identification of systems containing linear dynamic and static nonlinear elements identification of nonlinear systems which can be represented by combinations of linear dynamic and static nonlinear elements are considered previous results by the authors based on correlation analysis are combined to provide a unified treatment for this class of systems it is shown that systems composed of cascade feedforward feedback and multiplicative connections of linear dynamic and zero memory nonlinear elements can be identified in terms of the individual component subsystems from measurements of the system input and output only

modeling identification and control of a class of nonlinear systems in this paper we propose a new fuzzy hyperbolic model for a class of complex systems which is difficult to model the fuzzy hyperbolic model is a nonlinear model in nature and can be easily derived from a set of fuzzy rules it can also be seen as a feedforward neural network model and so we can identify the model parameters by bp algorithm we prove that the stable controller can be designed based on linear system theory two methods of designing the controller for the fuzzy hyperbolic model are proposed the results of simulation support the effectiveness of the model and the control scheme

two stage orthogonal least squares methods for neural network construction a number of neural networks can be formulated as the linear in the parameters models training such networks can be transformed to a model selection problem where a compact model is selected from all the candidates using subset selection algorithms forward selection methods are popular fast subset selection approaches however they may only produce suboptimal models and can be trapped into a local minimum more recently a two stage fast recursive algorithm tsfra combining forward selection and backward model refinement has been proposed to improve the compactness and generalization performance of the model this paper proposes unified two stage orthogonal least squares methods instead of the fast recursive based methods in contrast to the tsfra this paper derives a new simplified relationship between the forward and the backward stages to avoid repetitive computations using the inherent orthogonal properties of the least squares methods furthermore a new term exchanging scheme for backward model refinement is introduced to reduce computational demand finally given the error reduction ratio criterion effective and efficient forward and backward subset selection procedures are proposed extensive examples are presented to demonstrate the improved model compactness constructed by the proposed technique in comparison with some popular methods

adaptive switching supervisory control of nonlinear systems with no prior knowledge of noise bounds the problem of controlling nonlinear noisy systems affected by parametric uncertainties is approached via the introduction of a supervisor which whenever needed switches on in feedback to the plant a controller selected from a finite set of predesigned controllers a lyapunov based falsification criterion allows one to ensure robust stability in the presence of uncertain constant parameters and exogenous bounded disturbances simulations are discussed in order to illustrate the merits of the proposed algorithms

multistability of neural networks with time varying delays and concave convex characteristics in this paper stability of multiple equilibria of neural networks with time varying delays and concave convex characteristics is formulated and studied some sufficient conditions are obtained to ensure that an n neuron neural network with concave convex characteristics can have a fixed point located in the appointed region by means of an appropriate partition of the n dimensional state space when nonlinear activation functions of an n neuron neural network are concave or convex in 2k 2m 1 intervals this neural network can have 2k 2m 1 n equilibrium points this result can be applied to the multiobjective optimal control and associative memory in particular several succinct criteria are given to ascertain multistability of cellular neural networks these stability conditions are the improvement and extension of the existing stability results in the literature a numerical example is given to illustrate the theoretical findings via computer simulations

parameterized convergence bounds for volterra series expansion of narx models evaluation of the convergence bound in the frequency domain for volterra series expansion of nonlinear systems described by narx models is studied this provides new convergence criteria under which the nonlinear system of interest has a convergent volterra series expansion and the new criteria are expressed explicitly in terms of the input magnitude model parameters and frequency variable the new convergence criteria are firstly developed for harmonic inputs and then extended to multi tone and general input cases based on the theoretical analysis a general procedure for calculating the convergence bound is provided the results provide a fundamental basis for nonlinear signal processing using the volterra series theory

large discrete sets in stein manifolds we construct infinite discrete subsets of stein manifolds with remarkable properties these generalize results of rosay and rudin on discrete subsets of mathbb c n

