Block Diagram Algebra in control system

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.


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,


  • 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.


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


  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.


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


  • 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


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


  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


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,


  • 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.


  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.


  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.


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,


  • 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


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


  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.


  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.