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 article, a unified treatment of aggregability, lumpability, coherency, reversibility, partial balance and similar properties appearing in different fields including power systems and queueing theory is presented.
Abstract: This paper presents a unified treatment of aggregability, lumpability, coherency, reversibility, partial balance and similar properties appearing in different fields including power systems and queueing theory. A coherency condition, well known in power systems, implies the existence of a finite state filter for Markov chains, while aggregability and coherency yield a new condition for decentralized computation of the invariant measure.
20 citations
TL;DR: In this article, the authors proposed a new robust control technique which combines a model predictive control (MPC) and linear quadratic gaussian (LQG) approach to support the frequency stability of modern power systems.
Abstract: This work presents a new robust control technique which combines a model predictive control (MPC) and linear quadratic gaussian (LQG) approach to support the frequency stability of modern power systems. Moreover, the constraints of the proposed robust controller (MPC-LQG) are fine-tuned based on a new technique titled Chimp optimization algorithm (ChOA). The effectiveness of the proposed robust controller is tested and verified through a multi-area power system (i.e., single-area and two-area power systems). Each area contains a thermal power plant as a conventional generation source considering physical constraints (i.e. generation rate constraint, and governor dead band) in addition to a wind power plant as a renewable resource. The superiority of the proposed robust controller is confirmed by contrasting its performance to that of other controllers which were used in load frequency control studies (e.g., conventional integral and MPC). Also, the ChOA’s ingenuity is verified over several other powerful optimization techniques; particle swarm optimization, gray wolf optimization, and ant lion optimizer). The simulation outcomes reveal the effectiveness as well as the robustness of the proposed MPC-LQG controller based on the ChOA under different operating conditions considering different load disturbances and several penetration levels of the wind power.
20 citations
10 Dec 2012
TL;DR: Based on a notion of network clustering, a state aggregation method for positive systems evolving over directed networks, which are called positive networks is proposed and an ℌ2-error bound of the state discrepancy caused by the aggregation is derived.
Abstract: In this paper, based on a notion of network clustering, we propose a state aggregation method for positive systems evolving over directed networks, which we call positive networks. In the proposed method, we construct a set of clusters (i.e., disjoint sets of state variables) according to a kind of local uncontrollability of systems. This method preserves interconnection topology among clusters as well as stability and some particular properties, such as system positivity and steady-state characteristic (steady-state distribution). In addition, we derive an ℌ 2 -error bound of the state discrepancy caused by the aggregation. The efficiency of the proposed method is shown through the reduction of a chemical master equation representing the time evolution of the Michaelis-Menten chemical reaction system.
20 citations
TL;DR: In this article, an overall evaluation of the computational efficiency of two decentralized optimal control approaches, namely, the non-singular perturbation and the hierarchical control, is performed in order to provide the grounds for a quantitative comparison between them.
Abstract: An overall evaluation of the computational efficiency of two decentralized optimal control approaches, namely, the non-singular perturbation and the hierarchical control approaches, is performed in order to provide the grounds for a quantitative comparison between them. The linear-quadratic regulator problem is adopted as the vehicle to conduct the comparative study. Major conclusions drawn from this comparison are the computational superiority, wide applicability and potential flexibility of the hierarchical control approach over the non-singular perturbation approach.
20 citations
TL;DR: In this paper, the authors provide a methodology for designing reduced-order controllers for large-scale, linear systems represented by differential equations having time-periodic coefficients using Lyapunov-Floquet transformation.
Abstract: This paper provides methodology for designing reduced-order controllers for large-scale, linear systems represented by differential equations having time-periodic coefficients. The linear time-periodic sys tem is first converted into a form in which the system stability matrix is time invariant. This is achieved by the application of Lyapunov-Floquet transformation. Then a completely time-invariant auxiliary system is constructed and order reduction algorithms are applied to this system to obtain a reduced-order system. The control laws are calculated for the reduced-order system by minimizing the least square error between the auxiliary and the transformed system. These control laws are then transformed to obtain the desired control action in the original domain. The schemes formulated arc illustrated by designing full-state feedback and output feedback controllers for a five-mass inverted pendulum exhibiting parametric instability
19 citations
Cites methods from "Control of large-scale dynamic syst..."
...In this paper, we apply aggregation (Aoki, 1968) that comprises the linear transformation of auxiliary state z by a time-invariant transformation T....
[...]
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