Showing papers in "Mathematical Problems in Engineering in 1995"

Stability of conditionally invariant sets and controlleduncertain dynamic systems on time scales

Abstract: A basic feedback control problem is that of obtaining some desired stability property from a system which contains uncertainties due to unknown inputs into the system. Despite such imperfect knowledge in the selected mathematical model, we often seek to devise controllers that will steer the system in a certain required fashion. Various classes of controllers whose design is based on the method of Lyapunov are known for both discrete [4], [10], [15], and continuous [3–9], [11] models described by difference and differential equations, respectively. Recently, a theory for what is known as dynamic systems on time scales has been built which incorporates both continuous and discrete times, namely, time as an arbitrary closed sets of reals, and allows us to handle both systems simultaneously [1], [2], [12], [13]. This theory permits one to get some insight into and better understanding of the subtle differences between discrete and continuous systems. We shall, in this paper, utilize the framework of the theory of dynamic systems on time scales to investigate the stability properties of conditionally invariant sets which are then applied to discuss controlled systems with uncertain elements. For the notion of conditionally invariant set and its stability properties, see [14]. Our results offer a new approach to the problem in question.

Economic analysis of production bottlenecks

Abstract: The management of bottlenecks has become a central topic in the planning and control of production systems. In this paper, we critically analyze bottlenecks from an economic perspective. Using a queueing network model, we demonstrate that bottlenecks are inevitable when there are differences in job arrival rates, processing rates, or costs of productive resources. These differences naturally lead to the creation of bottlenecks both for facilities design and demand planning problems. To evaluate bottlenecks from an economic perspective, we develop the notion of an “economic bottleneck,” which defines resources as bottlenecks based on economic, rather than physical, characteristics.

Feedback control of time-delay systems with bounded control and state

Abstract: This paper is concerned with the problem of stabilizing linear time-delay systems under state and control linear constraints. For this, necessary and sufficient conditions for a given non-symmetrical polyhedral set to be positively invariant are obtained. Then existence conditions of linear state feedback control law respecting the constraints are established, and a procedure is given in order to calculate such a controller. The paper concerns memoryless controlled systems but the results can be applied to cases of delayed controlled systems. An example is given.

Mathematical theory of improvability for production systems

Abstract: A mathematical model for continuous improvement processes in production systems is formulated. Both constrained and unconstrained cases are addressed. A solution for the case of a serial production line with finite buffers and a Bernoulli model of machines reliability is given. In particular, it is shown that a production line is unimprovable under constraints if each buffer is on the average half full and each machine has equal probability of blockages and starvations. Based on this result, guidelines for continuous improvement processes are formulated.

System-theoretic analysis of due-time performance in production systems

Abstract: Along with the average production rate, the due-time performance is an important characteristic of manufacturing systems. Unlike the production rate, the due-time performance has received relatively little attention in the literature, especially in the context of large volume production. This paper is devoted to this topic. Specifically, the notion of due-time performance is formalized as the probability that the number of parts produced during the shipping period reaches the required shipment size. This performance index is analyzed for both lean and mass manufacturing environments. In particular, it is shown that, to achieve a high due-time performance in a lean environment, the production system should be scheduled for a sufficiently small fraction of its average production rate. In mass production, due-time performance arbitrarily close to one can be achieved for any scheduling practice, up to the average production rate.

A unified approach to fixed-order controller design via linear matrix inequalities

Abstract: We consider the design of fixed-order (or low-order) linear controllers which meet certain performance and/or robustness specifications. The following three problems are considered; covariance control as a nominal performance problem,𝒬-stabilization as a robust stabilization problem, and robust L∞ control problem as a robust performance problem. All three control problems are converted to a single linear algebra problem of solving a linear matrix inequality (LMI) of the type BGC

A modeling and control framework for operating large-scale electric power systems under present and newly evolving competitive industry structures

Abstract: This paper introduces a systematic, structure-based modeling framework for analysis and control of electric power systems for processes evolving over the mid-term and long-term time horizons. Much simpler models than the detailed dynamics specifically for control design at different hierarchical levels are obtained by applying both temporal and spatial separation. These simple models, or the aggregate models, represent the net effect of interactions among interconnected regions on specific hierarchical levels. They are exact, since no assumptions on weak interconnections among the subsystems are made. Moreover they are easily understood in terms of power flows among the regions. The approach is essential for improving present performance of the system. It is also potentially useful in a competitive utility environment in which it is critical to study the interplay between technical and economic processes.

Global asymptotic stability of a passive juggling strategy: A possible parts-feeding method

Abstract: In this paper we demonstrate that a passive vibration strategy can bring a one-degree-of-freedom ball to a specified periodic trajectory from all initial conditions. We draw motivation from the problem of parts feeding in sensorless assembly. We provide simulation results suggesting the relevance of our analysis to the parts feeding problem.

