Control of large-scale dynamic systems by aggregation
TL;DR: Using the quantitative definition of weak coupling proposed by Milne, a suboptimal control policy for the weakly coupled system is derived and questions of performance degradation and of stability of such suboptimally controlled systems are answered.
Abstract: A method is proposed to obtain a model of a dynamic system with a state vector of high dimension. The model is derived by "aggregating" the original system state vector into a lower-dimensional vector. Some properties of the aggregation method are investigated in the paper. The concept of aggregation, a generalization of that of projection, is related to that of state vector partition and is useful not only in building a model of reduced dimension, but also in unifying several topics in the control theory such as regulators with incomplete state feedback, characteristic value computations, model controls, and bounds on the solution of the matrix Riccati equations, etc. Using the quantitative definition of weak coupling proposed by Milne, a suboptimal control policy for the weakly coupled system is derived. Questions of performance degradation and of stability of such suboptimally controlled systems are also answered in the paper.
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TL;DR: In this paper, a simple iterative technique, which is free of certain shortcomings of the previous methods, is proposed for the approximation of large linear systems by a lower-order model, where the measure of the goodness of the approximate model is taken to be the value of the integral-square error between the step responses of the exact and the simplified systems.
Abstract: A simple iterative technique, which is free of certain shortcomings of the previous methods, is proposed for the approximation of large linear systems by a lower- order model. Here, the measure of the goodness of the approximate model is taken to be the value of the integral-square error between the step responses of the exact and the simplified systems. The proposed technique consists of a two-step iterative scheme. In the first step, the optimum residues are obtained by the minimization of the objective function, while the poles (or eigenvalues) are kept constant. In the second step, the poles are optimized while the residues remain fixed. This procedure is continued cyclically until the objective function is satisfactorily minimized. The necessary and sufficient conditions for existence of an optimum are satisfied in each step. The residues, poles and objective functions converge monotonically. The resulting reduced-order model obtained by this method is stable if the original system is stable. The method can also be applied to systems with repeated poles and to multivariable systems. The results are superior to those obtained previously in the steady-state, the point-by-point transient response, and the value of the integral-square error. Illustrative examples are presented.
10 citations
01 Jan 1997
TL;DR: In this paper, the impact of the Japanese stock market on the Finnish derivatives is investigated. And the authors provide new evidence on the influence of Japanese stock markets on Finnish derivatives by testing for the presence of unit roots among the observed input-output processes.
Abstract: In the present study we provide new evidence on the impact of the Japanese stock market on the Finnish derivatives. Initially, we test for the presence of unit roots among the observed input-output processes. Next, cointegration in the system is explicitly tested, to justify the estimated vector-valued state space model. The trend and cyclical components of the endogenous vector are extracted and analyzed. The content of the cyclical component is analyzed by spectral analysis. Finally, the error processes are subjected to some statistical tests and used as signal variables in nonlinear model building.
10 citations
18 Nov 2010
TL;DR: A novel method to construct power system dynamic equivalents for day-ahead stability studies is proposed, based on recent results in the topic of reduction which allow small size and robust (against daily variations of the grid situation) dynamic equivalents.
Abstract: A novel method to construct power system dynamic equivalents for day-ahead stability studies is proposed. It provides a reduced-order model of the neighbor areas of a Transmission System Operator (TSO). When connected to the detailed representation of the TSO zone to be studied, this model is adequate for quick transient-stability studies. The approach is based on recent results in the topic of reduction which allow small size and robust (against daily variations of the grid situation) dynamic equivalents. The methodology is shown on the situation treated by the French TSO (Reseau de Transport d'Electricite — RTE) for its North-Eastern border.
10 citations
Cites background from "Control of large-scale dynamic syst..."
...[5] and [2] by simply keeping the relevant machines with respect to a given chord (which by that time was the slow coherency one) and replacing the other ones (the less relevant machines) by correction factors....
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Dissertation•
21 Mar 2011
TL;DR: In this article, the authors developed a model that is capable of accurately predicting thermal behaviors of the ground-source heat pump systems with vertical ground heat exchangers (GHEs).
Abstract: Ground-source heat pump systems with vertical ground heat exchangers (GHE) are gaining popularity worldwide for their higher coefficients of performance and lower CO2 emissions. However, the higher initial cost of installing the borehole GHEs is a main obstacle to spread the systems. To reduce the required total GHE length and efficiently operate the systems, various systems such as hybrid ones (e.g. solar heat injection) have recently been introduced. Accurate prediction of heat transfer in and around boreholes of such systems is crucial to avoid costly overdesigns or catastrophic failures of undersized systems as it is for typical GCHP systems. However, unlike the traditional sizing methods, it is increasingly required to take into account detailed borehole configuration and transient effects (e.g. short circuit effects between U-tubes). Many of the existing GHE models have been reviewed. Some of these models have serious limitations when it comes to transient heat transfer, particularly in the borehole itself. Accordingly, the objective of this thesis is to develop a model that is capable to accurately predict thermal behaviors of the GHEs. A precise response to input variations even in a short time-step is also expected in the model. The model also has to account for a correct temperature and flux distribution between the U-tubes and inside the borehole that seems to be important in the solar heat injection case. Considering these effects in 3D with a detailed mesh used for describing the borehole configurations is normally time-consuming. This thesis attempts to alleviate the calculation time using state model reduction techniques that use fewer modes for a fast calculation but predict similar results. Domain decomposition is also envisaged to sub-structure the domain and vary the time-step sizes. Since the decomposed domains should be coupled one another spatially as well as temporally, new coupling methods are proposed and validated particularly in the FEM. For the simulation purpose, a hybrid model (HM) is developed that combines a numerical solution, the same one as the 3D-RM but only for the borehole, and well-known analytical ones for a fast calculation. An experimental facility used for validation of the model has been built and is described. A comparison with the experimental results shows that the relatively fast transients occurring in the borehole are well predicted not only for the outlet fluid temperature but also for the grout temperatures at different depths even in very short time-steps. Even though the current version of 3D-RM is experimentally validated, it is still worth optimizing the model in terms of the computational time. Further simulations with the 3D-RM are expected to be carried out to estimate the performance of new hybrid systems and propose its appropriate sizing with correspondent thermal impacts on the ground. Finally, the development of the model 3D-RM can be an initiation to accurately model various types of GHE within an acceptable calculation time.
