Dr. D.Y.Patil Institute of Engineering, Management and Research

1. Dr. D.Y.Patil Institute of Engineering, Management and Research Mechatronics - 302050 UNIT -V Mrs. Amruta Adwant

2. Syllabus Modelling and Analysis of Mechatronic System • System modeling (Mechanical, Thermal and Fluid), • Stability Analysis via identification of poles and zeros, • Time Domain Analysis of System and estimation of Transient characteristics: % Overshoot, Damping factor, Damping frequency, Rise time, • Frequency Domain Analysis of System and Estimation of frequency domain parameters such as Natural Frequency, Damping Frequency and Damping Factor.

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 Mathematics • Engineering Mathematics (I, II & III)

6. Reference Books • Golnaraghi & Kuo, Automatic Control System, 9th Ed, John Wiley & Sons, • Dorf & Bishop, Modern Control Systems, 12th Ed, Prentice Hall • Ogata, Modern Control Engineering, 4th Ed, Prentice Hall

7. System Modelling • To understand and control complex systems, one must obtain quantitative mathematical models of these systems to analyze the relationships between the system variables. • Because the systems under consideration are dynamic in nature, the descriptive equations are usually differential equations obtained by utilizing the physical laws of the process

8. System Modelling • The Laplace transform can be used to obtain a solution describing the operation of the system. • In practice, the complexity of systems and our ignorance of all the relevant factors necessitate the introduction of assumptions concerning the system operation.

9. System ModellingTransfer Function based Approach • Why Transfer Function? • Because it is easier / better to assess some things using classical techniques, such as gain and phase margin. • How to determine TF? • Derive the Governing Differential Equation • Assume I.C=Zero and • Take Laplace transform of output • Take Laplace transform of input • Transfer function = L (output) / L (input)

10. System Modelling: Mechanical System

11. System Modelling: Thermal System • A closed, insulated vessel filled with liquid and contains an electrical heater immersed in liquid. Heating element is contained within metal jacket that has a thermal resistance of RHL. Thermal resistance of vessel and its insulation is RLa. Heater has a thermal capacitance of CH, and liquid has a thermal capacitance of CL. Heater temperature is TH and that of liquid is TL (assumed to be uniform due to the mixer). Rate at which energy is supplied to the heating element is qi.

12. System Modelling: Thermal System • Take Laplace of Eq. (1) and determine the equation for TH(s) . • Take Laplace of Eq. (2), determine the equation for TL(s) and substitute the equation for TH(s) in it. • Transfer Function is: TL(s)/qi(s)

13. System Modelling: Hydraulic System

14. System Modelling: Hydraulic System

15. System Modelling: Hydraulic System

16. Poles and Zeros • Poles and zeros are properties of transfer function, which characterize the differential equation, and provide a complete description of the behavior of the system . • The poles of a transfer function are the roots of the characteristic equation in the denominator of the transfer function. • The zeros of a transfer function are the roots of the characteristic equation in the numerator of the transfer function.

17. Relation: Poles of System and Damping and Natural Frequency Relation between Pole Location, Damping (ζ) and Natural Frequency (ωn)

18. Relation: Pole Location and Response of System Response of Systems Pole Location for Different Systems

19. Example: Identification of Poles

20. Example: Identification of Zeros

21. Stability Analysis based on Location of Poles • For all initial conditions, if the response of a system decays to equilibrium, the system is presumed to be stable, at large. • Absolute Stability: Whether stable or not • Relative Stability: Stable under what conditions • The real part of the poles of a given system must be on the left side of the s-plane for the system to be stable at large.

22. Stability Analysis based on Location of Poles • In case the poles are a complex conjugate pair, their real part must be negative for the system to be stable. • If any of the poles have a value, zero, the system is deemed to be marginally stable • If the poles are positive / have a positive real part, the system is deemed to be un-stable.

23. Time Domain Analysis of System • Time is used as an independent variable in most systems. • It is usually of interest to evaluate the output response of the system with respect to time or, simply, the time response. • Time response is divided into: Transient Response, Steady State Response. • In the time domain analysis problem, a reference input signal is applied to a system, and the performance of the system is evaluated in the form of the time domain specifications w.r.t to the transient as well as the steady state response.