online identification of the bilinear model expansion on laguerre orthonormal bases this paper proposes a new representation of discrete bilinear model by developing its coefficients associated to the input to the output and to the crossed product on three independent laguerre orthonormal bases compared to classical bilinear model the resulting model entitled bilinear laguerre model ensures a significant parameter number reduction as well as simple recursive representation however this reduction is still subject to an optimal choice of the laguerre poles defining the three laguerre bases therefore we propose an analytical solution to optimise the laguerre poles which depend on fourier coefficients defining the bilinear laguerre model and that are identified using the regularised square error the identification procedures of the laguerre poles and fourier coefficients are combined and carried out on a sliding window to provide an online identification algorithm of the bilinear laguerre model the bilinear laguerre model as well as the proposed algorithm are illustrated and tested

difference equations versus differential equations a possible equivalence for the rossler system abstract when a set of nonlinear differential equations is investigated most of time there is no analytical solution and only numerical integration techniques can provide accurate numerical solutions in a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which the time variable is discrete in numerical investigations a fourth order runge kutta integration scheme is usually sufficient nevertheless sometimes a set of difference equations may be required and in this case standard schemes like the forward euler backward euler or central difference schemes are used the major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps in some cases it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps nonstandard schemes as introduced by mickens nonstandard finite difference models of differential equations world scientific 1994 allow to obtain more robust difference equations in this paper using such nonstandard scheme we propose some difference equations as discrete analogues of the rossler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used in particular it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the rossler system as long as the time step is less than the threshold value associated with the nyquist criterion

quality improvements in extruded meshes using topologically adaptive generalized elements in this dissertation a novel method to extrude near body meshes from surface meshes of arbitrary topology that exploits topologically adaptive generalized elements to improve mesh quality is presented specifically an advancing layer algorithm to generate near body meshes which are appropriate for viscous fluid flows is discussed first an orthogonal two layer algebraic reference mesh is generated the reference mesh is then smoothed using a locally three dimensional poisson type mesh generation equation that is generalized to smooth extruded meshes of arbitrary surface topology local quality improvement operations such as edge collapse face refinement and local reconnection are performed in each layer to drive the mesh toward isotropy an automatic marching thickness reduction algorithm is used to extrude from multiple geometries in close proximity a global face refinement algorithm is used to improve the transition from the extruded mesh to the void filling tetrahedral mesh a few example meshes along with quality plots are presented to demonstrate the efficacy of the algorithms developed

adaptive statistic tracking control based on two step neural networks with time delays this paper presents a new type of control framework for dynamical stochastic systems called statistic tracking control stc the system considered is general and non gaussian and the tracking objective is the statistical information of a given target probability density function pdf rather than a deterministic signal the control aims at making the statistical information of the output pdfs to follow those of a target pdf for such a control framework a variable structure adaptive tracking control strategy is first established using two step neural network models following the b spline neural network approximation to the integrated performance function the concerned problem is transferred into the tracking of given weights the dynamic neural network dnn is employed to identify the unknown nonlinear dynamics between the control input and the weights related to the integrated function to achieve the required control objective an adaptive controller based on the proposed dnn is developed so as to track a reference trajectory stability analysis for both the identification and tracking errors is developed via the use of lyapunov stability criterion simulations are given to demonstrate the efficiency of the proposed approach

the determination of optimum structures for the state space representation of multivariate stochastic processes when identifying a model for a multivariate stationary stochastic process an important problem is that of determining the structure of the state variable model several overlapping parameterizations can usually be fitted to a given process and the question arises as to which structure leads to the most accurate parameter estimates the accuracy of parameter estimates is often measured by the determinant of the fisher information matrix we show that all admissible structures will give asymptotically the same value to this criterion for finite data some structures may still be better than others and two heuristic structure estimation methods are analyzed some simulation results are also presented

multiobjective parameter estimation for non linear systems affine information and least squares formulation this paper defines a class of system information affine information that includes both the dynamic residuals and some types of auxiliary information that can be used in system parameter estimation as special cases the types of information that can be cast under the affine information format give rise to quadratic functions that measure the extent to which a model fits such information and that can be aggregated in a single weighted quadratic cost functional this allows the definition of a multiobjective methodology for parameter estimation in non linear system identification which allows taking into account any type of affine information the results are presented in terms of a set of efficient solutions of the multiobjective estimation problem such a solution set is more meaningful than a single model since any affine information leads to a convex quadratic functional the whole set of efficient solutions is exactly accessible via the minimization of the quadratic functional with different weighti

