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
TL;DR: In this paper, a large number of problems are available for complex higher-order systems, and it is needful to approximate higher order models to lower-order models to solve them.
Abstract: A large number of problems are available for complex higher order systems. These systems are difficult to analyze and synthesize. Therefore it is needful to approximate higher order models to lower...
6 citations
01 Feb 2019-Microsystem Technologies-micro-and Nanosystems-information Storage and Processing Systems
TL;DR: The proposed technique is based on Cuckoo search which is used to obtain unknown elements of the reduced system with an error criterion minimization and the obtained results are compared with other well-known order reduction methods existing in the literature.
Abstract: In this study, a new technique for discrete time system reduction is suggested which preserves the substructure of the higher order system in the reduced system. Motivated by various system reduction and optimization techniques available in the literature, the proposed technique is based on Cuckoo search which is used to obtain unknown elements of the reduced system with an error criterion minimization. The efficacy of the proposed technique is justified by reducing few benchmark systems and the obtained results are compared with other well-known order reduction methods existing in the literature.
6 citations
11 Dec 1996
TL;DR: In this article, the problem of computing an L/sub 2/-optimal reduced-order model for a given stable multivariable linear system is formulated as minimizing the model reduction cost over the Stiefel manifold.
Abstract: This paper deals with the problem of computing an L/sub 2/-optimal reduced-order model for a given stable multivariable linear system. By way of an orthogonal projection the problem is formulated as that of minimizing the L/sub 2/ model reduction cost over the Stiefel manifold so that the stability constraint on reduced-order models is automatically satisfied and thus totally avoided in the new problem formulation. The closed form formula for the gradient of the cost over the manifold is derived, from which a gradient flow is formed as an ordinary differential equation. A number of nice properties about such a flow are obtained. Among them are the decreasing property of the cost along the ODE solution and the convergence of the flow from any starting point in the manifold. Furthermore, an explicit iterative convergent algorithm is developed from the flow and inherits the properties that the iterates remain on the manifold starting from any orthogonal initial point and that the model-reduction cost decreases to minimums along the iterates.
6 citations
TL;DR: In this article, it was shown that stability is preserved under a general projection of the state matrix of a balanced realization provided that the Hankel singular values of the realization are distinct.
6 citations
TL;DR: In this paper , a higher-scale interconnected wind turbine is investigated to realize its analytical design and simulation performance in terms of reduced order model with reference to model order reduction studies, and the authors visualize the efficacy of the propounded harris hawks optimization (HHO) approach for different reduced order transfer functions.
Abstract: This paper illustrates the investigation of higher-scale interconnected wind turbine to realize its analytical design and simulation performance in terms of reduced order model. The present research is aimed to visualize the efficacy of the propounded harris hawks optimization (HHO) approach for different reduced order transfer functions with reference to model order reduction studies. Compared to the higher-scale model, the applied methodology aims to obtain better results in terms of steadiness and computational effort. Moreover, to realize a higher-dimensional power system, the implementation of the HHO may proffer more effectiveness and robustness as opposed to the state-of-the-art optimization techniques. The applied methodology aims to obtain better results in terms of stability and computational effort. Furthermore, to realize a higher dimensional system, implementation of the HHO may offer more effectiveness and robustness as opposed to the state-of-the-art optimization techniques.
6 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