Contents Introduction Workshop logbook Logbook organization Dates for mini-test Login access to PCs in the control lab Analysis of Open Loop Systems Open loop system definition 2nd order canonical model Derivation of Poles S-plane geometry Open Loop investigation Matlab/Simulink Open_loop.slx Model System 1 calculations How to define a Transfer function in Simulink System 2 calculations Systems 3,4,5,6, Table 1 and conclusions Final value theorem (open Loop) How to determine the Characteristic equation from Poles Model Approximation 3rd order to 2nd order example 4th and 5th order to 2nd order exercises Analysis of Closed loop systems using Root Locus Closed loop system definition Simulation of closed loop systems using Closed_loop.slx model Closed loop system results and Table 2 Final Value theorem for closed loop system Root locus plots ...... Forward path compensators Frequency Response Analysis (Open loop) Straight line asymptotic approximation Calculation of Log gain Calculation of Phase angle Investigation into sketching bode plots and verifying with Matlab Open loop frequency response Closed loop frequency response Exercises to determine Gain margins Exercise to determine Phase margins System identification from Bode plots Page 1 of 33 Page 12 17 19 14 23 2 27 32 3 Simulink models for workshop exercises can be downloaded from your Moodle site. Please note that MATLAB is available for download from a link on your Moodle site. Introduction The Principles of control workshop is designed to support and provide a practical application of the theory taught in class. The contents of this document are an introduction to basic control theory and involve application of mathematics to control systems. The mathematics covers Complex numbers, second order/ higher order Quadratic/polynomial equations and how to calculate their Roots and Pythagoras theorem. The workshop exercises will be coupled with demonstrations by Technical staff where necessary. Students are expected to be familiar with the basic maths theory and it is your responsibility to revise if you have forgotten. The time spent in the lab must be used efficiently to bring you up to date and ready for a Mini Test. Workshop Logbook and Mini Test The workshop practice carries 30% of the total module mark. The mark is divided between the logbook (15%) and the Mini test(15%). The final exam will carry a mark of 70% to bring the total mark of the module to 100%. Both the logbook and the test must be passed in order to get full marks for the Module. It is not possible to pass the module without the workshop mark. This should make it absolutely clear to you how important the workshop is. Logbook organization The logbook is a complete record of the all the work that you do in the workshop. It must be properly labelled with your Name and ID number, your Supervisor's name, your course code and course title, the laboratory number T405 and properly maintained with dates and contents page with titles of the topics covered as you progress through the semester. The work should include all calculations, graphs and conclusions, necessary to obtain the marks due for each topic. It must be intelligible and easy to read with neat hand writing. Each page must be numbered and all pages must be written into. DO NOT leave blank pages in your logbook. Your supervisor will sign your logbook every week at the end of the session to confirm that you have carried out the exercises. DO NOT write on separate pieces of paper and later copy them in the logbook. This will be observed very strictly in the workshop. Your logbook will be marked by your Supervisor only and not by Technicians If the supervisor is unable to read your hand writing or understand your results or graphs, then you will get a mark of Zero for the topic or questions. You should always make comments about your results, especially, calculations and graphs. Graphs should be sketched by hand. It is not necessary for these to be to scale but a clear approximation. DO NOT copy and paste images. Printed copies of graphs will not be accepted. You will use your logbook to revise for the mini test. Therefore, it is absolutely imperative that your logbook should be up to date. Write your work as you progress in the workshop. Date for the Mini test The date of your mini test is given in your Module guide. The logbook should be handed into the faculty office(in your supervisor's name) on the date given in the module guide. You must obtain a receipt for the logbook from the faculty office. Login access, Health and safety in the Control Laboratory Please note that your LSBU Password will not work in this laboratory. A common password will be given to you by staff. Audio and Video recordings and taking photographs are not allowed. Eating and drinking is not