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

非線形システムは、状態空間表現で優れたモデリングを見つけます。
質量中心が半分の長さにある状態空間で表される単純な振り子を考えてみましょう。重量と加えられるトルクは重要な要素です。
トルクを合計すると、微分方程式が得られます。状態変数が選択され、状態方程式が定式化されます。
平衡点を中心に方程式を線形化するには、平衡点を中心に摂動された状態変数を考えます。
テイラー級数近似を採用して解くと、線形状態方程式が得られます。
非線形ばねシステムについて考えます。微分方程式を定式化し、小さな摂動を導入します。
その後、この方程式は線形化されます。これを代入して微分を実行すると、線形化された中間微分方程式が得られます。
平衡力はxoを計算するために使用され、最終的に最終的な線形化微分方程式を形成します。
状態変数が選択され、状態方程式と出力方程式が書き込まれ、状態空間内のシステムの完全なモデルが提供されます。最後に、すべての方程式がベクトル行列形式に変換されます。

時間領域における線形近似

Linear Approximation in Time Domain

非線形システムでは、正確なモデリングと分析のために高度なアプローチが必要になることが多く、状態空間表現が特に効果的です。この方法は、単純な振り子や非線形ばねを備えた並進機械システムなど、変数とパラメータが時間や動作条件によって変化するシステムに特に役立ちます。 
質量が長さに沿って均等に分散され、質量の中心が振り子の長さの半分にある単純な振り子の場合、動的動作は、ピボット点の周りのトルクを合計することによって捉えることができます。結果として得られる微分方程式には、重力と適用されたトルクの影響が組み込まれています。これを状態空間形式で表すために、システムの位置と速度を表す状態変数が選択されます。これらの状態変数は、システムの時間的変化を表す状態方程式の定式化につながります。 
平衡点の周りでこれらの非線形状態方程式を線形化するには、小さな摂動を考慮する必要があります。状態変数を平衡値の周りで摂動させ、テイラー級数展開を適用することで、非線形項を線形項で近似できます。この近似により、線形システム理論を使用して分析できる線形状態方程式が得られます。 
非線形バネを備えた並進機械システムの場合、システムのダイナミクスは同様に、非線形バネ力を考慮した微分方程式によって制御されます。平衡位置の周囲に小さな摂動を導入すると、微分方程式の線形化が可能になります。x_0 (平衡位置) での平衡力が計算され、摂動された方程式が微分されて線形化された微分方程式が得られます。 
最終的な線形化された状態空間表現には、適切な状態変数の選択が含まれます。これには、多くの場合、質量の位置と速度が含まれます。次に、状態方程式と出力方程式が定式化されます。これらの方程式をベクトル行列形式に変換すると、システムの包括的な線形モデルが得られます。このモデルは、さまざまな線形制御および推定手法を使用して分析できるため、本質的に非線形なシステムの制御戦略の設計と実装が容易になります。

非線形システムは、状態空間表現で優れたモデリングを見つけます。
質量中心が...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Modeling in Time and Frequency Domain

時間領域および周波数領域におけるモデリング

KEY TERMS AND CONCEPTS

主要用語と概念

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

transfer function in control systems

Transfer Function from Input to Output

The transfer function describes how a linear time-invariant, or LTI, system maps an input to an output in the frequency domain. It gives a compact way to study system behavior without working only in the time domain. It also connects the differential equations of the system to a form that is easier to handle mathematically. For an nth-order LTI system, the behavior is written as a differential equation with constant coefficients. In that equation, r(t) is the input and c(t) is the output. The...

electrical systems

RLC Circuit Transfer Functions

RLC circuits use resistors, capacitors, and inductors to show how electrical signals change over time. In circuit analysis, these passive linear components are arranged so the link between an input and an output can be described with a transfer function. For this topic, the output is the voltage across the capacitor, compared with the input voltage. A series RLC circuit can be analyzed with Kirchhoff's Voltage Law, or KVL. KVL says that the sum of all voltages around a closed loop is zero.

mechanical systems

Transfer Functions in Mechanical Motion

Mechanical motion can be modeled with springs, masses, and viscous dampers. In these models, springs behave like inductors and masses act like capacitors in an electrical network. A viscous damper works like a resistor because it removes energy from the system. A force acts on the mass in the direction of motion. It is balanced by the spring force, the damping force, and the force from the mass’s acceleration. Newton’s second law is used to describe this balance, and the sum of the forces on...

electro-mechanical systems

DC Motor Modeling in Electromechanical Systems

DC motor modeling is a key part of electromechanical systems. These systems combine electrical parts and mechanical parts to produce a useful output. A DC motor is one common example because it turns electrical energy into mechanical motion. It is used in many places, from simple fans to complex robotic mechanisms. Inside the DC motor, the armature is a rotating circuit placed in a magnetic field. When current flows through the armature, it experiences a force from the magnetic field. That...

linear approximation in frequency domain

Linearizing Nonlinear RL Circuits

Linearizing nonlinear RL circuits helps turn a hard circuit problem into a simpler linear one. Linear systems follow superposition, which means the total response equals the sum of the responses to separate inputs. They also follow homogeneity, which means scaling the input scales the output by the same amount. Nonlinear systems do not automatically have those properties. For small changes around an operating point, though, a nonlinear system can often be treated as linear. This is done with a...

state space representation

RLC Circuit State Equations

RLC circuit state equations show how a circuit changes over time using state variables. This time-domain method is useful for linear, time-invariant systems, but it also helps when frequency-domain analysis is not enough. It is especially useful for nonlinear systems, time-varying systems, and multiple-input multiple-output systems. The state-space approach turns an nth-order system into a set of simultaneous first-order differential equations called state equations. For a common second-order...

transfer function to state space

Phase-Variable State Equations from Transfer Functions

Phase-variable state equations show how a transfer function can be rewritten for computer simulation. In state-space form, the system is described with state variables, which are the output and its first several derivatives. This makes it easier to model a physical system on a digital computer. The starting point is an nth-order linear differential equation with constant coefficients. Systems like an RLC circuit can be written this way. To build the state equations, the transfer function is...

state space to transfer function

Finding a Transfer Function from State Space

Finding a transfer function from state-space form is a key step in system analysis. It changes a time-domain model into a frequency-domain form. That makes it easier to study and design control systems. The process starts with the state-space equations. These include the state equation and the output equation. In these equations, x(t) is the state vector, u(t) is the input vector, y(t) is the output vector, and A, B, C, and D are the matrices that define the system dynamics. Next, the Laplace...

linear approximation in time domain

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 電気工学 は、電気回路の基本概念と回路解析テクニックを網羅したアニメーションビデオシリーズです。オペアンプと回路変換および 1 次・ 2 次回路の概念を網羅しています。実験を撮影した実写動画では、設計例や現代社会での応用が含まれており、複雑な概念も分かりやすく学習いただけます。

electrical engineering

Linearizing Nonlinear State-Space Models

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