Nonlinear Model Reduction of Differential Algebraic Equation (DAE) Systems

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1 Nonlinear Model Reduction of Differential Alebraic Equation DAE Systems Chuili Sun and Jueren Hahn Department of Chemical Enineerin eas A&M University Collee Station X hahn@tamu.edu repared for presentation at the 004 Annual Meetin Austin X November 7- Copyriht Chuili Sun and Jueren Hahn November 004 Unpublished AIChE shall not be responsible for statements or opinions contained in papers or printed in its publications

2 Abstract Most process models resultin from first principles consist of not only nonlinear differential equations but also contain nonlinear alebraic equations resultin in nonlinear DAE systems. Since lare-scale nonlinear DAE systems are too comple to be used for real-time optimiation or control model reduction of these types of models is a stratey that needs to be applied for online applications. However in the past model reduction techniques mainly focused on differential equations and no eneral model reduction methods specifically eared towards reducin DAE systems have been proposed. Since in most cases the number of alebraic equations by far eceeds the number of differential equations it is insufficient to simply reduce the differential equations by conventional methods OD balancin etc. and leave the rest of the model intact. his paper presents a novel technique for reducin nonlinear DAE systems. his method can reduce both the order of the model by eliminatin some of the differential equations as well as the number and compleity of the alebraic equations. his is achieved by a 3-step approach: performin order reduction of the differential equations and alebraic equations; identifyin correlation in the variables that connect the retainin differential equations to the alebraic ones; 3 reduction of the input-output behavior of the alebraic equations via system identification techniques. his procedure has the advantaes over other methods in that it addresses both reduction of the alebraic and the differential equations and that it results in a system where the alebraic equations can be represented by a feedforward neural network. his last property is important insofar as the reduced model does not require a DAE solver for its solution but can instead be computed by reular ODE solvers. A more detailed description of the model reduction procedure is provided net: in a first step the controllability and observability covariance matrices for the differential variables are computed. While a controllability covariance matri can be computed for the alebraic as well as the differential variables it is important to point out that the information contained in this matri is only meaninful for the description of the input-to-state behavior of the differential equations. At the same time the covariance matri can be used to determine the relationship between the alebraic variables. Since alebraic variables have no dynamics the perturbation on the alebraic variables cannot introduce output responses which are not already reflected in the perturbation of the differential states. herefore describin a state-to-output behavior for the alebraic variables is not meaninful. In a second step balancin is applied to compute the state transformation matri for the differential variables; and sinular value decomposition is applied to determine the deree of correlation between the alebraic variables. he two transformation matrices obtained from these computations can then be applied to the system. he resultin model still has the same order and identical input-output behavior to the oriinal system. However it has the advantae that it can easily be determined how much a model can be reduced without loosin the important parts for the input-output behavior of the system. he model reduction itself is performed by balanced truncation or residualiation for the differential equations and by replacin the alebraic equations with an eplicit epression obtained from system identification. Feedforward neural networks are used in this work for the reduction of the alebraic equations. his technique is illustrated with a case study. he behavior of reduced-order models of a distillation column with 3 differential equations and 3 alebraic equations is compared.

3 for for. Introduction Most process models derived from first principles consist of differential as well as alebraic equations resultin in differential-alebraic equation DAE systems [ ]. Models described by DAE systems are often of hih order resultin in difficulties for online control due to the etensive computational effort. Reducin the sie of the model while retainin important system properties for controller desin is the main oal of control-relevant model reduction. his paper presents a new technique for reducin nonlinear DAE systems for controller desin. he method reduces both the order of the model by eliminatin differential equations as well as the number and compleity of the alebraic equations. his technique addresses both reduction of the alebraic and the differential equations and results in a system where the alebraic equations can be represented by an eplicit epression e.. a feedforward neural network. his last property is important insofar as the reduced model does not require a DAE solver for its solution but can instead be computed by reular ODE solvers.. Model reduction of DAE systems he work presented in this paper focuses on DAE systems of the followin form & f u 0 y h he reduction is performed by first determinin state transformations that transform the system into a form appropriate for model reduction which is then followed by a truncation/system identification procedure... Computation of transformations For ordinary differential equation ODE systems the procedure of computin the transformation matri via balancin of controllability and observability covariance matrices was presented in Hahn and Edar [3]. However since alebraic equations do not represent dynamic behavior it is not meaninful to compute the controllability and observability covariance matrices for the entire systems differential states and alebraic variables. herefore a modified version of the balancin procedure is presented for application to DAE systems: the system is oriinally at steady state and is then ecited by chanes in the inputs alon the lines of the computation procedure for the controllability covariance matri; data are collected alon the trajectories enerated by these ecitations and a covariance matri d d d nd W W W R W R W nd d nd nd W W W R W R. is computed where WB Bis equal to WBCB the controllability covariance matri of the differential states and WB Bis the covariance matri of the alebraic variables; the observability covariance matri WBOB is computed for states described by the differential equations; 3 the transformation BB the states is computed from balancin BandB WBC BWBOB; 4 a sinular value decomposition of WBB is used to compute the transformation BB W * U Σ V 3 the alebraic variables where BB UBB.

