Closed-Loop Performance

These occur between trays 8 and 10. Remember that these results are for the 5/7/5 design (Ns = 5, Npx = 7, and Nr = 5). The effect of changing the number of reactive trays on these steady-state gains is discussed in a later section. 

---

Normally we design the feedback controller flj,) to give some desire closed-loop performance. For example, we might specify a desired closedloop damping coefficient. 

The amplitude of temperature fluctuations was controlled in a feedback loop by adjusting the relative phase between the primary and secondary forced air flows. A demonstration of the closed-loop performance is illustrated in Fig. 24.12. The controller converged on the optimum phase with a 1/e rise time of approximately 30 control steps (Fig. 24.12a). Figure 24.126 illustrates the difference between the power spectra with control off (i.e., neither primary nor secondary drivers) and control optimized. The response time necessary to reach the optimum phase was slowed by the large variations in the measured coherence (examples shown in Fig. 24.12a) which are attributed to the complex interactions between the inlet mode, the combustor modes, and the preferred mode of the jet. 

Tang et al. [20] have also investigated the same case study. The closed loop performance obtained using the two methodologies is compared in Fig. 7. A clear improvement in the performance is observed from Fig. 7 when the proposed technique is used. In Fig. 8 the same comparison is shown for the case of PID control. 

This paper presents a general mathematical programming formulation the can be used to obtain customized tuning for PID controllers. A reformulation of the initial NLP problem is presented that transforms the nonlinear formulation to a linear one. In the cases where the objective function is convex then the resulting formulation can be solved easily to global optimality. The usefulness of the proposed formulation is demonstrated in five case studies where some of the most commonly used models in the process industry are employed. It was shown that the proposed methodology offers closed loop performance that is comparable to the one... 

The use of a non-square controller (e.g., an MPC), such that the number of manipulated inputs is lower than the number of controlled variables, is certainly possible. While this approach eschews the use of cascaded configurations, it is intuitively detrimental to closed-loop performance due to the reduced number of manipulated variables. 

For the closed-loop performance bound given in Eq. 9.9, the variance of the output error is... 

The third class of techniques include a frequency-domain method based on the identification of the sensitivity function S s)) and the complementary sensitivity function T s)) from plant data or CPM of multivariable systems [140]. Robust control system design methods seek to maximize closed-loop performance subject to specifications for bandwidth and peak... 

The control computer/DCS system consists of controllers, A/D and D/A converters, and the signal conditioifing hardware and software, i.e., filtering and validation. Each of these components requires separate evaluation. Table 15.5 lists possible problems with the controller/DCS system. One way to initially check controller tuning is to place the control loop in manual (open the control loop) and observe whether the controlled variable lines out to a steady-state or near steady-state value. Comparing the open-loop and closed-loop performance indicates whether the controller is upsetting the process. If not, disturbances to the control loop in question are the primary source of the upsets. 

After each of the components has been evaluated and corrected wherever possible, the closed-loop system should be checked. From an overall point of view, there are three general factors that affect the closed-loop performance of a control loop (1) the type and magnitude of disturbances, (2) the lag associated with the components that compose the control loop, and (3) the precision to which each component of the control loop performs. Actuator deadband affects the variability in the controlled variable. The addition of lag to a control loop (e.g., sensor filtering) results in slower disturbance rejection, which can increase the variability in the controlled variable. Disturbance magnitude directly affects variability. 

The closed-loop deadband is an indication of the variability in the controlled variable that results from the combined effects of actuator deadband, sensor noise, and resolution of the A/D and D/A converters. The closed-loop settling time is an indication of the combined lags of the control loop components. The closed-loop performance assessment is a means of determining whether all the major problems within a control loop have been rectified. 

The following is a step-by-step troubleshooting process along with intermediate results for a temperature controller that was observed to result in sluggish closed-loop performance. 

