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: The results from the simulation study illustrate that the proposed QALO based MOR method performs preferably better than other compared techniques, effectively utilized for global optimization and model order reduction.
Abstract: The field of optimization science is proliferating that has made complex real-world problems easy to solve. Metaheuristics based algorithms inspired by nature or physical phenomena based methods have made its way in providing near-ideal (optimal) solutions to several complex real-world problems. Ant lion Optimization (ALO) has inspired by the hunting behavior of antlions for searching for food. Even with a unique idea, it has some limitations like a slower rate of convergence and sometimes confines itself into local solutions (optima). Therefore, to enhance its performance of classical ALO, quantum information theory is hybridized with classical ALO and named as QALO or quantum theory based ALO. It can escape from the limitations of basic ALO and also produces stability between processes of explorations followed by exploitation. CEC2017 benchmark set is adopted to estimate the performance of QALO compared with state-of-the-art algorithms. Experimental and statistical results demonstrate that the proposed method is superior to the original ALO. The proposed QALO extends further to solve the model order reduction (MOR) problem. The QALO based MOR method performs preferably better than other compared techniques. The results from the simulation study illustrate that the proposed method effectively utilized for global optimization and model order reduction.
2 citations
TL;DR: This paper focuses on models based on ordinary differential equations and reviews recent results where abstraction is achieved by aggregation of variables, reflecting on the shortcomings in the state of the art and setting out challenges for future research.
Abstract: Like with most large-scale systems, the evaluation of quantitative properties of collective adaptive systems is an important issue that crosscuts all its development stages, from design (in the case of engineered systems) to runtime monitoring and control. Unfortunately it is a difficult problem to tackle in general, due to the typically high computational cost involved in the analysis. This calls for the development of appropriate quantitative abstraction techniques that preserve most of the system's dynamical behaviour using a more compact representation. This paper focuses on models based on ordinary differential equations and reviews recent results where abstraction is achieved by aggregation of variables, reflecting on the shortcomings in the state of the art and setting out challenges for future research.
2 citations
Cites background from "Control of large-scale dynamic syst..."
...Here, starting with Aoki [1], numerous approaches have considered the transformation of the original model into a reduced one that preserves controllability, i.e., the capability of bringing the system to a desired state by an appropriate choice of control inputs [26]....
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...Here, starting with Aoki [1], numerous approaches have considered the transformation of the original model into a reduced one that preserves controllability, i....
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TL;DR: In this paper, a computation-effective control design procedure for eigenstructure assignment using aggregation is presented, where Dominant eigenvalues are placed at specified locations in the complex plane and the non-dominant eigen values are placed in a specified disk.
Abstract: Design procedures based on exact eigenstructure assignment are not suitable because of very high computational requirements. A computation-effective control design procedure for eigenstructure assignment using aggregation is presented. Dominant eigenvalues are placed at specified locations in the complex plane and the non-dominant eigenvalues are placed in a specified disk. The proposed design procedure is applied to the trajectory tracking problem of a robot manipulator. An error-pattern based payload estimation and compensation scheme is also proposed to improve performance robustness.
2 citations
27 Dec 2001
TL;DR: In this paper, the stability equation method and integral squared error criterion are used together to find the uncertain reduced order model, and lower and upper bounds ci-,ci+ and dj-,dj+ for (i=0,1,...r,-1) and (j=1,2,3,4) of the original uncertain system and uncertain reduced model are found from the coefficients of the error transfer functions.
Abstract: This paper presents a method of reduction for linear structured uncertain system using the integral squared error criterion. The four fixed Kharitonov's polynomials associated with the numerators nsI(s), nmI(s) and denominators dsI(s), dmI(s) of the original uncertain system and uncertain reduced model are obtained. By taking all combinations of the nsk(s), nmk(s) and dsh(s),dsh(s) for (k,h = 1,2,3,4), respectively, we obtain sixteen fixed Kharitonov's systems and sixteen fixed Kharitonov's reduced models. Stability equation method and integral squared error criterion are used together to find the uncertain reduced order model. The stability equation method is used to preserve the stability of the sixteen fixed Kharitonov's systems and original uncertain system by first determining the denominator coefficients of the sixteen fixed Kharitonov's reduced models and uncertain reduced model respectively. The numerators of the sixteen fixed Kharitonov's reduced models are determined so that the integral squared error between the unit step responses of the sixteen fixed Kharitonov's reduced models and the corresponding sixteen fixed Kharitonov's systems are minimum. The sixteen fixed Kharitonov's reduced models tend to approximate the transient portions of the corresponding sixteen fixed Kharitonov's systems in the sense of minimum squared error, while the steady portions of the sixteen fixed Kharitonov's reduced models are matched exactly with that of the corresponding sixteen fixed Kharitonov's systems. Instead of actually evaluating time responses of the sixteen fixed Kharitonov's systems and reduced models, a matrix formulas are used for calculating the integral squared error from the coefficients of the error transfer functions. Finally the lower and upper bounds ci-,ci+ for (i=0,1,...r,-1) and dj-,dj+ for (j=1,2...,r) of the uncertain reduced model are found from the coefficients of the sixteen fixed Kharitonov's reduced models. An illustrative example is included in order to demonstrate the main points.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
2 citations
01 Jan 2018
TL;DR: In this chapter, a variable structure observer is designed for a class of nonlinear large-scale interconnected systems in the presence of uncertainties and nonlinear interconnections based on the Lyapunov direct method.
Abstract: In this chapter, a variable structure observer is designed for a class of nonlinear large-scale interconnected systems in the presence of uncertainties and nonlinear interconnections. The modern geometric approach is used to explore system structure and a transformation is employed to facilitate the observer design. Based on the Lyapunov direct method, a set of conditions are developed such that the proposed variable structure systems can be used to estimate the states of the original interconnected systems asymptotically. The internal dynamical structure of the isolated nominal subsystems as well as the structure of the uncertainties are employed to reduce conservatism. The bounds on the uncertainties are nonlinear and are employed in the observer design to enhance robustness. A numerical example is presented to illustrate the results and simulation studies show that the proposed approach is effective.
2 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