аЯрЁБс>ўџ /1ўџџџ.џџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџьЅС)` №RПЋbjbj›U›U8(љ?љ?ЋџџџџџџЄДДДДДДДШllll€ ШG *˜˜˜˜˜˜Ў КЦ Ш Ш Ш Ш Ш Ш $q hй<ь ДТ˜˜ТТь ДД˜˜ тттТД˜Д˜Ц тТЦ тт:ю ,ДДv ˜Œ №/sZНЯЮlв Ц  0G $ Rтv Дv PТТтТТТТТь ь тТТТG ТТТТШШШЄlШШШlШШШДДДДДДџџџџ An Adaptive Order Reduction and Feedback Controller Synthesis methodology for Dissipative Partial Differential Equation Systems Antonios Armaou The Pennsylvania State University, Chemical Engineering Department Many industrially relevant diffusion-convection-reaction processes are characterized by the presence of strong spatial variations due to coupling between diffusive and convective transport mechanisms with chemical reactions. Typical examples include distillation and bed reactors in chemical, plasma reactors and chemical vapor deposition in microelectronics, and tin float process in glass industries. The feedback control problem for these processes is nontrivial owing to the spatially distributed nature of their dynamics, i.e. their mathematical descriptions are usually in the form of dissipative partial differential equations (DPDEs). This necessitates the use of model reduction in order to design practically implementable controllers. The key step in this approach is the computation of basis functions that are subsequently utilized to obtain finite dimensional approximations to the original infinite dimensional system. In principle, the spatial operator eigenfunctions could be combined with the method of weighted residuals to identify an ordinary differential equation (ODE) approximation. However, for processes with complex spatial domains or nonlinear operators such functions cannot be analytically derived. A common approach for this task is then the proper orthogonal decomposition (POD) combined with Galerkin’s method. However, this approach requires the a priori availability of a sufficiently large ensemble of PDE solution data (snapshots) which excites all possible dynamic behaviors of the PDE system, a requirement which may be difficult to satisfy. In this talk, we will be presenting our recent results on output-feedback control of distributed-parameter-systems. To circumvent the discussed issues, specifically for controller design, we initially compute basis functions off-line using a small ensemble of available data and continuously refine the basis functions on-line as more snapshots of the process become available. The computation of the (so-called) empirical eigenfunctions requires the solution of the eigenvalue problem of the covariance matrix of the snapshots, which might become expensive for online computations. We describe a procedure to recursively compute the empirical eigenfunctions of a given PDE system. The approach is based on the computation of an approximation of the dominant eigenspace of the spatial operator. This dominant eigenspace is updated recursively as new snapshots from the process are added to the ensemble, simultaneously increasing or decreasing its dimensionality if required. We maintain that as long as the dimensionality of the dominant eigenspace remains small, the computational burden for updating the dominant eigenspace remains small and can be easily performed online. The focus of this procedure is the derivation of low-order models specifically tailored for the design of feedback controllers and observers for distributed processes. Following the derivation, we present the synthesis of geometric and robust output feedback controllers. We illustrate the effectiveness of the proposed approach numerically through representative examples such as the Kuramoto Sivashinsky equation, where the synthesized controllers successfully regulate the system at open-loop unstable steady states. References: S. Pitchaiah, and A. Armaou, “Feedback control of dissipative distributed parameter systems using