[Paper Review] Balanced model order reduction for systems depending on a parameter

Carles Batlle, Néstor Roqueiro|arXiv (Cornell University)|Apr 27, 2016

Model Reduction and Neural Networks16 references3 citations

TL;DR

This paper presents a parameter-aware balanced model order reduction method for linear time-invariant systems with uncertain parameters, using power series expansions up to second-order corrections in the parameter. The approach preserves system structure and stability while explicitly incorporating the parameter in the reduced model, enabling robust controller design across a range of parameter values, with novel second-order singular subspace corrections derived analytically.

ABSTRACT

We provide an analytical framework for balanced realization model order reduction of linear control systems which depend on an unknown parameter. Besides recovering known results for the first order corrections, we obtain explicit novel expressions for the form of second order corrections for singular values and singular vectors. The final result of our procedure is an order reduced model which incorporates the uncertain parameter. We apply our algorithm to the model order reduction of a linear system of masses and springs with parameter dependent coefficients.

Motivation & Objective

• To develop a model order reduction framework that preserves the parameter dependence of linear systems with uncertain coefficients.
• To overcome the limitation of standard balanced truncation, which requires numerical parameter values and loses symbolic parameter dependence.
• To derive analytical expressions for first- and second-order corrections in singular values and singular subspaces under parameter variation.
• To construct a reduced-order model that explicitly includes the parameter as a polynomial, enabling use in controller design over a parameter range.
• To validate the method on a mass-spring system with parameter-dependent stiffness and mass coefficients.

Proposed method

• Expand the controllability and observability Gramians as power series in the uncertain parameter, solving Lyapunov equations perturbatively.
• Compute the square roots of the Gramians via power series expansion to obtain the balanced transformation matrix.
• Apply singular value decomposition (SVD) to the balanced system, deriving explicit second-order corrections for singular values and singular subspaces.
• Truncate the balanced system to a reduced order, retaining polynomial dependence on the parameter up to second order.
• Implement the algorithm in MATLAB and validate it on a 20-state mass-spring system with 10 masses and parameter-dependent stiffness and mass.
• Use Bode plots to compare the accuracy of zeroth-, first-, and second-order polynomial approximations against the exact reduced model.

Experimental results

Research questions

• RQ1 Can balanced model order reduction be extended to systems with symbolic, uncertain parameters while preserving stability and structure?
• RQ2 What are the analytical expressions for first- and second-order corrections to singular values and singular subspaces under parameter variation?
• RQ3 How does the accuracy of the reduced-order model improve with increasing order of polynomial approximation in the parameter?
• RQ4 Can the resulting parametric reduced model be used effectively for controller design across a range of parameter values?
• RQ5 What is the trade-off between computational cost and approximation error in the parametric reduced model?

Key findings

• The method successfully computes a reduced-order model that explicitly depends on the uncertain parameter as a polynomial, enabling parametric controller design.
• Second-order corrections to singular subspaces are derived analytically for the first time in this context, providing a novel contribution to SVD perturbation theory.
• For the mass-spring system with parameter-dependent coefficients, the second-order approximation yields significantly better Bode plot agreement with the exact reduced model than first- or zeroth-order approximations.
• The zeroth-order approximation corresponds to fixed parameter value m=0, and higher-order terms improve accuracy across the parameter range.
• Simulations indicate that truncation errors from polynomial approximations decrease with increasing order, suggesting improved approximation quality.
• The method is limited to stable systems and does not currently provide error bounds for the full approximation chain, though simulations suggest convergence with higher-order terms.