Modern Wiener-Hopf design of optimal controllers--Part II: The multivariable case

In many modern-day control problems encountered in the fluid, petroleum, power, gas and paper industries, cross coupling (interaction) between controlled and manipulated variables can be so severe that any attempt to employ single-loop controllers results in unacceptable performance. In all these situations, any workable control strategy most take into account the true multivariable nature of the plant and address itself directly to the design of a compatible multivariable controller. Any practical design technique most be able to cope with load disturbance, plant saturation, measurement noise, process lag, sensitivity and also incorporate suitable criteria delimiting transient behavior and steady-state performance. These difficulties, when compounded by the fact that many plants (such as chemical reactors) are inherently open-loop unstable have hindered the development of an inclusive frequency-domain analytic design methodology. However, a solution based on a least-square Wiener-Hopf minimization of an appropriately chosen cost functional is now available. The optimal controller obtained by this method guarantees an asymptotically stable and dynamical closed-loop configuration irrespective of whether or not the plant is proper, stable, or minimum-phase and also permits the stability margin of the optimal design to be ascertained in advance. The main purpose of this paper is to lay bare the physical assumptions underlying the choice of model and to present an explicit formula for the optimal controller.