matrix computations , 1988 .

[95]  Biswa Nath Datta,et al.  An algorithm for the multiinput eigenvalue assignment problem , 1990 .

[96]  A. Laub,et al.  Computation of Frequency Response of Descriptor Systems by Rational Interpolation , 1993 .

[97]  John R. Rice,et al.  Numerical methods, software, and analysis , 1983 .

[98]  V. Kučera,et al.  Efficient Algorithm for Spectral Factorization , 1984 .

[99]  C. Paige,et al.  An algorithm for pole assignment of time invariant linear systems , 1982 .

[100]  D. Heller A Survey of Parallel Algorithms in Numerical Linear Algebra. , 1978 .

[101]  Webb Miller,et al.  Software for Roundoff Analysis of Matrix Algorithms , 1980 .

[102]  H. T. Kung,et al.  Matrix Triangularization By Systolic Arrays , 1982, Optics & Photonics.

[103]  M. Sain Finite dimensional linear systems , 1972 .

[104]  Paul Van Dooren,et al.  Algorithm 590: DSUBSP and EXCHQZ: FORTRAN Subroutines for Computing Deflating Subspaces with Specified Spectrum , 1982, TOMS.

[105]  G. Forsythe,et al.  Computer solution of linear algebraic systems , 1969 .

[106]  J. Doyle,et al.  Practical computation of the mixed μ problem , 1992, 1992 American Control Conference.

[107]  Edmond A. Jonckheere,et al.  On computing the spectral radius of the Hankel plus Toeplitz operator , 1988 .

[108]  A. Tits,et al.  Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics , 1991 .

[109]  Robert A. Walker,et al.  Remark on “Algorithm 506: HQR3 and EXCHNG: Fortran Subroutines for Calculating and Ordering the Eigenvalues of a Real Upper Hessenberg Matrix” , 1982, TOMS.

[110]  P. Dooren,et al.  An improved algorithm for the computation of Kronecker's canonical form of a singular pencil , 1988 .

[111]  J. Pearson,et al.  Iterative computation of minimal H∞ norm , 1985, 1985 24th IEEE Conference on Decision and Control.

[112]  P. Dooren A Generalized Eigenvalue Approach for Solving Riccati Equations , 1980 .

[113]  Tsui An algorithm for the design of multi-functional observers , 1984 .

[114]  M. Kreĭn,et al.  ANALYTIC PROPERTIES OF SCHMIDT PAIRS FOR A HANKEL OPERATOR AND THE GENERALIZED SCHUR-TAKAGI PROBLEM , 1971 .

[115]  G. Stewart,et al.  An Algorithm for Generalized Matrix Eigenvalue Problems. , 1973 .

[116]  Paul Van Dooren,et al.  A systolic algorithm for riccati and lyapunov equations , 1989, Math. Control. Signals Syst..

[117]  N. Nichols,et al.  Robust pole assignment in linear state feedback , 1985 .

[118]  J. K. Reid Proc. IFIP TC 2 working conference on The relationship between numerical computation and programming languages , 1989 .

[119]  G. Stewart On the Perturbation of Pseudo-Inverses, Projections and Linear Least Squares Problems , 1977 .

[120]  A. Laub,et al.  Rational iterative methods for the matrix sign function , 1991 .

[121]  Petko H. Petkov,et al.  A computational algorithm for pole assignment of linear single-input systems , 1984 .

[122]  Chia-chi Tsui An algorithm for computing state feedback in multi-input linear systems , 1985, 1985 24th IEEE Conference on Decision and Control.

[123]  A. Varga A Schur method for pole assignment , 1981 .

[124]  David Y. Y. Yun,et al.  Fast Solution of Toeplitz Systems of Equations and Computation of Padé Approximants , 1980, J. Algorithms.

[125]  Gene H. Golub,et al.  Matrix computations , 1983 .

[126]  Åke Björck,et al.  Numerical Methods , 2020, Markov Renewal and Piecewise Deterministic Processes.

[127]  C. Loan The Sensitivity of the Matrix Exponential , 1977 .

[128]  J. H. Wilkinson,et al.  Handbook for Automatic Computation. Vol II, Linear Algebra , 1973 .

[129]  A. Laub Invariant Subspace Methods for the Numerical Solution of Riccati Equations , 1991 .

[130]  C. Kenney,et al.  Numerical integration of the differential matrix Riccati equation , 1985 .

[131]  Alan J. Laub,et al.  On Scaling Newton's Method for Polar Decomposition and the Matrix Sign Function , 1990, 1990 American Control Conference.

