Control Response Patterns

1. Control Response Patterns Engr. R. L. Nkumbwa May, 2009

2. System Metrics and Time-Domain Analysis • System Metrics • Time-Domain Analysis • Time Response • Poles and Zeros • Transient Response Eng. R. L. Nkumbwa @CBU 2010

3. System Metrics • So, What are System Metrics? • When a system is being designed and analyzed, it doesn't make any sense to test the system with all manner of strange input functions or to measure all sorts of arbitrary performance metrics. • Instead, it is in everybody's best interest to test the system with a set of standard, simple, reference functions. Eng. R. L. Nkumbwa @CBU 2010

4. Control Systems Eng. R. L. Nkumbwa @CBU 2010

5. Eng. R. L. Nkumbwa @CBU 2010

6. System Metrics • Once the system is tested with the reference functions, there are a number of different metrics that we can use to determine the system performance. • So, what are the examples of such metrics? Eng. R. L. Nkumbwa @CBU 2010

7. Time-Response Analysis • Since time is used as an independent variable in most control systems, it is usually of interest to evaluate the state and the output responses with respect to time or simply, the Time-Response. • In control system design analysis, a reference input signal is applied to a system and the performance of the system is evaluated by studying the system response in the time-domain. Eng. R. L. Nkumbwa @CBU 2010

8. Time-Response • The time-response of a control system is usually divided into two parts namely; the Steady-State Response and the Transient Response. • In other words, the output response of a system is the sum of two responses: the forced response (steady-state response) and the natural response (zero-input response). Eng. R. L. Nkumbwa @CBU 2010

9. Time-Response of an Elevator Eng. R. L. Nkumbwa @CBU 2010

10. Transient Response • Defined as the part of the time response that goes to zero as time goes to infinity. Eng. R. L. Nkumbwa @CBU 2010

11. Steady-State Response • Defined as the part of the total response that remains after the transient has died out. Eng. R. L. Nkumbwa @CBU 2010

12. Poles and Zeros • The poles of a transfer function are: • (1) The values of the Laplace transform variable, s , that cause the transfer function to become infinite, or • (2) Any roots of the denominator of the transfer function that are common to roots of the numerator. • The zeros of a transfer function are: • (1) The values of the Laplace transform variable, s , that cause the transfer function to become zero, or • (2) Any roots of the numerator of the transfer function that are common to roots of the denominator. Eng. R. L. Nkumbwa @CBU 2010

13. Eng. R. L. Nkumbwa @CBU 2010

14. Response blueprint • A pole of the input function generates the form of the forced response. • A pole of the transfer function generates the form of the natural response. • A pole on the real axis generates an exponential response. • The zeros and poles generate the amplitudes for both the forced and natural responses. Eng. R. L. Nkumbwa @CBU 2010

15. Natural response Forced response Eng. R. L. Nkumbwa @CBU 2010

16. Standard Input Signals • All of the standard inputs are zero before time zero. All the standard inputs are causal. • So, what is causal? • Causal: A system whose output does not depend on future inputs. All physical systems must be causal. Eng. R. L. Nkumbwa @CBU 2010

17. Standard Input Signals • There are a number of standard inputs that are considered simple enough and universal enough that they are considered when designing a control system. • These inputs are known as a unit step, a ramp, and a parabolic input functions. Eng. R. L. Nkumbwa @CBU 2010

18. Unit Step Function • A unit step function is defined piecewise as such: • The unit step function is a highly important function, not only in control systems engineering, but also in signal processing, systems analysis, and all branches of engineering. Eng. R. L. Nkumbwa @CBU 2010

19. Unit Step Function Eng. R. L. Nkumbwa @CBU 2010

20. Ramp Input Function • A unit ramp is defined in terms of the unit step function, as such: r(t) = tu(t). • It is important to note that the ramp function is simply the integral of the unit step function: Eng. R. L. Nkumbwa @CBU 2010

21. Ramp Input Function Eng. R. L. Nkumbwa @CBU 2010

22. Parabolic Input Function • A unit parabolic input is similar to a ramp input: • Notice also that, the unit parabolic input is equal to the integral of the ramp function: Eng. R. L. Nkumbwa @CBU 2010

23. Parabolic Input Function Eng. R. L. Nkumbwa @CBU 2010

24. Steady State • To be more precise, we should have taken the limit as t approaches infinity. However, as a shorthand notation, we will typically say "t equals infinity", and assume the reader understands the shortcut that is being used. Eng. R. L. Nkumbwa @CBU 2010

