Patent ID: 11860078
Assignee: NORTHWESTERN UNIVERSITY
Field: Measurement (Instruments)
Classification: CPC B  G | IPC B  G

Claim 3:
4. The control method according to claim 3, wherein according to a SDC theory, a modelling and control method of a stochastic distribution system is described as follows:
y∈[α, ξ] represents a uniformly-bounded stochastic variable for describing output of a dynamic stochastic distribution system, and is an output random variable; and u(k)∈Rm represents a control input of the stochastic distribution system at the time k, which indicates that the output random variable y is described through the PDF shaping at any sampling time k, and a definition formula is shown as follows:, P
      ⁡
      (
      
       
        a
        ≤
        y
        <
        ξ
       
       ,
       
        u
        ⁡
        (
        k
        )
       
      
      )
     
     =
     
      
       ∫
       0
       ξ
      
      
       
        γ
        ⁡
        (
        
         y
         ,
         
          u
          ⁡
          (
          k
          )
         
        
        )
       
       ⁢
       dy
      
     
    
   
   
    
     (
     1
     )
    
   
  
 

wherein in the formula (1), γ(y,u(k)) represents the PDF of the output random variable y, and is an output PDF; and P(a≤y<ξ, u(k)) represents a probability that the output random variable y of the stochastic distribution system falls in a range [α,ξ] under the action of the control input u(k) at the time k, and an output PDF shaping γ(y,u(k)) is controlled by the control input u(k);
the control input u(k) represents the screw feeding amount, the mill disc gap and the mill speed; and
the output random variable y represents the powder particle sizes, and the output PDF γ(y,u(k)) represents the shape of the distribution of the powder particle sizes;
a neural network is adopted for approximating the output PDF at any moment, and the neural network adopting a fixed structure is adopted for approximating the output PDF, and the neural network comprises a B-spline neural network and an RBF (radial basis function) neural network; weight of the neural network is related to the control input u(k), and the output PDF is controlled by controlling the weight of the neural network;
the B-spline neural network is adopted for approximating the output PDF, so as to obtain the following formula:, γ
       (
       
        y
        ,
        
         u
         ⁡
         (
         k
         )
        
       
       )
      
      =
      
       
        
         ∑
         
          i
          =
          1
         
         n
        
        
         
          
           ω
           i
          
          (
          
           u
           ⁡
           (
           k
           )
          
          )
         
         ⁢
         
          
           B
           i
          
          (
          y
          )
         
        
       
       +
       
        e
        ⁡
        (
        
         y
         ,
         
          u
          ⁡
          (
          k
          )
         
        
        )
       
      
     
     ;
     
      ∀
      
       y
       ∈
       
        [
        
         a
         ,
         b
        
        ]
       
      
     
    
   
   
    
     (
     2
     )
    
   
  
 

wherein in the formula (2), ωi(u(k)) represents weight of the B-spline neural network at the time k, and Bi(y) represents a corresponding B-spline basis function; and e(y, u(k)) represents an approximate error and is ignored;
an integral of the output PDF within a definition domain is always equal to 1, which indicates that n−1 weights in n weights are independent mutually, so as to obtain the following formula:

γ(y,u(k))=C(y)V(k)+h(V(k))Bn(y)  (3)

wherein in the formula (3), C(y)=[B1(y), B2(y), . . . , Bn-1(y)], V (k)=[ω1(k), ω2 (k), L, ωn-1(k)]T and h(V(k)) represent function expressions of front n−1 weights; and a following dynamic relation between the following neural network approximate weight and the control input u(k) is considered:

V(k+1)=ƒ(V(k),u(k))  (4)

wherein in the formula (4), ƒ(·) represents a function relation between the control input and the weight and is a conventional linear function or non-linear function, and a system formed by the formula (3) and the formula (4) is used for describing a relation between the control input u(k) and the output PDF of the stochastic distribution system; and therefore, the output PDF shaping is controlled by designing appropriate control input u(k);
based on the formula (3) and the formula (4), a dynamic model of the output PDF of the stochastic distribution system is shown as follows:, {
     
      
       
        
         
          V
          ⁡
          (
          
           k
           +
           1
          
          )
         
         =
         
          f
          ⁡
          (
          
           
            V
            ⁡
            (
            k
            )
           
           ,
           
            u
            ⁡
            (
            k
            )
           
          
          )
         
        
       
      
      
       
        
         
          γ
          ⁡
          (
          
           y
           ,
           
            u
            ⁡
            (
            k
            )
           
          
          )
         
         =
         
          
           
            C
            ⁡
            (
            y
            )
           
           ⁢
           
            V
            ⁡
            (
            k
            )
           
          
          +
          
           
            h
            ⁡
            (
            
             V
             ⁡
             (
             k
             )
            
            )
           
           ⁢
           
            
             B
             n
            
            (
            y
            )
           
          
         
        
       
      
     
    
   
   
    
     (
     5
     )
    
   
  
 

according to a SDC principle, based on a tracking PDF error shown in the formula (6), the set target PDF shaping of the distribution of the powder particle sizes is controlled by designing different control inputs u(k);

e(y,u(k))=g(y)−γ(y,u(k))  (6)

wherein in the formula (6), g(y) represents the set target PDF shaping of the distribution of the powder particle sizes;
in order to obtain an appropriate control input u(k), the control input u(k) is obtained by adopting an optimal performance index shown in the formula (7):, J
     =
     
      
       
        ∫
        a
        b
       
       
        
         
          (
          
           
            g
            ⁡
            (
            y
            )
           
           -
           
            γ
            ⁡
            (
            
             y
             ,
             
              u
              ⁡
              (
              k
              )
             
            
            )
           
          
          )
         
         2
        
        ⁢
        d
        ⁢
        y
       
      
      +
      
       
        R
        1
       
       ⁢
       
        
         u
         2
        
        (
        k
        )
       
      
     
    
   
   
    
     (
     7
     )
    
   
  
 

wherein in the formula (7), J represents a performance index adopted for designing the control input u(k), and R1 represents weight of the control input.