Systems with Multiple Loops

Block diagram reduction Control systems with multiple loops... 

In such cases control systems with multiple loops may arise. Typical examples of such configurations, that we will study in the present chapter, are the following ... 

Chap. 20 Control Systems with Multiple Loops... 

CENTRAL MULTIPLE HEAT PUMP SYSTEM WITH GROUND LOOP... 

Figure 22. Central multiple heat pump system with ground loop and primary/secondary pumping...

As mentioned above, more efficient algorithms are needed to make real-time dynamic simulation a reality. This need is particularly great fw robotic systems with multiple chains and closed kinematic loops. Thus, a fundamental goal of this book is the development of better and more efficient algorithms for the dynamic simulation of multiple chain robotic systems. In particular, solutions to the Direct Dynamics problem fw simple closed-chain mechanisms are investigated. 

Recognize that these ranges are approximate and that it may not be possible to choose PI or PID controller settings that result in specified GM and PM values. Tan et al. (1999) have developed graphical procedures for designing PI and PID controllers that satisfy GM and PM specifications. The GM and PM concepts are easily evaluated when the open-loop system does not have multiple values of 0) or Wg. However, for systems with multiple o)g, gain margins can be determined from Nyquist plots (Doyle et al., 1992). 

The Nyquist stability criterion is similar to the Bode criterion in that it determines closed-loop stability from the open-loop frequency response characteristics. Both criteria provide convenient measures of relative stability, the gain and phase margins, which will be introduced in Section J.4. As the name implies, the Nyquist stability criterion is based on the Nyquist plot for GqiXs), a polar plot of its frequency response characteristics (see Chapter 14). The Nyquist stability criterion does not have the same restrictions as the Bode stability criterion, because it is applicable to open-loop unstable systems and to systems with multiple values of co or cOg. The Nyquist stability criterion is the most powerful stability test that is available for linear systems described by transfer function models. 

A control system may have several feedback control loops. For example, with a ship autopilot, the rudder-angle control loop is termed the minor loop, whereas the heading control loop is referred to as the major loop. When analysing multiple loop systems, the minor loops are considered first, until the system is reduced to a single overall closed-loop transfer function. 

A canonical set of structures for a system with more orbits than electrons is obtained by arranging all the orbits (including phantom orbits for 5>0) in a ring and then drawing non-intersecting bonds to a number determined by the number of electrons and the multiplicity. If two electrons occupy the same orbit, forming an unshared pair, a loop is drawn with its ends at the orbit. 

The multivariable Nyquist plots discussed above give one curve. These curves can be quite complex, particularly with high-order systems and with multiple deadtlmes. Loops can appear in the curves, making it difficult sometimes to see if the (— 1,0) point is being encircled. 

In between these tangencies, the curves R and L have three intersections, so the system has multiple stationary states (Fig. 7.3(b)). We see the characteristic S-shaped curve, with a hysteresis loop, similar to that observed with cubic autocatalysis in the absence of catalyst decay ( 4.2). 

Even with the largest values of y, corresponding to systems with very little temperature sensitivity, some limited multiplicity is possible. Condition (7.95) again gives the range within which the cooling temperature must lie and within which the system can display the inverse hysteresis loop for some combinations of 0ad and tn. 

Fig. 9.4. (a) The dependence of the stationary-state concentration of reactant A at the centre of the reaction zone, a (0), on the dimensionless diffusion coefficient D for systems with various reservoir concentrations of the autocatalyst B curve a, / = 0, so one solution is the no reaction states a0i>8 = 0, whilst two other branches exist for low D curves b and c show the effect of increasing / , unfolding the hysteresis loop curve d corresponds to / = 0.1185 for which multiplicity has been lost, (b) The region of multiple stationary-state profiles forms a cusp in the / -D parameter plane the boundary a corresponds to the infinite slab geometry, with b and c appropriate to the infinite cylinder and sphere respectively. 

We now consider the dependence of the stationary-state solution on the parameter d. To represent a given stationary-state solution we can take the dimensionless temperature excess at the middle of the slab, 0ss(p = 0) or 60,ss-With the above boundary conditions, two different qualitative forms for the stationary-state locus 0O,SS — <5 are possible. If y and a are sufficiently small (generally both significantly less than i), multiplicity is a feature of the system, with ignition on increasing <5 and extinction at low <5. For larger values of a or y, corresponding to weakly exothermic processes or those with low temperature sensitivity, the hysteresis loop becomes unfolded to provide... 

For a system with a fixed y, the steady-state multiplicity may be revealed by the curve of superficial velocity of solids Up versus Ap, as illustrated in Fig. 8.17, where a hysteresis loop is formed between two branch (upper and lower) curves [Chen et al., 1984], The upper branch corresponds to Regimes 2, 3, or 4, whereas the lower one corresponds to Regime 1. At Ap less than (Ap)i, the system is operated on the upper branch of the curves, while it is operated on the lower branch of the curves at Ap larger than (Ap)2- Between these two... 

We noted earlier in this chapter that many reactions in the chemical industries are exothermic and require heat removal. A simple way of meeting this objective is to design an adiabatic reactor. The reaction heat is then automatically exported with the hot exit stream. No control system is required, making this a preferred way of designing the process. However, adiabatic operation may not always be feasible. In plug-flow systems the exit temperature may be too hot due to a minimum inlet temperature and the adiabatic temperature rise. Systems with baekmixing suffer from other problems in that they face the awkward possibilities of multiplicity and open-loop instability. The net result is that we need external cooling on many industrial reactors. This also carries with it a control system to ensure that the correct amount of heat is removed at all times. 

In order to avoid problems with sample inhomogeneity, the entire oil sample from each sample of shale was dissolved in 1.5 to 2.5 mL of CS2 (about 1 g oil to 1.5 mL solvent). One pL of this solution was injected into a Hewlett-Packard Model 5880 Gas Chromatograph equipped with capillary inlet and a 50 m x 0.25 mm Quadrex "007" methyl silicone column. Injection on the column is made with a split ratio of approximately 1 to 100. The column temperature started at 60°C and increased at 4°C/min to 280°C where it remained for a total run time of 90 min. The carrier gas was helium at a pressure of 0.27 MPa flowing at a rate of 1 cm /min. The injector temperature was 325°C and the flame ionization detector (FID) temperature was 350°C. Data reduction was done using a Hewlett-Packard Model 3354 Laboratory Automation System with a standard loop interface. Identification of various components was based on GC/MS interpretation described previously (4). For multiple runs on the same shale, the relative standard deviations of the biomarker ratios were about 10%. 

Unfortunately, not enough time is spent teaching students to think critically about the models they are using. Most mathematics and statistics classes focus on the mechanics of a calculation or the derivation of a statistical test. When a model is used to illustrate the calculation or derivation, little to no time is spent on why that particular model is used. We should not delude ourselves, however, into believing that once we have understood how a model was developed and that this model is the true model. It may be in physics or chemistry that elementary equations may be true, such as Boyle s law, but in biology, the mathematics of the system are so complex and probably nonlinear in nature with multiple feedback loops, that the true model may... 

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