Performance measurement of a control system

The time domain characteristics are used to measure the control system' performance in a more realistic way. And the explanation for this is that most control system's performance is assessed on the basis of the time responses because of certain test signals. This is in distinction to the Communication system's analysis and design in which the frequency response is more important, as the maximum of the signals need to be processed are either composed of sinusoidal components or sinusoidal. In case of design problems, no unified methods of arriving at a designed system are there that fulfills the time-domain performance specifications, like rise time, maximum overshoot, settling time, delay time, and many more. Other than this, in the frequency domain, the graphical method's wealth is available that are not restricted to the low-order systems. 

It is necessary to understand that the time-domain performances and the frequency-domain are interconnected with each other in a linear system, hence the predictions on the system's time-domain properties can be done depending on the on the frequency-domain characteristics. The domain of frequency is very convenient for measuring the sensitivity of a system to parameter variations and noise. Having such concepts, we think that convenience and the availability of the existing analytical tools are main motivations for conducting control systems analysis and design in the frequency domain. Another reason is that it has given a second point of view to the problems of control-system, which usually gives important information in the control system's complex analysis and design. So, to conduct the linear control system's frequency-domain analysis does not mean that the system is only sinusoidal input's subject.

With the help of the frequency-response studies,we can project the system's time-domain performance. the system's schematic and demonstrated this step for a position control system. In order to get a schematic, the engineers of control system have to make many simplifying assumptions so as to keep the ensuing model manageable and also can approximate the physical reality. The other step is to make the mathematical models from the schematics of physical systems. Here we will be discussing two methods: (1) transfer functions in the frequency domain and (2) state equations in the time domain.

As we move further, we will see that the first step in every case for developing a mathematical model is the use of the fundamental physical  laws of science and engineering. For instance, when electrical networks are modeled, Kirchhoff's laws and ohm's law, which are the important basic electronic network's laws should be applied first. We will add voltages in a loop or will sum the currents at a node. At the time of studying the mechanical systems, we have to use Newton's laws as the fundamental guiding principles. Here we will add up all the torques and forces. With all these equations we will get the relationship between the input and output of the system.

The structure of the differential equation with its coefficients are the description of the system. Even though with the use of the differential equation the relation between the system's input and output can be made but, it is an unsatisfying representation with a system's perception. We can see that the parameters of system, like the coefficient, the inputs,r(t) and the output,c(t), are seen throughout the equation. It is better to go for the mathematical representation like in Figure 2.1(a), where the system, output and the input are the separate and distinct parts. Also we can represent easily the connection between many subsystems.

Like, we can show cascaded interconnections, as shown in Figure 2.1(b), where there will be a mathematical function, known as the transfer function, is within each block and the block functions can be combined to get Figure 2.1 (a) for analysis and design. This convenience cannot be obtained with the differential equation.

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