Block Diagram Algebra

Hello friends, in this blog article, we will learn Block diagram algebra in the control system. It will include block diagram reduction rules, some block diagram reduction examples and solutions.

We know that the input-output behavior of a linear system is given by its transfer function: G(s)=C(s)/R(s)

where R(s) = Laplace transform of the input variable and C(s) is Laplace transform of the output variable

A convenient graphical representation of such a linear system (transfer function) is called Block Diagram Algebra. 

A complex system is described by the interconnection of the different blocks for the individual components. Analysis of such a complicated system needs simplification of block diagrams by the use of block diagram algebra. Below table showing some of the rules for Block Diagram Reduction.

Block Diagram Reduction Rules

Block diagram reduction rules help you to minimize the block diagram thus solving the equations quickly. Below table represents block diagram reduction rules in the control system

Using the above rules you have to follow below simple steps to solve the block diagrams:

1. Combine all cascade blocks
2. Combine all parallel blocks
3. Eliminate all minor (interior) feedback loops
4. Shift summing points to left
5. Shift takeoff points to the right
6. Repeat steps 1 to 5 until the canonical form is obtained

Block diagram reduction examples

Now we will see some block diagram reduction examples. We will start with some simple examples and then will solve a few complex ones.

Example 1: In the below example, all the three blocks are in series (cascade). We just need to multiply them as G1(s)×G2(s)×G3(s).

 

Example 2: In this example, two blocks are in parallel but there is one summing point as well.

 

Example 3: Solve the below block diagram

Example 4: Simplify the block diagram shown in Figure below.

Solution:

Step 1: Moved H2 before G2

Step 2: H1 and G2 are in parallel, thus added them as below

Step 3: (H1+G2) and G3 are in series, thus multiplied them

Step 4: Moved takeoff point 2 after G3(G2+H1)

Step 5: Minimizing parallel block with a feedback loop

Step 6: Finally, we will get the minimized equation as below

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How PID Controller Works

Time is most precious thing human can ever have since we do not have the ability to reutilize it. Being human we always want work to be done automatically, accurately and timely, to do so we need an optimal control strategy which should have command over all tenses of time i.e. Present + Past + Future. One of such control strategy is PID Controller.

How PID Controller Works?

This mode of the controller is a complex combination of proportional-integral-derivative control modes.  PID control mode possesses zero steady state error, oscillations, and high stability. By the addition of the derivative term to PI control mode helps to reduce overshoot, reduce settling time as well as becomes capable of handling sluggish and fast dynamics higher order processes. In this mode, integral terms try to stabilize the lightly damped system, usually, only PD control mode can not do it easily.

Mathematically this is represented as,

Where,

  • P = PD controller’s output
  • KP = Proportional Gain
  • KI = Integral Gain (=1/Integral Time (Ti))
  • Ki = KP / Ti
  • ­KD= Derivative Gain (=Derivative Time (Td))
  • Kd = KP x Td
  • ep (t)= Desired Value of controlled variable – Measured Value
  • P­I(0) = Integral term initial value

Since, PID control mode can be utilized in many different ways as shown in above equation, which actually helps to define tunable parameters of PID controller.

Applications

PID controller has many industrial as well as domestic applications.
The example we are going to consider here is “maintaining the position of booster rocket at the time of taking off”. To replicate this problem in simplified terms, let’s consider launch pad as a cart and rocket as an inverted pendulum. Now, this a classic example of runaway process, i.e. pinch to the pendulum in normal condition will result in instability of the overall system. Such a system either only PI or only PD controller can not stabilize since one can not handle sudden disturbance and another can not handle initial instability. PID controller maintains it’s position by eliminating steady-state error and predicting error trend.

PID Controller

How to tune a PID Controller: Loop Tuning

Please visit this article to get more understanding on PID Controller Tuning.

Advantages: Why PID controller is used?

  1. No steady state error
  2. Improves overall stability of system
  3. Ability to handle nonlinear higher order unstable system such as CSTR runaway processes
  4. Accurate and quick desired value tracking

Disadvantages

  1. Due to its linear nature, it shows poor performance for hysterically nonlinear processes. (Such as HVAC system)
  2. To avoid high-frequency noise accentuation from derivative term we need to add the low pass to the measured value of the controlled variable.
  3. Single PID controller can control only one variable, hence not suitable for coupled system such as quadcopter.
  4. With fixed tuning of parameters, it can not handle processes which has variable dead time.

Video

Please watch below video for more understanding on PID Controller in a simple way.

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PD Controller (Proportional-Derivative) Controller in Control System

How does a PD controller work?

