SISO

The original formulations of MPC (i.e., DMC and IDCOM) were based on empirical hnear models expressed in either step-response or impulse-response form. For simphcity, we will consider only a singleinput, single-output (SISO) model. However, the SISO model can be easily generalized to the MIMO models that are used in industrial applications. The step response model relating a single controlled variable y and a single manipiilated variable u can be expressed as... 

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

As with a SISO system, a sensitivity function may be defined... 

A SISO eontroller was designed (for mean erystal size), even though relative gain array analysis showed possible interaetions between all of the three eontrol... 

Compound Si-O (A) both ring types Si-O-Si (°) 51404 rings Si-O-Si (°) SisOs rings Reference... 

If you are working with only single-input single-output (SISO) problems, it would be more appropriate to replace the notation B by b and C by cT, and write d for D. 

We use a simple liquid level controller to illustrate the concept of a classic feedback control system.1 In this example (Fig. 5.1), we monitor the liquid level in a vessel and use the information to adjust the opening of an effluent valve to keep the liquid level at some user-specified value (the set point or reference). In this case, the liquid level is both the measured variable and the controlled variable—they are the same in a single-input single-output (SISO) system. In this respect, the controlled variable is also the output variable of the SISO system. A system refers to the process which we need to control plus the controller and accompanying accessories such as sensors and actuators.2... 

Figure 5.4. Block diagram of a simple SISO closed-loop system.

We first establish the closed-loop transfer functions of a fairly general SISO system. After that, we ll walk through the diagram block by block to gather the thoughts that we must have in synthesizing and designing a control system. An important detail is the units of the physical properties. 

In a SISO system, we manipulate only one variable, so we must make a decision. Since our goal is to control the tank temperature, it would be much more sensible to manipulate the steam temperature TH instead of the inlet temperature. We can arrive at this decision with physical... 

Our analyses of SISO systems seldom take into account simultaneous changes in set point and load.2 We denote the two distinct possibilities as... 

After we have chosen the controlled and manipulated variables, the remaining ones are taken as load variables in a SISO system. 

We now return to the use of state space representation that was introduced in Chapter 4. As you may have guessed, we want to design control systems based on state space analysis. State feedback controller is very different from the classical PID controller. Our treatment remains introductory, and we will stay with linear or linearized SISO systems. Nevertheless, the topics here should enlighten( ) us as to what modem control is all about. 

There are many advanced strategies in classical control systems. Only a limited selection of examples is presented in this chapter. We start with cascade control, which is a simple introduction to a multiloop, but essentially SISO, system. We continue with feedforward and ratio control. The idea behind ratio control is simple, and it applies quite well to the furnace problem that we use as an illustration. Finally, we address a multiple-input multiple-output system using a simple blending problem as illustration, and use the problem to look into issues of interaction and decoupling. These techniques build on what we have learned in classical control theories. 

It is apparent from Eq. (10-22) that with interaction, the controller design of the MIMO system is different from a SISO system. One logical question is under what circumstances may we make use of SISO designs as an approximation Or in other words, can we tell if the interaction may be weak This takes us to the next two sections. 

After proper pairing of manipulated and controlled variables, we still have to design and tune the controllers. The simplest approach is to tune each loop individually and conservatively while the other loop is in manual mode. At a more sophisticated level, we may try to decouple the loops mathematically into two non-interacting SISO systems with which we can apply single loop tuning procedures. Several examples applicable to a 2 x 2 system are offered here. 

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. 

We will simply state that the SISO system design tool sisotool, as explained in Session 6, can be used to do frequency response plots. Now, we want to use the default view, so we just need to enter ... 

Figure 6. Argon-39 ages vs. depth for a borehole at Station Crite in central Greenland. Key -X-. 3tAr-age ---, SisO (H. Clausen, Copenhagen) and-, linear regression.

In principle, any type of process model can be used to predict future values of the controlled outputs. For example, one can use a physical model based on first principles (e.g., mass and energy balances), a linear model (e.g., transfer function, step response model, or state space-model), or a nonlinear model (e.g., neural nets). Because most industrial applications of MPC have relied on linear dynamic models, later on we derive the MPC equations for a single-input/single-output (SISO) model. The SISO model, however, can be easily generalized to the MIMO models that are used in industrial applications (Lee et al., 1994). One model that can be used in MPC is called the step response model, which relates a single controlled variable y with a single manipulated variable u (based on previous changes in u) as follows ... 

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 ... 

In the previous chapter we discussed the elements of a conventional single-input-single-output (SISO) feedback control loop. This configuration forms the backbone of almost all process control structures. 

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... 

The DMC discussed in this chapter is for a SISO system. We will say more about DMC in Chap. 17 since this methodology is fairly easily extended to multi-variable systems, which is where its real potential usefulness occurs. 

One final comment should be made about model-based control before we leave the subject. These model-based controllers depend quite strongly on the validity of the model. If we have a poor model or if the plant parameters change, the performance of a model-based controller is usually seriously affected. Model-based controllers are less robust than the more conventional PI controllers. This lack of robustness can be a problem in the single-input-single-output (SISO) loops that we have been examining. It is an even more serious problem in multi-variable systems, as we will find out in Chaps. 16 and 17. 

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