Riemannian Optimal Identification Method for Linear Systems With Symmetric Positive-Definite Matrix

This article develops identification methods for linear continuous-time symmetric systems, such as electrical network systems, multiagent network systems, and temperature dynamics in buildings. To this end, we formulate three system identification problems for the corresponding discrete-time systems. The first is a least-squares problem in which we wish to minimize the sum of squared errors between the true and model outputs on the product manifold of the manifold of symmetric positive-definite matrices and two Euclidean spaces. In the second problem, to reduce the search dimensions, the product manifold is replaced with the quotient set under a specified group action by the orthogonal group. In the third problem, the manifold of symmetric positive-definite matrices in the first problem is replaced by the manifold of matrices with only positive diagonal elements. In particular, we examine the quotient geometry in the second problem. We propose Riemannian conjugate gradient methods for the three problems, and select initial points using a popular subspace method. The effectiveness of our proposed methods is demonstrated through numerical simulations and comparisons with the Gauss–Newton method, which is one of the most popular approach for solving least-squares problems.

[1]  Michel Verhaegen,et al.  Identification of the deterministic part of MIMO state space models given in innovations form from input-output data , 1994, Autom..

[2]  Toshiharu Sugie,et al.  Iterative identification method for linear continuous-time systems , 2008, Proceedings of the 45th IEEE Conference on Decision and Control.

[3]  Ralf Peeters,et al.  System identification based on riemannian geometry : theory and algorithms , 1993 .

[4]  Peter C. Young,et al.  Direct Identification of Continuous-time Models from Sampled Data: Issues, Basic Solutions and Relevance , 2008 .

[5]  Shun-ichi Amari,et al.  Natural Gradient Works Efficiently in Learning , 1998, Neural Computation.

[6]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[7]  Benedikt Wirth,et al.  Optimization Methods on Riemannian Manifolds and Their Application to Shape Space , 2012, SIAM J. Optim..

[8]  Bart De Moor,et al.  N4SID: Subspace algorithms for the identification of combined deterministic-stochastic systems , 1994, Autom..

[9]  Stephen P. Boyd,et al.  Optimal wire and transistor sizing for circuits with non-tree topology , 1997, 1997 Proceedings of IEEE International Conference on Computer Aided Design (ICCAD).

[10]  Levent Tunçel,et al.  Optimization algorithms on matrix manifolds , 2009, Math. Comput..

[11]  D. Bernstein.  Matrix Mathematics: Theory, Facts, and Formulas , 2009 .

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

[13]  Bart De Moor,et al.  Continuous-time frequency domain subspace system identification , 1996, Signal Process..

[14]  Bart De Moor,et al.  A unifying theorem for three subspace system identification algorithms , 1995, Autom..

[15]  U. Helmke,et al.  Optimization and Dynamical Systems , 1994, Proceedings of the IEEE.

[16]  Lennart Ljung,et al.  On-line structure selection for multivariable state-space models , 1982, Autom..

[17]  Marion Gilson,et al.  The CONTSID Toolbox: A Software Support for Data-based Continuous-time Modelling , 2008 .

[18]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[19]  Brigitte Maier,et al.  Fundamentals Of Differential Geometry , 2016 .

[20]  Akira Ohsumi,et al.  Subspace identification for continuous-time stochastic systems via distribution-based approach , 2002, Autom..

[21]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[22]  Karolos M. Grigoriadis,et al.  Stabilization and H∞ control of symmetric systems: an explicit solution , 2001, Syst. Control. Lett..

[23]  Anders Helmersson,et al.  Data driven local coordinates for multivariable linear systems and their application to system identification , 2004, Autom..

[24]  René Vidal,et al.  Bundle Reduction and the Alignment Distance on Spaces of State-Space LTI Systems , 2017, IEEE Transactions on Automatic Control.

[25]  Anders Rantzer,et al.  Optimal H∞ state feedback for systems with symmetric and Hurwitz state matrix , 2016, 2016 American Control Conference (ACC).

[26]  Hiroyuki Sato,et al.  A Dai–Yuan-type Riemannian conjugate gradient method with the weak Wolfe conditions , 2014, Comput. Optim. Appl..

[27]  Patrick Dewilde,et al.  Subspace model identification Part 1. The output-error state-space model identification class of algorithms , 1992 .

[28]  Takeshi Hatanaka,et al.  Physics-integrated hierarchical/distributed HVAC optimization for multiple buildings with robustness against time delays , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[29]  Michiel Hazewinkel.  Moduli and canonical forms for linear dynamical systems II: The topological case , 2005, Mathematical systems theory.

[30]  Wallace E. Larimore,et al.  Canonical variate analysis in identification, filtering, and adaptive control , 1990, 29th IEEE Conference on Decision and Control.

[31]  Michel Verhaegen,et al.  Stochastic theory of continuous-time state-space identification , 1999, IEEE Trans. Signal Process..

[32]  H. Akaike.  A new look at the statistical model identification , 1974 .

[33]  Hiroyuki Sato,et al.  Structure-Preserving $H^2$ Optimal Model Reduction Based on the Riemannian Trust-Region Method , 2018, IEEE Transactions on Automatic Control.

[34]  John M. Lee.  Introduction to Smooth Manifolds , 2002 .

[35]  Si-Zhao Joe Qin,et al.  An overview of subspace identification , 2006, Comput. Chem. Eng..

[36]  Lennart Ljung,et al.  Experiments with Identification of Continuous Time Models , 2009 .

[37]  David F. Delchamps,et al.  Global structure of families of multivariable linear systems with an application to identification , 1985, Mathematical systems theory.

[38]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[39]  Francesco Bullo,et al.  Electrical Networks and Algebraic Graph Theory: Models, Properties, and Applications , 2018, Proceedings of the IEEE.

[40]  Hiroyuki Sato,et al.  Riemannian Optimal System Identification Algorithm for Linear MIMO Systems , 2017, IEEE Control Systems Letters.

[41]  片山 徹.  Subspace methods for system identification , 2005 .

[42]  Suvrit Sra,et al.  Conic Geometric Optimization on the Manifold of Positive Definite Matrices , 2013, SIAM J. Optim..

[43]  Jan C. Willems,et al.  REALIZATION OF SYSTEMS WITH INTERNAL PASSIVITY AND SYMMETRY CONSTRAINTS , 1976 .

[44]  Toshiharu Sugie,et al.  Projection-based identification algorithm for grey-box continuous-time models , 2013, Syst. Control. Lett..

[45]  Ralf Peeters,et al.  On the Riemannian interpretation of the Gauss-Newton algorithm , 1993 .

[46]  Michel Verhaegen,et al.  N2SID: Nuclear norm subspace identification of innovation models , 2016, Autom..

[47]  Brett Ninness,et al.  On Gradient-Based Search for Multivariable System Estimates , 2008, IEEE Transactions on Automatic Control.

[48]  R. Stern,et al.  Some Geometric Questions in the Theory of Linear Systems , 2022 .

[49]  Alan Edelman,et al.  The Geometry of Algorithms with Orthogonality Constraints , 1998, SIAM J. Matrix Anal. Appl..

[50]  H. V. Trees,et al.  Covariance, Subspace, and Intrinsic CramrRao Bounds , 2007 .

[51]  Magnus Egerstedt,et al.  Graph Theoretic Methods in Multiagent Networks , 2010, Princeton Series in Applied Mathematics.

[52]  B. Hanzon,et al.  OVERLAPPING BLOCK-BALANCED CANONICAL FORMS FOR VARIOUS CLASSES OF LINEAR SYSTEMS , 1998 .