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
More filters
TL;DR: In this article, the impact of the U.S. economy on two Scandinavian economies (Finland and Sweden) was investigated by testing for the presence of unit roots among the observed input-output processes.
Abstract: We provide new evidence on the impact of the U.S. economy on two Scandinavian economies (Finland and Sweden). Initially, we test for the presence of unit roots among the observed input-output processes. Next, Granger causality and cointegra-tion of the system is explicitly tested, to justify the estimated vector-valued state space model. The trend and cyclical components of the endogenous vector are extracted by three alternative decomposition methods. Finally, the content of the cyclical component is analysed by spectral analysis.
7 citations
TL;DR: In this paper, a singular value decomposition (SVD) scheme for model reduction of LSS with a direct and simple procedure that provides a methodology for performance enhancement and spillover compensation is presented.
7 citations
Cites background from "Control of large-scale dynamic syst..."
...It is assumed that the system S, is observed through m (WI <n) vector related to x in the manner z=Dx (2) where D is the (WI x n) aggregation matrix, which, from the geometric point of view, is the projection operator from R” of x variables to the space R” of z-variables....
[...]
TL;DR: In this paper, a method combining model reduction and aggregation has been suggested as a means of assigning prescribed values to a subset of the eigenvalues of a linear fixed control system, which is generalised to include all projective model reduction techniques and approximate eigenvalue assignment by output feedback.
Abstract: A method combining model reduction and aggregation has been suggested as a means of assigning prescribed values to a subset of the eigenvalues of a linear fixed control system. This method is generalised to include all projective model reduction techniques and approximate eigenvalue assignment by output feedback.
7 citations
01 Dec 2011
TL;DR: In this paper, a stable lower order interval model was obtained from a stable higher order interval plant by using Kharitonov's theorem and Routh approximation method, and the reduced order interval numerator and denominator polynomials were determined by using the Routh Approximation Method.
Abstract: This paper describes a technique to obtain a stable lower order Interval model from its stable higher order interval plant. The reduced order interval numerator and denominator polynomials are determined by using Kharitonov's theorem and Routh Approximation Method. The algorithm is simple, rugged and generates stable reduced order interval model for stable original high order interval plants. A numerical example illustrates the effectiveness of proposed algorithm.
7 citations
Cites methods from "Control of large-scale dynamic syst..."
...Among them, the familiar and important methods are aggregation method [1] , Pade approximation[2] , Routh approximation [3], moment matching technique[18] , and recently L ∞ optimization technique[4] ....
[...]
TL;DR: A brief discussion of various model simplification techniques currently available in the literature can be found in this paper, where the main intention is to highlight some of the unifying features of the different models simplification approaches.
7 citations
References
More filters
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