Introduction:
The closed loop transfer function is a mathematical representation of a control system that relates the output to the input. In this case, we are given the closed loop transfer function T(s) which is a ratio of two polynomials in the Laplace variable s. We need to determine the number of poles in the left-hand, right-hand, and on the imaginary axis.

Step 1: Finding the Poles
To find the poles of the system, we need to solve the denominator polynomial of the transfer function T(s) for s such that the denominator becomes zero. The roots of this polynomial will give us the poles of the system.

Step 2: Factoring the Denominator Polynomial
The denominator polynomial of T(s) is given as:
s^5 - 2s^4 + 3s^3 - 6s^2 + 2s - 4

We can factor this polynomial using techniques such as polynomial long division or synthetic division. After factoring, we obtain the polynomial in the form:
(s - a)(s - b)(s - c)(s - d)(s - e)

Step 3: Determining the Poles
The poles of the system are the values of s for which the denominator polynomial becomes zero. From the factored form obtained in the previous step, we can determine the values of a, b, c, d, and e.

Step 4: Classifying the Poles
Once we have the values of the poles, we can classify them into three categories: left-hand, right-hand, and on the imaginary axis.

- Left-hand Poles: These are the poles with negative real parts. In other words, the real parts of these poles are less than zero.

- Right-hand Poles: These are the poles with positive real parts. In other words, the real parts of these poles are greater than zero.

- Poles on the Imaginary Axis: These are the poles with zero real parts. In other words, the real parts of these poles are equal to zero, and the imaginary parts are non-zero.

Conclusion:
To determine the number of poles in the left-hand, right-hand, and on the imaginary axis, we need to find the roots of the denominator polynomial of the closed loop transfer function. By factoring the polynomial and analyzing the real parts of the roots, we can classify the poles accordingly.

2 Poles on LHS