MIMO system

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

In this section, we analyze a multiple input-multiple output (MIMO) system. 

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. 

Before we design the MIMO system, we need to check the paring of variables. The steady state gain matrix is... 

From Eq.(23) it is possible to deduce the (2x2) transfer function matrix, G(s), of the linearized MIMO system. 

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. 

P. Psarris and C. A. Floudas. Improving dynamic operability in MIMO systems with time delays. Chem. Eng. Sci., 45(12) 3505,1990. 

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). 

In the past few years, a number of workable pole-placement algorithms have been published. However, their application to MIMO systems with incomplete state variable feedback are often unsatisfactory in that ... 

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. 

In the chemical industry most of the processing systems are multiple-input, multiple-output systems. Since the design of SISO systems is simpler, we will start first with them and progressively cover the design of MIMO systems. 

What is a SISO system and what is a MIMO system Give examples from the chemical engineering field for both. 

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. 

In Chapter 24 we will examine the selection of the appropriate measurements and manipulations in order to close the loops. Furthermore, we will study the design of "decoupled loops for MIMO systems. 

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. 

The input/output dependency in a linear MIMO system can be described by the following relation ... 

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 system is stable because it has only one pole at -1/7.9 in LHP. 12.4.2 Directions in MIMO systems... 

For MIMO systems, the inputs and outputs are vectors, and therefore we need to sum up the magnitudes of the elements. If we select the vector 2-norm ... 

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