1 / 40

Modern Control Theory (Digital Control)

Modern Control Theory (Digital Control). Lecture 2. Outline. Signal analysis and dynamic response Discrete signals Discrete time – discrete signal plot z-Transform – poles and zeros in the z-plane Correspondence with continuous signals Step response Effect of additional zeros

kyle-jensen
Download Presentation

Modern Control Theory (Digital Control)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Modern Control Theory (Digital Control) Lecture 2

  2. Outline • Signal analysis and dynamic response • Discrete signals • Discrete time – discrete signal plot • z-Transform – poles and zeros in the z-plane • Correspondence with continuous signals • Step response • Effect of additional zeros • Effect of additional poles • s-Plane specifications • z-Plane specifications • Frequency response

  3. Signal analysis – discrete signals • Analysis • look at different characteristic signals • z-transform, poles and zeros • signals • unit pulse • unit step • exponential • general sinusoid

  4. Signal analysis – discrete signals • The z transform

  5. Signal analysis – discrete signals • The Unit Pulse

  6. Signal analysis – discrete signals • The unit Step Zeros : z=0 Poles : z=1

  7. Signal analysis – discrete signals • Exponential Zeros : z=0 Poles : z=r

  8. Signal analysis – discrete signals • General Sinusoid (let us look at the terms, one by one, and use linearity)

  9. Signal analysis – discrete signals Plots shown for Zeros : z=0, z=r cos(q) Poles : z=r exp(jq) , z=r exp(-jq)

  10. Signal analysis – discrete signals • Transients • r > 1, growing signal (unstable) • r = 1, constant amplitude signal • r < 1, decreasing signal (the closer r is to 0 the shorter the settling time. In fact, we can compute settling time in terms of samples N.) • Conclusions • General sinusoid

  11. Signal analysis – discrete signals • Samples per oscillation (cycle) • number of samples in a cycle is determined by q • or, N = samples/cycle depends on q • pole placements depend onq 4 5 3 We have 2 1 k=0 dependence of q

  12. Signal analysis – discrete signals • Samples per oscillation (cycle), cont.

  13. Signal analysis – discrete signals

  14. Signal analysis – discrete signals Pole placements

  15. Correspondence with cont. signals • Continuous signal Poles: s = -a + jb, s = -a - jb Pole map • Discrete signal Poles: z = exp(-aT - jbT) z = exp(-aT + jbT)

  16. Correspondence with cont. signals Pole map

  17. Correspondence with cont. signals • Recall, poles in the s-plane

  18. Correspondence with cont. signals Fixed z, varying wn Pole map Fixed z, varying wn

  19. Correspondence with cont. signals Fixed wn, varying z Fixed z, varying wn

  20. Correspondence with cont. signals • Notice, in the vicinity of z = 1, the map of z and wn looks like the s-plane in the vicinity of s = 0.

  21. Signal analysis – step response • Investigate effect of zeros • fix z1 = p1, and explore effect of z2 • a (delayed) second order sys is obtained • z = {0.5, 0.707} (by adjusting a1 and a2) • q = {18°,45°,72°} (by adj. a1 and a2) • a unit step U(z) = z/(z-1) is applied to the system (pole, z=1, and zero, z=0)

  22. Signal analysis – step response Discrete step responses for q = 18° Overshoot increases with the zero Z2

  23. Signal analysis – step response The zero has little infuence on the negative axis, large influence near +1

  24. Signal analysis – step response

  25. Signal analysis – step response • Investigate effect of extra pole • fix z1 = z2 = -1, and explore effect of moving singularity p1(from -1 to 1) • z = 0.5 • q = {18°,45°,72°} • a unit step is applied to the system

  26. Signal analysis – step response Mainly effect on rise time Rise time expressed as number of samples. The rise time increases with the pole

  27. Signal analysis – step response • Conclusions • Addition of a pole or a zero between -1 and 0 • Only small effect • Addition of a zero between 0 and +1 • Increasing overshoot when the zero is moving towards +1 • Addition of a pole between 0 and +1 • Increasing rise time when the pole is moving towards +1 (the pole dominates)

  28. s-Plane specifications • Spec. on transients of dominant modes • dominant first order • time constant t (related to 3 dB bandwidth) • dominant second order • rise time tr (related to natural frequency wn≈ 1.8/tr ) • settling time ts(related to real part s = 4.6 ts ) • overshoot Mp, or damping ratio z. • Spec. on reference tracking • typically step or ramp input specification • i.e. specifications on Kp and Kv , ess = r0 /Kv • ess is the steady state error for a ramp input of slope r0

  29. s-Plane specifications Example We have system with dominant 2. order mode Specifications: Notice, spec. on wn not shown

  30. z-Plane specifications • Discrete system • similar specifications • in addition, sample time T Example (continued) Notice, sample time T must be chosen. If fixed wn

  31. z-Plane specifications Specifications are 1) Overshoot Mp less than 16% 2) Settling time ts (1%) less than 10 sec. 3) Chose sample time T such that Example (7.2 and 7.5) A system is given by

  32. z-Plane specifications 1) Overshoot Mp less than 16% 2) Settling time ts (1%) less than 10 sec.

  33. z-Plane specifications Damping, radius r z wn Possible region Also, we might have an additional specification on rise time tr

  34. z-Plane specifications • Steady-state errors • ZOH of plant transfer function, i.e. G(s) to G(z) • Transfer function from R(z) to E(z), for investigating the error. R(z) E(z) controller D(z) U(z) plant G(z) Y(z) + -

  35. z-Plane specifications • Now, if r(kT) is a step, then

  36. Frequency response • Frequency response methods • Gain and phase can easily be plotted. • Freq. response can be measured directly on a physical plant. • Nyquist's stability criterion can be applied. • Error constants can be seen on gain plot. • Corrections to gain a phase by additional poles and zeros. Effect can easily be observed – in terms of cross over frequency, gain margin, phase margin. • Frequency response methods can also be applied for discrete systems (example).

  37. Frequency response Discrete Bode Plot, Example (7.8) Plot the discrete frequency response corresponding to Transform to z-domain by ZOH, with sample time T = 0.2, 1 and 2. Solution. Use Matlab c2d(sys,T). Matlab sysc = tf([1],[1 1 0]); sysd1 = c2d(sysc,0.2); sysd2 = c2d(sysc,1); sysd3 = c2d(sysc,2); bode(sysc,'-',sysd1,'-.', sysd2,':', sysd3,'-',)

  38. Frequency response Half sample frequency Primary effect, Additional lag Approx. phase lag Df = wT/2

  39. Frequency response Approx. phase lag Df = wT/2 Accurate up to wT = p/2

  40. Discrete Equivalents - Overview r(t) e(t) controller D(s) u(t) plant G(s) y(t) + - Translation to discrete plant Zero order hold (ZOH) Translation to discrete controller (emulation) Numerical Integration • Forward rectangular rule • Trapeziod rule (Tustin’s method, bilinear transformation) • Bilinear with prewarping Zero-Pole Matching Hold Equivalents • Zero order hold (ZOH) • Triangle hold Lecture 3

More Related