Valery Y. Glizer

SOLUTION OF A DELAYED INFORMATION LINEAR PURSUIT-EVASION GAME WITH BOUNDED CONTROLS

A class of linear pursuit-evasion games with first-order acceleration dynamics and bounded controls is considered, where the evader has perfect information and the pursuer has delayed information on the lateral acceleration of the evader. The other state variables are perfectly known to the pursuer. This game can be transformed to a perfect information delayed control game with a single state variable, the centre of the uncertainty domain created by the information delay. The delayed dynamics of the game is transformed to a linear first-order partial differential equation coupled with an integral-differential equation, both without delay. These equations are approximated by a set of K + 1 ordinary differential equations of first order, creating an auxiliary game. The necessary conditions of optimality derived for the auxiliary game lead to the solution of the delayed control game by a limit process as K → + ∞. The solution has the same structure as the other, already solved, perfect information linear pursuit-evasion games with bounded controls and indicates that the value of the delayed information pursuit-evasion game is never zero. Asymptotic expressions of the value of the game for small and large values of the information delay are derived.

Asymptotic Analysis and Solution of a Finite-Horizon H∞ Control Problem for Singularly-Perturbed Linear Systems with Small State Delay

A finite-horizon H∞ state-feedback control problem for singularly-perturbed linear time-dependent systems with a small state delay is considered. Two approaches to the asymptotic analysis and solution of this problem are proposed. In the first approach, an asymptotic solution of the singularly-perturbed system of functional-differential equations of Riccati type, associated with the original H∞ problem by the sufficient conditions of the existence of its solution, is constructed. Based on this asymptotic solution, conditions for the existence of a solution of the original H∞ problem, independent of the small parameter of singular perturbations, are derived. A simplified controller with parameter-independent gain matrices, solving the original H∞ problem for all sufficiently small values of this parameter, is obtained. In the second approach, the original H∞ problem is decomposed into two lower-dimensional parameter-independent H∞ subproblems, the reduced-order (slow) and the boundary-layer (fast) subproblems; controllers solving these subproblems are constructed. Based on these controllers, a composite controller is derived, which solves the original H∞ problem for all sufficiently small values of the singular perturbation parameter. An illustrative example is presented.

Asymptotic Solution of a Boundary-Value Problem for Linear Singularly-Perturbed Functional Differential Equations Arising in Optimal Control Theory

The Hamiltonian boundary-value problem, associated with a singularly-perturbed linear-quadratic optimal control problem with delay in the state variables, is considered. A formal asymptotic solution of this boundary-value problem is constructed by application of the boundary function method. The justification of this asymptotic solution is done. The asymptotic solution of the Hamiltonian boundary-value problem is constructed and justified assuming boundary-layer stabilizability and detectability.

Euclidean space controllability of singularly perturbed linear systems with state delay

Abstract A singularly perturbed linear time-dependent control system with small point and distributed delays in state variables is considered. Connections between the properties of controllability of the reduced-order and boundary-layer systems, associated with the original one, and such a property of the original system itself are established.

Optimal Evasion from a Pursuer with Delayed Information

A class of prescribed duration pursuit–evasion problems with first-order acceleration dynamics and bounded controls is considered. In this class, the pursuer has delayed information on the lateral acceleration of the evader, but knows perfectly the other state variables. Moreover, the pursuer applies a strategy derived from the perfect information pursuit–evasion game solution. Assuming that the evader has perfect information on all the state variables as well as on the delay of the pursuer and its strategy, an optimal evasion problem is formulated. The necessary optimality conditions indicate that the evader optimal control has a bang–bang structure. Based on this result, two particular cases of the pursuer strategy (continuous and piecewise continuous in the state variables) are considered for the solution of the optimal evasion problem. In the case of the continuous pursuer strategy, the switch point of the optimal control can be obtained as a root of the switch function. However, in the case of the piecewise continuous (bang–bang) pursuer strategy, this method fails, because of the discontinuity of the switch function at this very point. In this case, a direct method for obtaining the switch point, based on the structure of the solution, is proposed. Numerical results illustrating the theoretical analysis are presented leading to a comparison of the two cases.

CONTINUOUS FEEDBACK CONTROL STRATEGY WITH MAXIMAL CAPTURE ZONE IN A CLASS OF PURSUIT GAMES

An interception problem of a highly maneuverable target is considered using a linearized kinematical model with first order acceleration dynamics of the interceptor and the target. The problem is interpreted as a differential game of pursuit. An admissible pursuer (interceptor) feedback strategy, continuous with respect to the state variables and having a maximal capture zone, is constructed. This strategy is the saturated version of a linear feedback control, obtained from the solution of an auxiliary linear-quadratic differential game with cheap controls. This strategy is evaluated by Monte-Carlo simulation of the interception with noisy measurements.

On stabilization of nonstandard singularly perturbed systems with small delays in state and control

A nonstandard singularly perturbed linear time-invariant system with a general type of small delay in state and control variables is considered. Two much simpler parameter-free systems (the reduced-order and the boundary-layer ones), associated with the original system, are introduced. Since the original system is nonstandard, the reduced-order one is descriptor (differential-algebraic). A composite feedback control exponentially stabilizing the original system is constructed based on stabilizing controls of the reduced-order and boundary-layer systems. An illustrative example is presented.

Robust Solution Of A Time-Variable Interception Problem: A Cheap Control Approach

A planar interception problem of a maneuverable target is considered using the linearized kinematic model with variable velocities and first-order dynamics of the interceptor and target. The maneuverabilities of the interceptor and target are assumed to be variable. By using an auxiliary zero-sum linear-quadratic differential game with cheap controls of both players, an interception guidance law, linear with respect to the state vector, is derived. An analytical description of the set of initial positions (capture set) is obtained, from which this guidance law provides zero miss distance, subject to given maneuverabilities of the interceptor and target. A numerical example illustrating the analytical results is presented.

Blockwise estimate of the fundamental matrix of linear singularly perturbed differential systems with small delay and its application to uniform asymptotic solution

A singularly perturbed system of linear differential equations with a small delay is considered. Estimates of blocks of the fundamental matrix solution to this system uniformly valid for all sufficiently small values of the parameter of singular perturbations are obtained in the cases of time-independent and time-dependent coefficients of the system. In the first case the system is considered on an infinite time-interval, while in the second case it is considered on a finite one. These estimates are applied to justify a uniform asymptotic solution of an initial-value problem for this system in both cases.

Infinite horizon quadratic control of linear singularly perturbed systems with small state delays: an asymptotic solution of Riccati-type equations

An infinite horizon linear-quadratic optimal control problem for a singularly perturbed system with multiple point-wise and distributed small delays in the state variable is considered. The set of Riccati-type equations, associated with this problem by the control optimality conditions, is studied. Since the system in the control problem is singularly perturbed, the equations of this set are also perturbed by a small parameter of the singular perturbations. The zero-order asymptotic solution to this set of equations is constructed and justified. Based on this asymptotic solution, parameter-free sufficient conditions for the existence and uniqueness of solution to the original optimal control problem are established.