4 .. ransformed system ransformations BB and BB d R d n R 4 can be applied to the oriinal model equation resultin in the transformed system 0 h h y u f u f & 5 where h f represent nonlinear functions of the transformed system. his transformed system has the same number of differential and alebraic equations as the oriinal system and identical input-output behavior. However the differential states as well as the alebraic variables are ordered in descendin order with their importance to the control-relevant behavior of the model. Essentially the system 5 is in a set of coordinates suitable for reducin the sie of the model..3. Order reduction of differential and alebraic equations Once the system is transformed into a form suitable for model reduction the number of differential equations and alebraic equations are reduced by truncation resultin in 0 h y u f ss ss & 0] [ 0] [ m m k k I I 6 where contains the states of the reduced system represents the states that are reduced and k is the number of differential equations in the reduced-order model..4. Further reduction of alebraic equations via system identification So far the number of differential and alebraic equations has been reduced separately by the described technique. It is possible to obtain a more suitable system of even smaller sie and lesser compleity. Consider equation 7 which is part of the DAE system where and which allows to rewrite equation 8 as ˆ 0 8 where ĝ represents a new nonlinear function containin variables and only. he relationship between and needs to be retained in the reduced model to achieve a ood approimation of the behavior of the oriinal system. Since the inputs and the outputs of the

5 alebraic equations are dependent in a static manner it is sufficient to identify this static relationship between and as shown in equation 9 ~ 9 resultin in the followin system: & f u ss ~ ss y h hˆ fˆ It has to be taken into account for the identification procedure that the relationship between and is usually nonlinear. One type of model that is able to take this property into account is artificial neural network ANN. he resultin reduced system is iven by & fˆ u ~ here : y hˆ u ANN 0 3. Case study 3.. Model description Consider a distillation column with 30 trays for separation of a binary miture. he column has 3 differential states concentrations of component A. he Wilson equation is used for computation of the vapor-liquid equilibrium resultin in a model with 3 differential equations and 3 alebraic equations. he reflu ratio is set to 3.0 and serves as the manipulated variable while the concentration of the distillate is the output of the system. 3.. Order reduction of differential and alebraic equations When balancin is applied for the reduction of the differential equations the main criterion to determine the number of states to be retained is based on the manitude of the Hankel sinular values of the balanced covariance matrices. Sorted by the manitude from lare to small the first 5 Hankel sinular values are shown in Fi. and able lists the values of the first 6. For this eample truncated systems that contain 3 and 5 states were investiated. he sinular values in the matri Σ Equations indicate that a system with 3 alebraic variables is sufficient for models reduced by balancin. It can also be concluded that systems with more states will more closely approimate the oriinal system. However there is always a tradeoff between the performance i.e. the quality of the approimation and efficiency i.e. the required computational effort as is illustrated in able. Based on Fi. and able it can be concluded that the reduced system with 3 differential equations results in a very ood approimation to the oriinal system with a relatively small computational burden Further reduction of alebraic equations via neural network

6 he neural net contains hidden layer and output layer with 5 nodes in the hidden layer and 3 nodes in the output layer. Hyperbolic tanent functions were used in the hidden layer and linear functions in the output layer. he network was trained usin the Levenber- Marquardt alorithm [4 5]. After order reduction has been performed the remainin differential states serve as the inputs to the neural network while the outputs are iven by the remainin alebraic variables. hree cases are compared to illustrate the performance of the presented method for model reduction of DAE system: a linearied system with 3 differential equations and 3 alebraic variables; only the differential equations are reduced by balanced truncation while the alebraic equations remain unchaned resultin in a reduced system with 3 differential equations and 3 alebraic variables; 3 the system is reduced by the presented procedure i.e. differential equations as well as alebraic equations are reduced by truncation and the effect that the states have on the remainin alebraic variables is identified by a neural network. his last reduced system contains 3 differential equations and a neural network with three inputs and three outputs. Fi. 3 shows a comparison of the performance of these three reduced-order systems for step chanes in the input of -0% and +0%. Several observations can be made based upon Fi. 3: the upper and the lower trajectories are not symmetric which illustrates the nonlinearity of the oriinal system; the performance of the linearied system is not as ood as the ones usin a nonlinear reduced model; 3 the reduced model includin a neural network provides a ood approimation to the full-order system. Althouh the reduced DAE system is a fairly small model it ehibits better performance than case and performance comparable to model and to the oriinal system. 4. Conclusions his paper presents a new approach for the reduction of nonlinear DAE systems. he investiated technique performs order-reduction of the differential equations and reduces the sie and compleity of the model. One of the stron points of this approach is that reduction of the differential and the alebraic equations is not performed independently from one another but the interplay between the states and the alebraic variables is taken into account. he procedure has been illustrated by applyin it to a model of a distillation column. he alebraic equations were reduced by identifyin a feedforward neural network resultin in a model of sinificantly smaller sie that is also easier to simulate since the alebraic variables can be computed via an eplicit epression. References [] K. E. Brenan S. L. Campbell L. R. etold Numerical Solution of Initial-Value roblems in Differential-Alebraic Equations North-Holland 989 [] G.D. Byrne.R. oni Differential-alebraic systems their application and solutions Computer and Chemical Enineerin [3] J. Hahn.F. Edar Balanced approach to minimal realiation and model reduction of stable nonlinear systems Ind. En. Chem. Res [4] K. Levenber A method for the solution of certain problems in least squares Quart. Appl. Math [5] D. Marquardt An alorithm for least-squares estimation of nonlinear parameters SIAM J. Appl. Math

7 able : Hankel sinular values for states State Sinular Value % of sum 94.9% 4.5% 0.4% 0.% 0.078% 0.09% able : Comparison of computation times for reduced-order models States ime Seconds Hankel Sinular Value State Fi.. Hankel sinular values of the distillation column model

8 Fi.. Comparison of reduced systems with different number of remainin differential equations Note: the curves represent the offsets between the values of each case and the correspondin values of the full-order system Distillate Composition A Oriinal system Linearied system Only Diff. Eq. are reduced resented method for DAE ime min Fi. 3. erformance of presented method for DAE model reduction

Reduction of stable differential algebraic equation systems via projections and system identification

Journal of Process Control 5 (25) 639 65 www.elsevier.com/locate/jprocont Reduction of stable differential algebraic equation systems via projections and system identification Chuili Sun, Juergen Hahn