Frequency-Domain Control Systems Eng R. L. Nkumbwa Copperbelt University School of Technology 2010

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7/3/2015Eng R. L. CBU Its all Stability of Control Systems

7/3/2015Eng R. L. CBU Frequency Response Roadmap We will cover the following: – General frequency analysis in Control Engineering

7/3/2015Eng R. L. CBU Introduction In practice, the performance of a control system is more realistically measured by its time domain characteristics. The reason is that the performance of most control systems is judged based on the time response due top certain test signals. In the previous chapters, we have learnt that the time response of a control system is usually more difficult to determine analytically, especially for higher order systems.

7/3/2015Eng R. L. CBU Introduction In design problems, there are no unified methods of arriving at a designed system that meets the time-domain performance specifications, such as: – Maximum overshoot, – Rise time, – Delay time, – Settling time and so on.

7/3/2015Eng R. L. CBU Introduction On the other hand, in frequency domain, there is a wealth of graphical methods available that are not limited to low order systems. It is important to realize that there are correlating relations between frequency domain performance in a linear system, Such that the time domain properties of the system can be predicted based on the frequency-domain characteristics.

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7/3/2015Eng R. L. CBU Example: Gun Positional Control

7/3/2015Eng R. L. CBU Why use Frequency-Domain? With the previous concepts in mind, we can consider the primary motivation for conducting control systems analysis and design in the frequency domain to be convenience and the availability of the existing analytical tools. Another reason, is that, it presents an alternative point of view to control system problems, which often provides valuable or crucial information in the complex analysis and design of control systems.

7/3/2015Eng R. L. CBU Characteristics of Frequency Response Frequency response methods are a good complement to the root locus techniques:– – Can infer performance and stability from the same plot – Can use measured data rather than a transfer function model – Design process can be independent of the system order – Time delays are handled correctly – Graphical techniques (analysis and synthesis) are quite simple.

7/3/2015Eng R. L. CBU Frequency-Domain Analysis The starting point for frequency-domain analysis of a linear system is its transfer system.

7/3/2015Eng R. L. CBU Time & Frequency-Domain Specs. So, what are time-domain specifications by now? Am sure u all know what they are? Ok, what of frequency domain specifications? What are they? Lets look at the pictorials views…

7/3/2015Eng R. L. CBU Time-Domain Specifications

7/3/2015Eng R. L. CBU Frequency-Domain Specifications

7/3/2015Eng R. L. CBU Wrap Up… The frequency response of a system directly tells us the relative magnitude and phase of a system’s output sinusoid, if the system input is a sinusoid. What about output frequency? If the plant’s transfer function is G (s), the open-loop frequency response is G (jw).

7/3/2015Eng R. L. CBU Further Frequency Response In previous sections of this course we have considered the use of standard test inputs, such as step functions and ramps. However, we will now consider the steady- state response of a system to a sinusoidal input test signal.

7/3/2015Eng R. L. CBU Further Frequency Response The response of a linear constant-coefficient linear system to a sinusoidal test input is an output sinusoidal signal at the same frequency as the input. However, the magnitude and phase of the output signal differ from those of the input sinusoidal signal, and the amount of difference is a function of the input frequency.

7/3/2015Eng R. L. CBU Further Frequency Response We will now examine the transfer function G(s) where s = jw and graphically display the complex number G(jw) as w varies. The Bode plot is one of the most powerful graphical tools for analyzing and designing control systems, and we will also consider polar plots and log magnitude and phase diagrams.

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7/3/2015Eng R. L. CBU Further Frequency Response How is this different from Root Locus? The information we get from frequency response methods is different than what we get from the root locus analysis. In fact, the two approaches complement each other. One advantage of the frequency response approach is that we can use data derived from measurements on the physical system without deriving its mathematical model.

7/3/2015Eng R. L. CBU Further Frequency Response What is the Importance of Frequency methods? They are a powerful technique to design a single-loop feedback control system. They provide us with a viewpoint in the frequency domain. It is possible to extend the frequency analysis idea to nonlinear systems (approximate analysis).

7/3/2015Eng R. L. CBU Who Developed Frequency Methods? Bode Nyquist Nichols And others In the 1930s and 1940s. Existed before root locus methods.

7/3/2015Eng R. L. CBU Variety of Frequency domain Analysis Bode Plot – Log |G(jω)| and Phase of G(jω) vs. Log frequency. – Simplest tool for visualization and synthesis – Typically plot 20log|G| which is given the symbol dB Polar (Nyquist) plot – Re vs.Im of G(jω) in complex plane. – Hard to visualize, not useful for synthesis, but gives definitive tests for stability and is the basis of the robustness analysis.

