PI-Lead Control

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1 Introduction to Proportional-Integral-Lead Controller (PI-Lead) Design

For applications such as Laser Communications a Fast Steering Mirror (FSM) is often often used to get laser light into a fiber optic cable. This often means rejecting high to mid-frequency disturbances.

The rejection of high to mid-frequency requires a high closed loop bandwidth. A PI-Lead controller is a proportional gain in parallel with an integrator; both in series with a lead controller. The proportional gain provides fast error response. The integrator drives the system to a 0 steady-state error. The lead controller provides for better phase margin - less closed-loop peaking and smaller oscillations.

2 Proportional-Integral-Lead Controller (PI-Lead) Details

Figure 1: PI-Lead Examples

The proportional-integral-lead controller is a combination of a PI and Lead controller in series. Eqn. 1 and Eqn. 2 (below) have the math behind the PI-Lead controller. Examples of the controller with different PI zeros are at the right.

The closed loop bandwidth is key to performance. Typically the closed loop bandwidth is 1.2x to 2x the open loop crossover. In order to achieve the closed loop bandwidth an open loop crossover frequency is chosen and the gain of the open loop transfer function is adjusted in order to achieve that crossover. This requires a good plant transfer function as well as a compensator in series. In MATLAB the gain adjustment is

 [mag] = bode(OLsys, xo * (2*pi));
 OLsys = (1/mag) * OLsys;

where

OLsys is the open loop system (LTI object),
xo is the crossover frequency in Hz, and
mag is the magnitude of the open loop system at the crossover frequency.

Finally, system disturbance rejection typically looks like the inverse of the open loop transfer function (the mirror image about the 0 dB line). So an examination of Figure 1 leads to the conclusion that the higher the zero of the PI controller relative to the natural frequency the better the disturbance rejection.

The following sections provide more details on the individual components of this controller.

2.1 Proportional-Integral (PI) Controller Design

Figure 2: PI Controller Example

The Proportional-Integral (PI) Controller is a proportional controller (simple gain LaTeX: k_{p}) and an integrator LaTeX: \left(\frac{k_{i}}{s}\right). Examples of PI controllers with different zeros are on the right.

LaTeX: K\left(s\right)=k_{p} + \frac{k_{i}}{s}



LaTeX: K\left(s\right)=\frac{k_{p}s}{s} + \frac{k_{i}}{s}



LaTeX: K\left(s\right)=\frac{k_{p}s + k_{i}}{s}



LaTeX: K\left(s\right)=k_{p}\frac{s + \frac{k_{i}}{k_{p}}}{s} Eqn. 1



The MATLAB command to create a PI controller is

 PIcomp = @(z) tf([1 z],[1 0]);

2.2 Lead Controller Design

Figure 3: Lead Controller Example

The purpose of the lead controller is to provide better phase margin. Phase margin is determined at the open loop crossover frequency. So the peak phase addition should be at or near the open loop crossover. Examples of Lead controllers with different separations are to the right.

The lead controller is created by the following (see Standard Controller Forms)

LaTeX: K\left(s\right)=\frac{\left(s+a\right)}{\left(s+b\right)} Eqn. 2


where

LaTeX: a < b

The maximum phase addition is at the geometric mean of the lead controller. The amount of phase lead is determined by the separation of the zero, a, and the pole, b. The optimal lead controller is defined by the frequency of the peak phase and the separation of the zero and pole.

The separation factor is defined as

LaTeX: a=\Chi b


where

LaTeX: \Chi is the separation factor.

In order to achieve the maximum phase at the geometric mean the zero and pole are determined with the following equations

LaTeX: a=\frac{f}{\sqrt{\Chi}}


where

LaTeX: f is the desired frequency of the maximum phase in (rad/s).
LaTeX: b=f\sqrt{\Chi}



3 See Also

4 External Sites

Wikipeida article on Laser Communications