Non-singular locally optimal ellipsoidal approximation of the estimate of the states of linear systems

A problem which arises when estimating the attainability domains of linear dynamical systems by ellipsoids is investigated in a short time interval in the case when the initial position of the system in phase space is known precisely for some at least coordinates. A method is proposed which allows one to avoid problems associated with the degeneracy of the right-hand sides of the differential equations of the locally optimal ellipsoidal approximation. The mathematical meaning of these equations is made more precise in the case of the minimization of the phase volume. An example is given.     

Properties of the Optimal Ellipsoids Approximating the Reachable Sets of Uncertain Systems

The ellipsoidal estimation of reachable sets is an efficient technique for the set-membership modelling of uncertain dynamical systems. In the paper, the optimal outer-ellipsoidal approximation of reachable sets is considered, and attention is paid to the criterion associated with the projection of the approximating ellipsoid onto a given direction. The nonlinear differential equations governing the evolution of ellipsoids are analyzed and simplified. The asymptotic behavior of the ellipsoids near the initial point and at infinity is studied. It is shown that the optimal ellipsoids under consideration touch the corresponding reachable sets at all time instants. A control problem for a system subjected to uncertain perturbations is investigated in the framework of the optimal ellipsoidal estimation of reachable sets.     

The optimal choice of control constraints

The problem of the optimal choice of the limits of a set of possible values of the control during motion for the purpose of obtaining the required form of the attainability set of a linear dynamical system in a specified time interval is considered. Using the method, in which these sets are approximated by ellipsoids, the problem of controlling the parameters of the ellipsoid containing the control vector is solved. Then a functional, which depends on the matrix of the ellipsoid, containing the phase vector, reaches its maximum. The order in which the corresponding formulae are used is illustrated using the example of a simple mechanical system. The results obtained are suitable for systems in which, instead of the control vector, there is an interference vector with controllable boundaries of possible changes and can be extended to stochastic systems.     

Computation of alternated reachability tubes for linear control systems under uncertainty

The problem of constructing the reachability domain for linear control system in a presence of geometrically bounded unknown disturbance is considered. A high actuality of the problem for engineering applications requires an efficient calculation technique for the reach sets in a class of closed-loop control. The technique suggested in the article is based on the ellipsoidal approximations developed by A.B. Kurzhansky for the alternated reachability domains. In the article these estimates are complemented with an adaptive regularization to guarantee the continuability of the ellipsoidal estimates. The quadratic structure of the regularization combines well with an ellipsoidal nature of the estimates thus making it possible to adjust the existing ellipsoidal estimation schema in a transparent fashion for achieving the continuable and non-singular estimates via adaptive choice of regularization parameters.     

A dynamic programming method for controlled linear systems with delays

A linear system with permanent delay is considered. A method of dynamic programming for constructing attainability sets and solving the problem of target control for the systems is used. The expressions for value functionals described by solutions to the corresponding Hamilton-Jacobi-Bellman equation are obtained. It is proved that these value functionals calculated by means of convex analysis satisfy the above equations. Strategies for synthesized control for the problem of hitting on the target set are given.     

Ellipsoidal Bounds on Reachable Sets of Dynamical Systems with Matrices Subjected to Uncertain Perturbations1

Linear dynamical systems described by finite-difference or ordinary differential equations are considered. The matrix of the system is uncertain or subject to disturbances, and only the bounds on admissible perturbations of the matrix are known. Outer ellipsoidal estimates of reachable sets of the system are obtained and equations describing the evolution of the approximating ellipsoids are derived. An example is presented.     

Ellipsoidal approximation of attainability sets of a linear system with indeterminate matrix

Linear dynamical systems described by finite-difference or differential equations are considered. It is assumed that the matrix of the system is either completely known or is subject to uncontrollable perturbations, so that each element is known only to within a certain possible interval. Outer approximations, by means of ellipsoids, are constructed for the attainability sets of such systems. The equations of evolution of the approximating ellipsoids are obtained. An example is presented.     

