a numerical algorithm for optimal control of a class of hybrid systems differential transformation based approach a novel numerical algorithm based on differential transformation is proposed for optimal control of a class of hybrid systems with a predefined mode sequence from the necessary conditions for optimality of hybrid systems the hybrid optimal control problem is first converted into a two point boundary value problem tpbvp with additional transverse conditions at the switching times then we propose a differential transformation algorithm for solving the tpbvp which may have discontinuities in the state and or control input at the switching times using differential transformation the hybrid optimal control problem reduces to a problem of solving a system of algebraic equations the numerical solution is obtained in the form of a truncated taylor series by taking advantage of the special properties of the linear subsystems and a quadratic cost functional the differential transformation algorithm can be further simplified for the switched linear quadratic optimal control problem we analyse the error of

numerical solution of the coupled mode equations in duct acoustics we present a fast solver for the coupled mode equations in duct acoustics it is based on a partitioning of the resulting system of ordinary differential equations into separate subsystems the number of which increases by the separability of the problem this structure is obtained by transformations of the dependent variable such that weakly interacting modes are separated the decoupling process requires the numerical solution of algebraic riccati equations however these computations can be done on a spatial scale much larger than the characteristic wavelength of the problem 23 refs 7 figs 1 tab

brief paper linear quadratic switching control with switching cost we study in this paper the linear quadratic lq optimal control problem of discrete time switched systems with a constant switching cost for both finite and infinite time horizons we reduce these problems into an auxiliary problem which is an lq optimal switching control problem with a cardinality constraint on the total number of switchings based on the solution structure derived from the dynamic programming dp procedure we develop a lower bounding scheme by exploiting the monotonicity of the riccati difference equation integrating such a lower bounding scheme into a branch and bound bnb framework we offer an efficient numerical solution scheme for the lq switching control problem with switching cost

an explicit fourth order orthogonal curvilinear staggered grid fdtd method for maxwell s equations the explicit fourth order staggered finite difference time domain scheme previously proposed for a cartesian grid is extended to maxwell s equations in an orthogonal curvilinear coordinate system and applied to electromagnetic wave problems a simple technique is also presented for generating orthogonal curvilinear grids that conform to the material boundaries and interfaces of the problem numerical experiments are presented to illustrate the efficiency and accuracy of the method

transition time optimization for switched mode dynamical systems this note considers the problem of determining optimal switching times at which mode transitions should occur in multimodal hybrid systems it derives a simple formula for the gradient of the cost functional with respect to the switching times and uses it in a gradient descent algorithm much of the analysis is carried out in the setting of optimization problems involving fixed switching mode sequences but a possible extension is pointed out for the case where the switching mode sequence is a part of the variable numerical examples testify to the viability of the proposed approach

optimal control of switched systems based on parameterization of the switching instants this paper presents a new approach for solving optimal control problems for switched systems we focus on problems in which a prespecified sequence of active subsystems is given for such problems we need to seek both the optimal switching instants and the optimal continuous inputs in order to search for the optimal switching instants the derivatives of the optimal cost with respect to the switching instants need to be known the most important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives based on the solution of a two point boundary value differential algebraic equation formed by the state costate stationarity equations the boundary and continuity conditions along with their differentiations this method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed an extension of the method is also applied to problems with internally forced switching examples are shown to illustrate the results in the paper

helmholtz and parabolic equation solutions to a benchmark problem in ocean acoustics the helmholtz equation he describes wave propagation in applications such as acoustics and electromagnetics for realistic problems solving the he is often too expensive instead approximations like the parabolic wave equation pe are used for low frequency shallow water environments one persistent problem is to assess the accuracy of the pe model in this work a recently developed he solver that can handle a smoothly varying bathymetry variable material properties and layered materials is used for an investigation of the errors in pe solutions in the he solver a preconditioned krylov subspace method is applied to the discretized equations the preconditioner combines domain decomposition and fast transform techniques a benchmark problem with upslope downslope propagation over a penetrable lossy seamount is solved the numerical experiments show that for the same bathymetry a soft and slow bottom gives very similar he and pe solutions whereas the pe model is far from accurate for a hard and fast bottom a first attempt to estimate the error is made by computing the relative deviation from the energy balance for the pe solution this measure gives an indication of the magnitude of the error but cannot be used as a strict error bound

optimal control of continuous time switched affine systems this paper deals with optimal control of switched piecewise affine autonomous systems where the objective is to minimize a performance index over an infinite time horizon we assume that the switching sequence has a finite length and that the decision variables are the switching instants and the sequence of operating modes we present two different approaches for solving such an optimal control problem the first approach iterates between a procedure that finds an optimal switching sequence of modes and a procedure that finds the optimal switching instants the second approach is inspired by dynamic programming and identifies the regions of the state space where an optimal mode switch should occur therefore providing a state feedback control law

