ࡱ> (*'{ bjbjzz 0< q8>DLc=8(LBB444<<<<<<<$>MA=5q3455=B=8885RBL<85<88:^;,;@@MV5v; <3=0c=;.A5A;;A;4484 4S444==7444c=5555A444444444 : Homework Set #1 - Solution Spring 2013 Control Systems I Aux_01 The objective of this problem is to calculate a linear approximation around the operating point of a non linear component. This linear model becomes the transfer function representing the component. A Temperature sensor has the following overall nonlinear response to changes in temperature. Solution Aux_01   Aux_02 A common example of a two-input control system is a home shower with separate valves for hot and cold water. The objective is to obtain a desired temperature of the shower water and a desired flow of water. You are to draw a block diagram of the closed loop control system. Note there will be two feedback loops and two command inputs. Solution Aux_02 One variation of the block diagram is shown below.  Aux-03 The figure below shows a closed loop control system. The feed back path is provided by the Speed Transducer. Also shown is a disturbance function labeled as TL. You are to predict what the output of the system will be for a disturbance input of constant amplitude for the control system without feedback (no feedback path), and for the system with the feedback path. Show what you think the response will be on the two plots shown below. Both plots show the output for some desired input idle speed (r. Show what will happen to the output plot for the disturbance input.  Aux-03 Solution:  Aux-04 Future Closed Loop Control Systems Let your imagination run wild and dream up a control system concept to control a process that currently does not exist (as far as you know). You are to essentially; a) Concisely define what it is you want to control, b) How you would sense or measure what it is you want to control, and c) How you would conceptually implement the closed loop system. For example when you bring your automobile to a stop at a red light, and there is no cross traffic at all, you are forced to wait until the cross light turns red and your light turns green before you can start. This is wasting your time. Consider the following control system: Maximize traffic flow at a 4 way intersection. By changing the traffic lights based on traffic volume and not by an arbitrary time interval. You have 4 TV cameras scanning each of the 4 incoming traffic lanes. The imagery is sent a computer with software that detects and counts the moving objects in the imagery along each incoming lane. Based on the number of objects and the time the lights have been red a decision is made as to which lights are to be red and green to maximize traffic throughput. If there are no automobiles in your cross lane, why have your light red. This would be a suitable answer. Aux_05. The objective of this problem is to demonstrate using the Final Value Theorem. Suppose you have a control system with the transfer function given below. In this problem the design requires that the magnitude of the final value for a unit step input to be 100. Based on this value what must the magnitude of the parameter B be set at to meet the final value requirement? In solving this problem refer to Section 2.4 in the text book discussing the Final Value Theorem.  EMBED Equation  Aux_05 Solution.  EMBED Equation   Aux_06 The objective of this problem is to become familiar with using Masons signal-flow gain formula. Solve for the transfer function between the input R(s) and the output Y(s) of the block diagram shown in Figure P2.36 on page 133 of the text book. Note, use the Masons signal-flow gain formula, Equation (2.96) in Section 2.7 in the text book. Write your answer with the denominator factored on decreasing powers of s.       FILENAME Homework_Set_SP_10_01_Solutions Page  PAGE 1 of  NUMPAGES 6 Last printed  PRINTDATE \@ "M/d/yyyy h:mm am/pm" 1/21/2010 1:40 PM  EMBED Equation   EMBED Equation  Answer  EMBED Equation  Valve adjust + - Hot Water Hot water system Human visual & touch Valve adjust - Cold Water Cold water system + - + + Water flow Water flow Desired water temp Desired water flow rate Actual water temp and flow rate     Answer B= 400 Answer  EMBED Equation  !$%&9:;BDhq  ! 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