Process controllability analysis using linear programming

Controllability analysis is concerned with determining the limitations on achievable dynamic performance. This paper proposes the use of linear programming to determine the best linear controller and corresponding dynamic performance for problems of the form: . That is, a controller, K, and a reference operating point, u₀, are selected to minimise a specified objective, J, while ensuring feasibility for all disturbances, w, within a specified set, W. When K is a linear time invariant (LTI) controller and the objective function J and the constraints c can be expressed as linear functions then the above problem can be solved by linear programming. This formulation encompasses a wide range of problems ranging from minimising the maximum deviation in the regulated outputs subject to disturbances of magnitude less than one (the l₁ optimal control problem) to optimising the expected value of a linear economic objective (the Optimal Linear Dynamic Economics (OLDE) problem). The relationship of this work to other approaches to controllability analysis is discussed. A highly flexible framework for addressing typical process performance requirements through appropriate selection of J, c and W is presented. The relative merits of alternative approaches for defining the achievable closed loop transfer functions based on the -parameterisation are carefully discussed. The feasibility of the proposed approach is demonstrated on an industrial reactor example. The needs for further work are discussed.