[Paper Review] H2-optimal approximation of MIMO linear dynamical systems

Paul Van Dooren, Kyle A. Gallivan+1 more|ArXiv.org|Jul 30, 2008

Model Reduction and Neural Networks11 references3 citations

TL;DR

This paper presents an ℋ₂-optimal model reduction method for multiple-input multiple-output (MIMO) linear dynamical systems by characterizing stationary points of the ℋ₂-norm error via tangential interpolation conditions in both continuous- and discrete-time settings. The key contribution is a parameterization of optimal reduced-order models using interpolation constraints tied to Jordan block structure, enabling gradient-based optimization despite non-unique state-space realizations.

ABSTRACT

We consider the problem of approximating a multiple-input multiple-output (MIMO) $p imes m$ rational transfer function $H(s)$ of high degree by another $p imes m$ rational transfer function $\hat H(s)$ of much smaller degree, so that the ${\cal H}_2$ norm of the approximation error is minimized. We characterize the stationary points of the ${\cal H}_2$ norm of the approximation error by tangential interpolation conditions and also extend these results to the discrete-time case. We analyze whether it is reasonable to assume that lower-order models can always be approximated arbitrarily closely by imposing only first-order interpolation conditions. Finally, we analyze the ${\cal H}_2$ norm of the approximation error for a simple case in order to illustrate the complexity of the minimization problem.

Motivation & Objective

• To develop an ℋ₂-optimal approximation method for MIMO linear dynamical systems with reduced order.
• To characterize stationary points of the ℋ₂-norm error in terms of interpolation conditions for both continuous- and discrete-time systems.
• To address the non-uniqueness of state-space realizations by deriving non-redundant, parameter-independent stationarity conditions.
• To analyze the existence and nature of local minima in the ℋ₂ approximation problem through low-degree system examples.
• To provide a numerically robust framework for ℋ₂-optimal model reduction using Sylvester equation formulations instead of Jordan canonical forms.

Proposed method

• Derives the gradient of the squared ℋ₂-norm error using Wilson's formula, enabling optimization over state-space parameters.
• Characterizes ℋ₂-optimal approximants via tangential interpolation conditions involving the transfer function and its dual, expressed in terms of Jordan block parameters.
• Uses Sylvester equations to express interpolation conditions in a numerically stable form, avoiding ill-conditioning from Jordan forms.
• Applies the dual transfer function $ H_*(z) = z^{-1}H^T(z^{-1}) $ to reformulate conditions in the discrete-time case.
• Employs matrix equations with polynomial vectors to derive conditions for higher-order poles, generalizing first-order results.
• Analyzes the structure of the error norm for McMillan degree-one and degree-two systems to demonstrate the existence of multiple local minima.

Experimental results

Research questions

• RQ1 Can ℋ₂-optimal MIMO approximations be characterized by interpolation conditions that are independent of state-space parameterization?
• RQ2 How do interpolation conditions change when the reduced-order model has higher-order Jordan blocks rather than simple poles?
• RQ3 Is it possible to achieve arbitrarily close approximation using only first-order interpolation conditions for all lower-order models?
• RQ4 What is the role of the Sylvester equation formulation in improving numerical stability over Jordan canonical forms?
• RQ5 How many local minima can exist in the ℋ₂ approximation problem for low-order systems?

Key findings

• Stationary points of the ℋ₂-norm error correspond to tangential interpolation conditions involving the transfer function and its dual, with order dependent on Jordan block size.
• For first-order poles, the conditions reduce to matching the transfer function and its first derivative at the reduced-order poles, generalizing the SISO result of [ML67].
• The ℋ₂ norm of the error for systems of McMillan degree one and two reveals that multiple local minima can exist, indicating non-convexity of the optimization problem.
• The use of Sylvester equation-based interpolation conditions provides a numerically robust alternative to Jordan-based formulations, which become ill-conditioned near higher-order poles.
• There is no globally smooth parameterization of the set of rational transfer functions of degree $ n $ when $ \min(p,m) > 1 $, but local smooth parameterizations exist via the Sylvester formulation.
• The discrete-time case is fully characterized using the dual transfer function $ H_*(z) $, with analogous interpolation conditions derived via polynomial expansions in $ (\lambda - z) $.