An unconstrained H 2 model order reduction optimisation algorithm based on the Stiefel manifold for bilinear systems

ABSTRACT In this paper, the optimal H 2 model order reduction (MOR) problem for bilinear systems is explored. The orthogonality constraint of the cost function generated by the H 2 MOR error makes it is posed not on the Euclidean space, but can be discussed on the Stiefel manifold. Then, the H 2 optimal MOR problem of bilinear systems is turned into the unconstrained optimisation on the Stiefel manifold. The explicit expression of the gradient for the cost function on this manifold is derived. Full use of the geometry properties of this Stiefiel manifold, we propose a feasible and effective iterative algorithm to solve the unconstrained H 2 minimisation problem. Moreover, the convergence of our algorithm is rigorously proved. Finally, two practical examples related to bilinear systems demonstrate the effectiveness of our algorithm.

[1]  Robert E. Mahony,et al.  Convergence of the Iterates of Descent Methods for Analytic Cost Functions , 2005, SIAM J. Optim..

[2]  K. Gallivan,et al.  Convergence analysis of Riemannian trust-region methods , 2006 .

[3]  Chin Hsu,et al.  Realization algorithms and approximation methods of bilinear systems , 1983, The 22nd IEEE Conference on Decision and Control.

[4]  Abderrahman Bouhamidi,et al.  A note on the numerical approximate solutions for generalized Sylvester matrix equations with applications , 2008, Appl. Math. Comput..

[5]  Hamid Reza Shaker,et al.  Control configuration selection for bilinear systems via generalised Hankel interaction index array , 2015, Int. J. Control.

[6]  Xiaolong Wang,et al.  Model reduction of bilinear systems based on Laguerre series expansion , 2012, J. Frankl. Inst..

[7]  A. Isidori,et al.  Realization and Structure Theory of Bilinear Dynamical Systems , 1974 .

[8]  Peter Benner,et al.  Lyapunov Equations, Energy Functionals, and Model Order Reduction of Bilinear and Stochastic Systems , 2011, SIAM J. Control. Optim..

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

[10]  Yao-Lin Jiang,et al.  H2 optimal model order reduction by two-sided technique on Grassmann manifold via the cross-gramian of bilinear systems , 2017, Int. J. Control.

[11]  Peter Benner,et al.  Interpolation-Based H2-Model Reduction of Bilinear Control Systems pdfsubject , 2011 .

[12]  Zhaojun Bai,et al.  A projection method for model reduction of bilinear dynamical systems , 2006 .

[13]  Tobias Breiten,et al.  Krylov subspace methods for model order reduction of bilinear control systems , 2010, Syst. Control. Lett..

[14]  James Lam,et al.  An approximate approach to H2 optimal model reduction , 1999, IEEE Trans. Autom. Control..

[15]  R. Adler,et al.  Newton's method on Riemannian manifolds and a geometric model for the human spine , 2002 .

[16]  Yaguang Yang.  Globally Convergent Optimization Algorithms on Riemannian Manifolds: Uniform Framework for Unconstrained and Constrained Optimization , 2007 .

[17]  Ping Yang,et al.  $$\mathscr{H}_{2}$$ model order reduction for bilinear systems based on the cross gramian , 2017, IMA J. Math. Control. Inf..

[18]  Yao-Lin Jiang,et al.  Time domain model order reduction using general orthogonal polynomials for K-power bilinear systems , 2016, Int. J. Control.

[19]  Serkan Gugercin,et al.  Multipoint Volterra Series Interpolation and H2 Optimal Model Reduction of Bilinear Systems , 2013, SIAM J. Matrix Anal. Appl..

[20]  Hamid Reza Shaker,et al.  Frequency-Interval Model Reduction of Bilinear Systems , 2014, IEEE Transactions on Automatic Control.

[21]  James Lam,et al.  On H2 model reduction of bilinear systems , 2002, Autom..

[22]  P. Absil,et al.  Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation , 2004 .

[23]  Min-Shin Chen,et al.  Exponential stabilization of a class of unstable bilinear systems , 2000, IEEE Trans. Autom. Control..

[24]  Yao-Lin Jiang,et al.  Two-sided projection methods for model reduction of MIMO bilinear systems , 2013 .

[25]  Hamid Reza Shaker,et al.  Generalized Hankel Interaction Index Array for control structure selection for discrete-time MIMO bilinear processes and plants , 2014, 53rd IEEE Conference on Decision and Control.

[26]  S. Sastry.  Nonlinear Systems: Analysis, Stability, and Control , 1999 .

[27]  H. T. Dorissen.  Canonical forms for bilinear systems , 1989 .

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

[29]  Ubaid M. Al-Saggaf,et al.  New model reduction scheme for bilinear systems , 1994 .