Closed-loop Formulation Research Articles

Vibration control of flexible structures with an Active Modal Tuned Mass Damper

AbstractIn this paper an active modal TMD control logic for the vibration suppression is presented. Starting from the traditional TMD theory, a closed loop formulation is proposed to calculate the active control force. The possibility to describe every mechanical systems or civil structures with a set of modal coordinates allows to act independently on each modal degree of freedom. The technique is compared with different solutions already present in literature, such as the IMSC and the PPF. Numerical simulations, based on a FEM linear model, are carried out to investigate the pro and con of each logic.

Dynamic pricing and inventory control: robust vs. stochastic uncertainty models—a computational study

In this paper, we consider a variety of models for dealing with demand uncertainty for a joint dynamic pricing and inventory control problem in a make-to-stock manufacturing system. We consider a multi-product capacitated, dynamic setting, where demand depends linearly on the price. Our goal is to address demand uncertainty using various robust and stochastic optimization approaches. For each of these approaches, we first introduce closed-loop formulations (adjustable robust and dynamic programming), where decisions for a given time period are made at the beginning of the time period, and uncertainty unfolds as time evolves. We then describe models in an open-loop setting, where decisions for the entire time horizon must be made at time zero. We conclude that the affine adjustable robust approach performs well (when compared to the other approaches such as dynamic programming, stochastic programming and robust open loop approaches) in terms of realized profits and protection against constraint violation while at the same time it is computationally tractable. Furthermore, we compare the complexity of these models and discuss some insights on a numerical example.

Formulation of Closed-Loop Min–Max MPC as a Quadratically Constrained Quadratic Program

In this note, we show that min-max model predictive control (MPC) for linearly constrained polytopic systems with quadratic cost can be cast as a quadratically constrained quadratic program (QCQP). We use the rigorous closed loop formulation of min-max MPC, and show that any such min-max MPC problem with convex costs and constraints can be cast as a finite dimensional convex optimization problem, with the QCQP arising from quadratic costs as a special case. At the base of the proof is a lemma showing the convexity of the dynamic programming cost-to-go, which implies that the worst case on an infinite polytopic set is assumed on one of its finitely many vertices. As the approach is based on a scenario tree formulation, the number of variables in this problem grows exponentially with the horizon length. Fortunately, the QCQP is tree structured, and can thus be efficiently solved by specially tailored interior-point methods whose computational costs are linear in the number of variables. The new formulation as a tree sparse QCQP promises to facilitate online solution of the rigorous min-max MPC problem with quadratic costs

Design and analysis of nonlinear digital controllers-based two-level hierarchy for electric utility industry

This paper suggests a control strategy of coordinating multiple dynamic-braking units during the transients ensuing major disturbances. The control strategy considered in this study is a two-level hierarchy. The proposed two-level structure results from the decomposition of the overall problem into parallel subproblems. This allows the retention of the closed-loop controls associated with each subsystem, which together constitute the lower level (Level I). The central coordinating controller forms the upper level (Level II). The coordination of the local controllers by the central controller accounts for nonlinear terms and interconnections and yields the global optimization of the overall system transient performance. The local controllers are not dependent on one another and are robust to any changes in the network configuration, due to the feedback or closed-loop control formulation inherent in the proposed strategy. To ensure physical realizability of the local controllers, the input was restricted to locally measurable signals. The methodology was implemented in a prototype software program, which was tested on a single machine connected to a very large network approximated by an infinite bus, and then on the IEEE four-generator test system. These studies considered fault-clearing times greater than the critical, assuring an unstable condition. The well-damped optimal state and control trajectories illustrate the successful solution of the problem, indicating that the technique is a valuable tool dealing with transient control problems for large-scale systems.

On minimum finite length control problem

The article discusses a control problem when the manipulated variable is constrained to be a finite length sequence, i.e. it can modify the behaviour of the controlled system only in a limited number of steps and is zero afterwards. The basic criterion considered is to minimize the number of control steps or to minimize l2 norm of the control signal. Open-loop and closed-loop formulations are presented.

Output feedback system design via inverse systems

Application of polynomial matrices to the representation of multivariable closed-loop control systems has found highly successful results. Many design methodologies have formulated the open-loop and desired closed-loop systems in terms of transfer function matrices, and choice of the proper compensation required for desired objectives result in a synthesis problem. A difficulty in direct application of synthesis methodologies to the feedback problem is in calculation of matrix inverses. This problem may be alleviated by working with the inverse open-loop and desired closed-loop system formulation. This paper presents some results on the inverse systems approach when applied to the output feedback system configuration. Conditions for the existence of compensators necessary to obtain desired overall system objectives are developed and applied to decoupling, model matching and pole placement problems. Examples are provided to illustrate application of the conditions to various design cases.

A sampled data equilibrium Stackelberg coordination scheme for the multicontroller problem

In this paper we consider a multicontroller problem in which each control agent has a different objective function. The actions of the controllers are coordinated by the influence of one controller who plays according to a Stackelberg strategy relative to the remaining players. We investigate this formulation when the information flow is restricted by constraining it to be in the form of sampled data acquisition. The computational advantage of the sampled data formulation is quite significant. Of equal importance is the relationship among the sampled data, closed-loop, and open-loop problems. The existence of and solution for the closed-loop Stackelberg solution for the continuous-time game are, at present, unresolved problems. The primary motivation for considering the sampled data formulation is to obtain a solution which maintains the computational simplicity of the tractable open-loop solution while gaining the responsiveness of a state feedback solution, avoiding the complications of the purely closed-loop formulation. The linear quadratic problem is considered in detail, and an efficient solution algorithm is derived which takes advantage of certain characteristics of the sampled data solution.

