Name: linearizing nonlinear state-space models
Uploaded: 2024-11-21T14:54:05.000Z
Duration: 1 min 21 s
Channel: JoVE

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 xo, 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.

linear approximation in time domain

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&#39;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&#39;s position and velocity. These state variables lead to the formulation of state equations, which describe the system&#39;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&#39;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.

Linear Approximation in Time Domain

Nonlinear systems find a superior modeling in state-space representation.
Consider a simple pendulum represented in state-space with the center of mass at ...

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KEY TERMS AND CONCEPTS

linearizing nonlinear state-space models

Linearizing nonlinear state-space models helps describe systems that change with time or operating conditions. State-space representation is often effective for this kind of modeling. It is especially useful for systems such as a simple pendulum or a translational mechanical system with nonlinear springs.

A simple pendulum can be modeled by summing torques around the pivot point. In this case, the mass is evenly distributed along the pendulum, and the center of mass is located at half its length. The resulting differential equation includes gravitational force and any applied torque. State variables are then chosen to represent position and velocity.

These state variables lead to state equations, which describe how the system changes over time. To study the system near an equilibrium point, small perturbations are added to the state variables. A Taylor series expansion is then used to approximate the nonlinear terms with linear ones. The result is a set of linear state equations that can be analyzed with linear system theory.

The same idea applies to a translational mechanical system with a nonlinear spring. Its motion is governed by a differential equation that includes the nonlinear spring force. A small perturbation around the equilibrium position allows the equation to be linearized. 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 model uses appropriate state variables, often position and velocity of the mass. State equations and output equations are written in vector-matrix form. This linear model makes it easier to apply linear control and estimation methods to systems that are inherently nonlinear.

Linearizing Nonlinear State-Space Models

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Linearizing nonlinear state-space models helps describe systems that change with time or operating conditions. State-space representation is often effective for this kind of modeling. It is especially useful for systems such as a simple pendulum or a translational mechanical system with nonlinear springs. A simple pendulum can be modeled by summing torques around the pivot point. In this case, the mass is evenly distributed along the pendulum, and the center of mass is located at half its...

key terms and concepts

Learn how nonlinear systems like a pendulum or spring-mass setup are linearized with small perturbations, Taylor series, and state-space equations.

Linearizing Nonlinear State-Space Models | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Linearizing Nonlinear State-Space Models

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