lineariseerimist

Lineariseerimist, or linearization, is the process of approximating a nonlinear function or system by a linear one in the neighborhood of a chosen point or operating point. The aim is to simplify analysis, prediction, and control by using a model that is easier to study while still capturing the nearby behavior of the original system.

In mathematical terms, for a scalar function f: R → R and a point x0, the linear approximation is f(x) ≈ f(x0) + f′(x0)(x − x0). For a function f: R^n → R^m, the linearization around x0 uses the Jacobian Jf(x0): f(x) ≈ f(x0) + Jf(x0)(x − x0). In dynamical systems, linearization around an equilibrium x* yields a linear system dξ/dt = Aξ with A = Df(x*); the eigenvalues of A determine local stability.

Applications span multiple fields. In engineering and physics, small-signal analysis linearizes nonlinear components or equations around a working point to study responses to small disturbances. In control theory, linearized models enable the design and analysis of controllers for nonlinear plants. In economics and other sciences, local linearization helps understand behavior near steady states or operating points. In numerical analysis, linearization underpins methods such as Newton’s method, which uses tangent (linear) approximations to locate zeros of nonlinear functions.

Limitations are important: linearization is a local approximation and may be valid only for small perturbations around the chosen point. Far from the point, nonlinear effects can dominate, potentially invalidating conclusions drawn from the linear model.