adaptive model reduction”, Ind. & Eng. Chem. Res., 55, 906-918, 2010. A. Varshney, S. Pitchaiah and A. Armaou, “Feedback control of dissipative PDE systems using adaptive model reduction”, AICHE Journal., 55, 906-918, 2009. In case the original abstract is too long here is a short version: The problem of feedback control of spatially distributed processes described by dissipative partial differential equations (PDEs) is considered. Typically, this problem is addressed through model reduction, where finite dimensional approximations to the original PDE system are derived and used for controller design. The key step in this approach is the computation of basis functions that are subsequently utilized to obtain finite dimensional ordinary differential equation (ODE) models using the method of weighted residuals. A common approach to this task is Galerkin’s method combined with the method of snapshots (called proper orthogonal decomposition method, POD). To circumvent the hurdle of a priori availability of a sufficiently large ensemble of PDE solution data for the successful application of POD, the focus is on the recursive computation of eigenfunctions as additional data from the process becomes available. Initially, an ensemble of eigenfunctions is constructed based on a relatively small number of snapshots, and the covariance matrix is computed. The dominant eigenspace of this matrix is then utilized to compute the empirical eigenfunctions required for model reduction. This dominant eigenspace is recomputed with the addition of each snapshot with possible increase or decrease in its dimensionality; due to its small dimensionality the computational burden is relatively small. Subsequently, we derive low-order models specifically tailored for the design of feedback controllers and observers for distributed processes. Following the derivation, we present the synthesis of geometric and robust output feedback controllers. We illustrate the effectiveness of the proposed approach numerically through representative examples such as the Kuramoto Sivashinsky equation, where the synthesized controllers successfully regulate the system at open-loop unstable steady states. €гЖ Ф Э н і ј 9 : ч э љ  1 Q f g Ф Щ є žН*Tv‰“˜šЈкфћ'8DE” JK†‡№њ \ЙсљЄЏёщощощощощощощощощощощощощоЯУЯУДУДУДУДУДУДУДУДУДУДУДУДУДУДУІh‰XghpМ5OJQJ\h›mThpМB*OJQJph$ hpМB*OJQJph$ h‰XghpМB*OJQJph$ h‰XghpМOJQJhpМOJQJhpМ5CJOJQJ\aJ<€гЄАXђѓ67ЊЋїїїячпдПЌ››“‚$d№Є7$8$H$a$gdХ 3$a$gdХ 3$d№Є7$8$H$a$gdШwB$d№Є№Є7$8$H$a$gd›mT$ & Fd№Є7$8$H$a$gd›mT $ & Fa$gd›mT$a$gdШwB$a$gdŠF\$a$gd›mT$a$gd‰Xg Ћ§ЏАђѓ67ту=>xyзи*+k|Ђё§ў@f…†Яа()‚ƒзи)*ŒЛШЩгbПчџЊЋ№хнЬОЬВЃВЃВЃВЃВЃВЃВЃВЃВЃВЃВЃВЃВЃВЃВЃВЃВЃВЃВЃВЃВЃВЃВхh›mThpМB*OJQJph$ hpМB*OJQJph$ hpМ5B*OJQJph$ hШwBhpМ5B*OJQJph$ hpМOJQJh‰XghpМOJQJhШwBhpМB*OJQJph$ 461hP:pиАа/ Ар=!А "А # $ %ААаАа а†œB@ёџB иck‡e dЄШCJ_HaJmH sH tH $A@ђџЁ$ иžЄ‹Еk=„W[SOBiѓџГB nfhˆШwB5?L<>OфPЗeQќ;SчiT№iT›mTл_[ŠF\Иx^Bwbj?]}ЂЈАЗЧЩ    & 0 2 3 : !6"000 0 0 0000006ў:ў>ў@ўDўZў\ў^ўџџџџ џ џџџџџ=џ@џ\џ]џ^џрџ$([{ЃЅЗ  0 0 000000Yў[ў]ўџџџ;џ[џсџхџ  ДД  2ƒ№HX №џ$PџџџџџџџџџџџџџџџџџџџџџО"З2џџAn Adaptive Order Reduction and Feedback Controller Synthesis methodology for Dissipative Partial Differential Equation SystemsDavoodЎ_o(u7b  ўџр…ŸђљOhЋ‘+'Гй0ш˜ ,<HTd x„ Є А МШаирЈ€An Adaptive Order Reduction and Feedback Controller Synthesis methodology for Dissipative Partial Differential Equation SystemsDavoodNormal ЮЂШэгУЛЇ2Microsoft Office Word@FУ#@вЃYНЯЮ@вЃYНЯЮbIўџеЭеœ.“—+,љЎ0Ќ X`lt|„ Œ”œЄЈ)  '  ўџџџўџџџ !"#$%ўџџџ'()*+,-ўџџџ§џџџ0ўџџџўџџџўџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџRoot Entryџџџџџџџџ РFА§ŠZНЯЮ2€1Tableџџџџџџџџ)WordDocumentџџџџџџџџ8(SummaryInformation(џџџџDocumentSummaryInformation8џџџџџџџџџџџџ&CompObjџџџџџџџџџџџџmџџџџџџџџџџџџџџџџџџџџџџџџўџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџўџ џџџџ РFMicrosoft Office Word ЮФЕЕ MSWordDocWord.Document.8є9Вq