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NONLINEAR MODEL REDUCTION AND ITS APPLICATION TO MODEL PREDICTIVE CONTROL Juergen Hahn Department of Chemical Engineering Lehrstuhl für Prozesstechnik Texas A&M University RWTH Aachen College Station, TX 77843-3122 D - 52056 Aachen U.S.A. Germany Overview • Introduction • Balanced model reduction Linear & nonlinear systems • Nonlinear model predictive control scheme • Case studies Compare performance of controllers • Discussion • Conclusions Introduction • Considerations for real time process control: Time span between two control moves is fixed Control move has to be computed within the sampling time Model reduction can facilitate real time control of large scale processes For linear model predictive control (MPC) the computation time scales with n3 • Focus on balanced model reduction because it approximates input-output behavior by reducing number of states Balanced model reduction • For stable linear time invariant systems x(t ) Ax(t ) Bu (t ) y (t ) Cx(t ) the gramians characterize the input-to-state and state-to-output behavior • Linear controllability gramian (u→x) W e BB e dt C At T AT t 0 • Linear observability gramian (x→y) W e C Ce dt O AT t T At 0 Balanced model reduction Input Controllability State Observability Output u WC e BB e At T AT t dt x WO e A t C T Ce A t dt T y 0 0 x(t ) Ax(t ) Bu (t ) y (t ) Cx(t ) Balanced model reduction • Find a coordinate transformation for the states that balances the system, i.e. make the gramians identical and equal to a diagonal matrix • The entries along the diagonal of the balanced gramians are called Hankel singular values • The largest entry along the diagonal corresponds to the state of the transformed system that is influenced the most by control moves and the outputs react the most to changes of this state • Reduce states corresponding to small Hankel singular values Balanced model reduction • Find balancing transformation x Tx x TAT 1 x TBu A x B u y CT 1 x C x 1 0 0 0 0 2 0 0 WC WO 0 0 3 0 1 2 3 n 0 0 0 n 0 Balanced model reduction • Apply balancing transformation and partition system x1 A11 A12 x1 B1 u x A 2 21 A22 x2 B2 x1 y C1 C2 x 2 • Truncated system x1 A11x1 B1u y C1 x1 Balanced model reduction Original WC Compute Compute System of Balancing Gramians WO Equations Transformation x Ax Bu y Cx T TWC T T σi » σi+1 Determine Size T T WOT 1 Balance Reduced- of Reduced Gramians and Order System System System x1 A11x1 B1u x TAT 1 x TBu y C1 x1 y CT 1 x Balanced model reduction • Extension of balancing algorithms to nonlinear systems x(t ) f ( x(t ), u (t )) y (t ) h( x(t )) • Nonlinear gramians and balancing have been introduced in theory, but no general purpose algorithm exists • Hybrid method is required Balanced model reduction • Introduce covariance matrices for nonlinear systems • Reduce to linear gramians for linear systems (and impulse inputs) • Capture much of the behavior of the nonlinear system • Can be computed from simulation (or experimental) data for realistic operating conditions Balanced model reduction • Controllability covariance matrix p r s 1 WC 2 (t )dt ilm i 1 l 1 m 1 rscm 0 ilm(t) = (xilm(t)-x0ilm(v(t)))( xilm(t)-x0ilm(v(t)))T xilm(t) is the state of the nonlinear system corresponding to the input u(t) = cmTleiv(t)+uss(0) i, l, and m correspond to the orientation, direction, and magnitude of the excitation Balanced model reduction • Observability covariance matrix r s 1 WO rscm 2 Tl lm (t )Tl T dt l 1 m 1 0 lmij(t) = (yilm(t)-y0ilm)T( yjlm(t)-y0jlm) yilm(t) is the output of the nonlinear system corresponding to the initial condition x(0) = cmTlei+xss Balanced model reduction x Tx 1 0 0 0 0 0 0 WC TWCT T 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 WO (T 1 )T WO T 1 0 0 3 0 0 0 0 0 Balanced model reduction • The reduced model is given by x1 (t ) PTf (T 1 x (t ), u (t )) f ( x (t ), u (t )) x2 (t ) x2, ss (0) y (t ) h(T 1 x (t )) h ( x (t )) where x1 x x P I 0 2 (Hahn and Edgar, Comp. Chem. Eng., 2002) Balanced model reduction Original Compute WC Compute System of Covariance Balancing Equations Matrices WO Transformation x(t ) f ( x(t ), u (t )) y (t ) h( x(t )) T TWC T T σi » σi+1 Determine Size T T WOT 1 