Linear Lumped-parameter Systems Research Articles

Robust Control Strategy for a Conduction–Convection System Based on the Scenario Optimization

This paper deals with the robust control of an uncertain conduction–convection system in the framework of probabilistic control design based on both the geometric control and the scenario optimization approach. Thus, a robust control strategy that copes with parameter uncertainties is proposed for a heated rod taken as an application example of a conduction–convection system. The design approach consists in two steps. In the first step, assuming a nominal model, a state feedback that yields a stable linear lumped parameter system, of first order, in closed loop is designed by means of geometric control theory. The stability of the resulting closed-loop system is demonstrated based on the perturbation theorem from semigroup theory. The second step consists in defining the input reference of the designed state feedback by a structured robust controller. The parameter tuning of the structured controller is formulated as a semi-infinite (or robust) optimization problem which is, then, relaxed using the scenario approach leading to a standard finite optimization problem. The solution of this scenario optimization problem is achieved using a genetic algorithm. The proposed control strategy is adopted to cope with parameter uncertainties in the problem of heating a steel rod. The effectiveness of the proposed robust control strategy is demonstrated by simulation.

Optimal time-delayed boundary control of beams using wavelets

Time delays exist in active control systems. Although a time delay in most cases is small, it results in the actuator applying energy to the control system with a time delay. This might cause instability of the dynamic systems and degrade the performance of the control systems. The objective of the present work is to propose a feasible methodology that can achieve good control performance for a dynamic beam structure system by time-delayed boundary torque actuators. The control problem is to determine optimal time-delayed boundary control by minimizing a given performance criterion. The performance criterion is specified as a weighted quadratic functional of the dynamic responses of the beam which is to be minimized at a specified terminal time using continuous time-delayed torque actuators. The modal expansion approach is used to convert the optimal control problem of a distributed parameter system into the optimal control problem of a linear lumped parameter system (LPS). A direct state-control parameterization approach is proposed where wavelets are employed to solve time-delayed LPS. The operational matrices of integration and delay are utilized to reduce the solution of linear time-varying delayed systems to the solution of algebraic equations. A numerical simulation is presented to illustrate the efficiency of the proposed control algorithm.

Parametric state feedback design of linear distributed-parameter systems

The parametric approach for the design of state feedback controllers has been formulated so far only for linear lumped-parameter systems. It yields an explicit parametric expression for the state feedback gain given the closed-loop eigenvalues and the set of corresponding parameter vectors. This contribution presents a parameterisation of state feedback controllers for linear distributed-parameter systems with scalar state and distributed control. By introducing the closed-loop eigenvalues and the parameter vectors as design parameters, an explicit expression for the state feedback is obtained. In contrast to the pure eigenvalue assignment, the parameterisation allows the assignment not only of the closed-loop eigenvalues but also of the closed-loop eigenfunctions. The usefulness of the proposed parametric approach is demonstrated by decoupling the transfer behaviour of a MIMO diffusion system with respect to its dominant modes.

Study of the Human Body Vibrations for the Seated Vehicle Driver

AbstractThe paper presents some aspects about the dynamic response of the human body subjected to the vertical vibration in the travel inside the auto vehicle. There is a model presented in the specialized literature. This is transformed for the given study because was necessary to realize the vertical vibration along the human body. There are two different studies: using a linear lumped parameter systems seat ‐ human body model; using a nonlinear lumped parameter systems seat‐human body model. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Fundamentals of applied dynamics

Our Niche in the Cosmos. Design, Modeling, and Formulation of Equations of Motion. Kinematics. Momentum Formulation for Systems of Particles. Variational Formulation for Systems of Particles. Dynamics of Systems Containing Rigid Bodies. Dynamics of Electrical and Electromechanical Systems. Vibration of Linear Lumped-Parameter Systems. Dynamics of Continuous Systems. Bibliography. Appendices. Answers to most of the Odd-Numbered Problems. List of Tables. Index.

Recursive methods for parameter estimation of linear lumped systems using Hermite polynomials

A recursive method for estimation of the parameters of a linear lumped parameter system is developed. An example is given to illustrate the algorithm, and results of the estimation in the presence of white noise are also given.

Design criteria for predicting damping in underdamped linear lumped-parameter systems

Simple design criteria that determine whether the modal damping ratios of an underdamped linear lumped-parameter system are greater than, equal to, or less than a given non-negative real scalar are presented. The modal damping ratio expression is an extension of the usual single degree-of-freedom relation. The damping ratio conditions are derived from the modal damping ratio expression and the definiteness of certain combinations of coefficient matrices. An example illustrates how the damping ratio prediction relations are used in design.

A Design Criterion for Avoiding Resonance in Lumped Mass Normal Mode Systems

A simple design criterion that determines whether a normal mode multiple degree of freedom damped linear lumped parameter system can or cannot resonate is presented. The relations are derived based on criteria for resonance in the single degree of freedom case, and on the definiteness of certain combinations of coefficient matrices. An example follows that both numerically verifies the derivation and illustrates the simplicity of implementing the result as a design criterion.

