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Because of the widespread existence of unsymmetrical system in the production 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.

As the most promising control theory, nonlinear system control theory has become the theme of modern control theory [

Most of the early research about unsymmetrical system focus on system linearization, but pay less attention to the design of control algorithms. With the advantages of multi-step prediction, rolling optimization and feedback correction, Regression Optimal Control (ROC) [

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 [

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.

Consider a class of unsymmetrical system:

( x ( k + 1 ) = f σ ( x ( k ) , u ( k ) ) y ( k ) = h σ ( x ( k ) ) , (1)

where x ( k ) ∈ R n is the system state, u ( k ) ∈ R m is the control input, y ( k ) ∈ R q is the system output. The control input u ( k ) satisfies the following condition: ‖ u ( k ) ‖ 2 ≤ u max , ‖ ⋅ ‖ denotes the Euclidean norm. f and h are smooth functions with appropriate dimensions. And the determining function of the positive-negative model is given as follows:

σ = ( + , if u ( k ) increases − , if u ( k ) decreases , (2)

This paper focuses on the control algorithm design for unsymmetrical system, so one of the linearization method in [

( x ( k + 1 ) = ∂ f σ ∂ x T x ( k ) + ∂ f σ ∂ u T u ( k ) y ( k ) = ∂ h σ ∂ x T x ( k ) , (3)

The above linearization method is considered to obtain a linear representation of the unsymmetrical system (1) in the positive and negative directions:

( x ( k + 1 ) = A σ x ( k ) + B σ u ( k ) y ( k ) = C σ x ( k ) , (4)

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:

[ x ( k + 1 ) x ( k + 2 ) ⋮ x ( k + P ) ] = [ A σ ( A σ ) 2 ⋮ ( A σ ) P ] x ( k ) + [ B σ 0 0 ⋯ 0 A σ B σ B σ 0 ⋯ 0 ⋮ ⋮ ⋮ ⋱ 0 ( A σ ) P − 1 B σ ( A σ ) P − 2 B σ ⋯ ⋯ B σ ] [ u ( k ) u ( k + 1 ) ⋮ u ( k + P − 1 ) ] , (5)

[ y ( k + 1 ) y ( k + 2 ) ⋮ y ( k + P ) ] = [ C σ A σ C σ ( A σ ) 2 ⋮ C σ ( A σ ) P ] x ( k ) + [ C σ B σ 0 0 ⋯ 0 C σ A σ B σ C σ B σ 0 ⋯ 0 ⋮ ⋮ ⋮ ⋱ 0 C σ ( A σ ) P − 1 B σ C σ ( A σ ) P − 2 B σ ⋯ ⋯ C σ B σ ] [ u ( k ) u ( k + 1 ) ⋮ u ( k + P − 1 ) ] . (6)

where P is regression optimal step.

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:

( x ^ ( k + 1 ) = f ^ σ ( x ( k ) , u ( k ) ) y ^ ( k ) = h ^ σ ( x ( k ) ) , (7)

2) Rolling optimization

Minimizing suitable performance indicator to calculate the control law:

m i n J σ ( k + j ) where j = 0,1, ⋯ , P . (8)

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 u ( k + 1 ) .

According to the idea of regression optimal functional control, we choose controller as:

u σ ( k ) = ∑ l = 1 L μ l σ g l ( k ) , (9)

where μ l σ is the weight coefficient of the linear combination of basis functions, g l is a set of known basis functions, L is number of basis functions, l = 1,2, ⋯ , L .

Let Γ σ ( k + i ) = [ μ 1 σ ( k + i ) μ 2 σ ( k + i ) ⋯ μ L σ ( k + i ) ] , T ( k + i ) = [ g 1 ( k + i ) g 2 ( k + i ) ⋯ g L ( k + i ) ] T , i = 0,1, ⋯ , P − 1 .

