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
31 Mar 1996
TL;DR: A method for structured linear uncertain system reduction is presented and it is shown that the reduced model is obtained by retaining the components which are contributing most to impulse response energy.
Abstract: In this paper a method for structured linear uncertain system reduction is presented. The reduced model is obtained by retaining [/spl beta//sub 1//sup -/, /spl beta//sub 1//sup +/] and [/spl alpha//sub 1//sup -/, /spl alpha//sub 1//sup +/] (for the steady state of reduced model and original system be equal) along with [/spl beta//sub 1//sup -/, /spl beta//sub 1//sup +/] and [/spl alpha//sub 1//sup -/, /spl alpha//sub 1//sup +/] for (i=2, 3, ..., r-1) which are contributing most to impulse response energy [I/sub s//sup -/, I/sub s//sup +/]. A numerical example illustrates the procedure.
4 citations
01 Jan 2014
TL;DR: In this article, an enhanced modal dominancy approach for reduction of second-order systems is presented, where a modal reduction approach is combined with optimality considerations such that the difference between the frequency response function of the full and reduced modal model is minimized in \(\mathcal{H}_{2}\) sense.
Abstract: The strength of the modal based reduction approach resides in its simplicity, applicability to treat moderate-size systems and also in the fact that it preserves the original system pole locations. However, the main restriction has been in the lack of reliable techniques for identifying the modes that dominate the input-output relationship. To address this problem, an enhanced modal dominancy approach for reduction of second-order systems is presented. Briefly stated, a modal reduction approach is combined with optimality considerations such that the difference between the frequency response function of the full and reduced modal model is minimized in \(\mathcal{H}_{2}\) sense. A modal ranking process is performed without solving Lyapunov equations. In the first part of this study, a literature survey on different model reduction approaches and a theoretical investigation of the modified modal approach is presented. The error analysis of the proposed dominancy metric is carried out. Furthermore, the performance of the method is validated for a lightly damped structure and the results are compared with other dominancy metrics. Finally the optimality of the obtained reduced model is discussed and the results are compared with the optimum solution.
4 citations
TL;DR: In this paper, the authors explored the application of balanced truncation technique to obtain a reduced order model from the original high-order model of the Advanced Heavy Water Reactor (AHWR).
4 citations
Cites methods from "Control of large-scale dynamic syst..."
...Thereafter, we calculate the similarity transformation T in (10) such that the reachability and observability Gramians in the transformed coordinate system are diagonal and equal....
[...]
...In the past few decades, several analytical model reduction techniques have been proposed, such as retaining of the dominant modes (Davison, 1966; Marshall, 1966), model reduction by aggregation (Aoki, 1968) and decomposition of higher order model into slow and fast systems by two-time-scale methods and singular perturbation analysis (Kokotovic et al., 1976), etc. These methods dealt with the eigenvalues of the system and require the assessment of dominant modes present in the model. Various other methods such as balanced truncation, balancing free technique, etc., are also available for model order reduction. For the state-space models, model order reduction method based on the assessment of degree of controllability and observability has been suggested in Moore (1981) and Pernebo and Silverman (1982) which is popularly known as balanced truncation....
[...]
...Reachability and observability Gramians play a major role in obtaining system balancing transformation....
[...]
...This approach may turn out to be numerically inefficient and ill-conditioned as the Gramians WR and WO often have numerically low rank i.e., the eigenvalues ofWR and WO decay rapidly....
[...]
...It can be achieved by simultaneously diagonalizing the reachability and the observability Gramians (Laub et al., 1987), which are the solutions to reachability and observability Lyapunov equations. The positive decreasing diagonal entries in the diagonal reachability and observability Gramians in the new basis are called the Hankel singular values of the system. The reduced order model is obtained simply by truncation of the states corresponding to the smallest singular values. The number of states that can be truncated depends on how accurate the approximate model should be. There are some other techniques to obtain the balanced truncation viz., Schur method (Safonov and Chiang, 1989), balance square root method (Varga, 1991) similar to Moore (1981), however, they differ in the algorithms to obtain the balancing transformation....
[...]
11 Dec 2019
TL;DR: A heuristic algorithm is provided to identify clusters that are not only suboptimal but are also connected, that is, each cluster forms a connected induced subgraph in the network system.
Abstract: A model reduction technique is presented that identifies and aggregates clusters in a large-scale network system and yields a reduced model with tractable dimension. The network clustering problem is translated to a graph reduction problem, which is formulated as a minimization of distance from lumpability. The problem is a non-convex, mixed-integer optimization problem and only depends on the graph structure of the system. We provide a heuristic algorithm to identify clusters that are not only suboptimal but are also connected, that is, each cluster forms a connected induced subgraph in the network system.
4 citations
Cites background from "Control of large-scale dynamic syst..."
...We associate this condition to [12], which to our knowledge is the earliest to provide it....
[...]
...system are projected on a lower-dimensional state space, [12], which yields a projected system whose state vector contains the aggregated states of the clusters....
[...]
01 Jan 2021
TL;DR: In this article, a new approach of model reduction for complex linear time-invariant (LTI) systems is offered by using the merits of big bang big crunch (BBBC) algorithm and modified eigen permutation (MEP) approach.
Abstract: A new approach of model reduction for complex linear time-invariant (LTI) systems is offered in this chapter by using the merits of big bang big crunch (BBBC) algorithm and modified eigen permutation (MEP) approach. The BBBC optimization utilizes big bang phase followed by the big crunch phase. The random points are formed in the big bang phase, and these random solutions converge towards a representative point in the big crunch phase. The proposed approach ensures the stability of the approximant for a stable higher-order model (HOM). The numerical examples from multiple inputs–multiple outputs (MIMO) and single input–single output (SISO) models are included to show the superiority of the proposed approach over prevailing approaches.
4 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