24. Time Domain Analysis of System • Stable systems exhibit transient phenomena to some extent before the steady state is reached. • e.g. in Mechanical systems Inertia is unavoidable and, thus, response cannot follow sudden changes in input, instantaneously, and transients are usually observed. • Control of transient response, which leads to deviation between output response and the input response, before the steady state is reached, must be closely controlled. • Control of the steady state response is also very important since it determines the accuracy of the system • More so W.R.T position control system

25. Time Domain Specifications Unit Step Response of Second Order System

26. Time Domain 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

27. Time Domain Specifications • 2% 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

28. 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

29. Effect of Damping and Natural Frequency in Time Domain • Response of system depends on damping ζand natural frequency ωn • Settling time and rise time of the system reduces with increase in the natural frequency, ωn • As damping decreases below 1, the response overshoots and oscillates about final value • Smaller the value of damping: larger the overshoot and longer it takes for the oscillations to die

30. Effect of Damping on System Response

31. Effect of Natural Frequency on System Response

32. Frequency Domain Analysis of System • The frequency domain analysis of a system is defined as the steady-state response of the system to a sinusoidal input signal. • The sinusoid is a unique input signal, and the resulting output signal for a linear system, as well as signals throughout the system, is sinusoidal in the steady state. • Output differs from the input only in amplitude and phase.

33. Frequency Domain Analysis of System • Frequency domain analysis is a better option w.r.t to higher order system • Time response of a higher order system is difficult to determine, analytically. • Frequency domain analysis is better suited when it comes to determining sensitivity of system to uncertainty (parameter/process variation, mechanical / electrical noise) • Frequency domain analysis is better suited when it comes to accessing relative stability of a system.

34. Frequency Domain Analysis of System • Input applied is some form of sine wave. • The output of the system is also some form of sine wave given by: • In above: A is the amplitude of the sine wave, ω is the frequency of the sine wave, Y is the magnitude of output sine wave and Ø is the phase shift.

35. Frequency Domain Analysis using Bode Plot • Steady state performance can be characterised in the form of magnitude and phase shift w.r.t to frequency (ω)

36. Rules for Drawing Bode Plot: Effect of Gain

37. Rules for Drawing Bode Plot: Effect of Real Pole

38. Rules for Drawing Bode Plot: Real Zero

39. Rules for Drawing Bode Plot: Effect of Pole at Origin

40. Rules for Drawing Bode Plot: Effect of Zero at Origin

41. Example: Bode Plot for Typical Transfer Function with single Pole at origin

42. Example: Bode Plot for Typical Transfer Function with two poles at origin

43. Example: Bode Plot for Typical Transfer Function with one Pole on negative real axis

44. Example: Bode Plot for Typical Transfer Function with one Pole at origin and one Pole on negative real axis

45. Example: Bode Plot for Typical Transfer Function with both poles on negative real axis

46. Frequency Domain Specifications • Resonant Peak (Mr): It is the maximum value of the magnitude. • Mrgives indication on the relative stability of a stable closed-loop system. Normally, a large Mr corresponds to a large maximum overshoot of the step response. For most control systems, it is generally accepted in practice that the desirable value of Mr should be between 1.1 and 1.5. • Resonant Frequency (ωr): It is the frequency at with peak resonance, Mr, occurs. • Bandwidth (BW): It is the frequency range over which the magnitude drops 3 decibels (dB) from its zero frequency value. • BW gives indications of the transient response properties in time domain. A large bandwidth corresponds to a faster rise time.

47. Frequency Domain Specifications

48. Gain & Phase Margin from Bode Plot • Simply knowing that system is stable is not enough • Important to access relative stability • Stability Margins help accessing the relative stability • Gain Margin: It is the factor by which system gain can be increased before the system becomes un-stable in closed loop. • Gain Margin should be > 1 for system to be stable in closed loop. • Gain Margin is determined at phase cross over • Phase Margin: It is the amount by which the phase exceeds -1800 • Phase Margin should be > 0 for system to be stable in closed loop • Phase Margin is determined at gain cross over

49. Closed Loop Stability of System based on Gain & Phase Margin

50. Effect of Damping on Bode Plot