SISO controllers

This MPC formulation for SISO control problems can easily be extended to MIMO problems. 

In this illustration, we do not have to detune the SISO controller settings. The interaction does not appear to be severely detrimental mainly because we have used the conservative ITAE settings. It would not be the case if we had tried Cohen-Coon relations. The decouplers also do not appear to be particularly effective. They reduce the oscillation, but also slow down the system response. The main reason is that the lead-lag compensators do not factor in the dead times in all the transfer functions. 

The minimization of the quadratic performance index in Equation (16.2), subject to the constraints in Equations (16.5-16.7) and the step response model such as Equation (16.1), forms a standard quadratic programming (QP) problem, described in Chapter 8. If the quadratic terms in Equation (16.2) are replaced by linear terms, a linear programming program (LP) problem results that can also be solved using standard methods. The MPC formulation for SISO control problems described earlier can easily be extended to MIMO problems and to other types of models and objective functions (Lee et al., 1994). Tuning the controller is carried out by adjusting the following parameters ... 

Avoid control-loop interaction if possible, but if not, make sure the controllers are tuned to make the entire system stable. Up to this point we have discussed tuning only single-input-single-output (SISO) control loops. Many... 

Chapter 16 covers the analysis of multivariable processes stability, robustness, performance. Chapter 17 presents a practical procedure for designing conventional multiloop SISO controllers (the diagonal control structure) and briefly discusses some of the full-blown multivariable controller structures that have been developed in recent years. 

The first part of this chapter deals with the conventional diagonal structure multiloop SISO controllers. FuU-blown multivariable controllers are briefly discussed at the end of the chapter. 

In addition to the visual observations of the dynamic responses, a quantitative measure is needed to provide a better comparison. With such an objective, lAE values were evaluated for each closed-loop response. The PUL option shows the lowest lAE value of 5.607 x 10 , while the value for the Petlyuk column turns out to be 2.35 x 10. Therefore, the results of the test indicate that, for the SISO control of the heaviest component of the ternary mixture, the PUL option provides the best dynamic behavior and improves the performance of the Petlyuk column. Such result is consistent with the prediction provided by the SVD analysis. 

Figure 7 shows the dynamic responses obtained when the set point for the intermediate component was changed from 0.98 to 0.984. One may notice the better response provided by the Petlyuk column in this case, which is faster than the other two systems and without oscillations. When the lAE values were calculated, a remarkable difference in favor of the Petlyuk system was observed 2.87 x 10 for the Petlyuk column, compared to 0.0011 for the PUL system and 0.0017 for the PUV system. The results from this test may seem unexpected, since the new arrangements have been proposed to improve the operation capabilities of the Petlyuk column. The SISO control of the intermediate component, interestingly, seems to conflict with that of the other two components in terms of the preferred choice from dynamic considerations. 

With the on-line CSD-measurement equipment available, it becomes relevant to investigate and design control schemes for CSD s. A number of schemes were investigated by simulation in the past. Most of these control schemes use one proportional single-input-single-output (SISO) controller with the nucleation rate (or a... 

To keep the plant at its middle unstable steady state can be achieved by stabilizing the unstable steady state with a simple feedback control loop. For the sake of simplicity, we use a SISO (single input single output) proportional feedback control, in which the dense-phase temperature of the reactor is the controlled measured variable, while the manipulated variable can be any of the input variables of the system Yfa, FCD, etc. We use Yfa as the manipulated variable here. The set-point of the proportional controller is the dense-phase reactor temperature at the desired middle steady state in this case. Our simple SISO control law is... 

The control loop affects both the static behavior and the dynamic behavior of the system. Our main objective is to stabilize the unstable saddle-type steady state of the system. In the SISO control law (7.72) we use the steady-state values Yfass = 0.872 and Yrdss = 1.5627 as was done in Figures 7.14(a) to (c). A new bifurcation diagram corresponding to this closed-loop case is constructed in Figure 7.20. 

The use of control structures is an advanced flowsheeting feature. Flowsheet controllers are particularly useful in plant operation. An important application is the simulation of the steady state behaviour of SISO controllers. With respect to control action we can distinguish between two basic types, feedback and feedforward control. 

The controllability tools presented in here are based on the theory of linear systems, which is valid for relatively small disturbances around the stationary state. A non-linear approach, more suited for investigating the effect of large variations, will be developed in Chapter 13. The chapter starts with a brief introduction in process dynamics, followed by the properties of linear systems. The controllability analysis begins with SISO (single input/single output) systems and reviews the major concepts in feedback control. Then, the analysis is extended to MIMO (multi input/multi output) systems, with emphasis on decentralised control systems (multi SISO control loops), which is the most encountered in plantwide applications. 

