Balanced Model Order Reduction Tutorial

Overview

Balanced model order reduction is a systematic approach to approximating high-dimensional dynamical systems with lower-dimensional models while preserving key system properties. This technique is particularly useful in control theory and computational modeling where computational efficiency is crucial.

Mathematical Foundation

Consider a linear time-invariant system:

ẋ = Ax + Bu
y = Cx + Du

Where:

x ∈ ℝⁿ is the state vector
u ∈ ℝᵐ is the input vector
y ∈ ℝᵖ is the output vector
A, B, C, D are system matrices

Key Concepts

Controllability and Observability Gramians

The controllability Gramian Wc measures how much energy is required to reach each state:

AWc + WcAᵀ + BBᵀ = 0

The observability Gramian Wo measures how well each state can be observed from the output:

AᵀWo + WoA + CᵀC = 0

Hankel Singular Values

The Hankel singular values σᵢ are the square roots of the eigenvalues of WcWo:

σᵢ = √λᵢ(WcWo)

These values indicate the “importance” of each state - larger values correspond to states that are both highly controllable and observable.

Balanced Realization Algorithm

Solve Lyapunov equations to find Wc and Wo
Compute the Cholesky decomposition: Wc = RRᵀ
Perform SVD on RᵀWoR = UΣVᵀ

Form transformation matrices:

T = R⁻ᵀUΣ⁻¹/²
T⁻¹ = Σ⁻¹/²VᵀRᵀ

Transform the system:
```
Ab = T⁻¹AT,  Bb = T⁻¹B,  Cb = CT
```

Model Reduction

After balancing, truncate the system by keeping only the r most significant states (largest Hankel singular values):

Ar = Ab(1:r, 1:r)
Br = Bb(1:r, :)
Cr = Cb(:, 1:r)
Dr = D  (unchanged)

Error Bounds

The approximation error is bounded by:

||G - Gr||∞ ≤ 2∑(i=r+1 to n) σᵢ

Where G and Gr are the transfer functions of the original and reduced systems.

Advantages

Preservation of stability: Balanced truncation preserves system stability
Error bounds: Provides computable error bounds
Systematic approach: No trial-and-error required
Physical insight: Hankel singular values provide insight into system behavior

Applications

Control system design: Reducing controller complexity
Structural dynamics: Simplifying finite element models
Circuit analysis: Reducing large-scale circuit models
Fluid dynamics: Reducing CFD models for real-time applications

Example Workflow

% MATLAB example
sys = ss(A, B, C, D);           % Original system
[sysr, info] = balred(sys, r);   % Balanced reduction to order r
sigma(sys, sysr);               % Compare frequency responses

Conclusion

Balanced model order reduction provides a principled way to reduce system complexity while maintaining essential dynamics. The method’s theoretical guarantees and practical effectiveness make it a cornerstone technique in modern control theory and computational modeling.

Thomas Soares Mullen