Reachable Set Bounding for Linear Discrete-Time Systems with Delays and Bounded Disturbances

This paper addresses the problem of reachable set bounding for linear discrete-time systems that are subject to state delay and bounded disturbances. Based on the Lyapunov method, a sufficient condition for the existence of ellipsoid-based bounds of reachable sets of a linear uncertain discrete system is derived in terms of matrix inequalities. Here, a new idea is to minimize the projection distances of the ellipsoids on each axis with different exponential convergence rates, instead of minimization of their radius with a single exponential rate. A smaller bound can thus be obtained from the intersection of these ellipsoids. A numerical example is given to illustrate the effectiveness of the proposed approach.     

Parallel algorithm for calculating the invariant sets of high-dimensional linear systems under uncertainty

The development of efficient computational methods for synthesizing controls of high-dimensional linear systems is an important problem in theoretical mathematics and its applications. This is especially true for systems with geometrical constraints imposed on the controls and uncertain disturbances. It is well known that the synthesis of target controls under the indicated conditions is based on the construction of weakly invariant sets (reverse reachable sets) generated by the solving equations of the process under study. Methods for constructing such equations and corresponding invariant sets are described, and the computational features for high-dimensional systems are discussed. The approaches proposed are based on the previously developed theory and methods of ellipsoidal approximations of multivalued functions.     

External Ellipsoidal Estimation of the Union of Two Concentric Ellipsoids and Its Applications

For an arbitrary set representable as the convex hull formed by the union of two concentric ellipsoids we propose a method to construct a family of external undominated ellipsoidal approximations and represent the estimated set as the intersection of all estimates from a given family. A sufficient condition of undominated guaranteed ellipsoidal approximation of a convex compactum is derived. A method is described that for certain classes of sets (such as the intersection of an ellipsoid or a cone with two halfspaces) constructs a family of internal undominated ellipsoidal approximations using the previous formulas for the external estimates of the union of concentric ellipsoids.     

Estimation of statistical characteristics of attainability sets of controllable systems

We study an expansion of the notion of invariance for sets with respect to controllable systems and differential inclusions. Namely, we study statistically invariant sets and statistical characteristics of attainability sets of controllable systems. We obtain a lower bound for the lower relative frequency of the absorption of the attainability set of a system by a given set and establish new sufficient conditions of the statistical invariance of the set with respect to the controllable system. We give examples of the calculation of statistical characteristics for the linear Cauchy problem and a linear controllable system with almost periodic coefficients.     

On the problem of output feedback control synthesis under uncertainty and finite-time observers

We consider the problem of controlling a linear system of ordinary differential equations with a linear observable output. The system contains uncertain items (disturbances), for which we know only “hard” pointwise constraints. The problem of synthesizing a control that brings the trajectories of the system into a given target set in finite time is solved under weakened conditions without assuming that the control and the disturbance are of the same type. To this end, we suggest an approach that amounts to constructing an information set and a weakly invariant set with subsequent “aiming” of the first set at the second. Both stages are carried out in a finite-dimensional space, which permits one to use an efficient algorithm for solving the synthesis problem approximately on the basis of the ellipsoidal calculus technique. The results are illustrated by an example in which the control of a linear oscillation system is constructed.     

Guidance problem for a distributed system with incomplete information on the state coordinates and an unknown initial state

We study the problem of guaranteed positional guidance of a linear partially observable control system with distributed parameters to a convex target set at a given time. The problem is considered under incomplete information. More precisely, we assume that the system is subjected to an unknown disturbance; in addition, the initial state is assumed to be unknown as well. Further, the sets of admissible disturbances and the set of admissible initial states, which is assumed to be finite, are known. An algorithm for solving the problem is suggested.     