7/3/2015Eng R. L. CBU Variety of Frequency domain Analysis Nichols Plot – |G(jω)| vs. Phase of G(jω), which is very handy for systems with lightly damped poles.

7/3/2015Eng R. L. CBU What are the advantages? We can study a system from physical data and determine the transfer function experimentally. We can design compensators to meet both steady state and transient response requirements. We can determine the stability of nonlinear systems using frequency analysis. Frequency response methods allow us to settle ambiguities while drawing a root locus plot. A system can be designed so that the effects of undesirable noise are negligible.

7/3/2015Eng R. L. CBU What are the disadvantages? Frequency response techniques are not as intuitive as root locus. Find more cons

7/3/2015Eng R. L. CBU Concept of Frequency Response The frequency response of a system is the steady state response of a system to a sinusoidal input. Consider the stable, LTI system shown below.

7/3/2015Eng R. L. CBU Characteristics of Frequency Domain

7/3/2015Eng R. L. CBU Concept of Frequency Response The input-output relation is given by:

7/3/2015Eng R. L. CBU Concept of Frequency Response

7/3/2015Eng R. L. CBU Obtaining Frequency Response

7/3/2015Eng R. L. CBU Concept of Frequency Response Obtaining Magnitude M and Phase Ø

7/3/2015Eng R. L. CBU Concept of Frequency Response For linear systems, M and Ø depend only on the input frequency, w. So, what are some of the frequency response plots and diagrams?

7/3/2015Eng R. L. CBU Frequency Response Plots and Diagrams There are three frequently used representations of the frequency response: Nyquist diagram: a plot on the complex plane (G(jw)-plane) where M and Ø are plotted on a single curve, and w becomes a hidden parameter.

7/3/2015Eng R. L. CBU Frequency Response Plots and Diagrams Bode plots: separate plots for M and Ø, with the horizontal axis being w is log scale. The vertical axis for the M-plot is given by M is decibels (db), that is 20log 10 (M), and the vertical axis for the Ø -plot is Ø in degrees.

7/3/2015Eng R. L. CBU Plotting Bode Plots

7/3/2015Eng R. L. CBU Amplitude Ratio (AR) on log-log plot – Start from steady-state gain at ω=0. If G OL includes either integrator or differentiator it starts at infinity or 0. – Each first-order lag (lead) adds to the slope –1 (+1) starting at the corner frequency. – Each integrator (differentiator) adds to the slope –1 (+1) starting at zero frequency. – A delays does not contribute to the AR plot.

7/3/2015Eng R. L. CBU Phase angle on semi-log plot Start from 0°or -180°at ω =0 depending on the sign of steady- state gain. Each first-order lag (lead) adds 0°to phase angle at ω =0, adds -90°(+90°) to phase angle at ω = ∞, and adds -45°(+45°)to phase angle at corner frequency. Each integrator (differentiator) adds -90°(+90°)to the phase angle for all frequency. A delay adds -ωθ to phase angle depending on the frequency.

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7/3/2015Eng R. L. CBU Try Solving the Following Using Bode Technique

7/3/2015Eng R. L. CBU Nyquist Diagram or Analysis The polar plot, or Nyquist diagram, of a sinusoidal transfer function G(jw) is a plot of the magnitude of G(jw) versus the phase angle of G(jw) on polar coordinates as w is varied from zero to infinity. Thus, the polar plot is the locus of vectors |G(jw)| LG(jw) as w is varied from zero to infinity.

7/3/2015Eng R. L. CBU Nyquist Diagram or Analysis The projections of G(jw) on the real and imaginary axis are its real and imaginary components. The Nyquist Stability Criteria is a test for system stability, just like the Routh-Hurwitz test, or the Root-Locus Methodology.

7/3/2015Eng R. L. CBU Nyquist is an alternative representation of frequency response Compact (one plot) Wider applicability of stability analysis than Bode plot High frequency characteristics will be shrunk near the origin. Inverse Nyquist diagram: polar plot of G(jw) Combination of different transfer function components is not easy as with Nyquist diagram as with Bode plot.

7/3/2015Eng R. L. CBU Nyquist Diagram or Analysis Note that in polar plots, a positive (negative) phase angle is measured counterclockwise (clockwise) from the positive real axis. In the polar plot, it is important to show the frequency graduation of the locus. Routh-Hurwitz and Root-Locus can tell us where the poles of the system are for particular values of gain.

