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.

Now by combining the finite element analysis and interval mathematics, a new, non-probabilistic, set-theoretical models, that is interval analysis method is being developed in scientific and engineering communities to predict the variability or uncertainty resulting from the unavoidable scatter in structural parameters and the external excitations as an alternative to the classical probabilistic approaches. Interval analysis methods of uncertainty were developed for modeling uncertain parameters of structures, in which bounds on the magnitude of uncertain parameters are only required, not necessarily knowing the probabilistic distribution densities. Instead of conventional optimization studies, where the minimum possible response is sought, here an uncertainty modeling is developed as an anti-optimization problem of finding the least favorable response and the most favorable response under the constraints within the settheoretical description. In this study, we presented the non-probabilistic interval analysis method for the dynamical response of structures with uncertain-but-bounded parameters. Under the condition of the interval vector, which contains the uncertain-but-bounded parameters, determined from probabilistic statistical information or stochastic sample test, through comparing between the non-probabilistic interval analysis method and the probabilistic approach in the mathematical proof and the numerical examples, we can see that the region of the dynamical response of structures with uncertain-but-bounded parameters obtained by the interval analysis method contains that produced by the probabilistic approach. In other words, the width of the maximum or upper and minimum or lower bounds on the dynamical responses yielded by the probabilistic approach is tighter than those produced by the interval analysis method. This kind of results is coincident with the meaning of the probabilistic theory and interval mathematics.

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.

In this paper, by combining the finite element analysis and non-probabilistic convex models, we present the numerical algorithm of non-probabilistic convex models and interval analysis method for the static displacement of structures with uncertain-but-bounded parameters. Under the condition of the box or interval vector determined from the ellipsoid of the uncertain-but-bounded structural parameter vector, by comparing the numerical algorithm of non-probabilistic convex models and the interval analysis method in the mathematical proof and the numerical example, we can see that the width of the maximum or upper and minimum or lower bounds on the static displacement yielded by the numerical algorithm of non-probabilistic convex models is tighter than those produced by the interval analysis method. Copyright.