Lab 7 Rev. 2 Open Lab Due COB Friday April 27, 2018
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1 EE314 Sytem Spring Semeter 2018 College of Engineering Prof. C.R. Tolle South Dakota School of Mine & Technology Lab 7 Rev. 2 Open Lab Due COB Friday April 27, 2018 In a prior lab, we et up the baic hardware and the oftware framework for thi lab. A general overview of thi etup i given below: Matlab 2017b wa ued in the production of thi lab handout. When appropriate, utilize Matlab publihing feature to produce your lab report a much a i reaonable. Ueful Matlab command or Simulink block: imulink, contant block, gain block, Arduino PWM block, Arduino Analog in Block, Scope Block, Tranfer Function Block, PWM Generator (DC-DC) Block, Matlab and Simulink Arduino Support Package. Electronic component: Arduino Mega 2560 Breadboard Male to Male jumper wire pleae ue proper jumper wire in your breadboard to help increae the breadboard life time! The ue of the wrong ized wire within the breadboard damage them. 9 Volt Battery connector 2N3904BU NPN BJT Tranitor Two 22Ω 5% 0.25 Watt Reitor (if the two reitor overheat, replace them with three 15Ω 5% 0.25 Watt Reitor.) 20kΩ 1% 0.25Watt Reitor BC2301-ND NTC Thermitor 10kΩ 5% Bead Thi lab i a imple extenion from our previou Lab 5 and 6, thu our hardware and initial oftware ha already beendeveloped. Inthilab wewill bemodifyingtheoftware fromthelat labinordertoimplement api-controller uing the ame hardware etup a lat time. A copy of the hardware deign i given below: +5V Pin (Arduino) 20kΩ 9V PWM Pin 13 2N3904 Taped to Together V ADC Pin A0 10kΩ Thermitor 44Ω Two: 22Ω W in erie GND Common Arduino Pin & Battery 1
2 PI Controller: Deign Veru Tuning In thi lab we will be tudying a PI controller, hown a in the below block diagram: + R() C x () G() Y() The controller i denoted above a C x (), where x further denote the mean by which the PI controller ha been implemented, i.e. deigned or tuned. A part of thi introduction, let take a high level look at the difference between controller deign and controller tuning. Firt, conider the PI deign proce at a conceptual level (becaue it i beyond the content of thi cla). Deigning a PI controller i imple. Firt, one place a pole at the origin then calculate the location of an added zero that would move the cloed-loop root locu through our deired operating point uing the root locu angle criteria: (2k +1)180 = zero angle pole angle; where k Z (1) Thi proce i formally taught in chapter 9 of our book a well a in our enior level control cla EE/ME 453. Next, the overall ytem feedback gain i calculated to move the cloe-loop pole to that operating point uing the root locu magnitude criteria: pole length K =, (2) zero length where the pole and zero length are defined a the ditance from the open loop pole and zero location of C(S)G() in the -plane to our deired operating point on the newly tranformed cloe-loop root locu (once again beyond the content of the cla). Overall the deigned controller can be ummed up a: C D () = K (+z). (3) Thi i in tark contrat to a PID tuning procedure which i far more heuritic. There are numerou publihed tuning method. Perhap the mot famou in the Ziegler-Nichol tuning method. Let ummerize it quickly baed on Dorf and Bihop preentation in their 12 th Ed. Modern Control Sytem book page , Copyrighted by Prentice Hall, In thi cae we conider the controller in thi form: C TPID () = K P + K I +K D; (4) but in our implified PI cae thi reduce to: C T () = K P + K I (5) where K D ha been aumed to be zero at all time. The firt tep in tuning i to et K I = 0 and K D = 0. Next adjut K P to achieve a marginally table repone to a tep input, i.e. the ytem output remain ocillating in a table inuoidal pattern (thi implie that the dominate ytem pole are on the jω-axi). Next reduce K P to it quarter amplitude decay, roughly one quarter of the maximum overhoot. A rule of thumb, thi i about one half the K P found at the edge of tability. Next, adjut K I and K D a needed to achieve the deired tep repone. Dorf and Bihop provide the following table to help in thi adjutment procee: PID Gain Percent Overhoot Setting Time Steady-State Error K P Minimal Effect K I Zero SSE K D No Effect In ummary, we can directly compare thee two method by relating the deign parameter to each other. In the 2
3 cae of PI controller deign, we are chooing K and z. Mean while, during tuning we are chooing K P and K I. By etting C D (S) = C T (), we can ee the relationhip between thee to et of parameter: The reult of thi relationhip i: C D () = Relationhip = C T () (6) ( ) ( ( )) (+z) + K I K K = K P P = K P + K I (7) Deign Parameter Tuning Parameter K = K P (9) z = K I K P. (10) What are the reult of thee relationhip? Well, a one change K P in the tuning procedure, one i both tranforming the locu of root a well a liding the operating point around on the locu. The ame i true for K I. Thi can be explained by conidering major bifurcation point within each of thee technique. What i meant by Bifurcation? A bifurcation i a major change in the ytem dynamic, i.e. in our cae a large change in how the root locu look, not jut a mall cale change of current hape of the locu. Non-bifurcation point exhibit mall locu cale change for mall parameter change. So, our bifurcation point aociated with our root locu are not clearly defined or predicted within the tuning proce and thereby urprie the engineer while tuning. Contrary to the