Designing a Lead Compensator with Root Locus

hey everyone welcome back to control system lectures let's continue our discussion on lead and lag compensators and in this video we're going to focus specifically on designing a phase lead compensator using the root Locus method and if you haven't already watched the introduction to lead lag compensators I'd recommend you start there first before watching this video all right let's get to it let's say that you're designing a system and you already have a model of your open loop plant now you might have developed your model from your real physical system using a method called system

identification or you might have just been given a transfer function from your professor as a good example either way the methods for designing a lead compensator are the same the first thing that you need to do is determine your design requirements these are the performance goals that you hope your system's going to meet things like rise time settling time phase margin and gain margin damping ratio and so on but whatever your requirements are you determine at this point that your open loop system can't meet them so you decide that you need to implement a closed

loop control system around this plant and once you do that as the control system designer you get to choose any compensation technique that you want as long as it meets the system requirements and your control technique of choice is a phase lead compensator so let's design one wait a second we're getting ahead of ourselves how in the world did we determine that we wanted phase lead and not phase lag or PID compensation or some other compensation method well to answer that question let's look at how phase lead compensators affect the root Locus for your system

and then from there we can deduce which situations call for a phase lead compensator let me draw the root Locus for a simple two pole system and I'll arbitrarily put one pole at minus one and one pole at minus 3 so we'll start the root Locus by placing the poles at minus1 and -3 and then we know from the video on sketching a root Locus by hand that the locus exists to the left of the odd number of poles and zeros which would be right between the two and that when these two locai meet each

other they leave the real line at 90° and then they tend towards Infinity along two asmp tootes one at 90° and one at 270° and the location of where the ASM toote crosses the real line is at a point called the center of gravity or the centroid which is simply the sum of the location of the finite poles minus the sum of the locations of the finite Zer divid the number of poles minus the number of zeros and in our case we have two poles and no zeros so the center of gravity or the centroid

is directly between the two of them at minus 2 but what if we added a phase lead compensator to the system and we saw how that affected the root Locus and I'll choose s +4 over s+ 5 and again this is just an arbitrary lead compensator but you can tell it's phase lead because the zero is closer to the origin than the pole let me redraw the root Locus down here so it's a little bit cleaner we still have the two poles from the plant at minus1 and minus 3 but now we've added a 0er

at -4 and another pole at minus5 again we draw the locus to the left of the odd number of poles and zeros so it mostly looks the same so far and we still have two lines tending towards Infinity at Plus 90 and plus 270° since we added both a pole and a zero so that's the same but what's happened to the centroid well the denominator is the same since we still only have two lines tending towards Infinity but in the numerator we're subtracting an additional five because of the extra pole and we're only adding an

additional four because of the extra zero so that means that our net gain in the numerator is minus1 and that our new centroid is now atus 2.5 or as You' probably guessed by now by adding a lead compensator we've essentially dragged the asmm tootes further into the left half plane so now as we increase the gain of our system the closed loop poles will follow these lines instead so all else being equal we've added stability to our system by moving the closed loop poles further into the left half plane and we've done all of that

with a lead compensator all right so now that we've done that you might be thinking to yourself well how does that help our situation well typically when you use the root Locus method you usually have a position in mind where you want your dominant closed loop poles to reside so you basically you want to move them from some uncompensated closed loop position to some desired location that's going to allow your system to meet its requirements let me briefly go on a side note here and explain what a dominant pole is so that hopefully this will

make a little bit more sense if you have a system with a single pole then the trans response to an Impulse dies off based on the equation E to Theus St and as you move that pole further into the left half plane that impulse response is going to die off quicker since you are increasing the negative exponent in that equation e to the minus St and if you have a system with both poles that I've drawn here then the transient response will die off at some combination of the two rates and since the pole that's

closer to the imaginary line is going to die off slow than the other pole the response of the system will tend to follow that pole more than the others and if it is much much closer to the imaginary line than the others then that's considered a dominant pole and the system behaves so much like it that you can pretty much ignore the other poles and have very little error so you can have a system with five poles that look like this but since these two poles are much closer to the imaginary axis than the other

ones it's going to behave like a second order system as though those are the only two two poles in the system okay that leads me now to a side side note why do you hope that your system behaves like a second order system well because when you're talking about performance measures like damping ratio for example this is only defined in the traditional sense for a second order system let me explain that a little differently when you're given your design requirements like rise time and settling time and so on you need to convert these into a

closed loop pole location in the S plane for example example if you're given a rise time requirement then the equation to determine the damping ratio and natural frequency of an undamped second order system is this which can be approximated by this simpler equation and if you're given that the damping ratio needs to be 7 then you can solve for the natural frequency and figure out where your closed loop poles need to be in order to meet those requirements however you can place those poles there with a compensator but you can only guarantee that the system

