Derivatives measure how tracking error changes when a model quantity changes by a small amount. A derivative with respect to a parameter indicates local influence, while derivatives with respect to incoming data show how data perturbations affect performance. Comparing these rates helps identify the quantities that most strongly control accuracy and provides a mathematical basis for sensitivity functions and perturbation analysis.
Poor conditioning means that relatively small changes in parameters, model assumptions, or incoming data can produce comparatively large changes in tracking error. This behavior signals that the model or tracking task may respond unreliably to uncertainty. Sensitivity analysis exposes such weaknesses, helping researchers distinguish well-behaved performance from responses that may indicate instability or numerical difficulty.
The analysis can vary model parameters and incoming data separately, then compare their effects on tracking error. Parameter perturbations reveal which characteristics of the model influence accuracy, whereas data perturbations show how strongly the system responds to changes in its input information. This distinction supports more precise interpretation and helps target improvements where they matter most.
A typical analysis first identifies the prescribed signal or trajectory and the quantities that may influence tracking. It then measures the corresponding tracking error, introduces controlled changes to parameters or incoming data, and compares the resulting errors. Derivatives, perturbation analysis, or sensitivity functions can express those comparisons and reveal the dominant sources of performance variation.
Tracking sensitivity is useful when a controller must follow a prescribed signal reliably despite changes in the model or incoming data. By showing which quantities have the greatest effect on tracking error, the analysis guides robust controller design. It also helps reveal poorly conditioned or unstable behavior before that behavior undermines tracking accuracy.
Sensitivity results indicate how strongly predictions or tracking outcomes depend on uncertain quantities. Numerical algorithms can use this information to identify poorly conditioned behavior and improve reliability, while model interpretation can focus attention on influential parameters or data. The same analysis supports optimization by showing which changes are most likely to improve accuracy or performance.