# Control Systems/Controllers and Compensators

## Controllers

There are a number of different standard types of control systems that
have been studied extensively. These controllers, specifically the P,
PD, PI, and PID controllers are very common in the production of
physical systems, but as we will see they each carry several drawbacks.

## Proportional Controllers

center\|framed\|A Proportional controller block
diagram

Proportional controllers are simply gain values. These are essentially
multiplicative coefficients, usually denoted with a *K*. A P controller
can only force the system poles to a spot on the system\'s root locus. A
P controller cannot be used for arbitrary pole placement.

We refer to this kind of controller by a number of different names:
proportional controller, gain, and zeroth-order controller.

## Derivative Controllers

center\|framed\|A Proportional-Derivative controller block
diagram

In the Laplace domain, we can show the derivative of a signal using the
following notation:

$$D(s) = \mathcal{L} \left\{ f'(t) \right\} = sF(s) - f(0)$$

Since most systems that we are considering have zero initial condition,
this simplifies to:

$$D(s) = \mathcal{L} \left\{ f'(t) \right\} = sF(s)$$

The derivative controllers are implemented to account for future values,
by taking the derivative, and controlling based on where the signal is
going to be in the future. Derivative controllers should be used with
care, because even small amount of high-frequency noise can cause very
large derivatives, which appear like amplified noise. Also, derivative
controllers are difficult to implement perfectly in hardware or
software, so frequently solutions involving only integral controllers or
proportional controllers are preferred over using derivative
controllers.

Notice that derivative controllers are not proper systems, in that the
order of the numerator of the system is greater than the order of the
denominator of the system. This quality of being a non-proper system
also makes certain mathematical analysis of these systems difficult.

### Z-Domain Derivatives

We won\'t derive this equation here, but suffice it to say that the
following equation in the Z-domain performs the same function as the
Laplace-domain derivative:

$$D(z) = \frac{z - 1}{Tz}$$

Where T is the sampling time of the signal.

## Integral Controllers

center\|framed\|A Proportional-Integral Controller block
diagram

To implemenent an Integral in a Laplace domain transfer function, we use
the following:

$$\mathcal{L}\left\{ \int_0^t f(t)\, dt \right\} = {1 \over s}F(s)$$

Integral controllers of this type add up the area under the curve for
past time. In this manner, a PI controller (and eventually a PID) can
take account of the past performance of the controller, and correct
based on past errors.

### Z-Domain Integral

The integral controller can be implemented in the Z domain using the
following equation:

$$D(z) = \frac{z + 1}{z - 1}$$

## PID Controllers

!A block diagram of a PID
controller

PID controllers are combinations of the proportional, derivative, and
integral controllers. Because of this, PID controllers have large
amounts of flexibility. We will see below that there are definite
limites on PID control.

{{-}}

### PID Transfer Function

The transfer function for a standard PID controller is an addition of
the Proportional, the Integral, and the Differential controller transfer
functions (hence the name, PID). Also, we give each term a gain
constant, to control the weight that each factor has on the final
output:

$$D(s) = K_p + {K_i \over s} + K_d s$$

Notice that we can write the transfer function of a PID controller in a
slightly different way:

$$D(s) = \frac{A_0 + A_1s}{B_0 + B_1s}$$

This form of the equation will be especially useful to us when we look
at polynomial design.

### PID Signal flow diagram

!Signal flow diagram for a PID
controller

### PID Tuning

The process of selecting the various coefficient values to make a PID
controller perform correctly is called **PID Tuning**. There are a
number of different methods for determining these values:[^1]

1\) Direct Synthesis (DS) method

2\) Internal Model Control (IMC) method

3\) Controller tuning relations

4\) Frequency response techniques

5\) Computer simulation

6\) On-line tuning after the control system is installed

7)Trial and error

**Notes:**

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### Digital PID

In the Z domain, the PID controller has the following transfer function:

$$D(z) = K_p + K_i \frac{T}{2} \left[ \frac{z + 1}{z - 1} \right] + K_d \left[ \frac{z - 1}{Tz} \right]$$

