Design of Retarded Fractional Delay Differential Systems Using the Method of Inequalities

Methods based on numerical optimization are useful and effective in the design of control systems. This paper describes the design of retarded fractional delay differential systems （RFDDSs） by the method of inequalities, in which the design problem is formulated so that it is suitable for solution by numerical methods. Zakian＇s original formulation, which was first proposed in connection with rational systems, is extended to the case of RFDDSs. In making the use of this formulation possible for RFDDSs, the associated stability problems are resolved by using the stability test and the numerical algorithm for computing the abscissa of stability recently developed by the authors. During the design process, the time responses are obtained by a known method for the numerical inversion of Laplace transforms. Two numerical examples are given, where fractional controllers are designed for a time-delay and a heat-conduction plants.

Computation of Peak Output for Inputs Restricted in L_2 and L_∞ Norms Using Finite Difference Schemes and Convex Optimization

Control systems designed by the principle of matching gives rise to problems of evaluating the peak output. This paper proposes a practical method for computing the peak output of linear time-invariant and non-anticipative systems for a class of possible sets that are characterized with many bounding conditions on the two- and/or the infinity-norms of the inputs and their derivatives. The original infinite-dimensional convex optimization problem is approximated as a large-scale convex programme defined in a Euclidean space, which are associated with sparse matrices and thus can be solved efficiently in practice. The numerical results show that the method performs satisfactorily, and that using a possible set with many bounding conditions can help to reduce the design conservatism and thereby yield a better match.

Design of Non-rational MIMO Feedback Systems with Inputs and Outputs Satisfying Certain Bounding Conditions by Using Rational Approximants

When a feedback system has components described by non-rational transfer functions, a standard practice in designing such a system is to replace the non-rational functions with rational approximants and then carry out the design with the approximants by means of a method that copes with rational systems. In order to ensure that the design carried out with the approximants still provides satisfactory results for the original system, a criterion of approximation should be explicitly taken into account in the design formulation. This paper derives such a criterion for multi-input multi-output（MIMO） feedback systems whose design objective is to ensure that the absolute values of every error and every controller output components always stay within prescribed bounds whenever the inputs satisfy certain bounding conditions. The obtained criterion generalizes a known result which was derived for single-input single-output（SISO） systems; furthermore, for a given rational approximant matrix, it is expressed as a set of inequalities that can be solved in practice. Finally, a controller for a binary distillation column is designed by using the criterion in conjunction with the method of inequalities. The numerical results clearly demonstrate that the usefulness of the criterion in obtaining a design solution for the original system.