This paper presents an algebraic theory for the design of a decoupling Compensator for linear time-invariant multiinput multioutput systems. The design method uses a two-input one-output compensator, which gives a convenient parametrization of all diagonal input-output (I/ 0) maps and all disturbance-to-output (D/O) maps achievable by a stabilizing compensator for a given plant. It is shown that this method has two degrees of freedom: any achievable diagonal 110 map and any achievable D/O map can be realized simultaneously by a choice of an appropriate compensator. The difference between all achievable diagonal and nondiagonal I/O maps and the "cost" of decoupling is discussed for some particular algebraic settings. recommended bv Associate Editor. S. P. Bhattacharvva. This work was Manuscript received April 29, 1985; revised February 14, 1986. Paper supported by~thgNationa1 Science Foundation under Grant ECS-8119763, Comuuter Sciences and the Electronics Research Laboraton;. Universih, of The authors are with the Department of Electrical Engineering and ~~ G i h n i a , Berkeley, CA 94720. IEEE Log Number 8609227.

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Conditions are presented for closed-loop stabilizability of linear time-invariant (LTI) multi-input, multi-output (MIMO) plants with I/O delays (time delays in the input and/or output channels) using PID (Proportional + Integral + Derivative) controllers. We show that systems with at most two unstable poles can be stabilized by PID controllers provided a small gain condition is satisfied. For systems with only one unstable pole, this condition is equivalent to having sufficiently small delay-unstable pole product. Our method of synthesis of such controllers identify some free parameters that can be used to satisfy further design criteria than stability. ᭧

Recently (Gündeş et al., 2006) obtained stabilizing PID controllers for a class of MIMO unstable plants with time delays in the input and output channels (I/O delays). Using this approach, for plants with one unstable pole, we investigate resilient PI and PD controllers. Specifically, for PD controllers, optimal derivative action gain is determined to maximize a lower bound of the largest allowable controller gain. For PI controllers, optimal proportional gain is determined to maximize a lower bound of the largest allowable integral action gain.

Abstract-Closed-loop stabilization using proportional-integral-derivative (PID) controllers is investigated for linear multiple-input-multiple-output (MIMO) plants. General necessary conditions for existence of PID-controllers are derived. Several plant classes that admit PID-controllers are explicitly described. Plants with only one or two unstable zeros at or "close" to the origin (alternatively, at or close to infinity) as well as plants with only one or two unstable poles which are at or close to origin are among these classes. Systematic PID-controller synthesis procedures are developed for these classes of plants.