Simulation of continuous Systems

Simulation of Continuous Systems

A continuous system is a system in which state variables change continuously over time. These systems are usually described using differential equations.

Key Characteristics

• Time is continuous
• State variables change smoothly
• Modeled using differential equations
• Common in physical and engineering systems

Examples

• Water level in a reservoir
• Speed of a motor
• Temperature variation
• Aircraft motion

Simulation of Continuous Systems

Simulation of continuous systems involves studying the behavior of such systems over time using mathematical models and computational tools.

General Steps

• Identify system variables
• Formulate differential equations
• Choose simulation method
• Solve equations numerically
• Analyze system behavior

General Continuous System Diagram

Input → [ Continuous System ] → Output
           (Differential Eq.)

Analog Simulation vs Digital Simulation

• Analog Simulation: Uses physical analogs like electrical circuits to represent system variables.
• Digital Simulation: Uses computers and numerical methods to simulate the system.

Comparison Table: Analog vs Digital Simulation

Basis	Analog Simulation	Digital Simulation
Representation	Continuous physical signals	Discrete numerical values
Accuracy	Limited	High
Flexibility	Low	Very high
Modification	Difficult	Easy
Cost	High	Low
Speed	Fast for simple systems	Depends on computation
Example	Electrical circuit	MATLAB / Python

Analog vs Digital Diagram

Analog:   Physical Circuit → Output
Digital:  Mathematical Model → Computer → Output

Simulation of Water Reservoir System

System Description: A water reservoir system controls the water level based on inflow and outflow.

Variables

• Inflow rate (I)
• Outflow rate (O)
• Water level (H)

Mathematical Model

$$\frac{dH}{dt}=I-O$$

Where:

• H = Water height
• I = Inflow rate
• O = Outflow rate

Reservoir Simulation Diagram

        Inflow (I)
           ↓
      ┌──────────┐
      │ Reservoir│
      │  Water   │
      └──────────┘
           ↓
        Outflow (O)

Applications

• Dam management
• Flood control
• Water supply planning

Table: Reservoir Parameters

Parameter	Meaning
H	Water level
I	Inflow rate
O	Outflow rate
t	Time

Simulation of a Servo System

A servo system is a feedback control system used to control position, speed, or acceleration.

Examples

• Robotic arm
• Antenna positioning
• CNC machines

Servo System Components

• Reference input
• Error detector
• Controller
• Motor
• Feedback device

Servo System Block Diagram

Reference → (+) → Controller → Motor → Output
              ↑                         ↓
              └────── Feedback ─────────┘

Mathematical Representation

$$Error=Reference-Output$$

System dynamics are expressed using differential equations.

Why Servo Simulation is Important

• Stability analysis
• Performance improvement
• Design optimization

Simulation of an Auto-Pilot System

An auto-pilot system automatically controls an aircraft’s altitude, direction, and speed without continuous human intervention.

Main Functions

• Maintain altitude
• Control direction
• Stabilize aircraft

Auto-Pilot System Components

• Sensors (gyroscope, altimeter)
• Controller
• Actuators
• Aircraft dynamics
• Feedback loop

Auto-Pilot Block Diagram

Desired Path → Controller → Actuators → Aircraft
                   ↑                         ↓
                   └──── Sensors / Feedback ─┘

Simulation Purpose

• Test safety before real flights
• Reduce risk
• Improve fuel efficiency
• Study response to disturbances

Table: Auto-Pilot Variables

Variable	Description
Pitch	Up-down motion
Roll	Side motion
Yaw	Direction control
Speed	Velocity

Summary Table

System	Type	Main Variable
Water Reservoir	Continuous	Water level
Servo System	Continuous	Position / speed
Auto-Pilot	Continuous	Aircraft motion

Discrete System Simulation

A discrete system is a system in which state variables change only at specific points in time, not continuously.

Key Characteristics

• Time advances in steps or events
• State changes occur instantaneously
• Modeled using events, queues, counters
• Widely used in business and computer systems

Examples

• Bank queue system
• Railway reservation system
• Inventory management
• CPU scheduling

Discrete System Simulation

Discrete system simulation models the operation of a system as a sequence of events over time.

Each Event

• Occurs at a specific time
• Causes a change in system state

General Discrete Simulation Diagram

Event 1 → Event 2 → Event 3 → Event 4
   t1        t2        t3        t4

Fixed Time Step vs Event-to-Event Model

A. Fixed Time Step Simulation

In this approach, time advances in fixed intervals regardless of whether an event occurs.

Example

• Checking inventory every 1 hour
• Temperature sampled every second

Diagram: Time → |Δt|Δt|Δt|Δt|Δt|

B. Event-to-Event Simulation

Time jumps directly from one event to the next event.

Example

• Customer arrival in bank
• Machine breakdown

Diagram: Time → Event1 -------- Event2 ------ Event3

Comparison Table: Fixed Time Step vs Event-to-Event

Basis	Fixed Time Step	Event-to-Event
Time advancement	Constant interval	Event based
Computation	High	Efficient
Accuracy	Moderate	High
Suitable for	Continuous-like systems	Discrete systems
Example	Inventory check	Queue simulation

Generation of Random Numbers

Why Random Numbers are Needed?

In simulation, randomness represents uncertainty, such as:

• Customer arrival time
• Service time
• Demand variability

Properties of Good Random Numbers

• Uniform distribution
• Independent
• Non-repetitive
• Long cycle

Common Method: Linear Congruential Method (LCM)

$$X_{n+1}=(a{X}_n+c)\mathrm{m}\mathrm{o}\mathrm{d}m$$

Where:

• X = random number
• a = multiplier
• c = increment
• m = modulus

Random Number Generation Diagram

Seed → Formula → Random Number → Next Seed

Test of Randomness

Random numbers must be tested to ensure reliability.

Frequency (Chi-Square) Test

Checks whether numbers are uniformly distributed.

Class	Observed	Expected
0–0.2	O1	E
0.2–0.4	O2	E
...	...	...

Runs Test

Checks independence of random numbers.

Types of Runs

• Runs above and below mean
• Runs up and down

Autocorrelation Test

Checks correlation between successive numbers.

Randomness Test Summary Table

Test	Purpose
Frequency test	Uniformity
Runs test	Independence
Autocorrelation	Dependence detection

Monte-Carlo Computation

Monte-Carlo method uses random sampling to solve deterministic problems.

Key Features

• Based on probability
• Repeated random sampling
• Numerical solution

Example: Estimating the value of π using random points.

Monte-Carlo Diagram: Random Inputs → Mathematical Model → Numerical Result

Stochastic Simulation

A stochastic simulation models systems that have random inputs and random behavior.

Examples

• Bank queue simulation
• Inventory with random demand
• Traffic flow

Stochastic Simulation Diagram

Random Variables → System Model → Performance Measures

Monte-Carlo Computation vs Stochastic Simulation

Basis	Monte-Carlo Computation	Stochastic Simulation
Purpose	Solve mathematical problems	Model real systems
System dynamics	No time evolution	Time-based
Randomness	Input sampling	System behavior
Output	Numerical result	Performance measures
Example	Estimating π	Bank queue

Summary Table

Topic	Key Point
Discrete simulation	Event-based
Fixed time step	Regular intervals
Event-to-event	Efficient for discrete systems
Random numbers	Represent uncertainty
Randomness tests	Ensure quality
Monte-Carlo	Mathematical estimation
Stochastic simulation	Real-world system modeling

Exam Writing Tips

• Start with definition
• Draw diagrams
• Use tables for comparison
• Write examples
• Highlight key formulas