**Q1.** Define the following:

(i) zero input response (ii) zero state response (iii) characteristic polynomial (iv) characteristic equation and (v) eigen values

*Ans.:*

**(i) Zero Input Response (ZIR):** The response which occurs even without input is known as zero input response and it is due to initial condition.

ZIR = Φ(*k*) **x**(0) = **A**^{k} **x**(0)

where Φ(*k*) = **A** * ^{k}*is termed as the state transition matrix.

**(ii) Zero State Response:** The response which occurs when all initial conditions are zero is known as zero state response.

**(iii) Characteristic Polynomial:** The characteristic polynomial of the matrix **A** of order

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