Control of beams of trajectories of automation type stationary systems given discrete inaccurate measurements

Аuthors

Nemychenkov G. I.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: grigorian_05@list.ru

Abstract

The article considers the problem of optimal control for deterministic discrete stationary automation-type systems under parametric uncertainty in the presence of discrete inaccurate measurements.

The discrete automation system (SAT) is described by recurrent equations and serves as a mathematical model of control devices in the form of a memory automation. On a continuous time period of the SAT operation a finite number its state changes occurs. At the switching moments, when the changes of the state occur, the system trajectory has jumps. The system maintains its state between switches. In contrast to the classical models of discrete systems [1, 2], changes in the states, occurring at specified (clock) moments of time, the SAT switching might be arbitrary, not predetermined by moments of time [3, 4]. The quality of one trajectory control of the system is estimated by the functional, which takes account of the switching costs. Selection of the number of switches and clock moments is one of the control resources and is a subject to optimization [5]. This does not exclude multiple switching at a fixed time [3]. Thus, the problem of optimal SAT synthesis generalizes the problem of the discrete controlled system control [1, 2].

The problems in which suboptimal beam control is optimal are of interest. The well-known results relate to the linear-quadratic problems where the linear systems control is evaluated by a quadratic quality functional. It is shown in [6] that the optimal control of the beam of trajectories of a continuous system coincides with the optimal control of one (isolated) trajectory coming from the geometric center of gravity of the set of possible initial states.

Generally speaking, the principle of separation is not fulfilled in the linear-quadratic problem of controlling the beams of the set of trajectories In the examples given in [7], the optimal control does not coincide with the optimal control for the trajectory coming from the geometric center of gravity of the set of possible initial states. The reason for this consists in the fact that the price function in the linear-quadratic control problem of non-stationary CAT is not quadratic. It is shown in [18] that for the linear-quadratic control problem of stationary CAT, the price function is piecewise quadratic. This circumstance allows prove the validity of the principle of separation (with some modification) [19].

Based on the sufficient conditions the algorithms of synthesis of suboptimal and conditional suboptimal control of beams of set trajectories are developed. The algorithm efficiency the is demonstrated by the academic example of a linear-quadratic problem.

Keywords:

automatic type system, sub-optimal control, separation principle, beam of trajectories

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