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: It is shown that the proposed SMC guarantees the stability of the full-order plant with the reduced-order triggering mechanism, which does not admit a triggering sequence with Zeno behavior, and the transmission of the reduce-order vector can outperform theFull-order based design owing to the severe challenges that persistently occur in the data network.
18 citations
31 Mar 1996
TL;DR: In this article, a multipoint Pade approximation for discrete interval systems is presented, where the expansion points used in the approximation could be a mixture of real, imaginary, complex and multiple points, many of the computational difficulties for such a combination of points are eliminated.
Abstract: This paper presents multipoint Pade approximation for discrete interval systems The numerator and denominator of the reduced model are obtained such that G/sub m/(z) to be a Pade approximant of G/sub s/(z), about 2r points The expansion points used in the approximation could be a mixture of real, imaginary, complex and multiple points, many of the computational difficulties for such a combination of points are eliminated by this method Numerical examples illustrate the procedure
18 citations
TL;DR: A network systems consist of subsystems and their interconnections and provide a powerful framework for the analysis, modeling, and control of complex systems.
Abstract: Network systems consist of subsystems and their interconnections, and provide a powerful framework for analysis, modeling and control of complex systems. However, subsystems may have high-dimensional dynamics, and the amount and nature of interconnections may also be of high complexity. Therefore, it is relevant to study reduction methods for network systems. An overview on reduction methods for both the topological (interconnection) structure of the network and the dynamics of the nodes, while preserving structural properties of the network, and taking a control systems perspective, is provided. First topological complexity reduction methods based on graph clustering and aggregation are reviewed, producing a reduced-order network model. Second, reduction of the nodal dynamics is considered by using extensions of classical methods, while preserving the stability and synchronization properties. Finally, a structure-preserving generalized balancing method for simplifying simultaneously the topological structure and the order of the nodal dynamics is treated.
18 citations
Cites background from "Control of large-scale dynamic syst..."
...The pioneering work on clustering-based model reduction of dynamic networks in (24, 49) introduces a notion of cluster reducibility, which is relevant to the classic notions exact aggregation and approximate aggregation from the control and model reduction literature (75, 76)....
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01 Dec 2013
TL;DR: In this paper, a direct truncation methodology for reducing the order of large scale interval systems is presented, which is computationally simple and intuitively appealing, and is shown to reduce the complexity of large-scale interval systems.
Abstract: This paper presents a direct truncation methodology for reducing the order of large scale interval systems. The algorithm is computationally simple, and intuitively appealing. Numerical examples illustrating the effectiveness of the proposed method are included.
18 citations
Cites methods from "Control of large-scale dynamic syst..."
...3 3 2 [1, 2] [3,4] [8,10] ( ) [6,6] [9,9....
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...Typical methods for continuous systems are Aggregation technique [1], Pade approximation [2], Routh approximation [3-4], Moment matching [5], truncation method [6] and mixed methods [7]....
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01 Jan 1988
17 citations
Cites background from "Control of large-scale dynamic syst..."
...Another remark concerning AOKI's work is that it focuses on the aggregation procedure itself and not at all on the "diretion" along which aggregation is performed: in [4] and [5], the aggregate variable sets are arbitrary....
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...The particular case of a quadratic objective and a feedback control law u=Kx* based on the aggregate state vector is then investigated by AOKI....
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...For instance, a model described by x = Fx + Gu + Dy, where y is the observed output of the real system is proposed in [4]....
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...AOKI [4],[5], [94] proposes a concrete but restricted formulation for the concept of aggregation in control and explores the problems arising when one tries to reduce the dimensionality of a model (i.e. if one tries to determine a control based on a reduced-size model)....
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...The aggregation techniques proposed by AOKI are mostly intended to retain the dominant modes of the detailed model in the aggregate one and, in that respect, they are perfectly well suited to the needs of multilayer systems....
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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