21.7
Funzione di Trasferimento allo Spazio di Stato
La rappresentazione dello spazio di stato, è un potente strumento per simulare sistemi fisici su computer digitali, che richiede la conversione della…
La rappresentazione dello spazio degli stati viene utilizzata per simulare sistemi fisici su computer digitali. A tale scopo, la funzione di trasferimento deve prima essere convertita nello spazio degli stati.
Si consideri un'equazione differenziale lineare di ordine ncon coefficienti costanti.
L'output e le sue derivate n-1 sono scelte come variabili di stato. Differenziando queste serie di equazioni e sostituendole nelle equazioni originali, si ottengono le equazioni di stato.
Ogni variabile di stato successiva è definita come la derivata della precedente.
Le equazioni risultanti vengono quindi rappresentate in forma di matrice vettoriale, creando un modello distinto di 1 e 0 insieme ai coefficienti negativi dell'equazione differenziale. Questa struttura unica è la forma variabile di fase delle equazioni di stato.
Si consideri una funzione di trasferimento. L'equazione viene moltiplicata in modo incrociato e l'equazione differenziale corrispondente viene quindi trovata prendendo la trasformata di Laplace inversa, assumendo condizioni iniziali pari a zero.
Le variabili di stato vengono scelte come derivate successive.
La differenziazione viene quindi applicata a entrambi i lati dell'equazione, ottenendo le equazioni di stato e l'equazione di output.
Queste equazioni vengono poi presentate in forma di matrice vettoriale.
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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.