Control multivariable

Recall the example of the shower in Sect. 3.2. The action was the ability to change the cold water flow and the target the shower temperature (more specifically the pleasant feeling on the skin). But there may be situations in which, although the temperature is right, one cannot take a shower because the water flow is too low or too high. Implicitly there appears a second objective, which is adequate water flow. Intuitively, you can put together two control loops as indicated in Fig. 9.9. 

Suppose that the temperature is fine, but the water flow is less than desired. If FIC-I03 increases the cold water flow, it will increase the total water flow however, the water temperature will go down. Consequently, TIC-103 will detect that the temperature has decreased and will try to compensate by increasing the flow of hot water. Of course, adding more hot water will lead to an increase in the total flow, but if both drivers (FlC-103 and TIC-103) are well tuned, the process will end with both converging to the desired values. 

The problem that occurs when a control loop disturbs the stability of the other control loop is called coupling, and it can be a difficult problem to address. 

Alternatively, one might try controlling the total flow by manipulating the hot water flow and the temperature by manipulating the cold water flow. Given multiple options for designing the control loops, the choice is usually based on specialized techniques. When variables cannot be linked, to avoid important couplings, uncouplers can be used. Uncouplers are elements that compensate actions on the loops in which no action has been taken. 

This technique has been applied quite successfully to reaction processes in the petroleum industry including fluid catalytic cracking units and catalytic reformers. 

In our example the two CVs are both compositions, either inferentials or analysers. Our MVs are reflux and reboil. Plant testing gave the process gain matrix shown in Table 12.9. 

The MVC continuously updates the bias terms as required so that the predicted CV matches the actual CV. It does this using the full dynamic model whereas our example is written on a steady-state basis. 

If we assume that when both CVs are at their targets of 5 %, the reflux is 56.5 and the reboiler steam rate 14. We can therefore write Equations (12.111) and (12.112) as 

What drives the MVC to select a node are the objective coefficients (or cost coefficients). These are applied to each MV (P) and, in some MVC packages, also to each CV (0. 

If m is the number of CVs and n the number of MVs then the MVC objective function or cost function (C) is given by 

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Multivariable control strategies utilize multiple input—multiple output (MIMO) controUers that group the interacting manipulated and controlled variables as an entity. Using a matrix representation, the relationship between the deviations in the n controlled variable setpoints and thek current values,, and the n controUer outputs, is... 

Three examples of simple multivariable control problems are shown in Fig. 8-40. The in-line blending system blends pure components A and B to produce a product stream with flow rate w and mass fraction of A, x. Adjusting either inlet flow rate or Wg affects both of the controlled variables andi. For the pH neutrahzation process in Figure 8-40(Z ), liquid level h and the pH of the exit stream are to be controlled by adjusting the acid and base flow rates and w>b. Each of the manipulated variables affects both of the controlled variables. Thus, both the blending system and the pH neutralization process are said to exhibit strong process interacHons. In contrast, the process interactions for the gas-liquid separator in Fig. 8-40(c) are not as strong because one manipulated variable, liquid flow rate L, has only a small and indirec t effect on one controlled variable, pressure P. 

Control Strategies for Multivariable Control Problems If a conventional multiloop control strategy performs poorly due to control loop interactions, a number of solutions are available ... 

Use a decouphng control system d. Use a multivariable control scheme (e.g., model predictive control)... 

The current widespread interest in MFC techniques was initiated by pioneering research performed by two industrial groups in the 1970s. Shell Oil (Houston, TX) reported their Dynamic Matrix Control (DMC) approach in 1979, while a similar technique, marketed as IDCOM, was published by a small French company, ADERSA, in 1978. Since then, there have been over one thousand applications of these and related MFC techniques in oil refineries and petrochemical plants around the world. Thus, MFC has had a substantial impact and is currently the method of choice for difficult multivariable control problems in these industries. However, relatively few applications have been reported in other process industries, even though MFC is a veiy general approach that is not limited to a particular industiy. 

One important class of nonlinear programming techniques is called quadratic programming (QP), where the objective function is quadratic and the constraints are hnear. While the solution is iterative, it can be obtained qmckly as in linear programming. This is the basis for the newest type of constrained multivariable control algorithms called model predic tive control. The dominant method used in the refining industiy utilizes the solution of a QP and is called dynamic matrix con-... 

