Deterministic Learning Theory Of Distributed Parameter Systems And Their Applications

Fluid systems and vibration systems are widely found in nature and industry. Withthe rapid development of modern industry, the improved performance of light, intelligence,precision of the equipment is requested, which makes ?uid systems and vibration systemsof ?exible body are concerned by the scienti?c and engineering community. Due to thetexture of materials, the system has an in?nite number of degrees of freedom, and thesystem state unlike the rigid-body system can be description by ?nite parameters butmust use a ?eld for complete description, which meas that the system state is not only afunction of time variable but also a function of spatial variables, thus the system is calledas in?nite-dimensional distributed parameter systems (DPS). If one wants to carry outor good design for DPS, a adequate understanding of dynamic behavior of the systemis necessary. However, in uncertain dynamical environments, to accurately acquaint thedynamics of DPS is a challenging task. Recently, Wang etc. propose the deterministiclearning theory (DLT) by utilizing results from concepts and tools of adaptive controland dynamical systems.This thesis studies the DLT of nonlinear DPS with unknown dynamics. Since thein?nite-dimensional essential feature, to establish complete and accurate dynamics is verydi?cult, evenmore it is a impossible task under current conditions. Using the DLT, weprovide a framework and methodology to approximate/identify the dynamics of nonlinearDPS within a certain accuracy. Therefore, we ?rst establish an approximation systemfor DPS in a ?nite-order accuracy; then by using the deterministic learning algorithm,we obtain an accurate radial basis function (RBF) neural network (NN) approximationof the ?nite-dimensional dynamical system (FDDS) along the trajectories; ?nally theidenti?cation of dynamics of the original DPS is achieved in some accuracy. The resultingsystem model can re?ects the nature of the original DPS in the accuracy, and it is easyto be work. The contents contain: ?nite-dimensional approximation method, systemidenti?ability conditions, and identi?cation algorithms, etc. The main contents of thethesis are as follows:1. We investigate the identi?cation of a class of parabolic DPS. The parabolic DPSusually arises from physical phenomena such as heat conduction, ?ow ?eld and particledi?usion. The considered DPS describe a homogeneous and isotropic object with circulargeometric structure. Based on this feature, we ?rst reduce DPS into a FDDS by thediscrete Fourier transform (DFT) method. Secondly, some important properties of FDDS,including the discrete symmetry and the partial dominance of system dynamics accordingto point-wise observations, are analyzed. Due to the coe?cient matrix is introduced by DFT in discrete process, the coe?cient matrix is a circulant matrix by the connectionbetween DFT and circulant matrix. The system geometry decides the dominant dynamicsof FDDS along the diagonal of coe?cient matrix. Finally, by using the deterministiclearning algorithm, it is show that locally relatively accurate NN approximation of maindynamics of the FDDS is achieved in local region along the recurrent trajectories. Then,a locally identi?cation of the dynamics of the parabolic DPS is accomplished.2. We investigate the modeling and rapid detection of rotating stall via deterministiclearning. Rotating stall is one distinct aerodynamic instabilities in axial ?ow compressorof turbofan engine. Once a compressor enters fully developed rotating stall, rotor andstator blades are under tremendous stress caused by stalled ?ow to damage the com-pressor, the non-axisymmetric ?ow can result internal overtemperatures in combustionchamber and turbine to burn the wall and blades. Rotating stall makes the mass ?owand press rise through the compressor is decreased which leads to the thrust sudden dropgreatly in engine, and the function is to increase the pressure of the ?ow. Thus, thiscondition severely limit the compressor performance and reduce the e?ciency of the en-gine. Evenmore, the blade channel blockage caused by the rotating stall ?ow can leadsurge which is another typical unsteady ?ow phenomena/fault in compressor. For thesereasons, rapid detection of rotating stall is very important for improving the performanceand preventing surge. To resistant the rotating stall, the key recognizes the rotatingstall precursor (RSP). Rapid and accurate detection method is based on precision modelof RSP. Therefore, the detection process for rotating stall consists of two phases: themodeling/identi?cation phase and the recognition phase. In the identi?cation phase, wearrange multiple sensors at circumferential and axial of compressor to measure ?ow andpressure signals. Then, based on the measured signals, use dynamical RBF NN to i-dentify the dynamics of RSP according to DLT. We obtain the RBF NN model, andaverage of the NN weights in a time segment after convergence process (constant NNweights). The constant NN weights is stored as the pattern for RSP. Using the sameprocedure, a pattern library of compressor can be established for rotating stall. In therecognition phase, based on the established model of RSP, we use pattern library storedin the identi?cation phase to rapid detection RSP. For monitored compressor system, aseries dynamical estimators is constructed by the constant NN weights. Since RFB NNmodel can present the dynamics of RSP, the dynamical estimator embeds the dynamicsof RSP. By comparing the monitored compressor system and dynamical estimators, aseries residual systems is obtained. By the rapid dynamical pattern recognition method,residual can measure the similarity for dynamical patterns. Therefore, we use an averageof l1 norm to evaluate the residuals. The smallest average norm of all residuals is the pattern coming in compressor. If the norm of the RSP is smallest, which is suggest therotating stall is coming; rapid detection of the RSP is achieved. Due to rapid detectioncompletes in rotating stall inception, rotating stall can be predicted, and the surge canbe prevented because rotating stall is a symptom of surge and alert engine.3. We investigate the DLT of completely resonant wave systems with Dirchiletboundary conditions. According to Lagrangian stress theorem, Newton's second lawand the Dirchilet boundary conditions, considered completely resonant wave system isdescribed the ?exible vibrating string with ?xed endpoints. Two types of wave systemare discussed: 1) the unknown part of dynamics within the system; 2) the unknown allof dynamics within the system. We use ?nite di?erence method to discretize the wavesystem into a higher-dimensional nonlinear dynamic system in the space domain. Then,Gronwall inequality theorem and Leary-Schauder ?xed point theorem are used to provethe existence and uniqueness of solution of FDDS, and the solution of FDDS convergeto the solution of wave system. Therefore, these ensure that the approximation is valid,and makes the FDDS keeps the essence dynamics of the wave system. Finally, accordingto DLT, we employ the dynamical RBF NN to accurately identify the FDDS along thetrajectory. Then the accurate approximation of dynamics of the wave system is obtainedin some accuracy. Since it is di?erence that unknown dynamics of the ?rst and secondtype systems associated with the discrete points, the input dimension of RBF NN isdi?erent. For the ?rst type system, the input of RBF NN is only the discrete pointitself; while the second type system, the input of RBF NN is not only the discrete pointitself but also adjacent two discrete points. Analysis the input dimension, the input ofRBF NN is decide by the most order of spatial di?erential term. Therefore, if truncationerror accuracy is same, increasing the low-order di?erential item will not increase theinput dimension of RBF NN. To improve the approximation accuracy can be succeed byincreasing the ?nite-dimensional truncation error accuracy, the number of space discretepoints and the number of neurons.