Use Analytical Techniques to Form Models of Engineering Systems & ProcessesControl Systems and AutomationBy Brendan BurrBrendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and AutomationTable of ContentsTABLE OF CONTENTS ........................................................... 2 TASK 1 ................................................................................ 6 1.1(a) Define two types of control systems and give at least 2 examples of each type. ........................................................ 6 Solution:- .........................................................................................6 1.1(b) Draw a well labelled diagram showing the basic elements of a closed loop control system. ............................ 7 Solution:- .........................................................................................7 1.2(a) Derive from first principles the general feedback equation for a simple closed loop control system in canonical form................................................................................... 8 Solution:- .........................................................................................8 1.2(b) Modify the transfer function for unity negative feedback. ............................................................................ 9 Solution:- .........................................................................................9 1.3(a) Using block diagram algebra reduce the multi-loop control system shown in Fig.1 to a single loop diagram....... 10 Solution:- .......................................................................................10 1.3(b) Hence determine the closed loop transfer function of the system........................................................................ 13 Solution:- .......................................................................................13 1.4 Determine the overall transfer function of the following system in Fig 2 :- ............................................................... 14 Solution:- .......................................................................................14 1.5 The following Fig 3. represents a multi-input system :-. . 15 Determine an expression for the single transfer function for the resultant output C....................................................... 15 Solution:- .......................................................................................15 TASK 2 .............................................................................. 17 2.1(a) State the s-plane stability criterion. ......................... 17 Solution:- .......................................................................................17 2.1(b) Show the location of the poles and zeroes on an S-plane diagram for the following transfer function. ............... 18 Solution:- .......................................................................................18 2Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and Automation2.1(c) Apply the S plane stability criterion to determine whether the system is stable or unstable............................ 18 Solution:- .......................................................................................18 2.2 Plot the P-Z diagram corresponding to the following transfer functions. In each case state whether the corresponding system is stable or unstable........................ 19 2.2(a) ............................................................................ 19 Solution:- .......................................................................................19 2.2(b) ............................................................................. 20 Solution:- .......................................................................................20 2.2(c) ............................................................................ 21 Solution:- .......................................................................................21 2.2(d) ............................................................................. 22 Solution:- .......................................................................................22 2.3 State the Nyquist stability criterion for an Open Loop System............................................................................. 23 Solution:- .......................................................................................23 2.4. A control system is characterised by the open loop transfer function:- ............................................................. 24 ...................................................................................... 24 Plot the Nyquist diagram for the system using values of w as follows:- ............................................................................ 24 2.4(a) Determine the gain and phase margins using the Nyquist Plot...................................................................... 24 Solution:- .......................................................................................24 2.4(b)Comment upon the relative closed-loop stability of this system.............................................................................. 24 Solution:- .......................................................................................24 2.5 The open loop frequency response of a control system has the following data obtained practically :- ............................ 26 2.5(a) Plot the Nyquist diagram......................................... 26 Solution:- .......................................................................................26 2.5(b) Determine the Gain margin in dB from the Nyquist Plot......................................................................................... 27 Solution:- .......................................................................................27 2.5(c) Determine the Phase margin from the Nyquist Plot... 27 Solution:- .......................................................................................27 3Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and Automation2.5(d) Comment upon the relative closed-loop stability of the system............................................................................. 27 Solution:- .......................................................................................27 2.6 Determine the value of K for a system with the following open loop transfer function:-............................................. 28 ..................................................................................... 28 Which will give:-................................................................ 28 2.6(a)A marginally stable system. ................................... 28 Solution:- .......................................................................................28 2.6(b) A gain margin of 6 dB.............................................. 31 Solution:- .......................................................................................31 2.7 Determine the phase margin for a system having the following open loot transfer function:- ................................ 32 ..................................................................................... 32 Solution:- .......................................................................................32 2.8 A control system has the following open loop transfer function:........................................................................... 35 ..................................................................................... 35 2.8(a)Re-arrange the transfer function in Bode form and hence state the break point. .............................................. 35 Solution:- .......................................................................................35 2.8(b) Draw Bode Plots for the Log Modulus and Phase Angle using asymptotic approximation. ....................................... 36 Solution:- .......................................................................................36 Solution:- .......................................................................................38 TASK 3 .............................................................................. 39 3.1 Explain what is meant by Compensation in a control system............................................................................. 39 Solution:- .......................................................................................39 3.2 State 2 types of compensation techniques commonly used................................................................................. 39 Solution:- .......................................................................................39 3.3 Compare and discuss their relative merits.................... 39 Solution:- .......................................................................................39 EVALUATION..................................................................... 42 4Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and AutomationCONCLUSION..................................................................... 42 Books ............................................................................... 43 Catalogues ........................................................................ 43 Websites........................................................................... 43 5Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and AutomationTask 11.1(a) Define two types of control systems and give at least 2 examples ofeach type.Solution:-Open Loop Control System An electric fire system:Irrigation Sprinkler:Closed Loop Control System An electric fire system with thermostat:6Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and Automation1.1(b) Draw a well labelled diagram showing the basic elements of a closed loop control system. Solution:-7Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and Automation1.2(a) Derive from first principles the general feedback equation for a simple closed loop control system in canonical form.Solution:-Definition:G = Forward Transfer FunctionH = Feedback Transfer FunctionG . H = Loop Transfer FunctionB R EH C BandE G CCBHECGt ..So:( )( )H C G R G CH C R G CB R G C. . ..t t t Signs SwitchR G H G C C . . . 1 . Factorise( ) G R H G C . . 1 H GGRC. 18Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and Automation1.2(b) Modify the transfer function for unity negative feedback.Solution:-B R EC BandE G CC BECG .So:( )( )C G R G CC R G CB R G C. . Signs SwitchR G G C C . . 1 . +Factorise( ) G R G C . 1 +GGRC+1Unity Negative Feedback9Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and Automation1.3(a) Using block diagram algebra reduce the multi-loop control system shown in Fig.1 to a single loop diagram.Solution:-Step 1:G3G2++Merging gives:G2+G3Step 2:H1 H2Merging gives:H1xH210Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and AutomationStep 3:G2+G3H1xH2-+Merging gives:G2+G31+(G2+G3)x(H1xH2)Therefore:-+ G1G2+G31+(G2+G3)x(H1xH2)++ G4RCStep 4:G1G2+G31+(G2+G3)x(H1xH2)Merging gives:G1x(G2+G3)1+(G2+G3)x(H1xH2)11Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and AutomationStep 5:++ G4Merging gives:G41-G4Step 6:G1x(G2+G3)1+(G2+G3)x(H1xH2)G41-G4Merging gives:G1x(G2+G3)xG41+(G2+G3)x(H1xH2)x(1-G4)Step 8:G1x(G2+G3)xG41+(G2+G3)x(H1xH2)x(1-G4) -+CR12Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and Automation1.3(b) Hence determine the closed loop transfer function of the system.Solution:-( )( ) ( ) ( )( )( )( ) ( ) ( ) ( )

