Explanation:

A system in control theory is defined by its transfer function, which represents the relationship between the input and output of the system. The transfer function is a mathematical representation of the system's dynamics. It can be written in the form of a ratio of polynomials, where the numerator represents the zeros of the system and the denominator represents the poles.

Definition of Poles and Zeros:

- Poles: The poles of a system are the values of 's' (complex numbers) for which the transfer function becomes infinite. In other words, the poles are the roots of the denominator polynomial. Each pole represents a point in the complex plane where the system's response becomes unbounded or unstable.

- Zeros: The zeros of a system are the values of 's' for which the transfer function becomes zero. In other words, the zeros are the roots of the numerator polynomial. Each zero represents a point in the complex plane where the system's response becomes zero or cancels out.

Relationship between Type of System and Poles:

The type of a system is determined by the number of poles at the origin (s=0) in the transfer function. The poles at the origin arise from the integrators in the system.

- Type 0: If the transfer function has no poles at the origin, i.e., the denominator polynomial does not have any factors of 's', then the system is said to be of type 0.

- Type 1: If the transfer function has one pole at the origin, i.e., the denominator polynomial has one factor of 's', then the system is said to be of type 1.

- Type 2: If the transfer function has two poles at the origin, i.e., the denominator polynomial has two factors of 's', then the system is said to be of type 2.

And so on.

Answer:

In the given options, the correct answer is option 'B', which states that the type of the system denotes the number of poles at the origin. This means that the type of the system is determined by the number of integrators present in the system. The more integrators in the system, the higher the type of the system.

The number of zeros or poles at infinity does not affect the type of the system. These infinity poles or zeros represent the behavior of the system at very high frequencies or for large input values.

Therefore, the number of poles at the origin (s=0) is directly related to the type of the system, and that is why the correct answer is option 'B'.