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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04 Sep 2012
TL;DR: In this paper, label equivalence, a relation which is shown to induce an exactly lumped fluid model, a potentially smaller ODE system which can be exactly related to the original one, is introduced.
Abstract: We study behavioural relations for process algebra with a fluid semantics given in terms of a system of ordinary differential equations (ODEs). We introduce label equivalence, a relation which is shown to induce an exactly lumped fluid model, a potentially smaller ODE system which can be exactly related to the original one. We show that, in general, for two processes that are related in the fluid sense nothing can be said about their relationship from stochastic viewpoint. However, we identify a class of models for which label equivalence implies a correspondence, called semi-isomorphism, between their transition systems that are at the basis of the Markovian interpretation.
39 citations
Cites background from "Control of large-scale dynamic syst..."
...This problem has motivated work on ODE aggregations in diverse contexts such as control theory [1], theoretical ecology [12], and economics [11]....
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TL;DR: In this article, a necessary and sufficient condition for a bilinear time-invariant dynamical system to admit an aggregated reduced-order model is given, along with a sufficient condition.
Abstract: A necessary and sufficient condition for a bilinear time-invariant dynamical system to admit an aggregated reduced-order bilinear model is given.
39 citations
30 Mar 2013
TL;DR: In this paper, a mixed method of interval systems is proposed for model order reduction of systems with uncertain parameters, where two separate methods are used for finding parameters of the numerator and denominator.
Abstract: Mixed method of interval systems is a combination of classical reduction methods and stability preserving methods of interval systems. This paper proposed a new method for model order reduction of systems with uncertain parameters. The bounds on the uncertain parameters are known a priori. Two separate methods are used for finding parameters of the numerator and denominator. The numerator parameters are obtained by either of these methods such as differentiation method, factor division method, cauer second form, moment matching method or Pade approximation method. The denominator is obtained by the differentiation method in all the cases. A numerical example has been discussed to illustrate the procedures. From the above mixed methods, differentiation method and cauer second form as resulted in better approximation when compared with other methods. The errors between the original higher order and reduced order models have also been highlighted to support the effectiveness of the proposed methods.
39 citations
TL;DR: Two modified minimal realization methods are suggested for getting reduced order models in time domain with the help of modified Hankel matrices, computationally simple, efficient and applicable to Multiple-Input–Multiple-Output (MIMO) systems as well.
Abstract: Two modified minimal realization methods are suggested in this paper for getting reduced order models in time domain with the help of modified Hankel matrices. The methods are computationally simple, efficient and also applicable to Multiple-Input–Multiple-Output (MIMO) systems as well. The proposed methods are validated with the help of a few examples from literature.
38 citations
TL;DR: In this paper, a linear dynamic equivalent model for large-scale wind power plants on the basis of data obtained at the point of interconnection in simulation is developed. But the model is difficult to model because of the large number of turbines that operate stochastically, with little information exchanged between the wind energy plants and the serving utilities.
Abstract: The dynamic performance of wind power plants is difficult to model because of the large number of turbines that operate stochastically, with little information exchanged between the wind power plants and the serving utilities. In this study, we develop a linear dynamic equivalent model for large-scale wind power plants on the basis of data obtained at the point of interconnection in simulation. The modeling technique is based on adaptive parameter estimation of an equivalent model representing the dynamic performance of the plant. The developed model is tested against a large-scale plant model to confirm the effectiveness of the proposed technique.
36 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