optimal control of switching systems this paper considers an optimal control problem for a switching system for solving this problem we do not make any assumptions about the number of switches nor about the mode sequence they are determined by the solution of the problem the switching system is embedded into a larger family of systems and the optimization problem is formulated for the latter it is shown that the set of trajectories of the switching system is dense in the set of trajectories of the embedded system the relationship between the two sets of trajectories 1 motivates the shift of focus from the original problem to the more general one and 2 underlies the engineering relevance of the study of the second problem sufficient and necessary conditions for optimality are formulated for the second optimization problem if they exist bang bang type solutions of the embedded optimal control problem are solutions of the original problem otherwise suboptimal solutions are obtained via the chattering lemma

one way large range step methods for helmholtz waveguides a useful approach for long range computation of the helmholtz equation in a waveguide is to re formulate it as the operator differential riccati equation for the dirichlet to neumann map for waveguides with slow range dependence the piecewise range independent approximation is used to derive a second order range stepping method for this one way re formulation the range marching formulas are exact for each range independent piece and a large range step is possible if the range dependence is gradual based on a fourth order conservative exponential method for linear evolution equations a fourth order method that admits even larger range steps is developed for the one way re formulation numerical examples are used to demonstrate the improved accuracy of the fourth order method

mmse interference suppression for direct sequence spread spectrum cdma we consider interference suppression for direct sequence spread spectrum code division multiple access cdma systems using the minimum mean squared error mmse performance criterion the conventional matched filter receiver suffers from the near far problem and requires strict power control typically involving feedback from receiver to transmitter for acceptable performance multiuser detection schemes previously proposed mitigate the near far problem but are complex and require explicit knowledge or estimates of the interference parameters in this paper we present and analyze several new mmse interference suppression schemes which have the advantage of being near far resistant to varying degrees depending on their complexity and can be implemented adaptively when interference parameters are unknown and or time varying numerical results are provided that show that these schemes offer significant performance gains relative to the matched filter receiver we conclude that mmse detectors can alleviate the need for stringent power control in cdma systems and may be a practical alternative to the matched filter receiver

analysis of axial vibration of compound bars by differential transformation method although there are solutions for the axial vibration of compound bars i e bars of different cross section connected in mechanical series this note is to illustrate another solution the primary objective is to analyze this problem by a relatively new technique originated in 1986 and known as the differential transformation dt method the numerical solutions are compared with classical exact solution and other studies in the literature the accuracy simplicity and effectiveness of the dt approach are demonstrated

piecewise linear quadratic optimal control the use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control lower bounds on the optimal control cost are obtained by semidefinite programming based on the bellman inequality this also gives an approximation to the optimal control law an upper bound to the optimal cost is obtained by another convex optimization problem using the given control law a compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy

convex dynamic programming for hybrid systems a classical linear programming approach to optimization of flow or transportation in a discrete graph is extended to hybrid systems the problem is finite dimensional if the state space is discrete and finite but becomes infinite dimensional for a continuous or hybrid state space it is shown how strict lower bounds on the optimal loss function can be computed by gridding the continuous state space and restricting the linear program to a finite dimensional subspace upper bounds can be obtained by evaluation of the corresponding control laws

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

linear multiuser detectors for synchronous code division multiple access channels under the assumptions of symbol synchronous transmissions and white gaussian noise the authors analyze the detection mechanism at the receiver comparing different detectors by their bit error rates in the low background noise region and by their worst case behavior in a near far environment where the received energies of the users are not necessarily similar optimum multiuser detection achieves important performance gains over conventional single user detection at the expense of computational complexity that grows exponentially with the number of users it is shown that in the synchronous case the performance achieved by linear multiuser detectors is similar to that of optimum multiuser detection attention is focused on detectors whose linear memoryless transformation is a generalized inverse of the matrix of signature waveform crosscorrelations and on the optimum linear detector it is shown that the generalized inverse detectors exhibit the same degree of near far resistance as the optimum multiuser detectors the optimum linear detector is obtained

spectral efficiency of cdma with random spreading the cdma channel with randomly and independently chosen spreading sequences accurately models the situation where pseudonoise sequences span many symbol periods furthermore its analysis provides a comparison baseline for cdma channels with deterministic signature waveforms spanning one symbol period we analyze the spectral efficiency total capacity per chip as a function of the number of users spreading gain and signal to noise ratio and we quantify the loss in efficiency relative to an optimally chosen set of signature sequences and relative to multiaccess with no spreading white gaussian background noise and equal power synchronous users are assumed the following receivers are analyzed a optimal joint processing b single user matched filtering c decorrelation and d mmse linear processing

a fast algorithm for the electromagnetic scattering from a large cavity a fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane the medium inside the cavity is assumed to be vertically layered by introducing a transparent artificial boundary condition the problem in the open cavity is reduced to a bounded domain problem a simple finite difference method is then applied to solve the model helmholtz equation the fast algorithm is designed for solving the resulting discrete system in terms of the discrete fourier transform in the horizontal direction a gaussian elimination in the vertical direction and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system the existence and uniqueness of the finite difference solution are established for arbitrary wave numbers our numerical experiments for large numbers of mesh points up to 16 million unknowns and for large wave numbers e g between 100 and 200 wavelengths show that the algorithm is extremely efficient the cost for calculating the radar cross section which is of significant interest in practice is o m2 for an m times m mesh the proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

