The state space representation of a linear time invariant system is:

̇Ẋ (t) = A X(t) + Bu(t)

Ẏ (t) = C X(t)

What is the transfer function H(s) of the system?

This question was previously asked in

ESE Electronics 2016 Paper 1: Official Paper

Option 1 : \(C {[ sI - A]^{ - 1}} B\)

__Explanation:__

**State** gives the future behaviour of the system with present input and history which is nothing but the initial condition described by the state variables.

**NOTE: **Resistive circuit doesn’t have any State variable, called a “Memoryless system”

**Number of State variables:**

1) The number of capacitors and inductors for the RLC circuit.

2) For the differential equation, it is the order.

**A standard form of State-space matrix:**

\(\dot X = AX + BU\)

It is called as “State equation or Dynamic equation”

Y = CX + DU

It is known as “Output equation”

\(\dot X\): Differential State Vector

Y: Output vector

U: Input vector

A: State matrix

B: Input matrix

C: Output matrix

D: Transition matrix

**Transfer Function:**

For the state model defined by we have to write the Transfer function for the analysis.

It is given by

\(TF = C{\left[ {sI - A} \right]^{ - 1}}B + D\)

\(TF = C\frac{{adj\left[ {sI - A} \right]}}{{\det \left[ {sI - A} \right]}}B + D\)

Characteristic equation is defined by

**|sI – A| = 0**