A decentralized model predictive controller applicable for some systems which exhibit different dynamic characteristics in different channels was presented in this paper. These systems can be regarded as combinations of a fast model and a slow model, the response speeds of which are in two-time scale. Because most practical models used for control are obtained in the form of transfer function matrix by plant tests, a singular perturbation method was firstly used to separate the original transfer function matrix into two models in two-time scale. Then a decentralized model predictive controller was designed based on the two models derived from the original system. And the stability of the control method was proved. Simulations showed that the method was effective.

Applications of singular perturbation in control theory can be traced back to 1970s [

In actual processes, model predictive control (MPC) was regarded as “the only advanced control methodology which has made a significant impact on industrial control engineering” [

Therefore, we focused on designing an MPC for systems with two-time scale characteristic in this paper. First, we introduced the background of this field. Then we described the two-time scale decomposition of a transfer function matrix with different dynamics in different channels. At part 3, we presented a kind of two-time scale decentralized MPC algorithm step by step and proved its stability. At last, we gave several simulations to test the validity of the two-time scale decentralized MPC algorithm.

In some systems, the dynamics varies with different channels. And the response speeds of those different channels vary so much even in different time scales. The characteristics can be got from the transfer function matrix intuitively. We simply took a two-in-two-out first-order transfer function matrix, for example and gave the following definition.

A Two-in-two-out first-order transfer function matrix

Considering the system

Some papers [

Based on the method mentioned in literature[

So we can rewrite (

This form can be regarded as

We denoted the transfer function of the slow model

Equation (

We had expressions of the fast model

We took system (

Here

Then we can design a decentralized controller based on characteristics of the fast model

MPC is the only advanced control methodology which has made a significant impact on industrial control engineering [

For the two-in-two-out system mentioned above with two-time scale characteristic, we designed a decentralized controller based on different time scales. We took the model without delay

Model prediction based on slow model

On the t-time scale, the fast dynamic achieved a steady state. The model can be fully expressed by information not so necessary as

Feedback Correction based on slow model

Let error vector be

Model prediction based on fast model

On the

Feedback Correction based on fast model

The input

Rolling horizon optimization based on fast model

Let objective function be

Framework of the control system.

We introduced the algorithm step by step in the above sections. And we would like to discuss the stability of the controller in this section. First we put forward a sufficient condition of the controller with one inner circulate.

A two-time scale decentralized MPC with the control parameters

Let the prediction horizon

The response speed of

So

When the inner circulate is large than one, that is,

We considered a two-in-two-out system. Two streams flow into a reactor, and

The response of output

Response of

Response of

Response of

Response of

Response of

Response of

Chen studied the nonlinearity of a CSTR and modeled the CSTR by the following nonlinear equations [

We chose

We chose a steady state and identified the input-output model to design a model predictive controller.

The input-output model is

The respond speed of

Perfect soft-sensing,

Without soft-sensing

Decentralized MPC

For the systems with little dynamic differences in different channels, if the controlled variables can provide sufficient reliable information in high frequency, the standard MPC can be applied and good control quality can be achieved (Figures

Values of parameters.

2 | 1.1 | 1.3 | 1.5 | 1 | 2 | 1000 | 900 | 0.5 | 0.7 | 10 | 8 |

Parameter values.

Variable | Definition | Value |
---|---|---|

Feed concentration of species A | 5.1 mol/L | |

Feed temperature | 104.9 C | |

Collision factor for reaction 1: | ||

Collision factor for reaction 2: | ||

Collision factor for reaction 3: | ||

Normalized activation energy for reaction 1 | ||

Normalized activation energy for reaction 1 | ||

Normalized activation energy for reaction 1 | ||

Enthalpies of reaction 1 | 4.2 kj/mol A | |

Enthalpies of reaction 2 | ||

Enthalpies of reaction 3 | ||

Heat transfer coefficient for cooling jacket | 4.032 kj/(h | |

Surface of cooling jacket | 0.215 | |

Reactor volume | 0.01 | |

Coolant mass | 5.0 kg | |

Heat capacity of coolant | 2.00 kj/(kg | |

Heat capacity | 3.01 kj/(kg | |

Density | 0.9342 kg/L |

Stable states.

2.4308 mol/L | 1.0802 mol/L | 115.4559 C | 114.9944 C | 20 |

Response of

Response of

Response of

Response of

Response of

Response of

In this article, we focused on a kind of special system and designed a decentralized model predictive controller for it. This kind of system has different dynamics in different channels and exhibits two-time scale. A centralized MPC controller cannot satisfy the fast and the slow channels simultaneously. We used singular perturbation method to get the fast and the slow model from the original system. In actual processes, input-output models that can be obtained easily by identification were usually used to describe the real system. We demonstrated the singular perturbation method applying in transfer function matrix. Then we presented a decentralized model predictive controller based on the fast and the slow model and provided a sufficient condition for the algorithm stability when

The algorithm is based on the idea of fully using the information of the system. For the systems with two-time scale characteristics, the fast and slow channels are controlled, respectively, in the decentralized algorithm. This algorithm makes best use of the transition information of the fast channels and the slow channels and reduces the computation burden, which provides short control interval and increases the response speed. For those systems without two-time scale characteristics, this algorithm also works well. MPC has intensively been applied in the industrial process. The two-time scale MPC algorithm which is presented in this paper extends the applying scopes of MPC.

The authors gratefully acknowledge the financial support of 863 Program of China (no. 2007AA041402), National Key Scientific and Technical Project of China (no. 2007BAF22B05) and National Science Foundation of China (no. 60804023).