Mechatronics

1. UNIT –VI • Control Systems MrManojRajale Mechatronics

2. Syllabus Control Systems • P, I and D control actions, • P, PI, PD and PID control systems, • Transient response:- Percentage overshoot, Rise time, Delay time, Steady state error • PID tuning (manual)

3. Objectives Understand key elements of Mechatronics system, representation into block diagram Understand concept of transfer function, reduction and analysis Understand principles of sensors, its characteristics, interfacing with DAQ microcontroller Understand the concept of PLC system and its ladder programming, and significance of PLC systems in industrial application Understand the system modeling and analysis in time domain and frequency domain. Understand control actions such as Proportional, derivative and integral and study its significance in industrial applications.

4. Outcomes Identification of key elements of mechatronics system and its representation in terms of block diagram Understanding the concept of signal processing and use of interfacing systems such as ADC, DAC, digital I/O Interfacing of Sensors, Actuators using appropriate DAQ micro-controller Time and Frequency domain analysis of system model (for control application) PID control implementation on real time systems Development of PLC ladder programming and implementation of real life system

5. Assumed Knowledge Dynamics: • Engineering Mechanics Electrical & Electronics • Elements of Electrical Engineering Mathematics • Engineering Mathematics (I, II & III)

6. Reference Books Astrom & Hagglund, PID Controllers: Theory, Design & Tuning, Chapter 2, 2nd Ed, Instrument Society of America, 1995. Golnaraghi & Kuo, Automatic Control System, Chapter 1/5/9, 9th Ed, John Wiley & Sons, 2009

7. Why is Controller Necessary? • Blue response resembles an un-controlled system. This response is oscillatory as well as it takes much longer to settle down. • For a mechanical system, this could be due to Inertia effect, friction, backlash etc • The red response is of a controlled system. This response contains no oscillations and it settles to equilibrium / steady state in lesser time. • Job of a control system is to “generate a control input / effort that can be used to drive the un-controlled system, albeit externally, to achieve the desired performance”.

8. Illustration: What does Controller do? -imaginary X Undesirable Open Loop Pole Location X Desired Closed Loop Pole Location X u -real X +real Control is all about shifting of system poles from un-desirable to desirable location. This shifting is done by the control signal, u, provided the system allows it i.e. the system is “controllable” u X +imaginary

9. Analysis of Response: Transient Specifications Unit Step Response of Second Order System

10. Transient Response Specifications • Percentage Overshoot (% O.S): It is the amount that the response overshoots the steady state, or final, value at the peak time, expressed as a percentage of the steady-state value. • Rise Time (Tr): Time required for the step response to rise from 10% to 90% of its final value. • Delay Time (Td): Time required for the step response to reach 50% of final value • Settling Time (Ts): Time required for the step response to decrease and stay within ±2% of its final value • Steady State Error (ess): It is the difference between the output and the reference input after the steady state has reached

11. Feedback Controller Block Diagram of Feedback Controller Feedback controller generates an control signal / effort / external disturbance based on the input signal it receives. The input signal is error; difference between measured value and desired value, or set point. Feedback counters disturbance as well as variation in process

12. ControllabilityAdvanced Learning (Out of Syllabus) • Before a controller is implemented it is necessary to determine is the system is controllable • Test the “Controllability” of the system • Controllability is the ability of the system to be controlled provided an external disturbance is available.

13. Proportional Integral Derivative Control e u + ∑ Input PID Plant Output _ Block Diagram of PID Controller PID stands for Proportional Integral Derivative Control. Being robust & easy to implement, it is one of the most widely used closed loop control for precise operation of industrial applications and processes.

14. Proportional Control • In Proportional Control, the control signal, u, is directly proportional to the error, e. • As the gain is increased the system responds faster to changes in set-point but becomes progressively under damped and eventually unstable.

15. Proportional Control Action P Control Signal

16. Proportional Control Advantages: • Simple and easy to design and tune • Rapid Response / Reduces Rise Time • Reduces Steady State Error Disadvantages: • Not possible to eliminate Steady State Error / Offset • Could lead to instability / rise in overshoot/ oscillations Applications: • Float Valve, Thermostat etc

17. Derivative Control • Derivative control produces a control signal proportional to the rate at which the error is changing. • Also known as rate controller. • While sudden/rapid change in error leads to a control signal of larger magnitude, gradual change leads to small magnitude. • Even if the error is huge, the derivative control will generate no signal if the error is constant • Thus, not used alone; used with P control

18. Derivative Control Action D Control Signal

19. Derivative Control Advantages: • Reduces Settling time; Adds lead • Reduces Overshoot; Adds more stability Disadvantages: • Not possible to eliminate Steady State Error / Offset • Not possible to use alone • Excessive use may make the system slow • Amplifies Noise Applications: • In conjunction with P Control

20. Integral Control • Rate of change of integral control signal is proportional to error. • Control signal proportional to integral of error. • When the error is zero, the control signal is a constant value. • When the error is constant, the control signal varies at constant rate.

