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I. Concepts and Tools

I. Concepts and Tools. Mathematics for Dynamic Systems Differential Equation Transfer Function State Space Time Response Transient Steady State Frequency Response Bode and Nyquist Plots Stability and Stability Margins Extensions to Digital Control. A Differential Equation of Motion.

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I. Concepts and Tools

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  1. I. Concepts and Tools • Mathematics for Dynamic Systems • Differential Equation • Transfer Function • State Space • Time Response • Transient • Steady State • Frequency Response • Bode and Nyquist Plots • Stability and Stability Margins • Extensions to Digital Control

  2. A Differential Equation of Motion • Newton’s Law: • A Linear Approximation:

  3. Laplace Transform and Transfer Function

  4. State Space Description

  5. From State Space to Transfer Function

  6. Linear System Concepts • States form a linear vector space • Controllable Subspace and Controllability • Observable Subspace and Observability • The Linear Time Invariance (LTI) Assumptions • Stability • Lyapunov Stability (for linear or nonlinear systems) • LTI System Stability: poles/eigenvalues in RHP

  7. A Motion ControlProblem

  8. From Differential Eq. To Transfer Function

  9. Transfer Function model of the motion plant

  10. Unity Feedback Control System

  11. Common Nonlinearities

  12. Linearization

  13. Time Response

  14. Open Loop Transient Response • How parameters of transfer functions affect output • Terminologies for 1st and 2nd order systems

  15. Basis of Analysis

  16. First Order Transfer Function

  17. Pure 2nd Order Transfer Functions

  18. Pure 2nd Order Transfer Functions

  19. Pure 2nd Order Transfer Functions

  20. Terminologies

  21. Calculations

  22. Pure 2nd Order Transfer Functions

  23. Pure 2nd Order Transfer Functions

  24. Pure 2nd Order Transfer Functions

  25. Steady State Response Steady state response is determined by the dc gain: G(0)

  26. Steady State Error

  27. Frequency Response: The MOST useful concept in control theory • Performance Measures • Bandwidth • Disturbance Rejection • Noise Sensitivity • Stability • Yes or No? • Stability Margins (closeness to instability) • Robustness (generalized stability margins)

  28. Frequency Response

  29. Bode Plot (Magnitude and Phase vs. Frequency) G(s) =1/(s +2)

  30. Polar Plot: imaginary part vs. real part of G(jw) G(s) =1/(s +2)

  31. Bandwidth of Feedback Control • -3dB Frequency of CLTF • 0 dB Crossing Frequency (wc) of Gc(jw)Gp(jw) • Defines how fast y follows r

  32. Disturbance Rejection

  33. Noise Sensitivity

  34. Nyquist Plot Using G (jw)to determine the stability of

  35. The Idea of Mapping

  36. Nyquist Contour RHP

  37. Nyquist Stability Criteria • Determine stability by inspection • Assume G(s)H(s) is stable, let s complete the N-countour The closed-loop system is stable if G(s)H(s) does not encircle the (-1,0) point • Basis of Stability Robustness • Further Reading: unstable G(s)H(s), # of unstable poles

  38. Nyquist Stability Criteria Stable Unstable

  39. Stability Robustness • The (-1,0) point on the GH-plane becomes the focus • Distance to instability: G(s)H(s)-(-1)=1+G(s)H(s) • Robust Stability Condition Distance to instability > Dynamic Variations of G(s)H(s) • This is basis of modern robust control theory

  40. Gain and Phase Margins (-1,0) is equivalent of 0dbÐ(-180)° point on Bode plot |G(jw)H(jw)| ÐG(jw)H(jw)

  41. Stability Margins: Nyquist Plot Bode Plot

  42. Digital Control

  43. Discrete Signals

  44. Digital Control Concepts • Sampling • Rate • Delay • ADC and DAC • Resolution (quantization levels) • Speed • Aliasing • Digital Control Algorithm Difference equation

  45. Discrete System Description • Discrete system h[n]: impulse response • Difference equation h[n] y[n] u[n]

  46. Discrete Fourier Transform and z-Transform

  47. Discrete Transfer Function and Frequency Response

  48. Application of Basic Concepts to Previously Designed Controllers

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