Design of Resilient Processing Plants--ix. Effect of Model Uncertainty on Dynamic Resilience

The achievable quality of control for a particular system (its dynamic resilience) is limited by the nonminimum phase characteristics of the plant, constraints on the manipulated variables and model uncertainty. Model uncertainty requires that the controller be detuned and performance be sacrificed. The goal of this paper is to quantify this well-known qualitative statement. The closed-loop system must remain stable for all possible plants as defined by the uncertainty description. This robust stability requirement is used to derive simple bounds on the nominal performance for some specific cases. These bounds are relatively easy to evaluate and should be effective tools for screening alternative designs in terms of their resilience characteristics. The RGA and the minimized condition number are accurate measures with respect to element uncertainty, provided the relative errors of the transfer matrix elements are independent (uncorrelated) and have similar magnitude bounds. t. INTRODIJCXION Most chemical plants are designed on the basis of steady state considerations, and the control system is designed separately in a subsequent stage of the project. This separation is acceptable provided that there exist suitable design-stage methods which can assess the ‘*controllability” of the plant. That is, it must be determined u priori whether the design of a control system offering “reasonable” closed-loop response will subsequently be feasible. Until recently, such methods were not available. As a result, the expected performance often was not achieved in the opei’ating plant. In some instances, a minor change at the initial design stage could have resulted in a “controllable” plant. Previously, the controllability assessment has been based on simulations. This approach is complex and requires a complete dynamic model of the plant. Usually a number of case studies are performed with different choices of inputs, disturbances, operating conditions, controller structures and controller parameters. All those choices could bias the controllability assessment in an erroneous manner. Morari (1983) suggested making the problem of controllability assessment independent of the controller selection problem. This is done by finding a plant’s best achievable closed-loop control performance. for all possible constant parameter linear controllers. This target, the upper bound on the achievable closed loop performance, is defined as the plant’s dynamic resilience. Thus, “dynamic resilience” is an expression of the plant’s inherent limitation on the closed-loop system’s dynamic response which is not biased by specific choices of controllers. The limitations imposed by non-minimum phase elements and constraints have been discussed in quantitative detail by Morari (1983) and Holt and Morari (1985a, b). Fundamentally, perfect control can only be achieved if the plant is invertible. Non-minimum phase elements [Right Half Plane (RHP) zeros and time delays] make it impossible to invert the plant and retain (internal) stability of the closed-loop system. The effect of constraints on performance is also related to a plant’s closeness to singularity. If the minimum singular value of a plant P,@(P)) is small then the plant is nearly singular. This means that the plant has a very small gain for a particular input direction. To achieve tight control, the controller would have to provide very large input signals in this direction, possibly violating input size constraints. The objective of this paper is to study the effects of model uncertainty on dynamic resilience. Model uncertainty requires that the controller be detuned and performance be sacrificed. The primary goal is to quantify this well-known qualitative statement by deriving expressions relating achievable closed-loop performance and uncertainty. The first (and most important) step is to quantify the model uncertainty. This is usually not a trivial problem, and very misleading results may arise if an inappropriate uncertainty description is used. Another goal of this paper is to demonstrate some of these pitfalls. Therefore, the design engineer encounters a difficult situation: simple achievable performance bounds may be obtained with a crude uncertainty description but such bounds are often misleading. On the other hand, a detailed description of the model uncertainty is needed to find more meaningful bounds. Such descriptions are normally not available. A first step in resolving this dilemma is to identify for specific problem classes (e.g. distillation columns) the sources of model uncertainty which are likely to cause complications. The engineer can then concentrate on these when quantifying the uncertainty. Some of the examples in this paper will be helpful in this respect.