Then, u σ ( k + i ) can be expressed as:

u σ ( k + i ) = ∑ l = 1 L μ l σ ( k + i ) g l ( k + i ) = Γ σ ( k + i ) T ( k + i ) , (10)

Substituting Equation (10) into the regression optimal model (5) (6), and the regression optimal model can be written as:

( x m = H σ x ( k ) + Γ σ Ψ σ y m = F σ x ( k ) + Γ σ Φ σ , (11)

with

x m = [ x ( k + 1 ) x ( k + 2 ) ⋮ x ( k + P ) ] , y m = [ y ( k + 1 ) y ( k + 2 ) ⋮ y ( k + P ) ] , H σ = [ A σ ( A σ ) 2 ⋮ ( A σ ) P ] , Ψ σ = [ φ 11 φ 12 ⋯ φ 1 L φ 21 φ 22 ⋯ φ 2 L ⋮ ⋮ ⋱ ⋮ φ L 1 φ L 1 ⋯ φ L L ] , F σ = C σ H σ , Φ σ = C σ Ψ σ .

and

φ 11 = B σ g 1 ( k ) , φ 12 = B σ g 2 ( k ) , φ 1 L = B σ g L ( k ) ,

φ 21 = A σ B σ g 1 ( k ) + B σ g 1 ( k + 1 ) ,

φ 22 = A σ B σ g 2 ( k ) + B σ g 2 ( k + 1 ) ,

φ 2 L = A σ B σ g L ( k ) + B σ g L ( k + 1 ) ,

⋯

φ L 1 = ( A σ ) P − 1 B σ g 1 ( k ) + ( A σ ) P − 2 B σ g 1 ( k + 1 ) + ⋯ + B σ g 1 ( k + P − 1 ) ,

φ L 2 = ( A σ ) P − 1 B σ g 2 ( k ) + ( A σ ) P − 2 B σ g 2 ( k + 1 ) + ⋯ + B σ g 2 ( k + P − 1 ) ,

φ L L = ( A σ ) P − 1 B σ g L ( k ) + ( A σ ) P − 2 B σ g L ( k + 1 ) + ⋯ + B σ g L ( k + P − 1 ) .

According to regression optimal model (11), the expected output is defined as: y r = [ r T ( k ) r T ( k + 1 ) … r T ( k + P ) ] T , the optimization problem of performance index is proposed:

m i n J σ

J σ = ( y r − y m ) T Q ( y r − y m ) + ( u σ ) T R u σ , (12)

where Q = d i a g [ Q 1 , Q 2 , ⋯ , Q P ] and R = d i a g [ R 1 , R 2 , ⋯ , R P ] are the weight matrices of appropriate dimensions, and Q = Q T > 0 , R = R T > 0 .

Substituting Equation (10), (11) into Equation (12):

J σ = ( y r − y m ) T Q ( y r − y m ) + ( u σ ) T R u σ = ( y r − ( F σ x ( k ) + Γ σ Φ σ ) ) T Q ( y r − ( F σ x ( k ) + Γ σ Φ σ ) ) + ( Γ σ T ) T R ( Γ σ T ) = ( y r − F σ x ( k ) ) T Q ( ( y r − F σ x ( k ) ) − 2 ( Γ σ ) T ( Φ σ ) T Q ( y r − F σ x ( k ) ) ) + ( Γ σ ) T ( ( Φ σ ) T Q Φ σ + T T R T ) Γ σ . (13)

The necessary condition of minimum performance index is ∂ J σ ∂ u σ = 0 , then:

− 2 ( Φ σ ) T Q ( y r − F σ x ( k ) ) + 2 ( ( Φ σ ) T Q Φ σ + T T R T ) Γ σ = 0 , (14)

Γ σ = ( ( Φ σ ) T Q Φ σ + T T R T ) − 1 ( Φ σ ) T Q ( y r − F σ x ( k ) ) , (15)

Then, we select that

μ 1 σ ( k ) = ( 1,0, ⋯ ,0 ) ( ( φ 11 ) T Q 1 φ 11 + T T ( k ) R 1 T ( k ) ) − 1 ( φ 11 ) T Q 1 ( r ( k ) − C σ A σ x ( k ) ) = Θ r 1 σ r ( k ) − Θ x 1 σ x ( k ) , (16)

