Alexander N. Daryin

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.

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.

Estimation of reachability sets for large-scale uncertain systems: From theory to computation

The estimation of reachability sets for systems of high dimensions is a challenging issue due to its high computational complexity. For linear systems, an efficient way of calculating such estimates is to find their set-valued approximations provided by ellipsoidal calculus. The present paper deals with various aspects of such approach as applied to systems of high dimensions with unknown but bounded input disturbances. We present an innovative technique based on parallel computation that involves on-line mixing of ellipsoidal tubes found in parallel. This improves robustness of the ellipsoidal estimates. Finally discussed is an implementation of the algorithm intended for supercomputer clusters.

On the Theory of Fast Controls under Disturbances

Abstract Fast controls are those that act within a very small time horizon. They are treated as bounded approximations of generalized impulse controls (belonging to the class of higher-order distributions). In this paper, we investigate the fast controls in feedback form under unknown-but-bounded disturbances.

Nonlinear Feedback Types in Impulse and Fast Control

Abstract It is well-known that a system with linear structure subjected to bounded control inputs for optimal closed-loop control yields nonlinear feedback of discontinuous bang-bang type. This paper investigates new types of nonlinear feedback in the case of optimal impulsive closed-loop control which may naturally generate discontinuous trajectories. The realization of such feedback under impulsive inputs that are allowed to use δ-functions with their higher derivatives requires physically realizable approximations. Described in this paper is a new class of realizable feedback inputs that also allows to produce smooth approximation of controls. Such approach also applies to problems in micro time scales that require so-called fast or ultra-fast controls.

Impulse Control Inputs and the Theory of Fast Feedback Control

Abstract This paper deals with problems of impulse control which allow control inputs consisting not only of delta functions but also of their higher derivatives (impulses of higher order). The controls are sought for in the form of feedback strategies which leads to the application of respective generalized dynamic programming techniques, where the role of traditional Hamilton–Jacobi–Bellman equations is taken by respective variational inequalities of similar structure. Further proposed are physically realizable approximations which converge to these ideal solutions. Since the ideal solutions allow to transfer a controllable system from one given position to another in zero time, their approximations lead us to physically realizable “fast” controls with piecewise constant realizations. Such feedback control inputs are then compared with traditional bang-bang type strategies and turn out to be more robust. Computational schemes for related problems of reachability and control synthesis are further described with examples of damping oscillating systems of high order in minimal time being demonstrated.

Numerical algorithm for solving the problem of synthesis of impulse controls under uncertainty

Feedback control problem for a linear system under an unknown but bounded disturbance is considered. A numerical algorithm for the synthesis of controls in the problem of impulse control based on the representation of the value function using methods of convex analysis in terms of the conjugate function and the subsequent approximation of the conjugate function by piecewise affine convex functions is described.

On the problem of impulse measurement feedback control

A problem of impulse measurement feedback control is considered with noisy observations. The solution scheme is based on dynamic programming techniques in the form of analogs of Hamiltonian formalism equations, and the solution is a sequence of delta functions. The sets of state vectors compatible with a priori data and current measurements are considered as the information state of the system. Observation models are considered either as continuous with “uncertain“ disturbances, for which there is no statistical description, or as stochastic and discrete ones coming from a communication channel in the form of a Poisson flow with disturbances that are distributed uniformly over a given set. All the results are obtained by means of operations in a finite-dimensional space. Computation schemes are discussed. Examples of numerical modeling are presented.

Closed-loop impulse control of oscillating systems

Abstract This paper deals with the problem of damping the oscillations of a finite cascade of springs through impulsive controls in finite time. The control problem is treated as one of specifying a closed-loop (feedback) strategy and is solved by applying Hamiltonian technique in its Dynamic Programming version. A numerical algorithm is indicated and the limit case with control time tending to infinity is described.

Attenuation of Uncertain Disturbances Through Fast Control Inputs

This paper introduces a new class of controls that ensure an effect similar to that produced by conventional matching conditions between control and disturbance inputs in a linear system, but now for a broader class of such inputs. Namely, this is due to an application of piecewise-constant control functions with varying amplitudes, generated by approximations of “ideal controls,” which are linear combinations of delta functions and their higher order derivatives. Such a class allows to calculate feedback control solutions by solving problems of open-loop control, thus reducing the overall computation burden.