The Utilization of MATLAB and RVSL in Establishing the Root Locus of the System - Assignment Example

1 Introduction As noticed in Lab 3 the stability, response as well as all features of a systems response are governed by system poles’ location. Poles’ location can differ if a factor in the system is altered (mostly gain (K).The poles usually move with a pattern as the gain is varied in the system, which can then be used in establishing system features for every value of K. This is facilitated by Root Locus which enables us to calculate and graphically plot all likely K values within a system. System pole movement plotting facilitates response features for the definite gain to be read straight off the graph, which is extremely helpful as it indicates system’s response for all potential K values. For a basic feedback control system as indicated below, the root locus is determined; where the closed-loop transfer function is represented by N(s) is the numerator polynomial and D(s) is the denominator polynomial or the distinctive equation. From the above equations, any K value can be determined for any s value. Making use of these plotting of the root locus and much information regarding the system can be acquired. As indicated by figure 1 below, a system can be designed to consistently respond and as needed by use of root locus. Radial lines radiating from the source are lines of steady dampening whereby . This, therefore, can be utilised in determining percentage overshoot which gives the ratio of real to imaginary elements; this implies the lines are also lines of constant overshoot. The settling time increases with an increase in the real component of the pole, which is indicated by vertical lines. The system can be managed by adding a potentiometer on the load as well as a feedback loop. This is indicated in RSVL as a screenshot below; Figure 1- A screenshot of RSVL, showing the system and virtual oscilloscope The system so modelled can be represented using a differential equation made. A transfer function can be produced for the system by using a Laplace transform on the differential equation. Making an assumption that the coupling of the load and motor is perfect, then the friction co-efficient (bs)=0 as well as the spring constant (K), modelling of the drive can be done by the following equation; Provides a system transfer function of; This is the system transfer function which is being considered in section 1 of this laboratory. Section B of the first Experiment needs the utilisation of a filter in the control system which is utilized in the reduction of noise. RSVL has alternatives for no filter, first as well as second order filters. The first order filter utilized in this lab has a function; 1.3 Test Results PART A 1A RSVL step response, showing the steady state error in green PART B 1.4 Discussion and Conclusions Q1) Discuss any differences between the responses generated by MATLAB and those produced by simulation in RSVL. Variations between RSVL and MATLAB emanated from correctness in values. This is mostly due to the utilization of the virtual oscilloscope in measurement of the features together with the noise and indefinite simulated measurement in RSVL while MATLAB determines exact response values. Q2: Discuss the effect of adding a filter to the system on: a) The root locus? Normally the root locus has minimal values in the negative side of the jω-axis whereby the system will be steady. The range comprises of all gain values starting from 0 to 123.However, devoid of the filter the stability gain range is all positive K. b) The gain corresponding to a 10% overshoot? A much bigger gain was needed with the filter in position to attain the 10 percent overshoot needed. The bigger increase might make the system safer since the gain to attain the obligatory overshoot is far from 0, where the system is not stable beneath. c) Other characteristics of the response itself? There is a similarity between both the unfiltered and filtered systems, with filter addition only somewhat raising the time consumed for there to be a response from the system. Thus, the peak time, settling time and rise time of the response are all somewhat bigger. d) closed-loop pole locations? At -10, a pole is added to the system by the filter system-effectively turning the entire closed-loop transfer function into a third order system. With the filter now being in place, the system’s complex poles move somewhat inside towards the source of the s-plane. e) The stability region of the system? The stability range reduces when the filter is added to the system, stabilizing only for the gain values more than zero and less than about 125.On the other hand the filter less system is stable for every value of positive gain. Q3: Is a second-order approximation valid for the 3rd order system considered in experiment 1B? Justify your answer. In some specific conditions second-order estimation can be regarded as suitable for the system. This might get rid of the pole at -10 for deliberation and slightly impact the response features; however, the entire system response would be the same. Q4: What advantages and disadvantages resulted from the inclusion of the filter in the system? What significance may these have in real systems such as the application being simulated in the RSVL? The main benefits of using a filter comprise a bigger gain value for the needed 10% overshoot, raising the margin where the gain can be altered by noise, which might be advantageous in a system such as RSVL, where the components and system can be harmed by a quicker response time. Experiment 2 2.1 Requirements of Experiment 2 2.2 Introduction Just like experiment 1, experiment 2 needs a more correct model of the system from 1.1 introduction. A more precise system representation in the time domain comprises: Taking the Lapace transform of both: And: This transfer function facilitates the utilization of MATLAB in establishing the root locus of the system more precisely and with additional parameters incorporated in calculations. RVSL shaft sets the nominal mode utilized in the first experiment where there is no friction within the shaft. The shaft is set to model 1 by this experiment, which facilitates correct system representation. 2.3 Solution Description Experiment 1 and experiment 2 are similar. MATLAB code was just extended to function with a higher order system with the shaft model set to incorporate friction. 2.4 Results MATLAB Results: 2.5 Discussion and Conclusions Q1: Discuss any differences between the response generated by MATLAB and that produced by simulation in the RSVL? RSVL generated responses are less precise. This may be the case due to less definitive reading off the virtual oscilloscope, since RSVL brings about noise or due to the introduced inexactness inside RSVL simulation itself. RSVL response times were generally longer. Q2: Discuss the effect of using the more complex (but accurate) model on: a) The root locus? The most complicated system has extra poles which raise the intricacy as well as the number of lines on the root locus plot. This therefore indicates that the impact of the difference in gain with a more complicated system is most variable. b) The gain corresponding to a 10% overshoot? The gain that corresponds to a ten percent overshoot is slightly raised for the more intricate system. The tiny change signifies system refinement as well as the increased accuracy in utilizing a more complicated transfer function. c) Other characteristics of the step response itself? Additionally the increased accurateness of modeling the system slightly varies response features. However, with only very small variations, the typical variations are insignificant. d) Closed-loop pole (and zero) locations? The number of poles in the system is increased by the complexity as well as higher order transfer function. In addition, two poles are put together much further out from the source than the less complicated system. The real pole stays in place at -10 on the real axis with the poles from the lower order estimation varying very insignificantly because of a different gain that is used. e) The stability region of the system? The system’s stability range is increased with more complicated function. In addition, the stable gain’s lower limit is removed ensuring the system is stabilized for any gain value more than 120. Q3: Is a second-order approximation valid for the 3rd order system considered in experiment 2? Justify your answer. Second-order estimation is applicable since the system is very alike in response features. Q4: From your results, is the simpler, nominal model sufficiently accurate to be used in the design of this system? Justify your answer. Yes, the results of the nominal model are almost the same to the system’s better estimation. If extra accurateness were needed, the third-order system estimation would be utilized. Q5: Discuss the advantages and disadvantages of using a simpler model for control system design. What significance may these have in real systems such as the application being simulated in the RSVL? The merits of utilizing a less complicated model is that it makes the system easy to design utilize as well as predict how it will react to various inputs. This might imply a reduction in the system’s cost size and thus might be worth the less exact results. 2.7 MATLAB Code Read More