Frequency Domain Analysis of Control Systems and Time Domain Analysis - Term Paper Example

Frequency domain analysis of control systems is better than time domain analysis Department, institution Acknowledges: Contact information: Abstract Signal processing has the primary goal of providing the underlying information on the specific problems for the purposes of decision making. In control systems, control theory is normally used both in the analysis and design of feedback systems. Such systems include those that regulate temperature, fluid flow, force, motion, pressure, voltage, tension and current. The control theory when skillfully used, guides engineers in all the phases of the product and process design cycles. Additionally, it helps the engineers to predict performance and anticipate problems thus providing solutions. The main role of using the control systems is to improve the process or a working machine. This paper explores the two main types of analysis used in control systems. They are frequency domain analysis and time domain analysis. Discussion is made on the two analysis methods, to show how frequency domain analysis is better in regard to the output power of the system, system behavior and signal transformations. Key words: control systems, frequency domain analysis, time domain analysis, control theory Frequency domain analysis of control systems is better than time domain analysis Control system refers to the use of algorithms and feedback in engineered systems. Control system includes such examples as feedback loops in set point controllers and even router protocols controlling traffic flow on the internet. Emerging applications in control systems include high-confidence software systems, as noted by (Achanta, Hemami, Estrada, & Susstrunk, 2009). Control is, therefore, an information science including the use of data in both analog and digital representations (Achanta et al., 2009). In these control systems, electric signals have both time and frequency domain representations. Frequency domain analysis is a cornerstone of signal and system analysis, and in most cases it is analyzed using Fourier transforms. Frequency domain analysis is the most commonly applied in fields such as electrical engineering, science as well as several areas of mathematics (Takarada et al., 2010). Literature review Both terms are associated with the study of signals and systems. The terms refer to an independent variable(x in the functional relationship and y=f(x)) used in the two-dimensional identification of the signal. Most of the industrial control systems are designed using frequency response methods. Frequency response refers to the steady-state response of the control system to a sinusoidal input. For a system with sinusoidal input r(t)=A sin wt. A steady state output could be written as c(t)=B sin. Frequency response refers to the magnitude and the phase relationship occurring between the sinusoidal input and the steady-state output of a system (Takarada et al., 2010). For example, a signal r(t)=3 sin 4t-2cos 5t is represented as a graph of values of r against time(t), the representation is in the time domain. r(t)=3 sin 4t-2 cos wt Correlation between the two systems is defined by bandwidth while overshoot correlates to peaking. It is worth noting that in the real systems, time domain is more difficult to interpret. In control systems, many phenomena will occur in combinations of frequencies; for instance there may be a mild resonance of 400Hz and peaking at 60Hz. Feedback resolution, on the other hand, may have a limitation on the interpretation of the response to small amplitude steps. Moreover, time-domain measures rely so much on the step command (Suja & Jerome 2010). This is because the step command has rich frequency content. It is characterized by high frequencies from the steep edges and low frequencies due to the fixed value between edges. Like a Bode plot, step commands will excite the system across a broad range of frequencies. Resultantly, the step response offers an alternative to the frequency response (Takarada et al., 2010). Steady-state response of the system is the frequency response of the system to a sinusoidal/harmonic input. In linear systems, the resulting output is harmonic, differing from the input in amplitude and phase only. An illustration is given below β represents the multiplication factor for the magnitude, and represents a relative phase shift between the input and the output. If β is greater than 1.the system will amplify the input, and when β is less than 1, the system will attenuate the input (Takarada et al., 2010). A common way for representing the frequency response of a linear system is by use of a Bode diagram. The bandwidth (BW) is the frequency at which the system is 3 Decibels which is down from its low-frequency value. The system responds at 71% of its low-frequency value. The illustration is given in the figure below. For a system with a second order and a damping ratio less than 0.707, the resonant frequency and resonant magnitude will be estimated using the equations below. The equations can be employed for any pair of complex poles, in higher ranking systems that are separated from other poles and zeros. In this particular case, Mr represents a rise in magnitude from the pre-resonance value. It is important to note that the characteristics in the frequency-domain correlate with the behavior of