Implementation Examples of Two Methods for Linearizing Nonlinear Dynamic Equations

In this technical documentation, we present two distinct methodologies for linearizing nonlinear dynamic equations: exact linearization and feedback linearization. Before examining these techniques in depth, it is crucial to understand their application context. Nonlinear dynamic equations often present computational challenges due to their inherent complexity, but linearization enables simplification and tractable solutions through systematic approximation. The first methodology we examine is exact linearization (also known as Jacobian linearization). This technique involves computing a linear approximation of the nonlinear system around a specific operating point using Taylor series expansion. The implementation typically requires calculating partial derivatives to construct the Jacobian matrix, which represents the system's local linear behavior. While this method provides high accuracy near the operating point, it demands significant computational resources for derivative calculations and may become inefficient for large-scale systems. A typical MATLAB implementation would involve using the jacobian() function or manually computing partial derivatives symbolically/numerically. The second approach we explore is feedback linearization, a control-theoretic technique that transforms nonlinear dynamics into equivalent linear systems through nonlinear state feedback. This method designs a control law that actively cancels nonlinear terms, resulting in a linear input-output relationship. Implementation requires careful selection of coordinate transformations and validation of relative degree conditions. Unlike exact linearization, this method provides global linearization rather than local approximation, but necessitates thorough understanding of differential geometric control theory. Code implementation often involves symbolic math toolbox operations for Lie derivative calculations and control law derivation. By mastering these two linearization techniques, engineers can effectively transform challenging nonlinear systems into manageable linear formulations, enabling application of powerful linear control theories and simplification of complex dynamic analyses. Both methods require different mathematical prerequisites and offer distinct advantages depending on system characteristics and application requirements.