# 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
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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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