Adaptive band-limited disturbance rejection in linear discrete-time systems

Abstract: The problem of adaptively rejecting a disturbance consisting of a linear combination of sinusoids with unknown and/or time varying frequencies for SISO LTI discrete-time systems is considered. The rejection of the disturbance input is achieved by constructing the set of stabilizing controllers using the Youla parametrization and adjusting the Youla parameter to achieve asymptotic disturbance rejection. The first main result in this paper concerns off-line controller design where a controller that achieves regulation is numerically designed off-line based on the assumption that only the sequence of discrete disturbance input values (as opposed to a model of the disturbance) is available. A least squares based optimization algorithm is used in the controller design. As expected, it is shown, under some mild assumptions, that if the off-line designed controller achieves regulation, then it must include a model of the disturbance input. The second main result concerns on-line controller design where recursive versions of the off-line algorithm used above for controller design are presented and their convergence properties analyzed. Conditions under which the on-line algorithms yield an asymptotic controller that achieves regulation are presented. Conditions both for the case where the disturbance input properties are constant but unknown and for the case where they are unknown and time-varying are given. The on-line controller construction amounts to an adaptive implementation of the Internal Model Principle. The performance robustness of the off-line designed controller in the face of plant model uncertainties is investigated. It is shown, under some mild assumptions, that performance robustness is realized provided internal stability is maintained. The performance of the adaptation algorithms is illustrated through a simulation example.

A nested decomposition algorithm for parallel computations of very large sparse systems

Abstract: In this paper we present a generalization of the balanced border block diagonal (BBD) decomposition algorithm, which was developed for the parallel computation of sparse systems of linear equations. The efficiency of the new procedure is substantially higher, and it extends the applicability of the BBD decomposition to extremely large problems. Examples of the decomposition are provided for matrices as large as 250 , 000 × 250 , 000 , and its performance is compared to other sparse decompositions. Applications to the parallel solution of sparse systems are discussed for a variety of engineering problems.

Necessary conditions of optimality for infinite dimensional uncertain systems

Abstract: In this paper we consider optimal control problem for infinite dimensional uncertain systems. Necessary conditions of optimality are presented under the assumption that the principal operator is the infinitesimal generator of a strongly continuous semigroup of linear operators in a reflexive Banach space. Further, a computational algorithm suitable for computing the optimal policies is also given.

A high-precision algorithm for axisymmetric flow.

Abstract: We present a new algorithm for highly accurate computation of axisymmetric potential flow. The principal feature of the algorithm is the use of orthogonal curvilinear coordinates. These coordinates are used to write down the equations and to specify quadrilateral elements following the boundary. In particular, boundary conditions for the Stokes' stream-function are satisfied exactly. The velocity field is determined by differentiating the stream-function. We avoid the use of quadratures in the evaluation of Galerkin integrals, and instead use splining of the boundaries of elements to take the double integrals of the shape functions in closed form. This is very accurate and not time consuming.

Mathematical problems in modeling artificial heart

Abstract: In this paper we discuss some problems arising in mathematical modeling of artificial hearts. The hydrodynamics of blood flow in an artificial heart chamber is governed by the Navier-Stokes equation, coupled with an equation of hyperbolic type subject to moving boundary conditions. The flow is induced by the motion of a diaphragm (membrane) inside the heart chamber attached to a part of the boundary and driven by a compressor (pusher plate). On one side of the diaphragm is the blood and on the other side is the compressor fluid. For a complete mathematical model it is necessary to write the equation of motion of the diaphragm and all the dynamic couplings that exist between its position, velocity and the blood flow in the heart chamber. This gives rise to a system of coupled nonlinear partial differential equations; the Navier-Stokes equation being of parabolic type and the equation for the membrane being of hyperbolic type. The system is completed by introducing all the necessary static and dynamic boundary conditions. The ultimate objective is to control the flow pattern so as to minimize hemolysis (damage to red blood cells) by optimal choice of geometry, and by optimal control of the membrane for a given geometry. The other clinical problems, such as compatibility of the material used in the construction of the heart chamber, and the membrane, are not considered in this paper. Also the dynamics of the valve is not considered here, though it is also an important element in the overall design of an artificial heart. We hope to model the valve dynamics in later paper.

Development of a Finite-Element-Based Design Sensitivity Analysis for Buckling and Postbuckling of Composite Plates

Abstract: A finite element based sensitivity analysis procedure is developed for buckling and postbuckling of composite plates. This procedure is based on the direct differentiation approach combined with the reference volume concept. Linear elastic material model and nonlinear geometric relations are used. The sensitivity analysis technique results in a set of linear algebraic equations which are easy to solve. The procedure developed provides the sensitivity derivatives directly from the current load and responses by solving the set of linear equations. Numerical results are presented and are compared with those obtained using finite difference technique. The results show good agreement except at points near critical buckling load where discontinuities occur. The procedure is very efficient computationally.

Optimality conditions for systems driven by nonlinear evolution equations

Abstract: Using the Dubovitskii-Milyutin theory we derive necessary and sufficient conditions for optimality for a class of Lagrange optimal control problems monitored by a nonlinear evolution equation and involving initial and/or terminal constraints. An example of a parabolic control system is also included.

Nonlinear singularly perturbed systems of differential equations: A survey

Abstract: In this paper a survey of the most effective methods in singular perturbations is presented. Many applied problems can be modeled by nonlinear singularly perturbed systems, as, for example, problems in kinetics, biochemistry, semiconductors theory, theory of electrical chains, economics, and so on. In this survey we consider averaging and constructive methods that are very useful from the point of view of their numerical and computer realizations.