Prediction of the behavior of error will always result in better stability. In order to avoid effects of the sudden change in load, the derivative of the error signal taken in this mode to predict the trend of a controlled variable. So let us see in detail, how does PD Controller work.

Almost all physical processes have transportation lag (Dead Time) in their system (usually due to improper allocation of the sensor) since only proportional controller’s output will react after some time to sudden change in load and which may result in a huge transient error.  But, with the addition of a derivative controller, this mode becomes capable of predicting error with consideration of dead time. So that, sudden jerks or spike signals are not given to actuator, hence improves the life span of actuators.

Mathematically this is represented as,

PD Controller

Where,

  • P = PD controller’s output
  • KP = Proportional Gain
  • ­KD= Derivative Gain
  • ep (t)= Desired Value of controlled variable – Measured Value
  • P­(0) = Controller’s output when error is zero

From the equation, we can say that this mode cannot eliminate the steady state error of proportional controller. However, It can handle fast process load changes as long as the load change steady state error is acceptable.

Applications of PD Controller

Maintaining a level of liquid inside the tank is a sluggish and integrating process and many cases due to improper allocation of level sensor (in this case which is measured as a function of flow) result into the significant addition of transportation lag. PD controller mode has the capability to predict future of error, hence the effect of additional dead time is reduced. A sudden change in desired value of level will result in high overshoot in the case of PI control mode, but in the case of PD control mode, this integrating effect will be reduced by addition derivative term with the proportional term.

PD Controller

Advantages

  1. Overall stability of system improves
  2. Capable of handling processes with time lag
  3. Reduces settling time by improving damping and reducing overshoot

Disadvantages

  1. Not suited for fast responding systems which are usually lightly damped or initially unstable.
  2. Amplifies noise at higher frequencies which result in improper handling of actuators.
  3. Does not eliminate steady state error

Video

Please watch below video for more understanding on PD Controller:

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PI Controller (Proportional Integral) Controller in Control System

How does PI Controller Work?

The control mode of PI Controller has a one-to-one correspondence of the proportional mode as well as the integral mode which eliminates that inherent offset. This controller is mostly used in areas where the speed of the system is not an issue. Since proportional controller can not provide new nominal controller’s output in case of new load condition, but in this new configuration necessity of fixed (offset) error has been replaced by the accumulation of error term i.e integral term. Mathematically, this can be represented as,

Where,

  • P = PI controller’s output
  • KP = Proportional Gain
  • ­KI= Integral Gain
  • ep (t)= Desired Value of controlled variable – Measured Value
  • I(0) = Integral term initial value

Form PI controller’s equation we can say that when an error is zero, but the controlled variable is oscillating about desired value, then integral action tries to eliminate error and reaches desired value.

When an error is not zero and only accumulated error is not sufficient for resulting in the quick ramp up, in that case, the proportional controller reduces rise time and tries to achieve optimal controller’s output at new load conditions.

Applications of PI Controller

Flow control of any liquid is a dynamic process, improper prediction of error might result in control value saturation or extended flow of liquid which usually happens when we apply derivative controller to such a system.

In this case, the Proportional controller gives proper ramp up to achieve desired value quickly as well as the occurrence of offset error or steady state error about desired value has been eliminated by the integral term.

Advantages

  1. Desired value can be achieved accurately.
  2. Ease to apply for fast response processes as well as processes in which load change is large and frequent.
  3. Removes steady state error.

Disadvantages

  1. The speed of response of system becomes sluggish due to the addition of integral term.
  2. During start-up of a batch process, the integral action causes an overshoot.
  3. Since PI controller doesn’t have the ability to predict the future errors of the system, therefore it cannot eliminate steady state oscillations and reduces settling time. Hence, overall stability system is comparatively low.

Video

Please watch below video for more understanding on PI Controller. In this video, they have explained how we can eliminate steady state error using PI Controller.

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Derivative Controller in Control System

How does Derivative Controller work?

With an integral controller, we can calculate accumulated error, but with the derivative control, we can calculate the ratio of error change per unit time, hence act as a predictor. Derivative controller action responds to the rate at which the difference between desired value and the measured value is changing that is derivative of the error.  Mathematically represented as below,

Where,

  • KD­ = Derivative gain
  • The derivative controller is also known as Rate action controller or anticipatory controller.
  • ep (t)= Desired Value of controlled variable – Measured Value

Applications

The derivative controller is not used alone because it provides no output when the error is constant.

Advantages

  1. Effect of transportation lag occurred due to the remote allocation of the sensor can be minimized.
  2. Accumulation of error which will go to affect on actuator saturation can be minimized.
  3. Peak errors are minimized.

