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6. Dynamic Response Characteristics of More Complicated Systems. Contents 1. Poles and Zeros and Their Effects on System Response. 2. Time Delays. 3. Approximation of Higher-Order Systems. 4. Interacting and Noninteracting Processes. 5. Multiple-Input, Multiple-Output Processes. Example).
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1. Poles and Zeros and Their Effects on System Response.
2. Time Delays.
3. Approximation of Higher-Order Systems.
4. Interacting and Noninteracting Processes.
5. Multiple-Input, Multiple-Output Processes.
Where .6.1 Poles and Zeros and Their Effect on System Response
- Response characteristics of the processes are determined by the factors of the transfer function denominator, the characteristic polynomial.
Use the partial fraction and inverse Laplace transform.
terms resulting from the factor.
Figure 6.1. Roots of the denominator of plotted in the complex plane.
Response modes ; and .
Pole ; .
Response mode ; .
If , the transfer function simplifies to as a result cancellation of numerator and denominator elements Pole-Zero Cancellation.
Example. Calculate the response of the lead-lag element(6.3) to a step change of magnitude in its input.
(6.3) can be expanded into partial fractions.
Pole determines the response mode!
Zero determines the weighting factor of the response mode!
Solution The response of this system is
Note as expected; hence, the effect of including the single zero dose not change the final value nor dose it change the number or location of the response modes.
But the zero does affect how the response mode are weighted in the solution, (6.8).
Example. For the case of a single zero in an overdamped second-order transfer function,
Example A fluid is transported through a pipe in plug flow.
Figure 6.4. Transportation of fluid in a pipe.
If is some fluid property at point 1 and is the same property at point 2, then and are related by a simple time delay .
Thus the output is simply the same input function shifted back in time by the amount of the delay.
The transfer function of a time delay of units can be obtained by applying the Laplace transform to (6.10).
Figure 6.5. The effect of a pure delay is a translation of the function in time.
1. Taylor series expansion.
; Only good approximation for is small.
2. Padé approximation.
This transfer function can represent a series of n stages, each described by a first-order transfer function, with a total residence time equally divided among the stages.
(6.17) is well approximated by a time delay of magnitude , as shown in Figure 6.6.
6.3Approximation of Higher-Order Systems
Figure 6.6. A pure time delay element can approximate a large number of first-order systems in series.
Examplenth-order process( ) composed of first-order processes in series ( ) is dominated by two of these process( and ), then an approximation transfer function
for the system is
Figure 6.7. A series configuration of two noninteracting tanks.
Tank 1 level depends on tank 2 level ( and vice versa ) as a result of the interconnecting stream with flow rate .
Much more complicated expression than (6-21) !
Figure 6.8. Two tanks in series whose liquid levels interact.
Figure 6.9. A multi-input, multi-output mixing process.