University:
Kentucky State University
Course:
MAT 172 | Accelerated Contemp. Math
Academic year:

2020
Views:

491

Pages:

13
Author:

Athena Harrison

Transfer Function Approach to Modelling Dynamic Systems In the field of system dynamics, transfer functions are frequently used to characterize the input−output relationships of components or systems that can be described by linear timeinvariant differential equations. Transfer Function The transfer function of a linear timeinvariant differential equation system is defined as the ratio of the Laplace transform of the output (response function) to the Laplace transform of the input (driving function) under the assumption that all initial conditions are zero. 1 Transfer Function Approach to Modelling Dynamic Systems By using the concept of a transfer function, it is possible to represent system dynamics by algebraic equations in 𝑠. If the highest power of 𝑠 in the denominator of the transfer function is equal to 𝑛, the system is called an nth-order system. 2 Transfer Function Approach to Modelling Dynamic Systems Comments on the Transfer Function The applicability of the concept of the transfer function is limited to linear time-invariant differential-equation systems. Still, the transfer-function approach is used extensively in the analysis and design of such systems. 1. The transfer function of a system is a mathematical model of that system, in that it is an operational method of expressing the differential equation that relates the output variable to the input variable. 3 Transfer Function Approach to Modelling Dynamic Systems 2. The transfer function is a property of a system itself, unrelated to the magnitude and nature of the input or driving function. 3. The transfer function includes the units necessary to relate the input to the output; however, it does not provide any information concerning the physical structure of the system. (The transfer functions of many physically different systems can be identical.) 4 Transfer Function Approach to Modelling Dynamic Systems 4. If the transfer function of a system is known, the output or response can be studied for various forms of inputs with a view toward understanding the nature of the system. 5. If the transfer function of a system is unknown, it may be established experimentally by introducing known inputs and studying the output of the system. Once established, a transfer function gives a full description of the dynamic characteristics of the system, as distinct from its physical description. 5 Transfer Function Approach to Modelling Dynamic Systems Block diagrams of dynamic systems A block diagram of a dynamic system is a pictorial representation of the functions performed by each component of the system and of the flow of signals within the system. Such a diagram depicts the interrelationships that exist among the various components. Differing from a purely abstract mathematical representation, a block diagram has the advantage of indicating the signal flows of the actual system more realistically. 6 Transfer Function Approach to Modelling Dynamic Systems In a block diagram, all system variables are linked to each other through functional blocks. The functional block, or simply block, is a symbol for the mathematical operation on the input signal to the block that produces the output. The transfer functions of the components are usually entered in the corresponding blocks, which are connected by arrows to indicate the direction of the flow of signals. Note that a signal can pass only in the direction of the arrows. 7 Transfer Function Approach to Modelling Dynamic Systems An element of a block diagram The arrowhead pointing toward the block indicates the input to the block, and the arrowhead leading away from the block represents the output of the block. X(s) Transfer function G(s) Y(s) 8 Transfer Function Approach to Modelling Dynamic Systems The advantages of the block diagram representation of a system lie in the fact that it is easy to form the overall block diagram for the entire system merely by connecting the blocks of the components according to the signal flow and that it is possible to evaluate the contribution of each component to the overall performance of the system. In general, the functional operation of a system can be visualized more readily by examining a block diagram of the system than by examining the physical system itself. 9 Transfer Function Approach to Modelling Dynamic Systems Summing point The figure below shows a circle with a cross, the symbol that stands for a summing operation. The plus or minus sign at each arrowhead indicates whether the associated signal is to be added or subtracted. Branch point A branch point (pickoff point) is a point from which the signal from a block goes concurrently to other blocks or summing points. 10 Transfer Function Approach to Modelling Dynamic Systems Block diagram of a closed-loop system The figure below shows a block diagram of a closed-loop system. The output 𝐶(𝑠) is fed back to the summing point, where it is compared with the input 𝑅(𝑠). 11 Transfer Function Approach to Modelling Dynamic Systems Any linear system can be represented by a block diagram consisting of blocks, summing points, and branch points. When the output is fed back to the summing point for comparison with the input, it is necessary to convert the form of the output signal to that of the input signal. 12 Transfer Function Approach to Modelling Dynamic Systems This conversion is accomplished by the feedback element whose transfer function is H(s). 13

Transfer Function Approach to Modelling Dynamic Systems

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