,_

+ +++ + ++4 1 2 1 3 2 14 3 2 114 1 2 1 3 2 14 3 2 1'G H H G GG G G GG H H G GG G G GG( )( )( ) ( )( ) ( ) ( ) ( ) 4 3 2 1 4 1 2 1 3 2 14 3 2 1'G G G G G H H G GG G G GG+ + + ++13Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and Automation1.4 Determine the overall transfer function of the following system in Fig 2 :-Fig 2Solution:--+ 10C2S-11S+1RStep 1:102S-1Merging gives:20S-1Step 2:( )( ) ( ) 20 1 11 20'111201120'+ + +

,_

+

,_

+s ssGs ssG( )191 20'2++s sG14Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and Automation1.5 The following Fig 3. represents a multi-input system :-Fig 3.Determine an expression for the single transfer function for the resultantoutput C.Solution:-Step 1:Initially u equal zero.Therefore:-+1S1S+1R1S+2CRStep 2:( )( )( )( ) ( )( )( ) ( ) ( ) 1 1 21 11121112112121 121+ + ++

,_

++ ++++++ +s s ssRCss sRCs s ss sRCs s s sRRR15Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and Automation( ) ( )( )Rs s ssCR1]1

+ + ++1 1 21Step 3:Now let R equal zero.++U1SCU1S+21S+1Merging gives:( ) ( )( ) ( )( )( ) 2 1112 11 1112 112111+ ++ + + +++s ssRCs s ssRCs s s sUU( )( )( ) ( )( )Us s ss sCU1]1

