Positive zero

The term b — Aac), called the discriminant, may be positive, zero or negative which will make the roots real and unequal, real and equal or complex. This gives rise to the three different types of transient response described in Table 3.4. 

If we put Inp = x, hiKi = /, the curve InK = f(lnp) will be a straight line having a positive, zero, or negative gradient according... 

According as is = 0 there will be positive, zero, or negative adsorption, respectively. 

This is a form that serves many purposes. The term in the denominator introduces a negative pole in the left-hand plane, and thus probable dynamic effects to the characteristic polynomial of a problem. The numerator introduces a positive zero in the right-hand plane, which is needed to make a problem to become unstable. (This point will become clear when we cover Chapter 7.) Finally, the approximation is more accurate than a first order Taylor series expansion.1... 

When you repeat this exercise with MATLAB in the Review Problems, check that xz is negative in case (c). More commonly, we say that this is the case with a positive zero. After we have learned frequency response, we ll see that this is an example of what we refer to as nonminimum phase. 

The controller function will take on a positive pole if the process function has a positive zero. It is not desirable to have an inherently unstable element in our control loop. This is an issue which internal model control will address. 

There is still one unfinished business. We do not know how to choose G c yet. Before we make this decision, we may recall that in direct synthesis, the poles of Gc are inherited from the zeros of Gp. If Gp has positive zeros, it will lead to a Gc function with positive poles. To avoid that, we split the approximate function as a product of two parts ... 

We use the first order Pade approximation for the dead time and isolate the positive zero term as in Eq. (6-33) ... 

The result is an ideal PD controller with the choice of xD = xp. See that you can obtain the same result with IMC too. Here, take the process function as the approximate model and it has no parts that we need to consider as having positive zeros. There is no offset the integrating action is provided by Gp. 

To begin with, this is a second order system with no positive zeros and so stability is not an issue. Theoretically speaking, we could have derived and proved all results with the simple second order characteristic equation, but we take the easy way out with root locus plots. 

Power supply designers are usually aware that the most stable ceramic capacitance comes from materials dubbed COG material, also called NPO (for negative positive zero, referring to its near perfect temperature coefficient). But this is a low dielectric constant material, and unsuitable for modern miniaturization. So the common materials in use today are called X7R, X5R, and so on. There are others, starting with a Y or Z prefix, which no power supply designer worth his or her salt will ever use. 

Mathematically, inverse response can be represented by a system that has a transfer function with a positive zero, a zero in the RHP. Consider the system sketched in Fig. ll.lOn. There are two parallel first-order lags with gains of opposite sign. The transfer function for the overall system is... 

Keep in mind that the positive zero does not make the system openloop unstable. Stability depends on the poles of the transfer function, not on the zeros. Positive zeros in a system do, however, affect closedloop stability as the example below illustrates. 

Remember that in Example 10.8 adding a lead or a negative zero made the closedloop system more stable. In this example we have shown that adding a positive zero has just the reverse effect. 

Since this eigenvalue is negative it corresponds to a bound state and, if y is the smallest positive zero of the determinant, to the ground state. The next smallest zero of the determinant will correspond to the first excited state and so on. 

Williams Trans. Farad. Soo. I. 1914) bas shown that on the assumption that both solvent and solute are adsorbed by the adsorbing agent we may obtain positive, zero or negative adsorption as the solute is adsorbed more strongly, equally or less strongly than the solvent and that as an alteration in concentration of the solution takes place the adsorption may pass through all these separate phases. 

From the equation for (cR/c,)2 in the literature [36], s = S2 is a positive zero point of the following cubic polynomial,... 

The table below lists a series of proteins along with their respective isoelectric points (p/). By completing the table with positive (+) or negative (—) signs or zeros (0) indicate for each protein whether the net charge will be positive, zero or negative at the three values of pH specified. 

It is different from the curvature defined earlier in this section, since boundary vertices contribute differently to it. If (r, q) is elliptic, parabolic, and hyperbolic, then the curvature of interior vertices is positive, zero, and negative, respectively. 

There will be only some technical difficulties if we try to obtain the catalog of elementary (R, Q)-potycycles, i.e. the generalization of (R, )-polycycles allowing the set Q for values of degree of interior vertices. Such polycycle is called elliptic, parabolic, or hyperbolic if -f — (where r = maxiGfi i, q = max,-6g r) is positive, zero, or negative, respectively. The decomposition and other main notions could be applied directly. 

The interesting question is to enumerate, if possible, those elementary (R,q)gen-polycycles. Call an (R, q)gen-polycycle elliptic, parabolic, or hyperbolic if the number i + T — i (where r = max C/ i) is positive, zero, or negative, respectively. In Theorem 7.2.1, we will see that the number of elementary (r, hyperbolic pairs (r, q). But in [DeStOl] and [DeSt02b], all elliptic elementary (r, [Pg.76]

When reactor capacity is limited by heat removal, an often-recommended control structure is to run with maximum coolant flow and manipulate feed flowrate to control reactor temperature (Tr F0 control). This control scheme has the potential to achieve the highest possible production rate. However, if the feed temperature is lower than the reactor temperature, the transfer function between temperature and feed flowrate contains a positive zero, which degrades dynamic performance, as we demonstrate quantitatively in this section. The choice of a control structure for this process presents an example of the often encountered conflict between steady-state economics and dynamic controllability. 

Note that the b2i coefficient is negative when the feed temperature T0 is less than the reactor temperature TR. This produces a positive root of the numerator polynomial given in Eq. (3.47), so the openloop transfer function has a positive zero. 

If the design value for the feed temperature is increased to 320 K, the jacket temperature is only slightly lower (320.2 K), the heat transfer rate is slightly higher (2.21 x 106 J/s) and the cooling water flowrate increases to 20.1 kg/s. The positive zero becomes +0.0322. Controller settings become Kc = 5.5 and Tj = 440 min. Figure 3.51 shows that the response of this system is also very slow, but the peak deviation in reactor... 

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