Alexander B. Kurzhanski

Ellipsoidal Techniques for Reachability Analysis

This report describes the calculation of the reach sets and tubes for linear control systems with time-varying coefficients and hard bounds on the controls through tight external and internal ellipsoidal approximations. These approximating tubes touch the reach tubes from outside and inside respectively at every point of their boundary so that the surface of the reach tube is totally covered by curves that belong to the approximating tubes. The proposed approximation scheme induces a very small computational burden compared with other methods of reach set calculation. In particular such approximations may be expressed through ordinary differential equations with coefficients given in explicit analytical form. This yields exact parametric representation of reach tubes through families of external and internal ellipsoidal tubes. The proposed techniques, combined with calculation of external and internal approximations for intersections of ellipsoids, provide an approach to reachability problems for hybrid systems.

Dynamic optimization for reachability problems

This paper uses dynamic programming techniques to describe reach sets andrelated problems of forward and backward reachability. The original problemsdo not involve optimization criteria and are reformulated in terms ofoptimization problems solved through the Hamilton–Jacobi–Bellmanequations. The reach sets are the level sets of the value function solutionsto these equations. Explicit solutions for linear systems with hard boundsare obtained. Approximate solutions are introduced and illustrated forlinear systems and for a nonlinear system similar to that of theLotka–Volterra type.

On Ellipsoidal Techniques for Reachability Analysis. Part I: External Approximations

The paper describes the calculation of the reach sets and tubes for linear control systems with time-varying coefficients and ellipsoidal hard bounds on the controls and initial states. This is achieved by parametrized families of external and internal ellipsoidal approximations constructed such that they touch the reach sets at every point of their boundary at any instant of time (both from outside and inside, respectively). The surface of the reach tube would then be entirely covered by curves that belong to the approximating tubes. It is further shown how such approximations may be expressed through ordinary differential equations with coefficients given in explicit analytical form. This allows exact parametric representation of reach tubes through families of external and internal ellipsoidal tubes as compared with earlier methods based on constructing one or several isolated approximating tubes. The approach opens new routes to the arrangement of efficient numerical algorithms. The present Part I deals with external approximations.

Ellipsoidal techniques for reachability analysis: internal approximation

For the reach tube of a linear time-varying system with ellipsoidal bounds on the control variable consider the following approximation problem. Find a tight ellipsoid-valued tube inside the reach tube that touches it along any prespecified smooth curve on the boundary. We construct this ellipsoidal tube, and show that if the given curve is itself a system trajectory, the tube can be calculated recursively, with minimum computational burden.

On Reachability Under Uncertainty

The paper studies the problem of reachability for linear systems in the presence of uncertain (unknown but bounded) input disturbances that may also be interpreted as the action of an adversary in a game-theoretic setting. It defines possible notions of reachability under uncertainty emphasizing the differences between reachability under open-loop and closed-loop control. Solution schemes for calculating reachability sets are then indicated. The situation when observations arrive at given isolated instances of time leads to problems of anticipative (maxmin) or nonanticipative (minmax) piecewise open-loop control with corrections and to the respective notions of reachability. As the number of corrections tends to infinity, one comes in both cases to reachability under nonanticipative feedback control. It is shown that the closed-loop reach sets under uncertainty may be found through a solution of the forward Hamilton--Jacobi--Bellman--Isaacs (HJBI) equation. The basic relations are derived through the investigation of superpositions of value functions for appropriate sequential maxmin or minmax problems of control.

Ellipsoidal Techniques for Reachability Under State Constraints

The paper presents a scheme to calculate approximations of reach sets and tubes for linear control systems with time-varying coefficients, bounds on the controls, and constraints on the state. The scheme provides tight external approximations by ellipsoid-valued tubes. The tubes touch the reach tubes from the outside at each point of their boundary so that the surface of the reach tube is totally covered by curves that belong to the approximating tubes. The result is an exact parametric representation of reach tubes through families of external ellipsoidal tubes. The parameters that characterize the approximating ellipsoids are solutions of ordinary differential equations with coefficients given partly in explicit analytical form and partly through the solution of a recursive optimization problem. The scheme combines the calculation of external approximations of infinite sums and intersections of ellipsoids, and suggests an approach to calculate reach sets of hybrid systems.

On Ellipsoidal Techniques for Reachability Analysis. Part II: Internal Approximations Box-valued Constraints

Following Part I, this article continues to describe the calculation of the reach sets and tubes for linear control systems with time-varying coefficients and ellipsoidal hard bounds on the controls and initial states. It deals with parametrized families of internal ellipsoidal approximations constructed such that they touch the reach sets at every point of their boundary at any instant of time. The reach tubes are thus touched internally by ellipsoidal tubes along some curves. The ellipsoidal tubes are chosen here in such a way that the touching curves do not intersect and that the boundary of the reach tube would be entirely covered by such curves. This allows exact parametric representation of reach tubes through unions of tight internal ellipsoidal tubes as compared with earlier methods based on constructing one or several isolated approximating tubes. The method of external and internal ellipsoidal approximations is then propagated to systems with box-valued hard bounds on the controls and initial states. It appears that the proposed technique may well work for nonellipsoidal, box-valued constraints. This broadens the range of applications of the approach and opens new routes to the arrangement of efficient numerical algorithms.

Dynamic programming for impulse controls

Abstract This paper describes the theory of feedback control in the class of inputs which allow delta-functions and their derivatives. It indicates a modification of dynamic programming techniques appropriate for such problems. Introduced are physically realizable bang-bang-type approximations of the “ideal” impulse-type solutions. These may also serve as “fast” feedback controls which solve the terminal control problem in arbitrary small time. Examples of damping high-order oscillations in finite time are presented.

Reachability under uncertainty

The paper studies the problem of reachability for linear systems in the presence of uncertain (unknown but bounded) input disturbances, which may also be interpreted as the action of an adversary in a game-theoretic setting. It defines possible notions of reachability under uncertainty emphasizing the differences between open-loop and closed-loop control. Solution schemes for calculating reachability sets are indicated. The situation when observations arrive at given isolated instances of time leads to problems of anticipative (maxmin) and nonanticipative (minmax) piecewise open-loop control with corrections and to corresponding notions of reachability. As the number of corrections tends to infinity, one comes in both cases to reachability under nonanticipative feedback control. It is shown that the closed-loop reach sets under uncertainty may be found through a solution of the forward Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation. The basic relations are derived through the investigation of superpositions of value functions for appropriate sequential maxmin or minmax problems of control.

Optimization of Output Feedback Control Under Set-Membership Uncertainty

This paper presents a description of solution approaches to the problem of output feedback control under unknown but bounded disturbances with hard bounds on the controls and the uncertain items. The problem is treated within a finite horizon which requires to track the system dynamics throughout the whole time interval rather than through asymptotic properties. The demand for such solutions is motivated by increasing number of applications. The suggested approaches are designed as a combination of Hamiltonian techniques in the form of generalized dynamic programming with those of set-valued analysis and problems on minimax. The paper indicates the crucial role of properly selecting the on-line generalized state of the system in the form of information states or information sets. The description ranges from theoretical schemes to computational routes with emphasis on the possibility of treating the overall problem through only finite-dimensional methods. The results apply to nonlinear systems, with more details for linear models. It turns out that in the last case, while moving through calculations, one may avoid the fairly difficult stage of integrating HJB equations. The procedures are here confined to ordinary differential equations and ellipsoidal or polyhedral techniques. The suggested schemes are also quite appropriate for parallel computation.