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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, the authors show how certain structured feedback gains necessarily emerge as the optimal controller gains in two linear optimal control formulations for multi-agent systems, i.e., linear optimal synchronization and linear optimal centroid stabilization.

14 citations

Journal ArticleDOI
TL;DR: The improved algorithm is utilized to improve the frequency response in a hybrid renewable power grid by fine-tuning the proportional-integral-derivative (PID) controller parameters and is called I_Rao_3.
Abstract: In this paper, an improved optimization algorithm is proposed to overcome the original Rao algorithm limitations (i.e., different characteristics in exploration and exploitation) and enhance the performance of the original Rao algorithm. In the improved algorithm, the self-adaptive multi-population and Levy flight methods are utilized in the original Rao algorithm. The improved algorithm is called I_Rao_3. The improved algorithm’s efficiency is confirmed by comparing it to the original Rao algorithm utilizing various standard benchmark test functions. Moreover, the proposed I_Rao_3 algorithm is utilized to improve the frequency response in a hybrid renewable power grid by fine-tuning the proportional-integral-derivative (PID) controller parameters. The targeted system used for this study is a hybrid power grid, which encompasses conventional generating stations (i.e., thermal power plants), renewable power stations (i.e., PV and wind power stations) for the analysis of the load frequency control (LFC) issue. Unlike other previously published works, this study considers the impact of DC links in parallel to AC links to interconnect the two-hybrid renewable power system area. In addition, the nonlinearities effects (i.e., generation rate constraint and a governor dead band) are applied to each area in order to achieve a more realistic study. The superiority of the proposed PID controller-based I_Rao_3 algorithm is endorsed by comparing its performance with many other optimization algorithms.

14 citations

Journal ArticleDOI
TL;DR: In this paper, the problem of the construction of stable lower-order models and the system-order determination is addressed by using a measure of reducibility defined in connection with the minimal realization algorithm.
Abstract: This paper is concerned with (1) the problem of the construction of lower-order models and (2) the Telated problem of the order determination of a real system based upon an estimated model with an overestimated order. Methods of the construction of stable lower-order models and the system-order determination are proposed. The approach adopted is to obtain a minimal realization of the original system by taking the principal components of the predictors of the outputs as the state and then to construct reduced models based upon a measure of reducibility defined in connection with the minimal realization algorithm. The measure of reducibility is useful to get a priori information about how small the order of the reduced model can be without much deterioration. Simulation studies are also carried out to demonstrate the effectiveness of the measure of reducibility and the proposed methods.

14 citations

Journal ArticleDOI
01 Nov 1989
TL;DR: In this paper, a computationally simple sequential algorithm for designing an optimal modal controller is presented, which is capable of systematically calculating a set of appropriate weighting matrices for the cost functional in the quadratic performance index, so that the resulting optimal control law would not only minimise the cost, but also place the eigenspectrum of the closed-loop system at some predefined locations.
Abstract: A computationally simple sequential algorithm for designing an optimal modal controller is presented. The approach is capable of systematically calculating a set of appropriate weighting matrices for the cost functional in the quadratic performance index, so that the resulting optimal control law would not only minimise the cost functional, but also place the eigenspectrum of the closed-loop system at some predefined locations.

14 citations

Journal ArticleDOI
TL;DR: In this paper, a theorem from the nonlinear vibrations theory is applied to derive analytical coherency-criteria for a classical model of the system and results for a sample system are shown.

13 citations

References
More filters
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