# Control Engineering

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INTRODUCTION
Control engineering is the discipline that applies control theory to design systems with predictable behaviors. The practice uses sensors to measure the output performance of the device being controlled and those measurements can be used to give feedback to the input actuators that can make corrections toward desired performance. There are two major divisions in control theory, namely, classical and modern, which have direct implications over the control engineering applications. Classical Control Theory

The scope of classical control theory is limited to single-input and single-output (SISO) system design. The system analysis is carried out in time domain using differential equations, in complex-s domain with Laplace transform or in frequency domain by transforming from the complex-s domain. All systems are assumed to be second order and single variable, and higher-order system responses and multivariable effects are ignored. A controller designed using classical theory usually requires on-site tuning due to design approximations. Modern Control Theory

Modern control theory is carried out strictly in the complex-s or the frequency domain, and can deal with multi-input and multi-output (MIMO) systems. This overcomes the limitations of classical control theory in more sophisticated design problems, such as fighter aircraft control. In modern design, a system is represented as a set of first order differential equations defined using state variables. Nonlinear, multivariable, adaptive and robust control theories come under this division. MATLAB:

MATLAB stands for "Matrix Laboratory" and is a numerical computing environment and fourth-generation programming language. Developed by Math Works, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages, including C, C++, and Fortran. QUESTION: 1

Ships at sea undergo motion about their roll axis, as shown in Figure Q1 below,

Figure Q1
Fins called stabilizers are used to reduce this rolling motion. The stabilizers can be positioned by a closed-loop roll control system that consists of components, such as fin actuators and sensors, as well as the ship’s roll dynamics. Assume the rolling dynamics, which relates the roll-angle output, θ(s), to a disturbance-torque input, TD(s), is

Obtain the following:

a.Evaluate the natural frequency, damping ratio, peak time, settling time, rise time, and percent overshoot.

b.Evaluate the analytical expression for the output response to a unit step input.

c.Use MATLAB to solve (a) and (b) and plot the response found in (b).

(a)
Assuming a second-order approximation:
ωn2 = 2.25 Therefore ωn = 1.5
2ζωn = 0.5 Therefore ζ = 0.167
TS = 4/ζωn = 16
TP = π/ωn = 2.12
%OS = e-ζπ / x 100 = 58.8%
ωnTr = 1.169 therefore, Tr = 0.77.

(b)

Taking the inverse Laplace transform

(c)

Program:
'(a)'
numg=2.25;
deng=[1 0.5 2.25];
G=tf(numg,deng)
omegan=sqrt(deng(3))
zeta=deng(2)/(2*omegan)
Ts=4/(zeta*omegan)
Tp=pi/(omegan*sqrt(1-zeta^2))
pos=exp(-zeta*pi/sqrt(1-zeta^2))*100
t=0:.1:2;
[y,t]=step(G,t);
Tlow=interp1(y,t,.1);
Thi=interp1(y,t,.9);
Tr=Thi-Tlow

'(b)'
numc=2.25*[1 2];
denc=conv(poly([0 -3.57]),[1 2 2.25]);
[K,p,k]=residue(numc,denc)

'(c)'
[y,t]=step(G);
plot(t,y)
title('Roll Angle Response')
xlabel('Time(seconds)')
ylabel('Roll Angle(radians)')
Computer Response:
ans =
(a)
Transfer function:
2.25
------------------
s^2 + 0.5 s + 2.25
omegan = 1.5000
zeta = 0.1667
Ts = 16
Tp = 2.1241
pos = 58.8001
Tr = 0.7801

ans =
(b)
K = 0.1260
-0.3431 + 0.1058i
-0.3431 - 0.1058i
0.5602
p = -3.5700
-1.0000 + 1.1180i
-1.0000 - 1.1180i
0
k = []

ans =
(c)

QUESTION: 2

The block diagram of a video laser disc recording system is shown in Figure Q2...