+ ++ +1 2 12 1Step 4:( ) ( ) ( )( )( )( ) ( ) ( )Us s ss sRs s ssCC C CU R1]1

+ ++ ++1]1

+ + +++ 1 2 12 11 1 21( )( )( )( )( )( )( ) 1 2 12 11 1 21 + ++ +++ + ++s s ss s Us s ss RC16Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and AutomationTask 22.1(a) State the s-plane stability criterion. Solution:-The S-Plane is the complex frequency plane, representing all the values of j s + on a two dimensional diagram.The transfer function of a linear system is in general a ratio of two polynomial expressions, which are expressed in terms of s in a factorised form.( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )NMP s P s P s P sZ s Z s Z s Z sK s G ......3 2 13 2 1K is a constant which represents system Gain (K>1) or system Loss/Attenuation (Kz.This equation can be rearranged to give:( )

,_

+

,_

+

,_

pszspzKs G11The terms K and z/p are constant gain terms, whereas the 1+s/z is a real zero term with a time constant of 1/z, and a real pole term of 1+s/p with a time constant of 1/p.Since p>z then 1/z > 1/p.Therefore the Bode plot is in the form shown below.The values given are for K=1 and various ratios of z/p.However when z=p and the pole and zero terms cancel to five G(s)=1, then the magnitude is a straight line along the 0dB axis and the phase becomes a straight line along the 0 degree axis.The effect of introducing a lead compensator is to lower the magnitude plot at low frequencies and raise the overall phase angle of the output relative to the input.40Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and AutomationLag CompensatorThe transfer function of a cascade lead compensator is as follows:( )( )( ) p sz s Ks G++Where z>p.This equation can be rearranged to give:( )

,_

+

,_

+

,_

pszspzKs G11The terms K and z/p are constant gain terms, whereas the 1+s/z is a real zero term with a time constant of 1/z, and a real pole term of 1+s/p with a time constant of 1/p.Since z>p then 1/p > 1/z.Therefore the Bode plot is in the form shown below.The values given are for K=1 and various ratios of z/p.However when z=p and the pole and zero terms cancel to five G(s)=1, then the magnitude is a straight line along the 0dB axis and the phase becomes a straight line along the 0 degree axis.The effect of introducing a lag compensator is to lower the magnitude plot at high frequencies and raise the overall phase angle of the output relative to the input.41Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and AutomationEvaluationThe first three sections of Task 1 were relatively straight forward.We have been covering open and closed loop circuits for the past three years, so it was easy to get to a conclusive answer almost immediately for 1.1.There were excellent notes to deriving the first principles for a simple closed loop control system in canonical form from class, however I also did a bit of background reading in the Control Engineering Book, referenced in my bibliography.This helped me understand the theory before writing the equations, and also allowing me to easily modify the function for unity negative feedback.Moving onto the block diagram algebra, I took this subject in relatively quickly. It didnt take me long to work out the answers for tasks 1.3, 1.4 and 1.5.Task 2 explored a completely new subject to me, which was Nyquist Diagrams and Bode plots.It was a straight forward, but confusing subject for me, and required a clear head to remember previous notes during class.I used notes and research in the Control Engineering Book to iron out any confusion I had, and then found the assignment fairly straight forward.I was able to work the answers out quite quickly it was the typing up the answers which took hours!For this reason it is definitely recommended that the assignment be worked on throughout the year, rather than leaving it until the end of the year (which I was fortunate enough to avoid!).I was quite pleased with the excel spreadsheet I produced to help me answer Task 2.5.It allowed me to enter the equation and produce an answer in Polar Form with ease, then all I had to do was convert the Polar answers into Rectangular answers and then plot this in Graphmatica.I was pleased to be able to produce the Bode Plots for Task 2.8 using Graphmatica.I spent a while trying to set up the Graph Paper and axis, but I think the result was worth it.I should note that I had to export the graph to MS Paint, so that the points could be joined with lines, as this function didnt seem to be available on Graphmatica.Task 3 involved using the explanation from class as well as some brief research to compile my own description of Compensation in Control Engineering.I found this relatively straightforward.ConclusionThis assignment has been worked on throughout the entire academic year, so evaluating points after the entire assignment has been complete, proved a bit difficult.I will take this into account so that in future I can evaluate a Task after I have completed it rather than waiting until the end.I have learnt a lot from this assignment as there have been many new subjects.The unit as a whole has been a mix of multiple mathematical techniques and has definitely tested the knowledge and understanding gained from previous years.42Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and AutomationBibliographyThrough guidance from my lecturer, the following text books, catalogues and websites I was able to complete this assignment:BooksControl Engineering (W. Bolton) ISBN: 0-582-32773-3CataloguesN/AWebsitesN/A43