NONLINEAR MODEL REDUCTION AND ITS APPLICATION TO MODEL PREDICTIVE

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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  Qyk  ysp  d          u          Ruk
                                                 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
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mit einer enthaltenen Vors chau.                   mit einer enthaltenen Vors chau.
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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