Application to Control Design

2. Chapter 6 The Frequency-Response Design Method Application to Control Design Root locus of G(s) = 1/[s(s+1)2] Bode plot for KG(s) = 1/[s(s+1)2], K = 1.

3. Chapter 6 The Frequency-Response Design Method Application to Control Design II II I I IV IV III III • The detour of C1includes the pole on the imaginary axis. • The detour of C1excludes the pole on the imaginary axis. will be done now already done last time

4. Chapter 6 The Frequency-Response Design Method Application to Control Design • In this new detour of C1, only the section IV-I is changed. • As in previous evaluation, this path can be evaluated by replacing II I IV • Performing it, III • Evaluation of G(s) forms a half circle with a very large radius. • The half circle starts at 90° and ends at –90°, routing in counterclockwise direction.

5. Chapter 6 The Frequency-Response Design Method Application to Control Design • Combining all four sections, we will get the complete Nyquist plot of the system. IV II III I

6. Chapter 6 The Frequency-Response Design Method Application to Control Design • There is one encirclement of –1 in counterclockwise direction(N=–1). • There is one pole enclosed by C1 since we choose the contour to enclose the pole at origin (P=1). • The number of closed-loop pole in RHP Z = N + P = –1 + 1 = 0 • → No unstable closed-loop pole (the same result as when encircling the pole at origin to the right).

7. Chapter 6 The Frequency-Response Design Method Application to Control Design • If we choose K > 2, the plot will encircle –1 once in clockwise direction ( N = 1 ). • Since P = 1, Z = N + P = 2→ The system is unstable with 2 roots in RHP. • The matter of choosing a path to encircle poles on the imaginary axis does not affect the result of analyzing Nyquist plot.

8. Chapter 6 The Frequency-Response Design Method Stability Margin • A large fraction of control systems behave in a pattern roughly similar to the system we just discussed: stable for small gain values and becoming unstable if the gain increases past a certain critical point. • Two commonly used quantities that measure the stability margin for such systems are gain margin (GM) and phase margin (PM). • They are related to the neutral stability criterion described by • If |KG(jω)|< 1 at G(jω) = –180°, then the system is stable. “ ”

9. Chapter 6 The Frequency-Response Design Method Stability Margin • Gain margin (GM) is the factor by which the gain can be raised before instability results. • A GM ≥ 1 is required for stability. • Phase margin (PM) is the amount by which the phase of G(jω) exceeds –180° when |KG(jω)| = 1. • A positive PM is required for stability. • PM and GM determine how far the complex quantity G(jω) passes from –1. • Within stable values of GM and PM, as can be implied from the figure, there will be no Nyquist encirclements.

10. Chapter 6 The Frequency-Response Design Method Stability Margin • Crossover frequency, ωc, is referred to as the frequency at which the gain is unity, or 0 db.

11. Chapter 6 The Frequency-Response Design Method Stability Margin • PM is more commonly used to specify control system performance because it is most closely related to the damping ratio of the system. • The relation between PM, ζ, and furthermore Mp for a second-order system can be summarized in the following two figures.

12. Chapter 6 The Frequency-Response Design Method Stability Margin • Conditionally stable system: a system in which an increase in the gain can make it stable. • Several crossover frequencies exist, and the previous definition of gain margin is not valid anymore.

13. Chapter 6 The Frequency-Response Design Method Stability Margin • As one example, determine the stability properties as a function of gain K for the system with the open-loop transfer function

14. Chapter 6 The Frequency-Response Design Method Application to Control Design • We again choose path IV-I as a half circle with a very small radius, routing from negative imaginary axis to positive imaginary axis in counterclockwise direction. • This path can be evaluated by replacing • Performing it, • Evaluation of G(s) forms an one-and-a-half circle with a very large radius. • Inserting the value of θ, the half circle starts at 270° and ends at –270°, routing in clockwise direction.

15. Chapter 6 The Frequency-Response Design Method Stability Margin • The Nyquist plot was drawn for a stable value K = 7. • There is one cc encirclement and one ccw encirclement of the –1 point • Net encirclement equals zero  the system is stable.