7/3/2015Eng R. L. CBU Nyquist Diagram or Analysis By altering the gain of the system, we can determine if any of the poles move into the RHsP, and therefore become unstable. However, the Nyquist Criteria can also give us additional information about a system. The Nyquist Criteria, can tell us things about the frequency characteristics of the system.

7/3/2015Eng R. L. CBU Nyquist Diagram or Analysis For instance, some systems with constant gain might be stable for low-frequency inputs, but become unstable for high-frequency inputs. Also, the Nyquist Criteria can tell us things about the phase of the input signals, the time-shift of the system, and other important information.

7/3/2015Eng R. L. CBU Nyquist Kuo’s View Kuo et al (2003) suggests that, the Nyquist criterion is a semi-graphical method that determines the stability of a closed loop system by investigating the properties of the frequency domain plot, the Nygmst plot of L(s) which is a plot of L(jw) in the polar coordinates of M [L(jw)] versus Re[L(jw)] as ω varies from 0 to ∞.

7/3/2015Eng R. L. CBU Nyquist Xavier’s View While, Xavier et al (2004) narrates that, the Nyquist criterion is based on “Cauchy’s Residue Theorem” of complex variables which is referred to as “Principle of Argument”.

7/3/2015Eng R. L. CBU The Argument Principle If we have a contour, Γ (capital gamma), drawn in one plane (say the complex laplace plane, for instance), we can map that contour into another plane, the F(s) plane, by transforming the contour with the function F(s). The resultant contour, Γ F(s) will circle the origin point of the F(s) plane N times, where N is equal to the difference between Z and P (the number of zeros and poles of the function F(s), respectively).

7/3/2015Eng R. L. CBU Nyquist Criterion Let us first introduce the most important equation when dealing with the Nyquist criterion: Where: – N is the number of encirclements of the (-1, 0) point. – Z is the number of zeros of the characteristic equation. – P is the number of poles of the open-loop characteristic equation.

7/3/2015Eng R. L. CBU Nyquist Stability Criterion Defined A feedback control system is stable, if and only if the contour ΓF(s) in the F(s) plane does not encircle the (-1, 0) point when P is 0. A feedback control system is stable, if and only if the contour ΓF(s) in the F(s) plane encircles the (-1, 0) point a number of times equal to the number of poles of F(s) enclosed by Γ.

7/3/2015Eng R. L. CBU Nyquist Stability Criterion Defined In other words, if P is zero then N must equal zero. Otherwise, N must equal P. Essentially, we are saying that Z must always equal zero, because Z is the number of zeros of the characteristic equation (and therefore the number of poles of the closed-loop transfer function) that are in the right-half of the s plane.

7/3/2015Eng R. L. CBU Nyquist Manke’s View While Manke (1997) outlines that, the Nyquist criterion is used to identify the presence of roots of a characteristic equation of a control system in a specified region of s-plane. He further adds that although the purpose of using Nyquist criterion is similar to RHC, the approach differs in the following respect:

7/3/2015Eng R. L. CBU Nyquist Manke’s View Cont… – The open loop transfer G(s) H(s) is considered instead of the closed loop characteristic equation 1 + G(s) H(s) = 0 – Inspection of graphical plots G(s) H(s) enables to get more than YES or NO answer of RHC pertaining to the stability of control systems.

7/3/2015Eng R. L. CBU Kuo’s Features of Nyquist Criterion Kuo also outlines the following as the features that make the Nyquist criterion an attractive alternative for the analysis and design of control systems: – In addition to providing the absolute stability, like the RHC, the NC also gives information on the relative of a stable system and the degree of instability. – The Nyquist plot of G(s) H(s) or of L (s) is very easy to obtain.

7/3/2015Eng R. L. CBU Kuo’s Features of Nyquist Criterion – The Nyquist plot of G(s) H(s) gives information on the frequency domain characteristics such as Mr, Wr, BW and others with ease. – The Nyquist plot is useful for systems with pure time delay that cannot be treated with the RHC and are difficult to analyze with root locus method.

7/3/2015Eng R. L. CBU Benefits of Frequency Response Frequency responses are the informative representations of dynamic systems Example of an Audio Speaker

7/3/2015Eng R. L. CBU Benefits of Frequency Response Lets now look at a Mechanical or Civil Engineering example of frequency domain, say a structure like a bridge.

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7/3/2015Eng R. L. CBU Frequency Stability Tests Want tests on the loop transfer function L(s)=Gc(s)G(s) that can be performed to establish stability of the closed-loop system

7/3/2015Eng R. L. CBU Frequency Stability Tests

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7/3/2015Eng R. L. CBU Any more worries about freqtool…

7/3/2015Eng R. L. CBU