Closed-loop zeros

The more complicated methods of compensation, such as this, allow the designer much more control over the final closed-loop bode response of the system. The poles and zeros can be located independently of one another. Once their frequencies are chosen, the corresponding component values can be easily determined by the step-by-step procedure below. The zero and pole pairs can be kept together in pairs, or can be separated. The high-frequency pole pair appear to yield better results if they are separated and placed as below. The zero pair are usually kept together, but can be separated and placed either side of the output filter pole s corner frequency to help minimize the gain effects of the Q of the T-C filter (refer to Figure B-23). 

For the condition / = 0, the Nyquist criterion is A closed-loop control system is stable if, and only if, a contour in the G s)H s) plane does not encircle the (—1, jO) point when the number of poles of G s)H s) in the right-hand. v-plane is zero . 

In practice, only the frequencies lu = 0 to+oo are of interest and since in the frequency domain. v = jtu, a simplified Nyquist stability criterion, as shown in Figure 6.18 is A closed-loop system is stable if, and only if, the locus of the G(iLu)H(iuj) function does not enclose the (—l,j0) point as lu is varied from zero to infinity. Enclosing the (—1, jO) point may be interpreted as passing to the left of the point . The G(iLu)H(iLu) locus is referred to as the Nyquist Diagram. 

SafeChem, a subsidiary of Dow, has developed a handling system for chlorinated solvents that allows them to be used in closed-loop degreasing systems. The Safe-Tainer system uses two dedicated double wall containers one to hold fresh solvent and the other used solvent. The containers are connected to the cleaning equipment with zero dead volume, leak-free connections that prevent spills, leaks or vapour emissions during use. Used solvent is collected for recycling and professional disposal of any residues. The system minimises solvent use and release to the environment. A study carried out by Dow during a trial in... 

For review after the chapter on root locus with the strategy in Fig. 5.3, the closed-loop characteristic polynomial and thus the poles remain the same, but not the zeros. You may also... 

The controller setting is different depending on which error integral we minimize. Set point and disturbance inputs have different differential equations, and since the optimization calculation depends on the time-domain solution, the result will depend on the type of input. The closed-loop poles are the same, but the zeros, which affect the time-independent coefficients, are not. 

A couple of quick observations First, Gc is the reciprocal of Gp. The poles of Gp are related to the zeros of Gc and vice versa—this is the basis of the so-called pole-zero cancellation.1 Second, the choice of C/R is not entirely arbitrary it must satisfy the closed-loop characteristic equation ... 

The idea is that we may cancel the (undesirable open-loop) poles of our process and replace them with a desirable closed-loop pole. Recall in Eq. (6-20) that Gc is sort of the reciprocal of Gp. The zeros of Gc are by choice the poles of Gp. The product of GcGp cancels everything out—hence the term pole-zero cancellation. To be redundant, we can rewrite the general design equation as... 

Since the system characteristic equation is 1 + GcGp = 0, our closed-loop poles are only dependent on our design parameter xc. A closed-loop system designed on the basis of pole-zero cancellation has drastically different behavior than a system without such cancellation. 

This is the idea behind the plotting of the closed-loop poles—in other words, construction of root locus plots. Of course, we need mathematical or computational tools when we have more complex systems. An important observation from Example 7.5 is that with simple first and second order systems with no open-loop zeros in the RHP, the closed-loop system is always stable. 

There will be m root loci, matching the order of the characteristic polynomial. We can easily see that when Kc = 0, the poles of the closed-loop system characteristic polynomial (1 + KCG0) are essentially the same as the poles of the open-loop. When Kc approaches infinity, the poles of the closed-loop system are the zeros of the open-loop. These are important mathematical features. 

The system in (e) can become unstable, while a proper addition of an open-loop zero, as in (f), can help stabilize the system (Fig. E7.6c). In (e), the two loci from -1 and -2 approach each other (arrows not shown). They then break away and the closed-loop poles become unstable. The... 

A locus (closed-loop root path) starts at an open-loop pole and either terminates at an open-loop zero or extends to infinity. 

Note 2 As we reduce the integral time constant from Xi = 3 min to exactly 2 min, we have the situation of pole-zero cancellation. The terms in the closed-loop characteristic equation cancel... 

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. 

For a first order function with deadtime, the proportional gain, integral and derivative time constants of an ideal PID controller. Can handle dead-time easily and rigorously. The Nyquist criterion allows the use of open-loop functions in Nyquist or Bode plots to analyze the closed-loop problem. The stability criteria have no use for simple first and second order systems with no positive open-loop zeros. 

There is a single root. It lies on the real axis in the z plane and its location depends on the value of the feedback controller gain. When the feedback controller gain is zero (the openloop system), the root lies at z = b. As is increased, the closed-loop root moves to the left along the real axis in the z plane. We will return to this example in the next section. 

This practice is undesirable because the addition of makeup water often results in the discharge of phossy water, unless an auxiliary tank collects phossy water overflows from the storage tanks, thus ensuring zero discharge. A closed-loop system is then possible if the phossy water from the auxiliary tank is reused as makeup for the main phosphorus tank. 

The method used for the localization of the orbitals is to be carefully chosen. It is natural to expect that if the orbitals are localized into different spatial regions, for the matrix elements ij kf) the zero differential approximation can be applied all terms containing at least one factor ij kl) in which tj/itj/,-and/or are localized to different spatial regions can be neglected. Thus the summation in a closed loop in evaluating a perturbation correction should only be extended over indices of orbitals which are localized into the same region of space. 

The zero-emission energy recycling system (ZEROS) is a closed-loop thermal oxidation process that incinerates waste and recycles flue gas emissions for electrical co-generation. The technology uses a two-stage plasma torch combustion system, energy recovery system, and combustion gas cleanup systems. 

The law of conservation of energy, which states that the sum of the potential differences around any closed loop is zero, can also be applied to this system if the potential differences between the ions can be calculated. To determine these, it is convenient to recognize that each bond acts as a capacitor, C,y, with the atoms acting as the plates that carry the charges and the bond providing the field linking them. This capacitor then supports the potential difference, P,y, according to the capacitor eqn (2.8) ... 

Since the law of conservation of energy requires that the sum of the potentials, Py, around any closed loop be zero,... 

The process was then developed on a closed loop of nitrogen, circulating in three different zones, thus reducing the nitrogen consumption to zero during continuous operation. 

The loci typically drawn out by these equations as 0 varies are shown in Fig. 4.8(a). For y closed loop near the origin. Inside this loop, the stationary state is an unstable node. The larger outer loop separates stable focal character (inside curve) from stable nodal states (outside curve). As y increases beyond i the small inner loop shrinks to zero size the outer loop still exists. Stable focal character exists over some values of the parameters n and k for any value of y. 

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