10 citations
Cites methods from "Control of large-scale dynamic syst..."
...Linear Aggregation method Linear aggregation (LA) method proposed by Aoki [Aoki 1968] has been applied and optimized by several authors ( [Michailesco 1979] [Oulefki 1993] [Jaafar et al....
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TL;DR: A reduced-order model for complex multifeed gas networks is developed which is based on an aggregation procedure and yields good steady-state data over a range of operating conditions, and hence a robust estimation scheme has been developed.
Abstract: A reduced-order model for complex multifeed gas networks has been developed which is based on an aggregation procedure. The reduced model is then used with an estimation scheme to determine the pressure distribution in the network from a limited number of measurements. The results obtained show that the model yields good steady-state data over a range of operating conditions, and hence a robust estimation scheme has been developed.
10 citations
References
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TL;DR: A technique is presented for the decomposition of a linear program that permits the problem to be solved by alternate solutions of linear sub-programs representing its several parts and a coordinating program that is obtained from the parts by linear transformations.
Abstract: A technique is presented for the decomposition of a linear program that permits the problem to be solved by alternate solutions of linear sub-programs representing its several parts and a coordinating program that is obtained from the parts by linear transformations. The coordinating program generates at each cycle new objective forms for each part, and each part generates in turn from its optimal basic feasible solutions new activities columns for the interconnecting program. Viewed as an instance of a “generalized programming problem” whose columns are drawn freely from given convex sets, such a problem can be studied by an appropriate generalization of the duality theorem for linear programming, which permits a sharp distinction to be made between those constraints that pertain only to a part of the problem and those that connect its parts. This leads to a generalization of the Simplex Algorithm, for which the decomposition procedure becomes a special case. Besides holding promise for the efficient computation of large-scale systems, the principle yields a certain rationale for the “decentralized decision process” in the theory of the firm. Formally the prices generated by the coordinating program cause the manager of each part to look for a “pure” sub-program analogue of pure strategy in game theory, which he proposes to the coordinator as best he can do. The coordinator finds the optimum “mix” of pure sub-programs using new proposals and earlier ones consistent with over-all demands and supply, and thereby generates new prices that again generates new proposals by each of the parts, etc. The iterative process is finite.
2,281 citations
01 Jan 1960
TL;DR: In this article, the authors considered the problem of least square feedback control in a linear time-invariant system with n states, and proposed a solution based on the concept of controllability.
Abstract: THIS is one of the two ground-breaking papers by Kalman that appeared in 1960—with the other one (discussed next) being the filtering and prediction paper. This first paper, which deals with linear-quadratic feedback control, set the stage for what came to be known as LQR (Linear-Quadratic-Regulator) control, while the combination of the two papers formed the basis for LQG (Linear-Quadratic-Gaussian) control. Both LQR and LQG control had major influence on researchers, teachers, and practitioners of control in the decades that followed. The idea of designing a feedback controller such that the integral of the square of tracking error is minimized was first proposed by Wiener [17] and Hall [8], and further developed in the influential book by Newton, Gould and Kaiser [12]. However, the problem formulation in this book remained unsatisfactory from a mathematical point of view, but, more importantly, the algorithms obtained allowed application only to rather low order systems and were thus of limited value. This is not surprising since it basically took until theH2-interpretation in the 1980s of LQG control before a satisfactory formulation of least squares feedback control design was obtained. Kalman’s formulation in terms of finding the least squares control that evolves from an arbitrary initial state is a precise formulation of the optimal least squares transient control problem. The paper introduced the very important notion of c ntrollability, as the possibility of transfering any initial state to zero by a suitable control action. It includes the necessary and sufficient condition for controllability in terms of the positive definiteness of the Controllability Grammian, and the fact that the linear time-invariant system withn states,
1,451 citations
TL;DR: A method is proposed for reducing large matrices by constructing a matrix of lower order which has the same dominant eigenvalues and eigenvectors as the original system.
Abstract: Often it is possible to represent physical systems by a number of simultaneous linear differential equations with constant coefficients, \dot{x} = Ax + r but for many processes (e.g., chemical plants, nuclear reactors), the order of the matrix A may be quite large, say 50×50, 100×100, or even 500×500. It is difficult to work with these large matrices and a means of approximating the system matrix by one of lower order is needed. A method is proposed for reducing such matrices by constructing a matrix of lower order which has the same dominant eigenvalues and eigenvectors as the original system.
614 citations
TL;DR: In this article, a constructive design procedure for the problem of estimating the state vector of a discrete-time linear stochastic system with time-invariant dynamics when certain constraints are imposed on the number of memory elements of the estimator is presented.
Abstract: The paper presents a constructive design procedure for the problem of estimating the state vector of a discrete-time linear stochastic system with time-invariant dynamics when certain constraints are imposed on the number of memory elements of the estimator. The estimator reconstructs the state vector exactly for deterministic systems while the steady-state performance in the stochastic case may be comparable to that obtained by the optimal (unconstrained) Wiener-Kalman filter.
68 citations
67 citations