Georgy V. Kostin

The optimization of the motion of an elastic rod by the method of integro-differential relations

The possibility of constructing a solution for given cost functionals and the optimization of the motion of elastic systems with distributed parameters are investigated. The regular integro-differential approach to a broad class of boundary value problems is developed, and a cost functional for the solution obtained is proposed. For the two-dimensional motions of a uniform straight elastic rod, the case of polynomial control is considered. An algorithm for forming the optimal control, which steers the system to the state of minimum total energy at a final time instant, is developed. The parameters of the problem are adjusted so that the period of the lower order mode is comparable with the interval in which the motions are investigated. The results obtained by using the method of separation of variables and the method of integro-differential relations are analyzed and compared.

Integrodifferential relations in linear elasticity

This work treats the elasticity of deformed bodies, including the resulting interior stresses and displacements.It also takes into account that some of constitutive relations can be considered in a weak form. To discuss this problem properly, the method of integrodifferential relations is used, and an advanced numerical technique for stress-strain analysis is presented and evaluated using various discretization techniques. The methods presented in this book are of importance for almost all elasticity problems in materials science and mechanical engineering.

Modeling of Controlled Motions of an Elastic Rod by the Method of Integro-Differential Relations

The possibilities of controlling elastic systems with distributed parameters and constructing a solution corresponding to some given quality criteria are studied. The regular integro-differential approach to the construction of solutions for a wide class of boundary-value problems is developed and a quality criterion for the obtained solution is proposed. The case of the relay control, which induces substantial elastic oscillations in the system and results in significant computational difficulties when applying classic approaches, is considered for plane motions of the homogeneous straight rod. The problem parameters are chosen so that the period of the lower mode of oscillation is comparable with the interval on which the motions are studied. The results obtained by the variable separation method and the integro-differential approach are analyzed and compared.

Approaches to control design and optimization in heat transfer problems

In this paper, boundary control problems are considered for a distributed heating system. The dynamical model of the heating system under consideration is given by a parabolic partial differential equation. In the first stage, the implementation of the Fourier method is discussed for the problem of heat convection and conduction. In the second stage, two alternative solutions to the design of tracking controllers are discussed. On the one hand, an optimal control problem is solved based on the method of integrodifferential relations. On the other hand, this procedure is used to verify the quality of a flatness-based control strategy. The results obtained by the integrodifferential approach are compared with finite-node Fourier approximations. After derivation of suitable, general-purpose solution procedures for the design of open-loop as well as closed-loop boundary control strategies, experimental results are presented. These results highlight the applicability of these procedures in a real-world experiment. For the experimental validation, a test setup at the University of Rostock has been used.

Adaptive Control Strategies in Heat Transfer Problems with Parameter Uncertainties Based on a Projective Approach

Control problems for distributed heating systems described by parabolic partial differential equations are considered in this paper. This type of mathematical model is also a common description for other distributed parameter systems involving diffusion as well as heat and mass transfer. The goal of the paper is to develop an adaptive strategy including online parameter identification for efficient control of heat transfer systems. The developed strategy is based on the method of integrodifferential relations, a projective approach, and a suitable finite element technique. An adaptive control algorithm with predictive estimates of the desired output trajectories is proposed and its specific features are discussed. We use the parameters, geometry, and actuation principles of a real test setup available at the University of Rostock for the numerical simulation and verification. The test setup consists of a metallic rod equipped with a finite number of Peltier elements which are used as distributed control inputs allowing for active cooling and heating. A validation of the control laws derived in this contribution is performed taking into account the explicit local and integral error estimates resulting directly from the method of integrodifferential relations.

An integrodifferential approach to modeling, control, state estimation and optimization for heat transfer systems

Abstract In this paper, control-oriented modeling approaches are presented for distributed parameter systems. These systems, which are in the focus of this contribution, are assumed to be described by suitable partial differential equations. They arise naturally during the modeling of dynamic heat transfer processes. The presented approaches aim at developing finite-dimensional system descriptions for the design of various open-loop, closed-loop, and optimal control strategies as well as state, disturbance, and parameter estimation techniques. Here, the modeling is based on the method of integrodifferential relations, which can be employed to determine accurate, finite-dimensional sets of state equations by using projection techniques. These lead to a finite element representation of the distributed parameter system. Where applicable, these finite element models are combined with finite volume representations to describe storage variables that are—with good accuracy—homogeneous over sufficiently large space domains. The advantage of this combination is keeping the computational complexity as low as possible. Under these prerequisites, real-time applicable control algorithms are derived and validated via simulation and experiment for a laboratory-scale heat transfer system at the Chair of Mechatronics at the University of Rostock. This benchmark system consists of a metallic rod that is equipped with a finite number of Peltier elements which are used either as distributed control inputs, allowing active cooling and heating, or as spatially distributed disturbance inputs.

