Open-Loop Oscillations

The steady state is evidently dynamically unstable since the eigenvalues have positive real parts. In addition, the eigenvalues are complex, indicating that the system will move away from its unstable steady state in an oscillatory fashion. 

TABLE 4.1 Reactor Operating Parameters for Open-Loop Oscillations 

Heat capacity of feed Heat capacity of product 

Normalized activation energy (.E fR) Heat of reaction Mass of metal wall in reactor Heat capacity of metal wall Heat losses 

Steady-state temperature Coolant temperature Feed flowrate Feed concentration Pre-exponential factor Heat transfer parameter (UAi Residence time Effective liquid volume 

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To prevent surges, a well-trained operator would put the controller in manual mode and freeze the valve in an open position. This stops the control loop oscillations and decreases the compressor discharge resistance, thus breaking the surge cycle. Unfortunately, the operator has no way of knowing how much to open the valve and, subsequently, how much to close it. 

To find the new state feedback gain is a matter of applying Eq. (9-29) and the Ackermann s formula. The hard part is to make an intelligent decision on the choice of closed-loop poles. Following the lead of Example 4.7B, we use root locus plots to help us. With the understanding that we have two open-loop poles at -4 and -5, a reasonable choice of the integral time constant is 1/3 min. With the open-loop zero at -3, the reactor system is always stable, and the dominant closed-loop pole is real and the reactor system will not suffer from excessive oscillation. 

A periodically forced system may be considered as an open-loop control system. The intermediate and high amplitude forced responses can be used in model discrimination procedures (Bennett, 1981 Cutlip etal., 1983). Alternate choices of the forcing variable and observations of the relations and lags between various oscillating components of the response will yield information regarding intermediate steps in a reaction mechanism. Even some unstable phase plane components of the unforced system will become apparent through their role in observable effects (such as the codimension two bifurcations described above where they collide and annihilate stable, observable responses). 

This indicates that the oscillation, once set in motion, will be maintained with constant amplitude around the closed-loop for =. % = 0. If, however, the open-loop gain or AR of the system is greater than unity, the amplitude of the sinusoidal signal will increase around the control loop, whilst the phase shift will remain unaffected. Thus the amplitude of the signal will grow indefinitely, i.e. the system will be unstable. 

This heuristic argument forms the basis of the Bode stability criterion(22,24) which states that a control system is unstable if its open-loop frequency response exhibits an AR greater than unity at the frequency for which the phase shift is —180°. This frequency is termed the cross-over frequency (coco) for reasons which become evident when using the Bode diagram (see Example 7.7). Thus if the open-loop AR is unity when i/r = —180°, then the closed-loop control system will oscillate with constant amplitude, i.e. it will be on the verge of instability. The greater the difference between the open-loop AR (< I) at coc and AR = 1, the more stable the closed-loop... 

General Second-Order Element Figure 8-3 illustrates the fact that closed-loop systems can exhibit oscillatory behavior. A general second-order transfer function that can exhibit oscillatory behavior is important for the study of automatic control systems. Such a transfer function is given in Fig. 8-15. For a unit step input, the transient responses shown in Fig. 8-16 result. As can be seen, when t, < 1, the response oscillates and when t, < 1, the response is S-shaped. Few open-loop chemical processes exhibit an oscillating response most exhibit an S-shaped step response. 

If the inductance L in the simple circuit shown in Fig. 4.13 has a d.c. resistance of 100 Q, the current through it with the switch closed is 0.24 A. Opening the switch sets the charge in the LC loop oscillating, and the peak instantaneous current is 0.24 A. Because the maximum energy stored in the capacitor (jCU2) must be equal to that stored in the inductor (jLI2), it follows that... 

We show the reactor and jacket temperatures (T and T ) along with CA, the concentration of component A in the reactor. Initially, when the simulation started, the heat transfer area was sufficiently large to maintain open-loop static and dynamic stability. However, a few minutes into the simulation, we reduced UA by 20 percent. This creates dynamic instability with complex eigenvalues as in Eq. (4.25). The reactor temperature and composition start oscillating with a growing amplitude. However, the amplitude growth stops roughly 5 hours after the onset of instability and the reaction enters into a limit cycle of constant period and amplitude. 

Figure 18.3 (a) Open-loop system with sinusoidal input (set point) (b) corresponding closed-loop system with sustained oscillation (zero input). 

At some instant of time the set point is set to zero, while at the same time we close the loop (Figure 18.3b). Under these conditions the comparator inverts the sign of the ym, which now plays the same role as that played by the set point in the open loop. Notice that the error e remains the same. Theoretically, the response of the system will continue to oscillate with constant amplitude, since AR = 1, despite the fact that both the load and the set point do not change. 

How would you select the sampling rate for (a) the response of a general underdamped open-loop system, and (b) the oscillating response of a closed-loop system ... 

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