Decentralized integral controllability and D-stability

Abstract

This paper studies the integral controllability of decentralized control systems. Conventionally, the design philosophy is trying to preserve the closed-loop stability under failure conditions by pairing the input and output variables appropriately. The relative gain array (RGA) and Niederlinski index (NI) are generally employed to eliminate unworkable pairings. The design problem (failure tolerance) of multivariable control systems is explored. New results on the RGA, NI and the block relative gain are derived to understand failure tolerance in a transparent manner and to eliminate additional unworkable pairings. The operational problem is addressed with the definition of the decentralized integral controllability (DIC): “whether positive feedback will occur when each loop is detuned independently?”. The definition of DIC is similar to the well-known mathematical problem D-stability. A necessary and sufficient condition for DIC is derived for 3 × 3 systems for the first time. Furthermore, the criterion can be expressed in terms of the well-known interaction measure RGA. Despite the limitation on the system size (multivariable systems with the size less than 4 × 4), the result is generally applicable to most multivariable systems in chemical process control. Finally, the rules for variable pairing based on the new results are summarized. The mathematical results in this paper offer a solution to the practical important problem (DIC) in multivariable process control.