μ 2 σ ( k ) = ( 0,1, ⋯ ,0 ) ( ( φ 12 ) T Q 1 φ 12 + T T ( k ) R 1 T ( k ) ) − 1 ( φ 12 ) T Q 1 ( r ( k ) − C σ A σ x ( k ) ) = Θ r 2 σ r ( k ) − Θ x 2 σ x ( k ) , (17)

⋯

μ L σ ( k ) = ( 0,0, ⋯ ,1 ) ( ( φ 1 L ) T Q 1 φ 1 L + T T ( k ) R 1 T ( k ) ) − 1 ( φ 1 L ) T Q 1 ( r ( k ) − C σ A σ x ( k ) ) = Θ r L σ r ( k ) − Θ x L σ x ( k ) . (18)

And the current moments control input can be expressed as:

u σ ( k ) = ∑ l = 1 L μ l σ ( k ) g l ( k ) = ∑ l = 1 L Θ r l σ r ( k ) g l ( k ) − ∑ l = 1 L Θ x l σ x ( k ) g l ( k ) (19)

Define the increment of the control input as:

Δ u σ ( k ) = u σ ( k ) − u ( k − 1 ) , σ = + or − (20)

The principle of the system selects controller is as follows

(1) If Δ u + ( k ) ≥ 0 , Δ u − ( k ) ≥ 0 , then u ( k ) = u + ( k ) ;

(2) If Δ u + ( k ) < 0 , Δ u − ( k ) < 0 , then u ( k ) = u − ( k ) ;

(3) If Δ u + ( k ) ≥ 0 , Δ u + ( k ) < 0 or Δ u + ( k ) < 0 , Δ u + ( k ) ≥ 0 , then the system selects the controller that makes the performance index small: if J + < J − , then u ( k ) = u + ( k ) ; if J + > J − , then u ( k ) = u − ( k ) .

Theorem 1. Consider unsymmetrical system (1), which can be approximated by positive-negative model (4), regression optimal functional controller (19) and switching law can ensure the closed loop system is stable.

Proof: According to the regression optimal functional control algorithm, the analytical solution of the controller u σ ( k ) is obtained by satisfying condition of ∂ J σ ∂ u σ = 0 , which can ensure that the subsystems performance indicator J σ is minimized, respectively.

If u ( k ) = u + ( k ) (or u ( k ) = u − ( k ) ), continue to calculate u σ ( k + 1 ) according to the above algorithm.

(1) When Δ u + ( k + 1 ) ≥ 0 , Δ u − ( k + 1 ) ≥ 0 , then u ( k + 1 ) = u + ( k + 1 ) ;

(2) When Δ u + ( k + 1 ) < 0 , Δ u − ( k + 1 ) < 0 , then u ( k + 1 ) = u − ( k + 1 ) ;

(3) When Δ u + ( k + 1 ) ≥ 0 , Δ u − ( k + 1 ) < 0 or Δ u + ( k + 1 ) < 0 , Δ u − ( k + 1 ) ≥ 0 , the system selects the controller that makes the performance index smaller:

If

( y r − y m ) T Q ( y r − y m ) + ( u + ) T R u + ≤ ( y r − y m ) T Q ( y r − y m ) + ( u − ) T R u − ,

then J + ≤ J − , u ( k ) = u + ( k + 1 ) ;

If

( y r − y m ) T Q ( y r − y m ) + ( u + ) T R u + > ( y r − y m ) T Q ( y r − y m ) + ( u − ) T R u − ,

then J + > J − , u ( k ) = u − ( k + 1 ) .

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 k = 1 , the analytical solution of u σ ( k ) is obtained by satisfying condition of ∂ J σ ∂ u σ = 0 .