a domain decomposition method for the helmholtz equation in a multilayer domain the two dimensional helmholtz equation for problems where the physical domain consists of layers with different material properties is studied an efficient preconditioner for iterative solution of the problem is constructed the problem is discretized with fourth order accurate finite difference operators for the construction of the radiation boundary conditions a fourth order finite element method also is used the large sparse complex indefinite and ill conditioned system of equations that arises is solved with preconditioned restarted gmres a domain decomposition method is used in which the preconditioning is based on the schur complement algorithm with fast poisson type preconditioners for the subdomains the memory requirements for the preconditioner are nearly linear in the number of unknowns the arithmetic complexity for each iteration is low whereas the construction of the preconditioner is a bit more expensive electromagnetic wave propagation in a three layered waveguide is used as a model problem numerical experiments show that convergence is achieved in a few iterations compared with banded gaussian elimination which is a standard solution method the iterative method shows significant gain in both memory requirements and arithmetic complexity furthermore the relative gain grows when the problem size increases

on solving the initial value problems using the differential transformation method in this paper initial value problems are solved by the differential transformation method the differential transformation method of fixed grid size is used to approximate solutions of linear and nonlinear initial value problems an adaptive grid size mechanism based on the fixed grid size technique is also proposed the proposed adaptive grid size procedure provides concise adjustment policy and raises computational efficiency of using differential transformation method

separation of variables in deformed cylinders 327 abstract we study the laplace operator subject to dirichlet boundary conditions in a two dimensional domain that is one to one mapped onto a cylinder rectangle or infinite strip as a result of this transformation the original eigenvalue problem is reduced to an equivalent problem for an operator with variable coefficients taking advantage of the simple geometry we separate variables by means of the fourier decomposition method the ode system obtained in this way is then solved numerically yielding the eigenvalues of the operator the same approach allows us to find complex resonances arising in some noncompact domains we discuss numerical examples related to quantum waveguide problems the aim of these experiments is to compare the method based on the separation of variables with the standard finite volume procedure for the most computationally difficult examples related to domains with narrow throats one can clearly see the advantages of the proposed method

on the average near far resistance for mmse detection of direct sequence cdma signals with random spreading the performance of a near far resistant finite complexity minimum mean squared error mmse linear detector for demodulating direct sequence ds code division multiple access cdma signals is studied assuming that the users are assigned random signature sequences we obtain tight upper and lower bounds on the expected near far resistance of the mmse detector averaged over signature sequences and delays as a function of the processing gain and the number of users since the mmse detector is optimally near far resistant these bounds apply to any multiuser detector that uses the same observation interval and sampling rate the lower bound on near far resistance implies that even without power control linear multiuser detection provides near far resistant performance for a number of users that grows linearly with the processing gain

two dimensional differential transform for partial differential equations the differential transform is a numerical method for solving differential equations in this paper we present the definition and operation of the two dimensional differential transform a distinctive feature of the differential transform is its ability to solve linear and nonlinear differential equations partial differential equation of parabolic hyperbolic elliptic and nonlinear types can be solved by the differential transform we demonstrate that the differential transform is a feasible tool for obtaining the analytic form solutions of linear and nonlinear partial differential equation

acoustic transmission in curved ducts with varying cross sections methods for predicting the acoustic propagation properties of ducts are developed and analysed for ducts that are asymptotically straight with constant crosssection with application to splitter si

a unified framework for hybrid control model and optimal control theory we propose a very general framework that systematizes the notion of a hybrid system combining differential equations and automata governed by a hybrid controller that issues continuous variable commands and makes logical decisions we first identify the phenomena that arise in real world hybrid systems then we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems evolving on continuous variable state spaces and subject to continuous controls and discrete transitions the model captures the identified phenomena subsumes previous models yet retains enough structure to pose and solve meaningful control problems we develop a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework in particular we demonstrate the existence of optimal relaxed and near optimal precise controls and derive generalized quasi variational inequalities that the associated value function satisfies we summarize algorithms for solving these inequalities based on a generalized bellman equation impulse control and linear programming

cdma systems in fading channels admissibility network capacity and power control we study the admissibility and network capacity of imperfect power controlled code division multiple access cdma systems with linear receivers in fading environments in a cdma system a set of users is admissible if their simultaneous transmission does not result in violation of any of their quality of service qos requirements the network capacity is the maximum number of admissible users we consider a single cell imperfect power controlled cdma system assuming known received power distributions we identify the network capacities of single class systems with matched filter mf receivers for both the deterministic and random signature cases we also characterize the network capacity of single class systems with linear minimum mean square error mmse receivers for the deterministic signature case the network capacities can be expressed uniquely in terms of the users signal to interference ratio sir requirements and received power distributions for multiple class systems equipped with mf receivers we find a necessary and sufficient condition on the admissibility for the random signature case but only a sufficient condition for the deterministic signature case we also introduce the notions of effective target sir and effective bandwidth which are useful in determining the admissibility and hence network capacity of an imperfect power controlled system