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11.4 Calculating PI Gains Based On Stability and Bandwidth

At the end of last section, we discussed the possibility of using a single parameter that could help tune the speed PI loop in a motor control system. To develop this parameter, let's review the open-loop transfer function for the entire speed loop:

Equation 49.

Where:

K is the coefficient that contains several terms related to the motor and load
and are the PI coefficients for the speed loop
L is the motor inductance
is one of the PI coefficients for the current loop
s is the Laplace frequency variable

Assuming that the pole at occurs at a higher frequency than the zero at , and that the unity gain frequency occurs somewhere in-between these two frequencies, we should end up with a Bode plot that looks something like Figure 12-7.

Figure 11-7 Bode Plot

The reason the shape of this curve is so important is because the phase shift at the 0 dB frequency determines the stability of the system. In order to achieve maximum phase margin (phase shift: 180°) for a given separation of the pole and zero frequencies, the 0 dB frequency should occur exactly half way in-between these frequencies on a logarithmic scale. In other words,

Equation 50.

and,

Equation 51.

Combining Equation 50 and Equation 51 we can establish that:

Equation 52.

From Equation 49, we see that ω_pole and ω_zero are already defined in terms of the PI coefficients. Therefore,

Equation 53.

Where "δ" we will define as the damping factor. The larger δ is, the further apart the zero corner frequency and the current loop pole will be. And the further apart they are, the phase margin is allowed to peak to a higher value in-between these frequencies. This improves stability at the expense of speed loop bandwidth. If δ = 1, then the zero corner frequency and the current loop pole are equal, which results in pole/zero cancellation and the system will be unstable. Theoretically, any value of δ > 1 is stable since phase margin > 0. However, values of δ close to 1 result in severely underdamped performance.

We will talk more about δ later, but for now, let's turn our attention towards finding the last remaining coefficient: . From Equation 50 we see that the open-loop transfer function of the speed loop from Equation 49 will be unity gain (0 dB) at a frequency equal to the zero inflection point frequency multiplied by δ. In other words,

Equation 54.

By performing the indicated substitution for "s" in Equation 54 and solving, we obtain:

Equation 55.

Finally, we can solve for :

Equation 56.

At this point, let's step back and try to see the forest for the trees. We have just designed a cascaded speed controller for a motor which contains two separate PI controllers: one for the inner current loop and one for the outer speed loop. In order to get pole/zero cancellation in the current loop, we chose as follows:

Equation 57.

sets the bandwidth of the current controller:

Equation 58.

Once we have defined the parameters for the inner loop current controller, we select a value for the damping factor (δ) which allows you to precisely quantify the tradeoff between speed loop stability and bandwidth. Then it is a simple matter to calculate and :

Equation 59.

Equation 60.

The benefit of this approach is that instead of trying to empirically tune four PI coefficients which have seemingly little correlation to system performance, you just need to define two meaningful system parameters: the bandwidth of the current controller and the damping coefficient of the speed loop. Once these are selected, the four PI coefficients are calculated automatically.

The current controller bandwidth is certainly a meaningful system parameter, but in speed controlled systems, it is usually the bandwidth of the speed controller that we would like to specify first, and then set the current controller bandwidth based on that. In the next section, let's take a closer look at the damping factor, and we will come up with a way to set the current loop bandwidth based on the desired speed loop bandwidth.