Optimal Real-Time Control of Flexible Rack Feeders Using the Method of Integrodifferential Relations

Abstract Rack feeders are of high practical importance as automated conveying systems. In this paper, control-oriented models are derived for an experimental setup representing the structure of a typical high bay rack feeder. On the basis of these models, feedforward control strategies are designed. The rack feeder is a viscoelastic structure consisting of two identical flexible beams which are attached to a horizontally movable carriage. The beams are rigidly connected at their tip by a pulley block which is necessary for the vertical positioning of a payload. To develop a real-time applicable control algorithm, a frequency analysis is performed for the original double-beam structure. As a consequence, a simplified Bernoulli beam model is derived with specific boundary conditions. The control objective under consideration is the positioning of the flexible beam structure at a desired position in such a way that the terminal mechanical energy stored in the beam is minimized. A modification of the Galerkin method which is based on an integrodifferential approach and a suitable finite element technique are employed to describe the viscoelastic structural vibrations and to design optimal control strategies. Results of numerical simulations are presented and compared with measured data.

The construction of optimal control for the motion of elastic bodies by using the method of integro-differential relations

The opportunities of modeling and optimization of motion of elastic systems with distributed parameters are investigated. A regular integro-differential approach, which reduces a wide class of linear initialboundary value problems to a conditional minimization of non-negative quadratic functionals is developed, and a cost function of approximate solutions obtained is proposed. For longitudinal motions of a uniform straight elastic rod, the case of polynomial control of the motion of its end is considered. An algorithm of constructing an optimal control that steers the system to the state of a minimal mechanical energy at the final time instant. The parameters of the problem are adjusted so that the time of transition processes would be comparable with the interval, on which the motions are investigated. The analysis and comparison of the results obtained by using the method of integro-differential relations for a one-dimensional model of a thin elastic rod and a proposed approximate three-dimensional model of a prismatic beam.

Numerical validation of order reduction techniques for finite element modeling of a flexible rack feeder system

As shown in previous work, there exist several options for modeling of flexible mechanical systems that are readily applicable for the design of observer-based closed-loop control strategies. Especially, finite element techniques, which are derived by means of an early-lumping approach, provide a good compromise between accurate modeling and real-time capability of the resulting control and estimation procedures. However, an increase of the degrees of (polynomial) ansatz functions for the representation of characteristic quantities such as bending deflections as well as an increase of the number of finite elements lead to rapidly growing system orders. On the one hand, this order increase brings the drawback that an excessive computational effort may be required. In the worst case, this results in a loss of real-time capability. On the other hand, it is also possible that system models are obtained which contain information that is redundant for a feedback control design. The latter holds for cases in which eigenvalues are included in finite-dimensional state-space representations with features of the system dynamics that are sufficiently faster than the available actuators. Hence, order reduction techniques become necessary in practical control tasks. In this paper, suitable modeling and validation approaches are described which allow for limiting the system order to a value that still guarantees sufficient accuracy for the control task at hand. Representative simulation results are discussed in this paper which are the basis for a future experimental validation.

Reliable finite-dimensional models with guaranteed approximation quality for control of distributed parameter systems

A large variety of technical systems is characterized by process variables which do not only depend on time but which also depend on at least one spatial coordinate. Such processes are typically contained in systems for heat and mass transfer as well as in flexible mechanical structures. Mathematical descriptions for such systems are generally given in terms of partial differential equations. To derive control and state estimation approaches which can be evaluated in real time, these infinite-dimensional models are commonly reduced to a finite-dimensional description. In this paper, different strategies are derived which, on the one hand, allow for a derivation of finite-dimensional representations on the basis of the method of integrodifferential relations and, on the other hand, allow for a reliable quantification of the resulting approximation quality. As application scenarios, an energy-optimal heating-up strategy based on Pontryagins maximum principle as well as state and disturbance estimation procedures are presented for a one-dimensional heat transfer process.