Multiple-criterion Control: A Convex Programming Approach

The performance of linear dynamic systems with respect to a non-linear increasing value function that aggregates convex functionals is investigated within a convex programming framework. The methodology developed in this paper combines fundamental properties of convex sets in order to decompose the multicriterion control problem into a two-level structure. The upper level of this structure comprises only decision-making aspects of the problem, and involves the solution of a relaxed and static multicriterion problem. The lower level of the structure comprises, in turn, only optimal control aspects of the problem. The overall procedure is easily implemented. The solution of the aggregated problem is obtained as the limit of a sequence of optimal control problems that preserve their original solution properties. Relationships between the approach adopted here and existing solution techniques are discussed. The solutions of the classical minimax and Salukvadze problems are obtained as special cases of the proposed approach. © 1995.31710251029Athans, Falb, (1966) Optimal Control, , MacGraw-Hill, NYBoyd, Barratt, (1991) Linear Controller Design-Limits of Performance, , Prentice-Hall, Englewood Cliffs, NJCohon, Marks, A review and evaluation of multiobjective programming techniques (1975) Water Resources Research, 11, pp. 208-220Ferreira, Geromel, An interactive projection method for multicriteria optimization problems (1990) IEEE Trans. Syst. Man Cybernetics, 20 SMC, pp. 596-605Geoffrion, Duality in nonlinear programming a simplified applicationsâ oriented development (1971) SIAM Review, 13, pp. 1-37Haimes, Hall, Freedman, (1975) Multiobjective Optimization in Water Resources Systems, , Elsevier, AmsterdamHwang, Masud, Multiple Objective Decision-Making Methods and Applications (1979) Lecture Notes in Economics and Mathematical Systems, 164. , Springer-Verlag, BerlinKhargonekar, Rotea, Multiple objective optimal control of linear systems: the quadratic norm case (1991) IEEE Trans. Autom. Control, 36, pp. 14-24Kirk, (1970) Optimal Control Theory—An Introduction, , Prentice-Hall, Englewood Cliffs, NJKoussoulas, Leondes, The multiobjective linear quadratic Gaussian problem (1986) Int. J. Control, 43, pp. 337-347Kresselmeier, Steinhauser, Application of vector performance optimization to a robust control loop design of a fighter aircraft (1983) International Journal of Control, 37, pp. 251-284Kuvshinov, Salyuchenko, Practical multiobjective design method for aircraft control system (1991) Proc. 1st IFAC Symp. on Design Methods of Control Systems, pp. 637-641. , ZurichLasdon, (1970) Optimization Theory for Large Systems, , Macmillan, LondonLi, On the minimax solution of multiple linear-quadratic problem (1990) IEEE Transactions on Automatic Control, 35 AC, pp. 1153-1156Li, A new solution approach to Salukvadze problem (1990) Proc. American Control Conf., pp. 409-414. , San Diego, CALi, On general multiple linear-quadratic control problems (1993) IEEE Trans. Autom. Control, 38 AC, pp. 1722-1727Li, Hierarchical control of large-scale systems with general multiple linear-quadratic structure (1993) Automatica, 29, pp. 1451-1461Medanić, Andjelić, On a class of differential games without saddle-point solution (1971) J. Optim. Theory Applies, 8, pp. 413-430MedaniÄ , Andjelić, Minimax solution of the multiple-target problem (1972) IEEE Transactions on Automatic Control, 17 AC, pp. 597-604Salukvadze, An approach to the solution of the vector optimization problem of dynamic systems (1982) J. Optim. Theory Applies, 38, pp. 409-422Toivonen, A multiobjective linear quadratic Gaussian control problem (1984) IEEE Transactions on Automatic Control, 29 AC, pp. 279-280Toivonen, Mäkilä, Computer-aided design procedure for multiobjective LQG control problems (1989) Int. J. Control, 49, pp. 655-666Yu, (1985) Multiple Criteria Decision Making, , Plenum Press, NYZadeh, Optimality and non-scalar valued performance criteria (1963) IEEE Transactions on Automatic Control, 8 AC, pp. 59-6