Control of multivariable systems

The concepts of controllability and observability were introduced by Kalman (1960) and play an important role in the control of multivariable systems. 

Gegov, A., Distributed Fuzzy Control of Multivariable Systems, Kluwer, 1996. 

This chapter presents fundamental elements from the process control theory necessary in assessing the controllability of a design. It prepares also the computational tools used in Chapter 13 devoted to plantwide control. Prior knowledge is welcome. If the material is too difficult, the reader should go back to specialized undergraduate books. A concise presentation of the process control essentials viewed from process engineer s perspective has been recently published by Luyben Luyben (1998). In the field of feedback control of multivariable systems with emphasis on controllability analysis we recommend the book of Skogetsad and Postletwhite (1998). 

This tutorial looks at how MATLAB commands are used to convert transfer functions into state-space vector matrix representation, and back again. The discrete-time response of a multivariable system is undertaken. Also the controllability and observability of multivariable systems is considered, together with pole placement design techniques for both controllers and observers. The problems in Chapter 8 are used as design examples. 

J. Eemandez de Canete, S. Gonzalez-Perez and P. del Saz Orozco. A development of tools for monitoring and control of multivariable neurocontrolled systems wifti application to distillation columns , Proc. EANN 2007 Int. Conf., Thesaloniki, Greece, (2007), pp. 296-305. 

The third class of techniques include a frequency-domain method based on the identification of the sensitivity function S s)) and the complementary sensitivity function T s)) from plant data or CPM of multivariable systems [140]. Robust control system design methods seek to maximize closed-loop performance subject to specifications for bandwidth and peak... 

The states of a dynamic system are simply the variables that appear in the time differential. The time-domain differential equation description of multivariable systems can be used instead of Laplace-domain transfer functions. Naturally, the two are related, and we derive these relationships below. State variables are very popular in electrical and mechanical engineering control problems, which tend to be of lower order (fewer differential equations) than chemical engineering control problems. Transfer function representation is more useful in practical process control problems because the matrices are of lower order than would be required by a state variable representation. For example, a distillation column can be represented by a 2X2 transfer function matrix. The number of state variables of the column might be 200. 

Mathematical equations that are used on state variables of multivariable systems and associated with modern control theory... 

Gilbert, E.G. Decoupling of multivariable systems by state feedback. SIAM Journal of Control, 7(l) 50-63, February, 1969. 

Sheikholeslami, N., Shahmirzadi, D., Semsar, E., Lucas, C., Yazdanpanah, M. J. (2006). Applying brain emotional learning algorithm for multivariable control of HVAC systems. Journal of Intelligent Fuzzy Systems, 17, 35-46. 

U. ZUhlke, and A.S. Hauksdottir, Implementation of decoupling controllers for multivariable systems with a time delay, submitted to The 42nd IEEE Conference on Decision and Control, to be held in Maui, Hawaii, 2003. 

To get good control of the entire PRT, not only should the expander be controlled, but a completely integrated control system for this application should be designed. Most conventional control systems consist of individual control loops that only consider their specific tasks. The PRT—from a control perspective—is a multivariable system that requires integration between the different control loops. Further, some of the disturbances on the PRT are so fast that closed-loop control is too slow to keep the train under control. 

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. 

In recent years a number of commercial programs have been developed that produce root locus plots (and provide other types of analysis tools). These software packages can speed up controller design. Some of the most popular include CC, CONSYD, and MATRIX-X. We will refer to these packages again later in the book since they are also useful in the frequency and z domains, as well as for handling multivariable systems. /... 

In Chap. 15 we reviewed a tittle matrix mathematics and notation. Now that the tools are available, we will apply them in this chapter to the analysis of multivariable processes. Our primary concern is with closedloop systems. Given a process with its matrix of openloop transfer functions, we want to be able to see the effects of using various feedback controllers. Therefore we must be able to find out if the entire closedloop multivariable system is stable. And if it is stable, we want to know how stable it is. The last question considers the robustness of the controller, i.e., the tolerance of the controller to changes in parameters. If the system becomes unstable for small changes in process gains, time constants, or deadtimes, the controller is not robust. 

Interaction among control loops in a multivariable system has been the subject of much research over the last 20 years. Various types of decouplers were explored to separate the loops. Rosenbrock presented the inverse Nyquist array (INA) to quantify the amount of interaction. Bristol, Shinskey, and McAvoy developed the relative gain array (RGA) as an index of loop interaction... 

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... 

Based on the linearized models around the equilibrium point, different local controllers can be implemented. In the discussion above a simple proportional controller was assumed (unity feedback and variable gain). To deal with multivariable systems two basic control strategies are considered centralized and decentralized control. In the second case, each manipulated variable is computed based on one controlled variable or a subset of them. The rest of manipulated variables are considered as disturbances and can be used in a feedforward strategy to compensate, at least in steady-state, their effects. For that purpose, it is t3q)ical to use PID controllers. The multi-loop decoupling is not always the best strategy as an extra control effort is required to decouple the loops. 

For effective control of crystallizers, multivariable controllers are required. In order to design such controllers, a model in state space representation is required. Therefore the population balance has to be transformed into a set of ordinary differential equations. Two transformation methods were reported in the literature. However, the first method is limited to MSNPR crystallizers with simple size dependent growth rate kinetics whereas the other method results in very high orders of the state space model which causes problems in the control system design. Therefore system identification, which can also be applied directly on experimental data without the intermediate step of calculating the kinetic parameters, is proposed. 

Van Den Hof, P.J.M. DUMSI-package for off line multivariable system identification Laboratory for Measurement and Control, Delft University of Technology, Delft, I989. 

Even after linearization, the state-space model often contains too many dependent variables for controller design or for implementation as part of the actual control system. Low-order models are thus required for on-line implementation of multivariable control strategies. In this section, we study the reduction in size, or order, of the linearized model. 

The behaviour of many control loop components can be described by first-order differential equations provided that certain simplifying assumptions are made. Great care should be taken that the assumptions made are reasonable under the conditions to which the component is subjected. Two examples of a first-order system are described—a measuring element and a process. An illustration of a multivariable system which approximates to first order with respect to each input variable can be found in Example 7.11. 

A bioprocess system has been monitored using a multi-analyzer system with the multivariate data used to model the process.27 The fed-batch E. coli bioprocess was monitored using an electronic nose, NIR, HPLC and quadrupole mass spectrometer in addition to the standard univariate probes such as a pH, temperature and dissolved oxygen electrode. The output of the various analyzers was used to develop a multivariate statistical process control (SPC) model for use on-line. The robustness and suitability of multivariate SPC were demonstrated with a tryptophan fermentation. 

Section 2.2, in that two layers of control action involving separate controllers are proposed, whereas composite control relies on a single (possibly multivariable) controller with two components, a fast one and a slow one. Thus, the hierarchical control structure accounts for the separation of the flow rates of the process streams into two groups of inputs that act upon the dynamics in the different time scales. On the other hand, composite controller design (Figure 2.9) presupposes that the available manipulated inputs impact both the fast and the slow dynamics and relies on one set of inputs to regulate both components of the system dynamics. 

Rosenbrock, H. H., "Design of Multivariable Control Systems Using the Inverse Nyquist Array," Proc. IEEE, 1969, 116, 1929. 

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