21.7
Função de Transferência para Espaço de Estado
A representação do espaço de estado é uma ferramenta poderosa para simular sistemas físicos em computadores digitais, necessitando da conversão da fun…
A representação do espaço de estados é usada para simular sistemas físicos em computadores digitais. Para isso, a função de transferência deve primeiro ser convertida em espaço de estados.
Considere uma equação diferencial linear de nésima ordem com coeficientes constantes.
A saída e seus derivados n-1 são escolhidos como variáveis de estado. Diferenciar essas séries de equações e substituí-las de volta nas equações originais fornece as equações de estado.
Cada variável de estado subsequente é definida como a derivada da anterior.
As equações resultantes são então representadas na forma de matriz vetorial, criando um padrão distinto de 1 e 0 junto com os coeficientes negativos da equação diferencial. Essa estrutura única é a forma de variável de fase das equações de estado.
Considere uma função de transferência. A equação é multiplicada cruzada e a equação diferencial correspondente é então encontrada tomando a transformada de Laplace inversa, assumindo condições iniciais zero.
As variáveis de estado são escolhidas como derivadas sucessivas.
A diferenciação é então aplicada a ambos os lados da equação, produzindo as equações de estado e a equação de saída.
Essas equações são então apresentadas na forma de matriz vetorial.
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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.