Interval Mathematics Techniques for Control Theory Computations (original) (raw)
Related papers
Interval Identification and Robust Control Design: A New Perspective
Design Methods of Control Systems, 1992
This paper aims initially to present a n ide nt if icati on methcd for constructing an interval model IVhich ca n be aggregated to robust control design. Mathematical programming is used as a tool to achieve this goal. We. then. formulate necessary c onditions for robust stability of con tinu ous a nd d iscrete interval plants and seek low order co ntr oller candida te s fo r both cases. Decision making strategies based on Kharitonov and Mansour-Kraus results are proposed to effectively construct and solve standard linear programming problems instead of in t e rva l ones. The so obtained candidates are cheked for guaran te ed robust stabilizers using (generalized) edge or box theorems.
Interval Constraints: Results and Perspectives
Lecture Notes in Computer Science, 2000
Reliably solving non-linear real constraints on computer is a challenging task due to the approximation induced by the resort to floating-point numbers. Interval constraints have consequently gained some interest from the scientific community since they are at the heart of complete algorithms that permit enclosing all solutions with an arbitrary accuracy. Yet, soundness is beyond reach of present-day interval constraint-based solvers, while it is sometimes a strong requirement. What is more, many applications involve constraint systems with some quantified variables these solvers are unable to handle. Basic facts on interval constraints and local consistency algorithms are first surveyed in this paper; then, symbolic and numerical methods used to compute inner approximations of real relations and to solve constraints with quantified variables are briefly presented, and directions for extending interval constraint techniques to solve these problems are pointed out.
Reliable Computing, 1996
One of the main problems of interval computations is, given a function f(x 1 ; :::; x n ) and n intervals x 1 ; :::; x n , to compute the range y = f(x 1 ; :::; x n ). This problem is feasible for linear functions f, but for generic polynomials, it is known to be computationally intractable. Because of that, traditional interval techniques usually compute the enclosure of y, i.e., an interval that contains y. The closer this enclosure to y, the better. It is desirable to describe cases in which we can compute the optimal enclosure, i.e., the range itself.
A unified framework for interval constraints and interval arithmetic
1998
We are concerned with interval constraints: solving constraints among real unknowns in such a way that soundness is not a ected by rounding errors. The contraction operator for the constraint x + y = z can simply be expressed in terms of interval arithmetic. An attempt to use the analogous de nition for x y = z fails if the usual de nitions of interval arithmetic are used. We propose an alternative to the interval arithmetic de nition of interval division so that the two constraints can be handled in an analogous way. This leads to a uni ed treatment of both interval constraints and interval arithmetic that makes it easy to derive formulas for other constraint contraction operators. We present a theorem that justi es simulating interval arithmetic evaluation of complex expressions by means of constraint propagation. A naive implementation of this simulation is ine cient. We present a theorem that justi es what we call the totality optimization. It makes simulation of expression evaluation by means of constraint propagation as e cient as in interval arithmetic. It also speeds up the contraction operators for primitive constraints.
Stabilility analysis of a nonlinear system using interval analysis
2006
Consider a given dynamical system, described byẋ = f (x) (where f is a nonlinear function) and [x 0 ] a subset of R n. We present an algorithm, based on interval analysis, able to show that there exists a unique equilibrium state x ∞ ∈ [x 0 ] which is asymptotically stable. The effective method also provides a set [x] (subset of [x 0 ]) which is included in the attraction domain of x ∞ .
Computing, 1991
Decomposition of Arithmetic Expressions to Improve the Behavior of Interval Iteration for Nonlinear Systems. Interval iteration can be used, in conjunction with other techniques, for rigorously bounding all solutions to a nonlinear system of equations within a given region, or for verifying approximate solutions. However, because of overestimation which occurs when the interval Jacobian matrix is accumulated and applied, straightforward linearization of the original nonlinear system sometimes leads to nonconvergent iteration. In this paper, we examine interval iterations based on an expanded system obtained from the intermediate quantities in the original system. In this system, there is no overestimation in entries of the interval Jacobi matrix, and nonlinearities can be taken into account to obtain sharp bounds. We present an example in detail, algorithms, and detailed experimental results obtained from applying our algorithms to the example.
Scientific Research and Essays, 2011
The problem of tightening computable overestimation error bounds arising from the solution of linear interval system of equations by some interval methods is considered. This study describes the quality of enclosure of these methods with a view of diminishing the overestimation bounds due to excessive use of interval operations. In particular, two different interval multiplication operations are examined and applied on the Krawczyk's method. It also compare note with results obtained from the well known Hansen-Bliek-Rohn method. It is shown that our results are never worse than the bounds afforded by the Bauer-Skeel and Hansen-Bliek-Rohn bounds. MSC(2000): 65G20, 65G30.
Interval estimations of solution sets to real-valued systems of linear or non-linear equations
Reliable computing, 2002
This is a second paper devoted to present the Modal Interval Analysis as a framework where the search of formal solutions for a set of simultaneous interval linear or non-linear equations is started on, together with the interval estimations for sets of solutions of real-valued systems in which coefficients and right-hand sides belong to certain intervals. The main purpose of this second paper is to show that the modal intervals are a suitable tool to approach problems where logical references appear, for example, to find interval estimates of a special class of generalized sets of solutions of real-valued linear and non-linear systems, the UE-solution sets.
A class of problems that can be solved using interval algorithms
2011
The paper discusses several theoretical and implementational problems of interval branch-and-bound methods. A trial to define a class of problems that can be solved with such methods is done. Features and variants of the method are presented. Useful data structures and shared-memory parallelization issues are considered.