Name: finding a transfer function from state space
Uploaded: 2024-11-21T14:54:05.000Z
Duration: 1 min 21 s
Channel: JoVE

يمكن أيضا تحويل تمثيل حيز الحالة إلى دالة نقل في تحليل النظام.
يبدأ التحويل بالحالة المعطاة ومعادلات الإخراج.
يتم تطبيق تحويل لابلاس هنا ، بافتراض أن الظروف الأولية صفرية. هذا يحول المعادلات من مجال الوقت إلى مجال التردد.
يتم حل معادلة الحالة ل X (s) ، حيث يمثل I مصفوفة الهوية. ثم يتم استبدال هذا الحل في معادلة الإخراج.
تربط المصفوفة الناتجة ، والمعروفة باسم مصفوفة دالة النقل ، متجه الإخراج بمتجه الإدخال.
عندما تكون هذه المتجهات قياسية ، يصبح من الممكن العثور على دالة النقل النهائية ، وبالتالي إكمال التحول من تمثيل مساحة الحالة إلى وظيفة النقل.
ضع في اعتبارك نظاما محددا بواسطة مصفوفات ذات أبعاد مختلفة تشكل معادلات الحالة والإخراج.
في حين أن جميع المصطلحات الأخرى في معادلة دالة النقل محددة بالفعل ، إلا أن مصطلحا واحدا لا يزال مجهولا.
للعثور على هذا المصطلح ، يتم استخدام قيم المصفوفة المعروفة من معادلة الحالة. علاوة على ذلك ، يتم حساب المعكوس.
عند الاستبدال ، يتم تحويل تمثيل مساحة الحالة إلى دالة نقل.

مساحة الحالة لدالة نقل

State Space to Transfer Function

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إن تحويل تمثيل مساحة الحالة إلى دالة نقل هو عملية أساسية في تحليل النظام. فهو يوفر طريقة للانتقال من وصف المجال الزمني إلى تمثيل المجال الترددي، وهو أمر بالغ الأهمية لتبسيط تحليل وتصميم أنظمة التحكم.

تبدأ عملية التحويل بتمثيل مساحة الحالة، والذي يتميز بمعادلة الحالة ومعادلة المخرجات. وعادةً ما يتم تمثيل هذه المعادلات على النحو التالي:

<img alt="Equation1" src="/files/ftp_upload/16102/16102_PT_equation_001.svg" style="height: 14px;">

<img alt="Equation2" src="/files/ftp_upload/16102/16102_PT_equation_002.svg" style="height: 14px;">

حيث x(t) هو متجه الحالة، وu(t) هو متجه الإدخال، وy(t) هو متجه الإخراج، وA وB وC وD هي مصفوفات تحدد ديناميكيات النظام.

لتحويل هذه المعادلات إلى مجال التردد، يتم تطبيق تحويل لابلاس، بافتراض أن الشروط الأولية صفرية. ثم يتم حل معادلة الحالة لـ X(s).

لنفترض وجود نظام به مصفوفات معينة A وB وC وD. تتضمن عملية التحويل حساب معكوس (SI−A)، واستبدال القيم المعروفة، وتبسيط التعبير للحصول على دالة النقل. يعد هذا التحويل محوريًا لتحليل سلوك النظام وتصميم وحدات التحكم وفهم استجابة التردد للنظام.

وفي الختام، فإن تحويل تمثيل مساحة الحالة إلى دالة نقل يتضمن تطبيق تحويل لابلاس، وحل معادلة الحالة في مجال التردد، واستنتاج مصفوفة دالة النقل، والتي يتم تبسيطها إلى دالة نقل قياسية لأنظمة الإدخال الفردي والإخراج الفردي (SISO).
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يمكن أيضا تحويل تمثيل حيز الحالة إلى دالة نقل في تحليل النظام.
يبدأ التحول بالحالة والمخرجات المعطاة ...

Education

JoVE Core

Electrical Engineering

الهندسة الكهربائية

Unit 1

الوحدة الأولى

Modeling in Time and Frequency Domain

النمذجة في النطاقين الزمني والترددي

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

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 transform is applied. This step assumes zero initial conditions. The state equation is then solved for X(s), the state vector in the Laplace domain.

For a system with known A, B, C, and D matrices, the next step is to calculate the inverse of (sI−A). The known values are substituted into the expression, and the result is simplified to obtain the transfer function. This result is especially useful for studying system behavior and frequency response.

The same method gives a transfer function matrix for multi-input, multi-output systems. For single-input, single-output, or SISO systems, that matrix reduces to a scalar transfer function. This conversion is an important tool for control design and system analysis.

Finding a Transfer Function from State Space

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Phase-Variable State Equations from Transfer Functions

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state space to transfer function

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...

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Linearizing Nonlinear State-Space Models

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key terms and concepts

Learn how state-space equations are converted into a transfer function using the Laplace transform, zero initial conditions, and the inverse of (sI−A).

Finding a Transfer Function from State Space | 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

Finding a Transfer Function from State Space

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