Question

' 1. Review Question a) Name three applications for feedback control systems. b) Functionally, how do closed-loop systems differ from open-loop systems? c) Name the three major design criteria for control systems. d) Name the performance specification for first-order systems. e) Briefly describe how the zeros of the open-loop system affect the root locus and the transient response. What does the Routh-Hurwitz criterion tell us? f) 2. Given the electric network shown in Figure. a) Write the differential equation for the network if u(t) = u(t), a unit step. b) Solve the differential equation for the current, i(t), if there is no initial energy in the network. c) Make a plot of your solution if 1 3. Solve the following differential equations [analytically by using Laplace transform] d2χ dt2 4x(02-0 where x(0)-0, dx) where x(0)-2· 4x(0)- dt 습+2 +2x = 5e-2t + t b) 1 dt2 5. For the following transfer functions +2+43++4 Cs) 6+7s5+31+2s3+2+5 R(s) Figure (a) R(s) C(s) 100 s4 +20s3 + 10s2+7s + 100 Figure (b) i. How many poles in RHS (USE MATLAB) ii. Is the system stable? (USE ROUTH STABILITY) 6. For the following mechanical system:- a) Find a mathematical model b) Find the transfer function, G(s)= c) Find impulse, step and ramp response by using MATLAB functions d) Find harmonic response by using MATLAB SIMULINK F(S) x1(t) 1 N/m Ar) 1 kg 1 N-s/m1 kg Frictionless 7. For each of the transfer functions shown below, Find the locations of the poles and zeros, plot them on the s-plane, Write an expression for the general form of the step response without solving for the inverse Laplace transform. State the nature of each response (overdamped, underdamped, and so on): a. b. c. s(s2+6s+34) 225 ii.G(S)+30s+225 il G()3025) s(s+5)

Answer 1

Solution:

1.(a): Three applications of the feedback control system are given below:

1. Control of pressure in a vessel containing air
2. Toilet tank level control system
3. conveyor speed control system

1.(b): In an open loop control system, we insert an input signal of some form ( mechanical, electrical, etc. ) to the system which converts it into the desired output signal. In the process, the output is directly a function of input and process parameters but it has no effect on the input signal or the process parameters.

open toop closed lo o p

Suppose, we are running a dc motor and we desire a certain speed range, let's say 180-200 rpm. The initial setup gives us the speed in the given range but as time passes, due to several factors like poor components of dc motor, current being supplied or change in voltage, etc., speed goes below 180 rpm. But if we don't supervise the system the whole time and make the changes accordingly, it can not function properly. There comes the use of a feedback loop system which is described below:

Foowand P open Loop R. Cin feedba ck Fachon (H) ACS Feedback Path

A closed loop control system is also called a feedback control system. In a feedback control system, instead of input, the output of the system is recorded and modified according to the need. The main concept behind the feedback control system may be described by the following three steps:

• Sensing
• Controlling and
• Actuation.

1.(c): The three major design criteria for the control systems are given below:

1. Stability
2. Transient response and
3. steady-state error.

1.(d): The performance specifications for the first order system are listed below:

• Time constant: The time taken for the step response of a system to rise up to 63 % or 0.63 of its final value is called time constant. It is represented by 't'.

$t=1/a$

where 'a' is the exponential frequency.

• Rise time: It is defined as the time duration in which the waveform rises from 10 % to 90 % or 0.1 to 0.9 of its final value.

$T_r=\frac{2.2}{a}$

• Settling time:  It is time taken by the response to reach and stay at 2% of its final value. We can increase this parameter to up to 5%. It can be calculated by using the formula given below:

Ts

1.(e): The root locus plot or root locus technique is a method to determine the stability of a given control system. In this technique, we find the range of values of K for which the complete performance for the system will be satisfactory and the operation is stable.
Effect of zeroes of the open loop system on the root locus:

All the root loci start from the poles where k = 0 and terminates at the zeros where K tends to infinity. The difference between the number of poles & number of zeros of G(s)H(s) gives us the number of branches terminating at infinity.

Addition of zeroes tends to pull the root locus to the left of s-plane and hence improve the stability.

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The presence of a zero in the transfer function alters the transient performance. The positioning of the zero relative to the additional pole is very important. If this zero lies on the left of the pole, the system behaves as if it has only complex poles but with smaller peak overshoot. If the zero lies on the right of the pole the overshoot is greater than that of the system with complex poles.

1.(f): The Routh-Hurwitz criterion is a mathematical test to check the stability of linear time-invariant control systems. It gives us necessary and sufficient conditions for the stability of LTI control systems.

The significance of this criterion also lies in the fact that the roots p of the characteristic equation of a linear system with negative real parts represent solutions of the system that are stable (bounded). Thus, this criterion can be used as a way to determine if the equations of motion of a linear system have only stable solutions, without solving the system directly.

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