First-order lead

These are first-order systems where the phase of the output (in steady-state) leads the phase of the input. The transfer funetion of a first-order lead system is... 

Fig. 6.10 Bode gain and phase for a first-order lead system.

A phase lead eompensator is different from the first-order lead system given in equation (6.35) and Figure 6.10 beeause it eontains both numerator and denominator first-order transfer funetions. 

The so-called lead-lag element is a semi-proper function with a first order lead divided by a first order lag ... 

Example 8.4. What are the Bode and Nyquist plots of a first order lead G(s) = (xps + 1) ... 

To help understand MATLAB results, a sketch of the low and high frequency asymptotes is provided in Fig. E8.9. A key step is to identify the comer frequencies. In this case, the comer frequency of the first order lead is at 1/5 or 0.2 rad/s, while the two first order lag terms have their comer frequencies at 1/10, and 1/2 rad/s. The final curve is a superimposition of the contributions from each term in the overall transfer function. 

The MATLAB statements are essentially the same as the first order lead function ... 

On the Bode plot, the comer frequencies are, in increasing order, l/xp, Zq, and p0. The frequency asymptotes meeting at co = l/xp and p0 are those of a first-order lag. The frequency asymptotes meeting at co = z0 are those of a first-order lead. The largest phase lag of the system is -90° at very high frequencies. The system is always stable as displayed by the root locus plot. 

Kinetic schemes involving sequential and coupled reactions, where the reactions are either first-order or pseudo-first order, lead to expressions for concentration changes with time that can be modeled as a sum of exponential functions where each of the exponential functions has a specific relaxation time. More complex equations have to be derived for bimolecular reactions where the concentrations of reactants are similar.19,20 However, the rate law is always related to the association and dissociation processes, and these processes cannot be uncoupled when measuring a relaxation process. 

When H is small, the exponential may be developed to first order, leading to... 

Notice that the G s are ratios of polynomials in s. The s — on and s — a 2 terms in the numeratois are called first-order leads. Notice also that the denominators of all the G s are exactly the same. 

This is a first-order lead. It is physically unrealizable i.e., a real device cannot be built that has exactly this transfer function. 

This is called a lead-lag element and contains a first-order lag and a first-order lead. See Table 9.1 for some commonly used transfer function elements. 

Thus the transfer function for a PI controller contains a first-order lead and an integrator. It is a function of s, having numerator and denominator polynomials of order one. 

The feedforward controller contains a stcadyslate gain and dynamic terms. For this system the dynamic element is a first-order lead-lag. The unit step reaponae of this lead-lag is an initial change to a value that is (—followed by an exponential rise or decay to the final steadystate value... 

The Bode plot of this combination of an integrator and a first-order lead is shown in Fig. 13.13h, At low frequencies, a PI controller amplifies magnitudes and contributes —90° of phase-angle lag, This loss of phase angle is undesirable from a dynamic standpoint since it moves the Gj B polar plot closer to the ( — 1,0) point. 

The reaction rates in Equations 7 and 8 are assumed to be first order, leading to the expression ... 

Nichols plots, (a) First-order lag. (b) First-order lead, (c) Deadtime. id) Deadtime and lag. ie) Integrator. (/) Integrator and lag. ig) Second-order underdamped lag. 

In (35) the first index (M + 2i — 1) gives the vibrational quantum number and the second index M > 0 the total angular momentum the aUowed values for the vibrational levels i = 0,1,2,... For t = 0 the perturbation disappears in first order. While the perturbation in first order leads to a splitting of the energy terms E and E, the perturbation through second order in e for both cases yields the same value as the same sign. In general... 

This is the expression for a first-order lead-lag function. For a unit step change in the input, the output shows a peak value equal to T1/T2 and a final value of 1.0. 

In work carried out at the University of Delaware a few years ago, Rippin and Lamb showed that most of the benefit of feedforward is from the static gain term. Fairly simple feedforward dynamic functions such as first-order lead-lag are usually adequate and may be calculated off line. 

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