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 1986
TL;DR: In this article, two new approaches for reducing the order of large scale continuous time systems are presented, one of which is a modified version of the aggregation technology and the other is based on the singular value decomposition of the controlability Grammian of the system.
Abstract: Two new approaches for reducing the order of large scale continuous time systems are presented. The first approach is a modified version of the aggregation tech nique. It uses the matching properties of the steady state output covariance and the Markov parameters of the high fcll order system to those of the r order model. This approach also uses a new algorithm derived for the compu tation of the controllability and observability Grammians, to produce controllable, observable and stable low order models. There is no unique method available for evaluating the aggregation matrix or the matrix relating the system state vector to the model state vector. A procedure, based on the singular value decomposition of the control lability Grammian of the system, is provided for the computation of the aggregation matrix. This approach is also extended to design low order deterministic continuous time varying linear models.
3 citations
Cites methods from "Control of large-scale dynamic syst..."
...To overcome this difficulty, Aoki [5, 6] introduced the aggregation method....
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TL;DR: In this article, a review examines the recent developments related to the linear-quadratic control problem (LQP), providing a comprehensive coverage for both the lumped and distributed parameter LQP's.
Abstract: The linear-quadratic control problem (LQP) assumes central importance in control theory, and therefore it has been extensively studied in the past decade. This review examines the recent developments related to the LQP, providing a comprehensive coverage for both the lumped and distributed parameter LQP's. The existing theory and practice of the lumped LQP are presented in detail, while the distributed case is considered from a different viewpoint, since its applications and solution techniques are still in a stage of development. The generalized treatment of this subject should prove useful not only to the systems engineer but also the engineer who specializes in other areas of chemical engineering.
3 citations
01 Dec 2018
TL;DR: The design of the robust PI/PID controller for the higher order interval system via its reduced order model using the differential evolution (DE) algorithm is described and the results were successfully implemented.
Abstract: This paper describes the design of the robust PI/PID controller for the higher order interval system via its reduced order model using the differential evolution (DE) algorithm. A stable reduced interval model is generated from a higher order interval system using the DE in order to minimize the cost and reduce the complexity of the system. This reduced order interval numerator and denominator polynomials are determined by minimizing the Integral Squared Error (ISE) using the DE. Then, using reduced order interval model, a robust PI/PID controller is designed based on the stability conditions for determining robust stability of interval system. Finally, using these stability conditions, a set of inequalities in terms of controller parameters is obtained from the reduced order closed loop characteristic polynomial. Then these inequalities are solved to obtain robust controller parameters with the help of a DE algorithm. The designed, robust controller from the reduced order interval model will be attributed to the higher order interval system. The designed PI/PID controller from our proposed method not only stabilizes the reduced order model, but also stabilizes the original higher order system. The viability of the proposed methodology is illustrated through the numerical example of its successful implementation. The efficacy of the proposed methodology is also evaluated against the available approaches presented in the literature and the results were successfully implemented.
3 citations
Cites background from "Control of large-scale dynamic syst..."
...proposed control the large scale dynamic system [1]....
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TL;DR: In this article , the optimal control of network-coupled subsystems with coupled dynamics and costs is investigated, and the spectral decomposition of these three coupling matrices is used to decompose the overall system into decoupled Riccati equations.
Abstract: In this article, we investigate the optimal control of network-coupled subsystems with coupled dynamics and costs. The dynamics coupling may be represented by the adjacency matrix, the Laplacian matrix, or any other symmetric matrix corresponding to an underlying weighted undirected graph. Cost couplings are represented by two coupling matrices which have the same eigenvectors as the coupling matrix in the dynamics. We use the spectral decomposition of these three coupling matrices to decompose the overall system into $({L+1})$ $L$ $({L_\text{dist}+1})$ $L_\text{dist}\,(L_\text{dist}\leq L)$
3 citations
01 Feb 1992
TL;DR: A control system has been designed and implemented to provide more effective energy management of low-pressure gas distribution networks and a modelling technique has been developed which provides reduced order models that adequately describe the characteristics of multi-feed gas networks.
Abstract: A control system has been designed and implemented to provide more effective energy management of low-pressure gas distribution networks. The key to this is the provision of a control scheme that maintains low pressures across a network.The work was approached from first principles and a modelling technique has been developed which provides reduced order models that adequately describe the characteristics of multi-feed gas networks. The models were then used for the control system design, which in this case also included the selection of the optimal measurement points for most effective control.Following extensive design studies a relatively straightforward control scheme resulted which has been implemented and proved to be effective.
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