Ultimate period

A preview We can derive the ultimate gain and ultimate period (or frequency) with stability analyses. In Chapter 7, we use the substitution s = jco in the closed-loop characteristic equation. In Chapter 8, we make use of what is called the Nyquist stability criterion and Bode plots. 

B. Tuning relations based on closed-loop testing and the Ziegler-Nichols ultimate-gain (cycle) method with given ultimate proportional gain Kcu and ultimate period Tu. 

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. 

Ultimate gain and ultimate period (Pu = 2tt/(0u) that can be used in the Ziegler-Nichols continuous cycling relations. Result on ultimate gain is consistent with the Routh array analysis. Limited to relatively simple systems. 

The Ultimate Periodic Table (Poster). Wiley-VCH, Weinheim, Germany (2007)... 

An infinite tape w is ultimately periodic if w = uvw. .. for finite tapes u,v with v e, i.e. if w can be written as a finite (possibly enpty) initial string u followed by arbitrarily many repetitions of a nonempty finite string v. We need the next result which relates Turing machines and two-tape acceptors with special attention to behavior on ultimately periodic input tapes. 

THEOREM 6.3 The following problems are not partially decidable for two-tape oneway deterministic finite state acceptors M even if M is guaranteed not to diverge on (w,w) for w ultimately periodic ... 

This was because in our construction of Mj, from Turing machine T, we forced Hp to halt on (t,t) unless t encoded an infinite computation of T on the blank initial tape and by placing extra s in the encodement we made sure that such a tape t could never be ultimately periodic. Hence we can assume that M halts on (t,t) for t ultimately periodic. 

Again we examine Pg(M). As we did in (6) we notice that we can assume that M halts on (t,t) whenever t is ultimately periodic and this implies that Pg(M) enters qa or for all finite interpretations. Hence Pg(M) halts under all finite interpretations as does Z2. So Pg(M) is finitely equivalent to Z2 if and only if no computation ever reaches qa if and only if. 

The isotherma] three-CSTR process of Sec 5.2 has, as we will prove in Chap. 10, an ultimate gain of 64 and an ultimate period of 3.63 minutes. The ZN settings for this system are given in Table 7.2. The response of the closedloop system to a step load disturbance in Cas is shown in Fig. 7.16 with P, PI, and PID controllers and the ZN settings. 

The period of the limit cycle is the ultimate period (PJ for the transfer function relating the controlled variable x and the manipulated variable m. So the ultimate frequency is... 

To quantify this effect, jacket thickness hj is changed from 0.1 down to 0.025 m. The 350 K reactor temperature with 85% conversion is the case studied. Figure 3.21 shows the effect on the Nyquist plot. The improvement in controllability is indicated by the curves dropping more into the third quadrant as hj is decreased. The ultimate gain increases from 8.51 to 17.1 to 24.8. The ultimate period decreases from 2711 to 1846 to 1429 s. These indicate improved closedloop performance. Figure 3.22 shows this to be true. The Tyreus-Luyben settings are used in the three cases. 

The 1-CSTR process has a conversion of 98% in the single reactor with a reactant concentration of 0.16 kmol/m3. The reactor volume is high (262 m3), and the jacket heat transfer area is large (190 m2). The resulting jacket temperature is 339 K. Linear analysis gives an ultimate gain of 52.6 (dimensionless) and an ultimate period of 1419 s. 

Click the Closed loop ATV bullet, start the simulation mnning, and click the Start test button. After several cycles, click the Pause button to stop the simulation and click the Finish test button at the bottom of the Tune window (see Fig. 3.79). The ultimate gain (3.73) and the ultimate period (4.8 min) are displayed, as shown in the left side of Figure 3.80. To calculate the controller tuning constants, click the Tuning parameters page tab on the Tune window and select either Ziegler-Nichols or Tyreus-Luyben. 

Numerous empirical correlations have been developed to determine PID tuning parameters for load responses of processes. These correlations are based either on closed-loop procedures, which directly identify the ultimate gain and ultimate period of the loop, or on open-loop procedures, which identify the time... 

The closed-loop procedure requires tuning a controller with only gain and increasing that parameter until sustained oscillations are observed. When this occurs, the gain is called the ultimate gain (Ku) and the time between successive cycles is called the ultimate period (Tu). 

An alternative closed-loop approach called the relay method uses temporary narrow limits on the controller output and toggles between output limits each time the controller error changes sign. The ultimate period is determined as before and the ultimate gain is computed as Ku = 4process measurement, both in percent. 

The ultimate periodic symmetry is determined in both cases by the nanophase surfactant-packing requirements, so that similar space group and lattice symmetries may be observed by XRD and TEM. However, the XRD peaks of the two phases for a given surfactant have clearly different diffraction intensities, indicating different pore and wall structures. SBA-3 (see Figure 8.18) and other mesoporous silica from acidic synthesis systems have regular crystal morphology, even curved shapes. 

To initiate an ATV test, the process should be at steady-state or near steady-state conditions, Cq and Jo- Next, the controller output is set io Cq + h (or Cg - h) until y deviates significantly from Jq. At that point, the controller output is set to Cq - / (or Cq + h), which will turn the process back toward jo- Then, each time y crosses yo. the controller output is switched from Cq + h to Cq - h or from Cq - h to Cg + h. The process is also referred to as a relay feedback experiment. A standing wave is established after 3 to 4 cycles therefore, the values of a and the ultimate period, P , can be measured directly, and the ATV test is concluded. The ultimate gain, K, is calculated by... 

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