ENGG4420 END OF CHAPTER 1 QUESTIONS AND PROBLEMS
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1 CHAPTER 1 By Radu Muresan University of Guelph Page 1 ENGG4420 END OF CHAPTER 1 QUESTIONS AND PROBLEMS September :45 PM QUESTIONS SET 1 1. Give 3 advantages of feedback in control. 2. Give 2 disadvantages of feedback in control. 3. A temperature control system is found to have zero error to a constant tracking input and an error of 0.5 C to a tracking input that is linear in time, rising at the rate of 40 C/sec. What is the system type of this control system. 4. What is the main objective of introducing integral control? 5. What is the major objective of adding derivative control? 6. Why might a designer whish to put the derivative term in the feedback path rather than in the error path? 7. Give two reasons to use a digital controller rather than an analog controller. 8. Give two disadvantages of using a digital controller. 9. Give the substitution in the discrete operator z for the Laplace operator s if the approximation to the integral in the equation is take to be the rectangle of height e(kts) and the base Ts. 10. What is the advantage of having a tuning rule for PID controllers?
2 CHAPTER 1 By Radu Muresan University of Guelph Page 2 QUESTIONS SET You have been asked to design a computer-based system to control all the operations of a retail petrol (gasoline) station (control of pumps, cash receipts, sales figures, deliveries, etc.). What type of real-time system would you expect to use? An automatic bank teller works by polling each teller in turn. Some tellers are located outside buildings and others inside. How could the polling system be organized to ensure that the waiting time at the outside locations was less than at the inside locations? Would you classify any of the following systems as real-time? In each case give reasons for your answer and classify the real-time systems as hard or soft. (a) A simulation program run by an engineer on a personal computer. (b) An airline seat-reservation system with online terminals. (c) A microprocessor-based automobile ignition and fuel injection system. (d) A computer system used to obtain and record measurements of force and strain from a tensile strength testing machine (e) An aircraft autopilot For (a) the petrol (gas) station system in exercise 1.1 and (b) the automatic bank teller in exercise 1.2, list the activities which gave to be performed and estimate the time requirements of each activity. For the hot-air blower described in section 1.2, estimate the precision required for the analog-to-digital and digital-to-analog converters. In the passage given below identify the different types of time constraints. The boiler pressure alarm lamp must be lit within 0.6 seconds of the boiler pressure high signal being set true. The temperature of the steam leaving the boiler must be read every 0.5 seconds. The operator display should be updated on average at 2-second intervals on no occasion should the update interval exceed 10 seconds. The response to a request for printout of the current plant status should typically be competed within 2 minutes. Explain the difference between a real-time program and a non-real-time program. Why are real-time programs more difficult to verify than non-real-time programs?
3 CHAPTER 1 By Radu Muresan University of Guelph Page 3 PROBLEMS SET 1 1. Write the equations of motion for a body of mass M suspended from a fixed point by a spring with a constant k. Carefully define where the body s displacement is zero. 2. For the car suspension discussed in our example plot the position of the car and the wheel after the car hits a unit bump (i.e., r is a unit step) using MATLAB. Assume that m1 = 10 kg, m2 = 350 kg, Kw = 500,000 N/m, Ks = 10,000 N/m. Find the value of b that you would prefer if you were a passenger in the car. 3. Automobile manufacturers are contemplating building active suspension systems. The simplest change is to make shock absorbers with a changeable damping, b(u1). It is also possible to make a device to be placed in parallel with the springs that has the ability to supply an equal force, u2, in opposite directions on the wheel axle and the car body. Modify the equations of motion in the example to include such control input. Is the resulting system linear? Is it possible to use the force u2 to completely replace the springs and shock absorber? Is this a good idea? 4. Modify the equation of motion for the cruise control in Example so that it has a control law; that is, let: u = K(vr v), where vr = reference speed, K = constant. This is a proportional control law in which the difference between vr and the actual speed is used as a signal to speed the engine up or slow it down. Revise the equations of motion with vr as the input and v as the output and find the transfer function. Assume that m = 1000 kg and b = 50 N*sec/m, and find the response for a unit step in vr using MATLAB. Using a trial and error, find a value of K that you think would result in a control system in which the actual speed converges as quickly as possible to the reference speed with no objectionable behavior.
4 CHAPTER 1 By Radu Muresan University of Guelph Page 4 PROBLEMS SET 2 Optional 1. Consider the block diagram shown in this figure. Note that ai and bi are constants. Compute the transfer function for this system. This special structure is called the control canonical form. 2. Find the transfer functions for the block diagrams below:
5 CHAPTER 1 By Radu Muresan University of Guelph Page 5 CONT... NOTE: for solving the problems set 2 you need to study the supplemental material related to block diagram operations. The midterm will use some simple block diagram operations, so these exercises will help you.
