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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TL;DR: A recently developed framework for exploring the structure of linear time-invariant models of large systems, and for constructing interpretable or physically-based, reduced-order models that reproduce selected modes of the original systems to a desired accuracy is described.
Abstract: We describe a recently developed framework for exploring the structure of linear time-invariant models of large systems, and for constructing interpretable or physically-based, reduced-order models that reproduce selected modes of the original systems to a desired accuracy. Application of this framework to constructing lumped approximations for interconnections of lumped and distributed systems is briefly explored.
7 citations
Cites background from "Control of large-scale dynamic syst..."
...A more precise definition of the above class of problems is deferred to the sections that follow, but it may be noted here that several well known procedures fall into this class: modal reduction (Davison [4], Hickin and Sinha [5]), aggregation (Aoki [ 2 ]), singular perturbation methods applied to LTI systems (Kokotovic et al. [6])....
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...The above procedure underlies all that we discuss here, and in fact encompasses modal reduction, [4], [5], and aggregation, [ 2 ]....
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TL;DR: In this paper, a companion-type realisation of a rational transfer function is introduced, which can simultaneously match time moments, Markov parameters and to retain desired poles, thus combining the methods of partial realisation (Pade approximation) and aggregation.
Abstract: A new companion-type realisation of a rational transfer function is introduced. This form is then used for obtaining a reduced-order model. It is possible, using this approach, to simultaneously match time moments, Markov parameters and to retain desired poles, thus combining the methods of partial realisation (Pade approximation) and aggregation.
7 citations
01 Feb 2016
TL;DR: In this paper, a hybrid approach for model order reduction is proposed, where the approximants for denominator polynomial are derived by matching both Markov parameters and Time moments, whereas numerator poynomial derivation and error minimization is done using Fminsearch algorithm.
Abstract: A hybrid approach for model order reduction is proposed in this paper. The approximants for denominator polynomial are derived by matching both Markov parameters and Time moments, whereas numerator polynomial derivation and error minimization is done using Fminsearch algorithm. The efficiency of the proposed method can be investigated in terms of closeness of the response of reduced order model with respect to that of higher order original model and a comparison of the integral square error as well.
7 citations
01 Dec 2015
TL;DR: In this paper, an extended approach for order reduction of complex discrete uncertain systems is proposed using Interval arithmetic Routh Stability arrays are formed to obtained numerator and denominator of reduced order model, which preserves the stability aspect of reduced system if higher order uncertain system is stable.
Abstract: An extended approach for order reduction of complex discrete uncertain systems is proposed. Using Interval arithmetic Routh Stability arrays are formed to obtained numerator and denominator of reduced order model. The developed approach preserves the stability aspect of reduced system if higher order uncertain system is stable. A numerical example is included to illustrate the proposed algorithm along with the comparison with existing techniques.
7 citations
TL;DR: The use of lower-order models in obtaining approximate solutions in optimal control problems has been of much interest during the past few years as discussed by the authors, and two new procedures for order reduction, based on interpreting the system impulse response (or transfer function) as an input-output map, are presented.
Abstract: The use of lower-order models in obtaining approximate solutions in optimal control problems has been of much interest during the past few years Two new procedures for order reduction, based on interpreting the system impulse response (or transfer function) as an input–output map, are presented The relationship between the states of the original system and the states of its reduced-order model is investigated The lower-order model is used to obtain suboptimal controllers in linear regulator problems Several computational examples are given
7 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