CONTROL SYSTEM: BODE DIAGRAMS AND TRANSFER FUNCTIONS
A dynamic system could be stable or unstable. Even if such a system is stable, its performance might not be satisfactory. Therefore, in general one can say that all such systems would need some regulation, regulatory mechanism, or control of the variables to improve and enhance the performance of the system. A system that functions under partial of full supervision or control of some ‘‘control mechanism’’ (which could be regulatory control, feedback control, or feed forward control) is in every sense called a control system. In general control system analysis, say for an airplane, can be carried out using frequency-domain or time-domain methods. Frequency-domain methods are based on Bode diagrams, root locus, Nichols charts, and transfer functions. Direct time-domain analysis can be done using state-space methods. Due to easy availability of high-speed computers, any time-consuming technique can be easily adapted for analysis, design, and evaluation of control systems. The MATLAB/SIMULINK SW tool and its tool kit are convenient ways for such analysis. These tools can be easily learned and practiced and can save a lot of time and effort in carrying out detailed analysis and validation of control system designs.
The transfer function (Equation 2.1) for linear systems is defined in terms of Laplace transforms (LT). For a function/(f), the LT is defined as L(s) = J0° f (t)e~st df. For example, the LT of unit step input u is given as 1/s; s is the complex frequency s = s + jv. We can see that the function f(t) need not be periodic. The inverse LT is defined as f (t) = щ 1^°° L(s)est ds. If s in the TF is replaced by jv, we obtain the complex numbers with respect to frequency. If the input is a sign wave, then we get the steady-state output as a sign wave of the same frequency, but with different magnitudes and phase angle differences between output and input waves. The plot of these complex numbers with respect to frequency is called the frequency response.
Bode diagram is a plot of magnitude (or amplitude ratio) and the phase angle (phase difference between the output and input) of a TF vs. frequency (Figure 2.4). In definition, the Bode plot is the result of the plot of magnitude and phase of poles and zeros of the TF with respect to frequency in logarithmic coordinates. The simplification in Bode plots is because in logarithmic representation, multiplication and division are replaced by addition and subtraction, respectively. In a filter or control system’s Bode diagram, the cutoff frequency is the point where the response is 3 dB down in amplitude from the level of the pass band. Beyond this frequency, the filter will attenuate (the amplitude) at all other frequencies.