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6. Dynamic Response Characteristics of More Complicated Systems

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- 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)

Where .

- 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.

- The response of the system to any input contains the following functions of time.
- A constant term resulting from the factor.
- An term resulting from the factor.
- and
terms resulting from the factor.

Figure 6.1. Roots of the denominator of plotted in the complex plane.

- Additional terms determined by the specific input forcing term will also appear in the response, but the intrinsic dynamic features of the process, the so-called response models or natural modes, are determined by the process itself.
- Each of the above response modes is determined from the factors of the denominator polynomial(roots of the characteristic equation);
- Poles !

Two poles ; .

Response modes ; and .

- decays slower than .

- Pole;The roots of the characteristic polynomial. It decide the stability if the system, and the swiftness of the response and oscillations.
- The effect of pole for the process.
- Pole at origin : the presence of an integrating element.
- Pole closer to the imaginary axis : slower response mode.

- Pole to the right of the imaginary axis : unstable response mode.

Pole ; .

Response mode ; .

- grows without bound as becomes large, characteristic of unstable systems..

- Complex Pole( always appear as part of a conjugate) : oscillatory mode.

- Zero;The roots of the numerator of the transfer function.

- The effect of zero for the process.
- It has no effect on the number and location of the poles and their associated response modes unless there is an exact cancellation of a pole by a zero ïƒ¨ It has no effect on the stability of the process.
- It exert a profound effect on the coefficients of the response modes in the system response. Such coefficients are found by partial fraction expansion ïƒ¨ It decide the weighting factors of each response modes.

- If , the transfer function simplifies to as a result cancellation of numerator and denominator elements ïƒ¨ Pole-Zero Cancellation.

Example. Calculate the response of the lead-lag element(6.3) to a step change of magnitude in its input.

Solution

(6.3) can be expanded into partial fractions.

Pole determines the response mode!

Zero determines the weighting factor of the response mode!

Figure 6.2. Step response of (6.3) with various .

- The presence of a zero in the first-order system causes a jump discontinuity in at when the step input is applied.

- Case a:
- Case b:
- Case c:

calculate the response to a step change of magnitude .

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,

Figure 6.3. Step response of (6.7) with various .

- Case a:
- Case b:
- Case c:

- Case a shows that overshoot can occur if the zero( ) is large enough , that is , if is sufficiently large.
- Case b looks much like an ordinary first-order process response as a result of the single zero.
- Case c exhibits a so-called inverse response initially.

The transportation time between points 1 and 2 is given by

6.2Time Delays

- Whenever material or energy is physically moved in a process or plant there is a time delay associated with the movement.
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.

- First-Order Plus Time Delay(FOPTD) transfer function.

- This transfer function can be used to approximate various overdamped system.
- The simple exponential form is some what deceptive since is it a non-rational transfer functionïƒ¨ The presence of a time delay in a process means that we cannot factor the process transfer function in terms of simple poles and zeros only.
- Polynomial approximation technique !

- Polynomial approximation to .
1. Taylor series expansion.

; Only good approximation for is small.

2. PadÃ© approximation.

Long division

1st orderapproximation.

2nd orderapproximation.

- 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.

- The system response to a step of magnitude .

(6.17) is well approximated by a time delay of magnitude , as shown in Figure 6.6.

6.3Approximation of Higher-Order Systems

- As discussed earlier, a time delay can be used to approximate high-order model dynamics.

- Consider the step response of a hypothetical nth-order system with n equal time constants.

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

- Many processes that do not contain an explicit time delay consist of a large number of process units connected in series( for example trays in a distillation or adsorption column).
- A time-delay term can be used in a process model as a approximation for a number of small time constants.

Where

6.4 Interacting and Noninteracting Processes

- Processes with variables that interact with each other or that contain internal feedback of material or energy(recycle streams) will exhibit so-called interacting behavior.
- Noninteracting Processes.

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) !

- Interacting Processes.

Figure 6.8. Two tanks in series whose liquid levels interact.

- Process inputs - flow rates of hot and cold streams( and ).
- Process outputs - the level in the tank and temperature .
- Potential disturbances(or load variables) - the temperatures of the inlet streams and .

- MIMO system is an industrial process control applications involve a number of input(manipulated) and output (controlled) variables. ï‚«SISO system.
- Example)

Figure 6.9. A multi-input, multi-output mixing process.