21.7
Aktarım Fonksiyonundan Durum Uzayına
Durum uzayı gösterimi, dijital bilgisayarlarda fiziksel sistemleri benzetmek için güçlü bir araçtır ve aktarım fonksiyonunun durum uzayı formuna dönüş…
Durum uzayı gösterimi, dijital bilgisayarlarda fiziksel sistemleri simüle etmek için kullanılır. Bunun için transfer fonksiyonunun önce durum uzayına dönüştürülmesi gerekir.
Sabit katsayılara sahipn'inci dereceden, doğrusal bir diferansiyel denklemi düşünün.
Çıktı ve n-1 türevleri durum değişkenleri olarak seçilir. Bu denklem serilerini farklılaştırmak ve bunları orijinal denklemlere geri koymak, durum denklemlerini sağlar.
Sonraki her durum değişkeni, bir öncekinin türevi olarak tanımlanır.
Elde edilen denklemler daha sonra vektör matrisi biçiminde temsil edilir ve diferansiyel denklemin negatif katsayıları ile birlikte 1'ler ve 0'lardan oluşan farklı bir model oluşturulur. Bu benzersiz yapı, durum denklemlerinin faz değişkenli şeklidir.
Bir transfer fonksiyonu düşünün. Denklem çapraz çarpılır ve karşılık gelen diferansiyel denklem daha sonra sıfır başlangıç koşulları varsayılarak ters Laplace dönüşümü alınarak bulunur.
Durum değişkenleri ardışık türevler olarak seçilir.
Diferansiyel daha sonra denklemin her iki tarafına da uygulanır, durum denklemleri ve çıktı denklemi elde edilir.
Bu denklemler daha sonra vektör matrisi biçiminde sunulur.
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Q1: Why is state-space representation necessary for simulating physical systems on digital computers?
State-space representation enables digital simulation of physical systems by converting transfer functions into a matrix form suitable for computational processing. This vector-matrix format allows computers to efficiently model and analyze dynamic system behavior. The structured representation with state variables and their derivatives provides a clear mathematical framework for numerical computation and system analysis.
Q2: How do you select state variables when converting a transfer function to state space?
State variables are chosen as the output and its n-1 derivatives for an nth-order system. In practical applications like RLC circuits, the voltage across the capacitor and current through the inductor serve as state variables. Each subsequent state variable is defined as the derivative of the previous one, creating a systematic hierarchy that captures the system's dynamic behavior.
Q3: What is the phase-variable form in state-space representation?
The phase-variable form is the unique matrix structure created when state equations are represented in vector-matrix form. It displays a distinct pattern of 1's and 0's along with the negative coefficients from the original differential equation. This standardized structure provides a clear and concise method for simulating and analyzing the dynamic behavior of linear systems.
Q4: What steps are involved in converting a transfer function to state-space form?
First, cross-multiply the transfer function and apply the inverse Laplace transform assuming zero initial conditions to obtain the differential equation. Next, select state variables as successive derivatives of the output. Differentiate both sides of the equation to generate state equations, then represent these equations in vector-matrix form to complete the conversion.
Q5: How does the state-space representation relate to the inverse conversion process?
State-space representation and its inverse conversion form a complementary pair in control systems analysis. While state-space converts transfer functions into matrix form for time-domain simulation, the reverse process converts state-space back to transfer functions. Understanding both transformations is essential for comprehensive system modeling and analysis in control engineering applications.
Q6: Why are zero initial conditions assumed when converting transfer functions to state space?
Zero initial conditions simplify the mathematical conversion by allowing direct application of the inverse Laplace transform without accounting for initial energy stored in the system. This assumption creates a standard baseline for the conversion process and ensures the resulting state equations accurately represent the system's response to external inputs from a relaxed initial state.
Q7: How does an RLC circuit example illustrate state-space conversion?
In a series RLC circuit, the voltage across the capacitor and current through the inductor are selected as state variables. The transfer function is cross-multiplied to obtain the differential equation, then converted to state-space form using successive derivatives. This practical example demonstrates how physical circuit components map to state variables and how the resulting matrix equations model the circuit's dynamic response.