Stability of control systems in frequency domain

Stability of control systems in frequency domain

1. Unit -6 Stability Analysis in Frequency Domain 1 By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering

2. Control Systems Analysis and Design by Frequency Response Method By the term frequency response, we mean the steady-state response of a system to a sinusoidal input. In frequency-response methods, we vary the frequency of the input signal over a certain range and study the resulting response. An advantage of the frequency-response approach is that frequency-response tests are, in general, simple and can be made accurately by use of readily available sinusoidal signal generators and precise measurement equipment. Often the transfer functions of complicated components can be determined experimentally by frequency-response tests. In addition, the frequency-response approach has the advantages that a system may be designed so that the effects of undesirable noise are negligible and that such analysis and design can be extended to certain nonlinear control systems. By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 2

3. Obtaining Steady-State Outputs to Sinusoidal Inputs. We shall show that the steady-state output of a transfer function system can be obtained directly from the sinusoidal transfer function—that is, the transfer function in which s is replaced by j ω , where ω is angular frequency. Suppose that the transfer function G(s) of the system can be written as a ratio of two polynomials in s; that is The Laplace-transformed output Y(s) of the system is then By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 3

4. For a stable system, –s1, –s2 ,.. – sn have negative real parts. Therefore, as t approaches infinity, the terms , approach zero. Thus the steady-state response become s: By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 4 Y(s) The angle f may be negative, positive, or zero. Similarly, we obtain the following expression for G(– jv ):

5. Where Y=X |G(j ω )| .We see that a stable, linear, time-invariant system subjected to a sinusoidal input will, at steady state, have a sinusoidal output of the same frequency as the input. But the amplitude and phase of the output will, in general, be different from those of the input. By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 5

6. By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 6 It can be seen that for small ω , the amplitude of the steady-state output yss (t) is almost equal to K times the amplitude of the input. The phase shift of the output is small for small ω . For large ω , the amplitude of the output is small and almost inversely proportional to ω . The phase shift approaches –90° as ω approaches infinity. This is a phase-lag network. Example: Consider the given transfer function

7. Bode Plots Bode Diagrams or Logarithmic Plots. A Bode diagram consists of two graphs: One is a plot of the logarithm of the magnitude of a sinusoidal transfer function; the other is a plot of the phase angle; both are plotted against the frequency on a logarithmic scale. Basic Factors of G( j ω )H(j ω ). As stated earlier, the main advantage in using the logarithmic plot is the relative ease of plotting frequency-response curves. The basic factors that very frequently occur in an arbitrary transfer function G(j ω )H(j ω ) are: 1. Gain K 2. Integral and derivative factors (j ω ) 3. First-order factors (1+j ω T) 4. Quadratic factors Once we become familiar with the logarithmic plots of these basic factors, it is possible to utilize them in constructing a composite logarithmic plot for any general form of G(j ω )H(j ω ) by sketching the curves for each factor and adding individual curves graphically, because adding the logarithms of the gains corresponds to multiplying them together. By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 7

8. The Gain K. A number greater than unity has a positive value in decibels, while a number smaller than unity has a negative value. The log-magnitude curve for a constant gain K is a horizontal straight line at the magnitude of 20 logK decibels. The phase angle of the gain K is zero. The effect of varying the gain K in the transfer function is that it raises or lowers the log-magnitude curve of the transfer function by the corresponding constant amount, but it has no effect on the phase curve. By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 8

9. Integral and Derivative Factors ( j ω ). The logarithmic magnitude of 1/j ω in decibels is- The phase angle of 1/j ω is constant and equal to –90°. In Bode diagrams, frequency ratios are expressed in terms of octaves or decades. An octave is a frequency band from ω 1 to 2 ω 1 ,where ω 1 is any frequency value. A decade is a frequency band from ω 1 to 10 ω 1 ,where again v1 is any frequency. If the log magnitude –20 logv dB is plotted against v on a logarithmic scale, it is a straight line. If the transfer function contains the factor the log magnitude becomes, respectively, By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 9

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13. Quadratic Factors : Control systems often possess quadratic factors of the form- The magnitude and phase of the quadratic factor depend on both the corner frequency and the damping ratio . The asymptotic frequency-response curve may be obtained as follows: By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 13

14. By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 14 above.

15. By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 15 If |G( jw )|as a peak value at some frequency, this frequency is called the resonant f requency. Since the numerator of is constant, a peak value of will occur when denominator is minimum.

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17. Log-magnitude curves, together with the asymptotes, and phase-angle curves of the quadratic transfer function By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 17

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19. Note that when the individual asymptotic curves are added at each frequency, the slope of the composite curve is cumulative. Below ω =√2 the plot has the slope of –20 dB/decade. At the first corner frequency ω =√2 the slope changes to –60 dB/decade and continues to the next corner frequency ω =2, where the slope becomes –80 dB/decade. At the last corner frequency ω =3, the slope changes to –60 dB/decade By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 19

20. Initial magnitude of Bode plot By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 20

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23. Nyquist criteria The Nyquist stability criterion determines the stability of a closed-loop system from its open-loop frequency response and open-loop poles. Let closed loop TF is By: Dr.Ankita Malhotra SVKM'S D J Sanghvi College of Engineering 23 must lie in the left-half s plane. [It is noted that, although poles and zeros of the open-loop transfer function G(s)H(s) may be in the right-half s plane, the system is stable if all the poles of the closed-loop transfer function (that is, the roots of the characteristic equation) are in the left-half s plane.] The Nyquist stability criterion relates the open-loop frequency response G( jw )H( jw ) to the number of zeros and poles of 1+G(s)H(s) that lie in the right-half s plane.

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