the system in the time-domain. As a result, the resonant magnitude Mr will paint a picture of the relative stability. Large values of Mr indicate low damping, which suggests oscillatory response with large overshoots. Systems with large bandwidths will exhibit faster response systems than the ones with smaller bandwidths. This may however have a higher sensitivity and much noise. A combination of bandwidth and the rate of decay of the magnitude at high frequencies will determine the sensitivity to noise (Suja & Jerome, 2010). Response can as well be measured in the frequency domain by inspection of a gain plot. Most of the control systems will demonstrate a good command response at low frequencies but become very unresponsive at higher frequencies. At low frequencies, the controller will be able to govern the system but once the frequency increases, the controller would no longer have the control of the system, according to (Suja & Jerome, 2010). At low frequencies, the gain will be one but at higher frequencies, it will be much higher than one. For instance, a power supply set to produce sine wave voltages of up to 100Hz, would be expected that the power supply is almost ideal at low frequencies such as 1Hz. As the frequency approaches 100Hz, it will produce almost no voltage at higher frequencies of about 10kHz.When this is plotted on a Bode plot. The gain would be expected to be unity (0dB) at low frequencies, and it will then start to fall to very low values at high frequencies (Sato et al., 2009). It offers an alternative to time domain system identification for linear time-invariant systems. It is, therefore, preferred over time domain techniques since it is straightforward in obtaining good results in the presence of noise. It is useful when doing control design directly in the frequency domain. That is Lead Design, Lag Design, Nyquist Stability, Phase & Gain Margin (Achanta et al., 2009). Discussion From time to time, delays are experienced downloading web pages from the internet. This is as a result of factors such as traffic on the internet and the amount of data on the requested page. Images on the other hand require a large amount of data, making downloading of images to be slow. Therefore, when downloading a collection of images from a database of images can take be time-consuming. To solve the problems of downloading, frequency domain ideas are employed. The notion of hierarchical representation can be formalized by using ideas from the frequency domain analysis (Sato et al., 2009). Radio and TV transmit information via electromagnetic waves. The various sources may be transmitting simultaneously and in the same geographic region but when tuning, instead of hearing a specific radio station or television program one hears a jumble of all radio stations put together. This is because different radio transmissions use different frequencies and therefore even though all the signals are mixed in the time domain, and they will be distinct in the frequency domain. With basic frequency domain processing, it will be easy to separate signals and tune into the desired frequency. Something seemingly very complicated in the time domain can be rather simple in the frequency domain. Fourier transforms and frequency domain analysis are very useful in frequency representations of longer signals (Achanta et al., 2009). Conclusion To sum this discussion up, frequency domain provides an intuition on practical subjects cutting across the field of control systems. It will be used to quantify the responsiveness, stability, tuning in gain and rejection to disturbances by the control systems (Suja & Jerome 2010). A control engineer should be fluent in both frequency-domain and time-domain analysis for him/her to decide on the analysis to adopt. References Achanta, R., Hemami, S., Estrada, F., & Susstrunk, S. (2009, June). Frequency-tuned salient region detection. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on (pp. 1597-1604). IEEE. Available online at http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=5206596&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D5206596. DOI: 10.1109/CVPR.2009.5206596 Sato, J. R., Takahashi, D. Y., Arcuri, S. M., Sameshima, K., Morettin, P. A., & Baccalá, L. A. (2009). Frequency domain connectivity identification: an application of partial directed coherence in fMRI. Human brain mapping, 30(2), 452-461. Available online at http://onlinelibrary.wiley.com/doi/10.1002/hbm.20513/abstract?deniedAccessCustomisedMessage=&userIsAuthenticated=false. DOI: 10.1002/hbm.20513 Suja, S., & Jerome, J. (2010). Pattern recognition of power signal disturbances using S Transform and TT Transform. International journal of electrical power & energy systems, 32(1), 37-53. Available online at http://www.sciencedirect.com/science/article/pii/S014206150900101X. DOI: 10.1016/j.ijepes.2009.06.012 Takarada, S., Imanishi, T., Liu, Y., Ikejima, H., Tsujioka, H., Kuroi, A., ... & Akasaka, T. (2010). Advantage of next‐generation frequency‐domain optical coherence tomography compared with conventional time‐domain system in the assessment of coronary lesion. Catheterization and Cardiovascular Interventions, 75(2), 202-206. Available online at http://onlinelibrary.wiley.com/doi/10.1002/ccd.22273/abstract;jsessionid=E36A0705A987C36070C96B2A9C192483.f04t03?deniedAccessCustomisedMessage=&userIsAuthenticated=false. DOI: 10.1002/ccd.22273 Read More