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

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.

funzione di trasferimento allo spazio di stato

Transfer Function to State Space

La rappresentazione dello spazio di stato, è un potente strumento per simulare sistemi fisici su computer digitali, che richiede la conversione della funzione di trasferimento, in forma di spazio di stato. Si consideri un'equazione differenziale lineare di ordine n, con coefficienti costanti, come quelli riscontrati in un circuito RLC. Sono selezionate come output, le variabili di stato e le sue derivate n−1. La differenziazione di queste variabili e la loro sostituzione nell'equazione originale, produce le equazioni di stato.

In un circuito RLC, la tensione attraverso il condensatore e la corrente attraverso l'induttore, possono essere utilizzate come variabili di stato. Ad esempio, in un sistema di secondo ordine, come un circuito RLC in serie, possiamo definire la tensione attraverso il condensatore Vc, come prima variabile di stato e la corrente attraverso l'induttore iL, come seconda variabile di stato. La funzione di trasferimento per un circuito RLC, viene prima moltiplicata in modo incrociato per ottenere l'equazione differenziale. Applicando la trasformata inversa di Laplace e assumendo condizioni iniziali pari a zero, si può derivare la forma nel dominio del tempo dell'equazione.

<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;">

Vengono quindi scelte come derivate successive dell'output le variabili di stato e viene applicata la differenziazione a entrambi i lati dell'equazione, per generare le equazioni di stato.

<img alt="Equation3" src="/files/ftp_upload/16101/16101_Equation_003.svg" style="height: 25px;">

Le equazioni differenziali del sistema vengono quindi rappresentate in forma di matrice vettoriale, producendo uno schema distinto di 1 e 0, insieme ai coefficienti negativi dell'equazione differenziale originale.

<img alt="Equation4" src="/files/ftp_upload/16101/16101_Equation_004.svg" style="height: 90px;">

Questa è nota come forma a fase variabile delle equazioni di stato. La struttura a matrice, fornisce un metodo chiaro e conciso per simulare e analizzare il comportamento dinamico del sistema.

Funzione di Trasferimento allo Spazio di Stato

La rappresentazione dello spazio degli stati viene utilizzata per simulare sistemi fisici su computer digitali. Per questo, la funzione di trasferimento deve prima essere convertita in ...

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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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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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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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