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: Which aspects of the system are measurable, which have a significant effect, and whether the system is a discrete-time process, continuous time, or a hybrid of both are defined.
Abstract: 1. System Description what we know about the system. Boundary Conditions. Assumptions about the system (behaviour, environment) 2. Inputs, Outputs, Parameters, States Here we define which aspects of the system are measurable, which have a significant effect – What can we set, what can we measure, which do not change over time. Implicitly we also define the time-scale of interest (micro-seconds to decades), and whether the system is a discrete-time process, continuous time, or a hybrid of both. 3. Purpose of the Model What information do we want to represent and extract from the system.
3 citations
TL;DR: The authors put forward a new approach of structural reduced order, which is similar to the constrained substructural method in dynamics, and is also the extension of the method of aggregation raised by Aoki (1968).
Abstract: After introducing the concept and criteria of controllability and degree of controllability about structural wave control in this paper, the authors put forward a new approach of structural reduced order, which is similar to the constrained substructural method in dynamics, and is also the extension of the method of aggregation raised by Aoki (1968). >
3 citations
Dissertation•
01 Dec 2016
3 citations
Cites methods from "Control of large-scale dynamic syst..."
...Second, the matrix is reduced by aggregation procedure as proposed by Aoki (1968) to minimize the matrix size....
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01 Jan 1972
3 citations
Cites background from "Control of large-scale dynamic syst..."
...Aoki has proposed that large-scale dynamic systems be controlled by aggregating their relations according to certain criteria, and considering the optimal design problem for the aggregated system of reduced state dimension. Aggregated control has clear economic relevance and, since Aoki explicitly uses the aggregation concept developed by economists, incentive exists for pursuing this topic further. Recently, Mesarovic et_ aj_ have developed the notion of hierarchical optimal control. The economic significance of hierarchical control has already been established by Tinbergen’s analysis [1954] of centralised and decentralised policy-making; and is confirmed by such topical issues as coordination of Federal-State policy-making and policy harmonisation in customs unions....
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TL;DR: In this paper, a model reduction method based on partition of the state space into two orthogonal subspaces is proposed, which works for both stable and unstable systems but requires knowledge of the system solution in order to apply.
Abstract: Theoretical limitations for model reduction systems described by ordinary differential equations are investigated through use of the system solution rather than the system state equations. The general case is discussed first and the specialized to linear time-varying systems and finally to linear time-invariant systems. The distance between the original and reduced systems is measured by an error norm corresponding to energy. The reduction method is based on partition of the state space into two orthogonal subspaces. It is an effective procedure which works for both stable and unstable systems but requires knowledge of the system solution in order to be applied. In general the reduced-order model cannot be separated from the initial conditions, but this is possible for linear systems. If there is a driving function acting on the system, it will affect the reduced-order model in an essential way, and its order then cannot in general be reduced. >
3 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