Name: servo system response in second-order models
Uploaded: 2024-11-21T14:54:10.000Z
Duration: 1 min 20 s
Channel: JoVE

서보 시스템은 2차 시스템의 한 예입니다. 이는 비례 컨트롤러와 출력 위치를 주어진 입력 위치에 정렬하는 하중 요소로 구성됩니다.
2차 미분 방정식은 이러한 요소 간의 관계를 간략하게 설명합니다. 0의 초기 조건에서 라플라스 변환을 적용하면 입력이 출력으로 변환되는 방법을 보여주는 전달 함수가 제공됩니다.
이 시스템에 대한 새로운 해석은 closed-loop 전달 함수의 유도로 이어집니다. 이 전달 함수에서 두 개의 극점을 갖는 시스템은 2계 시스템으로 정의됩니다.
closed-loop 전달 함수는 주요 매개변수, 즉 감쇠 계수, 감쇠되지 않은 고유 주파수 및 감쇠 비율을 나타내기 위해 재배열 및 재작성할 수 있습니다.
댐핑 비율은 임계 댐핑에 대한 실제 댐핑의 비율로 정의됩니다. 이러한 파라미터를 사용하면 폐루프 전달 함수를 2차 시스템을 나타내는 표준 형식으로 다시 작성할 수 있습니다.
댐핑 비율이 1보다 작지만 0보다 크면 시스템은 언더댐핑된 것으로 간주됩니다.
댐핑 비율이 1이면 시스템이 임계 감쇠되고 1보다 크면 과감쇠된 것입니다.

2차 시스템 i

서보 시스템은 2차 시스템을 예시하며, 비례 컨트롤러와 출력 위치가 입력 위치와 일치하도록 보장하는 부하 요소를 특징으로 합니다. 이러한 구성 요소 간의 관계는 2차 미분 방정식으로 설명됩니다. 초기 조건이 0인 라플라스 변환을 적용하면 전달 함수가 생성되어 입력이 시스템에서 출력으로 변환되는 방식을 보여줍니다.

시스템을 재해석하면, 2차 시스템을 두 개의 극을 가진 시스템으로 특징짓는 폐루프 전달 함수를 도출할 수 있습니다. 이 폐루프 전달 함수는 필수 매개변수인 감쇠되지 않은 고유 진동수와 감쇠 비율을 강조하도록 재배열할 수 있습니다.

감쇠비(ζ)는 실제 감쇠와 임계 감쇠의 비율로 정의된 중요한 매개변수입니다. 이는 시스템의 동작과 안정성에 대한 통찰력을 제공합니다. 이러한 매개변수가 식별되면 폐쇄 루프 전달 함수는 종종 2차 시스템을 포괄적으로 표현하기 위해 표준 형식으로 다시 작성됩니다.

<img alt="Equation1" src="/files/ftp_upload/16113/16113_PT_Equation_001.svg" style="height: 44px;">

여기서, ωn는 감쇠되지 않은 자연 주파수를 나타내고, ζ는 감쇠 비율을 나타냅니다. 감쇠 비율은 시스템의 응답을 다음과 같이 분류합니다.

감쇠 부족(0 &lt; ζ &lt; 1): 시스템은 점차 진폭이 감소하는 진동적 행동을 보이며 안정되기 전에 평형점 주변에서 진동이 일어남을 나타냅니다.

임계 감쇠(ζ = 1): 시스템은 진동 없이 가능한 한 빨리 평형으로 돌아가며, 이는 빠르고 원활한 위치 지정이 필요한 서보 시스템에서 종종 필요합니다.

과감쇠(ζ &gt; 1): 시스템은 진동 없이 평형으로 돌아가지만 임계 감쇠 시스템에 비해 느리게 돌아가므로 반응 시간이 느려지므로 빠른 조정이 필요한 응용 분야에서는 바람직하지 않을 수 있습니다.

이러한 매개변수와 시스템 반응에 미치는 영향을 이해하는 것은 특정 성능 기준을 충족하도록 서보 시스템을 설계하고 조정하는 데 매우 중요합니다. 전달 함수와 감쇠 특성을 통해 2차 시스템을 분석하면 엔지니어가 다양한 작동 조건에서 시스템 동작을 예측하고 최적화할 수 있습니다.

2차 시스템 I

서보 시스템은 2차 시스템의 한 예입니다. 그것은 비례 컨트롤러와 출력 위치를 정렬하는 부하 요소로 구성됩니다 ...

Education

교육

JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

Transient and Steady-state Response Analysis

과도 및 정상 상태 응답 분석

transient-and-steady-state-response-analysis

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 transfer function, which is a common way to describe a second-order system.

Second-order systems have two poles. Their closed-loop transfer function is often rearranged to show two key parameters: the undamped natural frequency and the damping ratio. The damping ratio, written as zeta, is the ratio of actual damping to critical damping. It helps describe system behavior and stability.

The standard form of the closed-loop transfer function uses omega n for the undamped natural frequency and zeta for the damping ratio. These values help explain how the system responds over time. They are also useful when designing and tuning servo systems for specific performance goals.

The damping ratio divides the response into three cases. An underdamped system has 0 1 and does not oscillate, but it returns to equilibrium more slowly than a critically damped system.

Servo System Response in Second-Order Models

servo-system-response-in-second-order-models

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

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

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 servo systems act as second-order models, how transfer functions are formed, and how damping ratio shapes underdamped, critical, and overdamped response.

Servo System Response in Second-Order 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

Servo System Response in Second-Order Models

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