recursive hybrid algorithm for non linear system identification using radial basis function networks recursive identification of non linear systems is investigated using radial basis function networks a novel approach is adopted which employs a hybrid clustering and least squares algorithm the recursive clustering algorithm adjusts the centres of the radial basis function network while the recursive least squares algorithm estimates the connection weights of the network because these two recursive learning rules are both linear rapid convergence is guaranteed and this hybrid algorithm significantly enhances the real time or adaptive capability of radial basis function models the application to simulated real data are included to demonstrate the effectiveness of this hybrid approach

brief paper nonlinear multivariable adaptive control using multiple models and neural networks in this paper a multivariable adaptive control approach is proposed for a class of unknown nonlinear multivariable discrete time dynamical systems by introducing a k difference operator the nonlinear terms of the system are not required to be globally bounded the proposed adaptive control scheme is composed of a linear adaptive controller a neural network based nonlinear adaptive controller and a switching mechanism the linear controller can assure boundedness of the input and output signals and the neural network nonlinear controller can improve performance of the system by using the switching scheme between the linear and nonlinear controllers it is demonstrated that improved performance and stability can be achieved simultaneously theory analysis and simulation results are presented to show the effectiveness of the proposed method

dynamical realizations of finite volterra series in this paper realizations of finite volterra series are viewed as nonlinear analytic input output systems with state space described by an analytic manifold for a minimal realization guaranteed by h j sussmann the state space which is unique up to diffeomorphism is shown to have the homogeneous space structure of a nilmanifold the quotient of two nilpotent lie groups the structure of nilmanifolds as described by a malcev is used to show that for these systems the state space has a vector space structure as a consequence of this result it is shown that a minimal realization of a finite volterra series can be described as a cascade of linear subsystems with polynomial link maps in which the dimension o f each linear subsystem is independent of the realization considered

nonlinear adaptive control using neural networks and multiple models in this paper adaptive control of a class of nonlinear discrete time dynamical systems with boundedness of all signals is established by using a linear robust adaptive controller and a neural network based nonlinear adaptive controller and switching between them by a suitably defined switching law the linear controller when used alone assures boundedness of all the signals but not satisfactory performance the nonlinear controller may result in improved response but may also result in instability by using a switching scheme it is demonstrated that improved performance and stability can be simultaneously achieved

an adaptive generalized predictive control method for nonlinear systems based on anfis and multiple models in this paper an adaptive generalized predictive control method using adaptive network based fuzzy inference system anfis and multiple models is proposed for a class of uncertain discrete time nonlinear systems with unstable zero dynamics the proposed controller consists of a linear and robust generalized predictive adaptive controller a nonlinear generalized predictive adaptive controller based on anfis and a switching mechanism it has been shown that the linear generalized predictive adaptive controller can ensure the boundedness of the input and output signals and the nonlinear generalized predictive controller can improve the transient performance of the system by switching between the two earlier described controllers the switching mechanism can simultaneously improve the performance and ensure the closed loop stability moreover the method has relaxed the global boundedness assumption of the higher order nonlinear term and established the analysis of stability and convergence of the closed loop system in the proposed controller anfis is adopted to estimate and compensate the unmodeled dynamics which avoids some possible flaws of a backpropagation bp neural network simulation results have demonstrated the superiority of the proposed method and verified the theoretical analysis

control of nonlinear dynamical systems using neural networks ii observability identification and control for pt i see ibid vol 4 1993 this paper considers the problems of regulation and tracking of a dynamical system when the state variables of the dynamical system are not accessible the existence of the nonlinear maps describing the identifier and controller are first established and the implications for neural network realizations are described simulation results are included to complement the theoretical discussions

