Regression Optimal Functional Control for a Kind of Unsymmetrical System

Because of the widespread existence of unsymmetrical system in the produc-tion process, its research is getting more and more attention. In this paper, a regression optimal functional control method is proposed for a class of unsymmetrical system. For the positive-negative model of the unsymmetrical system, a regression optimal functional controller is designed, which can make the system stable. The proposed algorithm has less computation and good control effect. Finally, three simulation examples are given to verify the effectiveness of the proposed algorithm.

linearized model control of the whole system in [14] [15]. The regression optimal control remedies the error caused by the linearization process, but it is difficult to obtain the analytical solution of the controller.
As one of the classic ROC algorithms, Regression Optimal Functional Control (ROFC) has received wide attention. Different from other control algorithms, regression optimal functional control algorithm designs the controller as a linear combination of known basis functions, and the analytic solution of the controller can be obtained easily [16]. In [17], the regression optimal functional control based on multi-model switching is proposed, which greatly reduces the calculation and achieves good control effects. In [18], a Proportional Integral Derivative (PID) control algorithm based on regression optimal functional control idea is proposed, which not only contains the advantages of the regression optimal functional control algorithm, but also has the simple structure of the traditional PID controller. Considering that the calculation of ROFC is small and the analytical solution is easy to obtain, a regression optimal functional control algorithm for unsymmetrical system is proposed in this paper.
At the end of this section, the main contributions of this paper are summarized: (a) The regression optimal functional control algorithm is applied to unsymmetrical system. (b) The analytical solution of the controller is obtained by using the Minimum Principle.

Problem Statement and Preliminaries
Consider a class of unsymmetrical system: , , This paper focuses on the control algorithm design for unsymmetrical system, so one of the linearization method in [15] is considered to obtain a linear representation of the unsymmetrical system. It is assumed that the origin is the equilibrium point of the system and that f and h are continuously differentiable at the origin. Expand the unsymmetrical system (1) at equilibrium point: The above linearization method is considered to obtain a linear representation of the unsymmetrical system (1) in the positive and negative directions: where A σ , B σ and C σ are constant parameter matrices with appropriate dimensions.
According to the recursive relationship, regression optimal model (4) can be written as:

Theory of Regression Optimal Functional Control
The design of the regression optimal functional control algorithm consists three steps as follows:

1) Multi-step prediction
Select appropriate regression optimal model for multi-step prediction: , , 2) Rolling optimization Minimizing suitable performance indicator to calculate the control law: By solving equation (8), we can get ( ) u k j + . Only ( ) u k is applied to the system.

3) Feedback correction
At the next time k + 1, according the error of the expected output and the systems output, the performance indicator J is reoptimized to calculate ( )

Design of Regression Optimal Functional Controller
According to the idea of regression optimal functional control, we choose controller as: where l σ µ is the weight coefficient of the linear combination of basis functions, l g is a set of known basis functions, L is number of basis functions, 1, 2, , Then, can be expressed as: Substituting Equation (10) into the regression optimal model (5) (6), and the regression optimal model can be written as:

Analysis of Performance Index
According to regression optimal model (11), the expected output is defined as: , the optimization problem of performance index is proposed: are the weight matrices of appropriate dimensions, and Substituting Equation (10), (11) into Equation (12): The necessary condition of minimum performance index is 0 J u σ σ ∂ = ∂ , then: Then, we select that And the current moments control input can be expressed as:

Design of Switching Law
Define the increment of the control input as: The principle of the system selects controller is as follows (2) If (2) When ( ) To sum up, when switching does not occur, the regression optimal controller can stabilize the system; when switching occurs, the switching law is designed according to the principle of positive and negative model matching. Under the constraint of switching law, the controller switches reasonably between positive controller and negative controller. Finally, the stability of the closed-loop system is guaranteed by feedback correction and rolling optimization.
The executive strategy of the regression optimal functional control algorithm in this paper is given as follows: Step 1: When 1 k = , the analytical solution of ( ) Step 2: ( ) Step 4: Let 2 k k = + , go to the Step 1.

Simulation Examples
Consider three examples to verify the effectiveness of the algorithm in this paper: Example 4.1 Consider the pH control of acid-alkali neutralization with strong nonlinearity and strong unsymmetry. The neutral reaction of strong acid and alkali is conducted in the continuous stirring reactor. Assuming that the reaction level is stable, a mathematical model of the reaction process in CSTR is established:   The weight coefficients of the performance index Q I = and R I = , regression optimal step 8 P = . The simulation results are shown in Figure 2, Figure   3.
The solid lines and the dotted lines represent the system response trajectories under regression optimal functional control and regression optimal control, respectively. When the system adopts the regression optimal functional control method, the system switches at 1.74 s t = .
When the system adopts the regression optimal control method, the system switches at 3.16 s t = . Simulation results indicate that the regression optimal functional control algorithm can ensure the nonlinear unsymmetrical system state is ultimately stable. And compared with regression optimal controller, the regression optimal functional controller can bring the system state to steady faster.   The solid lines and the dotted lines represent the system response trajectories under regression optimal functional control and regression optimal control, respectively. Both two control methods make the system switches at The simulation results show that the regression optimal functional control algorithm can ensure the system state is ultimately stable. And compared with regression optimal controller, the regression optimal functional controller has better control effect.       optimal functional control can track the expected output in a short time.

Conclusion
In this paper, the regression optimal functional control algorithm for a class of

Date Availability
No data were used to support this study.