Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.

For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the dynamic behavior can be captured by summing torques around the pivot point. The resulting differential equation incorporates the effects of gravitational force and applied torque. To represent this in state-space form, state variables are chosen to describe the system's position and velocity. These state variables lead to the formulation of state equations, which describe the system's time evolution.

Linearization of these nonlinear state equations around an equilibrium point involves considering small perturbations. By perturbing the state variables around their equilibrium values and applying a Taylor series expansion, the nonlinear terms can be approximated by their linear counterparts. This approximation yields linear state equations that can be analyzed using linear system theory.

In the case of a translational mechanical system with a nonlinear spring, the system's dynamics are similarly governed by a differential equation that accounts for the nonlinear spring force. Introducing a small perturbation around the equilibrium position allows for the linearization of the differential equation. The equilibrium force at x₀ (the position of equilibrium) is calculated, and the perturbed equation is differentiated to obtain a linearized differential equation.

The final linearized state-space representation involves selecting appropriate state variables, which often include the position and velocity of the mass. State equations and output equations are then formulated. Converting these equations into vector-matrix form provides a comprehensive linear model of the system. This model can be analyzed using various linear control and estimation techniques, facilitating the design and implementation of control strategies for systems that are inherently nonlinear.