Feedback controllers 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. 

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. 

Analysis of full sheet data is useful for process performance evaluations and product value calculations. For feedback control or any other on-line application, it is necessary to continuously convert scanner data into a useful form. Consider the data vector Y ,k) for scan number k. It is separated into its MD and CD components as Y( , A ) = yM )( )+Yc )( , k) where Ymd ) s the mean of Y ,k) as a scalar and YcD -,k) is the instantaneous CD profile vector. MD and CD controllers correspondingly use these calculated measurements as feedback data for discrete time k. Univariate MD controllers are traditional in nature with only measurement delay as a potential design concern. On the other hand, CD controllers are multivariate in form and must address the challenges of controller design for large dimensional correlated systems. 

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). 

Skogestad, S., I. Postletwaite, 1998, Multivariable Feedback Control, Wiley Stephanopoulos, G., 1984, Chemical Process Control, Prentice Hall Zheng, A. R., R. V. Mahajanam, J. M. Douglas, 1999, Hierarchical procedure for plantwide control system synthesis, AlChEJ, 45,1255... 

Feedforward control is probably used more in chemical engineering systems than in any other field of engineering. Our systems are often slow-moving, nonlinear, and multivariable, and contain appreciable deadtime. All these characteristics make life miserable for feedback controllers. Feedforward controllers can handle all these with relative ease as long as the disturbances can be measured and the dynamics of the process are known. 

Remember that the inverse of a matrix has the determinant of the matrix in the denominator of each element. Therefore, the denominators of all of the transfer functions in Eq. (12.21) contain Det[/ + ji(j)Gc(. )] Now we know that the characteristic equation of any system is the denominator set equal to zero. Therefore, the closedloop characteristic equation of the multivariable system with feedback controllers is the simple scalar equation... 

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

MacFarlane, A.G.J. and Kouvaritakis, B. (1977) A design technique for linear multivariable feedback systems. International Journal of Control, 25, pp. 837-874. 

Glover (1986) studied the robust stabilization of a linear multivariable open-loop unstable system modelled as (G+A) where G is a known rational transfer function and A is a perturbation (or plant uncertainty). G is decomposed as G/+G2, where G, is antistable and G2 is stable (Figure 1). The controller and the output of the feedback system are denoted by K and y respectively. Gi is strictly proper and K is proper. Glover (1986) argued that the stable projection G2 does not affect the stabilizability of the system, since it can be exactly cancelled by feedback. The necessary and sufficient condition for G to be robustly stabilized is to stabilize its antistable projection G/. 

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