Closed-loop equations

With the given choices of Gc (P, PI, PD, or PID), Gp, Ga and Gm, plug their transfer functions into the closed-loop equation. The characteristic polynomial should fall out nicely. 

From this step, we derive the closed-loop equation (l G cGp)GL... 

Example 7.2B Do the root locus plot and find the ultimate gain of Example 7.2 (p. 7-5). The closed-loop equation from that example is ... 

Example 7.6 Construct the root locus plots of some of the more common closed-loop equations with numerical values. Make sure you try them yourself with MATLAB. 

However, MATLAB allows us to get the answer with very little work—something that is very useful when we deal with more complex systems. Consider a numerical problem with values of the process gain Kp = 1, and process time constants X = 2 and x2 = 4 such that the closed-loop equation is... 

Nyquist stability criterion Given the closed-loop equation 1 + Gol (joi) = 0, if the function G0l(J ) has P open-loop poles and if the polar plot of GOL(](o) encircles the (-1,0) point... 

For the present problem, and based on all the settings provided by the different methods, we may select Xj = 3 s and xD = 0.5 s. We next tune the proportional gain to give us the desired response. The closed-loop equation with an ideal PID controller is now ... 

With the choice of x = 0.5 s, but without the inner loop nor the secondary controller, the closed-loop equation is... 

What if we click a point not exactly on a root locus When we select a point s, MATLAB calculates the value k = -p(s )/q(s ), which will only be a real positive number if s satisfies the closed-loop equation. Otherwise, k is either complex, or negative if the pole is a real number. In this case, MATLAB calculates the magnitude of k, uses it as the gain and computes the corresponding closed-loop poles. Thus we find the chosen points are always right on the root loci no matter where we click. 

Substituting Eq.(50) into Eq.(49) the closed loop equations of the system can be written as ... 

Pole and zero placement using a dynamic compensator for an SISO system can be accomplished by specifying analytically the closed loop servo response (e.g., first or second order with deadtime). Suppose that the specified response is defined by P(s) solving the closed loop equation (5) yields an analytical... 

If ysp = Ku[) at the final steady state and the gain K is known, the appropriate preload expression is = yspIK. Then the closed-loop equation becomes... 

In the former USSR, there reportedly are two technologies in use one is old anthrahydroquinone autoxidation technology and the other is closed-loop isopropyl alcohol oxidation technology. Production faciUties include several smaller, 100-150-t/yr isopropyl alcohol oxidation plants and a larger, 15,000-t/yr plant, which reportedly is being expanded to 30,000-t/yr. Differences in this technology as compared to the Shell Chemical Co. process are the use of oxygen-enriched air in the oxidation step and, catalytic reduction of the coproduct acetone back to isopropyl alcohol per equation 21. 

In principle, ideal decouphng eliminates control loop interactions and allows the closed-loop system to behave as a set of independent control loops. But in practice, this ideal behavior is not attained for a variety of reasons, including imperfect process models and the presence of saturation constraints on controller outputs and manipulated variables. Furthermore, the ideal decoupler design equations in (8-52) and (8-53) may not be physically realizable andthus would have to be approximated. 

For field-oriented controls, a mathematical model of the machine is developed in terms of rotating field to represent its operating parameters such as /V 4, 7, and 0 and all parameters that can inlluence the performance of the machine. The actual operating quantities arc then computed in terms of rotating field and corrected to the required level through open- or closed-loop control schemes to achieve very precise speed control. To make the model similar to that lor a d.c. machine, equation (6.2) is further resolved into two components, one direct axis and the other quadrature axis, as di.sciis.sed later. Now it is possible to monitor and vary these components individually, as with a d.c. machine. With this phasor control we can now achieve a high dynamic performance and accuracy of speed control in an a.c. machine, similar to a separately excited d.c. machine. A d.c. machine provides extremely accurate speed control due to the independent controls of its field and armature currents. 

To determine the amount of gain needed to boost the closed loop function to OdB at the gain cross-over frequency (see Equation B.24) ... 

The Process Reaction Method assumes that the optimum response for the closed-loop system occurs when the ratio of successive peaks, as defined by equation (3.71), is 4 1. From equation (3.71) it can be seen that this occurs when the closed-loop damping ratio has a value of 0.21. The controller parameters, as a function of R and D, to produce this response, are given in Table 4.2. 

Comparing the closed-loop transfer function given in equation (4.113) with the standard form given in (3.42)... 

Assuming that the temperature of the surroundings O it) remains constant, the closed-loop transfer function (using equation (4.130)) for the temperature control system, is... 

The response to a step change in the desired temperature of 0-20 °C for the closed-loop transfer function given by equation (4.134) is shown in Figure 4.35. 

The closed-loop transfer function for any feedback control system may be written in the factored form given in equation (5.41)... 

Closed-loop poles (For K = 11.35) Since the closed-loop system is third-order, there are three closed-loop poles. Two of them are given in equation (5.81). The third lies on the real locus that extends from —5 to —oo. Its value is calculated using the magnitude criterion as shown in Figure 5.15. 

A frequency domain stability criterion developed by Nyquist (1932) is based upon Cauchy s theorem. If the function F(s) is in fact the characteristic equation of a closed-loop control system, then... 

Alternatively, the closed-loop frequency response can be obtained from a Nyquist diagram using the direct construction method shown in Figure 6.25. From equation (6.73)... 

Since many closed-loop systems approximate to second-order systems, a few interesting observations can be made. For the case when the frequency domain specification has limited the value of Mp to 3 dB for a second-order system, then from equation (6.72)... 

In equation (8.94) the matrix (A - BK) is the closed-loop system matrix. 

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 roots of equation (8.96) are the closed-loop poles or eigenvalues. 

Robust performance then means that the closed-loop system will meet the performance specification given in equation (9.145) if and only if the nominal system is closed-loop stable (equation (9.141)) and that the sensitivity function Sm(jar) and complementary sensitivity function for the nominal system satisfy the rela-... 

From the partitioned matrix in equation (9.161), the closed-loop transfer function matrix relating yi and uj is... 

The continuous and discrete closed-loop systems are shown in Figures 7.22(a) and (b). The digital compensator is given in equation (7.128). Script file examp78.m produces the step response of both systems (Figure 7.25) and prints the open and closed-loop continuous and pulse transfer functions in the command window... 

Rlccatl matrix, see equation (9.45) %Closed-loop eigenvalues... 

This chapter has provided a brief overview of the application of optimal control theory to the control of molecular processes. It has addressed only the theoretical aspects and approaches to the topic and has not covered the many successful experimental applications [33, 37, 164-183], arising especially from the closed-loop approach of Rabitz [32]. The basic formulae have been presented and carefully derived in Section II and Appendix A, respectively. The theory required for application to photodissociation and unimolecular dissociation processes is also discussed in Section II, while the new equations needed in this connection are derived in Appendix B. An exciting related area of coherent control which has not been treated in this review is that of the control of bimolecular chemical reactions, in which both initial and final states are continuum scattering states [7, 14, 27-29, 184-188]. 

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