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.
Citations
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TL;DR: It is demonstrated that the maximum conditional independence method is not only applicable to stable systems, but also applicable to unstable systems.
Abstract: By analysing information descriptions in state space models of linear stochastic systems, this paper proposes two model reduction methods via principles of maximizing independence and conditional independence among the reduced state vector, respectively. These methods are based on state aggregation. The independence and conditional independence are measured by the Kullback-Leibler information distance. It is demonstrated that the maximum conditional independence method is not only applicable to stable systems, but also applicable to unstable systems. Simulation results illustrate the efficiency of the present methods.
TL;DR: A fourth-order macroeconomic model, with typical parameters, which demonstrates the margin of power of the governmental body can exercise on the various sectoral activities, is used to illustrate some of the concepts presented in this paper.
Abstract: The paper gives a conceptual framework for robustness analysis of large-scale economic systems, and its realization through interactive computer-aided software. The mismatch between the economic system and the corresponding mathematical model is discussed. The computer-aided system combines algorithmic and expert system techniques. An important feature of the present system is the modularization of the software package which allows a distributed problem solving approach. A fourth-order macroeconomic model, with typical parameters, which demonstrates the margin of power of the governmental body can exercise on the various sectoral activities, is used to illustrate some of the concepts presented in this paper.
01 Jan 1986
TL;DR: In this article, the authors considered the sensitivity of analog and digital filters and showed that the filter response is critically dependent on the internal (state-space) structure of the filter, which is not a proven mathematical result but one which has been observed through numerous design studies.
Abstract: The signal processing community has long considered the problem of sensitivity in respect of the design of analog and digital filters. In analog filters, it is the component (i.e. R, L or C) tolerances and amplifier sensitivities (i.e. internal amplifier noise levels) which are significant whereas in digital filtering the corresponding considerations are in respect of the coefficient (i.e. choice of wordlength) and arithmetic sensitivities (numerically roundoff noise). In both cases, the nett effect of these inaccuracies on the filter response is critically dependent on the internal (state-space) structure of the filter. If one optimizes the filter structure with respect to the internal noise (amplifier or arithmetic) then the resulting so-called minimum noise gain structures have been shown to exhibit low sensitivity with respect to component or coefficient sensitivity. This is not a proven mathematical result but one which has been observed through numerous design studies.
TL;DR: In this paper, it was shown that for nearly aggregable systems, there exists a reduced-order model such that, for an appropriate initial condition, the trajectories of the reduced order model are near a linear combination of the full order model.
Abstract: The basic idea of aggregation is that there exists a reduced-order model such that, for an appropriate initial condition, the trajectories of the reduced-order model are linear combinations of the trajectories of the full-order model. The authors study systems which do not aggregate exactly, but which 'nearly aggregate'. It is shown that for nearly aggregable systems, there exists a reduced-order model such that, for an appropriate initial condition, the trajectories of the reduced-order model are near a linear combination of the trajectories of the full-order model. >
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