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									Design a Multivariable Controller for
the Quadruple-Tank Process:


               Characteristic Loci
                    Method



           mbarkhordary@ee.iust.ac.ir
(commutative)


                                                                  G(s)
  Gs   W s s W     1
                                s 
  s   diag 1 s , 2 s ,..., m s 

                                        W s                       G(s)

                                                 i s  : G(s)




  K  s  W  s  M  s W        1
                                        s 
  M s   diag 1 s ,  2 s ,...,  m s 
                                  (commutative)


                                                        :(Return ratio)

 Gs K s   W s s M s W 1 s   W s N s W 1 s 



  N s   diag  1 s , 2 s ,..., m s 
                
  i s   i s  i s 
                       (commutative)


                                                 W s 



     K s   As M s Bs 


                                 Bs    As 
W 1 s    W s 

                                                          :Align
                            Gs K s   W s diag{ i s }W 1 s 


                           1                                                               s 
S  s  W  s  diag {             }
                                    W   1
                                             s                 T  s  W  s  diag { i           W
                                                                                                     }   1
                                                                                                              s 
                       1  i  s                                                      1  i  s 


                                                                                                         

                       i s 
 i  s   1                  1                T s   I
                    1  i  s 
               Characteristic Loci


                      Align                         K h   G 1  j  b    1

 m  b              Align                         K m (s)                  2
K m  j    I
             
                                                          G s K h



 l   m              Align                          K l (s )               3
 K l  j    I
              
                  


                              K s   K h K m ( s) K l ( s)                 4
                               MFD Toolbox

                                                 MATLAB MFD

                                                   mfddemo
1)   Generating and Displaying an MVFR Matrix.
2)   Basic Mathematical Functions.
3)   Plotting Functions.
4)   Characteristic Locus Design Method.
5)   Direct Nyquist Array Design Method.
6)   Inverse Nyquist Array Design Method.
7)   Quasi Classical Design Method.
8)   Block Building and Connection.

0) Quit.

Select a demo number:
                                                       MFD Toolbox

mv2fr :   MV2FR   Frequency response of MIMO system
                  MV2FR(A,B,C,D,W) calculates the MVFR matrix of the system:
                            .
                            x = Ax + Bu                        -1
                            y = Cx + Du                G(s) = C(sI-A) B + D

                  Vector W contains the frequencies, in radians, at which the
                  frequency response is to be evaluated.



              MV2FR(NUM,COMDEN,W) calculates the MVFR matrix from the
                         transfer function description G(s) = NUM(s)/COMDEN(s)
                                              MFD Toolbox



FMULF       Multiply two MVFR matrices.
FCGERSH     Generate column Gershgorin circles.
FRGERSH     Generate row Gershgorin circles.
FGETF       Get component matrices from MVFR matrix.
ALIGN      Real alignment
PHLAG       design a phase lag compensator
PHLEAD      design a phase lead compensator
PLOTBODE    Plot Bode diagrams.
PLOTDB      Plot Magnitude Bode diagrams.
PLOTNIC     Plot Nichols chart.
PLOTNYQ     Plot Nyquist diagrams.
"The Quadruple-Tank Process: A Multivariable Laboratory
Process with an Adjustable Zero"



   "IEEE Transaction on Control Systems Technology"
               i                  Ai
                                  ai

                                  hi


v2   v1


          y2       y1


                        k1 , k2
                         1 ,  2  (0,1)
                   P-            P+



h , h 
  0
  1
       0
       2
               (12.4,12,7)   (12.6,13.0)    [cm]



 h , h 
  0
  3
        0
        4
                (1.8,1.4)     (4.8,4.9)     [cm]



  v , v 
   0
   1
        0
        2
               (3.00,3.00)   (3.15,3.15)     [V]



k1 , k 2     (3.33,3.35)   (3.14,3.29)   [cm3/Vs]



 1 ,  2    (0.70,0.60)   (0.43,0.34)      -
                        u i  vi  vi0   xi  hi  hi0




    1                    A3             1k1                        
    T            0
                         A1T3
                                  0  
                                             A1
                                                              0        
    1                                                               
    0             1              A4                       2 k2 
                           0                 0
dx                T2            A2T4                      A2        
                                    x               (1   2 )k 2 
                                                                         u
dt  0                      1
                   0             0         0                        
                           T3                              A3       
                                  1   (1   1 )k1                  
    0             0        0                              0        
                                  T4   A4                           


  k      0        0 0                                      A         2hi0
y c                                                    Ti  i
                   0 0
                        x
  0      kc                                                ai         g
                                               1c1                1   2 c1    
                                            1  sT1           (1  sT3 )(1  sT1 ) 
                            G s                                                 
                                           1   1 c2               2 c2        
                                      (1  sT4 )(1  sT2 )
                                                                    1  sT2        
                                                                                    


                   2.6                      1.5           
                 1  62 s           1  23 s 1  62 s 
G s                                                   
                   1. 4                     2.8           
           1  30 s 1  90 s 
                                          1  90 s        
                                                           

                                                                          1.5                     2.5           
                                                                        1  63 s          1  39 s 1  63 s 
                                                      G  s                                                  
                                                                          2.5                     1.6           
                                                                  1  56 s 1  91s 
                                                                                                1  91s         
                                                                                                                 
                                Kh


                                          5 rad/sec



K h   G 1  j  b         Kh=align(g_wh)



                     123.92     - 0.0067449
K h  G 1  j5  
                    - 0.0025492    167.03 


                                             h  b
Kh
Kh
Kh
Kh
           Kh




-1+j0           C.L.




 G  j h .K h
                                           Km


     G(j0.1)Kh  WW -1
                                             Align                 B      A

                                          Lag              M

                                                    s  10           
       1  sT       4.5977                                     0   
 
     1   sT                             M(s)   s 2.175
                                                               s  10 
                  T=0.1                                 0            
                                                             s 2.175

KA = align(inv(W));
KM=tf({[0.1 1],[0];[0],[0.1 1]},{[0.1 0.2175],[1];[1],[0.1 0.2175]});
KB = align(W);

                          K m ( s)  AM ( s) B
                     Km


G (s )K h K m (s )        Characteristic Loci
                     Km


G (s )K h K m (s )
                           Km




          -1+j0




                             M(s)


abs [G  j m  K h K m ( j m )]
                         Kl




                                      1  sT
                          K l (s)           I
                                        sT



Kl

                                                 M
             K l  sI
K L ( s)                      :
                 s
                             Kl

                                  Characteristic Loci
G (s )K h K m (s )K L (s )
                             Kl


G (s )K h K m (s )K L (s )
                  K s   K h K m ( s) K l ( s)




   3.1152 2.8438         0        0                  - 35.43    0 
  - 5.7952 - 5.2902                                  - 45.234    0 
                          0        0              B                
A                                    
                                                       0       9.3105
   0           0     0.032005 - 8.6308
                                                                   
   0           0    0.0081842 - 2.207                0       56.698



   - 12.904 - 11.508 0.0034438 - 0.71633            123.75 - 5.1816
C                                                D                
  0.52497 0.47905 - 0.27253      23.233 
                                                    - 5.1454 166.9 

								
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