where $c$ and $x_1$ are the concentrations of the input and reactant substance, $x_2$ is the concentration of the desired output product, $u$ is the normalized input flow rate of the reactant substance. The output $y$ here reflects the grade of the final product. The parameters $k_1, k_2, k_3$ and $c$ are positive constants under the isothermal considered conditions.
The issue of optimal output transition control for nonlinear differential flat systems with constraints is investigated. Of special interest is the generation of a state reference trajectory and a feedforward input, which are essential to the two-degree-of-freedom design. The proposed approach is to transfer the transition problem in the real output space into a trajectory planning issue in the flat output space. The distinguishing feature of our approach is the generation of a noncausal trajectory for the flat output. This approach is shown to be highly effective in creating performance improvements. It should also be noted that the proposed methodology guarantees that the planned trajectories are feasible for all nonlocal transitions. This allows the application of stable inversion in the planning of optimal output transitions. The proposed method is illustrated on a benchmark system, the isothermal continuous stirred tank reactor, although its applicability is much wider.