is going to meet these requirements if it's a second order system or if it has a pair of dominant poles that makes it act like a second order system if it doesn't then determining where to place your poles to make the system behave the way you want can be a bit more of an art form and possibly some iteration on the design until you're happy with the result but that's for another video let's get back to designing the lead compensator so you can see that after you've determined where you want your dominant poles to reside

then you can just go through the process of designing a compensator that places the poles there and if the poles are to the left of the current root Locus the uncompensated root Locus then you need to add lead to move the root Locus to the left and if they're to the right then you're going to need to add lag and if the roots already exist on the root Locus then you don't need to move it at all and all you have to do is pick the correct gain although you still might have a steady state

error problem and I'm going to discuss that in the lag compensator design video all right while I was talking I drew up this examp example it's the same plant as before with a root at minus1 and-3 and again I'll draw the root Locus for this system but I'm going to say based on my requirements I want my dominant poles to be where these green X's are which is minus three on the real line and plus or minus two on the imaginary line and I can tell from this that I need to add lead compensation because

I need to move the roots further into the left half plane so now at this point designing a lead compensator is really just an ex exercise in trigonometry just solving for the angles of a few triangles let me show you why in order for a point to exist on the root Locus then the sum of the angles of all the poles minus the sum of the angles of all of the zeros has to equal 180° and the angle I'm talking about is the angle of the line from the open loop Pole to the closed loop

pole on the root Locus I'll demonstrate it real quick here with this blue dot the angle of that first pole will be this Theta 1 and the angle of the second pole will be Theta 2 you always take it from the positive side of the real line and we can just sum them together and we can see that they equal 180° if you did out the math yourself since we don't have any zeros there's no subtraction there and the 180° is expected because that point really does exist on the root Locus let me redraw this

here again but this time I'm going to draw my desired pole location instead of an actual pole location now I can redraw my lines and then recalculate my two angles the angle on the left is just 90° and the angle on the right which I'm calling Theta 1 is just 90° plus this angle in the triangle which I'll call Theta 2 and when we add all of these angles up we get 90° plus 90° plus whatever Theta 2 is which is just the inverse tangent of one and this whole thing sums up to 225° and

since 225 is not 180° we know that this point is not on the root Locus not surprising so how do we get it there well we need to take 225° and subtract 45° and we're going to do that by adding a single pole and a single zero remember that the Zer subtract and this combination of a single pole and single zero is our phase lead compensator now at this point I'm going to somewhat arbitrarily place my 0 at minus 4 I'm going to explain this a little bit later but when I do that the angle

that it makes up is 63.4 3° and now that I've placed this zero the pole can only go in one specific spot specifically the spot where it makes an angle of 18.4 De with the root that we want in this case the math works out really nicely and the pole needs to be at - 9 so my lead compensator for this particular problem would be S +4 over s +9 and if you plot the root Locus using mat lab or some other technique you'll see that it in fact does go through the points that I

want but there is one other small matter that you need to figure out what the gain is in order to actually Place The Roots at those specific points and you can do it by hand using the magnitude Criterion or you can just do it in mat lab but that's relatively straightforward and something that I'm going to cover in a future video all right let's get back to that question how in the world did I place that first zero where I did well unfortunately there isn't a hard fast rule for placing the first zero it is

possible to place the zero too far to the left and that's because you need to generate at least enough negative angle to bring the sum back down to 180° also you can move the zero too far to the right if you do that you're bringing the closed loop pole in the compensator along with it and then you've messed up your D dominant pole so in this case you have two poles exactly where you want them to be but you also have this third pole that's closer to the imaginary axis and that acts as your dominant

pole so the system isn't going to behave the way that you predicted it to so we have a leftmost bound where you need a certain amount of negative angle and you have a rightmost bound where you don't want your compensator interfering with your dominant pole so a general rule of thumb is just to place the zero at or to the left of the second real axis open Loop pole this however doesn't guarantee that your overshoot requirement will be met so this is one of the places where some iteration might be required okay just a few

more things to note when you design a lead compensator this way it doesn't ensure that the system's going to be stable only that your desired pole locations do in fact exist on the root Locus it is possible that with higher order systems another pole is wandered over to the right half plane and made the system go unstable also this might be something that you want to try if you can't get a PID controller to work I tend to think PID is a little bit more intuitive and so I usually start there and then go to

a lead lag compensator afterwards and finally there's a trade-off like in any design that a faster responding system which means poles are more in the left half plane has the disadvantage of responding to noise also and other high frequency inputs and it's difficult to model highfrequency modes accurately so it's hard to test without actually testing it on the real system but now that you know the benefits and drawbacks of using a phase lead compensator you should be better equipped to determine whether your system requires one or not in the next video we're going to tackle

this exact same problem but we're going to use the bod plot instead of the root Locust if you guys have any questions or comments leave them below and I'll do my best to answer them don't forget to subscribe so you don't miss any future videos also you can follow me on Twitter at brianb Douglas and as always thanks for watching