21.7
Función de transferencia al espacio de estados
La representación en el espacio de estados es una herramienta poderosa para simular sistemas físicos en computadoras digitales, lo que requiere la con…
La representación del espacio de estados se utiliza para simular sistemas físicos en ordenadores digitales. Para ello, primero hay que convertir la función de transferencia en espacio de estados.
Considere una ecuación diferencial lineal de nésimo orden con coeficientes constantes.
La salida y sus n-1 derivadas se eligen como variables de estado. Diferenciando estas series de ecuaciones y sustituyéndolas de nuevo en las ecuaciones originales, se obtienen las ecuaciones de estado.
Cada variable de estado subsiguiente se define como la derivada de la anterior.
Las ecuaciones resultantes se representan en forma de matriz vectorial, creando un patrón distinto de 1 y 0 junto con los coeficientes negativos de la ecuación diferencial. Esta estructura única es la forma de variable de fase de las ecuaciones de estado.
Considere una función de transferencia. La ecuación se multiplica de forma cruzada y la ecuación diferencial correspondiente se encuentra tomando la transformada inversa de Laplace, asumiendo condiciones iniciales cero.
Las variables de estado se eligen como derivadas sucesivas.
A continuación, la diferenciación se aplica a ambos lados de la ecuación, dando lugar a las ecuaciones de estado y a la ecuación de salida.
Estas ecuaciones se presentan en forma de matriz vectorial.
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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.