Control System – Compensator Design

July 29, 2018June 17, 2020 Techie Aric 0 Comments

We all know that control systems are designed to perform specific tasks. In order to provide suitable performances, it is required to alter or adjust the control system design.This alteration (adjustment) of a control system is called compensation. The device used for the same is called compensator.

Standard compensators

The three standard compensator designs (apart from gain adjustment) are

Lead compensator
Lag compensator
Lag-lead compensator

Lead compensator

The Lead compensator can be realized by an electric network as shown below:

Control System - Compensator Design

Let us now derive transfer function assuming impedance to be zero and the output load impedance to be infinite.

$V_i(t)\quad =\quad\frac { { R }_{ 1 }i(t)\quad \times \quad \frac { 1 }{ C } \int _{ 0 }^{ t }{ i(t) } }{ { R }_{ 1 }i(t)\quad +\quad \frac { 1 }{ C } \int _{ 0 }^{ t }{ i(t) } } \quad +\quad { R }_{ 2 }i(t)$

The equation for output is :

$V_o(t)\quad =\quad R_2i(t)$

Take Laplace transform of above equations,

$V_i$

(s) = $\frac { { R }_{ 1 }I(s)\quad \times \quad \frac { I(s) }{ sC } }{ { R }_{ 1 }I(s)\quad +\quad \frac { I(s) }{ sC } } \quad +\quad { R }_{ 2 }I(s)$ ,

$V_0$

(s) = $R_2$ I(s) ,

Taking ratio of $V_o$ (s) and $V_i$ (s) to obtain transfer function,

$\frac{V_o(s)}{V_i(s)}$ = $\frac { s{ R }_{ 1 }{ R }_{ 2 }\quad +\quad \frac { { R }_{ 2 } }{ C } }{ s{ R }_{ 1 }{ R }_{ 2 }\quad +\quad \frac { 1 }{ C } ({ R }_{ 1 }+{ R }_{ 2 }) }$ (on simplifying)

Dividing numerator and denominator by $R_1R_2$ and defining ‘ τ = $R_1C$ and α = $\frac{R_2}{R_1 + R_2} < 1$ we get,

$\frac{V_o(s)}{V_i(s)}$ = $\frac{s+(\frac{1}{\tau})}{s+(\frac{1}{\alpha\tau})}$ ,

Thus the transfer function is given as

$G_c$ = $\frac{s+(\frac{1}{\tau})}{s+(\frac{1}{\alpha\tau})}$ ,

Now for the equation of phase, we replace s by jω.

$G_c$ (jω) = $\frac{j\omega+(\frac{1}{\tau})}{j\omega+(\frac{1}{\alpha\tau})}$

Hence , phase = ∠ $G_c$ (jω) = $\frac { \tan ^{ -1 }{ \omega \tau } }{ \tan ^{ -1 }{ \omega \tau \alpha } }$ = $\tan ^{ -1 }{ \omega \tau }$ – $\tan ^{ -1 }{ \omega \tau \alpha }$ ,

Since $\alpha$ < 1 , the phase of (jω) will be positive for all ω $\ge$ 0 . Thus this network is called a phase lead network.

Advantages of lead compensator

Increases system bandwidth.
It improves the dynamic response of the system.

Now , next we shall study about Lag compensators.

Lag compensators

A lag compensator can be realized by an electrical network shown below:

Control System - Compensator Design

Now we will derive the transfer function of lag network:

On applying KVL, we can obtain input and output equations,

$V_i$ (t) = $R_2$ i(t) + $R_1$ i(t) + $\frac { 1 }{ C } \int _{ 0 }^{ t }{ i(t) }$ dt ,

The equation for output is

$V_o$ (t) = $R_2$ i(t) + $\frac { 1 }{ C } \int _{ 0 }^{ t }{ i(t) }$ dt

Taking Laplace transform of equations,

$V_i$ (s) = $R_2$ I(s) + $R_1$ I(s) + $\frac{I(s)}{sC}$ ,

$V_o$ (s) = $R_2$ I(s) + $\frac{I(s)}{sC}$

Taking ratio of output and input

$\frac{V_0(s)}{V_i(s)}$ = $\frac{R_2I(s) + \frac{I(s)}{sC}}{R_2I(s) + R_1I(s) + \frac{I(s)}{sC}}$

Now, defining τ = $R_2C$ and $\gamma$ = $\frac{R_1 + R_2}{R_2}$ ,

The transfer function for Lag compensator will become,

$G_c$ (s) = $\frac { s\tau +1 }{ s\gamma \tau +1 }$ ,

Now for the equation of phase, we replace s by jω.

$G_c$ (jω) = $\frac { \tau j\omega +1 }{ j\omega \gamma \tau +1 }$ ,

phase = ∠ $G_c$ (jω) = $\frac { \tan ^{ -1 }{ \omega \tau } }{ \tan ^{ -1 }{ \omega \tau \alpha } }$ = $\tan ^{ -1 }{ \omega \tau }$ – $\tan ^{ -1 }{ \omega \tau \gamma }$ ,

since $\gamma\tau > \tau$ , the phase of $G_c$ (jω) will be negative for all ω $\ge$ 0 . Thus this network is called a phase lag network.

Advantages of lag compensator

Improves steady state accuracy
Suppresses high frequency noise

Lag – lead Compensator

The lag – lead network can be realized by a single electric network (refer figure shown below). It is basically a combination of both lag and lead network.

