This paper deals with a problem from discrete-time robust control which requires the solution of constraints over the reals that contain both universal and existential quantifiers. For solving this problem we formulate it as a program in a (fictitious) constraint logic programming language with explicit quantifier notation. This allows us to clarify the special structure of the problem, and to extend an algorithm for computing approximate solution sets of first-order constraints over the reals to exploit this structure. As a result we can deal with inputs that are clearly out of reach for current symbolic solvers.
Apart from the method used in this paper @cite_8 , there there have been several successful attempts at solving special cases of first-order constraints, for example using classical interval techniques @cite_36 @cite_21 or constraint satisfaction @cite_0 , and very often in the context of robust control @cite_2 @cite_30 @cite_28 @cite_11 .
@cite_30 @cite_8 @cite_36 @cite_28 @cite_21 @cite_0 @cite_2 @cite_11
1988549558 1504204411 2403711553 1972645441 1542502870 2611367645 1596669356 68519879
Many design problems, e.g. in control theory, amount to tuning a parameter vector c so as to guarantee that specifications are met for all feasible values of some unknown perturbation vector p. A new prototype algorithm for solving this guaranteed-tuning problem is proposed, and its convergence properties are established. It applies when the design specifications translate into a finite number of (possibly nonlinear) inequalities. Three test cases taken from the field of control are considered, namely the design of a PID controller robust to structured uncertainty, the control of a nonlinear discrete-time model with uncertain parameters and initial state, and a problem of motion planning, with obstacles to be avoided. This paper applies interval methods to a classical problem in computer algebra. Let a quantified constraint be a first-order formula over the real numbers. As shown by A. Tarski in the 1930's, such constraints, when restricted to the predicate symbols <, = and function symbols +, ×, are in general solvable. However, the problem becomes undecidable, when we add function symbols like sin. Furthermore, all exact algorithms known up to now are too slow for big examples, do not provide partial information before computing the total result, cannot satisfactorily deal with interval constants in the input, and often generate huge output. As a remedy we propose an approximation method based on interval arithmetic. It uses a generalization of the notion of cylindrical decomposition—as introduced by G. Collins. We describe an implementation of the method and demonstrate that, for quantified constraints without equalities, it can efficiently give approximate information on problems that are too hard for current exact methods. In this paper, theidentification problem, thetolerance problem, and thecontrol problem are treated for the interval linear equation Ax=b. These problems require computing an inner approximation of theunited solution set Σ∃∃(A, b)= x ∈ ℝ n | (∃A ∈ A)(Ax ∈ b) , of thetolerable solution set Σ∀∃(A, b)= x ∈ ℝ n | (∀A ∈ A)(Ax ∈ b) , and of thecontrollable solution set Σ∃∀(A, b)= x ∈ ℝ n | (∀b ∈ b)(Ax ∈b) respectively. Analgebraic approach to their solution is developed in which the initial problem is replaced by that of finding analgebraic solution of some auxiliary interval linear system in Kaucher extended interval arithmetic. The algebraic approach is proved almost always to give inclusion-maximal inner interval estimates of the solutionsets considered. We investigate basic properties of the algebraic solutions to the interval linear systems and propose a number of numerical methods to compute them. In particular, we present the simple and fastsubdifferential Newton method, prove its convergence and discuss numerical experiments. Abstract Several robustness problems such as stability and performance robustness analysis of feedback systems and robust design of control systems in the presence of mixed nonlinear parametric and nonparametric perturbations can be solved by means of algorithms based on interval-arithmetic computation. Some of the main algorithms available in the literature are presented, and their efficiency is tested on some examples of robustness analysis and design of control systems. © 1997 Elsevier Science Ltd. The work advances a numerical technique for computing enclosures of generalized AE-solution sets to interval linear systems of equations. We develop an approach (called algebraic) in which the outer estimation problem reduces to a problem of computing algebraic solutions of an auxiliary interval equation in Kaucher complete interval arithmetic. Non-linear real constraint systems with universally and or existentially quantified variables often need be solved in such contexts as s control design or sensor planning. To date, these systems are mostly handled by computing a quantifier-free equivalent form by means of Cylindrical Algebraic Decomposition (CAD). However, CAD restricts its input to be conjunctions and disjunctions of polynomial constraints with rational coefficients, while some applications such as camera control involve systems with arbitrary forms where time is the only universally quantified variable. In this paper, the handling of universally quantified variables is first related to the computation of inner-approximation of real relations. Algorithms for solving non-linear real constraint systems with universally quantified variables are then presented along with the theoretical framework on inner-approximation of relations supporting them. These algorithms are based on the computation of outer-approximations of the solution set of the negation of involved constraints. An application to the devising of a declarative modeller for expressing camera motion using a cinematic language is sketched, and results from a prototype are presented. The following sections are included: introduction; quantifier elimination; Bernstein expansion; approximation of the solution set; algorithm; examples; conclusions; acknowledgment; and references. This paper aims to start exploring the application of interval techniques to deal with robustness issues in the context of predictive control. The robust stability problem is transformed into that of checking the positivity of a rational function. Modal intervals are presented as a useful tool to deal with this kind of function.