Chu, C.C. (1985) Hoo-Optimization and Robust Multivariable Control, PhD Thesis, University of Minnesota, Minneapolis, MN. 

Lehtomaki, N.A., Sandell, Jr., N.R. and Athans, M. (1981) Robustness Results in Linear-Quadratic Gaussian Based Multivariable Control Designs, IEEE Trans, on Automat. Contr., AC-26(1), pp. 75-92. 

The multivariable modeling/control package is able to hold more tightly against constraints and recover more quickly from disturbances. This results in an incremental capacity used to justify multivariable control. An extensive test run is necessary to measure the response of unit variables. 

Finally, process control systems allow the unit to operate smoothly and safely. At the next level, an APC package (whether within the DCS framework or as a host-based multivariable control system) provides more precise control of operating variables against the unit s constraints. It will gain incremental throughput or cracking severity. 

Multitubular falling film continuous SO3 sulfonation units, 23 547 Multi tubular reactors, 23 544 Multiunit pilot plants, 19 459 Multiuse facility design, 11 47 Multiuse operation, for fermentation, 11 47 Multivariable control, 20 701... 

Override control (or "selective control as it is sometimes called) is a form of multivariable control in which a manipulated variable can be set at any point in time by one of a number of different controlled variables. 

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. 

It should be emphasized that the area of multivariable control is still in an early stage of development. Many active research programs are underway around the world studying this problem and every year brings many new developments. Therefore the methods and procedures presented in this book should be viewed as a summary of some of the practical tools developed so far. The weaknesses and limitations of the existing methods will be pointed out in the discussion. Improved methods will undoubtedly grow from current and future research. 

Controller structures that are not diagonal but have elements in all positions in the matrix are called multivariable controllers. 

This is the most important equation in multivariable control. It applies for any type of controller, diagonal (multiloop SISO) or full multivariable controller. If any of the roots of this equation are in the right half of the s plane, the system is closedloop unstable. 

Therefore the discussions of the RGA INA, and decoupling techniques will be quite brief I include them, not because they are all that useful, but because they are part of the history of multivariable control. You should be aware of what they are and of their limitations. 

Rosenbrock (Computer-Aided Control System Design, Academic Press, 1974) was one of the early woikers in the area of multivariable control. He proposed the use of INA plots to indicate the amount of interaction among the loops. 

Some of the earliest work in multivariable control involved the use of decouplers to remove the interaction between the loops. Figure 16.7 gives the basic structure of the system. The decoupling matrix is chosen such that each loop does not affect the others. Figure 16.8 shows the details of a 2 x 2 system. The decoupling element Dy can be selected in a number of ways. One of the most straightforward is to set I>n = D22 = 1 and design the 0 2 and Dji elements so that they cancel... 

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. 

So the multiloop SISO diagonal controller remains an important structure. It is the base case against which the other structures should be compared. The procedure discussed in this chapter was developed to provide a workable, stable, simple SISO system with only a modest amount of engineering effort. The resulting diagonal controller can then serve as a realistic benchmark, against which the more complex multivariable controller structures can be compared. 

One of the major questions in multivariable control is how to tune controllers in a diagonal multiloop SISO system. If PI controllers are used, there are 2N tuning parameters to be selected. The gains and reset times must be specified so that the overall system is stable and gives acceptable load responses. Once a consistent and rational tuning procedure is available, the pairing problem can be attacked. 

Several multivariable controllers have been proposed during the last few decades. The optimal control research of the 1960s used variational methods to produce multivariable controllers that rninirnized some quadratic performance index. The method is called linear quadratic (LQ). The mathematics are elegant but very few chemical engmeering industrial applications grew out of this work. Our systems are too high-order and nonlinear for successful application of LQ methods. 

In the last decade several other multivariable controllers have been proposed. We will briefly discuss two of the most popular in the sections below. Other multivariable controllers that will not be discussed but are worthy of mention are minimum variance controllers (see Bergh and MacGregor, lEC Research, Vol. 26, 1987, p. 1558) and extended horizon controllers (see Ydstie, Kershenbaum, and Sargent, AIChE J., Vol. 31, 1985, p. 1771). 

Undoubtedly the most popular multivariable controller is the multivariable extension of dynamic matrix control. We developed DMC for a SISO loop in Chap. 8. The procedure was a fairly direct least-squares computational one that solved for the future values of the manipulated variable such that some performance index was rninirnized. 

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