[132]  W. Enright On the efficient and reliable numerical solution of large linear systems of ODE's , 1979 .

[133]  P. Dooren The generalized eigenstructure problem in linear system theory , 1981 .

[134]  A. Bunse-Gerstner,et al.  Numerical Methods for Algebraic Riccati Equations , 1989 .

[135]  R. F. Sincovec,et al.  Analysis of descriptor systems using numerical algorithms , 1981 .

[136]  Jaroslav Kautsky,et al.  Robust pole assignment in singular control systems , 1989 .

[137]  G. Miminis Using deflation in the pole assignment problem with output feedback , 1989 .

[138]  J. Rissanen Algorithms for triangular decomposition of block Hankel and Toeplitz matrices with application to factoring positive matrix polynomials , 1973 .

[139]  A M Ostrowski On the spectrum of a one parametric family of matrices , 1952 .

[140]  S. Hammarling Numerical Solution of the Stable, Non-negative Definite Lyapunov Equation , 1982 .

[141]  R. V. Patel,et al.  A determinant identity and its application in evaluating frequency response matrics , 1988 .

[142]  L. M. Delves,et al.  Production and Assessment of Numerical Software , 1980 .

[143]  Alan J. Laub,et al.  Numerical Issues in Robust Control Design Techniques , 1993 .

[144]  P. Petkov,et al.  A computational algorithm for pole assignment of linear multi-input systems , 1984, The 23rd IEEE Conference on Decision and Control.

[145]  Paul Van Dooren Reduced order observers: A new algorithm and proof , 1984 .

[146]  A. Antoulas New results on the algebraic theory of linear systems: the solution of the cover problems , 1983 .

[147]  Peter R. Cappello,et al.  Systolic Computation Of Multivariable Frequency Response , 1988, Photonics West - Lasers and Applications in Science and Engineering.

[148]  A. J. Laub,et al.  Efficient matrix-valued algorithms for solving stiff Riccati differential equations , 1990 .

[149]  J. Makhoul,et al.  Linear prediction: A tutorial review , 1975, Proceedings of the IEEE.

[150]  Wayne Cowell,et al.  Portability of Numerical Software , 1977, Lecture Notes in Computer Science.

[151]  P. Khargonekar,et al.  State-space solutions to standard H/sub 2/ and H/sub infinity / control problems , 1989 .

[152]  J. Pearson Linear multivariable control, a geometric approach , 1977 .

[153]  P. Misra,et al.  Numerical algorithms for eigenvalue assignment by state feedback , 1984, Proceedings of the IEEE.

[154]  Zdenek Vostrý New algorithm for polynomial spectral factorization with quadratic convergence. II , 1975, Kybernetika.

[155]  G. Stewart Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems , 1973 .

[156]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[157]  Alan J. Laub,et al.  A stability-enhancing scaling procedure for Schur-Riccati solvers , 1989 .

[158]  C. Loan Computing integrals involving the matrix exponential , 1978 .

[159]  H. Zeiger,et al.  Approximate linear realizations of given dimension via Ho's algorithm , 1974 .

[160]  Alston S. Householder,et al.  The Theory of Matrices in Numerical Analysis , 1964 .

[161]  Biswa Nath Datta,et al.  An algorithm to assign eigenvalues in a Hessenberg matrix: Single input case , 1987 .

[162]  Michael A. Malcolm,et al.  Computer methods for mathematical computations , 1977 .

[163]  P. Khargonekar,et al.  State-space solutions to standard H2 and H∞ control problems , 1988, 1988 American Control Conference.

[164]  W. Arnold,et al.  Numerical Solution of Algebraic Matrix Riccati Equations. , 1984 .

[165]  Ralph Byers,et al.  Hamiltonian and symplectic algorithms for the algebraic riccati equation , 1983 .

[166]  Adhemar Bultheel,et al.  Recursive algorithms for the matrix Padé problem , 1980 .

[167]  G. Golub,et al.  A Hessenberg-Schur method for the problem AX + XB= C , 1979 .

[168]  S. P. Kumar,et al.  Solving Linear Algebraic Equations on an MIMD Computer , 1983, JACM.

[169]  Paul Van Dooren Deadbeat control: A special inverse eigenvalue problem , 1984 .

[170]  Petko Hr. Petkov,et al.  Invariants and canonical forms for linear multivariable systems under the action of orthogonal transformation groups , 1981, Kybernetika.

[171]  J. H. Wilkinson The algebraic eigenvalue problem , 1966 .

[172]  Alan J. Laub,et al.  Solution of the Sylvester matrix equation AXBT + CXDT = E , 1992, TOMS.

[173]  B. Moore Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .