The Convex Model Theory Of Vibration Control Systems With Uncertain Parameters

The vibration control theory for the systems with deterministic parameters has been well developed. However, in actual situations, the structural parameters are often uncertain, such as the inaccuracy of the measurement, errors in the manufacturing process, invalidity of some components, etc. Therefore, the concept of uncertainty plays an important role in the control problem of the vibration structures. Many studies have been done about the problems only from the view point of mathematics, which are difficult applied to solve the actual engineering problems. Generally speaking, during structural analysis and design, these uncertainties need be quantified by some uncertain methods. And also there are some researches made on the relations among these uncertainties. Nowadays, according to the mathematical models with uncertainties, there are some models as follows, probability models, where uncertainties are described as random or probability variables; interval models, which use the interval variables to represent uncertainties and get interval analysis to solve them; fuzzy models, which use the fuzzy statistic to describe uncertainties by fuzzy variables or functions. Fuzzy optimizations can draw conclusions in fuzzy field in design dimensions; convex models, which can, with certain sets (such as ellipsoid), describe those uncertainties by many convex optimizations. The most common methods for solving uncertainty problems are to model the structural parameters as a random vector or fuzzy set. Unfortunately, the probabilistic approaches can not give reliable results unless sufficient experimental data are available to validate the assumptions about the joint probability densities of the random variables or functions involved. Similarly, for the fuzzy model, uncertainties in the fuzzy statistic still exist such as the fuzzy statistical errors or uncertainties in the fuzzy statistics, and the choice of subjection degree functions has the artificial uncertainties. Therefore, it is necessary to develop a new model, non-probability and non-fuzzy model, which needs less information and can give more reliable results, to describe the uncertainty. Recently, the convex model was used to deal with the uncertain problems in robust analysis of control system and structural failure. For example, Ben-Haim, Elishakoff and Lindberg used the convex model to study the dynamic response and failure of structures with pulse loads. Shi and Gao used the convex model to solve the robustness of control system. At present dissertation, the convex model is used to deal with the control problems of systems with uncertain parameters. The uncertainties of the structural parameters are described by an ellipsoid. And by combining the perturbation theory and optimization technique, the upper and lower bounds of the eigenvalues and the responds of the closed-loop system with uncertain parameters are discussed in this paper. There are some details as follows:The method to research the vibration control problems of the systems with uncertain parameters is developed at present dissertation. The equations of the system with uncertain parameters are divided into two parts, the deterministic part and the uncertain part. The deterministic part is used to investigate the vibration control problem, for example, it is can be used to design the feedback control law. then the uncertain part is used to determine the effects of the uncertain parameters on the actual uncertain system.A non-probability unknown-but-bounded convex model is used to describe the uncertain parameters of the actual uncertain system. First the uncertain parameters are all included in a given N-dimension ellipsoid and the uncertainties of the system are described by convex model theory. Then by combining the perturbation theory and optimization technique, the method for estimating the upper and lower bounds of the eigenvalues of the closed-loop system with uncertain parameters is derived at present dissertation. A numerical example is given to illustrate...