a nonlinear control method based on anfis and multiple models for a class of siso nonlinear systems and its application this paper presents a novel nonlinear control strategy for a class of uncertain single input and single output discrete time nonlinear systems with unstable zero dynamics the proposed method combines adaptive network based fuzzy inference system anfis with multiple models where a linear robust controller an anfis based nonlinear controller and a switching mechanism are integrated using multiple models technique it has been shown that the linear controller can ensure the boundedness of the input and output signals and the nonlinear controller can improve the dynamic performance of the closed loop system moreover it has also been shown that the use of the switching mechanism can simultaneously guarantee the closed loop stability and improve its performance as a result the controller has the following three outstanding features compared with existing control strategies first this method relaxes the assumption of commonly used uniform boundedness on the unmodeled dynamics and thus enhances its applicability second since anfis is used to estimate and compensate the effect caused by the unmodeled dynamics the convergence rate of neural network learning has been increased third a one to one mapping technique is adapted to guarantee the universal approximation property of anfis the proposed controller is applied to a numerical example and a pulverizing process of an alumina sintering system respectively where its effectiveness has been justified

recovering map static nonlinearities from chaotic data using dynamical models abstract this paper is concerned with the estimation from chaotic data of maps with static nonlinearities a number of issues concerning model construction such as structure selection over parametrization and model validation are discussed in the light of the shape of the static non linearities reproduced by the estimated maps a new interpretation of term clusters and cluster coefficients of polynomial models is provided based on this approach the paper discusses model limitations and some useful principles to select the structure of nonlinear maps some of the ideas have been tested using several nonlinear systems including a boost voltage regulator map and a set of real data from a chaotic circuit

self organizing adaptive fuzzy neural control for a class of nonlinear systems this paper proposes a self organizing adaptive fuzzy neural control safnc via sliding mode approach for a class of nonlinear systems the proposed safnc system is comprised of a computation controller and a supervisory controller the computation controller including a self organizing fuzzy neural network sofnn identifier is the principal controller the sofnn identifier is used to online estimate the controlled system dynamics with the structure and parameter learning phases of fuzzy neural network fnn simultaneously the structure learning phase possesses the ability of online generation and elimination of fuzzy rules to achieve optimal neural structure and the parameter learning phase adjusts the interconnection weights of neural network to achieve favorable approximation performance the supervisory controller is used to achieve the l2 norm bound tracking performance with a desired attenuation level moreover all the parameter learning algorithms are derived based on lyapunov function candidate thus the system stability can be guaranteed finally simulation results show that the safnc can achieve favorable tracking performances

analysis of the effects of time delay in nonlinear systems using generalised frequency response functions abstract a recursive algorithm for computing the generalised frequency response functions gfrfs of nonlinear time delay systems described by nonlinear differential difference equation models is derived using the operator e n and used to analyse the effects of time delay in nonlinear systems the algorithm shows an explicit relationship between the model parameters and the gfrfs such a relationship provides important insight into the behaviour of nonlinear systems the effect of delay on several properties of nonlinear systems such as harmonic generation gain compression expansion and desensitisation is studied by considering the example of a duffing oscillator with retarded damping results of the studies convincingly demonstrate that the system delay has a significant effect on several important properties of nonlinear systems

robust adaptive neural network control for a class of uncertain mimo nonlinear systems with input nonlinearities in this paper robust adaptive neural network nn control is investigated for a general class of uncertain multiple input multiple output mimo nonlinear systems with unknown control coefficient matrices and input nonlinearities for nonsymmetric input nonlinearities of saturation and deadzone variable structure control vsc in combination with backstepping and lyapunov synthesis is proposed for adaptive nn control design with guaranteed stability in the proposed adaptive nn control the usual assumption on nonsingularity of nn approximation for unknown control coefficient matrices and boundary assumption between nn approximation error and control input have been eliminated command filters are presented to implement physical constraints on the virtual control laws then the tedious analytic computations of time derivatives of virtual control laws are canceled it is proved that the proposed robust backstepping control is able to guarantee semiglobal uniform ultimate boundedness of all signals in the closed loop system finally simulation results are presented to illustrate the effectiveness of the proposed adaptive nn control