6 CHAPTER 1 By Radu Muresan University of Guelph Page 6 PROBLEMS SET 3 1. One possible representation of an automobile speed-control system with integral control is shown in this slide. (a) With a zero reference velocity input (vc = 0), find the transfer function relating the output speed v to the wind disturbance w. (b) What is the steady-state response of v if w is a unit-ramp function? 2. Consider a system with a plant transfer function G(s) = 1/s(s+1). Compute the discrete equivalents for the dynamic controllers below: (a) D1(s) = (s+2)/2; (b) D2(s) = 2*(s+2)/(s+4); (c) D3(s) = 5*(s+2)/(s+10). 3. Give the difference equations corresponding to each of the discrete controllers respectively found at problem 2.
7 CHAPTER 1 By Radu Muresan University of Guelph Page 7 PROBLEMS SET 4 2. Consider a system with a plant transfer function G(s) = 1/s(s+1). Compute the discrete equivalents for the dynamic controllers below: (a) D1(s) = (s+2)/2; (b) D2(s) = 2*(s+2)/(s+4); (c) D3(s) = 5*(s+2)/(s+10). 3. Give the difference equations corresponding to each of the discrete controllers respectively found at problem 2.
8 CHAPTER 1 By Radu Muresan University of Guelph Page 8 2. A paper machine has the transfer function: G(s) = exp(-2s)/(3s+1), Where the input is stock flow onto the wire and the output is basis weight or thickness. (a) Find the PID-controller parameters using the Z-N tuning rules. (b) The system becomes marginally stable for a proportional gain of Ku = 3.044, as shown by the unit-impulse response in this figure. Find the optimal PID-controller parameters according to the Z-N tuning rules. A person s reaction time can be measured by sending, at random intervals, a character to a VDU screen and asking the subject to press a key when the character appears. If you have access to a personal computer or some other small computer write a program to carry out such an experiment. Modify the code of Example 1 to incorporate the velocity subtraction method of preventing integral action wind-up. Assume that a logic signal is available to indicate when the control actuator is in saturation. (a) Draw a flowchart to show how bumpless transfer (Method 2 tracking of the manipulated variable) can be incorporated into the standard PID controller. (b) Based on the flowchart write a program (in any language) for the system. Write a program, in any language, to implement the velocity algorithm for the PID controller. How would you incorporate (a) in the standard PID digital controller and (b) into the velocity form of the PID controller, the requirement that ht manipulated variable should not change by more than 1% between two sample intervals? Discuss the problems of testing the computer implementation of a digital control algorithm. Work out a test scheme which would minimize the amount of time required. Work out a test scheme which would minimize the amount time required for test purposes not the actual plant. The scheme should show the various stages of testing and should be designed to eliminate coding errors and logic design errors prior to the connection of the controller to the plant.
9 CHAPTER 1 By Radu Muresan University of Guelph Page 9 9. The results of an open-loop response to a unit step for a plant are: Time (seconds) Output Find (a) the approximate plant model, (b) a suitable sampling interval for a digital PID controller and (C) estimates of the optimum controller settings for PI and PID control. 10. The analog system shown in figure 10 can be discretised using the z-transform plus zeroorder hold method. The resulting algorithm is where e(n) =r c(n) c(n) =(k/a 2 )[Ae(n-1) + Be(n-2)] + [Cc(n 1) Dc(n-2)] A =T s a 1 + exp( - at s ) B = 1 exp(-at s ) T s a exp(-at s ) C = 1 + exp(-at s ) D = exp(-at s ) T s = sampling interval Write a program which will enable you to calculate the change in output of the system, c(n), with time. It is suggested that 50 values are calculated. The program should enable different values of k, a, T s and r to be entered R(s) + k C(s) _ s( s a) Figure 10. Control system for exercise 10.
10 CHAPTER 1 By Radu Muresan University of Guelph Page A plant operating in a remote location is controlled by an embedded computer control system. The plant operates in two modes referred to as Amode and Bmode. The control algorithm for Amod is of the form m( n) Ae( n) Be( n 1) Ce( n 2) Dm( n 1) Em( n 2) and for Bmode m( n) K1e( n) K 2e( n 1) K 3 where e( n) c( n) R c( n) R set point the measured output e( n 2) of the plant at interval n. The change-over from Amode to Bmode is to be made when c(n) > ChangeAfor five successive readings. The change-over from Amode to Bmode is to be made when c(n) < ChangeB for five successive readings. The parameters ChangeA and ChangeB and the set point R can all be changed from a central station. A change to the value of R requires a change in the values of ChangeA and ChangeB. The controller parameters A, B, C, D, E and K 1, K 2 and K 3 also need changing. They must be changed as the set {A, B,C,D,E,K 1,K 2,K 3 } and not as individual elements. The data transmission link to the remote station has a slow transmission speed and is subject to frequent bursts of interference. You can assume that the data transmission system support software contains error checking software and organizes retransmission of erroneous data. Discuss the problems of designing the software for the embedded computer system and discuss possible ways of dealing with the slow and unreliable data transmission system.
11 CHAPTER 1 By Radu Muresan University of Guelph Page 11 ENGG4420. CHAPTER 1: Real-Time Computer Control. Developed by Radu Muresan, Univesity of Guelph MATERIAL COVERED BY CHAPTER 1 WILL BE TESTED IN MIDTERM 1!!!
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