Process Observers and Data Reconciliation Using Mass and Energy Balance Equations - Advanced Control and Supervision of Mineral Processing Plants - page 47

When the node imbalance estimation of the flowrates is followed by the recon-
ciliation of the species concentrations for these flowrate estimates, the method is
a sub-optimal two-step LQ method [34]. The next section shows that a similar hi-
erarchical technique can be used to find the true optimal solution to the bilinear
reconciliation problem.

2.8 The Non-linear Cases

As discussed above, when the constraints and the measurement equations are linear,
the solution of the reconciliation problem can be developed analytically. However,
in non-linear cases, it is normally impossible to derive an explicit expression for the
reconciled states. Several methods (for instance [26], [81-86]) are possible depend-
ing upon the approach selected to handle the constraints (substitution or Lagrange
multipliers techniques), and the optimization technique that is used to minimize the
criterion. The following optimization techniques are possible options:

• Any Nnon-linear programming (NLP) method, involving a search algorithm to

iteratively approach the optimal values. The substitution method is well adapted
to these procedures since it decreases the number of search variables to be ma-
nipulated by the NLP algorithm.

• A hierarchical minimization method where the criterion and the search variables

are split into blocks. These approaches allow hybrid minimization methods, in
which some parts are optimized by analytical methods, others by NLP algo-
rithms.

• Iterative numerical resolution of the system of equations expressing that the La-

grange criterion derivatives have zero values.

It would be too long to detail all the possible options for solving non-linear rec-
onciliation problems, but examples may help to illustrate some possible approaches.

2.8.1 An Example of Substitution Methods: Mass and Heat
Balance of a Thermal Exchanger

Let us consider the mixer-exchanger of Figure 2.8 [87], which heats an electrolyte
solution containing dissolved copper and nickel subsequently fed to an electro-
refinery plant. Part of the solution is directly recycled to the mixer-exchanger, and
another part is cleaned from impurities before being recycled. Data collected and
averaged for a time period of 3 h are presented in Table 2.3, together with an esti-
mation of their standard deviations. The process variables are the solution flowrates
F , densities ρ, temperatures T , copper and nickel concentrations c and n for the
streams 1, 2 and 3, and heating power P . The system parameters are the specific
heats of the solutions C , respectively 4400, 4210, 4350 J /( kg . K ) for streams 1, 2