a new notion of tameness abstract we generalize the notion of tame discrete sets introduced by rosay and rudin from complex euclidean space to arbitrary complex manifolds and establish their basic properties we show that complex linear algebraic groups different from the complex line or the punctured complex line contain tame discrete sets

solution adaptive grid generation using parabolic partial differential equations a solution adaptive parabolic grid generation scheme has been developed a new upwind finite volume formulation for space marching solution of the steady euler equations is also presented in addition a solution adaptive elliptic grid scheme developed previously and applied to unsteady flow solutions is applied in the present steady flow context solutions for hypersonic flow are computed with the present finite volume formulation coupled to both the parabolic and elliptic adaptive grid schemes the results show that the solution adaptive parabolic scheme produces grids that track the flow features of interest as well as the elliptic grid procedure this is significant because the parabolic procedure is much faster than an elliptic scheme producing a grid in an order of magnitude of less computational time

grid generation algorithms for complex glaze ice shapes reynolds averaged navier stokes simulations the paper presents the developments of novel mesh generation algorithms over complex glaze ice shapes containing multicurvature ice accretion geometries such as single double ice horns the twofold approaches tackle surface geometry discretization as well as field mesh generation first an adaptive curvilinear curvature control algorithm is constructed solving a one dimensional elliptic partial differential equation with periodic source terms this method controls the arc length grid spacing so that high convex and concave curvature regions around ice horns are appropriately captured and is shown to effectively treat the grid shock problem second a novel blended method is developed by defining combinations of source terms with two dimensional elliptic equations the source terms include two common control functions sorenson and spekreijse and an additional third source term to improve orthogonality this blended method is shown to be very effective for improving grid quality metrics for complex g

a novel estimation algorithm based on data and low order models for virtual unmodeled dynamics in this paper the challenging issue of estimating virtual unmodeled dynamics is addressed a novel estimation algorithm based on historical data and the output of low order approximation models for virtual un modeled dynamics is presented in particular the virtual un modeled dynamics are decomposed into known and unknown parts where only the unknown part is to be estimated the method effectively avoids the need to use the unknown control input directly and enables the estimation of the un modeled dynamics with a relatively simple algorithm moreover it is shown that the proposed algorithm overcomes the difficulty in obtaining the control solutions caused by the fact that the controller input is embedded in un modeled dynamics finally simulation studies are presented to demonstrate the effectiveness of the proposed method

input output parametric models for non linear systems part ii stochastic non linear systems recursive input output models for non linear multivariate discrete time systems are derived and sufficient conditions for their existence are defined the paper is divided into two parts the first part introduces and defines concepts such as nerode realization multistructural forms and results from differential geometry which are then used to derive a recursive input output model for multivariable deterministic non linear systems the second part introduces several examples compares the derived model with other representations and extends the results to create prediction error or innovation input output models for non linear stochastic systems these latter models are the generalization of the multivariable arm ax models for linear systems and are referred to as narmax or non linear autoregressive moving average models with exogenous inputs

existence and uniqueness of realizations of nonlinear systems necessary and sufficient conditions for the existence of either analytic or smooth symmetric realizations of nonlinear input output maps are given it is also shown that two minimal realizations are diffeomorphic

identification and control of a nonlinear discrete time system based on its linearization a unified framework this paper presents a unified theoretical framework for the identification and control of a nonlinear discrete time dynamical system in which the nonlinear system is represented explicitly as a sum of its linearized component and the residual nonlinear component referred to as a higher order function this representation substantially simplifies the procedure of applying the implicit function theorem to derive local properties of the nonlinear system and reveals the role played by the linearized system in a more transparent form under the assumption that the linearized system is controllable and observable it is shown that 1 the nonlinear system is also controllable and observable in a local domain 2 a feedback law exists to stabilize the nonlinear system locally and 3 the nonlinear system can exactly track a constant or a periodic sequence locally if its linearized system can do so with some additional assumptions the nonlinear system is shown to have a well defined relative degree delay and zero dynamics if the zero dynamics of the linearized system is asymptotically stable so is that of the nonlinear one and in such a case a control law exists for the nonlinear system to asymptotically track an arbitrary reference signal exactly in a neighborhood of the equilibrium state the tracking can be achieved by using the state vector for feedback or by using only the input and the output in which case the nonlinear autoregressive moving average narma model is established and utilized these results are important for understanding the use of neural networks as identifiers and controllers for general nonlinear discrete time dynamical systems

