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Underlying algorithms for designing multivariable decoupling and multiloop PI/PID controllers in a sequential fashion are addressed. A single-loop technique, composed of biased relay identification schemes and tuning formulae leading to the minimum weighted integral of square error, is developed to tune each loop in the predetermined sequence of loop closing. The proposed tuning technique is appropriate for a wide range of process dynamics in a multivariable environment. A method is then proposed to design decouplers to compensate for the effect of interactions and tune the resultant weakly interacting, single-loop PI/PID controllers sequentially. The decouplers, together with the single-loop controllers, constitute the multivariable decoupling controller. If the interactions are not significant, multiloop PI/PID controllers, which do not incorporate decouplers, could be employed. Simulation and comparative results are shown for one 2 × 2 and one 3 × 3 multivariable system from the literature. Despite its simplicity, the proposed design method yields superior multivariable designs on the basis of performance, robust stability, and integrity.

This paper examines particle trapping and release in confined microvortex flows, including those near a solid surface due to variations in the electrokinetic slip velocity and those at a liquid-gas interface due to an external momentum source. We derive a general analytical solution for a two-dimensional microvortex flow within a semicircular cap. We also use a bifurcation theory on the kinetic equation of particles under various velocity and force fields to delineate the conditions for a vortex trap, a point trap, a limit cycle trap, and the selective sorting of the particles into different traps. In the presence of only divergence-free forces on suspended particles, we find that two parameters, such as those related to Stokes drag, gravity, and flow vorticity, are sufficient to classify all the trap topologies for a given slip velocity distribution. We also show that nondivergence-free forces such as nonuniform repulsion or attraction can capture suspended particles in one trap and selectively sort a binary suspension into different traps.

Conditions for the onset of high Reynolds number roll waves on inclined interfaces are sought. The model equations of Dressler [Commun. Pure Appl. Math. 2, 149 (1949)] and Needham and Merkin [Proc. R. Soc. London Ser. A 394, 259 (1984)] are analyzed using dynamic singularity theory (normal form techniques) and numerical methods. A new family of roll-wave solutions is discovered. They provide bounds and averages of the velocities of all roll waves at a given Froude number. These are favorably compared to the data of Brock [Proc. Am. Soc. Civ. Eng. 12, 2565 (1970)] and Brauner and Maron [Int. J. Heat Mass Transfer 25, 99 (1982)]. The average wave velocity is demonstrated to be approximately 1.5 times the average fluid velocity.

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