Name: step response of second-order systems
Uploaded: 2024-11-21T14:54:10.000Z
Duration: 1 min 18 s
Channel: JoVE

減衰が不十分な場合、単位ステップ入力が適用されると、逆ラプラス法を使用して伝達関数方程式が解かれ、出力応答が得られます。
入力と出力の差は、減衰した正弦波振動を示すエラー信号です。定常状態では、エラーはありません。
減衰比がゼロに等しい場合、応答は減衰しなくなり、振動は無限に続きます。
臨界減衰のシナリオでは、システムの 2 つの極は同一です。単位ステップ入力の場合、出力方程式とその逆ラプラス変換が決定されます。
オーバーダンピング シナリオでは、システムの 2 つの極は負、実数、不等です。
単位ステップ入力の場合、出力方程式が定式化され、その逆ラプラス変換が計算され、2 つの減衰指数項が得られます。
減衰比が 1 よりも大幅に大きい場合、一方の指数関数的減衰はもう一方の指数関数的減衰よりもはるかに速く、無視できます。これにより、システムの単位ステップ応答を与える 1 次システムに似た近似伝達関数が得られます。

二次システム ii

減衰率 ζ が 0 から 1 の間にある振動減衰の二次システムでは、単位ステップ入力によって伝達関数が生成され、これを逆ラプラス法で変換すると出力応答が明らかになります。出力は減衰正弦波振動を示し、入力と出力の差は誤差信号と呼ばれます。この誤差信号は減衰振動動作も示します。最終的に、システムが定常状態に達すると、誤差はゼロに減少します。

ζ = 0 の場合、システムは減衰しなくなり、減衰のない永続的な振動が発生します。システムは無期限に振動し続け、安定状態に到達することはありません。

ζ = 1 の臨界減衰シナリオでは、システムの極は同一です。単位ステップ入力の場合、出力方程式が導出され、逆ラプラス変換によって応答が得られます。この応答は振動を示さず、オーバーシュートすることなく可能な限り迅速に平衡に戻ります。

ζ > 1 の過剰減衰シナリオでは、システムの 2 つの積分成分は実数で、負で、不等です。単位ステップ入力が適用されると、出力方程式が定式化されます。この方程式の逆ラプラス変換により、2 つの減衰指数項を特徴とする応答が得られます。減衰率が 1 よりはるかに大きい場合、これらの指数減衰の 1 つは、もう 1 つよりも大幅に速くなります。その結果、減衰が速い項は無視できることが多く、応答が一次システムのように単純化されます。この近似により、近似伝達関数を導出でき、分析と設計が単純化されます。

要約すると、さまざまな減衰条件下での単位ステップ入力に対する二次システムの動作を理解することは、システムの設計と分析にとって重要です。振動減衰の応答は減衰振動を特徴とし、振動減衰の応答は連続振動を特徴とします。臨界減衰システムは振動なしで速やかに平衡に達し、減衰過剰システムはより遅い非振動応答を示します。これらの洞察は、システムを調整して望ましいパフォーマンス特性を実現し、実際のアプリケーションで安定性と精度を確保するために不可欠です。

二次システム II

減衰が不十分な場合、単位ステップ入力が適用されると、伝達関数方程式が逆ラプラス法を使用して解かれ、出力が得られます...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Transient and Steady-state Response Analysis

過渡応答および定常状態応答解析

transient-and-steady-state-response-analysis

KEY TERMS AND CONCEPTS

主要用語と概念

Second Order systems II

step response of second-order systems

Second-order systems react to a unit-step input in different ways depending on the damping ratio. When the damping ratio, zeta, is between 0 and 1, the system is underdamped. In this case, inverse Laplace methods give an output that oscillates with a decreasing amplitude. The error signal, which is the difference between the input and output, also shows damped oscillations before it goes to zero at steady state.

When zeta equals 0, the system is undamped. The response keeps oscillating forever and does not fade out. It never reaches a steady state. When zeta equals 1, the system is critically damped. The poles are identical, and the step response returns to equilibrium as fast as possible without overshooting or oscillating.

For zeta greater than 1, the system is overdamped. Its two components are real, negative, and unequal. The unit-step response is found by inverse Laplace transformation and contains two decaying exponential terms. If the damping ratio is much greater than 1, one term dies away much faster than the other. That faster term can often be ignored, so the response can be treated like a first-order system and an approximate transfer function can be written. These damping cases show how second-order systems can be tuned for stable and accurate performance.

Step Response of Second-Order Systems

transient and steady-state response

Control System Test Signals and Stability

Control system test signals help measure how a system reacts to change, stability, and accuracy. A ramp function is useful when a system changes gradually. A step function is used when a system faces a sudden disturbance. An impulse function is the best test signal for a shock input. These signals are used to study two main performance features: transient response and steady-state response. Transient response is the short-term change from the initial state to the final state. It shows how fast...

first order systems

RC System Responses to Step Inputs

RC circuits and other first-order systems show how a simple input can shape system response over time. These systems are often studied through their transfer function, which links the input and output under zero initial conditions. That relationship makes it easier to analyze how the system behaves when the input changes. A unit-step input gives a response that starts at zero and rises toward one. The Laplace transform of the step input is applied to the transfer function, then the result is...

second order systems i

Servo System Response in Second-Order Models

A servo system is a second-order model that links an input position to an output position. It uses a proportional controller and load elements to keep the output aligned with the input. This behavior can be written as a second-order differential equation. When the equation is transformed with the Laplace transform under zero initial conditions, it becomes a transfer function. The transfer function shows how the input is converted into the output. It can also be rewritten as a closed-loop...

second order systems ii

Second-order systems react to a unit-step input in different ways depending on the damping ratio. When the damping ratio, zeta, is between 0 and 1, the system is underdamped. In this case, inverse Laplace methods give an output that oscillates with a decreasing amplitude. The error signal, which is the difference between the input and output, also shows damped oscillations before it goes to zero at steady state. When zeta equals 0, the system is undamped. The response keeps oscillating forever...

stability

System Response and Stability

System response and stability describe how a linear time-invariant (LTI) system reacts to a change in input. The time response has two parts: transient response and steady-state response. The transient response is the first reaction and it fades to zero over time. The steady-state response remains after the transient effects have disappeared. An LTI system is stable when it gives a bounded output for a bounded input. For these systems, stability depends on the roots of the characteristic...

routh-hurwitz criterion i

Checking Stability with the Routh Table

Checking stability with the Routh table is a useful part of analyzing a closed-loop transfer function. It helps determine whether a system will remain stable when load conditions change or when a fault occurs. In a power grid, this kind of check is important because instability can lead to blackouts. The Routh-Hurwitz criterion uses a table built from the denominator polynomial of the transfer function. The rows of the Routh table are labeled by powers of the complex variable s, starting with...

routh-hurwitz criterion ii

Routh Table Special Cases

Routh table special cases can make stability analysis harder when using the Routh-Hurwitz criterion. Two situations need extra care. One is a zero in the first column. The other is an entire row of zeros. When a single zero appears in the first column, division by zero would occur. The usual fix is to replace that zero with a very small number, epsilon (ε). The stability test then continues by assuming a sign for ε. If ε is positive, any sign change in the first column means the system is...

key terms and concepts

Learn how second-order systems respond to a unit-step input under underdamped, undamped, critically damped, and overdamped conditions.

Step Response of Second-Order Systems | 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

Step Response of Second-Order Systems

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