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.
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TL;DR: This paper proposes a new method for order reduction of higher-order linear time invariant systems based on stability equation method and particle swarm optimization algorithm and the results are compared with well-known methods available in the literature.
Abstract: Most of the physical systems can be represented by mathematical models. The mathematical procedure of system modeling often leads to a comprehensive description of a process in the form of higher-order differential equations which are difficult to use either for analysis or for controller synthesis. It is, therefore, useful and sometimes necessary to find the possibility of some equations of the same type but of lower order that may be considered to adequately reflect almost all essential characteristics of the system under consideration. This paper proposes a new method for order reduction of higher-order linear time invariant systems based on stability equation method and particle swarm optimization algorithm. Reduced-order model will definitely be stable if the original model is stable. The superiority of the proposed method is illustrated by numerical examples of single-input, single-output systems and multiple-input and multiple-output systems. The results are compared with well-known methods available in the literature.
60 citations
TL;DR: In this paper, a new projection method is described that matches q covariances and q Markov parameters of the original system while maintaining the correct dimension of the input vector, but this method requires a factorization to obtain the input matrix, and since the dimension of this matrix factor depends upon rank properties not known a priori, this method may not maintain the original dimension.
Abstract: Covariance equivalent realization theory has been used recently in continuous systems for model reduction [1]-[4] and controller reduction [2], [5] In model reduction, this technique produces a reduced-order model that matches q + 1 output covariances and q Markov parameters of the full-order model In controller reduction, it produces a reduced controller that is "close" to matching q + 1 output covariances of the full-order controller, and q Markov parameters of the closed-loop system For discrete systems, a method was devised to produce a reduced-order model that matches the q + 1 covariances [6], but not any Markov parameters This method requires a factorization to obtain the input matrix, and since the dimension of this matrix factor depends upon rank properties not known a priori, this method may not maintain the original dimension of the input vector Hence, this method [6] is obviously not suitable for controller reduction A new projection method is described that matches q covariances and q Markov parameters of the original system while maintaining the correct dimension of the input vector
58 citations
22 Apr 2017
TL;DR: ERODE supports two recently introduced, complementary, equivalence relations over ODE variables: forward differential equivalence yields a self-consistent aggregate system where each ODE gives the cumulative dynamics of the sum of the original variables in the respective equivalence class.
Abstract: We present ERODE, a multi-platform tool for the solution and exact reduction of systems of ordinary differential equations ODEs. ERODE supports two recently introduced, complementary, equivalence relations over ODE variables: forward differential equivalence yields a self-consistent aggregate system where each ODE gives the cumulative dynamics of the sum of the original variables in the respective equivalence class. Backward differential equivalence identifies variables that have identical solutions whenever starting from the same initial conditions. As back-end ERODE uses the well-known Z3 SMT solver to compute the largest equivalence that refines a given initial partition of ODE variables. In the special case of ODEs with polynomial derivatives of degree at most two covering affine systems and elementary chemical reaction networks, it implements a more efficient partition-refinement algorithm in the style of Paige and Tarjan. ERODE comes with a rich development environment based on the Eclipse plug-in framework offering: i seamless project management; ii a fully-featured text editor; and iii importing-exporting capabilities.
57 citations
Cites methods from "Control of large-scale dynamic syst..."
..., [32,2,27]) using techniques such as abstract interpretation [18,13] and bisimulation [39,19,26,9,12]....
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TL;DR: In this paper, the authors summarized the model reduction techniques used for the reduction of large-scale linear and nonlinear dynamic models, described by the differential and algebraic equations that are commonly used in control theory.
Abstract: This paper summarises the model reduction techniques used for the reduction of large-scale linear and nonlinear dynamic models, described by the differential and algebraic equations that are commonly used in control theory. The groups of methods discussed in this paper for reduction of the linear dynamic model are based on singular perturbation analysis, modal analysis, singular value decomposition, moment matching and methods based on a combination of singular value decomposition and moment matching. Among the nonlinear dynamic model reduction methods, proper orthogonal decomposition, the trajectory piecewise linear method, balancing-based methods, reduction by optimising system matrices and projection from a linearised model, are described. Part of the paper is devoted to the techniques commonly used for reduction (equivalencing) of large-scale power systems, which are based on coherency, synchrony, singular perturbation analysis, modal analysis and identification. Two (most interesting) of the described techniques are applied to the reduction of the commonly used New England 10-generator, 39-bus test power system.
54 citations
Cites methods from "Control of large-scale dynamic syst..."
...In [136], the construction of the newly defined border synchrony chord was combined with the aggregation techniques [11, 56] to provide a structurepreserving reduced model by balancing techniques....
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...The method may be regarded as the combination of SPA and aggregation [56]....
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...Starting from these classes, aggregation methods [11, 56] are used to provide a physical dynamic equivalent (a dynamic equivalent which consists in real machines, regulations, lines, etc....
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TL;DR: In this paper, the relation between the synchrony and the loss of controllability and observability is investigated and, from that, new aggregation methodologies are proposed for two distinct situations.
Abstract: This paper fills the gap between the well-known in control theory model reduction techniques based on the balanced realization and the structure preserving dynamic model equivalencing approaches used in power systems. The relations between the synchrony and the loss of controllability and observability are investigated and, from that, new aggregation methodologies are proposed for two distinct situations. The first one corresponds to the case, already treated in the literature, where a full model is available for the power system which must be reduced. For the second one, which is new, it is considered that part of the data of the power system is not available when the reduction is performed. Both small theoretic and large-scale realistic examples are considered.
52 citations
Cites methods from "Control of large-scale dynamic syst..."
...Structure-preserving reduced models were obtained using the aggregation techniques introduced in [21] and [22]....
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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