Name: phase-variable state equations from transfer functions
Uploaded: 2024-11-21T14:54:05.000Z
Duration: 1 min 23 s
Channel: JoVE

يستخدم تمثيل مساحة الحالة لمحاكاة الأنظمة المادية على أجهزة الكمبيوتر الرقمية. لهذا ، يجب أولا تحويل وظيفة النقل إلى مساحة حالة.
ضع في اعتبارك معادلة تفاضلية خطية من الدرجة n ذات معاملات ثابتة.
يتم اختيار المخرجات ومشتقاته n-1 كمتغيرات الحالة. إن التمييز بين هذه السلسلة من المعادلات واستبدالها مرة أخرى في المعادلات الأصلية ، يوفر معادلات الحالة.
يتم تعريف كل متغير حالة لاحق على أنه مشتق من متغير السابق.
ثم يتم تمثيل المعادلات الناتجة في شكل مصفوفة متجهة ، مما يؤدي إلى إنشاء نمط مميز من 1 و 0 جنبا إلى جنب مع المعاملات السالبة للمعادلة التفاضلية. هذا الهيكل الفريد هو شكل متغير الطور لمعادلات الحالة.
ضع في اعتبارك وظيفة النقل. يتم ضرب المعادلة بشكل متقاطع ، ثم يتم العثور على المعادلة التفاضلية المقابلة عن طريق أخذ تحويل لابلاس العكسي ، بافتراض أن الظروف الأولية صفرية.
يتم اختيار متغيرات الحالة كمشتقات متتالية.
ثم يتم تطبيق التمايز على كلا طرفي المعادلة ، مما ينتج عنه معادلات الحالة ومعادلة الإخراج.
ثم يتم تقديم هذه المعادلات في شكل مصفوفة متجهية.

نقل الدالة إلى فضاء الحالة

Transfer Function to State Space

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إن تمثيل فضاء الحالة هو أداة قوية لمحاكاة الأنظمة الفيزيائية على أجهزة الكمبيوتر الرقمية، مما يستلزم تحويل دالة النقل إلى شكل فضاء الحالة. لنفترض وجود معادلة تفاضلية خطية من الدرجة n ذات معاملات ثابتة، مثل تلك الموجودة في دائرة RLC. يتم اختيار متغيرات الحالة كمخرجات ومشتقاتها n−1. يؤدي التمييز بين هذه المتغيرات واستبدالها مرة أخرى في المعادلة الأصلية إلى إنتاج معادلات الحالة.


في دائرة RLC، يمكن استخدام الجهد عبر المكثف والتيار عبر المحث كمتغيرات حالة. على سبيل المثال، في نظام من الدرجة الثانية، مثل دائرة RLC المتسلسلة، يمكننا تعريف الجهد عبر المكثف V_c كمتغير حالة أول والتيار عبر المحث i_L كمتغير حالة ثانٍ. يتم أولاً ضرب دالة النقل لدائرة RLC بشكل متبادل للحصول على المعادلة التفاضلية. من خلال تطبيق تحويل لابلاس العكسي وافتراض الظروف الأولية صفر، يمكن للمرء استنباط شكل المجال الزمني للمعادلة.


<img alt="Equation1" src="/files/ftp_upload/16101/16101_Equation_001.svg" style="height: 140px;">

<img alt="Equation2" src="/files/ftp_upload/16101/16101_Equation_002.svg" style="height: 60px;">

يتم بعد ذلك اختيار متغيرات الحالة كمشتقات متتالية للإخراج، ويتم تطبيق التمايز على جانبي المعادلة لتوليد معادلات الحالة.

<img alt="Equation3" src="/files/ftp_upload/16101/16101_Equation_003.svg" style="height: 25px;">

يتم بعد ذلك تمثيل معادلات النظام التفاضلية في شكل مصفوفة متجهة، مما يؤدي إلى ظهور نمط مميز من 1 و0، إلى جانب المعاملات السلبية لمعادلة التفاضل الأصلية.


<img alt="Equation4" src="/files/ftp_upload/16101/16101_Equation_004.svg" style="height: 90px;">

يُعرف هذا باسم شكل متغير الطور لمعادلات الحالة. يوفر هيكل المصفوفة طريقة واضحة وموجزة لمحاكاة وتحليل السلوك الديناميكي للنظام.


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يستخدم تمثيل مساحة الحالة لمحاكاة الأنظمة المادية على أجهزة الكمبيوتر الرقمية. لهذا ، يجب أولا تحويل وظيفة النقل إلى ...

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

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 first cross-multiplied to form the differential equation. Then the inverse Laplace transform is applied, with zero initial conditions, to get the time-domain form.

Next, the state variables are chosen from the output and its successive derivatives. For a second-order system such as a series RLC circuit, the voltage across the capacitor, Vc, can be used as the first state variable. The current through the inductor, iL, can be used as the second state variable.

After the variables are selected, each one is differentiated and substituted back into the original equation. This process generates the state equations. The equations are then written in vector-matrix form.

In phase-variable form, the matrix has a clear pattern of 1s and 0s. It also contains the negative coefficients from the original differential equation. This structure gives a compact way to simulate and analyze the system's dynamic behavior.

Phase-Variable State Equations from Transfer Functions

phase-variable-state-equations-from-transfer-functions

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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...

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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 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...

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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

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 transfer functions are converted into phase-variable state equations using state variables, inverse Laplace transforms, and matrix form for RLC circuits.

Phase-Variable State Equations from Transfer Functions | 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

Phase-Variable State Equations from Transfer Functions

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