Interval Analysis And Its Applications To Minimax Optimization And Robust Control

It has been shown that interval analysis is very powerful in efficiently boundingthe range of a function and can provide mathematically rigorous results. Furthermoreintervals allow to express the uncertainty in input data. These capabilities areparticularly welcome to some control problems such as robust control, robustidentification, etc. It has attracted considerable attention of researchers from the fieldof control theory and engineering. Different methods based on interval analysis havebeen developed to deal with control problems. This dissertation is to do someresearch on this topic. The work focuses on minimax optimization and robust controlof linear systems involving parametric uncertainty. The contributions of thisdissertation are as follows.First of all, an overview of interval analysis and its applications to controlproblems is presented. Problems and further development of research on intervalmethods for control are point out.Secondly, two reliable algorithms based on interval constraint solving areproposed for the nonlinear minimax optimization problem. As for the first method, abetter solution is obtained by the valid constraint set solving with interval constraintpropagation. The process is repeated till the global optimal solution is found. As forthe second, bound constraints of functions are solved with consistency techniques andvariables' domains are tested by the monotonicity test with the more accurate gradientrange. The global optimal solution is found by successive bound constraints solvingand monotonicity test. The algorithms converge globally and are able to find all theglobal optimal solutions. Numerical results on benchmark functions show thereliability and the efficiency of the algorithms.Thirdly, an algorithm is presented for robust stability analysis of characteristicpolynomials in the presence of parametric uncertainties. It is based on the inspectionof the value set of the family of polynomials and interval constraint solvingtechniques. The numerical example demonstrates the effectiveness. It is applicable to a very general case where the polynomial coefficients may be any continuousfunction of the uncertain parameters.Fourthly, a new method is presented for the robust design of PID controller forinterval plants. It is based on the generalized Kharitonov theorem and the algorithmof set inversion via interval analysis. According to the generalized Kharitonovtheorem, robust PID controller design problem for interval plants can be solved bystabilizing the corresponding extreme plants. Then the Routh tables are establishedand the problem is formulated as a set of inequalities, which are solved by thealgorithm of set inversion via set inversion. Numerical examples show that theproposed method can character the set of all values of the controller parameters thatguarantees the robust stability with any supposed accuracy and its solution is reliable.Furthermore, it can find if the supposed controller is feasible or not.Fifthly, a new method is presented for the state feedback robust controller designfor the linear system involving parametric dependence of a general class. Firstly,according to the Lyaponov function, the control specifications defined in terms ofclosed-loop stability or stability degree can be described as a set of inequalities withinterval coefficients, which are solved by the feasible controller parameters searchingalgorithm. Then the optimal controller of the nominal system is designed according tothe performance demand. At last, the state feedback controller of the systeminvolving parametric uncertainty is determined by considering both the Lyaponov andthe Riccati function. Simulation results of its application to the pneumatic hydraulicvariable damping system using magnetorheological fluid show that the methodenables the close-loop system hold the robust stability and the optimality of thenominal system.Finally, the work of this dissertation is summarized and the prospective offurther research is discussed.