Chapter 2 Control Systems Mathematical Model of Systems

Answer True/False

Very few physical systems are linear within some range of the variables.

• True

• False

Answer True/False

The s-plane plot of the poles and zeros graphically portrays the character of the natural response of a system.

• True

• False

The roots of the characteristic equation are the zeros of the closed-loop system.

• True

• False

A linear system satisfies the properties of superposition and homogeneity.

Select one:

• True

• False

The transfer function is the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable, with all initial conditions equal to zero.

• True

• False

The transfer function is the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable, with all initial conditions equal to zero.

• True

• False

Consider the system in Figure 2.79 where

Gc(s) = 10, H(s) = l, and G(s) = (s + 50)/(s^2+60s_500)

If the input R(s) is a unit step input, Td(s) = 0, and N(s) = 0, the final value of the output

Y(s) is:

(use formula Yss=lim t->infinity y(t) = lim s->0 sY(s)

• 1

• 100

• 50

• none of above

Consider the system in Figure 2.79 with Gc(s)=20, H(s) = 1, and G(s) = (s+4)/(s^2-12s-65). When all initial conditions are zero, the input R(s) is an impulse, the disturbance Td(s) = 0, and the noise N(s) = 0, the output y(t) is ...

Hint: First, you need to find the transfer function T(s) of the overall system

T(s) = GcG/(1+GcGH)

After substitution and simplification, you need to factorise and apply partial fraction.

Then, use Laplase Table to convert to time domain

• y(t)=e^(-8t) + 10e^(-t)

• y(t)=10e^(-5t) + 10e^(-3t)

• y(t)=20e^(-8t) + 5e^(-15t)

• y(t)=10e^(-5t) - 10e^(-3t)

Consider a differential equation

d^2y/dt^2 + 2 dy/dt + y(t) = u(t)

where initials conditions are zero and u(t) is a unit step. The poles of the systems are:

Select one:

Hint: First, you need to convert to Laplase Domain. Whenever you see dy/dt, replace with Y(s)

Then rearrange your equation

Factorise the denominator in order to find the roots.

• s1=1j; s2=-1j

• s1=-1; s2=-1

• s1=-1; s2=-2

• None of the above

Choose the right terminologies that describe the transfer function T(s) = A(s)/B(s).

The roots of the numerator A(s) are called ........

The roots of the denominator B(s) are called .......

The denominator of a transfer function is called ......

• poles, zeros, characteristic equation

• zeros, poles, characteristic equation

• zeros, poles, transient

• transient, poles, zeros