And we can convert this into a canonical equation by manipulating the
above equation to obtain:

$$D(z) = \frac{a_0 + a_1 z^{-1} + a_2 z^{-2}}{1 + b_1 z^{-1} + b_2 z^{-2}}$$

Where:

$$a_0 = K_p + \frac{K_i T}{2} + \frac{K_d}{T}$$

$$a_1 = -K_p + \frac{K_i T}{2} + \frac{-2 K_d}{T}$$

$$a_2 = \frac{K_d}{T}$$

$$b_1 = -1$$

$$b_2 = 0$$

Once we have the Z-domain transfer function of the PID controller, we
can convert it into the digital time domain:

$$y[n] = x[n]a_0 + x[n-1]a_1 + x[n-2]a_2 - y[n-1]b_1 - y[n-2]b_2$$

And finally, from this difference equation, we can create a digital
filter structure to implement the PID.

## Bang-Bang Controllers

Despite the low-brow sounding name of the Bang-Bang controller, it is a
very useful tool that is only really available using digital methods. A
better name perhaps for a bang-bang controller is an on/off controller,
where a digital system makes decisions based on target and threshold
values, and decides whether to turn the controller on and off. Bang-bang
controllers are a non-linear style of control.

Consider the example of a household furnace. The oil in a furnace burns
at a specific temperature---it can\'t burn hotter or cooler. To control
the temperature in your house then, the thermostat control unit decides
when to turn the furnace on, and when to turn the furnace off. This
on/off control scheme is a bang-bang controller.

## Compensation

There are a number of different compensation units that can be employed
to help fix certain system metrics that are outside of a proper
operating range. Most commonly, the phase characteristics are in need of
compensation, especially if the magnitude response is to remain
constant. There are four major types of compensation 1. Lead
compensation 2. Lag compensation 3. Lead-lag compensation 4. Lag-lead
compensation

## Phase Compensation

Occasionally, it is necessary to alter the phase characteristics of a
given system, without altering the magnitude characteristics. To do
this, we need to alter the frequency response in such a way that the
phase response is altered, but the magnitude response is not altered. To
do this, we implement a special variety of controllers known as **phase
compensators**. They are called compensators because they help to
improve the phase response of the system.

There are two general types of compensators: **Lead Compensators**, and
**Lag Compensators**. If we combine the two types, we can get a special
**Lag-lead Compensator** system.(lead-lag system is not practically
realisable).

When designing and implementing a phase compensator, it is important to
analyze the effects on the gain and phase margins of the system, to
ensure that compensation doesn\'t cause the system to become unstable.
phase lead compensation:- 1 it is same as addition of zero to open loop
TF since from pole zero point of view zero is nearer to origin than pole
hence effect of zero dominant.

## Phase Lead

The transfer function for a lead-compensator is as follows:


$$T_{lead}(s) = \frac{s-z}{s-p}$$

To make the compensator work correctly, the following property must be
satisfied:

$$| z | < | p |$$

And both the pole and zero location should be close to the origin, in
the LHP. Because there is only one pole and one zero, they both should
be located on the real axis.

Phase lead compensators help to shift the poles of the transfer function
to the left, which is beneficial for stability purposes.

## Phase Lag

The transfer function for a lag compensator is the same as the
lead-compensator, and is as follows:

$$T_{lag}(s) = \frac{s-z}{s-p}$$

However, in the lag compensator, the location of the pole and zero
should be swapped:

$$| p | < | z |$$

Both the pole and the zero should be close to the origin, on the real
axis.

The Phase lag compensator helps to improve the steady-state error of the
system. The poles of the lag compensator should be very close together
to help prevent the poles of the system from shifting right, and
therefore reducing system stability.

## Phase Lag-lead

The transfer function of a **Lag-lead compensator** is simply a
multiplication of the lead and lag compensator transfer functions, and
is given as:

$$T_{Lag-lead}(s) = \frac{(s-z_1)(s-z_2)}{(s-p_1)(s-p_2)}.$$

Where typically the following relationship must hold true:

$$| p_1 | > | z_1 | > | z_2 | > | p_2 |$$


## External links

-   Standard Controller Forms on
    ControlTheoryPro.com
-   PID Control on
    ControlTheoryPro.com
-   PI Control on
    ControlTheoryPro.com

[^1]: Seborg, Dale E.; Edgar, Thomas F.; Mellichamp, Duncan A. (2003).
    Process Dynamics and Control, Second Edition. John Wiley & Sons,Inc.
    