Disadvantages

  1. Cannot be used when an error is constant. (Derivative of constant value is zero).
  2. A small change in error will affect largely on controller’s output. The high derivative gain will result in heavy overshoots and overall system’s stability.

Proportional controller In Control System

The proportional controller gives control signal in one to one correspondence with an error over some range. Each value of error about desired value will have corresponding controller output like controller output values are interpolated with respect to error values.

Mathematically this can be represented as

Where,

  • P = Controller’s output
  • ep (t)= Desired Value of controlled variable – Measured Value
  • Kp = Proportional gain
  • P0 = Controller output with no error

It indicates that, if the error is zero, the output is constant and equal to P0 and if there is an error, for every 1 % error, a correction of Kp % is added to or subtracted from P0, depending on the sign of error.

Application of proportional controller

Thermostat used in room temperature control expands or compress it’s bimetallic spring as per temperature variation in the room and gives corresponding control signal to heater coil.

Another example is heat exchanger system,

  • If there is sudden large flow of cold fluid enters into heat exchanger, since TT (temperature transmitter) detects this sudden change then,
  • TC (Temperature controller) compares measured value of temperature with desired temperature value,
  • a corresponding error is also large and if we multiply it with proper gain (proportional gain) and subtract from P0,
  • Then control signal gradually opens steam flow through the control valve and maintains the temperature inside the heat exchanger.

proportional controller

Advantages and disadvantages of proportional controller

Advantages:

  1. Processes with slower dynamics are controlled effectively.
  2. Helps to stabilize higher order processes which have transient instability.

Disadvantages:

  1. The residual error is always required to maintain the desired value of the controlled variable.

E.g. If the controller’s output will increase with falling outlet temperature of the heat exchanger until there is enough steam flow admitted to the heat exchanger to prevent the temperature from falling any further. But in order to maintain this greater flow rate of steam (for greater heating effect), an error must develop between the measured temperature and desired temperature. In other words, the process variable (temperature) must deviate from desired value in order for the controller to call for more steam, in order that the process variable does not fall any further than this. This necessary error between the measured value and the desired value is called ‘offset’ or residual error.

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Types of Controllers in Process Control System

Hello, friends in this article we are going to learn different types of controllers in control system. Classification of controllers is nothing but the different control modes in process control. We will learn different types of controllers in process control along with examples.

What is Controller?

In this universe, there are several types physical changes exists, among those changes, which are under our bounded observation is known as process and device by which we can obtain a desired response from that process is commonly known as the controller.

Types of the controllers

Types of Controllers in Process Control System
Types of Controllers in Process Control System

1. Discontinuous Controller

A. Two position mode controller

Working

Historically, two position controller was abundantly used among the controllers, since it has only two possible positions i.e. 0% or 100%. A simple example of this type of controller is a relay.

Whenever the measured value of the controlled variable is less than desired value i.e. potential difference between two terminal of the coil of relay then, normally open contact gets closed or normally closed contact gets opened.

Similarly, if the measured value is equal to desired value then, there is no change in state.

Applications

Room heater, if the temperature of the room goes below the desired temperature then, the heater turns ON and if the temperature is above the desired temperature then heater turns OFF.

Advantages

  1. Simple
  2. Easily adaptable to large-scale systems which have slower process rates

Disadvantages

  1. Neutral zone exists, due to which though the difference between measured and desired value exists, there is no change in controller output.

B. Floating Position Mode Controller

a. Single Speed floating Position Mode Controller

Working

This controller’s output changes at a fixed rate when the difference between desired value and the measured value exceeds neutral zone. Mathematically, it can be represented as,

Where,

  • dp/dt = rate change of controller output with respect to time
  • KF= rate constant
  • ∆ep = half of the neutral zone

By integrating equation (1) we get,

Where p(0) is initial controller’s output.

This indicates that current controller’s output keeps a history of previous control outputs. In many cases, such a kind of information is not available.

b. Multiple Speed floating mode controller

Unlike single speed controller, it’s ‘KF’ values increases or decreases as per deviation exceeds certain limits. It means that for large error (|desired value – measured value|) will have large ‘KF’ value and vice versa.

Application

In self-regulating processes such as liquid flow rate control in the pipe (as shown in below diagram) a single speed floating controller is used. The load is determined by the inlet and outlet pressures Pin and Pout, and the flow is determined in part by the pressure P, within the DP cell and control valve. This is an example of a system with self-regulation. We assume that small control valve opening has been found to maintain the desired flow rate inside the pipe. If larger than the neutral zone, the valve begins to open or close at a constant rate until an opening is found that supports the proper flow rate at the new load conditions. Clearly, the rate is very important, because especially fast process lags cause the valve to continue opening or closing beyond that optimum self-regulated position.