allowed in any lab or workshop throughout the university. Do not touch, move or interfere with any equipment in the lab. Page 2 of 33 Open Loop system definition An open loop system is a system in which the output is not controlled, but only observed. In open loop, a system's output response is measured and analysed in response to an input excitation signal. A system can be represented using a standard mathematical model known as the Second Order Canonical transfer function as shown below: ANALYSIS OF OPEN LOOP SYSTEM RELATIONSHIP OF POLE LOCATION TO TRANSIENT RESPONSE A second order system can be modelled by the following canonical transfer function: U(s) Y(s) G(s) = Y(s) Koo² = U(s) s² +25@₂s+w²/ Where, is the damping ratio and on is the undamped natural frequency and K is the system Gain or amplification, also known as the DC-Gain. The input signal is U(s) and the output signal Y(s). The Ratio between the Output and Input is called the Transfer function G(s). In any system, there exists resistance/friction or viscosity, termed Damping and is represented by (Zeta). If damping is removed, the system becomes undamped and the system is free to oscillate with simple harmonic motion at its natural resonant frequency w, (Omega-n). The denominator polynomial (also called the characteristic polynomial) is second order with roots which may be real numbers or Complex and are termed Poles. If the numerator contains roots, these are termed Zeros. A Zero may also have real or complex values. The values of the Poles and Zeros and their interaction determine the systems behaviour. Real Axis Therefore to find the Poles/roots of the characteristic equation, the denominator is set equal to zero and the equation is solved: s²+25w₁s+w²2²=0 As the polynomial is second order, we expect to have two Poles whose real or complex values are plotted on real and imaginary axes, which in control theory, is termed the S-plane. The relationship between the coefficients of the characteristic i.e the parameters and @,, are shown below on the S-plane. The complex roots (poles) are at locations S-Plane 50 n @n Κω s²+250,s+0 0 Imaginary Axis (0,0) √1-5² joo, √√1-2² s=-5w₁ ±ja, √1-5² S= The distance of each pole from the origin(0,0) of the s-plane is an (undamped natural frequency) The angle 0 made by each pole is such that the damping ratio = cos Page 3 of 33 Derivation of the Poles Since the characteristic equation is a second order equation, the standard solution for the Poles can be expressed in terms of its coefficients as s²+25w,s+w=as² +bs + c = 0 - b ± √b² - 4ac The solution is: $1,2 2a Where, a=1, b=25@,,, and c= w. Substitute these into the solution we get: -250, ± √(250)² - 4w/ 2 $1,2 Take the square-root of 4 and we get: - 25w₁ ±2w₁ √5²-1 2 $1,2 = Therefore, the Poles are s=- S-Plane geometry Real Axis S-Plane Now rearrange √2-1 to @√√-1(-5² +1), then the imaginary part of the Poles will be 00₁,√-1√√1-5². Then the √1=j i.e the imaginary component followed by its coefficient √1-5². - 50 n " @₂ $1,2 = s=-5w₁ ±jw√√1-5² 8 -250 ± √√45²²-40²/ -250 ± √√40² (5²-1) 11 2 اع - الهز (0,0) 2 - jo, √₁-5² Real Axis = H , $1.2 = Page 4 of 33 S-Plane The Right angle Triangle defined by the sides AOH has side lengths which are: A 500, A = 50₁, 0=0,√√1-5² and H=@₁. Then cos(0)= @₁ The length of side H can be calculated using the Pythagoras theorem: H = √√² +0². √A² +0². H 0 8 (0,0) -=, therefore cos(0) = 5. Therefore, @,,= √ Note: Pole locations are indicated on the S-plane by the letter 'x' and Zeros by the letter 'o'. Open Loop Investigation In the Open Loop analysis part of the investigation, six different open loop system models will be tested to examine their responses using Matlab and Simulink models. Investigation requirements the The workshop investigations for all systems include calculation of the Parameters K, , ,, Systems Poles, the Value of the system's Steady-state output, a sketch of the system's response from the Scope and a sketch of Pole positions on the S-plane. Finally for each system a short comment should be written to describe its behaviour based on the values of the Parameters and Pole Locations. In the following examples, you will be shown how to calculate the parameters for each system. As you calculate the parameters for each system, fill in the data into Table 1 on page 9. How to run Matlab/Simulink On the Desktop, you will find the Icon for Matlab 2014a. Double click the Icon and wait for Matlab to run. The following screen will be displayed: MATLAB R2014 HOME Open PLOTS Details Ready FLE 44442CUsers Find Fles Compare Import Save Current Folder Name POC Control Engineering Bytronic MAB APPS Data Workspace New Varable Open Variable Clear Workspace SASABLE CONTROL Documents MATLAB O Command Window Analyze Code Ran and Time Clear Commands CODE