Open-Loop Characteristics

Just as we approached reactor control in Chap. 4, we will start by exploring the open-loop effects of thermal feedback. Consider Fig. 5.19, which shows an adiabatic plug-flow reactor with an FEHE system. We have also included two manipulated variables that wall later turn out to be useful to control the reactor. One of these manipulated variables is the heat load to the furnace and the other is the bypass around the preheater. It is clear that the reactor feed temperature is affected by the bypass valve position and the furnace heat load but also by the reactor exit temperature through the heat exchanger. This creates the possibility for multiple steady states. We can visualize the different 

The next step in the analysis is to seek another functional relationship between the reactor exit temperature and the reactor feed temperature resulting from the heat exchange, bypass, and influence of the furnace. Once we find the second relation we can superimpose it on top of the reactor temperature rise expression shown in Fig. 5.20. Intersections between the two curves constitute open-loop, steady-state solutions to the combined reactor-FEHE system. 

Equation (.5.5) provides the starting point for our second functional relation. When we use the nomenclature of Fig. 5.19 we get the following simple relationship around the heat exchanger. 

This follows since the cold stream to the heat exchanger has the smaller total heat capacity due to the bypass [i.e., (mCf)m = irhCp k ]. 

We then add the effect of the bypass as a simple mixing operation  

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The vinyl acetate reactor we use in Chap. 11 has been designed to be insensitive to parameter variations under normal operating conditions. The hot-spot temperature is only 162°C with an exit temperature of 159°C. It is adequate to control the exit temperature instead of the hot spot. Since multiplicity, open-loop stability, and sensitivity are of no concern for this reactor, we can focus our attention on the open-loop characteristics relevant, to the control of exit temperature with jacket cooling. 

We start by exploring the open-loop characteristics of the autothermal system with a 12 percent bypass rate. We already showed the steady-... 

Consider the simple control loop of Fig. 2. The open-loop characteristics for a typical system exclusive of the controller, are a magnitude ratio and phase angle which fall off with increasing frequency. When the loop is closed through a controller, the magnitude ratio of the closed loop has a resonance peak if the controller gain is sufficiently great. 

We want to look at the stability of the closedloop system with a proportional controller Gc(j) = K- First, however, let us check the openloop stability of this system. The open-loop characteristic equation is the denominator of the openloop transfer function set equal to zero. 

Flowers, K.A. 1976. Visual closed-loop and open-loop characteristics of voluntary movement in patients with Parkinsonism and intention tremor. Brain 99 269-310. 

Hopefully, the reader has observed how the open-loop characteristics of a process determine its closed-loop response. And how little influence the controller has over this response. It is particularly true for processes of increasing difficulty, where problems begin to appear. 

But the closed-loop characteristics have three very important advantages over the corresponding open-loop characteristics, at relatively high values of Tbi. ... 

The open-loop characteristics can be obtained by means of an external digital-to-analog converter (DAC) and analog-to-digital converter (ADC) with a harmonic response scan. The frequency response of the open loop outside the operational range is essential to ensure the system stability. After the initial open-loop response is obtained, the filter correction is made to adjust the feedback amplification, to make a flat response in the passband and predetermined behavior at low-and high-cutoff frequencies. The process is iterative in the same way as it was for the non-feedback devices. 

The form of compensation is the 2-pole-2-zero method of compensation. This is to compensate for the effect of the double pole caused by the output filter inductor and capacitor. One starts by determining the control-to-output characteristic of the open-loop system. 

B.2 Defining the Open Loop Response of the Switching Power Supply—The Control-to-Output Characteristics... 

The schematic and Bode plot for the single-pole method of compensation are given in Figure B-16. At dc it exhibits the full open-loop gain of the op amp, and its gain drops at -20dB/decade from dc. It also has a constant -270 degree phase shift. Any phase shift contributed by the control-to-output characteristic... 

The Nichols chart shown in Figure 6.26 is a rectangular plot of open-loop phase on the x-axis against open-loop modulus (dB) on the jr-axis. M and N contours are superimposed so that open-loop and closed-loop frequency response characteristics can be evaluated simultaneously. Like the Bode diagram, the effect of increasing the open-loop gain constant K is to move the open-loop frequency response locus in the y-direction. The Nichols chart is one of the most useful tools in frequency domain analysis. 

The roots of equation (8.95) are the open-loop poles or eigenvalues. For the closed-loop system described by equation (8.94), the characteristic equation is... 

The key is to recognize that the system may exhibit underdamped behavior even though the open-loop process is overdamped. The closed-loop characteristic polynomial can have either real or complex roots, depending on our choice of Kc. (This is much easier to see when we work with... 

Example 7.2 If we have only a proportional controller (i.e., one design parameter) and real negative open-loop poles, the Routh-Hurwitz criterion can be applied to a fairly high order system with ease. For example, for the following closed-loop system characteristic equation ... 

We can now state the problem in more general terms. Let us consider a closed-loop characteristic equation 1 + KCG0 = 0, where KCG0 is referred to as the "open-loop" transfer function, G0l- The proportional gain is Kc, and G0 is "everything" else. If we only have a proportional controller, then G0 = GmGaGp. If we have other controllers, then G0 would contain... 

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. 

Let say we have a simple open-loop transfer function G0 of the closed-loop characteristic equation... 

INFICON s Auto Control Tune is based on measurements of the system response w/ith an open loop. The characteristic of the system response is calculated on the basis of a step change in the control signal. It is determined experimentally through two kinds of curve accordance at two points. This can be done either quickly w/ith a random rate or more precisely with a rate close to the desired setpoint. Since the process response depends on the position of the system (in our case the coating growth rate), it is best measured near the desired virork point. The process information measured in this vray (process amplification Kp, time constant T., and dead time L) are used to generate the most appropriate PID control parameters. 

Equation 7.119 is the characteristic equation of the system shown in Fig. 7.34 and is dependent only upon the open-loop transfer function G (j)H(j) and is therefore the same for both set point and load changes (equations 7.109 and 7.110). 

The roots of the characteristic equation may be real and/or complex, depending on the form of the open-loop transfer function. Suppose at to be complex, such that ... 

Fig. 12.5 One-dimensional bifurcation diagram for the single-nephron model obtained by varying the slope a of the open-loop response characteristics, r is the normalized arteriolar radius. The delay in the tubuloglomerular feedback is T = 16 s.

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