Standard Controller Forms
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## 1 Introduction to Standard Controller Forms

There are several standard controller types and there is no need to reinvent the wheel. This article details the form of proportional, integral, derivative, PID and various other controller forms.

Note: Controllers are not magic. It should not be assumed that every system can be controlled or that desired performance can be achieved for any given system if only the correct controller were employed. Proper system design is always required.

### 1.1 Proportional Action[1]

Proportional action provides an instantaneous response to the control error. This is useful for improving the response of a stable system but cannot control an unstable system by itself. Additionally, the gain is the same for all frequencies leaving the system with a nonzero steady-state error.

### 1.2 Integral Action[2]

Integral action drives the steady-state error towards 0 but slows the response since the error must accumulate before a significant response is output from the controler. Since an integrator introduces a system pole at the origin, an integrator can be detrimental to loop stability. Only controllers with integrators can wind-up where, through actuator saturation, the loop is unable to comply with the control command and the error builds until the situation is corrected.

### 1.3 Derivative Action[3]

Derivative action acts on the derivative or rate of change of the control error. This provides a fast response, as opposed to the integral action, but cannot accomodate constant errors (i.e. the derivative of a constant, nonzero error is 0). Derivatives have a phase of +90 degrees leading to an anticipatory or predictive repsonse. However, derivative control will produce large control signals in response to high frequency control errors such as set point changes (step command) and measurement noise.

In order to use derivative control the transfer functions must be proper. This often requires a pole to added to the controller (this pole is not present in the equations below).

## 2 Proportional Control

Feedback control signal is linearly proportional to the error.

 $LaTeX: C_P\left(s\right)=k_p$ P[4]

where

$LaTeX: k_p$ is the proportional control gain.

### 2.1 Limitations

Higher-order systems can become unstable with large values of $LaTeX: k_p$.[5] Simply turning up the gain will often lead to instability or undesirable ringing - i.e. unacceptable steady-state error.

Also, keep in mind that any noise in the system will get amplified by this same gain. Consider the reference command to be exactly 1 for all time. Then consider a noise sensor output of approximately 1.

$LaTeX: r=1$
$LaTeX: \hat{y}=1\plusmn0.01$
$LaTeX: e=r-\hat{y}=0\plusmn0.01$
$LaTeX: u=k_pe=k_p\left(0\plusmn0.01\right)$

where

$LaTeX: r$ is the reference command,
$LaTeX: \hat{y}$ is the sensor output,
$LaTeX: e$ is the error, and
$LaTeX: u$ is the controller output.

The sensor noise is only 1% of the desired output. However, for a large gain the 1% error becomes substantial.

## 3 Integral Control

An integrator is used to drive the steady-state error towards 0. However, this comes at a cost of worse transient response.[6] Often the transient response degradation comes in the for of a longer rise time and more overshoot.

 $LaTeX: C_I\left(s\right)=\frac{k_I}{s}$ I

where

$LaTeX: k_I$ is the integral control gain and
$LaTeX: s$ is the Laplace variable.

## 4 Derivative Control

Derivative control differentiates the error. With noise present this is usually undesirable. Also, if the error were to remain constant (and nonzero) then derivative control would output a 0 control command effectively turning off the system.

Derivative control can be used in combination with other controllers to increase damping and an anticipatory response (i.e. the phase is +90 deg).

 $LaTeX: C_D\left(s\right)=k_{D}s$ D

where

$LaTeX: k_{D}$ is the integral control gain and

## 5 PI Control

Proportional-Integral or PI control combines proportional control and integral control in parallel.

 $LaTeX: C_{PI}\left(s\right)=K_P\left(1+\frac{1}{T_{r}s}\right)$ PI[7]

where

$LaTeX: K_P$ is the PI control gain and
$LaTeX: T_{r}$ is the reset time.

To Eqn. PI equal to Eqn. I then

 $LaTeX: T_{r}=\frac{k_P}{k_I}$

## 6 PD Control

Proportional-Derivative or PD control combines proportional control and derivative control in parallel.

 $LaTeX: C_{PD}\left(s\right)=K_P\left(1+T_{d}s\right)$ PD[8]

where

$LaTeX: K_P$ is the PID control gain and
$LaTeX: T_{d}$ is the derivative gain or derivative time.

## 7 PID Control

Proportional-Integral-Derivative or PID control combines proportional control, integral control, and derivative control in parallel.

 $LaTeX: C_{PID}\left(s\right)=K_P\left(1+\frac{1}{T_{r}s}+T_{d}s\right)$ PID[9]

where

$LaTeX: K_P$ is the PID control gain,
$LaTeX: T_{r}$ is the reset time, and
$LaTeX: T_{d}$ is the derivative gain or derivative time.

