(139f) A Combined Lqr-MPC Robust Strategy With Input Saturation for the Control of a Deisobutanizer Distillation Column

Since the early applications of Model Predictive Control (MPC) more than three decades ago, this control method has shown a tremendous development and has been largely implemented in areas such as oil refining, chemical, food processing, automotive and aerospace industries (Qin and Badgwell, 2003) and nowadays continues to gain the interest of other fields such as in medical research (Lee and Bequete, 2009).

Based on a model of the system to be controlled, MPC calculates at each time step a sequence of manipulated inputs that optimizes the predicted behavior of the system, usually subject to constraints on the inputs and on the outputs.

In the unconstrained case, MPC becomes equivalent to the Linear Quadradic Regulator (LQR) where a state feedback control law can be expressed as u(k+i)=K(k)x(k+i), i≥0. In the constrained case, the approach does not usually lead to a linear control law and because the control moves are not only a function of the actual state of the plant, they are denominated free control moves. In real MPC implementations, stability can be achieved for the nominal system but not in the general case where there are uncertainties associated with the unstable modes of the system. For instance, González et al. (2007) achieved stability for a limited class of systems with uncertainties on the integrating modes.

Kothare et al. (1996) proposed a strategy that is an extension of the LQR approach to the constrained case. They include conservative LMI constraints on the inputs and outputs so that the solution to the constrained case becomes a state feedback control law as in the unconstrained case. With this approach, the closed-loop stability can be achieved for the uncertain stable and unstable systems. Although the approach has opened the gate to several developments in the field of robust MPC and was improved over time, it still suffers from some limitations as: the conservative way the constraints are implemented reduces the attraction domain of the controller; the zone control strategy cannot be directly addressed and the large number of decision variables impacts the computational burden of the robust MPC of large systems.

In this paper, the free control moves and the state feedback control law strategies are integrated in a control strategy that benefits from the advantages of both strategies. In the configuration proposed in this work, a sub-set of the manipulated inputs of the system is allocated to control the unstable outputs through a state feedback control law while the other manipulated inputs are left free to control the remaining stable states of the system. In this approach, the unstable outputs are controlled internally and only the stable outputs are actually explicitly controlled by the free control moves. 

Focusing on the implementation on real systems of the process industry, the so called zone control strategy (Maciejowski, 2002) is considered and it is assumed that an upper layer in the control structure defines targets for some of the inputs. In this strategy, the set-points of the stable outputs are treated as additional decision variables that can be varied inside the output zones, while the set-points of the unstable outputs actually become new manipulated inputs in place of the inputs chosen to control the unstable modes of the system.

Unlike the conventional state feedback control law approach, the controller developed here allows the saturation of the inputs resulting from the state feedback control law by manipulating the remaining inputs of the system and the set-points of the unstable outputs. However, to make sure that these constraints can always be satisfied, the set-points of the unstable outputs are allowed to leave their zone when the constraints on these inputs may turn the controller unfeasible. This was achieved by considering the method proposed in Ferramosca et al. (2010).

Based on this formulation, a controller that benefits from the advantages of both the free control moves and the state feedback control law strategies is then proposed here. Unlike the free control moves strategy, the method can easily deal with uncertainties on the unstable modes of the system, and unlike the state feedback control law strategy, the input target and zone control strategy can be adopted. Also, the saturation of the inputs is allowed and the domain of attraction of the controller is maximized by the use of slack variables. The convergence and stability of the controller is proved for both the nominal and multi-model cases.

To avoid the use of a state observer, one can use a state space model as the realigned state model in which the state is composed of the past measured outputs and inputs of the system (Maciejowski, 2002). Recent papers dealing with model predictive control based on such non-minimal model includes Wang and Young (2006), González et al. (2009) and Zhang et al. (2011). In this work, the model representation proposed in González et al. (2009) is adopted.

The proposed robust controller is tested through the simulation of the control of a deisobutanizer distillation column.

References

Ferramosca, A., Limon, D., González, A. H., Odloak, D., and Camacho, E. F. (2010), 'MPC for tracking zone regions', Journal of Process Control, 20 (4), 506-516.

González, A. H., Perez, J. M., and Odloak, D. (2009), 'Infinite horizon MPC with non-minimal state space feedback', Journal of Process Control, 19 (3), 473-481.

González, A. H., Marchetti, J. L., and Odloak, D. (2007), 'Extended robust model predictive control of integrating systems', AIChE Journal, 53 (7), 1758-1769.

Kothare, M. V., Balakrisnan, V., and Morari, M. (1996), 'Robust constrained model predictive control using linear matrix inequalities', Automatica, 32 (10), 1361-1379.

Lee, S. M. (2009), 'Robust model predictive control using polytopic description of input constraints', Journal of Electrical Engineering and Technology, 4 (4), 566-569.

Maciejowski, J. M. (2002), Model Predictive Control with Constraints, Essex (England): Pearson Education.

Qin, S. J., and Badgwell, T. A. (2003), 'A survey of industrial model predictive control technology', Control Engineering Practice,11 (7), 733-764.

Wang, L., and Young, P. C. (2006), 'An improved structure for model predictive control using non-minimal state space realisation', Journal of Process Control, 16 (4), 355-371.

Zhang, R., Xue, A., Wang, B., and Renc, Z. (2011), 'An improved model predictive control approach based on extended non-minimal state space formulation', Journal of Process Control, 21 (8), 1183-1192.