Control System - Compensator Design

Now we will find out the transfer function of the above network as shown below:

We first apply KVL to obtain input and output equations:

$V_i$ (t) = $\frac { { R }_{ 1 }i(t)\quad \times \quad \frac { 1 }{ { C }_{ 1 } } \int _{ 0 }^{ t }{ i(t)dt } \quad }{ { { R }_{ 1 }i(t)\quad +\quad \frac { 1 }{ { C }_{ 1 } } \int _{ 0 }^{ t }{ i(t)dt } } } \quad +\quad { R }_{ 2 }i(t)\quad +\quad \frac { 1 }{ { C }_{ 2 } } \int _{ 0 }^{ t }{ i(t)dt }$ ,

And The equation for output is

$V_o$ (t) = $R_2$ i(t) + $\frac { 1 }{ { C }_{ 2 } } \int _{ 0 }^{ t }{ i(t)dt }$ ,

Now we need to take Laplace transform for both the above equation:

$V_i$ (s) = [ $\frac { { R }_{ 1 }\quad \times \quad \frac { 1 }{ s{ C }_{ 1 } } }{ { R }_{ 1 }\quad +\quad \frac { 1 }{ s{ C }_{ 1 } } } \quad +\quad { R }_{ 2 }\quad +\quad \frac { 1 }{ s{ C }_{ 2 } }$ ]I(s) ,

And $V_o$ (s) = [ $R_2 + \frac{1}{sC_2}$ ] I(s) ,

Taking ratio of output and input equations,

$\frac{V_0(s)}{V_i(s)}$ = $\frac { R_{ 2 }+\frac { 1 }{ sC_{ 2 } } I(s) }{ [\frac { { R }_{ 1 }\quad \times \quad \frac { 1 }{ s{ C }_{ 1 } } }{ { R }_{ 1 }\quad +\quad \frac { 1 }{ s{ C }_{ 1 } } } \quad +\quad { R }_{ 2 }\quad +\quad \frac { 1 }{ s{ C }_{ 2 } } ]I(s) }$ ,

On solving and simplifying we get ,

$\frac{V_0(s)}{V_i(s)}$ = $\frac{(sR_2C_2 + 1)(sR_1C_1 + 1)}{(s^2C_1C_2R_1R_2 + sR_2C_2 + sR_1C_1 + sR_1C_2 + 1)}$ ,

Dividing by $R_1R_2C_1C_2$ to both numerator and denominator we get,

$\frac{V_0(s)}{V_i(s)}$ = $\frac { (s+\frac { 1 }{ { R }_{ 1 }{ C }_{ 1 } } )(s+\frac { 1 }{ { R }_{ 2 }{ C }_{ 2 } } ) }{ { s }^{ 2 }\quad +\quad (\frac { 1 }{ { R }_{ 1 }{ C }_{ 1 } } +\frac { 1 }{ { R }_{ 2 }{ C }_{ 1 } } +\frac { 1 }{ { R }_{ 2 }{ C }_{ 2 } } )s+(\frac { 1 }{ { R }_{ 1 }{ R }_{ 2 }{ C }_{ 1 }{ C }_{ 2 } } ) }$ ,

Now defining $R_1C_1$ = $\tau_1$ , $R_2C_2$ = $\tau_2$ ,

$R_1R_2C_1C_2$ = $\alpha\gamma\tau_1\tau_2$ ,

And $(\frac{1}{R_1C_1}+\frac{1}{R_2C_1}+\frac{1}{R_2C_2})$ = $\frac{1}{\gamma\tau_1}+\frac{1}{\alpha\tau_2}$ ,

We get the transfer function of lag – lead compensator as :

$G_c(s)$ = $\frac{(s+\frac{1}{\tau_1})(s+\frac{1}{\tau_2})}{(s+\frac{1}{\gamma\tau_1})(s+\frac{1}{\alpha\tau_2})}$ ; $\gamma$ > 1 ,

Design of Compensators using Bode plot:

For design purpose, it is best to work with Bode plot.We know that if open loop gain is altered , then the magnitude curve further moves down or up without altering the slope of the given curve and also the phase curve will remains as it is. The common approach to design compensators using bode plot involves following steps:

Plot the magnitude and phase curve of the uncompensated open loop system.
Check the specification of the phase margin and gain margin.
If the specifications of GM and PM are not satisfied then use a suitable compensator.
Determine reshaped open-loop transfer function using the selected compensator.
Check other requirements to meet the specification .
Plot Bode plot using reshaped open-loop transfer function.

Rate feedback Compensation

Root locus helps us to point out the capability of rate feedback . Rate feedback is given by the equation:

$H_d$ (s) = $k_{\tau}$ t

You may refer the below diagram:

Control System - Compensator Design

This feedback system provides the ability to vary three parameters namely : A, k and $k_{\tau}$ . For achieving better-specified performance, there must be a large number of variables inserted into a system. Design will become much more complex. There are two design methods for the design of the control system with rate feedback. One based on the single point loci and other on multiple parameter loci.

A tachometer is one of the rate feedback devices. And the other one is rate gyro. Rate gyro is commonly used in aircraft autopilot system. Velocity feedback using a tachometer is very commonly used in position servo systems.

Techie Aric

Aric is a tech enthusiast , who love to write about the tech related products and ‘How To’ blogs . IT Engineer by profession , right now working in the Automation field in a Software product company . The other hobbies includes singing , trekking and writing blogs .

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Techie Aric

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