Chapter 6

2. Feedback Control Systems Chapter 6 The Frequency-Response Design Method

3. Chapter 6 The Frequency-Response Design Method Frequency Response • A linear system’s response to sinusoidal inputs is called the system’s frequency response. • Frequency response can be obtained from knowledge of the system’s pole and zero locations. • Consider a system described by where • With zero initial conditions, the output is given by

4. Chapter 6 The Frequency-Response Design Method Frequency Response • Assuming all the poles of G(s) are distinct, a partial fraction expansion of the previous equation will result in Transient response Steady-state response where

5. Chapter 6 The Frequency-Response Design Method Frequency Response • Examine Magnitude Phase • But the input is • Therefore • Meaning?

6. Chapter 6 The Frequency-Response Design Method Frequency Response • A stable linear time-invariant system with transfer function G(s), excited by a sinusoid with unit amplitude and frequency ω0, will, after the response has reached steady-state, exhibit a sinusoidal output with a magnitude M(ω0) and a phase Φ(ω0) at the frequency ω0. • The magnitude M is given by |G(jω)| and the phase Φ is given by G(jω), which are the magnitude and the angle of the complex quantity G(s) evaluated with s taking the values along the imaginary axis (s = jω). • The frequency response of a system consists of the frequency functions |G(jω)| and G(jω), which describe how a system will respond to a sinusoidal input of any frequency.

7. Chapter 6 The Frequency-Response Design Method Frequency Response • Frequency response analysis is interesting not only because it will help us to understand how a system responds to a sinusoidal input, but also because evaluating G(s) with s taking on values along jω axis will prove to be very useful in determining the stability of a closed-loop system. • As we know, jω axis is the boundary between stability and instability. Therefore, evaluating G(jω) along the frequency band will provide information that allows us to determine closed-loop stability from the open-loop G(s).

8. Chapter 6 The Frequency-Response Design Method Frequency Response Given the transfer function of a lead compensation (a) Analytically determine its frequency response characteristics and discuss what you would expect from the result. Φ M = D • At low frequency, ω→0, M→K, Φ→0. • At high frequency, ω→∞, M→K/α, Φ→0. • At intermediate frequency, Φ> 0.

9. Chapter 6 The Frequency-Response Design Method Frequency Response (b) Use MATLAB to plot D(jω) with K = 1, T = 1, and α = 0.1 for 0.1 ≤ ω ≤ 100 and verify the prediction from (a). Using bode(num,den), in this case bode([1 1],[0.1 1]), MATLAB produces the frequency response of the lead compensation.

10. Chapter 6 The Frequency-Response Design Method Frequency Response • For second order system having the transfer function we already plotted the step response for various values of ζ. • The damping and rise time of a system can be determined from the transient-response curve.

11. Chapter 6 The Frequency-Response Design Method Frequency Response • The corresponding frequency response of the system can be found by replacing s = jω • This G(jω) can be plotted along the frequency axis, for various values of ζ.

12. Chapter 6 The Frequency-Response Design Method Frequency Response • The damping of a system can be determined from the peak in the magnitude of the frequency response curve. • The rise time can be estimated from the bandwidth, which is approximately equal to ωn. ? The transient-response curve and the frequency-response curve contain the same information.

13. Chapter 6 The Frequency-Response Design Method Frequency Response • Bandwidth is defined as the maximum frequency at which the output of system will track an input sinusoid in a satisfactory manner. • By convention, the bandwidth is the frequency at which the output is attenuated to a factor of 0.707 times the input. • The maximum value of the frequency-response magnitude is referred to as the resonant peak Mr.

14. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques • Advantages of working with frequency response in terms of Bode plots: • Dynamic compensator design can be based entirely on Bode plots. • Bode plots can be determined experimentally. • Bode plots of systems in series can be simply added, which is quite convenient. • The use of a logarithmic scale permits a much wider range of frequencies to be displayed on a single plot compared with the use of linear scales. • Bode plot of a system is made of two curves, • The logarithm of magnitude vs. the logarithm of frequency, log M vs. log ω, or also Mdb vs. log ω. • The phase versus the logarithm of frequencyΦ vs. log ω.

15. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques • For root locus design method, the open-loop transfer function is written in the form • For frequency-response design method, s is replaced with jω to write the transfer function in the Bode form

16. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques • To draw the Bode plot of a transfer function, it must be rewritten in magnitude equation and phase equation, i.e., • Then Phase Equation Magnitude Equation Magnitude Equation (log) Magnitude Equation (db)

17. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques • Examining all the transfer functions we have dealt with so far, all of them are the combinations of the following four terms: • Gain • Pole or zero at the origin • Simple pole or zero • Quadratic poles or zeros • Once we understand how to plot each term, it will be easy to draw the composite plot, since log M and Φ are the additive combination of the magnitude logarithms and the phases of all terms.

18. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques Gain Magnitude Phase

19. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques Pole or zero at the origin Magnitude Phase

20. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques • As example, Bode plot of a zero (jω) at origin will be as follows:

21. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques Simple pole or zero • The point whereωτ= 1 or ω = 1/τ is called the break point.

22. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques • As example, Bode magnitude plot of a simple zero (jωτ+1) is given below, with τ = 10. • The break point lies at ω = 1/τ = 0.1. • Correction of Asymptote

23. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques • The corresponding Bode phase plot of a simple zero (jωτ+1) is given as: • Corrections of Asymptotes by 11°,at ω = 0.02 and ω = 0.5. • Corresponds to 1/5ωbreak and 5ωbreak

24. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques Quadratic poles or zeros • Asymptotes can be used for rough sketch. • Afterwards, correction must be made according to the value of damping factor ζ.

25. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques

26. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques

27. Chapter 6 The Frequency-Response Design Method Bode Plot: Example Plot the Bode magnitude and phase for the system with the transfer function • ωb1=0.5, ωb2=10, and ωb3=50 Convert the function to the Bode form, • One pole at the origin 5 terms will be drawn separately and finally composited

28. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques ωb1 = 0.5 ωb2 = 10 ωb3 = 50 60 40 –20 db/dec 0 db/dec 20 –20 db/dec db 0 –40 db/dec –20 –40 : Rough composite

29. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques ωb1 = 0.5 ωb2 = 10 ωb3 = 50 60 40 +3 db 20 –3 db db –3 db 0 –20 –40 Final Result : Rough composite : 3-db-corrected composite

30. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques ωb3 = 50 ωb2 = 10 ωb1 = 0.5 2 2.5 –11° –11° +11° +11° 10 0.1 –11° 50 +11° 250

31. Chapter 6 The Frequency-Response Design Method Bode Plot Techniques ωb3 = 50 ωb2 = 10 ωb1 = 0.5 Final Result

32. Chapter 6 The Frequency-Response Design Method Homework 8 • No.1, FPE (5th Ed.), 6.3. • Hint: Draw the Bode plot in logarithmic and semi-logarithmic scale accordingly. • No.2. • Derive the transfer function of the electrical system given above. • If R1 = 10 kΩ , R2 = 5 kΩ and C = 0.1 μF, draw the Bode plot of the system. • Deadline: 06.11.2012, at 07:30.