Preferences Simulink Layout Set Path Library SMULINK DWPONMENT This is a Classroom License for instructional use only. Research and commercial use is prohibited. >> | Hele Community Request Support +Add-Ons- RESOURCES Page 5 of 33 Search Documentation The default Matlab folder shows a number of sub-folders as shown in the Folder view. You are required to work in the POC folder only. Double click the folder and locate the filename Open_loop.slx. Double click it to load the Simulink Model. The following windows are displayed:/n Straight line Asymptotic approximations It is possible to sketch Asymptotic approximations of the system's frequency response using the above equations 9 and 10. Then for the transfer function the LogGain of the Numerator expression is divided by the LogGain of the Denominator expression. If the system is: 10 G(s)=- (s+10) LogGain of G(S) in dB = 20log|numerator-20log|deno min ator Since the numerator or the denominator can be complex numbers, then their magnitudes can be expressed as: G(s): a + jb c+ jd If Log Gain Calculation Therefore, the magnitude of the numerator is numerator = √a² + b² And the magnitude of the denominator is deno min ator = √² +d² Reminder: If X Z=- y z = x*y 10 s+10 0= tan then log(z) = log(x) -log(y) LogGain of G(S) in dB = 20 log √a² + b² -20log √² +d² Substitute the values of a=10, c=10 and jd = s = @ LogGain of G(S) in dB = 20Log√10² -20log √² +10² ⇒Gain(dB)=20-20log √²+10² Phase angle Calculation The phase angle for numerator and denominator can be expressed as: b 0 = tan [-]-tan [²] C -1 a 0 10 ⇒Phase(0)=0-tan where, jb=0 in this case since the s-term is missing in the numerator. log(z) = log(x) + log(y) tan -1 @ 10 -1 @ 10 equation 11 equation 12 Therefore, as the constant terms or the s-terms in the numerator / denominator are multiplied or divided their logarithmic values are added or subtracted respectively. The phase angles of the individual terms are also calculated in the same way, where the phase angle of terms multiplied are added, and phase angles of terms which are divided in the expression are subtracted for each term. Page 25 of 33 Tabulating the results for frequencies of 0.1, 1, 10, 100 and 1000 we can obtain an approximate response and plot these using straight line asymptotes as shown below: Gain(dB) = 20-20log √² +10² (ap) aprique p Phase (deg) 0.1 1 10 100 1000 Output flat until Break frequency -3db -20 -40 Bode Diagram Output-3 dB at break Frequency=10 rads/s Phase angle starts to fall a decade before the Break frequency Output drops at 20dB/decade after break frequency 10 Frequency (radisec) Phase(0) Phase angle = -45 at break Frequency =10 rads/s =-tan 0 -5.7 -45 -84.3 -89.4 Page 26 of 33 Straight line Asymptotic approximation -1 @ 10 On the next page the system can be simulated over the full frequency range using Matlab commands. INVESTIGATION INTO SKETCHING BODE PLOTS AND VERIFYING WITH MATLAB PREDICTIONS 10 sketch the straight-line asymptotic Bode-plot and verify using Matlab. (s+10)' The following command must be entered from the Matlab command window at the prompt >>. It defines the transfer function(tf-command) for G(s) as the variable g. A. Given G(s) = >>g = tf([10], [1 10]) The following is the Bode command which plots the transfer function response for g and also plots a grid in the graph. >>bode(g),grid You should see this bode plot 10 s + 10 (gp) apri Phase (deg) -10 -90 Q.1. 0,2.03. 10 (increasing (0) Bode Diagram 10 10 Frequency (radisec) -3 dB -ve attenuation at Break frequency w 10 Phase angle(-45) at Break frequency w=10 Plotted on a logarithmic scale 100 200 300 1000 10² The Bode plot is divided into two parts. The top part is a plot of the dB gain values on its linear vertical axis. The horizontal axis is a plot of all the w frequency values in radians/sec. The horizontal axis has a Logarithmic scale. Look at the horizontal axis. The frequency values start from 0.1, 0.2 and increment up to 1, then increment by 10 to 20, 30 then increment by 100 to 200, 300 etc up to 1000. As the numbers increase at each stage the scale also shrinks. This is known as a Logarithmic scale increasing in Decades. The Bottom part of the graph is the Phase angle plot with the angles on its linear vertical axis plotted against the frequency values in Decades. Now look at the system's response. The system Gain in dB starts with a flat response until it reaches the Pole location S=-10. This is known as the Break Frequency of 10 in this case. You will notice a sudden drop in the dB gain from 0 down to -3 dB. This is indicative that the system has a Pole at this frequency with the value of 10. Page 27 of 33 From the Bode plot answer the following questions: Q1: What is the break frequency? Q2: If the input signal is a sinusoid of amplitude 1 volt at frequency of 1 