Closed-loop instability

Process Reaction Curve Method (Cohen-Coon Tuning). For some processes, it may be difficult or hazardous to operate with continuous cycling, even for short periods. The process reaction curve method obtains settings based on the open loop response and thereby avoids the potential problem of closed loop instability. The procedure is as follows ... 

For two-time-scale systems, it is well established that inversion-based controllers designed without explicitly accounting for the time-scale multiplicity are ill-conditioned and can lead to closed-loop instability. In order to avoid such issues, controller design must be addressed on the basis of the reduced-order representations of the slow and fast dynamics, an approach referred to as composite control (see, e.g., Chow and Kokotovic 1976, 1978, Saberi and Khalil 1985, Kokotovic et al. 1986, Christofides and Daoutidis 1996a, 1996b). 

The term Cb + 71,1 dCB/dt is statically equivalent to Cb, and corresponds to requesting a first-order response in Cb when using a standard input-output linearizing controller. However, such a controller would lead to closed-loop instability, and the output requires a statically equivalent addition that would allow one to overcome this limitation. 

Example 19.1 Dead Time as a Main Source of Closed-Loop Instability... 

Since most of the chemical processes exhibit an open-loop response which can be approximated by a first-order system with dead time, it is clear that the possibility for closed-loop instability will, almost always, be present. Therefore, the tuning of the feedback controller becomes a crucial task. 

Conversely, the addition of integral control action can produce closed-loop instability. Curve b in Fig. 14.10 indicates that an unstable closed-loop system occurs for Gc(s) = 0.4 (1 + l/0.2s), because ARql > 1 when ql = -180°. To find Kcm for t/ = 0.2 min, we note that (o<. depends on t/ but not on K, because Kc has no effect on curve... 

Does not introduce instability in the closed-loop response Sensitive to process/model error... 

The tuning of the controller in the feedback loop can be theoretically performed independent of the feedforward loop (i.e., the feedforward loop does not introduce instability in the closed-loop response). For more information on feedforward/feedback control appications and design of such controllers, refer to the general references. 

The PI controller, even when optimally tuned, is also unable to prevent surge. Furthermore, it is unable to stop surge once it occurs. In the above situation, the operator would correctly identify the problem as instability of the closed-loop PI controller. The only viable action would be to open the closed control loop by placing the controller in manual, thereby freezing the valve open. In this scenario, open-loop control will stop surge. 

The closer the open-loop frequency response locus G(ja )//(ja ) is to the (—l,j0) point, the nearer the closed-loop system is to instability. In practice, all control... 

In the Boost and the Buck-Boost, we see that the output capacitor is in the critical path. So this capacitor should be close to the control IC, along with the diode. A paralleled ceramic capacitor can also help, provided it does not cause loop instability issues (especially in voltage mode control). 

For active instability suppression, the approach taken in the present study is to pulse the liquid fuel at the instability frequency and adjust the timing using a simple closed-loop circuit. Because the empha-... 

Figure 21.10 Active instability suppression in Case lA (a) onset of active control, 1 — without control, 2 — with closed-loop control (6) comparison of pressure spectra. 1 — imcontroUed, 2 — controlled...

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... 

Acts before the effect of a disturbance has been felt by the system Is good for systems with large time constant or dead time Does not introduce instability in the closed-loop response Requires direct measurement of all possible disturbances Cannot cope with unmeasured disturbances Is sensitive to process/model error... 

Does not require identification and measurement of any disturbance for corrective action Does not require an explicit process model Is possible to design controller to be robust to process/model errors Control action not taken until the effect of the disturbance has been felt by the system Is unsatisfactory for processes with large time constants and frequent disturbances May cause instability in the closed-loop response... 

Step 3. The open-loop instability of the reactor acts somewhat like a constraint, since closed-loop control of reactor temperature is required. By design, the exothermic reactor heat is removed via cooling water in the reactor and product condenser. We choose to control reactor temperature with reactor cooling water flow because of its direct effect. There are no process-to-process heat exchangers and no heat integration in this process. Disturbances can then be rejected to the plant utility system via cooling water or steam. 

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