Advantages

  1. Effect of neutral zone can be minimized.
  2. Overshoots and undershoots are reduced compare to two position mode
  3. Gives better performances for self-regulating lower order small dead time systems.

Disadvantages

  1. When applied to large-scale systems such as room temperature control, this type of controller shows inevitable cycling i.e. measured value will fluctuate around desired value for a long time.

In upcoming articles, we will learn about Continuous controllers, till then if you like this article, share it with your friends, like our Facebook page and subscribe to our newsletter for future updates. Have a nice day!

How to draw root locus graph with simple steps

We know that the stability of the given closed loop system depends upon the location of the roots of the characteristics equation. That is the location of the closed loop poles. If we change some parameter of a system, then the location of closed loop pole changes in ‘s’ plane. The study of this locus (of moving pole in ‘s’ plane) because of variation of any parameter of the system is very important while designing any closed loop system.

This movement of poles in ‘s’ plane is called as ‘Root Locus’.

This method was invented by W.R. Evans in 1948.

Root Locus is a simple graphical method for determining the roots of the characteristic equation. It can be drawn by varying the parameter (generally gain of the system but there are also other parameters that can be varied) from zero to infinity.

Root Locus Method with step by step solution

General steps for drawing the Root Locus of the given system:

  1. Determine the open loop poles, zeros and a number of branches from given G(s)H(s).
  2. Draw the pole-zero plot and determine the region of real axis for which the root locus exists. Also, determine the number of breakaway points (This will be explained while solving the problems).
  3. Calculate the angle of asymptotes.
  4. Determine the centroid.
  5. Calculate the breakaway points (if any).
  6. Calculate the intersection point of root locus with the imaginary axis.
  7. Calculate the angle of departure or angle of arrivals if any.
  8. From above steps draw the overall sketch of the root locus.
  9. Predict the stability and performance of the given system by the root locus.

Solved Problem on Root Locus:

Question: For a unity feedback system, G(s) = K/[s(s+4)(s+2)]. Sketch the nature of root locus showing all details on it. Comment on the stability of the system.

Solution:

Given system is unity feedback system. Therefore H(s) = 1.

Therefore G(s) H(s) = K/[s(s+4)(s+2)].

Step 1:

Poles = 0, -4, -2. Therefore P=3.

Zeros = there are no zeros. Z=0.

So all (P-Z=3) branches terminate at infinity.

Step 2: Pole-zero plot and sections of the real axis.

The pole-zero plot of the system is shown in the figure below. Here RL denotes Root Locus existence region and NRL denotes the non-existence region of root locus.

These sections of real axis identified as a part of the root locus as to the right sum of poles and zeros is odd for those sections.

Step 3: Angle of asymptotes

‘A line to which root locus touches at infinity is called asymptotes.’

Number of asymptotes = P-Z = 3. Therefore 3 asymptotes are approaching to infinity.

Step 4: Centroid or Centre of asymptotes.

Asymptote touches real axis at a point called centroid.

Branches will approach infinity along these lines which are asymptotes.

centroid 2

Step 5: To find a breakaway point, we have the characteristic equation as,

As there is no root locus between -2 to -4, -3.15 can not be a breakaway point. it also can be confirmed by calculating ‘K’ for s = -3.15. It will be negative that confirms s = -3.15 is not a breakaway point.

For s = -3.15, K = -3.079 (Substituting in equation for K). But as there has to be breakaway point between’0’and ‘-2’, s = – 0.845 is a valid breakaway point.

For s = -0.845 K = +3.079. As K is positive s = – 0.845 is valid breakaway point.

Step 6: Intersection point with the imaginary axis.

Characteristic equation

s^3+6s^2 +8s+K = 0

Routh’s array:

Intersection of root locus with imaginary axis is at ±j2.828 and corresponding value of K(marginal) = 48.

Step 7: As there are no complex conjugate poles or zeros, no angles of departures or arrivals are required to be calculated.

Step 8: The complete root locus is as shown in the figure below.

Step 9: Prediction of stability:

For 0 < K < 48, all the roots are in left half of s-plane hence the system is absolutely stable.

For K(marginal) = +48, a pair of dominant roots on imaginary axis with remaining root in left half. So the system is marginally stable oscillating at 2.82 rad/sec. For 48 < K < 8, dominant roots are located in right half of s-plane hence system is unstable.

