Feedback Control Systems Lecture 8 Dr Ing Erwin

Feedback Control Systems Lecture 8 Dr. -Ing. Erwin Sitompul President University http: //zitompul. wordpress. com 2 0 1 3 President University Erwin Sitompul FCS 8/1

Feedback Control Systems Chapter 6 The Frequency-Response Design Method President University Erwin Sitompul FCS 8/2

Chapter 6 The Frequency-Response Design Method Frequency Response n A linear system’s response to sinusoidal inputs is called the system’s frequency response. n Frequency response can be obtained from knowledge of the system’s pole and zero locations. n Consider a system described by where n With zero initial conditions, the output is given by President University Erwin Sitompul FCS 8/3

Chapter 6 The Frequency-Response Design Method Frequency Response n 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 President University Erwin Sitompul FCS 8/4

Chapter 6 The Frequency-Response Design Method Frequency Response n Examine Magnitude Phase n But the input is n Therefore • Meaning? President University Erwin Sitompul FCS 8/5

Chapter 6 The Frequency-Response Design Method Frequency Response n 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 steadystate, exhibit a sinusoidal output with a magnitude M(ω0) and a phase Φ(ω0) at the frequency ω0. n 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). n 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. President University Erwin Sitompul FCS 8/6

Chapter 6 The Frequency-Response Design Method Frequency Response n 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. n 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). President University Erwin Sitompul FCS 8/7

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 President University • At low frequency, ω→ 0, M→ K , Φ→ 0. • At high frequency, ω→∞, M→ K/α , Φ→ 0. • At intermediate frequency, Φ > 0. Erwin Sitompul FCS 8/8

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. President University Erwin Sitompul FCS 8/9

Chapter 6 The Frequency-Response Design Method Frequency Response n 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. President University Erwin Sitompul FCS 8/10

Chapter 6 The Frequency-Response Design Method Frequency Response n The corresponding frequency response of the system can be found by replacing s = jω n This G(jω) can be plotted along the frequency axis, for various values of ζ. President University Erwin Sitompul FCS 8/11

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 frequencyresponse curve contain the same information. President University Erwin Sitompul FCS 8/12

Chapter 6 The Frequency-Response Design Method Frequency Response n Bandwidth is defined as the maximum frequency at which the output of system will track an input sinusoid in a satisfactory manner. n By convention, the bandwidth is the frequency at which the output is attenuated to a factor of 0. 707 times the input. n The maximum value of the frequency-response magnitude is referred to as the resonant peak Mr. President University Erwin Sitompul FCS 8/13

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques n Advantages of working with frequency response in terms of Bode plots: 1. Dynamic compensator design can be based entirely on Bode plots. 2. Bode plots can be determined experimentally. 3. Bode plots of systems in series can be simply added, which is quite convenient. 4. 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. n Bode plot of a system is made of two curves, 1. The logarithm of magnitude vs. the logarithm of frequency, log M vs. log ω, or also Mdb vs. log ω. 2. The phase versus the logarithm of frequency Φ vs. log ω. President University Erwin Sitompul FCS 8/14

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques n For root locus design method, the open-loop transfer function is written in the form n For frequency-response design method, s is replaced with jω to write the transfer function in the Bode form President University Erwin Sitompul FCS 8/15

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques n To draw the Bode plot of a transfer function, it must be rewritten in magnitude equation and phase equation, i. e. , n Then Phase Equation Magnitude Equation (log) Magnitude Equation (db) President University Erwin Sitompul FCS 8/16

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques n 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 n 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. President University Erwin Sitompul FCS 8/17

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques Gain Magnitude Phase President University Erwin Sitompul FCS 8/18

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques Pole or zero at the origin Magnitude Phase President University Erwin Sitompul FCS 8/19

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques n As example, Bode plot of a zero (jω) at origin will be as follows: President University Erwin Sitompul FCS 8/20

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques Simple pole or zero Approximation Magnitude Phase • The point where ωτ = 1 or ω = 1/τ is called the break point. President University Erwin Sitompul FCS 8/21

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques n 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 President University Erwin Sitompul FCS 8/22

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques n 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 President University Erwin Sitompul FCS 8/23

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. President University Erwin Sitompul FCS 8/24

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques President University Erwin Sitompul FCS 8/25

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques 5 President University Erwin Sitompul FCS 8/26

Chapter 6 The Frequency-Response Design Method Bode Plot: Example Plot the Bode magnitude and phase for the system with the transfer function Convert the function to the Bode form, • ωb 1 = 0. 5, ωb 2 = 10, and ωb 3 = 50 • One pole at the origin 5 terms will be drawn separately and finally composited President University Erwin Sitompul FCS 8/27

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques ωb 1 = 0. 5 ωb 2 = 10 ωb 3 = 50 60 – 2 0 d b/ de c 40 – 2 20 0 d b/ de c db 0 db/dec 0 – 4 0 db /d ec – 20 – 40 : Rough composite President University Erwin Sitompul FCS 8/28

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques ωb 1 = 0. 5 ωb 2 = 10 ωb 3 = 50 60 40 +3 db 20 db – 3 db 0 – 20 – 40 : Rough composite Final Result : 3 -db-corrected composite President University Erwin Sitompul FCS 8/29

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques ωb 1 = 0. 5 ωb 2 = 10 ωb 3 = 50 2 2. 5 – 11° +11° 0. 1 10 – 11° +11° 50 +11° 250 President University Erwin Sitompul FCS 8/30

Chapter 6 The Frequency-Response Design Method Bode Plot Techniques ωb 1 = 0. 5 ωb 2 = 10 ωb 3 = 50 Final Result President University Erwin Sitompul FCS 8/31

Chapter 6 The Frequency-Response Design Method Homework 8 n No. 1, FPE (6 th Ed. ), 6. 4. Hint: Draw the Bode plot in logarithmic and semi-logarithmic scale accordingly. n No. 2. (a) Derive the transfer function of the electrical system given above. (b) If R 1 = 10 kΩ , R 2 = 5 kΩ and C = 0. 1 μF, draw the Bode plot of the system. n Due: Thursday, 21 November 2013. President University Erwin Sitompul FCS 8/32