Anti-optimization technique, on the one hand, represents an alternative and complement to traditional probabilistic methods, and on the other hand, it is a generalization of the mathematical theory of interval analysis. In this study, in terms of interval analysis or interval mathematics, the arithmetic operations and the partial order relation of anti-optimization technique can be defined, and the convex model variables and the convex model extension function of convex models can also be introduced. The comparison of the Lagrange multiplier method with the convex model extension method for evaluation the region of static displacements of structures with uncertain-but-bounded parameters shows that the with of the upper the lower bounds on the static displacement of structures with uncertain-but-bounded parameters shows that the width of the upper and lower bounds on the static displacement yielded by the Lagrange multiplier method ofconvex models is tighter than those produced by the convex model externsion.

In this paper, we present a method for computing upper and lower bounds of natural frequencies of the structures with uncertain parameters. There parameters are unknown except for the fact that they belong to given bounded sets. The sel of possible system states can be described by interval matrices. By solving the interval matrix problem, we obtain the bounds on frequencies of the structure. The numerical results demonstrates the efficacy of the method.

This paper is concerned with the problem of robust stability of linear dynamic systems with structured uncertainty by means of ellipsoidal set-theoretic approach. In this paper, the uncertainty in the physical parameters is expressed in terms of an ellipsoidal set in appropriate vector space. Two ellipsoidal set-theoretic approaches are presented for giving sufficient conditions for robust stability property of the systems with structured uncertainty. The bound produced by the ellipsoidal extension function theorem is shown to be less conservative than the one predicted by the Lagrange multiplier method. In order to introduce the ellipsoidal extension function theorem, in Appendix A of this paper, we try to present the theory of ellipsoidal algebra, following the thought of interval analysis. First of all, we give the concept of ellipsoidal numbers and define their arithmetic operations. Based on them, we finally introduce ellipsoidal vectors and ellipsoidal functions. In terms of the inclusion monotonic property of ellipsoidal functions, we present and prove the ellipsoidal extension function theorem.

In this study, a new, deterministic method is discussed for estimating the maximum, or least favorable frequency, and the minimum, or best favorable frequency, of structures with uncertain but non-random parameters. The favorable bound estimate is actually a set in eigenvalue space rather than a single vector. The obtained optimum estimate is the smallest calculable set which contains the uncertain system eigenvaluse. This kind of engenvalue problem involves uncertain but non-random interval stiffness and mass matrices. If one views the deviation amplitude of the interval matrix as a perturbation around the nominal value of the interval matrix, one can solve the generalized eigenvalue problem of the uncertain but non-random interval matrices. By applying the interval extension matrix perturbation formulation, the interval perturbation approximating formula is presented for evaluating interval eigenvalues of uncertain but non-random interval stiffness and mass matrices. A perturbation method is developed which allows one to calculate eigenvalues of an uncertain but non-random interval matrix pair that always contains the system's true stiffness and mass matrices. Inextensive computational effort is a characteristic of the presented method. A numerical example illustrates the application of the proposed method. Copyright.

By a counter example, we show that there seem to be some problems in Ben-Haim's theory of robust reliability of dynamical systems based on convex models. We still point out that the property of the expansion of convex models is just the addition of a convex model and a real vector, and the property of the translation of convex models is just the scalar multiplication convex models. By means of the partial-order relation of the superscribed hyperrectangle or interval vectors of convex models, we present a correct criterion of reliability of the dynamical system with bounded uncertainty. Based on them, we propose the expansion function which is different from the one of Ben-Haim. Following Ben-Haim's thoughts, based on the new expansion function, we again define the input, failure, and overall reliability indices. By Ben-Haim's example, we obtain some results different from his. The conclusion and results may be thought of as to the further development of Ben-Haim's robust reliability.