Differential guidance game with incomplete information on the state coordinates and unknown initial state

We study the problem of guaranteed positional guidance of a linear partially observable control system to a convex target set at a given time. The problem is considered in the case of incomplete information. More precisely, it is assumed that the system is subjected to some unknown disturbance; in addition, the initial state is unknown as well. But the sets of admissible disturbances and the set of admissible initial states are known. The latter is assumed to be finite. We construct an algorithm for solving this problem.     

Enclosing ellipsoids and elliptic cylinders of semialgebraic sets and their application to error bounds in polynomial optimization

This paper is concerned with a class of ellipsoidal sets (ellipsoids and elliptic cylinders) in ${\mathbb{R}^m}$ which are determined by a freely chosen m × m positive semidefinite matrix. All ellipsoidal sets in this class are similar to each other through a parallel transformation and a scaling around their centers by a constant factor. Based on the basic idea of lifting, we first present a conceptual min-max problem to determine an ellipsoidal set with the smallest size in this class which encloses a given subset of ${\mathbb{R}^m}$ . Then we derive a numerically tractable enclosing ellipsoidal set of a given semialgebraic subset of ${\mathbb{R}^m}$ as a convex relaxation of the min-max problem in the lifting space. A main feature of the proposed method is that it is designed to incorporate into existing SDP relaxations with exploiting sparsity for various optimization problems to compute error bounds of their optimal solutions. We discuss how we adapt the method to a standard SDP relaxation for quadratic optimization problems and a sparse variant of Lasserre’s hierarchy SDP relaxation for polynomial optimization problems. Some numerical results on the sensor network localization problem and polynomial optimization problems are also presented.     

On dominance core and stable sets for cooperative ellipsoidal games

Abstract

The allocation problem of rewards or costs is a central question for individuals and organizations contemplating cooperation under uncertainty. The involvement of uncertainty in cooperative games is motivated by the real world where noise in observation and experimental design, incomplete information and further vagueness in preference structures and decision-making play an important role. The theory of cooperative ellipsoidal games provides a new game theoretical angle and suitable tools for answering this question. In this paper, some solution concepts using ellipsoids, namely the ellipsoidal imputation set, the ellipsoidal dominance core and the ellipsoidal stable sets for cooperative ellipsoidal games, are introduced and studied. The main results contained in the paper are the relations between the ellipsoidal core, the ellipsoidal dominance core and the ellipsoidal stable sets of such a game.     

Approximate construction of attainability sets of control systems with integral constraints on the controls

The approximate construction of attainability sets of control systems with quadratic integral constraints on the controls is considered. It is assumed that a control system is non-linear with respect to the phase variable and linear with respect to the variable which describes the controlling action. The approximation of the attainability sets of a control system is accomplished in several stages. The latter class of controls generates a finite number of trajectories of the system. The trajectories of the system are then replaced by Euler broken lines. An estimate of the accuracy of the Hausdorff distance between the attainability set and the set which has been approximately constructed is obtained.     

Method for Constructing Piecewise Quadratic Value Functions in a Control Problem for a Switched System

The problem of constructing internal approximations to solvability sets and the control synthesis problem for a piecewise linear system with control parameters and disturbances (uncertainties) are solved. The solution is based on the comparison principle and piecewise quadratic value functions of a special form. Relations defining such functions and, in particular, “continuous binding conditions” for the functions and their first derivatives are obtained. The results are used to construct numerical methods for solving the control synthesis problem for the class of switched systems under study. An example of approximate solution of the control synthesis problem in a target control problem for a nonlinear mathematical model of a pendulum with a flywheel is considered.     

Stabilization of Multiple-Input Switched Linear Systems with Operation Modes of Different Dynamical Orders

The problem of stabilization of multiple-input switched linear systems operating under the conditions of bounded coordinate disturbances is considered. It is assumed that the operation modes can have different dynamical orders. To solve this problem, an algorithm for constructing a variable-structure controller is proposed based on the dynamical order extension method.