marching grid generation using parabolic partial differential equations abstract the unique aspect of grid generation in computational fluid dynamics is that the grid generation equations have no physical meanings so any equation may be used for this purpose if the grids generated are useful this aspect provides a vast freedom in developing new methods of grid generation this paper explores feasibility of using parabolic partial differential equations for grid generation the advantages of using parabolic partial differential equations are as follows 1 parabolic equations are initial value problems so grids are generated by a marching algorithm like the hyperbolic grid generation method 2 the parabolic partial differential equations have most properties of the elliptic equations particularly the diffusion effect which smooths out any singularity of the inner boundary condition if any and 3 the prescribed outer boundary conditions may be satisfied the importance of the marching algorithm stated above is twofold first computational time required is only a very small fraction of that for the elliptic grid generation equations second the fast memory space required during grid generation can be substantially reduced from that required by the elliptic grid generation method the authors consider the grid generation for a two dimensional airfoil flow calculations for simplicity of discussions although the method described in this paper is not restricted to two dimensions or airfoil problems

a suboptimal bootstrap method for structure detection of non linear output error models with application to human ankle dynamics identification of non linear systems involves estimating unknown parameters and structure detection selection of a subset of candidate terms that best describe the observed output this is a necessary procedure to compute an efficient system description which may afford greater insight into the functionality of the system or a simpler controller design for non linear systems simple output additive noise can generate multiplicative terms between the input output and noise the terms associated with noise need to be modelled to obtain unbiased parameter estimates significantly increasing the number of candidate terms to be estimated and considered in special cases it may be possible to use an output error oe model structure and the instrumental variable iv estimator to obtain unbiased parameters without modelling the noise this significantly reduces the dimensionality of the structure computation problem therefore in this paper we propose a suboptimal bootstrap structure detection sobsd algo

an improved estimation method for unmodeled dynamics based on anfis and its application to controller design by representing nonlinear systems as a combination of linear part and unmodeled dynamics in this paper an improved estimation algorithm using an adaptive neuro fuzzy inference system anfis for unmodeled dynamics is presented at first the unmodeled dynamics is divided into two parts using the differential expansion of the control input at the last time instant then the two parts are estimated by the anfis it has been shown that the proposed algorithm overcomes the problem that the unknown control input is embedded in unmodeled dynamics which makes the true value of unmodeled dynamics difficult obtain moreover the method improves the precision of the estimation of unmodeled dynamics second under the assumption that the growth rate of unmodeled dynamics does not exceed its input vector the one to one mapping and regularization technique are adopted to deal with the input and output data and the unmodeled dynamics respectively as a result the data vector can be guaranteed to lie inside a compact set which ensures the use of the universal approximation property of the anfis on the other hand it has been shown that datum of a system can be fully used to obtain the parameters centers widths in membership functions and the network connection weights in the anfis by offline training these parameters are tuned online to improve the estimation convergence rate of the unmodeled dynamics the effectiveness of the proposed estimation method is illustrated by comparing it with the simulation results that are obtained from the other existing methods finally the proposed estimation method is applied to the nonlinear switching control design both simulation and theoretical analysis have confirmed that the nonlinear switching control which adopts the proposed estimation method cannot only guarantee the stability and convergence of the system but can exhibit a desired dynamic performance for the closed loop system as well

a multiple model mrac scheme for multivariable systems with matching uncertainties this paper develops a multivariable multiple model adaptive control scheme for adaptive state feedback state tracking control of systems whose plant model matching conditions are uncertain and parameters are unknown to deal with the uncertainty of plant model matching conditions needed for adaptive control multiple reference model systems are employed to generate multiple parameter estimation and feedback control signals from which a most suitable control input is selected by a control switching mechanism designed using multiple estimation errors such a new multiple model control design is based on an expanded control system parametrization which has the capacity to cover system structural uncertainties stability analysis and simulation results ensure and verify the desired adaptive control system stability and tracking performance

