Derivation of transfer function from state-space equations

 

Let's derive the transfer function formula from the state-space equations step by step.

Step 1: Start with the standard state-space equations

The state equation is:

$$\dot{x}(t)=Ax(t)+Bu(t)$$

The output equation is:

$$y(t)=Cx(t)+Du(t)$$

Where:

• $A$ = System matrix

• $B$ = Input matrix

• $C$ = Output matrix

• $D$ = Feedforward matrix

Step 2: Take the Laplace transform

Taking the Laplace transform of:

$$\dot{x}(t)=Ax(t)+Bu(t)$$

We know that:

$$\mathcal{L}\{\dot{x}(t)\}=sX(s)-x(0)$$

Therefore,

$$sX(s)-x(0)=AX(s)+BU(s)$$

For zero initial conditions:

$$x(0)=0$$

Hence,

$$sX(s)=AX(s)+BU(s)$$

Step 3: Rearrange the equation

Move $AX(s)$ to the left side:

$$sX(s)-AX(s)=BU(s)$$

Taking $X(s)$ common:

$$(sI-A)X(s)=BU(s)$$

Here, $I$ is the identity matrix.

Step 4: Find the state vector X(s)

Multiply both sides by the inverse of $(sI-A)$:

$$(sI-A)^{-1}(sI-A)X(s) = (sI-A)^{-1}BU(s)$$

Since:

$$(sI-A)^{-1}(sI-A)=I$$

We get:

$$\boxed{X(s)=(sI-A)^{-1}BU(s)}$$

Step 5: Take the Laplace transform of the output equation

Original output equation:

$$y(t)=Cx(t)+Du(t)$$

Taking the Laplace transform:

$$Y(s)=CX(s)+DU(s)$$

Substitute the value of $X(s)$ from Step 4:

$$Y(s)=C(sI-A)^{-1}BU(s)+DU(s)$$

Taking $U(s)$ common:

$$Y(s)= [C(sI-A)^{-1}B+D]U(s)$$

Step 6: Divide by U(s)

We know that the transfer function is:

$$G(s)=\frac{Y(s)}{U(s)}$$

Therefore,

Final derived formula

$$\boxed{ G(s)=C(sI-A)^{-1}B+D }$$

Understanding the derivation visually

State equation

$\dot{x}=Ax+Bu$

Apply Laplace transform

$sX(s)-x(0)=AX(s)+BU(s)$

Assume zero initial conditions

$(sI-A)X(s)=BU(s)$

Calculate X(s)

$X(s)=(sI-A)^{-1}BU(s)$

Substitute into output equation

$Y(s)=CX(s)+DU(s)$

Final result

$$\boxed{G(s)=C(sI-A)^{-1}B+D}$$

Summary

1. Always remember $G(s)=C(sI-A)^{-1}B+D$.

2. This formula assumes zero initial conditions.

3. $I$ represents the identity matrix of the same order as $A$.

4. If there is no direct transmission, $D=0$.

GATE CONTROL SYSTEMS

State Space to Transfer Function MCQ Practice

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