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
01 Jan 2016
TL;DR: The developed algorithm is observant towards stability and final state value of approximated model if complex parent uncertain system is stable and the step response and frequency response envelope of actual as well as derived approximate model show optimum closeness.
Abstract: This brief paper focuses on complexity reduction of uncertain discrete interval time system. The novelty of algorithm is that the step response and frequency response envelope of actual as well as derived approximate model show optimum closeness. The developed algorithm is observant towards stability and final state value of approximated model if complex parent uncertain system is stable.
2 citations
TL;DR: A simplified model of reduced dimensionality is presented for a class of linear gyroscopic systems with quadratic performance indices based on the concept of weakly coupled subsystems and can be used in the synthesis of suboptimal controllers.
Abstract: A simplified model of reduced dimensionality is presented for a class of linear gyroscopic systems with quadratic performance indices. This model is based on the concept of weakly coupled subsystems and can be used in the synthesis of suboptimal controllers. Controllers based on this model compare favorably with both optimal and conventional controllers.
2 citations
01 Dec 2018
TL;DR: This paper points out and study a connection between the recently flourishing consideration of Koopman operators and classical systems theoretic concepts such as aggregation and observability decompositions of nonlinear systems.
Abstract: In this paper, we point out and study a connection between the recently flourishing consideration of Koopman operators and classical systems theoretic concepts such as aggregation and observability decompositions of nonlinear systems. The exploration of this newly unveiled cross-connection promotes a cross-fertilization of different methodologies and ideas intrinsic to the two different frameworks, resulting in a deeper understanding of both domains. In particular, the insights established in the paper connect intuitive systems theoretic viewpoints with the framework of Koopman operators.
2 citations
Cites background or methods from "Control of large-scale dynamic syst..."
...Aoki introduced in 1968 in a more encompassing systems theoretic framework the concept of aggregation of complex and large-scale systems [3], [4]....
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...In [3], the concept of aggregation was considered in connection with the design of LQR controllers based on a cost functional of the form...
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TL;DR: A useful technique for order reduction of large-scale linear dynamic time-invariant systems using the dominant pole retention, Pole Spectrum Analysis, and Pade approximations for a stable higher-order system is proposed.
Abstract: In this article, authors proposed a useful technique for order reduction of large-scale linear dynamic time-invariant systems using the dominant pole retention, Pole Spectrum Analysis, and Pade approximations. The denominator dynamics of the simplified system is obtained by using PSA and pole dominance algorithm; numerator dynamics of the same is obtained by using Pade approximation. The approximation is based on the principle that the mean of the poles (pole centroid) and centroid-based system stiffness are same for both large-scale and simplified systems. For a stable higher-order system, the method promises the stability of the simplified system. To validate the proposed technique, some numerical illustrations have been considered from the literature with the comparisons of performance in terms of a quality check through performance index and response matching between original higher-order and simplified systems.
2 citations
TL;DR: The connection between properties of symmetry groups and controllability, observability and decoupling properties of a class of linear distributed parameter systems is discussed and illustrated by an example in this article.
Abstract: The well-known symmetry principles and group theoretic methods of mathematical physics are applied, from a control point of view, to the analysis of linear distributed parameter systems (LDS) endowed with some degree of geometric symmetry. The connection between properties of symmetry groups and controllability, observability and decoupling properties of a class of LDS is discussed and illustrated by an example.
2 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