(438b) Near-Optimal Control of Batch Processes by Tracking of Approximated Sufficient Conditions of Optimality for Dynamic Optimization

Control system design is a model-based activity, also for batch process systems. We consider batch process optimization and implementation, where the goal is to obtain optimal real-time behavior with very low real-time computational burden in spite of process uncertainties and disturbances. This can be done if a self-optimizing control structure can be identified. Self-optimizing control is when acceptable economic operation can be achieved using simple feedback control with pre-computed set points without the need to re-optimize when disturbances occur (Skogestad and Postlethwaite 2005).

In the recent years there have been several publications directed towards inherently optimal control systems for batch processes. An interesting approach in that respect is tracking of necessary conditions of optimality, which aims at satisfying the necessary conditions of optimality for the optimal control problem by adapting the inputs when changes occur in the process. The performance of NCO tracking can be excellent, but requires knowledge of the full state vector (Srinivasan, Bonvin et al. 2003; Srinivasan, Palanki et al. 2003).

Consider a batch optimization problem on the Mayer form, where the objective functional depends only on the final state and the plant dynamics;

                                                                                                                    (1)

subject to

                                                                                     (2)

and with measurements depending only on the state;

                                                           .                                                       (3)

Our approach is based on the sufficient conditions of optimality for a dynamic optimization problem without path constraints, such as given by (1)-(3). These conditions are given by the accessory minimum problem of calculus of variations (AMP) (Bryson 1998). The sufficient conditions of optimality take the form of a linear-quadratic regulator problem with time-varying coefficient matrices and a generalized performance index (including cross-terms);

                                                (4)

subject to

                                                                      (5)

The weighting matrix in (4) is known as the Hamiltonian matrix and is the Hessian of the Hamiltonian function for the optimal control problem (1)-(2). The Hamiltonian function for a Mayer problem takes the form

                                                                                                (6)

where  is a vector of time-varying Lagrange multipliers known as the co-states. If the state-feedback solution found by solving the AMP generates a finite input-signal for arbitrary initial state in the neighborhood of the nominal system trajectory, then the nominal trajectory is optimal (Bryson 1998).

The solution to the AMP is a state-feedback law. We wish to approximate the input as generated by this feedback law, but without the use of state information. Consider the case where the cross-terms in the Hamiltonian matrix are zero. We now wish to control  linear combinations of measurements y such that that the state-feedback solution is closely approximated. We define a new performance index based on the linear combinations of measurements, ;

                                                   (7)

In order for the state-feedback solutions generated by the two performance indices to be equal, we need to select the matrices S, Q_c(t) and R(t) to match the entries of the Hamiltonian matrix. Clearly, . Then we must select S and Q_c such that

                                                                                      (8)

We may select the weighting matrix Q_csuch that it scales each controlled variable c_i by its expected variation due to changes in initial condition (the disturbance) or implementation error. Then, near-optimal operation can be obtained by controlling the linear combinations

                                                                                                                      (9)

to track their nominally optimal trajectories. The matrix S can be found by solving for the best candidate matrix in (8) in a least squares sense.

The method is applied to a linear time-invariant problem with three states and two inputs, as well as a production maximization problem for a batch distillation column. Numerical experiments show that near-optimal performance can be obtained by output feedback tracking control of linear combinations of the available measurements. References

Bryson, A. E. (1998). Dynamic Optimization, Pearson Education.

Skogestad, S. and I. Postlethwaite (2005). Multivariable feedback control - analysis and design, John Wiley & Sons, Ltd.

Srinivasan, B., D. Bonvin, et al. (2003). "Dynamic optimization of batch processes - II. Role of measurements in handling uncertainty." Computers & Chemical Engineering 27(1): 27-44.

Srinivasan, B., S. Palanki, et al. (2003). "Dynamic optimization of batch processes - I. Characterization of the nominal solution." Computers & Chemical Engineering 27(1): 1-26