Download Modern Control System EKT 308

Modern Control System
EKT 308
 Modeling in state space
 Modeling Physical Systems
State Variable Model
• Modern Control Theory is based on state variable.
• Can handle
 multiple-input multiple output
 Linear, nonlinear
 Time variant or invariant.
State: The smallest set of variables (called state variables) such that
knowledge of these variables at t  t0 , together with knowledge of the input
for t  t 0 , completely determines the behavior of the system at any time t  t 0
State Vector : State variables
x1 , x2 ,........, xn
form a vector called state vector.
representing a system state
 x1 (t ) 
 x (t )

x (t )   2 
 : 


x
(
t
)
 n 
State Variable Model (contd…)
State-space: The n-dimensional space spanned by the n state
vectors is called the state-space. Any state can be represented by
a point in the state space.
State-Space Equation
Variables: Input variables, output variables and state variables
Assume for a system,
Number of state variables  n
Number of inputs  r
Number of outputs  m
Then the system can be described by,
x1 (t )  f1 ( x1 , x2 ,...., xn ; u1 , u2 ,..., ur ; t )
x2 (t )  f 2 ( x1 , x2 ,...., xn ; u1 , u2 ,..., ur ; t )
....
....
...
....
xn (t )  f n ( x1 , x2 ,...., xn ; u1 , u2 ,..., ur ; t )
State-Space Equation (contd…)
Outputs of the system may be given by,
y1 (t )  g1 ( x1 , x2 ,...., xn ; u1 , u2 ,..., ur ; t )
y2 (t )  g 2 ( x1 , x2 ,...., xn ; u1 , u2 ,..., ur ; t )
....
....
...
....
ym (t )  g m ( x1 , x2 ,...., xn ; u1 , u2 ,..., ur ; t )
Let us define
 x1 (t ) 
 x (t )

x (t )   2 
 : 


x
(
t
)
 n 
 y1 (t ) 
 y (t ) 

y (t )   2 
 : 


y
(
t
)
 m 
 u1 (t ) 
u (t )

u (t )   2 
 : 


u
(
t
)
 r 
State-Space Equation (contd…)
Also define,
 f1 ( x1 , x2 ,...., xn ; u1 , u2 ,..., ur ; t ) 
 f ( x , x ,...., x ; u , u ,..., u ; t ) 
 
n
1
2
r

f ( x, u , t )   2 1 2
 ....
....
... 


 f n ( x1 , x2 ,...., xn ; u1 , u2 ,..., ur ; t )
 g1 ( x1 , x2 ,...., xn ; u1 , u2 ,..., ur ; t ) 
 g ( x , x ,...., x ; u , u ,..., u ; t ) 
 
n
1
2
r

g ( x, u , t )   2 1 2
 ....
....
... 


g
(
x
,
x
,....,
x
;
u
,
u
,...,
u
;
t
)
n
1
2
r
 m 1 2

Then
  

x (t )  f ( x , u , t )

  
y (t )  g ( x , u , t )
(1)
(2)
System response can be described by first order differenti al equation,
x1  a11x1  a12 x2  .......  a1n xn  b11u1  b12u2  ...  b1mum
x2  a21x1  a22 x2  .......  a2 n xn  b21u1  b22u2  ...  b21mum
...
.....
.....
....
...
xn  an1 x1  an 2 x2  .......  ann xn  bn1u1  bn 2u2  ...  bmmum
Above set of equations can be further simplified using matrix notation
 x1   a11 a12
  
d  x2  a21 a22

..
dt  ...   ..
  
 xn  an1 an 2
.. a1n   x1 
 b11 .. b1m   u1 



.. a2 n   x2  
  .. .. ..   .. 
.. ..   ... 
   bm1 .. bmm  um 
.. ann   xn 
(3)
Equation (3) can be written in short form



x  Ax  Bu
(4)
Similarly output can be written as



y  C x  Du
(5)
Matrix A is called state matrix.
Matrix B is called input matrix
Matrix C is called output matrix
Matrix D is called direct tra nsmission matrix.
Example
x1  capacitor voltage vc (t )
x2  inductor current iL (t )
dv
ic  C c  u (t )  iL
(5)
dt
diL
L
  Ri L  vc
(6)
dt
Output : vo  Ri L (t )
Equations (5) and (6) can rewritten as,
dx1
1
1
  x2  u (t )
(7)
dt
C
C
dx2 1
R
 x1  x2
(8)
dt
L
L
y1 (t )  Rx2
1

1
  0  C     
x
x   C u (t )

1
R
0

 
 
L
L

y  [0 R ] x
If R  3, L  1, C  1 / 2
 0  2  2
x
x   u

1  3 0

y  [0 3]x
Scilab program
A = [0, -2;1, -3];
disp (A);
B=[2;0];
disp(B);
C=[0 3];
disp(C);
t = 0:0.01:10;
len = length(t);
//u = sin(t);
//u = t;
u = ones(1, len);
dt = 0.01;
x = [0;0];
y = zeros(1, len);
for idx=1:len
xdot = A*x+B*u(idx);
x = x + xdot * dt;
y(idx)=C * x;
end
figure (1);
plot(t, y);
y (t ) when u (t ) is a unit step
y (t ) when u (t ) is a unit ramp
y (t ) when u (t ) is sinusoidal
Mechanical Systems
Mechanical systems are governed by Newton's Laws of motion.
There are three basic elements that comprise a mechanical system.
These are
Mass,
Damping(friction), and
Spring.
Newton's Second law: F  ma Force = mass x acceleration