Optimal operation of chemical processes can in general be formulated as a dynamic optimization problem. For many economic optimization problems a quasi steady-state description is acceptable and steady-state models may be used. For example, this is the case with most real-time optimization implementations. Other processes though, are transient in nature, and the dynamic behavior must be considered (Schlegel et. al., 2005). Such transient processes may be start-up and shut-down of continuous plants, grade changes or batch operations. The solution of such a problem is discontinuous in nature and consists of a set of arcs (Srinivasan et. al., 2003a).

The optimal solution should not be implemented in an open-loop manner in most cases because of uncertainty and unknown disturbances. It may be assumed that for small disturbances, the structure of the optimal solution stays the same. With structure we mean the number of arcs and their qualitative shapes. The optimal solution is often very sensitive to the switching times. An open-loop implementation with time-controlled switches between the arcs may easily lead to infeasibility or severe loss with regard to the true optimal solution. Two paradigms exist for implementation of near optimal control:

1. on-line optimization with feedback from measurements mainly used to update the model (MPC)

2. off-line optimization and on-line implementation using feedback to track the optimal properties of the solution ("self-optimizing control")

We follow the second option here (Narasimhan and Skogestad, 2007), and concentrate on obtaining properties of the optimal solution that ensure near-optimal control in spite of disturbances.

For steady state processes, the maximum gain rule is a good screening method for finding promising controlled variables within the self-optimizing control paradigm (Skogestad and Postlethwaite, 2005). The maximum gain rule has previously only been available for steady-state processes with linear process descriptions. This work suggests an extension to a general unsteady-state system where the operating policy is dynamic in nature, as found in batch unit operations in the chemical process industries.

The idea is that the performance of the plant can be described (locally) by a convex objective function. This function can then be approximated in a region around the nominally optimal operating region by a quadratic Taylor polynomial. This can be done also for the Hamiltonian in  nonlinear dynamic systems by expansion around the nominal trajectories. For the selection problem it is assumed that the inputs and states are unconstrained, because for optimal operation the active constraints should be controlled first; then self-optimizing control can be used to assign objectives to the remaining degrees of freedom. In self-optimizing control for static systems, the loss is defined as the difference between the true value of the objective function and the value when the optimal input is applied for a given disturbance; this is a positive number. In dynamic systems, the loss is a function of time, and is defined as the difference between the value of the Hamiltonian function for the implemented input and the optimal value. The Hamiltonian takes a constant value along the optimal trajectory for problems where time is not explicitly occurring in the objective functional (Naidu, 2002).

For each arc in the open-loop solution of the dynamic optimization problem it is assumed that there is a corresponding self-optimizing control structure. This means, for each arc there should be as many self-optimizing variables available to control as there are remaining degrees of freedom when the active constraints have been taken care of. It is shown that using the theory of neighboring optimal control and the LQ regulator problem for time-varying systems, it is possible to determine a time-varying gain-matrix M which relates measurements and inputs. It follows from simple analysis of the Taylor expansion of the Hamiltonian that optimal variable selection minimizes the norm of a time-varying matrix incorporating the sensitivity of the Hamiltonian to measurement variations as well as the time-varying process gains. Because there should only be one self-optimizing control structure for each arch the minimization can be done over the time-averaged 2-norm of the scaled time-varying gain matrix, where optimization and implementation errors in the sense of (Skogestad and Postlethwaite, 2005) have been included through scaling.

The method for variable identification will be demonstrated on a fed-batch bioreactor problem with four states, where the objective is to maximize the product concentration at the end of the batch. The same example has been used in the context of NCO tracking (Srinivasan et. al., 2003b).

References

Martin Schlegel, Klaus Stockmann, Thomas Binder and Wolfgang Marquardt (2005): Dynamic optimization using adaptive control vector parameterization. Comp. Chem. Eng. (29) pp. 1731-1751.

B. Srinivasan, D. Bonvin, and S. Palanki (2003a): Dynamic optimization of batch processes I. Characterization of the nominal solution. Comp. Chem. Eng. (27) pp. 27-4

Sigurd Skogestad and Ian Postlethwaite (2005): Multivariable feedback control - analysis and design, Wiley & Son Ltd., Leicester.

Desineni Subbaram Naidu (2002): Optimal control systems, CRC Press, Boca Raton.

Sridharakumar Narasimhan and Sigurd Skogestad (2007): Implementation of optimal operation using off-line computations, DYCOPS07 (submitted).

B. Srinivasan, D. Bonvin, E. Visser and S. Palanki (2003b): Dynamic optimization of batch processes II. Role of measurements in handling uncertainty. Comp. Chem. Eng. (27) pp. 27-44

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