Mathematical Models of Systems
2.1. Introduction
In order to understand and control complex systems, one must obtain quantitative mathematical models of these systems. Therefore it is necessary to analyze the relationships between the systems variables and to obtain a mathematical model. Because the systems under consideration are dynamic in nature, the descriptive equations are usually differential equations. In practice, the complexity of systems and ignorance of all the relevant factors necessitate the introduction of assumptions concerning the system operation. Therefore, we shall often find it useful to consider the physical system, delineate some necessary assumptions, and linearize the system. Then by using the physical laws describing the linear equivalent system, we a set of linear differential equations can be obtained. Finally, utilizing mathematical tools, such as Laplace transform, classical or numerical methods, we obtain a solution describing the operation of the system. In summary, the approach to dynamic system problems can be listed as follows [1]:
1. define the system and its components;
2. formulate the mathematical model and list the necessary assumptions;
3. write the differential equations describing the model;
4. solve the equations for the desired output variables;
5. examine the solutions and the assumptions; and than
6. reanalyze or design.
Block Diagram Models
The mathematical representations of control systems are usually represented by block diagrams. These diagrams have the advantage of indicating more realistically the actual processes which are taking place, as opposed a purely abstract mathematical representation. In addition, it is easy to form the over-all block diagram for the entire system by merely combining the block diagrams for each component or part of the system. It is common to use a block diagram in which each element in the system is represented by a block, as shown in Fig. 2.1. Each block is labeled with the name of the component, and a line is drawn from one block to the next. Arrows are used to show the direction of the flow of information. The block represents the function or characteristics, which characterize either dynamic behavior or steady-state operation of the component.
To investigate the performance of a control system, it is necessary to obtain the mathematical relationship relating the controlled variable and actuating signal. The relationship between the actuating signal x(t), which enters the control system, and the controlled variable y(t), which is the output of the control, as shown in Fig. 2.1, is expressed by the equation [2,4]
, (2.1)
where W represents the operation of the control system. A box is the symbol for multiplication. In this case the input quantity x(t) is multiplied by the function in the box W to obtain the output y(t).
Fig. 2.1. Control system.
The quantity W could be obtained by writing the mathematical equations describing the operation of each component between x(t) and y(t) and then combining these individual equations algebraically to obtain the over-all relationship between x(t) and y(t). It is shown below how the actual values of W for the specific control systems can be obtained.
Control systems require the arithmetic manipulation of addition or subtraction. In a block diagram a circle is used as a junction point or comparator, as shown in Fig.2.2. The sign at each input arrowhead indicates whether the quantity is to be added or subtracted.
A general representation for a feedback control system is shown in Fig. 2.3. It is to be noticed that the command signal, or desired input, does not usually go directly to the comparator but must be converted to a suitable input for this device. Similarly, the controlled variable, or output, in a general case must also be changed by the feedback elements before it can be measured by the comparator. The actuating signal is amplified by control elements before entering the controlled system. An external disturbance, as shown in Fig.2.3, is a disturbance, which acts independently to affect the operation of the system. Although in Fig.2.3 the external disturbance is shown entering the system between the control elements and the controlled system, in general, the external disturbance may enter the system at any point. It is also to be noticed from this generalized representation of a control system that the controlled variable is not necessarily the quantity, which is desired to control.
Fig. 2.3. Generalized feedback control system.
A block diagram is not involved with physical characteristics of the system, but only with the functional relationship between various points in the system. So it is clearly that apparently different physical systems may be analyzed by the same techniques.