A Few Points to Make(Zeros/Poles, Root Locus,

Steady-state error)

Get roles of players rightI noticed the first day that some people had the roles of the different loop components wrong—the plant talking to the actuator or the controller talking directly to the plant. These roles have to be correct for the loop model to be correct.

System poles and zerosThe poles of a system are the values of s that make its denominator 0. The zeros of a system are the values of s that make its numerator 0.

System poles and zerosHow a system responds dynamically depends upon the location of its closed-loop poles, primarily. These can be gotten from the characteristic equation of the closed-loop system.Often, however, it is useful to deal with a system’s open-loop poles and zeros…

Poles and zeros – example

Open-loop poles ats = -1/5, -1/3, -4

Open-loop zeros at s = -1/7

Poles and zeros – example

Poles and zeros – example

Closed-loop zeros:

s = -1/7, -4

Closed-loop poles depend on the value of KP.

If KP = 0, s = -1/5, -1/3, -4

If KP = ∞, s = -1/7

If KP = 1, s = -3.38, -1, -0.158 (from Matlab roots() function)

CL-pole migration with changing KP

As KP changes, closed-loop poles change their location. Thus by changing KP, you can change the way the closed-loop system responds.

Root locusActually this migration of poles is covered in Chapter 7 of my book, which we are going to skip. You can see how the poles migrate quickly by using a Matlab tool called rltool(). “rl” stands for “root locus”. “locus” means “place” in Latin. So the root locus is “where the roots are”. It’s the path that they follow as you increase K.

rltool() exampleTo use rltool(), use the open-loop transfer function. With the previous example

You first create a system variable in Matlab:

>> s = tf(‘s’)

rltool() exampleThen>> gol = (7*s+1)/((3*s+1)*(5*s+1) *(0.25*s+1))Then>> rltool(gol)Try this and see what happens.

Root locusThus we can use the controller to drive the roots where we want them for a particular system response. A few tips:1. The further to the left the closed-loop

poles are, the faster the system is.2. If there are any roots in the right half-

plane, the system is unstable and will blow up.

Root locus3. A system oscillates if it has complex

roots.4. If it has no complex roots, it does not

oscillate.5. A system is no faster than its slowest

pole(s). So the right-most pole(s) generally govern the behavior of the system.

Second-order, closed-loop pole location

and system response

Steady-state error, ess

Sometimes you tell a plant where to go and it doesn’t exactly go there. Why is this?Give cruise-control example.Table 6.1 (use GOL):

Type 0 Type 1 Type 2 Input Kx, Kv, Ka ess constant ess ess constant ess ess constant ess

Step 0 0

Ramp 0

Parabola