Consistency Principles for Stability in Decentralized Control Systems

Structural Analysis and Stability Conditions of Decentralized Control Systems

“…Indeed, for 2 × 2 systems, for which simple closed form of interaction terms are available, we have, Equation 23 can be easily verified from eqs 5, 7, and 8 or 7, 9, and 11. Seborg et al (1989) and Zhu and Jutan (1995a) also demonstrated the final form of the characteristic equation, i.e., the third term in the above equation, by rather lengthy algebraic transformation of various transfer functions. In contrast, eq 23 is readily obtained directly on the basis of the structural information of the system.…”

“…It is worthwhile to note that, in the case of inherently large process interaction, the only choice to maintain stability by means of controller design is to reverse the control direction of one of the controllers (Zhu and Jutan, 1995a). However, this is undesirable since system integrity has to be sacrificed and, as a result, one of the control loops will become open loop unstable by theorem 1.…”

“…When the process interaction itself becomes inherently large, tuning the other controller may not be sufficient enough to counteract the interaction in order to maintain stability (see theorem 3). Instead, the direction of one of the two controllers may have to be reversed, leading to the loss of integrity (Zhu and Jutan, 1995a)! 4.…”

“…It is clear that loop 1 can be made stable by reversing the direction of controller 1 (k 1 ) -0.5) and properly tuning k 2 . However, reversing the direction of controller 1 leads to the loss of integrity of the overall system against failure of loop 2, since a local positive feedback is formed (Zhu and Jutan, 1995a). Also, loop 2 becomes open loop unstable as a result according to theorem 1.…”

“…Proof of Theorem 3. According to Zhu and Jutan (1995a), when the assumption in theorem 3 holds, a consistency principle for stability (necessary condition for stability) for any SISO system is given by where In eq c1 and c2, g(s) and c(s) denote the process and the controller transfer functions, respectively, and g(0) and c′(0) denote the steady state gain of their perspective transfer functions. Condition c1 actually specifies a feedback condition in any SISO system.…”