Ryszard Gessing

Two-level hierarchical control for stochastic optimal resource allocation†

In this paper a two-level hierarchical partially decentralized control for the optimal resource allocation is considered. It is assumed that the coordinator has available the information which has an essential meaning for the whole system, while the local controllers, some different, have more detailed information concerning the particular subsystems. The original problem statement is presented in which some elastic constraint of the control variable is proposed and used for coordination. Owing to this it is possible to decompose the calculations and to realize the decentralized optimal control. In the LQ case considered, all the optimal control laws take analytical form and the control law of the coordinator is certainly equivalent.

Comments on "A modification and the Tustin approximation" with a concluding proposition

In the original paper (K.B. Janiszowski, ibid., vol. 38, no. 8, p. 1313-16, 1993) there is presented a modification of the Tustin approximation. The author argues that the justification of this modification, by means of three examples, is not convincing, because the choice of the examples is questionable and some calculations are not sufficiently accurate. It is shown that some good properties of the proposed modification can be justified by using the Tustin approximation of a continuous-time (CT) model for the discrete-time (DT) system. It is stressed that the proposed modification has only a limited value, since from application point of view the use of CT models is more reasonable. As the conclusion the DT controller design procedure by using CT methods is proposed. >

Word length of pulse transfer function for small sampling periods

Discrete-time systems composed of sampler, zero order hold and continuous-time plants are investigated. The well-known Astrom et al.s (1984) theorem in a reformulated version is first proved and then utilized for estimating the length of a digital word needed for recording the pulse transfer function parameters, in the case of small sampling periods. The estimated length, assuring a prescribed accuracy of the model, is somewhat larger than that obtained from Kaisers (1965) condition of nondestabilization of the model. It is shown that only nonzero poles of the plant transfer function cause an increase of the word length; the zero poles and the zeros of the latter transfer function have no influence on this length. The calculations performed for several examples confirm the correctness of the proposed approach.

Parallel Compensator for Continuous and Relay Control Systems with Difficult Plants

New method of design of the transfer function (TF) of the replacement plant (RP), being a part of the parallel compensator proposed in [3] is presented in the paper. The RPTF with relative degree equal to one is chosen such that the system is stable for high gain of the regulator and the numerator polynomial of the RPTF determines the characteristic equation of the closed loop system. The choice of some fast modes of this polynomial give fast transient response. The control saturations are also accounted. It is shown that the system with parallel compensator and relay implementation of the control, with hysteresis of the relay tending to zero, has the same output response as the continuous system with high gain of the regulator and control saturation values equal to those of the relay switched outputs.

Implementability of Regulation and Partial Decoupling of MIMO Plants

Some necessary conditions of implementability of regulation of MIMO plant without and with integrators are formulated and proved in the paper. The notions of the join gain and degree of independence of the outputs of the plant in steady state are introduced. Partial decoupling of steady states and of fast transients is considered. The latter is based on an original approach consisting in transforming the decoupled matrix transfer function (MTF) of the plant to the form having on its diagonal the elements with smallest relative degree in each row. It is also shown that by the appropriate choice of the control saturations the decoupling of fast transients may be partially retained

Two-level stochastic control with periodic coordination for a resource allocation problem

The resource allocation problem in a large-scale system, in the presence of the resource shortages, is considered. A two-level control structure with the coordinator and the local controllers having different information is proposed. It is assumed that the decision-making frequency in the higher level is less than in the lower level. It is shown that the control laws of the local controllers have an analytical form while that of the coordinator results from numerical calculations. On the basis of simulation research, the influence of the demand variance and the reserve capacity in the storage reservoir on the quality of control is analysed.

WHETHER THE DELTA OPERATOR MODELS ARE REALLY BETTER FOR SMALL SAMPLING PERIODS

Abstract It is noted that the superiority of the delta operator (DO) models over the shift operator (SO) ones, for small sampling periods, concerns only the recording of the model coefficients and some analytical calculations. It is also noted that in the simulations, making it possible to obtain the time response of the output for any input, as well as in the model identification under limited measurement accuracy, the superiority of the DO over SO models, for small sampling periods, disappears. These observations have an essential meaning since the mentioned problems of simulation and identification are important for applications.

Sliding mode control with decreased chattering-model and simulations

The idea of the system with sliding mode control and adaptation of the switched /spl beta//sup +/ and /spl beta//sup -/ amplitudes of the relay output is proposed. This is possible owing to the observation that the sliding mode control is equivalent to the two-position relay control of the sliding surface signal on the level zero. The slopes of the increasing and decreasing parts of the sliding surface signal curve are used for independent adaptation of /spl beta//sup +/ and /spl beta//sup -/ values of the relay output, respectively. The adaptation causes that the difference (/spl beta//sup +/-/spl beta//sup -/) is decreased and both the values tend to be placed symmetrically with respect to the needed mean value of the control. Owing to this the chattering effect appearing in the sliding control is decreased significantly and the system remains very robust. Simplified model of the proposed solution is described and simulations are performed.

Two-level stochastic control for the linear quadratic problem related to a static system

Two-level, hierarchical, partially decentralized control is considered for a large-scale system composed of M linear static subsystems with an interaction and a quadratic performance index. The coordinator and the local controllers, using their different information, realize the control resulting from the global and local optimization, respectively. It is seen that the optimal control laws are linear functions of some estimates and are determined by the formulated two-level certainty equivalence principle. A discussion on the usefulness of the application of particular local controllers is described.

Decentralized, stochastic control for a static LQ problem

An original problem statement for the optimal control laws design is presented. A large scale system composed of M linear static subsystems with an interaction and quadratic performance index is considered. A two-level hierarchical control structure is assumed, in which a coordinator and local controllers have access to different information. The so-called elastic constraint (Gessing 1985) is used for coordination. For the problem the possibility of partial decomposition of calculations and a decentralization of the control, as well as an analytical form of the optimal laws are obtained. The influence of the particular subsystem on the control quality is investigated.