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: In this paper, a generalized quarter-car model is proposed, incorporating both the frame as well as other-wheel ground contacts, using Laplace transforms, involving vertical motions of key points of interest and has intermediate complexity with improved realism.
Abstract: Quarter-car models are popular, simple, unidirectional in kinematics and enable quicker computation than full-car models. However, they do not account for three other wheels and their suspensions, nor for the frame’s flexibility, mass distribution and damping. Here we propose a generalized quarter-car modelling approach, incorporating both the frame as well as other-wheel ground contacts. Our approach is linear, uses Laplace transforms, involves vertical motions of key points of interest and has intermediate complexity with improved realism. Our model uses baseline suspension parameters and responses to step force inputs at suspension attachment locations on the frame. Subsequently, new suspension parameters and unsprung mass compliance parameters can be incorporated, for which relevant formulas are given. The final expression for the transfer function, between ground displacement and body point response, is approximated using model order reduction. A simple Matlab code is provided that enables quick parametric studies. Finally, a parametric study and wheel hop analysis are performed for a realistic numerical example. Frequency and time domain responses obtained show clearly the effects of other wheels, which are outside the scope of usual quarter-car models. The displacements obtained from our model are compared against those of the usual quarter-car model and show ways in which predictions of the quarter-car model include errors that can be reduced in our approach. In summary, our approach has intermediate complexity between that of a full-car model and a quarter-car model, and offers corresponding intermediate detail and realism.
8 citations
TL;DR: In this paper, it was shown that for continuous-time systems the class includes an infinite set, whereas for discrete-time the class has only two members, and that such systems form a class, and parameterize the class.
8 citations
01 Jan 2012
TL;DR: This dissertation focuses on the study of distributed control of interconnected multi-agent systems, and a distributed control design methodology is proposed, which realizes neighbor-based local controller for each agent.
Abstract: From ecology and evolutionary biology to social sciences, and from systems and control theory to aerospace and wireless sensor networks, researchers have been trying to develop an understanding of how a group of moving objects such as flocks of birds, schools of fish and crowds of people can perform collective tasks such as reaching a consensus or moving in a formation without centralized coordination. Researchers in the fields of robotics and control theory have also become interested in cooperative control of multi-agent systems such as a group of unmanned vehicles due to its vast variety of applications. This dissertation focuses on the study of distributed control of interconnected multi-agent systems. Research is conducted towards designing and analyzing distributed control algorithms for a fixed interconnection pattern as well as in the presence of switching network topology, interconnection delays between agents, and random link failure. A decomposition approach is considered to design distributed controller for coupled dynamic systems for both fixed and switching network topology. A special similarity transformation constructed from interconnection pattern matrix along with the result of extended linear matrix inequality (LMI) formulation for continuous-time systems makes it possible to derive explicit expression for computing the parameters of distributed controller for both static state feedback and dynamic output feedback cases. The distributed controllers are designed under H2, H∞ and α-stability performances. In the presence of interconnection delays, a distributed control design methodology is proposed, which realizes neighbor-based local controller for each agent. By transforming the original system into a decomposed structure and taking advantage of Lyapunov-Krasovskii functional, LMI based conditions are derived to design both v static state feedback and dynamic output feedback controllers. Moreover, multi-agent dynamic systems with random link failure are modeled by a linear discrete-time system with multiplicative random coefficients. A distributed control algorithm which uses only local and available neighboring information is proposed to stabilize the system in mean-square sense. A sufficient condition for designing stabilizing distributed controller is provided that ensures prescribed disturbance attenuation in L2 gain sense. The design is carried out using linear matrix inequalities. The problem of designing decentralized PI-observer based controllers is also considered in the convex optimization context for interconnected systems with linear subsystems and nonlinear time-varying interconnections. When the interconnections are linear, distributed PI-observer is introduced for disturbance estimation and fault detection. Finally, distributed controller based on linear quadratic regulator (LQR) is proposed for identical decoupled linear systems with a common objective. Two types of distributed dynamic output feedback algorithms are developed with local observers and distributed observers. The design provides a straightforward way to construct controller and observer gains that are decoupled from the communication graph structure.
8 citations
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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