16. Chapter 6 The Frequency-Response Design Method Compensation • As already discussed before, dynamic compensation is typically added to feedback controllers to improve the stability and error characteristics when a mere proportional feedback alone is not enough. • In this section we discuss several kinds of compensation in terms of their frequency-response characteristics. • To this point, the closed-loop system is considered to have the characteristic equation 1+KG(s)=0. • With the introduction of compensation, the closed-loop characteristic equation becomes 1+KD(s)G(s)=0. • All previous discussion pertaining to the frequency response of KG(s) applies now directly to the compensated case, where the response of KD(s)G(s) is of interest. • We call this quantity L(s), the “loop gain” or open-loop transfer function, L(s) = KD(s)G(s).

17. Chapter 6 The Frequency-Response Design Method PD Compensation • The PD compensation transfer function is given by • The stabilizing influence is apparent by the increase in phase and the corresponding 20 dB/dec- slope at frequencies above the break point 1/TD. • We use this compensation by locating 1/TD so that the increased phase occurs in the vicinity of crossover (that is, where |KD(s)G(s)|=1 PM is increased. • Frequency-Response of PD Compensation

18. Chapter 6 The Frequency-Response Design Method PD Compensation • Note: The magnitude of the compensation continues to grow with increasing frequency. • This feature is undesirable because it amplifies the high-frequency noise. • Frequency-Response of PD Compensation

19. Chapter 6 The Frequency-Response Design Method Lead Compensation • In order to alleviate the high-frequency amplification of the PD compensation, a first order pole is added in the denominator at frequencies substantially higher than the break point of the PD compensator. • Thus, the phase lead still occur, but the amplification at high frequency is limited. • This resulting lead compensation has a transfer function of where 1/α is the ratio between the pole/zero break-point frequencies. • Frequency-Response of Lead Compensation

20. Chapter 6 The Frequency-Response Design Method Lead Compensation • Note: A significant amount of phase lead is still provided, but with much less amplification at high frequencies. • A lead compensator is generally used whenever a substantial improvement in damping of the system is required. • The phase contributed by the lead compensation is given by • It can be shown that the frequency at which the phase is maximum is given by • The maximum phase contribution, i.e. the peak of D(s) curve, corresponds to

21. Chapter 6 The Frequency-Response Design Method Lead Compensation • For example, a lead compensator with a zero at s=–2 (T=0.5) and a pole at s=–10 (αT=0.1, α=0.2) would yield the maximum phase lead at

22. Chapter 6 The Frequency-Response Design Method Lead Compensation • The amount of phase lead at the midpoint depends only on α and is plotted in the next figure. • We could increase the phase lead up to 90° using higher values of the lead ratio, 1/α. • However, increasing values of 1/α also produces higher amplifications at higher frequencies. • A good compromise between an acceptable PM and an acceptable noise sensitivity at high frequencies must be met. • How? • How?

23. Chapter 6 The Frequency-Response Design Method First Design Using Lead Compensation Find a compensation for G(s)=1/[s(s+1)] that will provide a steady-state error of less than 0.1 for a unit-ramp input. Furthermore, an overshoot less than 25% is desired. • KD(0) must be greater than 10, so we pick K = 10.

24. Chapter 6 The Frequency-Response Design Method First Design Using Lead Compensation 45° • Sufficient PM to accommodate overshoot requirement (25%) is taken to be 45°. • From the read of Bode Plot, the existing PM is 20°  additional phase of 25° at crossover frequency of ω=3 rad/s

25. Chapter 6 The Frequency-Response Design Method First Design Using Lead Compensation • However, adding a compensator zero would shift the crossover frequency to the right also  requires extra additional PM. • To be save, we will design the lead compensator that supplies a maximum phase lead of 40°. 5 • As can be seen, 1/α = 5 will accomplish the goal. • The maximum phase lead from the compensator must occur at the crossover frequency. • This requires trial-and-error in placing the compensator’s zero and pole. • The best result will be obtained when placing zero at ω=2 rad/sec and the pole at ω=10 rad/s. K=10

26. Chapter 6 The Frequency-Response Design Method Design Procedure of Lead Compensation • Determine the low-frequency gain so that the steady-state errors are within specification. • Select the combination of lead ratio 1/α and zero values (1/T) that achieves an acceptable PM at crossover. • The pole location is then at (1/αT)

27. Chapter 6 The Frequency-Response Design Method Homework 10 • No.1, FPE (5th Ed.), 6.17. • No.2 • Design a compensation for the system which will yield an overall phase margin of 45° and the same gain crossover frequency ω1 as the uncompensated system. Hint: Use Matlab to check the phase margin before and after the compensation. Submit also the two Bode plots. • Deadline: 20.11.2012, at 07:30.