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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01 Jan 2013
TL;DR: A new mixed method for reducing the order of interval systems having uncertain but bounded parameters is proposed and the denominator of the reduced order model is obtained by a table and numerator is derived by applying factor division and Cauer second form.
Abstract: The paper proposes a new mixed method for reducing the order of interval systems i.e., systems having uncertain but bounded parameters. The denominator of the reduced order model is obtained by a table and numerator is derived by applying factor division and Cauer second form. A numerical example has been discussed to illustrate the procedures. The errors between the original higher order and reduced order models have also been highlighted to support the effectiveness of the proposed methods.
3 citations
Cites background from "Control of large-scale dynamic syst..."
...A wide variety of model reduction techniques [1–9] have proposed by several authors in frequency domain, no one gives the best results....
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TL;DR: In this paper, the state controllers with structural limitations are determined based on balanced realizations of the plant or the closed loop and the approximation of the behavior of a compensator which is designed without considering structural limitations is strived for.
Abstract: In this paper methods are proposed to determine state controllers with structural limitations. All these method are based on balanced realizations of the plant or the closed loop. The approximation of the behaviour of a compensator which is designed without considering structural limitations is strived for. Regarded limitations are the number of measured variables to determine an output feedback and the order to determine a dynamic compensator consisting of state estimator and state feedback
3 citations
Cites background from "Control of large-scale dynamic syst..."
...Gewichtungsmatrix Wx und W l r ca beim Entwurf optimaler Rekonstruierer ergibt sich aufgrund des durch die Aggregationsmatrix [24], die im hier betrachteten Fall aus den ersten Zeilen der balancierenden Transformationsmatrix besteht, bekannten Zusammenhangs zwischen dem exakten Verlauf von z, und χ zu WUcd = Τλ Wx T[....
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...Gewichtungsmatrix Wx und W l r ca beim Entwurf optimaler Rekonstruierer ergibt sich aufgrund des durch die Aggregationsmatrix [24], die im hier betrachteten Fall aus den ersten Zeilen der balancierenden Transformationsmatrix besteht, bekannten Zusammenhangs zwischen dem exakten Verlauf von z, und χ zu...
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01 Dec 2007
TL;DR: This paper extends, in a linear framework, some preliminary results obtained for state stability implementation to the more general target of output zeroing and tracking implementation.
Abstract: The recently introduced concept of system abstraction appears to be a promising tool for control purposes in a hierarchical framework. A challenging field of research is the so-called control implementation, i.e. how a high-level control law designed for an abstraction can be executed by the low-level original system. In this paper we extend, in a linear framework, some preliminary results obtained for state stability implementation to the more general target of output zeroing and tracking implementation.
3 citations
Cites background or methods from "Control of large-scale dynamic syst..."
...Control system projection is based on an epic aggregation map y = Cx, and coincides with the well-known notion of aggregation [2], [10]....
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...which are the well-known [2] conditions for Σα to be an aggregation of Σo....
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01 Nov 2016
TL;DR: A novel method of model order reduction is presented by integrating the merits of Firefly Algorithm with Pattern Search algorithm, which acts as a global optimizer while PS performs the local search in the hybrid topology to minimize the time response based error criterion.
Abstract: In this paper, a novel method of model order reduction is presented by integrating the merits of Firefly Algorithm (FA) with Pattern Search (PS) algorithm. A linear-continuous single-input single-output system of higher order has been taken up and reduced to a lower order system. FA acts as a global optimizer while PS performs the local search in the hybrid topology to minimize the time response based error criterion. Step and frequency responses of the original and reduced systems are plotted. The superiority of the proposed method is advocated by means of a numerical example from the available literature. Further, the reduced systems are compared with some of the existing techniques in terms of Integral of Square Error (ISE). The efficacy of the proposed method is also highlighted over the existing soft computing techniques serving similar purpose.
3 citations
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