Naturally, the type of controller plays an important role. In this chapter we limit the analysis to classical PID controllers. These form over 90% of the control loops in industry. As mentioned, from a plantwide control viewpoint multi-SISO controllers are the most adapted. Naturally, we do not exclude more sophisticated MIMO control systems, as DMC or Model Based Control systems, but these are typically applied to stand-alone complex units, as FCC reactors, complex distillation units in refining, etc. Hence, the controllability analysis presented here aims more to get a conceptual insight in the dynamics of a process related to its design than to offer a high-performance control solution. 

In a multi-input multi-output (MIMO) control system (Fig. 12.14), there are several controlled variables (vector y) that should be kept on set-points (vector r) faced to disturbances (vector d) by means of appropriate manipulated variables (vector u). The feedback controller K provides the algorithm that will ensure the link between the manipulated (inputs) and controlled (outputs) variables. In this chapter we will consider a decentralised control system that makes use of multi-SISO control loops, which means that a single controlled variables is controlled by a single manipulated variable. This arrangement is typical for plantwide control purposes. However, there will be interactions between different loops. These Interactions can be detrimental, or can bring advantages. Therefore, the assessment of interactions is a central issue in the analysis of MIMO systems. 

The RGA vs. frequency can be used to measure the diagonal dominance in multi-SISO control systems, by means of the simple quantity ... 

A value close to zero means quasi-independent SISO controllers. 

The dynamic simulation model has been adapted to meet the constraints of a large scale problem and of the equation solving mode of Aspen Dynamics. The final model contained more than 6000 equations. Since the change in material balance (inventory) takes place at long time scales, some substantial simplifications of the local control of units can be considered. Finally, the plantwide control problem is reduced to analyse a 3x7 system, where three outputs (concentration of impurities li, I2, and I3) should be controlled with three among five inputs (D2, SS2, Q2, D4, and Q4), in the presence of two disturbances (Fdce, X ). Because of decentralised control, at most three SISO controllers should be physically implemented. 

The steady-state analysis indicated good sensitivity of the controlled variables to inputs and the possibility to reject disturbances with a diagonal multi-SISO control structures, as for example Q2-Ii, SS2-I2, D2-I3 (see also Table 4). Although there are some differences in the open loop behaviour of flowsheet alternatives as compared to the base-case, these do not justify a net preference. On the other hand, the steady state RGA analysis predicted that a control structure using D2 to control I3 should be more affected by interactions as by manipulating D4 or Q4, but the differences cannot be evaluated only by the inspection of numerical values. Because a steady-state analysis cannot predict how the real disturbances would be handled by the control system, a deeper controllability analysis is necessary in the frequency domain. The battery of indices has been tried in each case, as described by Groenendijk et al.(2000). Here we give only representative results. 

RGA-number, defined as RGA - I s ni, gives a quantitative measure of the interactions in a diagonal decentralised control structure. The lower RGA-number the more preferred is the control structure. A value close to zero means quasi-independent SISO controllers. Note that a good controllability can be obtained up to the frequency where the RGA number does not exceed 1. Consequently, graphical representations versus frequency enable to evaluate the robustness of control faced to a certain frequency range of disturbances. 

This process has 15 control valves, so there is an enormous number (16 factorial) of possible simple SISO control structures. The nine inventories (six levels and the three pressures) must be controlled. The three impurities in the two product streams must also be controlled. The production rate must also be set. This leaves 15-9 — 3 — 1=2 control valves that can be set to accomplish other objectives (typically economic objectives, such as minimizing energy costs). 

EXAM E LE 1 2. 8. Determine the closedloop characteristic equation for the system whose openloop transfer function matrix was derived in Example 12.6. Use a diagonal controller structure (two SISO controllers) that are proportional only. 

In the last chapter we developed some mathematical tools and some methods of analyzing multivariable closedloop systems. This chapter studies the development of control structures for these processes. Because of their widespread use in real industrial applications, conventional diagonal control structures are discussed. These systems, which are also called decentralized control, consist of multiloop SISO controllers with one controlled variable paired with one manipulated variable. The major idea in this chapter is that these SISO controllers should be tuned simultaneously, with the interactions in the process taken into account. 

Tuning of single-input, single-output (SISO) controllers (P, PI, and PID controllers). Note that Section 21.4 provides instruction on model-based Pl-controller tuning. 

Figure 7.14. Schematic representation of a MTBE RD column with a 4x4 SISO control structure. Remarks the coupling between controlled and manipulated variables for each control loop are listed in table 7.5 numbers represent tray location.

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