Model Order Reduction for Linear Systems in MATLAB - Pade Approximation Source Code

Pade approximation is a classical model order reduction technique primarily used to simplify high-order linear systems' transfer functions or state-space models while preserving their essential dynamic characteristics. In MATLAB, Pade reduction can be implemented using built-in functions, making it applicable for control system design and simulation optimization scenarios.

### Fundamental Principles of Pade Approximation The core concept of Pade approximation involves using rational functions to approximate the dynamic response of high-order systems, particularly for time-delay component approximations. The method constructs a lower-order transfer function by matching the Taylor expansion coefficients of the original system at specific points (typically at zero). MATLAB provides the `pade` function to facilitate this process, allowing users to specify the reduced order while automatically computing approximate numerator and denominator polynomials. Key implementation detail: The function syntax `[num,den] = pade(T,N)` generates Nth-order Pade approximation for time delay T, where num and den represent the resulting transfer function coefficients.

### MATLAB Implementation Approach Transfer Function Reduction: When the original system is given in transfer function form, the `pade` function directly approximates its delay component and generates the reduced-order transfer function. Implementation example: For a system with delay, use `sys_red = pade(sys,N)` where sys is the original TF object and N is the desired approximation order. State-Space Model Processing: For state-space models, MATLAB first converts them to transfer function form, applies Pade reduction, and can subsequently convert back to state-space representation. Algorithm detail: This involves using `ss2tf` conversion before approximation and `tf2ss` for reverse transformation. Order Selection: Users must balance reduction accuracy against computational complexity, typically starting with low orders (e.g., 2nd or 3rd order) and progressively validating approximation effectiveness through comparative analysis.

### Application Scenarios and Considerations Pade approximation is particularly useful in control system simplification and real-time simulation, but requires attention to: Systems sensitive to high-frequency dynamics may introduce errors through reduction; Pade approximation of delay components might produce non-minimum phase responses in time domain, necessitating validation through step response or frequency domain analysis. Code verification tip: Use `step(sys_original, sys_reduced)` for comparative time-domain validation and `bode` plots for frequency response checking.

Extension Consideration: Pade method can be combined with other reduction techniques (like balanced truncation or Hankel norm approximation) for comparative analysis to adapt to different engineering requirements. Implementation approach: Use `reduce` function with different method specifications for systematic comparison.