Modern Control Theory (Digital Control)

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