Controllability and observability of general linear lumped-parameter systems

Conclusion A Kalman filter bank is designed to enhance the dynamic response time for the radome error slope estimate with compensation for the seeker dynamic lag. In calculating the critical weighting coefficients (a posteriori probabilities) a measurepredict-measure technique is used when the semi-Markov statistics of a random starting process are used to make the intermediate predictive step. That is, the resulting estimated radome slope parameter is the statistical average weighted by time-varying, a posteriori hypothesis probability, which is calculated concurrently with the recursive filter scheme by using Bayesian rule. To reduce computation burden, the Kalman filter bank is digitally simulated and designed by tuning the noise processes, including the measurement and plant noise, to allow a one-time calculation of the Kalman filter gain. The simple one-state filter described above can be modified to include a correlation parameter for studying the cross-plane errors. The bank of Kalman filters can be increased to five to enlarge the dynamic range while reducing the system response time simultaneously and to estimate the cross-plane radome error slope simultaneously for three-dimensional engagement. The adaptive radome estimator design is intended to be an add-on compensation network that is independent of guidance computer and autopilot design. The objective is to permit relaxation of missile bandwidth requirements by reducing error due to radome at the guidance computer output, thus enhancing missile performance.

On the stability of general dynamic systems using a Liapunov's direct method approach

A technique is presented for studying the stability of equilibrium of linear lumped-parameter systems involving general types of forces such as dissipative, non-conservative, gyroscopic, and circulatory. A modified approach to solving the Liapunov equation is used to provide a function V which is exploited to present different stability criteria for the equilibrium of such systems. It was shown that this approach is advantagous to the solution of the Liapunov equation, since it is more computationally attractive. Additionally, this may be used to determine the effects of different parameters on the systems stability or to design a controller for an actively controlled system. Several previously developed related works are studied and compared with this work. Furthermore, examples are used to illustrate the presented approach and some of its applications.

Some Results on the Nature of Eigenvalues of Discrete Damped Linear Systems

An analysis of the conditions under which the modes of a damped linear lumped parameter system are either all critically damped, overdamped or underdamped is presented. These conditions are derived from the definiteness of certain combinations of the coefficient matrices. Some results concerning the completeness of the eigenvectors are stated. The conditions are compared to previous results and their usefulness is illustrated by numerical examples.

Synthesis of linear lumped-parameter systems in which a mode shape is partially prescribed

The paper presents a means of synthesizing the inertial elements of a linear lumped-parameter system so that the system will freely vibrate at a prescribed frequency with a mode shape which, although not completely defined, satisfies certain specified requirements. Although the problem cannot normally be solved to give a unique result, it is shown that the solution may be expressed in terms of a linear combination of vectors and associated scalar multipliers, which may assume arbitrary values providing that these lead to non-negative inertial elements. The procedure is illustrated by two numerical examples, the second of which demonstrates a means of extracting mode shape admissibility criteria for systems in which the stiffness matrix is known.

Optimal control of distributed-parameter systems with integral equation constraints

A necessary condition for the optimal control of a class of integral equation constraint systems is derived by use of variational method with finite perturbation of the control variable. It is shown that certain linear partial differential equation systems with performance index based on the output variable of the system can be transformed into integral equation constraint problems. An analytical result for the singular control of a distributed-parameter system is obtained. Computational results are given for the optimal control of the thin-wall heat exchanger with steam heating control. When the kernel of the integral equation is e A(t-τ) u(τ), the system can be transformed into a related linear lumped-parameter system. The well-known optimization techniques of lumped-parameter systems can be applied to construct optimal control. Using a quadratic performance index a linear feedback control law can be obtained. Only the measurement of the output variable is required to generate the feedback control law for the servo problem. For the regulator problem the state of the system must be known before the feedback control law can be constructed.

Optimal control for a class of distributed parameter systems where the cost functions are norms

Some problem in optimal control, for a class of linear distributed parameter systems, are posed. By using a theorem of approximation theory, a maximum principle, which is applicable to these optimal control problems, is given. It is shown, that for the problems posed in this paper, this maximum principle is an extension of Butkovskii's maximum principle and of Demyanov and Rubinov's theorem. In addition, two optimal control problems, in linear lumped parameter systems areposed. For these problems it is shown that the maximum principle given in this paper, is an extension of Pontryagin's maximum principle.

On the Stability of Solutions of Linear Differential Equations with Periodic Coefficients

Previous article Next article On the Stability of Solutions of Linear Differential Equations with Periodic CoefficientsI. W. SandbergI. W. Sandberghttps://doi.org/10.1137/0112041PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Jack K. Hale, Oscillations in nonlinear systems, McGraw-Hill Book Co., Inc., New York, 1963, 21– MR0150402 (27:401) 0115.07401 Google Scholar[2] I. W. Sandberg, On the properties of some systems that distort signals-II, BSTJ, (1964), 91–, January Google Scholar[3] J. J. Bongiorno, Jr., An extension of the Nyquist-Barkhausen stability criterion to linear lumped-parameter systems, IEEE-PGAC, (1963), 170–, April Google Scholar[4] R. E. Kalman, Lyapunov functions for the problem of Lur' e in automatic control, Proc. Nat. Acad. Sci. U.S.A., 49 (1963), 201–205 MR0151696 (27:1680) 0113.07701 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Instability of Periodically Time-Varying Linear Control SystemsSIAM Journal on Control, Vol. 6, No. 1 | 18 July 2006AbstractPDF (789 KB) Volume 12, Issue 2| 1964Journal of the Society for Industrial and Applied Mathematics249-497 History Submitted:23 September 1963Published online:28 July 2006 InformationCopyright © 1964 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0112041Article page range:pp. 487-496ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

Analysis of linear distributed systems with random parameters and inputs

This paper is concerned with the statistical analysis of first-order linear distributed-parameter systems with random parameters and random inputs. In both cases, Gaussian processes are assumed. This work essentially extends the statistical theory available for analyzing random linear lumped-parameter systems to a restricted class of distributed-parameter systems.

An extension of the Nyquist-Barkhausen stability criterion to linear lumped-parameter systems with time-varying elements

An extension of the Nyquist-Barkhausen stability criterion to linear lumped-parameter systems with time-varying elements