Closed loop oscillation

The Ziegler and Nichols closed-loop method requires forcing the loop to cycle uniformly under proportional control. The natural period of the cycle—the proportional controller contributes no phase shift to alter it—is used to set the optimum integral and derivative time constants. The optimum proportional band is set relative to the undamped proportional band P , which produced the uniform oscillation. Table 8-4 lists the tuning rules for a lag-dominant process. A uniform cycle can also be forced using on/off control to cycle the manipulated variable between two limits. The period of the cycle will be close to if the cycle is symmetrical the peak-to-peak amphtude of the controlled variable divided by the difference between the output limits A, is a measure of process gain at that period and is therefore related to for the proportional cycle ... 

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. 

Actually, a total closed-loop phase lag limit of 3i5 degrees is commonly used by designers any closer to 360 degrees would constitute a metastable system. This could result in the power supply breaking out into periods of oscillation when large loads or line transients are experienced. 

Ziegler-Nichols Continuous Cycling (empirical tuning with closed loop test) Increase proportional gain of only a proportional controller until system sustains oscillation. Measure ultimate gain and ultimate period. Apply empirical design relations. 

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. 

Gouse, S. W., Jr., and C. D. Andrysiak, 1963, Fluid Oscillations in a Closed Looped with Transparent, Parallel, Vertical, Heated Channels, MIT Eng. Projects Lab. Rep. 8973-2, Massachusetts Institute of Technology, Cambridge, MA. (6)... 

A proportional controller is used to control a process which may be represented as two non-interacting first-order lags each having a time constant of 600 s (10 min). The only other lag in the closed loop is the measuring unit which can be approximated by a distance/velocity lag equal to 60 s (1 min). Show that, when the gain of a proportional controller is set such that the loop is on the limit of stability, the frequency of the oscillation is given by ... 

Summary. In this chapter the control problem of output tracking with disturbance rejection of chemical reactors operating under forced oscillations subjected to load disturbances and parameter uncertainty is addressed. An error feedback nonlinear control law which relies on the existence of an internal model of the exosystem that generates all the possible steady state inputs for all the admissible values of the system parameters is proposed, to guarantee that the output tracking error is maintained within predefined bounds and ensures at the same time the stability of the closed-loop system. Key theoretical concepts and results are first reviewed with particular emphasis on the development of continuous and discrete control structures for the proposed robust regulator. The role of disturbances and model uncertainty is also discussed. Several numerical examples are presented to illustrate the results. 

In regard dynamics and control scopes, the contributions address analysis of open and closed-loop systems, fault detection and the dynamical behavior of controlled processes. Concerning control design, the contributors have exploited fuzzy and neuro-fuzzy techniques for control design and fault detection. Moreover, robust approaches to dynamical output feedback from geometric control are also included. In addition, the contributors have also enclosed results concerning the dynamics of controlled processes, such as the study of homoclinic orbits in controlled CSTR and the experimental evidence of how feedback interconnection in a recycling bioreactor can induce unpredictable (possibly chaotic) oscillations. 

Because the initial emphasis of this study was on extending ACC to liquid-fueled combustors, a simple closed-loop controller, which had been well tested in the previous studies involving gaseous fuel, was utilized. Such a controller, however, may not be effective in a combustor where the oscillation frequencies drift significantly with the control. The main problem was the frequency-dependent phase shift associated with the frequency filter. For such a case, it would be more useful to employ an adaptive controller that can rapidly modify the phase setting depending on the shift in the dominant oscillation frequencies. 

The outputs of the sensors were used in two closed-loop control strategies developed for combustor performance optimization [7]. The objective of the first strategy, based on an adaptive least-mean squares (LMS) algorithm, was to maximize the magnitude and coherence of temperature oscillations at the forcing frequency /o in the measured region. The LMS algorithm was used to determine... 

Fig. 16a symmetric limit cycles for the second-harmonic mode (GCL) and in Fig. 16b, an nonsymmetric phase portrait example for 7) = 0.5 for BCL. In both cases the phase point settles down into a closed-loop trajectory, although not earlier than about x > 200. An intricate limit cycle is usually related to multiperiod oscillations. For example, the cycle in Fig. 16a corresponds to five-period oscillations of the fundamental and SHG modes intensity, and the phase portrait in Fig. 16b resembles the four-period oscillations (see Fig. 17). Generally, for 7) > 0.5, we observe many different multiperiod (even 12-period) oscillations in intensity and a rich variety of phase portraits.

In the previous subsection, the forcing frequency was exactly twice the natural oscillatory frequency. Thus the motion around one oscillation gives exactly two circuits of the forcing cycle for one revolution of the natural limit cycle. The full oscillation of the forced system has the same period as the autonomous cycle and twice the forcing period. The concentrations 0p and 6r return to exactly the same point at the top of the cycle, and subsequent oscillatory cycles follow the same close path across the toroidal surface. This is known as phase locking or resonance. We can expect such locking, with a closed loop on the torus, whenever the ratio of the natural and forcing... 

If the quotient o>/a>0 is irrational, the path across the toroidal surface will return to a different point on the completion of each cycle. Eventually the trajectory will pass over every point on the surface of the torus without ever forming a closed loop. This is quasi-periodicity , and an example is shown in Fig. 13.11. The corresponding concentration histories do not necessarily give complex waveforms, as can be seen from the figure. However, the period of the oscillations is neither simply that of the natural cycle nor just that of the forcing term, but involves both. 

It is interesting to consider the shapes of the subharmonic trajectories that lock on the torus in the various entrainment regions of order p/q. The subharmonic period 4 at the 4/3 resonance horn is, for example, a three-peaked oscillation in time [Fig. 7(a)] and has three closed loops in its phase-plane projection [Fig. 7(b)], while the subharmonic period 4 at the 4/ 1 resonance is a single-peaked, single-loop oscillation [Figs. 7(d) and 7(e)]. A subharmonic period 2 at the 2/3 resonance is also included in Figs. 7(g) and 7(h). Multipeaked oscillations observed in chemical systems (Scheintuch and Schmitz, 1977 Flytzani-Stephanopoulos et al., 1980) may thus result from the interaction of frequencies of local oscillators. Such trajectories are the nonlin-... 

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... 

In Fig. 9 a number of responses to a unit step load disturbances are shown. The performance of the closed loop system is acceptable since the controller eliminates quickly the deviation with a reasonable amount of oscillation. 

The controller has tuning parameters related to proportional, integral, derivative, lag, dead time, and sampling functions. A negative feedback loop will oscillate if the controller gain is too high but if it is too low, control will be ineffective. The controller parameters must be properly related to the process parameters to ensure closed-loop stability while still providing effective control. This relationship is accomplished, first,... 

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. 

The period of oscillation of a closed loop depends on the loop dead time. The period of oscillation in flow loops is 1 to 3 seconds for level loops, it is 3 to 30 seconds (sometimes minutes) for pressure loops, 5 to 100 seconds for temperature loops, 0.5 to 20 minutes and for analytical loops, it ranges from 2 minutes to several hours. A proportional loop oscillates at periods ranging from two to five dead times. PI loops oscillate at periods of three to five dead times, and PID loops at around three dead time periods. 

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