If the 1 is replaced by a gain, k and replace $LaTeX: \frac{1}{T_{r}}=k_i$, then the PID control can also be thought of an all inclusive form of the other controllers.

 $LaTeX: C_{PID}\left(s\right)=K_P\left(k+\frac{k_{i}}{s}+T_{d}s\right)$

From this generic form we can do the following:

set $LaTeX: T_{d}=0$ and Eqn. PID becomes equal to Eqn. PI
set $LaTeX: k_i=0$ and Eqn. PID becomes equal to Eqn. PD
set $LaTeX: k=0$ and Eqn. PID becomes an integral-derivative controller
set $LaTeX: k=0, k_i=0$ and Eqn. PID becomes equal to Eqn. D
set $LaTeX: k=0, T_{d}=0$ and Eqn. PID becomes equal to Eqn. I

PID control is popular since it provides access to all the benefits of all the previously described controllers and through tuning many of the downsides of the previously described controllers can be mitigated.

### 7.1 Alternative forms of PID Control

Standard form similar to Eqn. PID except that a pole has been added to the derivative action to ensure a proper transfer function.

 $LaTeX: C_{PID}\left(s\right)=K_P\left(1+\frac{1}{T_{r}s}+\frac{T_{d}s}{\tau_{D}s+1}\right)$ PID Alt 1[10]

Note: Typically $LaTeX: 0.1T_{d}\le\tau_{D}\le0.2T_{d}$

Series Form

 $LaTeX: C_{PID}\left(s\right)=K_s\left(1+\frac{I_s}{s}\right)\left(1+\frac{D_{s}s}{\gamma_{s}D_{s}s+1}\right)$ PID Alt 2 - Series Form[11]

Parallel Form

 $LaTeX: C_{PID}\left(s\right)=K_p+\frac{I_p}{s}+\frac{D_{p}s}{\gamma_{p}D_{p}s+1}$ PID Alt 3 - Parallel Form[12]

## 8 Lead or Lag Control

Lead and lag control are used to add or reduce phase between 2 frequencies. Typically these frequencies are centered around the open loop crossover frequency. A lead filter typically has unity gain (0 dB) are low frequencies while the lag provides a nonunity gain at low frequencies.[13]

 $LaTeX: C\left(s\right)=\frac{\tau_{1}s+1}{\tau_{2}s+1}$ Lead[14]

where

$LaTeX: \tau_{1}>\tau_{2}$ makes this a lead filter and
$LaTeX: \tau_{1}<\tau_{2}$ makes this a lag filter.

This is an extension of the Lead and Lag network described above. We can always stick a gain in front of something so reformulate $LaTeX: C\left(s\right)$ above to look like this

 $LaTeX: C\left(s\right)=k\frac{\tau_{a}s+1}{\tau_{b}s+1}$

Then a lead-lag controller would have the form

 $LaTeX: C\left(s\right)=k\left(\frac{\tau_{a}s+1}{\tau_{b}s+1}\right)\left(\frac{\tau_{c}s+1}{\tau_{d}s+1}\right)$ [15]

where

$LaTeX: k$ is a gain,
$LaTeX: \tau_{a}>\tau_{b}$ makes the first part a lead filter and
$LaTeX: \tau_{c}<\tau_{d}$ makes the last part a lag filter.

## 10 References

Goodwin, G. C., Graebe, S. F., and Salgado, M. E. 2000 Control System Design. 1st. Prentice Hall PTR. ISBN 0139586539
Franklin, G. F., Emami-Naeini, A., and Powell, J. D. 1993 Feedback Control of Dynamic Systems. 3rd. Addison-Wesley Longman Publishing Co., Inc. ISBN 0201527472

### 10.1 Notes

1. Goodwin et all, pg. 160
2. Goodwin et all, pg. 161
3. Goodwin et all, pg. 161
4. Goodwin et all, pg. 160, Eqn. 6.2.1
5. Franklin et all, pg. 180
6. Franklin et all, pg. 181
7. Goodwin et all, pg. 160, Eqn. 6.2.2
8. extrapolation from Franklin et all, pg. 185, Eqn. 4.31
9. Franklin et all, pg. 185, Eqn. 4.31
10. Goodwin et all, pg. 160, Eqn. 6.2.4
11. Goodwin et all, pg. 160, Eqn. 6.2.5
12. Goodwin et all, pg. 160, Eqn. 6.2.6
13. Franklin et all, pg. 412
14. Goodwin et all, pg. 170, Eqn. 6.6.1
15. Goodwin et all, pg. 170, Eqn. 6.6.1