State space analysis numerical problems solved

State Space Model Numerical PYQs with Solutions

Below are important numerical problems on State Space Model frequently asked in GATE, SSC JE, BEL, ESE, and Polytechnic Lecturer (Electronics) exams.

Numerical 1: State Space Model from Differential Equation

Question:
Obtain the state space model of the system:

d²y/dt² + 4 dy/dt + 3y = u(t)

Solution:

Let the state variables be:

x₁ = y
x₂ = dy/dt

Then,

dx₁/dt = x₂
dx₂/dt = −3x₁ − 4x₂ + u

State equation:

[ dx/dt ] = [ 0 1 ; −3 −4 ] [ x ] + [ 0 ; 1 ] u

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Numerical 2: Transfer Function to State Space Model

Question:
Obtain the state space representation of:

G(s) = 2 / (s² + 5s + 6)

Solution:

Choose phase variables:

dx₁/dt = x₂
dx₂/dt = −6x₁ − 5x₂ + 2u

Output equation:

y = x₁

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Numerical 3: Controllability Test

Question:
Check the controllability of the system:

A = [ 0 1 ; −2 −3 ]
B = [ 0 ; 1 ]

Solution:

AB = [ 1 ; −3 ]

Controllability matrix:

C = [ B AB ] = [ 0 1 ; 1 −3 ]

det(C) ≠ 0

Conclusion: The system is completely controllable.

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Numerical 4: Observability Test

Question:
Check observability of the system:

A = [ 0 1 ; −4 −5 ]
C = [ 1 0 ]

Solution:

CA = [ 0 1 ]

Observability matrix:

O = [ 1 0 ; 0 1 ]

Rank(O) = 2

Conclusion: The system is observable.

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Numerical 5: Stability of State Space System

Question:
Determine stability of the system:

A = [ 0 1 ; −6 −5 ]

Solution:

Characteristic equation:

s² + 5s + 6 = 0

Roots: s = −2, −3

Conclusion: The system is asymptotically stable.

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Exam Tip

For Polytechnic Lecturer / SSC JE, focus on:

• Differential equation to state model
• Controllability and observability numericals
• Stability using eigenvalues

For GATE / BEL / ESE, practice advanced rank-based numericals.

PLC Ladder diagram programming with examples

PLC Ladder Diagram: Explanation with Examples (Easy Guide)

A PLC (Programmable Logic Controller) is widely used in industries to automate machines and processes. To control any machine, the PLC uses different programming languages, and the most popular one is the Ladder Diagram (LD).

In this post, you will learn:

• What is a Ladder Diagram?
• Basic symbols of Ladder Logic
• How Ladder Logic works
• Simple and real-time examples
• Advantages of Ladder Diagram

What is a PLC Ladder Diagram?

A Ladder Diagram is a graphical programming language that looks like an electrical relay circuit. It has two vertical power rails and multiple horizontal lines called rungs. Each rung represents one logic operation.

It is the easiest PLC programming language because it uses simple symbols like contacts, coils, timers, counters, etc.

Basic Symbols in Ladder Logic

Symbol	Name	Function
━|	Normally Open (NO) Contact	Allows current when input is ON
/	Normally Closed (NC) Contact	Allows current when input is OFF
( )	Output Coil	Energizes an output device
T	Timer	Used for delay ON or delay OFF
C	Counter	Used for counting events

How Ladder Logic Works

PLC scans the program left to right and top to bottom. When all conditions on a rung are TRUE, the PLC activates the output coil on that rung.

Example 1: Start/Stop Motor Control

Problem:

Start a motor with a Start button and stop it using a Stop button. Motor should remain ON even after releasing the Start button (latching).

Ladder Diagram Logic:

|----[ ]----[/]---------------( )-------|

     Start   Stop             Motor

        |                     |

        |----[ Motor ]--------|

  

Explanation:

• Start (NO Contact): Turns motor ON when pressed.
• Stop (NC Contact): Breaks the circuit when pressed.
• Motor (Output Coil): Runs the motor.
• Latching Contact: Keeps motor ON until Stop is pressed.

Example 2: Lamp ON with 5-Second Delay

Problem:

Turn the lamp ON 5 seconds after the switch is pressed.

Ladder Diagram:

|----[ ]-----------------(TIMER T1 5s)-------------|

     Switch|----[ T1 Done ]-----------------( Lamp )----------| 

Explanation:

• Switch activates Timer T1.
• After 5 seconds, timer output becomes TRUE.
• Lamp turns ON.

Example 3: Conveyor Motor with Object Counter

Problem:

Conveyor motor should run until 10 objects pass the sensor.

Ladder Diagram:

|----[ ]------------- ( Conveyor Motor )------------------|

     Start|----[ Sensor ]------------( Counter C1 )------------------|

|----[ C1 >= 10 ]----------(/ Conveyor Motor )-------------| 

Explanation:

• Start button runs conveyor.
• Sensor counts each object.
• When count reaches 10, the motor stops.

Advantages of Ladder Diagram

• Easy to understand
• Similar to electrical wiring
• Best for beginners and technicians
• Easy debugging and troubleshooting
• Works well for industrial automation

Conclusion

PLC Ladder Diagram is the simplest and most widely used method for programming PLCs. With just contacts, coils, timers, and counters, you can control any machine—from motors to conveyors and automation systems.