radian/second, what is the amplitude of the output signal? What is the phase angle between the output and input signals? Hint: Read the dB gain of the system by right clicking the mouse at 1 rads/s on the Gain plot and substitute the dB gain in equation 8 on page 24. Similarly, read the phase angle value directly on the phase plot at 1 rad/s. Repeat the same procedure for the following exercises. Q3: If the input signal is a sinusoid of amplitude 1 volt at frequency of 10 radian/second, what is the amplitude of the output signal? What is the phase angle between the output and input signals? If the input signal is a sinusoid of amplitude 1 volt at frequency of 100 radian/second, what is the amplitude of the output signal? What is the phase angle between the output and input signals? Q4: sketch the straight-line asymptotic Bode-plot and verify using MATLAB. B. Given G(s) = s+10 S+100 g= tf([1 10], [1 100]) bode(g),grid % this line creates a transfer function G(s) % this draws the Bode plot From the Bode plot answer the following questions: Q1: What are the break frequencies? Q3: Q2: If the input signal is a sinusoid of amplitude 1 volt at frequency of 1 radian/second, what is the amplitude of the output signal? What is the phase angle between the output and input signals? If the input signal is a sinusoid of amplitude 1 volt at frequency of 100 radian/second, what is the amplitude of the output signal? What is the phase angle between the output and input signals? Q4: If the input signal is a sinusoid of amplitude 1 volt at frequency of 1000 radian/second, what is the amplitude of the output signal? What is the phase angle between the output and input signals? 1000 (s+10)(s+100) C. Given G(s) = prediction. Q1: What are the break frequencies? Q3: Q2: If the input signal is a sinusoid of amplitude 1 volt at frequency of 1 radian/second, what is the amplitude of the output signal? What is the phase angle between the output and input signals? If the input signal is a sinusoid of amplitude 1 volt at frequency of 100 radian/second, what is the amplitude of the output signal? What is the phase angle between the output and input signals? Q4: If the input signal is a sinusoid of amplitude 1 volt at frequency of 1000 radian/second, what is the amplitude of the output signal? What is the phase angle between the output and input signals? sketch its straight-line Bode diagram and verify with a MATLAB 1 s(s+1) D. Given the transmittance G(s) = - MATLAB prediction. Q1: What are the break frequencies? Q2: If the input signal is a sinusoid of amplitude 1 volt at frequency of 0.1 radian/second, what is the amplitude of the output signal? What is the phase angle between the output and input signals? Q3: Is the system stable? sketch its straight-line Bode diagram and verify with a Page 28 of 33 Closed loop Frequency response demo program So far we have only considered open-loop plants. When we close the loop with a compensator H(s) in the feedback path and a gain Kp in the forward path we are interested in predicting the additional gain that could be applied before instability results. We can do so by plotting the Bode-plot for the transmittance product GH (called the Open-loop product) and measuring the Gain Margin (GM). This is equivalent to the Critical gain in time domain i.e the Gain at which the system becomes unstable. First find the Gain Cross Over Frequency (GMF). This is the frequency at which the phase angle of GH is ± 180 degrees. At the cross over frequency, the Gain Margin (GM) is the additional gain (in decibels) necessary to make the amplitude of GH unity (i.e. 0 dB). Start your investigation by predicting the Gain Margin (GM) of the following systems. We will find the Phase Margin later with other examples. Type the following command >> margin_gui in the MATLAB Command Window to run a demo that shows system stability and its link with GM and PM. Wait for a few seconds for Matlab to do its calculations, then it will display the following window. Use the slider to vary the gain Kp from zero to max value. Gain and Phase Margin Analysis File Edit View Insert Iools Desktop Window Help Magnitude (dB) (6ap) asend 60 40 20 0 -20 -40 -60 -45 -90 -135 -180 -225 10 Open Loop G-K 10° Frequency (rad/sec) 10 Amplitude Closed-Loop Step Response 1.5 0.5 10 Time (sec) Gain Margin: 8.8 dB Phase Margin: 45 deg Closed-loop Stable? Yes 20 Page 29 of 33 Feedback Loop: G(s) = K-G 0.1 Loop Gain K: 0.5 s +1.3 s+1.2s²+1.6s 1 X Close 10 This demo plots the system time response and the bode plot simultaneously and allows you to vary the Proportional Gain Kp to observe the system's response. The system has 1 Zero and 3 Poles. Vary the Gain of the system and observe the response, particularly the Gain and Phase margins. Make notes in your logbook about your observations.