# PowerPoint Presentation by NX0nU3

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```									Design a Multivariable Controller for

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
PLOTBODE    Plot Bode diagrams.
PLOTDB      Plot Magnitude Bode diagrams.
PLOTNIC     Plot Nichols chart.
PLOTNYQ     Plot Nyquist diagrams.
"The Quadruple-Tank Process: A Multivariable Laboratory

"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

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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