Using this model, adaptive posi-cast controllers were designed, and detailed numerical simulation studies were carried out. These studies consisted of (i) the closed-loop performance of the adaptive controller, (ii) comparison of the adaptive controller with an empirical phase-shift controller, (iii) robustness with respect to parametric uncertainties, (w) robustness with respect to unmodeled dynamics and uncertain delays, (i/) performance in the presence of noise. 

Use simple criteria such as the one-quarter decay ratio (see Example 16.1), minimum settling time, minimum largest error, and so on. Such an approach is simple and easily implementable on an actual process. Usually, it provides multiple solutions (see Example 16.1). Additional specifications on the closed-loop performance will then be needed to break the multiplicity and select a single set of values for the adjusted parameters. 

The comparison of open and closed-loop responses (12.37) and (12.38) reveals that the sensitivity function S gives the reduction of sensitivity to disturbances, achieved by feedback control. It is evident that S =0 and T = 1 are desirable. In this way, the output follows perfectly the setpoint, and the process is not affected by disturbances. Both can be achieved by large controller gain, that is oo. However, large controller gain leads to instability, which sets limits on the achievable closed-loop performance. 

Figure 17.21 Closed Loop Performance Gain a) controllers IrQ2,12- SS2 and I3-D2 b) Ii-Q2 I2-SS2 and I3-D4...

Firstly, the implementation of three PI controllers has been tried. Figure 17.22a shows the results obtained in attempting to control the good impurity I3 with the structure Q2-Ii, SS2-I2 and D2-I3. The attempt failed, the system cannot be stabilised because of heavy interactions. As shown, the impurity I3 accumulates and exceeds its bound. The input magnitude of D2 is indeed too small to control I3, as was indicated by the analysis of the closed loop performance. Changing D2-I3 with D4-I3 does not change fundamentally the situation. Thus, the simultaneous control of the three impurities is not possible. This result was not predictable from the steady-state analysis, but it has been foreseen by the dynamic controllability analysis. Thus, it was decided to let the loop SS2-I2 on manual. 

Guideline 4. Choose output variables that exhibit significant interactions with other output variables. Plantwide control must handle the potential interactions in the process. Improved closed-loop performance is achieved by stabilizing output variables that interact significantly with each other. 

Guideline 12. Select measurement points that minimize time delays and time constants. Large time delays and dynamic lags in the process limit the achievable closed-loop performance. These should be reduced, whenever possible, in the process design and the selection of measurements. 

Insensitivity to model uncertainty, that is, the ability to control easily, and to provide adequate closed-loop performance, with relative insensitivity to model inaccuracies. 

In this chapter, several methods are described to assist the designer in rejecting designs that do not provide acceptable closed-loop performance, using models linearized about a steady state. These are generated by expressing the open-loop response of the process outputs, y s, in terms of the variations of the inputs, m 5, and disturbances,... 

After the recommended controller tuning constants have been calculated, they need to be put into the controller and then tested in closed loop for a final evaluation. Closed-loop performance results, that is, with the controller in automatic output mode, can be evaluated by the pattern recognition methods presented in Chapter 9. 

Engell and co-workers in Chapter C4 deal with the control structure selection based on input/output controllability measures. The limitations imposed by non-minimum phase characteristics on the attainable closed-loop performance are considered in the evaluation of the candidate set of control structure configurations. The optimisation of the attainable performance over the set of all linear stabilizing controllers can refine the controller structure with input constraints and coupling properties directly accounted for. 

While the main thrust of these analyses are to provide a plant that exhibits satisfactory closed-loop performance, the assumptions regarding the control system vary considerably across the various methods proposed. The open-loop indicators are largely based on factors that limit achievable closed-loop performance independent of controller type, whereas most of the optimization based integrated design formulations assume a specific controller type such as multiloop PI, LQG and so forth. While this is not considered to be a problem per se, it is important that the implications of these assumptions are clear so that appropriate deductions may be drawn. This chapter attempts to at least in part address this issue. 

---