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273497 System Identification and Distributed Control for Multi-Rate Sampling System

Chemical processes usually have many controlled variables, which may have different sampling times. For example, concentration measurements usually have much longer sampling time than temperature measurements. When implementing multiple-input-multiple-output (MIMO) control, traditional system identification (SI) and control techniques need to be extended to handle multi-rate sampling process.

Simply applying standard SI technique to the multi-rate system by using a single rate (the greatest common factor of different sampling times) would yield poor prediction results for the variables with large sampling times. A better approach for SI for multi-rate systems is by using *lifted model*, in which the system inputs and outputs with slower sampling rate are lifted to single rate, which will generate larger dimensions of inputs and outputs for the system models. The benefit of this approach is that all the variables are sharing the same sampling time, thus the traditional control techniques can be applied. Moreover, the faster sampling rate model contains more information than the slower rate model. However, this approach has limitations mainly on two aspects: first the dimensionality of the model will increase due to lifting, and the worst situation happens when some of the variables have much slower sampling rates compared to the fast rate; second, even slight inaccuracy of the lifted model may cause noticeable error of single-rate model when extracting single-rate model from the lifted model.

We propose a new approach to solve the control problem of multi-rate sampling systems that still utilizes the lifted model, but we leverage the advantages of distributed control techniques. The system outputs will be assigned to different subsystems based on their sampling times (in each subsystem, all the controlled variables have the same sampling time, but it is not necessary to have all the variables that have the same sampling time in a single subsystem). Then when identifying the lifted model of one subsystem, just input lifting (to single rate) is required, with the output unchanged, which would further reduce the dimensions of the subsystem models. Moreover, in order to develop distributed control, the influence from neighbor subsystem also needs to be considered. Thus the SI should also include the related inputs of neighbor subsystems as inputs to this subsystem. Subspace identification will be utilized as the SI method to obtain state space model for each subsystem.

After the distributed models under different sampling times are available, distributed model predictive control could be designed to deal with the control problem of multi-rate system. Only at the sampling time of each subsystem, the states of subsystems are updated. At each single-rate sampling time, first the input trajectories from neighbor subsystems are acquired, as constant inputs to this subsystem, then the objective function of the whole system is optimized with respect to its own inputs; more iterations of exchange of optimal trajectory with neighbor controllers and optimization could improve the global optimality. Finally the control sequences are updated to each subsystem. Asymptotic stability could also be guaranteed from the design of distributed model predictive control algorithm. This is important for this approach and needs to be proved carefully via Lyapunov stability.

The case study is the Tennessee Eastman challenge problem. The continuous variables are sampled every 1 minute, while there are also concentrations that are sampled every 6 or 15 minutes. Base control is implemented to ensure the system stability by a set of PI controllers. Then the distributed lifted models are built by subspace identification, and distributed control is used to control the system.

In summary, this work focuses on the control problem of multi-rate sampling systems. The concept of distributed control helps the decomposition of the multi-rate control problem, then standard SI could be conducted and traditional control algorithms can be applied to local controllers. Furthermore, distributed control can lower the dimension of the control problem, which could be challenging when the lifted model approach is utilized and centralized control is implemented.

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