Name: system response and stability
Uploaded: 2024-11-21T14:54:10.000Z
Duration: 1 min 28 s
Channel: JoVE

線形時不変システムの時間応答は、過渡応答と定常状態応答に分けられます。
過渡応答は、時間の経過とともにゼロに減少します。定常状態の応答は、過渡応答がフェードアウトした後も持続します。
このようなシステムの安定性は、システムの特性方程式の根またはその極に直接関連しています。
有界入力に対して有界出力を維持し、特性方程式の根を左半分の S 平面とするシステムは安定しています。ただし、ルートが右半分のs平面にある場合、不安定になります。
絶対安定性はシステムの安定性を確認し、相対安定性はその程度を測定します。
振り子を考えてみましょう。邪魔されていないときは、静的な平衡状態にあり、平衡の周りの小さな動きに対して安定性を維持します。
外部力または摩擦力が加わると、システムの過渡応答は減衰の影響により時間の経過とともに徐々に減少します。振り子は、平衡位置を中心に安定した動きを実現します。
対照的に、逆振り子は本質的に不安定であり、倒れないように積極的に制御する必要があります。外力に乱されると元の位置に戻らないため、不安定性を示します。

安定性

線形時間不変 (LTI) システムの時間応答は、過渡応答と定常応答に分けられます。過渡応答は、入力の変化に対するシステムの初期反応を表し、時間の経過とともにゼロに減少します。対照的に、定常応答は、過渡効果が消えた後も持続する動作です。 
LTI システムの安定性は、極と呼ばれる特性方程式の根によって決まります。システムが安定するのは、有界入力に対して有界出力を生成する場合です。これは、特性方程式のすべての根が s 平面の左半分に位置する場合に発生します。いずれかの根が s 平面の右半分に位置する場合、システムは不安定になります。絶対安定性は、システムが安定しているか不安定であるかを示しますが、相対安定性は、安定性の度合いを測定します。 
これらの概念のわかりやすい例として振り子が挙げられます。振り子は、乱されていない状態では静的平衡状態にあり、この平衡位置の周りの小さな動きに対しては安定性を示します。この安定性は、振り子が変位して揺れても維持されます。この場合、外部力または摩擦力が存在します。これらの力によって減衰が生じ、システムの過渡応答が時間の経過とともに徐々に減少します。最終的に、振り子は平衡位置の周りで安定した動きを達成し、安定したシステムの特性を示します。 
対照的に、振り子が不安定な平衡位置からずれると、振り子は時間応答が指数関数的に増加しながら揺れ始めます。この制限のない動きは不安定性を意味し、システムは平衡状態に戻らず、振動の振幅は増し続けます。 
過渡応答と定常応答の違い、および安定性の条件を理解することは、LTI システムの分析と設計に不可欠です。システムのすべての極が左半分の s 平面にあることを確認することで、エンジニアは安定性を保証し、さまざまな入力に対するシステムの動作を予測して、より堅牢で信頼性の高いシステムパフォーマンスを実現できます。

線形時不変システムの時間応答は、過渡応答と定常状態応答に分けられます。
過渡応答はゼロオーバーに減少します...

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

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 equation, also called poles. The system is stable when all poles are in the left-half of the s-plane. If any pole is in the right-half of the s-plane, the system becomes unstable.

Engineers also use the terms absolute stability and relative stability. Absolute stability tells whether a system is stable or unstable. Relative stability describes how stable the system is. These ideas help show not only if a system will settle, but also how strongly it resists unwanted growth.

A pendulum gives a clear example of these ideas. When it is undisturbed, it rests in static equilibrium and is stable for small motions around that position. If it is displaced and allowed to swing, external or frictional forces add damping. The transient response then decreases over time, and the pendulum reaches stable motion around equilibrium.

A pendulum can also show instability. If it is displaced from an unstable equilibrium position, it begins to swing with exponential growth in its time response. The motion does not return to equilibrium. Instead, the amplitude of the oscillations keeps increasing, which shows an unstable system.

Knowing the difference between transient and steady-state responses is important for analyzing and designing LTI systems. Engineers use the location of the poles in the left-half s-plane to check stability and predict how a system will react to different inputs. This leads to more reliable system performance.

System Response and Stability

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

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...

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 LTI systems respond with transient and steady-state behavior, how poles affect stability, and how a pendulum illustrates stable and unstable motion.

System Response and Stability | 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

System Response and Stability

Description

Transcript

Tags