1. Abstract An important issue in control structure selection is plant ”stabilization”. The paper presents a way to select measurement combinations c as controlled variables, such that when c is controlled at a constant setpoint, the effects of disturbances on the states are minimized. Indirect control: control y 1 indirectly by controlling the ”secondary” variables c: where H is the combination of measurements. Making some algebraic manipulations: Ultimate goal: Perfect disturbance rejection: P d = 0 and P c = P c0 Assumptions: 1. #c = #y 1 = #u; 2. #y = #u + #d; 3. The matrix P c0 is invertible. Solution: When P c0 =I → G=G 1 and G d =G d1. 3. Extension: Minimum State Deviation In this case #y1 > #u, so P d =0 is not possible. Instead we want to minimize the norm of P x d. Consider the following linear model: The effect of disturbance in the states is: P x and P x d represent the effect of the disturbances and implementation errors in the states Define the primary variables as linear combinations of the states (y 1 =Wx): What is the optimal choice of W that minimizes the value of P x d in Eq. 8? Assuming W=G xT : In summary, the main result in this paper can be summarized as follows: Theorem: Let P x d denote the steady-state transfer function from d to x with c=Hy kept constant. Then ||P x d || 2 is minimized by selecting H=G x T Ğ 1 Ğ y †,, where † indicates the pseudo-inverse. 4. Application to Distillation Distillation column: 82 states (41 compositions and 41 liquid holdups). Manipulated variables: reflux flow rate (L) and vapor boilup (V) Disturbances: feed flow rate (F) and fraction of liquid in the feed (q F ) Measurements: flow rates (L, V, D, and B). Combinations of states used: -Combination 1: Bottom and top compositions are the primary variables. Most common choice. -Combination 2: W was selected as being the transpose of G x (W=G x T ). -Combination 3: W was calculated solving min||[P x P x d ]|| 2. For each combination, matrix H was determined using Eq. 5. The resulting values of the 2-norm of ||P x ||, ||P x d ||, and ||[P x P x d ]|| are presented in Table 1. ||P x* || ||P x* d || ||P x* P x* d || 1 48.8289 2.5182 48.8817 2 0.0252 1.0886 1.0886 3 0.2560 1.0886 1.0886 Table 1 - Values of ||P x* ||, ||P x* d ||, and ||P x* P x* d || for all 3 combinations. 5. Conclusions 1. It is possible to control perfectly (having perfect disturbance rejection and minimizing the implementation error effects) any combination of the states if we have enough measurements available. 2. It is shown the importance of the use of the combination of states as primary variables. 3. Although the choice of the top and bottom compositions of a distillation column is good to reject perfectly the disturbances, it fails in the rejection of the implementation error and also it doesn’t give a good control of the states in the middle of the column. 4. The choice of W=G x T proved to be the best choice if the objective is to keep the states as close as possible to their desired (nominal) values. 5. It rejects very well both disturbances and implementation errors, although it doesn’t give perfect control of the top and bottom compositions. References Skogestad, S, 2004, Control structure design for complete chemical plants. Comp. Chem. Eng. 28, 219. Skogestad, S., and I. Postlethwaite, 1996, Multivariable Feedback Control. John Wiley & Sons, London. Strang, G, 1980, Linear Algebra and its Applications. Academic Press, New York. Acknowledgments The financial support of The National Council for Scientific and Technological Development (CNPq/Brasil) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES/Brasil) is gratefully acknowledged. 1. Introduction Regulatory control layer: main objective: ”stabilize” the plant ”Stabilization”: Includes both modes which are mathematically unstable (modes with RHP poles) as well as ”drifting” modes which need to be kept within limits to avoid operational problems. By avoiding “drift” (keeping all states close to their nominal values we are able to avoid problems resulting from nonlinear effects. Goal: Select secondary controlled variables (c=y 2 ) such that we minimize the effect of disturbances (d) on the weighted states (y 1 =Wx). 2. Previous work: Perfect Indirect Control Consider that we have the following linear model: P d - effect of disturbances with closed-loop (partial) control of c P c - effect on y 1 of changes in c Although the choice of top and bottom compositions as primary variables (combination 1) is able to control perfectly these two variables (the closed-loop gains relating the disturbances to the bottom and top compositions are zero), the gains of the states in the middle of the column are very large (above 0.7) (see Table 2). This choice doesn’t give good rejection of the implementation error (see matrix P x in Table 2). As expected (session 3), the results presented in Table 1 confirm that the use of W=G x T is an optimum choice. Table 2 - Values of the matrices P x* and P x* d for the 3 combinations. Combination 1 Combination 2 Combination3 P x* P x* d P x* P x* d P x* P x* d - G x (G x T G x ) -1 G x T is called projection matrix. - The product G x (G x T G x ) -1 G x T G x d is the closest point to G x d. - Thus, the choice W=G x T gives the minimum value of P x d - W=G x T is optimum for any choice of P c0 non-singular, i.e., result is not restricted to P c0 =I. - W can be arbitrarily chosen by the designer according to the objective of the process.