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, different methods of approximating d'approximation dans l'espace d'etat de systemes dynamiques lineaires par des modeles de dimension reduite sont presentees et comparees.
1 citations
TL;DR: In this article, a method is developed for using reduced order system-models for deriving suboptimal controllers for the industrial regulator design problem with input derivative constraints, which implies a knowledge of relevant aggregation matrices and results in considerable saving in computational requirements.
1 citations
TL;DR: The authors review the main framework of MRC and Goodwin growth cycle (GGC) model between two countries and drive the employment rate to be approximate stable in a high level by controlling the workers' share in the national income automatically.
Abstract: A useful method in intelligent engineering, called model reference
control (MRC), is applied in an economic control problem. The authors review the
main framework of MRC and Goodwin growth cycle (GGC) model between two
countries and drive the employment rate to be approximate stable in a high level
by controlling the workers' share in the national income automatically. It is very
helpful to constitute economic policies for a country or an economic union.
1 citations
05 Nov 2007
TL;DR: In this article, a canonical form for larger systems (expansions) that are obtained by expanding smaller systems (contractions) is derived, which offers full freedom in selecting appropriate matrices for the expansion-contraction process.
Abstract: The inclusion principle provides a mathematical framework for comparing behavior of dynamic systems having different dimensions. Our main objective is to derive a canonical form for larger systems (expansions) that are obtained by expanding smaller systems (contractions). The form offers full freedom in selecting appropriate matrices for the expansion-contraction process. We will broaden the form to include feedback and propose an explicit characterization of contractible control laws subject to overlapping information structure constraints.
1 citations
Cites background from "Control of large-scale dynamic syst..."
...Expansion, being an intersection of aggregation [21] and restriction [1, 2, 3], raises a question: What system properties are retained after the expansion-contraction process has been completed [22]? Much progress has been made in identifying the conditions that ensure the invariancy of controllability, observability, and stabilizability in the expanded systems [23, 24, 25]....
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TL;DR: A technique is suggested as an approach to developing controls for large-scale dynamical systems that enables one to formulate a small-scale dynamic optimization problem based upon certain key indicators of behavior related to the large- scale system.
Abstract: A technique is suggested as an approach to developing controls for large-scale dynamical systems. The method has the advantage that it enables one to formulate a small-scale dynamic optimization problem based upon certain key indicators of behavior related to the large-scale system. The resultant control problem is algebraic.
1 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