Step 2: u ( 0 ) is given, define Δ u σ ( k ) = u σ ( k ) − u ( k − 1 ) . According to the principle of selecting positive or negative controller, u ( k ) = u + ( k ) or u ( k ) = u − ( k ) .

Step 3: If u ( k ) = u + ( k ) (or u ( k ) = u − ( k ) ), continue to calculate u σ ( k + 1 ) according to the above algorithm. (1) When Δ u + ( k + 1 ) ≥ 0 , Δ u − ( k + 1 ) ≥ 0 , then u ( k + 1 ) = u + ( k + 1 ) ; (2) When Δ u + ( k + 1 ) < 0 , Δ u − ( k + 1 ) < 0 , then u ( k + 1 ) = u − ( k + 1 ) ; (3) When Δ u + ( k + 1 ) ≥ 0 , Δ u − ( k + 1 ) < 0 or Δ u + ( k + 1 ) < 0 , Δ u − ( k + 1 ) ≥ 0 , the system selects the controller that makes the performance index smaller: if J + ≤ J − , then u ( k + 1 ) = u + ( k + 1 ) ; if J + > J − , then u ( k + 1 ) = u − ( k + 1 ) .

Step 4: Let k = k + 2 , go to the Step 1.

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:

( V d x ( t ) d t = u ( t ) ( b − x ( t ) ) − F ( t ) ( a + x ( t ) ) p H ( t ) = lg x ( t ) + x ( t ) 2 + 4 K w 2 K w , (21)

where u ( t ) is the flow rate of the alkaline solution, F ( t ) is the flow rate of the acid solution, V is the volume of the reactor, x ( t ) is the difference between [OH^{−}] and [H^{+}], a is the concentration of the acid solution and b is the concentration of the alkaline solution, the water balance constant K w = 10 − 14 .

With the flow rate of the alkaline solution u ( t ) as the control input and pH value p H ( t ) as the output variable, consider the system has only one equilibrium point, the sampling time T s = 0.02 s , establish the positive-negative model:

( x ( k + 1 ) = A σ x ( k ) + B σ u ( k ) y ( k ) = C σ x ( k ) , (22)

where A σ = 1 − T s V u 0 − T s V F 0 + k 1 σ , B σ = T s V ( b − x 0 ) + k 2 σ , C σ = 1 ln 10 x 0 2 + 4 K w , u 0 , x 0 , F 0 are the corresponding values of each variable in the equilibrium state. a = 0.01 kmol / L , b = 0.01 kmol / L , u 0 = 0.112 L / s , F 0 = 0.112 L / s , x 0 = 0.009 , V = 0.02 kL , k 1 σ , k 2 σ are the correction factor of the positive-negative model, k 1 + = 1.84 , k 1 − = − 0.25 , k 2 + = 0.12 , k 2 − = 0.07 .

Select the wavelet functions as the basis functions of the regression optimization functional controller: g ( t ) = 2 3 π ( 1 − t 2 ) exp ( − t 2 2 ) . The output of the three basis functions is shown in

The weight coefficients of the performance index Q = I and

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

Example 4.2 Consider a numerical example:

where

Next, the positive-negative model is discretized, and the sampling time is

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

Example 4.3 (Tracking Problem) The system in Example 4.2 is considered, and the sampling time is

coefficient matrices

In

optimal functional control can track the expected output in a short time.

In this paper, the regression optimal functional control algorithm for a class of unsymmetrical system is proposed. For the positive-negative model of unsymmetrical system, the regression optimal functional controller is designed. Based on the matching of the positive-negative model, the controller switches between positive controller and negative controller reasonably, which can make the system stable. The proposed algorithm has less computation and good control effect. Three simulation examples are used to verify the effectiveness of the proposed algorithm.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

This research was supported by the Natural Science Foundation of China (Grant Nos. 61374004, 61773237, 61473170).

Yue, X.X. and Su, B.L. (2020) Regression Optimal Functional Control for a Kind of Unsymmetrical System. Applied Mathematics, 11, 363-376. https://doi.org/10.4236/am.2020.115026