adaptive neural output feedback tracking control for a class of uncertain discrete time nonlinear systems this brief studies an adaptive neural output feedback tracking control of uncertain nonlinear multi input multi output mimo systems in the discrete time form the considered mimo systems are composed of n subsystems with the couplings of inputs and states among subsystems in order to solve the noncausal problem and decouple the couplings it needs to transform the systems into a predictor form the higher order neural networks are utilized to approximate the desired controllers by using lyapunov analysis it is proven that all the signals in the closed loop system is the semi globally uniformly ultimately bounded and the output errors converge to a compact set in contrast to the existing results the advantage of the scheme is that the number of the adjustable parameters is highly reduced the effectiveness of the scheme is verified by a simulation example

semistructured grid generation in three dimensions using a parabolic marching scheme shock impingement point it was observed that a reduction in ow separation behind the shock ensued as a result of weakened shock strength conclusions the ow eld inside transonic compressors involves high level ow physics and is very sensitive to changes in blade geometry therefore the design of an axial compressor blade requires careful maneuvering through complex nonlinear design space numerical optimization is attractive in that constraints can be imposed to prevent ill effectsof a design constraints however limits designspace and in the case of an axial compressor blade design a constraint on inletmass ow rate takes toomuchspaceaway makingoptimization extremely dif cult to perform this note shows that incorporating the constraint into an objective function can improve the design results also if needed constraints are related to the objective function in someways it would be possiblethat they all becomea part of the objective function while discouraging the design process from moving into certain directions the study also shows that the developed design method can improve the adiabatic ef ciency of a blade section signi cantly by reducing the losses from the passage shock and the ow separation the design method however cannot handle three dimensional effects because it employs quasi three dimensional ow physics and did not produce desired design results for the blade section near the hub region to ascertain the signi cance of the quasi threedimensional design results constructing and evaluating the performance of three dimensionalblades based on the designed sections will be conducted in the future references 1damle s dang t stringham j and razinsky e practical use of 3d inverse method for compressor blade design american society of mechanical engineers asme paper 98 gt 115 june 1998 2wang z cai r chen h and zhang d a fullythree dimensional inverse method for turbomachinery blading with navier stokes equations american society of mechanical engineers asme paper 98 gt126 june 1998 3sorenson r l a computer program to generate two dimensional grids about airfoils and other shapes by use of poisson s equation nasa tm 81198 1980 4chima r v explicit multigrid algorithm for quasi threedimensionalviscous flows in turbomachinery journalof propulsion and power vol 3 no 5 1987 pp 397 405 5wilcox d c comparison of two equation turbulence models for boundary layers with pressure gradient aiaa journal vol 31 no 8 1993 pp 1414 1421 6katsanis t andmcnally w d revised fortranprogramforcalculating velocities and streamlines on the hub shroudmidchannel stream surface of an axial radial and mixed flow turbomachine or annular duct part i user s manual nasa tn d 8430 1977 7 dot user s manual ver 3 00 vma engineering goleta ca 1992 8hicks r m and henne p a wing design by numerical optimization journal of aircraft vol 15 no 7 1978 pp 407 412

variable selection algorithm for the construction of mimo operating point dependent neurofuzzy networks an input variable selection procedure is introduced for the identification and construction of multi input multi output mimo neurofuzzy operating point dependent models the algorithm is an extension of a forward modified gram schmidt orthogonal least squares procedure for a linear model structure which is modified to accommodate nonlinear system modeling by incorporating piecewise locally linear model fitting the proposed input nodes selection procedure effectively tackles the problem of the curse of dimensionality associated with lattice based modeling algorithms such as radial basis function neurofuzzy networks enabling the resulting neurofuzzy operating point dependent model to be widely applied in control and estimation some numerical examples are given to demonstrate the effectiveness of the proposed construction algorithm