Stability is predicted by locations of dominant roots. Dominant roots are those which are located closest to the imaginary axis.

MATLAB Program for Root Locus Method:

%AIM: To plot the root locus of given transfer function
%Created by : myclassbook.org
clc;
clear all;
close all;
s=tf('s');
k=1;
G=k/(s*(s+4)*(s+2))
rlocus(G)
Root locus method Matlab program
Root locus method Matlab program
Root locus method matlab program 1
Root locus method Matlab program

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Effects of addition of poles and zeros to closed loop transfer function

Effect of addition of pole and zero to closed loop transfer function

Aim:

Study and plot unit step response of addition of pole and zero to the closed loop transfer function for a unity feedback system. Plot the response for four different values of poles and zeros. Comment on the effect of addition of poles and zeros to closed loop transfer function of a system.

Theory:

Addition of pole to closed loop transfer function:

The closed loop transfer function of general second order system is given by,

1

Addition of pole to closed loop transfer function:

When we add a pole, the transfer function becomes,

2

MATLAB PROGRAM:

addition of pole to closed loop transfer function
addition of pole to closed loop transfer function
addition of pole to closed loop transfer function
addition of pole to closed loop transfer function

Effect of addition of pole to closed loop transfer function:

1)      As the pole moves towards the origin in s plane, the rise time increases and the maximum overshoot decreases, as far as the overshoot is concerned, adding a pole to the closed loop transfer function has just the opposite effect to that of adding a pole to forward path transfer function as discussed in the last article.

2)      The addition of left half pole tends to slow down the system response.

3)      The effect of addition of pole becomes more pronounced as pole location drifts away from imaginary axis.

4)      Addition of right half pole will make overall system response to be an unstable one.

Addition of zero to closed loop transfer function:

Suppose a zero at s=-z is added to closed loop transfer function, then the resultant transfer function becomes,

3

MATLAB PROGRAM:

addition of zero to closed loop transfer function 1
addition of zero to closed loop transfer function 1
addition of zero to closed loop transfer function 1
addition of zero to closed loop transfer function 1

Conclusion:

Effect of addition of zero to closed loop transfer function

1)      Makes the system overall response faster.

2)      Rise time, peak time, decreases but overshoot increases.

3)      Addition of right half zeros means system response slower and system exhibits inverse response. Such systems are said to be non-minimum phase systems.

Minimum phase system: The system which doesn’t have zeros in right half of s plane is said to be minimum phase system.

Non-minimum phase system: If a transfer function has poles or zeros in right half of s plane then the system shows the non-minimum phase behavior.

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Addition of poles and zeros to the forward path transfer function

Effect of addition of pole and zero to forward path transfer function

Aim:

Study and plot unit step response of addition of pole and zero to the forward path transfer function for a unity feedback system. Plot the response for four different values of poles and zeros. Comment on the effect of addition of poles and zeros to forward path transfer function of a system.

Theory:

Addition of pole to forward path transfer function:

The forward path transfer function of general second order system is given by,

forward path transfer function 123

Addition of pole to forward path transfer function:

When we add a pole, the transfer function becomes,

forward path transfer function 1234

MATLAB PROGRAM:

addition of pole to forward path transfer function 1
addition of pole to forward path transfer function 1
addition of pole to forward path transfer function 1
addition of pole to forward path transfer function 1

Conclusion:

Effect of addition of pole to forward path transfer function:

1)      Unit step response of closed loop system when Wn=10, zeta=0.5, P=[1,5,10,20]. These responses show that addition of pole to forward path transfer function generally has the effect of increasing maximum overshoot of closed loop system.

2)      As the pole moves closer to the origin in s plane, the maximum overshoot increases.

3)      These responses also show that added pole increases rise time of step response.

Addition of zero to forward path transfer function:

The forward path transfer function of general second order system is given by,

forward path transfer function 123

Addition of zero to forward path transfer function:

When we add a zero the forward path transfer function becomes,

forward path transfer function 12

MATLAB PROGRAM:

addition of zero to forward path transfer function 1
addition of zero to forward path transfer function 1
addition of zero to forward path transfer function 1
addition of zero to forward path transfer function 1

Conclusion:

Effect of addition of zero to forward path transfer function:

1)      Addition of forward path zeros to left hand side of s plane increases speed of output response but a large overshoot may result if the zero is sufficiently lose to 0.

2)      The order of system does not increase with addition of zero to forward path transfer function.

3)      The zero in right half of s plane retards the system and produces an overshoot. The percentage overshoot decreases as the zero moves along positive real axis towards infinity. Again system oscillates for a large time.

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