This paper presents a new nonlinear model predictive control (MPC) algorithm for Hammerstein systems subject to constraints on the state, input, and intermediate variable. Taking no account of constraints, a desired linear controller of the intermediate variable is obtained by applying pole placement to the linear subsystem. Then, actual control actions are determined in consideration of constraints by online solving a finite horizon optimal control problem, where only the first control is calculated and others are approximated to reduce the computational demand. Moreover, the asymptotic stability can be guaranteed in certain condition. Finally, the simulation example of the grade transition control of industrial polypropylene plants is used to demonstrate the effectiveness of the results proposed here.

Hammerstein systems consist of the cascade connection of a static (memoryless) nonlinear function followed from a linear dynamic system. Under certain assumptions (such as fading memory), the Hammerstein model could approximately represent nonlinear dynamics of real systems and has been successfully applied to many kinds of industrial processes such as pH neutralization [

Model predictive control (MPC) is an effective control algorithm for handling constrained control problems, and various MPC algorithms have been proposed for control of Hammerstein systems with constraints [

In this paper, we consider the problem of the grade transition control of industrial polypropylene plants and propose an efficient NMPC scheme for Hammerstein systems with overall constraints based on the pole placement method [

Consider the discrete-time Hammerstein systems described by

Given constraints of system (

Consider

Define a finite horizon objective function

This algorithm comprises the following steps:

Given

Set

With

With the current state

Implement

For the algorithm here, only the first control action (i.e.,

It should be pointed out that the feasibility of optimization problem (

In terms of Algorithm MTMPC, the MPC law of constrained Hammerstein system (

Set

Let a set of points

Consider the system (

Consider the system (

Let function

This establishes the theorem.

From the proof of the above theorem, we know that the stability property of closed-loop system (

Instead of conventional two-step MPC algorithms, where the actual control was determined by solving nonlinear algebraic equations based on desired controllers of the intermediate variable resulted from linear MPC, actual control actions here are derived by solving a finite horizon optimal control problem (i.e., nonlinear MPC), which is defined as tracking the desired state trajectories driven by pole placement state feedback controller. Thus, overall constraints of Hammerstein systems are taken into account when designing the predictive controller, and then performance and stability properties are guaranteed. Meanwhile, the ability to handling constraints is enhanced in the algorithm proposed here. Note that the linear controller of intermediate variables in (

Taking an example of polypropylene (PP) grade transition control, we illustrate the effectiveness of the proposed algorithm.

In propylene polymerization industry, melt index (

The grade transition sequence considered here is A

Grade indices and transition constraints.

Grade | MI (g/10 min) | ||||
---|---|---|---|---|---|

A | 2.7 | — | 343.15 | 0.050 | — |

B | 39.0 | — | 343.15 | 0.330 | — |

C | 10.0 | 2.4 | 343.15 | 0.236 | 1.99 |

A | — | — | |||

— | — | — | |||

B | |||||

— |

In the simulation running, the closed-loop desired poles of system (

Melt index

Concentration of ethylene in polypropylene

Input profiles of grade transition processes.

Reaction temperature

Concentration ratio of hydrogen to propylene

Concentration ratio of ethylene to propylene

In order to calculate the quantity of off-specification polymer in grade transition process, we employ the criterion where on-specification polymer is defined as the polymer with

Together with the pole placement method, the paper presented an efficient MPC algorithm for control of Hammerstein systems subject to constraints on states, inputs, and intermediate variables. Applying pole placement to the linear subsystem of Hammerstein models yielded linear controllers of the intermediate variable and then the actual control action was determined by online solving an optimization problem. The size of the online optimization problem depended only on the number of inputs, not on the predictive horizon, which greatly reduced the computational demand of the algorithm. With condition of feasibility of the optimization problem, the asymptotical stability of the closed-loop system with constraints was guaranteed and the results on simulation example of grade transition control of polypropylene plants showed the effectiveness of the results obtained here.

This work is supported by the National Natural Science Foundation of China under Grant no. 60904040, Specialized Research Fund for Doctoral Program of Higher Education of China under Grant no. 20093317120002 and Natural Science Foundation of Zhejiang Province under Grant no. Y1100911. Finally, the authors are grateful to the reviews’ constructive comments. This paper includes revised and extended text of the first author’s presentation at the 29th CCC, July 27, 2010, Beijiang, China.