25. Steady State • When a unit-step function is input to a system, the steady state value of that system is the output value at time t = ∞. • Since it is impractical (if not completely impossible) to wait till infinity to observe the system, approximations and mathematical calculations are used to determine the steady-state value of the system. Eng. R. L. Nkumbwa @CBU 2010

26. Steady State • Most system responses are asymptotic, that is, the response approaches a particular value. Systems that are asymptotic are typically obvious from viewing the graph of that response. Eng. R. L. Nkumbwa @CBU 2010

27. Step Response • The step response of a system is most frequently used to analyze systems and there is a large amount of terminology involved with step responses. • When exposed to the step input, the system will initially have an undesirable output period known as the transient response. • The transient response occurs because a system is approaching its final output value. • The steady-state response of the system is the response after the transient response has ended. Eng. R. L. Nkumbwa @CBU 2010

28. Step Response • It is common for a systems engineer to try and improve the step response of a system. • In general, it is desired for the transient response to be reduced. Eng. R. L. Nkumbwa @CBU 2010

29. First-Order Systems Eng. R. L. Nkumbwa @CBU 2010

30. Initial Conditions are zero Eng. R. L. Nkumbwa @CBU 2010

31. First-Order Systems Response Eng. R. L. Nkumbwa @CBU 2010

32. System Response K (1 − e−t /τ ) System response. K = gain Response to initial condition Eng. R. L. Nkumbwa @CBU 2010

33. Unit Step Response Eng. R. L. Nkumbwa @CBU 2010

34. Unit Step Response • The time constant can be described as the time for to decay to 37% of its initial value. Alternately, the time is the time it takes for the step response to rise to 67% of its final value. • The reciprocal of the time constant has the units (1/seconds), or frequency. Thus, we call the parameter a the exponential frequency. Eng. R. L. Nkumbwa @CBU 2010

35. Time Constant Eng. R. L. Nkumbwa @CBU 2010

36. Second-Order Systems Response ζ = 0 Eng. R. L. Nkumbwa @CBU 2010

37. Eng. R. L. Nkumbwa @CBU 2010

38. Step Response Vs. Pole Location Eng. R. L. Nkumbwa @CBU 2010

39. System Response Eng. R. L. Nkumbwa @CBU 2010

40. Target Value • The target output value is the value that our system attempts to obtain for a given input. • This is not the same as the steady-state value, which is the actual value that the target does obtain. • The target value is frequently referred to as the reference value, or the "reference function" of the system. • In essence, this is the value that we want the system to produce. Eng. R. L. Nkumbwa @CBU 2010

41. Example of an Elevator • When we input a "5" into an elevator, we want the output (the final position of the elevator) to be the fifth floor. • Pressing the "5" button is the reference input, and is the expected value that we want to obtain. • If we press the "5" button, and the elevator goes to the third floor, then our elevator is poorly designed. Eng. R. L. Nkumbwa @CBU 2010

42. Time-Domain Specifications • So, what are Time-Domain Specifications? Eng. R. L. Nkumbwa @CBU 2010

43. Time-Domain Specifications Eng. R. L. Nkumbwa @CBU 2010

44. Rise Time • Is the amount of time that it takes for the system response to reach the target value from an initial state of zero. • Rise time is defined as the time for the waveform to go from 0.1 to 0.9 of its final value. • Rise time is typically denoted tr, or trise. • This is because some systems never rise to 100% of the expected, target value and therefore, they would have an infinite rise-time. Eng. R. L. Nkumbwa @CBU 2010

45. Settling Time • After the initial rise time of the system, some systems will oscillate and vibrate for an amount of time before the system output settles on the final value. • The amount of time it takes to reach steady state after the initial rise time is known as the settling time • Which is defined as the time for the response to reach and stay within, 2% (or 5%) of its final value. • Damped oscillating systems may never settle completely. Eng. R. L. Nkumbwa @CBU 2010

46. Settling time Eng. R. L. Nkumbwa @CBU 2010

47. Peak Time • The time required to reach the first or maximum peak. Eng. R. L. Nkumbwa @CBU 2010

48. Percent Overshoot • The amount that the waveform overshoots the steady-state or final value at the peak time, expressed as a percentage of the steady-state value. Eng. R. L. Nkumbwa @CBU 2010

49. Pole-Zero Plots Eng. R. L. Nkumbwa @CBU 2010

50. Step Response as Poles Moves Eng. R. L. Nkumbwa @CBU 2010