Design and Analysis of Bayesian Model Predictive Controller

In this article, a novel predictive controller based on a Bayesian inferring nonlinear model (BMPC) is presented and analyzed. In the construction of the BMPC, the Bayesian inferring model is selected as the predictive model with the characteristics of on-line tracing ability to the actual controlled object. The nonlinear programming method called the steepest gradient is set as the receding horizon optimization algorithm of the BMPC. The on-line controller output is obtained using this method. The convergence analysis of the proposed BMPC is given and the examples (nonminimum phase and nonlinear objects) are selected to validate the performance of the BMPC. The simulation results show that with the help of the presented BMPC algorithm, the closed loop control system demonstrates the abilities of anti-disturbance and robustness.


Introduction
The model predictive control (MPC) method, derived from the practice industrial application, has been widely used in many industrial fields such as petroleum industry, chemical industry and pharmaceutical manufacturing industry (Armando, 2013;Anders, 2013;Ali, 2012).The reason for the success of the MPC is that the MPC method owns three characteristics, namely, predictive model, feedback correction and rolling optimization.Among them, the predictive model is not limited to any special model framework.And any model, which can provide the ability to predict the system output, all can be used as the predictive model.So the extensity of the predictive model contributes to the research prosperity of the MPC (Aswani, 2013;Rodriguez, 2013;Giselsson, 2013).
The predictive model plays an essential role to the predictive accuracy of the MPC.In the past decades, the theory of linear model predictive control based on linear predictive model has already been fully researched.But in industrial control, the majority of controlled objects possess many kinds of nonlinear characteristic.So the research of nonlinear model predictive control is very meaningful and practical.Until now, there is no uniform model describing method for all kinds of nonlinear systems in nonlinear model predictive research.Commonly, three design ideas are presented till now.The first idea is to make the nonlinear model linearization and then to solve the problem based on the developed linear predictive model control method (Christofides, 2013;Lincoln, 2013;Tung, 2013).The second idea is to obtain the nonlinear system model using identification method and converted the controller design problem into nonlinear programming issue.And then the control variable is optimized with the aid of some receding horizon methods.The nonlinear first-principles modeling method is also been used (Mesbah, 2010).
In the second research idea of nonlinear model predictive control, the fuzzy model and neural network (Xiangjie, 2013;Hazil, 2014;Karim, 2013) are two kinds of frequently used nonlinear models.But the designs of membership functions and fuzzy rules all have a great influence on the predictive accuracy.In the design of neural network model, the number of hidden layers and nodes and the dynamic or static neural structure are all Where, in according with the sa The trainin algorithms

Recedi
The most obtained t formula.
Where P th i desire where r i object.
From the ( ),  [ to make the ( ) J k minimum.In the implementation of control, only the * ( ) u k is used to act on the controlled object at the k time moment.Then at the time moment 1 k  , the optimization progress is repeated as the moment k .And in this way, the whole dynamic process of the controlled object is optimized.
The above receding horizon of * ( ) u k belongs to the nonlinear programming problem with constraint conditions.In this work, we choose the steepest gradient method as the receding horizon algorithm.Taking the comprehensive consideration of the training of the Bayesian inferring predictive model and the working flow of the BMPC, we give the implementation procedures of the BMPC method.
(1) Obtain the sample data of the controlled object.Determine the input and output vector of the Bayesian inferring model.Utilize the PSO algorithm (Vahid, 2013;Cabrerizo, 2013) to train the parameters of the threshold matrix D .
(2) Determine the width of the sliding window N and validate the on-line predictive ability of the obtained Bayesian inferring model.
(3) Based on the trained Bayesian inferring model, the implementation of the BMPC is set as follows.

Convergence Analysis of Bayesian Model Predictive Controller
The objective function ( ) J k is described as the vector form.
According to the above formula Obtain the one first order derivative of the The approximation of the first order derivative is set as Take the formula 10 into the above equation and the obtain the following formula \begin{equation} www.ccsen In order t non-increa

Simulat
To investig the closed In the expe

Results
From the parameters model is u prediction structure u In the clos ability of r solution.F effectivene system als quickly.

Conclusion
In this note, the Bayesian model predictive control system is designed based on the Bayesian inferring model.The particle swarm optimization algorithm is used to the off-line training of the Bayesian inferring model.In the on-line implementation of the Bayesian inferring model, the sliding window method is utilized to achieve the structure updating so as to capture the nonlinear characteristics of system quickly.The receding horizon optimization in the design of the Bayesian model predictive controller adopts the steepest gradient method, which ensures the convergence ability of the proposed model predictive control algorithm.And the experiments implemented on the nonminimum phase and nonlinear systems show that the Bayesian model predictive control system owns good control effect, anti-disturbance ability and robust performance.
Figure 2. Il a) Obtain the current output ( ) y k and then compute the Bayesian inferring model output ˆ( ) y k .The error ( the Bayesian inferring model to computer the ˆ( ), 1,..., y k i i P   ; according to the formula 1, compute the corrected predictive output ( ) p y k i  ; at the same time, compute the desired system output ( ) r y k i  according to the formula 7. c) Use the steepest gradient method to solve the minimization of the objective function ( ) J k and the optimized control variable * ( ) u k is obtained.d) Take the * ( ) u k to implement on the controlled object.Then go to the step a) to continue the next optimization procedure at the time moment 1 k  .

Figure
Figure 4. O Fig