Lab(8) cotroller desig usig root locus I this lab we will lear how to desig a cotroller usig root locus but before this we eed to aswer the followig questios: What is root locus? What is the purpose of the cotroller? What is the relatio betwee close loop system poles ad the respose? What are the types of cotroller? How to select the right cotroller for the system? To aswer the first questio root locus is a graphical presetatio of the close loop poles movemet o the axes as the gai k chage from zero to ifiity. To aswer the secod questio the purpose of the cotroller is to adjust the system to meet the required specificatio as settlig time, over shoot, peak time, ad sse. Before aswerig the third questio we eed to kow that the respose has two part the trasiet part ad the steady state part Trasiet respose Sse respose
Whe talkig about trasiet respose we talk about settlig time, rise time ad overshoot ad we talkig about ss respose we talk about ss error. Rememberig the secod order trasfer fuctio formulas we have w 2 g ( s ) s1,2 w jw 1 s w s w 2 2 2 2 4 Ts O. s e w 1 2 Sice the over shoot deped o (dampig ratio ) ad the settlig time ad peak time deped o (dampig ratio ad the dampig frequecy w ) ad sice the poles formula cosist of dampig ratio ad dampig frequecy the chagig the poles will chage the dampig ratio ad dampig frequecy which will result i a chage i trasiet respose ad ss respose To aswer the fourth questio, there are may types of cotroller ad ca be divided i two categories the first is the passive cotroller ad secod is the active cotroller cotroller active cotroller passive cotroller pd pi PID lead lag lead-lag
The deferece betwee the two categories is that the active cotroller is more accurate sice it is implemeted usig computer ad microprocessors ad the passive cotroller is iaccurate compare to active sice it is implemeted usig resistor ad capacitors but oe advatage for the passive cotroller over active cotroller is the price. Before aswerig cosider this, if the cotroller chage the system respose parameter like settlig time rise time overshoot ad steady state error the the cotroller will have to chage the poles locatio to aother locatio which archives the require specificatio ad the questio here is this locatio is o the root locus or ot o the root locus? If the locatio is i the root locus the the required specificatios ca be achieved by usig ( p cotroller ) oly ad if the locatio is ot o the root locus the we eed to add zero (PD cotroller) for example i order to chage the root locus to pass through that locatio Aother method to choose the right cotroller is to see where is the required specificatio lies for example if we eed to chage the trasiet respose use PD cotroller ad if we eed to chage ss respose we eed PI cotroller. Cotroller ame Cotroller trasfer effects fuctio Proportioal C(s)=k All ( ss + trasiet) Proportioal + derivative (PD) C(s)=k(s+z) Trasiet ad small effect o ss Proportioal + itegral k ( s z ) Ss ad small effect o (PI) s trasiet Proportioal + itegral k ( s z 1)( s z 2) All derivative (PID) s What is the deferece betwee proportioal ad PID
Cotroller desig usig Matlab:- There is a mai tool i Matlab used to desig a cotroller usig root locus ad its ame is sisotool The sisotool gives you may thigs 1- System root locus 2- System bode plot 3- Close loop ad ope loop systems step resposes 4- The formula of the cotroller ad the required gai I order to use the sisotool write sisotool (sys) ad sys ca be state space of TF or ZPK Example :- Desig a PD cotroller for the followig system to get a dampig ratio of 0.707 ad settlig time of 2 sec Step 1 :- 1 gs ( ) s ( s 2)( s 5) we eed to fid the desire close loop locatio Ts 4 w 2 4 2 w 4 w 2.82 2*0.707 2 s1,2 s1,2 w jw 1 2 2 j
Step 2 :- We eed to eter the system o Matlab ad activate the sisotool Num=[1] De=[1 7 10 0] Sys=tf(um,de) Sisotool(sys) A siso tool cosists of the followig two figures: Figure 1 sisotool first figure
Step 3:- we eed to locate the desire poit ( 2 2j ) o the sisotool. To place a pair of complex poles o your diagram at a dampig ratio of 0.707, select Desig requiremet ad the New, the followig meu will appear Figure 3 desig requiremet meu
Form this meu isert the settlig time value which equal s 2 ad the followig chage will appear o the siso tool Figure 4chage o the sisotool Now do the previous step oe more tome put this time for dampig ratio The followig chage will appear o the sisotool
The itersectio poit The desire poit is the itersectio poit ad therefore we eed the root locus to pass through this poit by addig zero at -2.38 ad i order to do that righ click o the root locus ad choose add real zero the go to the secod scree ad chage its locatio
Step 4 Move the added zero util the root locus pass through the itersectio poit Figure 5 root locus pass through the itersectio poit Now drag the close loop poles to the itersectio poit i order to fid the rag of the gai require to reach the desire poit
Close loop poles Now go the secod scree ad you will fid the trasfer fuctio of the cotroller gai Zero locatio Figure 6trasfer fuctio of the cotroller
How to make sure that the cotroller gives the required specificatios? Go to the secod scree ad choose aalysis plots the choose the plot type which is step i plot 1 the choose close loop system r to y ad the step respose will appear Now right click o the step respose ad choose characteristics ad from characteristics choose settlig time peak time ad other characteristics
Figure 7 step respose If you eed to desig a PI cotroller for the system do the followig Add the PI cotroller trasfer fuctio before desigig the PD cotroller s 0.001 The PI cotroller trasfer fuctio is from this trasfer fuctio s the pole is used to icrease the system type ad the zero used to preserve the root locus shape.
Report exercise: Check your result usig Labview? Please add the block diagram i your report ad the frot pael ad the step respose ad also the parameter of the step respose like settlig time over shoot ad peak time for the system before ad after cotroller?