Design algorithms for a sensitivity constrained suboptimal regulator

The problem of finding the linear feedback control comprised of state and trajectory sensitivity terms which minimizes an infinite-time quadratic cost functional containing sensitivity variables is discussed. Since trajectory sensitivity cannot be accurately modelled in a closed-loop formulation the control feeds back approximate sensitivity terms. As the minimizing control is dependent on initial conditions, necessary conditions for a minimum are derived which satisfy various alternative sets of initial condition criteria. An efficient computer algorithm, based on a gradient search technique, is proposed for finding the feedback gains. By means of a numerical example it is shown that it is possible to reduce trajectory sensitivity while retaining certain desirable features of the linear optimal regulator design. Finally an algorithm is suggested for finding the linear feedback control comprised of state terms alone which minimizes the augmented cost functional.

Two-Point Boundary Value Problems of Linear Hamiltonian Systems

Previous article Next article Two-Point Boundary Value Problems of Linear Hamiltonian SystemsR. S. BucyR. S. Bucyhttps://doi.org/10.1137/0115121PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Gilbert A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Ill., 1946ix+296 MR0017881 (8,212e) 0063.00459 Google Scholar[2] Robert Hermann, On the accessibility problem in control theoryInternat. Sympos. Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, 1963, 325–332 MR0149402 (26:6891) 0142.36102 CrossrefGoogle Scholar[3] R. E. Kalman and , R. S. Bucy, New results in linear filtering and prediction theory, Trans. ASME Ser. D. J. Basic Engrg., 83 (1961), 95–108 MR0234760 (38:3076) CrossrefGoogle Scholar[4] J. Radon, Zum Problem von Lagrange, Abh. Math. Sem. Univ. Hamburg, 6 (1928), 273–299 CrossrefGoogle Scholar[5] William T. Reid, A matrix equation related to a non-oscillation criterion and Liapunov stability, Quart. Appl. Math., 23 (1965), 83–87 MR0176997 (31:1263) 0132.00701 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Reduction of linear Hamiltonian systems of ordinary differential equations1 Jan 2015 Cross Ref Conservative numerical method for solving static linear boundary value problems of elastic shells of revolutionComputational Continuum Mechanics, Vol. 5, No. 1 | 1 Jan 2012 Cross Ref Hamiltonian approach to studying thin shells of revolution made of composite materialsJournal of Mathematical Sciences, Vol. 176, No. 5 | 6 July 2011 Cross Ref The synthesis of two suboptimal electrohydraulic suspensions, active and semiactive, employing the receding horizon methodNonlinear Analysis: Theory, Methods & Applications, Vol. 30, No. 4 | 1 Dec 1997 Cross Ref Invariant Subspace Methods for the Numerical Solution of Riccati EquationsThe Riccati Equation | 1 Jan 1991 Cross Ref Decentralized control of a class of nonlinear time‐varying interconnected systemsJournal of the Chinese Institute of Engineers, Vol. 14, No. 1 | 1 Jan 1991 Cross Ref Estimate of boundary layer thickness for linear, singularly perturbed two-point boundary-value problemsJournal of Optimization Theory and Applications, Vol. 63, No. 1 | 1 Oct 1989 Cross Ref Asymptotic solution of one class of singularly perturbed optimal control problemsJournal of Applied Mathematics and Mechanics, Vol. 47, No. 3 | 1 Jan 1983 Cross Ref Singular perturbations and optimal controlMathematical Control Theory | 26 August 2006 Cross Ref A modified quadratic cost problem and feedback stabilization of a linear systemIEEE Transactions on Automatic Control, Vol. 22, No. 5 | 1 Oct 1977 Cross Ref Singular perturbations and singular arcs--Part IIIEEE Transactions on Automatic Control, Vol. 22, No. 3 | 1 Jun 1977 Cross Ref Asymptotic Expansions of Singularly Perturbed Quasi-Linear Optimal SystemsPeddapullaiah SannutiSIAM Journal on Control, Vol. 13, No. 3 | 18 July 2006AbstractPDF (1466 KB)Singular perturbations and singular arcs--Part IIEEE Transactions on Automatic Control, Vol. 20, No. 2 | 1 Apr 1975 Cross Ref The Singularly Perturbed Linear State Regulator Problem. IIR. E. O’Malley and C. F. KungSIAM Journal on Control, Vol. 13, No. 2 | 18 July 2006AbstractPDF (854 KB)Asymptotic series solution of singularly perturbed optimal control problemsAutomatica, Vol. 10, No. 2 | 1 Mar 1974 Cross Ref An Historical Survey of Computational Methods in Optimal ControlE. PolakSIAM Review, Vol. 15, No. 2 | 2 August 2006AbstractPDF (3413 KB)Closed Loop Formulations of Optimal Control Problems for Minimum SensitivityControl and Dynamic Systems | 1 Jan 1973 Cross Ref ReferencesRiccati Differential Equations | 1 Jan 1972 Cross Ref ReferencesComputational Methods in Optimization: A Unified Approach | 1 Jan 1971 Cross Ref Solution of Ill-Conditioned linear Two-Point Boundary Value Problems by the Riccati TransformationI. H. Mufti, C. K. Chow, and F. T. StockSIAM Review, Vol. 11, No. 4 | 18 July 2006AbstractPDF (311 KB)On optimal terminal controlIEEE Transactions on Automatic Control, Vol. 14, No. 5 | 1 Oct 1969 Cross Ref Global theory of the Riccati equationJournal of Computer and System Sciences, Vol. 1, No. 4 | 1 Dec 1967 Cross Ref Volume 15, Issue 6| 1967SIAM Journal on Applied Mathematics History Submitted:14 November 1966Published online:17 February 2012 InformationCopyright © 1967 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0115121Article page range:pp. 1385-1389ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics