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Journal ArticleDOI

Control of large-scale dynamic systems by aggregation

01 Jun 1968-IEEE Transactions on Automatic Control (IEEE)-Vol. 13, Iss: 3, pp 246-253
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
More filters
Journal ArticleDOI
TL;DR: In this paper, a model size reduction approach based on Moore's balanced method for two kinds of small-size concrete hollow blocks is presented, which can lead to simple "model libraries" within building simulation codes.

51 citations

Journal ArticleDOI
TL;DR: The theory of model simplification is presented as a means of increasing model understanding, issues in the application to nonlinear models are considered, and software that facilitates model Simplification is discussed.
Abstract: The theory of model simplification is presented as a means of increasing model understanding. Simplification is based on a selection of the behavior modes defined by the linearized representation of the model and results in a smaller and more easily understood model. To allow understanding, the variables in the simplified model must be easy to interpret relative to those in the original model. This interpretation is complete in an exact simplification, a concept used to derive measures of the importance of different variables in generating selected behavior modes. These measures are used to select which variables to retain and which to omit in forming the simplified model. Issues in the application to nonlinear models are considered, and software that facilitates model simplification is discussed.

48 citations

Journal ArticleDOI
TL;DR: In this paper, an original method for the identification of reduced models (RM) of nonlinear diffusive thermal systems is proposed, which derives from the Modal Identification Method developed for linear systems in previous works.

46 citations

Journal ArticleDOI
TL;DR: In this article, it is shown that a family of systems (the homotopy) can make a continuous transformation from some initial system to the final system with a carefully chosen initial problem, and a theorem guarantees that all the systems along this path will be asymptotically stable and controllable.
Abstract: The optimal projection approach to solving the H2 reduced order model problem produces two coupled, highly nonlinear matrix equations with rank conditions as constraints Due to the resemblance of these equations to standard matrix Lyapunov equations, they are called modified Lyapunov equations The algorithms proposed herein utilize probability-one homotopy theory as the main tool It is shown that there is a family of systems (the homotopy) that make a continuous transformation from some initial system to the final system With a carefully chosen initial problem a theorem guarantees that all the systems along the homotopy path will be asymptotically stable, controllable and observable One method, which solves the equations in their original form, requires a decomposition of the projection matrix using the Drazin inverse of a matrix It is shown that the appropriate inverse is a differentiable function An effective algorithm for computing the derivative of the projection matrix that involves solving a set of Sylvester equations is given Another class of methods considers the equations in a modified form, using a decomposition of the pseudogramians based on a contragredient transformation Some freedom is left in making an exact match between the number of equations and the number of unknowns, thus effectively generating a family of methods

46 citations


Cites background from "Control of large-scale dynamic syst..."

  • ...…Skelton and Yousuff 1983), balancing (Moore 1981, Pernebo and Silverman 1982), Hankel-norm approximation (Kung and Lin 1981 a, b), aggregation (Aoki 1968, Kwong 1982), non-minimal partial realization (Hickin and Sinha 1980), projection methods (De Villemagne and Skelton 1987) and the optimal…...

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References
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Journal ArticleDOI
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

Journal ArticleDOI
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

Journal ArticleDOI
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