tuning proce, the deign proce eparate locu change from movement along an exiting locu. Thi eparation enure that change in K will not allow root locu change in any way, while movement of the deign zero, z, implie direct change in the root locu. Moreover, change in z eaily help predict mot bifurcation within the locu, becaue thee event generally occur when the zero approache or pae another ytem pole or zero. The above dicuion i meant a an initial introduction to ome of the idioyncraie between thee method. The iue become even more complicated when conidering full PID control. In that cae, the deigner i generally placing 2 real zero intead of one real zero. However the tuning proce allow the engineer a much wider deign pace, ince tuning allow for complex zero olution (which i achieved when K I > 1 4 K P), while the traditional deign method require real axi placement of zero only. Lab Software Deign Starting with the oftware from Lab 6, re-ave the Simulink model under a new name, e.g. ee314 lab7. Next, update the new model file in order to implement a feedback PI-controller uing the ame hardware etup a lat (8) 3
4 time. The updated oftware i hown below: Alo update the ubytem model a follow: Many of the added block above are explained in note above each block. The majority of the above model i the ame a our lat lab, however an additional feedforward path including an integral of the error ha been added. Thi new path i repreented by an integral block, i.e. the 1 block, followed by our newly added integral gain block, K I = 0.1. We recap the other main idea next. Our input are unigned integer; however controller and control theory i baed on floating point calculation, thu the firt tep i to convert thee integer into real number o a finer gain than tep of 1 can be applied. Notice that our ytem exhibit a negative repone to our input. Thi i taken into account by adding a -1 gain block into the feed forward path. A aturation non-linearity ha been added to the feed forward path only to enure that our commanded range remain within the required limit of [0, 255]. We will center our control range to the center of the actuation range. Thi i accomplihed via a imple bae command offet, e.g. we make our zero operating point in the middle of the actuation range, 128. By moving our control effective origin (the zero point) to the center of the controller acquiition range, the controller can leverage the room a a heat ink to allow for active cooling by inking exce heat to the room. Thi in part compenate for the ytem lack of active cooling, at the cot of a maller effective et point range. The final tep i to convert the floating point number back into a number that the PWM unit can ue, e.g. an unigned 8 4
5 bit integer. Lab Evaluation & Report Now that the oftware and hardware are complete, we will ue them to analyze and collect data on our new control ytem. The partial goal of thi lab i to compare a P controller to a PI controller, we will accomplih thi by comparing the reult in thi lab to our lat lab. Below are ome experiment along with quetion to conider and fully dicu within your lab report: 1 Baed on your lat two lab, chooe the et point (mean ytem et point) to be the value aociated with PWM command of 128. Firt, et your K P gain to 1 and your K I gain to 0. Allow your ytem to tabilize, apply a tep functional change to your ytem by lowering the et point by 20 unit (tore the reult for plotting later). Once teady tate ha been achieved return the ytem to it original et point. 2 Next et your K P gain to 1 and your K I gain to 0.1. Allow your ytem to tabilize, apply a tep function change to your ytem by lowering the et point by 20 unit (tore the reult for plotting later). Once teady tate ha been achieved return the ytem to it original et point. 3 On a ingle figure, with multiple ubplot, plot the reult from the previou two experiment a follow on the ame graph: temperature output veru time with the commanded temperature veru time; on the econd plot, plot the controller error ignal; on the third plot, plot the controller effort veru time. (In order to make thee plot, one hould add a mux and to workpace block in parallel to your cope block. Add the ignal you wih to plot to the mux input. Alo enure that thi block return a tructure with time o that you have acce to all of the ignal and time you need. Pleae ue the legend function to label each experiment, i.e. clearly denoting the controller gain, K P & K I, utilized to make the plot. 4 Etimate the rie time for each tep. I it a contant? Why? 5 Etimate the teady tate error of each experiment. Are the the ame? Explain? 6 How enitive i each control ytem to thermal diturbance (once a teady tate i achieved blow on the tranitor/thermitor combo record and dicu the reult)? 7 What type ytem i thi (firt order, econd order, higher order)? Why? What about the tep repone tell you thi? 8 Build a Simulink idealized model of your ytem plot it veru the actual reult. 9 Ha the ytem repone changed from lat lab? How? Why? 10 Tune your PI controller to improve the ytem rie time by 10%, baed on your original PI controller given above, K P = 1, K I = 0.1. Plot the reult veru the ytem original a requeted above. 11 What have you learned within thi hardware lab? 12 What iue or problem did you encounter completing the lab? Pleae complete your formal lab report by due date provided at the top of thi lab aignment. 5
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