Controllability MIMO systems

Figure 6 - Comparison between EPC and >,-DMC for cylinder temperature control MIMO system...

Analyze a MIMO system with relative gain array, and assess the pairing of manipulated and controlled variables. 

With this scenario, the system may eventually settle, but it is just as likely that the system in Fig. 10.12 will spiral out of control. It is clear that loop interactions can destabilize a control system, and tuning controllers in a MIMO system can be difficult. One logical thing that we can do is to reduce loop interactions by proper pairing of manipulated and controlled variables. This is the focus of the analysis in the following sections. 

We now derive the transfer functions of the MIMO system. This sets the stage for more detailed analysis that follows. The transfer functions in Fig. 10.11 depend on the process that we have to control, and we ll derive them in the next section for the blending process. Here, we consider a general system as shown in Fig. 10.12. 

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. 

V is the boil-up rate. Design a 2x2 MIMO system with PI controllers and decouplers as in Fig. 10.14. 

The diagram above shows an interactive MIMO system, where the controlled variables, outlet flow temperature and concentration, both depend on the manipulated variables. In order to design a decentralized control, a pairing of variables should be decided. A look at the state Eq.(23) suggests the assignment of the control of the temperature to the cooling flow and the concentration control to the reactor inlet flow. In this case, the internal variable Tj may be used to implement a cascade control of the reactor temperature. Nevertheless, a detailed study of the elements of the transfer matrix may recommend another option (see, for instance, [1]). 

Interaction can be between two or more processes or between actions produced by different control loops applied to a single process. The former has already been discussed in Section 1.53. Some degree of interaction between control loops will nearly always occur in a multiple-input/multiple-output (MIMO) system. For example, consider the distillation process described in Section 7.3 (Fig. 7.9). Suppose it is desired to control simultaneously the compositions of both the overheads product stream (by manipulating the reflux flowrate) and the bottoms product stream (by regulating the steam flowrate to the reboiler). A typical arrangement is shown in Fig. 7.73. 

The primary interest in the pole placement literature recently has been in finding an analytical solution for the feedback matrix so that the closed loop system has a set of prescribed eigenvalues. In this context pole placement is often regarded as a simpler alternative than optimal control or frequency response methods. For a single control (r=l), the pole placement problem yields an analytical solution for full state feedback (e.g., (38), (39)). The more difficult case of output feedback pole placement for MIMO systems has not yet been fully solved(40). 

This section considers the application of PID controllers to coupled MIMO processes. A key issue when applying PID controllers to MIMO systems is deciding which manipulated variable should be used to control which controlled variable. This is referred to as choosing the manipu-lated/controlled variable pairings [(c, y) pairings] or the control configuration. The factors that affect the choice of (c, y) pairings are analyzed here. 

Chapter 23 will discuss the new questions that must be answered for the controller design of MIMO systems. It will also present a methodology for the development of alternative control configurations for such systems, based on their degrees of freedom. 

Finally, in Chapter 25 we will present an introduction to the design of control systems for complete plants, which constitute the most complex MIMO systems to be encountered by a chemical engineer. 

For MIMO systems there is a large number of alternative control configurations. The selection of the most appropriate is the central and critical question to be resolved. 

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. 

Again, poles and zeros are important for evaluating stability and controllability properties of the physical system. To find the poles of an open-loop MIMO system one can use the transfer function matrix or the state-space description. They are related by ... 

The methodology presented hereafter regards a MIMO system that can be handled by a combination of multi SISO loops. It is an input/output controllability being based on linear analysis tools. It can be applied to a stand-alone complex unit, as a distillation column, or to a flowsheet. In this later case it has the character of a decentralised (integral) plantwide control problem. 

In the examples of controlled processes that we have been discussing, we have worked with systems handling one resource and with one target variable, i.e., single-input/single-output (SISO) systems. However, in actual practice, the process is affected by multiple input variables and has multiple target variables, i.e., a multiple-input/multiple-output (MIMO) system. 

The previous analysis for SISO systems can be generalized to MIMO systems by using the Principle of Superposition. For simplicity, we first consider a process control problem with two outputs, y and yi, and two inputs, u and Ui The predictive model consists of two equations and four individual step-response models, one for each input-output pair ... 

The dynamic matrix A for the MIMO system of N controlled variables and M manipulated variables with a prediction horizon P is... 

Fig. 3.17 Leftlbottom Picture of the drive (cylindrical piece length 40 nun diameter 22 nun) and its control unit (quadratic box holding the cylindrical drive unit) this whole unit slides into the detector system unit (top part in the left picture) Right Transfer function of MIMOS II velocity transducer (see also Sect. 3.1.1). For details of the drive unit and the model describing its behavior see [36, 45]...

This chapter is organized in the following way. First, the general model of the CSTR process, based on first principles, is derived. A linearized approximate model of the reactor around the equilibrium points is then obtained. The analysis of this model will provide some hints about the appropriate control structures. Decentralized control as well as multivariable (MIMO) control systems can be designed according to the requirements. 

Model reference adaptive control is based on a Lyapunov stability approach, while the hyperstability method uses Popov stability analysis. All of the above methods have been tested on experimental systems, both SISO and MIMO (53), (54), (55). The selftuning regulator is now available as a commercial software package, although this method is not satisfactory for variable time delays, an important industrial problem. 

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