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, ? (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: ?n2 = 2. 25 Therefore ? n = 1. 5 2?? n = 0. 5 Therefore ? = 0. 167 TS = 4/?? n = 16 TP = ? /? n = 2. 12 %OS = e-?? / x 100 = 58. 8% ?nTr = 1. 69 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: . 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 below. Obtain the following, a. If the focusing lens needs to be positioned to an accuracy of ±0. 005 ? m, evaluate the value of K1 K2 K3 if the wrap on the disc yields a worst-case disturbance in the focus of 15t2 ? . b. Use the Routh-Hurwitz criterion to show that the system is stable when the conditions of (a) are met. c. Use MATLAB to show that the system is stable when the conditions of (a) are met. Figure Q2 (a) The Input 15t2, transforms into 30/s3. e (? ) = 30/Ka = 0. 005. Therefore e (? ) = 30/Ka = Therefore K1K2K3= 106 (b) Using K1K2K3= 106 Therefore, Making a Routh table, s312x105 s22x1041. 2×108 s11940000 s01200000000 We see that the system is stable ? (c) Program: numg=200000*[1 600]; deng=poly([0 0 -20000]); G=tf(numg,deng); ‘T(s)’ T=feedback(G,1) oles=pole(T) Computer response: ans = T(s) Transfer function: 200000 s + 1. 2e008 ———————————— s^3 + 20000 s^2 + 200000 s + 1. 2e008 poles =1. 0e+004 *-1. 9990 -0. 0005 + 0. 0077i -0. 0005 – 0. 0077I ? QUESTION: 3 ?Most manufacturing welding situations involve many uncertainties, including dimensions of the part, joint geometry, and the welding process itself. To ensure weld quality, sensors are therefore necessary. Some such systems, as described by Figure Q3, use a vision system to measure the geometry of the puddle of melted metal.
Here, it is assumed that the rate of feeding the wire to be melted is constant. Obtain the following by using Matlab, a. Determine a second-order model for the closed-loop system. b. Find the overshoot and peak time of the system with gain K = 10 using both the second-order model and original system, then compare the results. (Assume a step input. ) Figure Q3. (a) Given The closed-loop transfer function is Therefore, the characteristic equation is The Routh array is given by s40. 00252. 522+K s30. 51254. 010 s22. 502+K s13. 6-0. 205K0 s02+K
Examining the first column, we determine that for stability we require -2 ? K ? 17. 6 (b) Using K=9, the roots of the characteristic equation are s1= -200, s2, 3 = -0. 33±2. 23j and s4= -4. 35 Assuming the complex roots are dominant, we compute the damping ratio ? = 0. 15. Therefore, we estimate the percent overshoot as P. O. = 100e-? = 62% The actual overshoot is 27%, so we see that assuming that the complex poles are dominant does not lead to accurate the predictions of the system response. ? MATLAB PROGRAM FOR QUESTION 3: ‘using Routh Hurwitz method we found that the system is steable when k is -2
Cite this Control Engineering and Control Theory
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