State Space to Transfer Function Conversion (Complete Guide with Derivation & Example)

In Control Systems, one of the most important concepts is the relationship between state space representation and the transfer function.

1. Definition of State Space Representation

A state space model represents a system using a set of first-order differential equations. It describes the internal behavior of the system using state variables.

The standard form is:

State Equation:

dx/dt = Ax + Bu

Output Equation:

y = Cx + Du

where: A = system matrix, B = input matrix, C = output matrix, D = feedforward matrix.

2. Definition of Transfer Function

The transfer function is defined as the ratio of output to input in the Laplace domain, assuming zero initial conditions.

G(s) = Y(s) / U(s)

It represents the input-output behavior of a system without considering internal states.

3. Derivation of State Space to Transfer Function

Start with state equation:

dx/dt = Ax + Bu

Step 1: Take Laplace Transform

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

Step 2: Rearrange

(sI − A)X(s) = BU(s)

Step 3: Solve for X(s)

X(s) = (sI − A)^-1 B U(s)

Step 4: Output equation

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

Substitute X(s):

Y(s) = C(sI − A)^-1B U(s) + D U(s)

Step 5: Transfer Function

G(s) = Y(s)/U(s)

Therefore,

G(s) = C(sI − A)^-1B + D

4. Key Formula

The most important result is:

G(s) = C(sI − A)^-1B + D

This formula directly converts state-space models into transfer functions.

5. Example Problem

Consider the system:

A = [ 0  -2
      1  -3 ]

B = [1
     0]

C = [2  -5]

D = 0

Find the transfer function G(s).

6. Solution

Step 1: Compute (sI − A)

sI - A =
[ s   2
 -1  s+3 ]

Step 2: Determinant

|sI - A| = s(s+3) + 2
         = s² + 3s + 2
         = (s+1)(s+2)

Step 3: Final transfer function

G(s) = (2s + 1) / ((s+1)(s+2))

8. Final Answer

For the given problem, the correct equivalent representations are:

Option A and Option C

Difference Between B and U in State Space Control Systems

In state-space representation, students often confuse U and B, but both represent completely different concepts.

1. What is U (Input)?

U(t) is the external input signal applied to the system. In Laplace domain, it is written as U(s).

It is the signal that drives the system from outside.

Examples of U(t):

Voltage applied to a circuit
Force applied to a mechanical system
Temperature input in a thermal system

Key point: U(t) is controllable and can be changed externally.

2. What is B (Input Matrix)?

B is a constant matrix in the state-space model:

dx/dt = Ax + Bu

It defines how the input affects the state variables of the system.

Key point: B is a fixed system property, not a signal.

3. Physical Meaning of B

Which states are affected by input
How strongly input affects each state
Direction of input influence in the system

B describes the internal structure of the system.

4. Key Differences

Feature	U (Input Signal)	B (Matrix)
Type	Time-varying signal	Constant matrix
Nature	External input	System parameter
Control	User controlled	Fixed by system design
Role	Drives the system	Distributes input into states
Changeable?	Yes	No

5. Simple Analogy

Think of a car:

U(t) → accelerator pedal (external input)
B → engine design deciding how input affects motion

U is what you apply, B is how the system internally responds.

Final Summary

U = external input signal
B = system matrix that distributes input into states

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