Identification of Reduced-Oder Dynamic Models of Gas Turbines

							CSC Student Seminars
 (Spring/Summer, 2006)




       Identification of Reduced-
        Oder Dynamic Models of
              Gas Turbines
                         PhD Student: Xuewu Dai
                         Supervisor: Tim Breikin and Hong Wang
             Introduction
 1. Introduction
 2. Reduced-order Model
 3. Long-term Prediction
 4. Dynamic Gradient Descent
 5. Nonlinear Least-Squares Optimization
 6. Future Works
             1. Introduction
 Modlling of Gas Turbines
 Fault Detection
 Condition Monitoring
                 Aims
 Reducing Computational Complexity:
         Real time
 Improving Prediction Accuracy:
         Long-term prediction
 Robustness
                   2. Reduced Order
            Thermodynamic        models:
                1. High order : 26th
                2. Non-linear

           
Linearisation   Our ARX models :
                1. Reduced order: 1st, 2nd …
                2. Linear: y (t )   T  z (t )
                           ˆ
                       [a1 ... an b1 ... bm ]T
           z (t )  [ y(t  1)... y(t  n) u(t  1) ... u(t  m)]
                                                                T
           3. Long-term Prediction

u (t )                      y (t )    u (t )                       y (t )



              Model        ˆ
                           y (t )                    Model         ˆ
                                                                   y (t )


 a. One-step Ahead Prediction Model       b. Long-term Prediction Model
           Model Equations
1.One-step ahead prediction
  y(t )    z (t )
  ˆ         T

       z(t )  [ y(t  1)... y(t  n) u(t 1) ... u(t  m)]    T


2. Long-term prediction
   y(t )    z (t )
   ˆ           ˆT


      z(t )  [ y(t  1) ... y(t  n) u(t  1) ... u(t  m)]
      ˆ         ˆ            ˆ                                 T
                     Challenges
 Computational Burden
    How many iterations need to identify the parameters?
   Dependency of Prediction Errors (Non-Gaussian Noise)
                             MSE=9.1318



                   ( y(t )  y(t )) 2 / N
                              ˆ


                     Autocorrelation of prediction errors
  4. Dynamic Gradient Descent
 Objective     Function
           1 N            1 N
   E ( )   ( (t )) 2   ( y (t )   T  z (t )) 2
                                              ˆ
           2 t 1         2 t 1

 Global    Gradient and local gradient
      E ( ) 1 N  2 (t ) N            (t )
                             (t ) 
             2 t 1       i 1        
     (t )
              g (t )
     
    Dynamic Gradient Descent

 y (t )    z (t )
  ˆ        T
                  ˆ                                    (a)

       E ( )
                         

  k 1                                                (b)
                             
            k

                                    k


   E ( )         N

    ( y (t )  y (t ))  g (t )
                                  ˆ                    (c)
                   t 1

 g (t )  z (t )  [ g (t  1)... g (t  n) 0...0]  (d)
          ˆ
                                DGD SearchRoute
       0.8




       0.7




       0.6




       0.5




       0.4
   b




       0.3




       0.2




       0.1




        0
        0.3   0.4   0.5   0.6                     0.7   0.8   0.9   1

Results 1: deepest direction          a
                                                    DGD+BFGS SearchRoute




              0.7




              0.6




              0.5
          b




              0.4




              0.3             Traing Error
300


200
              0.2

100


 0
      0   10 0.1 20   30    40   50      60    70   80   90     100
               0.4           0.5
                           DGD+BFGS Iteration 0.6         0.7         0.8       0.9    1
                                                                a
                                                                            BFGS direction
        5. Nonlinear Least-squares
       Optimization (Gauss-Newton)
                                    E ( k )
    k 1   k   * [ R(k )] 
                                1
                                                                    (a)
                                      k
   R(k)  J T  J                                                   (b)

     E ( k )        N

                  [( y (t )  y (t )]  g (t , k )
                                     ˆ                                   (c)
       k            t 1
   g (t , k )  z (t  1)  [ g (t  1, k )...g (t  n, k ) 0...0]  (d)
                  ˆ

   J  [ g (1, k ) g (2, k )...... g ( N , k )]T                   (e)


    Search direction, step size and
             initial value
 Search direction:
   Deepest descent: inverse global gradient
   Nonlinear Least Squares: Gauss-Newton
 Step size:
   fixed, adjustable, line search
 Initial value:
   Blind guess: [0.5 0.5 0.5 0.5]
   LSE: [1.2805 -0.29191 0.10582 0.15903]
            Result 3 Gauss-Newton
                          Gauss-Newton + Bisection SearchRoute




      0.6



      0.5



      0.4
b




      0.3
                           Traing Error
200

150
      0.2
100

50
   0.1
 0   0.4         0.5               0.6               0.7        0.8   0.9   1
      1     2       3          4            5
                                                      a 6   7
                Gauss-Newton + Bisection Iteration
    Prediction of 1st Order Model
                    Real output(blue) vs Estimation output(red)
         50
output




          0




         -50
               0           500                     1000            1500
                                       time
                   Error Curve   The mean squared error is9.1318
         10

          5
error




          0

          -5

         -10
               0           500                     1000            1500
                                        time
Comparison of 1st Order Model
 Methods           MSE                a              b      Iterations

LSE             23.49449        0.987395       0.032551     1
ANFIS           22.2925         N/A            N/A          200
GD              11.0163         0.9809         0.0376       N/A
Exhausted       9.131926        0.977774       0.043542     10000
  Search
DGD1*           9.131816 0.977764 0.043568 1000
DGD2*           9.131786 0.777774 0.043544 101
DGD3*           9.131785 0.977776 0.043543 98
 •DGD1: Deepest descent direction and adjusting step size
 •DGD2: BFGS direction and adjusting step size
 •DGD3: Gauss-Newton and line search
                     High Order Model
y(t )  a1 y(t 1)  a2 y(t  2)  b1u(t 1)  b2u(t  2)
ˆ
                initial (by LSE) : [1.2805        -0.29191 0.10582              0.15903]
                final:              [1.8604 -0.8641           0.07045       -0.007475]
                             Engine output(dashed) vs Model Preditcion(solid)
          50
  NHP




           0




          -50
                0                      500                     1000                    1500
                                               sample Time
                         Long-term Prediction Error: The MSE is:3.31361166E+000
          10


           5
  Error




           0


           -5
                0                      500                     1000                    1500
                                              Sample Time
            6. Future Works
        value Problem:
 Initial
 Robustness Problem: ???
 Applying such learning algorithm to Neural
  Networks
 Model structure selection by autocorrelation
  of prediction errors
 NARMX models
CSC Student Seminars
 (Spring/Summer, 2006)




                         Thanks
Appendix
                       Initial value problem
                                          manual setting of initial value
setting initial value by LSE
[1.2805   -0.29191 0.10582     0.15903]   [0.5 0.5 0.5 0.5]

[1.8604 -0.8641 0.07045 -0.007475]        [1.8604 -0.8641 0.07045 -0.007475]
Final MSE=3.313612
                                          Final MSE=8.60188
                               appendix
 (t ) ( y (t )  y (t ))
                    ˆ          y (t )
                                ˆ
                           
                            

         ( y (t ))
            ˆ
g(t) 
           
       ( T  z (t , ))
                 ˆ
   
              
                    z (t , )
                     ˆ
    z (t , ) 
      ˆ                        
                      
    z (t , )  g(t  1 ) g(t-2 ) ... g(t-n) 0 ... 0
      ˆ