21.9 : 时域中的线性近似

Nonlinear systems find a superior modeling in state-space representation.

Consider a simple pendulum represented in state-space with the center of mass at its half-length. The weight and applied torque are significant factors.

Summing the torques leads to a differential equation. State variables are selected, and the state equations are formulated.

To linearize the equation about the equilibrium point, consider the state variables perturbed about the equilibrium point.

The Taylor series approximation is employed and solved, resulting in linear state equations.

Consider a nonlinear spring system. Formulate a differential equation and introduce a small perturbation.

This equation is then linearized. Substituting this and carrying out the differentiation gives the linearized intermediate differential equation.

The equilibrium force is used to calculate x_o, ultimately forming the final linearized differential equation.

The state variables are selected, and state and output equations are written, providing a complete model of the system in state space. Finally, all equations are converted into vector-matrix form.