21. Integral Control Action I Control Signal

22. Integral Control Advantages: • Eliminates steady state error/offset • Decreases Rise Time Disadvantages: • Causes Integral Wind Up • Leads to minor increase in overshoot • Could make the system less stable • Increases Settling time Applications: • In conjunction with P Control

23. Integral Wind UpAdvanced Learning (Out of Syllabus) Caused by actuator saturation. What Happens? Feedback loop is broken and the system runs in open loop because the actuator remains saturated. While the error is zero, the integral term will keep building and become very large over a period of time. This in turn would lead to saturation of control signal. The condition will prevail even when the error changes and it may take a long time before the integrator and the controller output comes inside the saturation range. The consequence is that there are large time delay.

24. PID: Series / Interacting Form D I + + e + + u P • Derivate Action interacts with Integral Action • Modification in derivative time constant affects integral action • Commercially used controller

25. Transfer Function of Series Form

26. Transfer Function of Series Form

27. PID: Parallel / Non-Interacting Form Ideal Form Derivative Action does not Interact with Integral Action

28. Transfer Function of Parallel Form

29. Parallel Form: PI Control • Proportional Integral (PI) Control helps minimise rise time, settling time as well as eliminate steady state error.

30. PI Control

31. Parallel Form: PD Control • Proportional Derivative (PD) Control helps reduce rise time, settling time as well as minimize overshoot.

32. Proportional Derivative Control

33. Response of P, I & D w.r.t Error

34. Effect of P, I & D on Transient Specifications

35. P, I & D Control Action

36. PID: Stepwise Procedure for Manual Tuning Obtain an open-loop response and determine what needs to be improved Add a proportional control to improve the rise time Add a derivative control to improve the overshoot Add an integral control to eliminate the steady-state error Adjust each of P, I & D until you obtain a desired overall response referring to the table shown previously to find out which controller controls what characteristics. It is not necessary to implement all three controllers (P, I & D) into a single system. For example, if a PI controller gives a good enough response, then you don't need to add D control to the system. Simple is better.

37. PID: Stepwise Procedure for Manual Tuning NOTE It is not necessary to implement all three controllers (P, I & D) into a single system. For example, if a PI controller gives a good enough response, then you don't need to add D control to the system. Simple is better!

38. Applications of PID Control 90% processes are controlled using PID. • Regulation of Processes in Industry; for e.g. • Flow • Temperature • Pressure etc • Servo / DC motor Control • Linear Position Control

39. Example: Time Domain Specifications • Using the values of the natural frequency= =1.414 and the damping factor=ζ=0.177, determine the values for overshoot, rise time and 2% settling time

40. Effect of P, I & D on Transient Specifications

41. PID: Stepwise Procedure for Manual Tuning Obtain an open-loop response and determine what needs to be improved Add a proportional control to improve the rise time Add a derivative control to improve the overshoot Add an integral control to eliminate the steady-state error

42. Need for Optimal Control • System’s performance vs control effort needs to be thought about • Actuators have limited bandwidth Leads to need for Optimal Control • Optimal control works towards providing the end user a optimal trade-off between performance and control effort • Performance vs cost! • Linear Quadratic Regulator (LQR) is a optimal control technique used to determine feedback gain • Feedback gain is used to determine control effort

43. Linear Quadratic Regulator • In order to determine feedback gain, k, LQR aims to minimize below performance index, J In the above quadratic cost function: • T = final time, t = initial time. If T = ∞, the LQR works like a regulator • X: Vector comprising of state variables • E.g. in a mass spring damper system, position and velocity are state variables • U: Vector comprising of inputs

44. Linear Quadratic Regulator • Q: is a positive definite performance weighting matrix that relates to achieving a “desired outcome” • It is a “penalty” for deviation of performance from its desired • R: is positive definite effort weighting matrix that relates to limiting the control effort / input • It is a “penalty” for using large control inputs

45. Effect of Q • Larger the value of Q: ratio of Q/R increases • Larger the value of gain k • Larger the control input-could saturate the actuator • Better performance - settling time improves • Damped system - implies less overshoot

46. Effect of R • Larger the value of R: ratio of Q/R reduces • Smaller the value of gain k • Smaller the control input - reduced control effort • Limited performance - design objectives not achieved • Trade off: Performance of system v/s control effort • Faster performance requires bigger control effort

47. LQR vs PID • Being an optimal control, LQR provides direct trade-off between performance and cost • Both LQR and PID require measured / observed feedback • Tuning of LQR requires an analytical model • In comparison, PID can be tuned without any analytical model • In comparison to PID, designing an LQR is more complicated