In The Name of Allah The Most Beneficent The Most Merciful 1

ECE4545: Control Systems Lecture: Mathematical Modeling of Systems- Frequency Domain Engr. Ijlal Haider UoL, Lahore 2

Control Talk  The 2003 Jaguar features a forward alert system from Delphi that advises a driver to brake in the approach of slow moving tra ffi c ahead [Sharke 2003]. If the driver does not adequately respond to warnings, collision avoidance systems might take control of the throttle, brakes, or steering, to maneuver the vehicle back to a safe state.  Nissan’s new gas pedal will lift itself to alert the driver of a possible collision, and the brakes are automatically applied if the radar sensor detects a possible collision ahead and when the driver’s foot is o ff the gas [Kageyama 2006]  Taking a step further, the 2006 Honda Acura RL will even actively brake the car if an imminent rear-end collision is sensed [Honda 2005] 3

Control Talk  The so-called Honda Collision Mitigation Braking System TM (CMBSTM) predicts potential collision situations with a millimeter wave radar unit, which monitors the distance between the RL and objects in front of the car, as well as closing rates. If CMBS determines the closing rate between the RL and the vehicle directly in front has increased beyond an acceptable level, visual and audible warnings prompt the driver to take preventive action. If the distance further diminishes, the system provides a tactile warning by gently retracting the seatbelt and then applies light braking. If an accident is determined to be unavoidable, the system applies strong braking and strong retraction of the front seatbelt to reduce the speed of impact and to mitigate the damage of a collision. 4

Control Talk  With funding from DARPA, researchers at Princeton have developed "a new type of sensor [that] can identify substances as small as a molecule by examining the light they reflect, potentially leading to sensors for a wide range of substances, from explosives to cancer”. ASEE Weekly Newsletter 5

Plate Dispenser 6

Mathematical Modeling- Definition  Determining equations and relationships governing the behavior of system ©copyright: M Junaid Khan 7

Mathematical Modeling- Significance  The power of a mathematical model lies in the fact that it can be simulated in hypothetical situations, be subject to states would be dangerous in reality and it can be used as a basis for synthesizing controllers 8

Model Development  Essential step of the “Design Procedure”  Three different choices for modeling:  Analysis: Mathematical models based on physical principles.  Grey-Box: Model is developed and then parameters are inferred from experiments.  Black-Box: Input / Output data is used to infer a dynamic relationship. (System Identification)  Nonlinear models can be linearized.  Linear models are most often used in analysis and design. 9

On Building Models  High order models can only be justified where there is little uncertainty  Control-relevant models are often quite simple compared to the true system and generally combine physical reasoning with experimental data  Actuators should be included as they often are nonlinear and have their own dynamic behavior  A model is never perfect, hence modeling errors are a fact of life  Simplicity versus Accuracy 10

Mathematical Models A model is any representation of a real system.  May involve words, diagrams, mathematical notation, physical structure  A mathematical model may be as simple as a single equation relating a single dependent variable (y) to another independent variable (x) such as: y = ax + b  May be multi-component involving the interaction of many equations having several mutually dependent variables 11

System  Engineering process (continuous) – chemical plants, aircraft, traffic system, Electrical, Mechanical, Thermal systems  Information process (computer-based or digital) – computer network, telephone system,  Economic process – Pakistan economy, world economy, software industry  Social systems – suburban culture, welfare system ©copyright: M Junaid Khan 12

Purpose of Modeling  To analyze systems  To analyze each element in the system  Mathematical Models are equations representing input/output relationships of a system ©copyright: M Junaid Khan 13

Types of Models Under Consideration Transfer Function Models---Output/Input ratio Models Differential Equation Models State Space Models-A set of first Order Differential Equations Flow Diagram Models Block Diagram Models----Simulink Models ©copyright: M Junaid Khan 14

Transfer Function 15

Engineering Systems Under Consideration  Mechanical  Electrical  Thermal  Fluid (Hydraulic and Pneumatic) ©copyright: M Junaid Khan 16

Ingredients of a Model Building Blocks  Each block has a single property or function  Example  Electrical Circuit  The building blocks are resistors, capacitors and inductors.  Mechanical System  The building blocks are Spring, Mass and dampers  Note: There may be similarities in the building blocks used for different type of systems (analogous systems) ©copyright: M Junaid Khan 17

Building Blocks of Electrical Systems 18

Building Blocks of Mechanical Systems (Translational) 19

Building Blocks of Mechanical Systems ( Rotational ) 20

Physical Laws Governing any System  Law of Conservation of Energy  KVL  KCL  Newton’s 2 nd Law 21

Thank You! 22