Balance Reduced- of Reduced Matrices and Order System System System x1 (t ) PTf (T 1 x (t ), u (t )) x (t ) Tf (T 1 x (t ), u (t )) x2 (t ) x2, ss (0) y (t ) h(T 1 x (t )) y (t ) h(T 1 x (t )) Nonlinear model predictive control scheme • Balanced model reduction reduces size of a model • Results are usually evaluated with simulations for the open-loop case • However: Ultimate goal of model reduction is designing simpler controllers that are easier to design, implement, and maintain What effect does a reduced model have when used for controller design? Nonlinear model predictive control scheme Solve NLP N 1 N C 1 J yk ysp d Qyk ysp d u Ruk T T k k 0 k 0 ui ,min ui ,k ui ,max Implement control move (real model) Reduced-order model x1 (t ) PTf (T 1 x (t ), u (t )) x(t ) f ( x(t ), u (t )) x2 (t ) x2, ss (0) y (t ) h( x(t )) y (t ) h(T 1 x (t )) Update disturbance estimate Nd 1 d Nd y k 1 k ,m yk , p Nonlinear model predictive control scheme • This NMPC scheme is designed for flexibility (different models, linear or nonlinear, can be used for the computation) can be used to test the results achieved by controllers based upon reduced-order models is not optimized for computation speed (optimization problem and solution of the model are done sequentially and not simultaneously) Case studies • Two CSTRs in series (6 states, 2x2) 2 inputs (valve position & cooling rate) 2 outputs (volume & temperature in reactor 2) • Three different models Full-order system (6 states) Truncated system (4 states) Linearized system (6 states) Case studies Dynamic response for set point change and disturbance rejection Titel: Titel: E:\Juergen\Dis sertation\fig3-2x.eps E:\Juergen\Dis sertation\fig3-2y.eps Ers tellt von: Ers tellt von: MATLAB, The Mathworks , Inc. MATLAB, The Mathworks , Inc. Vors chau: Vors chau: Diese EPS-Grafik wurde nicht ges peichert Diese EPS-Grafik wurde nicht ges peichert mit einer enthaltenen Vors chau. mit einer enthaltenen Vors chau. Komm entar: Komm entar: Diese EPS-Grafik wird an einen Diese EPS-Grafik wird an einen Pos tScript-Drucker gedruckt, aber nicht Pos tScript-Drucker gedruckt, aber nicht an andere Druckertypen. an andere Druckertypen. Case studies • Distillation column (32 states, 1x1) Binary distillation column with constant molar volatility (Horton et al., Comp. Chem. Eng., 1991) Reflux ratio only control Distillate concentration only measurement • Two different models Full-order system (32 states) Truncated systems (1 state) Case studies Set-point change and disturbance rejection » » Computation times: » 32 states: 151.3 s » 1 state : 27.8 s Discussion • Pro Significant reduction of number of states possible (especially for models that behave like distributed systems) Identification of “dynamic degrees of freedom” Preserves nonlinear components of the model Extension of balancing for linear systems Several other methods form special cases of the approach presented here Discussion • Con Method is based upon projection • Reduction of number of equations, but equations will become more complex • Physical meaning of states is lost • Model structure is not preserved model becomes less sparse • Due to this: reduction in computational effort is hard to predict (at least with this implementation of MPC scheme) Conclusions • Presented a nonlinear model reduction procedure • Presented an implementation of a nonlinear model predictive control scheme • Based upon the case studies: Controllers based upon reduced-order nonlinear models can result in good performance Reduced-order nonlinear controllers can result in better performance than linear controllers for some models Acknowledgments • Prof. Thomas F. Edgar • Prof. Wolfgang Marquardt

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model predictive control, model reduction, nonlinear model, nonlinear mpc, nonlinear systems, prediction horizon, nonlinear model predictive control, optimization problem, predictive control, objective function, linear model, distillation column, cost function, sensitivity analysis, control algorithm

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posted: | 7/11/2010 |

language: | German |

pages: | 27 |

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