Question

What does a transfer function represent?

What are the poles and zeros of a transfer function?

What is the characteristic polynomial?

What is stability?

What is the condition of stability when the system has been expressed in differential equation form?

What is the condition of stability when the system has been expressed in transfer function?

Answer 1

1)What does a transfer function represent?

The transfer function of a control system is defined as the ratio of the Laplace transform of the output variable to Laplace transform of the input variable assuming all initial conditions to be zero.That is, the transfer function of the system multiplied by the input function gives the output function of the system.

2)What are the poles and zeros of a transfer function?

Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively. The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Control systems, in the most simple sense, can be designed simply by assigning specific values to the poles and zeros of the system.

Let's say we have a transfer function defined as a ratio of two polynomials:

Where N(s) and D(s) are simple polynomials.

Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting N(s) = 0 and solving for s.

Poles are the roots of D(s) (the denominator of the transfer function), obtained by setting D(s) = 0 and solving for s. Because of our restriction above, that a transfer function must not have more zeros than poles, we can state that the polynomial order of D(s) must be greater than or equal to the polynomial order of N(s).

3)What is the characteristic polynomial?

Let A be an n×n matrix. The characteristic polynomial of A is the function f(λ) given by

f(λ)=det(A−λIn).

4)What is stability?

stability: Ability of a substance to remain unchanged over time under stated or reasonably expected conditions of storage and use.

A system is said to be stable, if its output is under control. Otherwise, it is said to be unstable. A stable system produces a bounded output for a given bounded input. The following figure shows the response of a stable system. This is the response of first order control system for unit step input.

5)What is the condition of stability when the system has been expressed in differential equation form?

In terms of the solution of a differential equation, a function f(x) is said to be stable if any other solution of the equation that starts out sufficiently close to it when x = 0 remains close to it for succeeding values of x. A given equation can have both stable and unstable solutions

6)What is the condition of stability when the system has been expressed in transfer function?

When the poles of the closed-loop transfer function of a given system are located in the right-half of the S-plane (RHP), the system becomes unstable. When the poles of the system are located in the left-half plane (LHP) and the system is not improper, the system is shown to be stable