Componentwise Determination of the Interval Hull Solution for Linear Interval Parameter Systems

In this paper, the problem of determining the interval hull (IH) solution x∗ to a linear interval parameter system A(p)x = b(p), p ∈ p is revisited. A new iterative method for computing x∗ is suggested, which is based on individually finding each interval component xk of x ∗. Each component xk = [x ∗ k, x ∗ k] is in turn found by separately determining the lower endpoint xk and upper end-point x ∗ k of x ∗ k, respectively. The lower end-point xk is located by an iterative method which, at each iteration, makes use of a respective outer solution x and an upper bound xk on x ∗ k. The upper end-point xk is located in a similar manner using relevant outer solutions x∗ and lower bounds x k on x ∗ k. In both cases, appropriate modified monotonicity conditions are checked and used. Such an approach results in better performance compared to similar methods employing standard monotonicity conditions. The method is capable of determining the solution x∗ if the modified monotonicity conditions are satisfied for all components of p; otherwise, it only provides a two-sided enclosure [xk, x u k ] ([xk, xk]) of x ∗ k(x ∗ k). The method is extended to a more general setting where the problem is to determine the IH y∗ of an output variable vector y which depends on x and p, p ∈ p. A numerical example illustrating the new method is also given.