know about control theory i.r sampled data

1. Major Assignment on
Control Theory
Sampled data control system
Submitted by Submitted to
Sagar Bhardwaj Mr. Somesh Chaturvedi
K10692 (Asst. prof. EE dept)

2. Sampled-Data Systems :

3. State Description of Sampled CT plants
• A model of an A/D converter:
Samplerf(t) f(k)
k
t 0 1 2 3

4. A model of D/A converter
ZOH
t
k0 1 2 3
f(k) f+(t)
f+(t)=f(k); kT <= t< (k+1)T

5. DT systemCTsystem sampler
Discrete time
system
ZOH
Interconnection of DT and CT system
u(k) u+(t) y(t) y(k)
Equivalent Discrete
Time system

6. x(k+1)=Fx(k)+gu(k)
y(k)=cx(k)+du(k)
bdθe
Aθg
T
0 
F = eAt
Find eigen values By equation | λI – A | = 0
Get λ1, λ2 …
elt = g(l) = b0 + b1 l
el2t = g(l2) = b0 + b1 l2
el1t = g(l1) = b0 + b1 l1
eAt = b0+b1A
Thus we find λ1, λ2.

7. Example :
• From the given BD find out state equation.
The state variable defined by:
x1(t)=q(t),
x2(t)=dq(t)/dt
State Eqn are given by:
dx(t)/dt=Ax(t)+bu+(t)
y(t)=cx(t)
13-May-16

8. .
13-May-16
Gh0(s)
1
s(s+5)
u(t)
-
+
T=1s
u(k)
ZOH
plant
Q(t)
u+(t)

9. • A= 0 1 b= 0 c=[1 0]
0 5 1
Find eigen values, l1 =0, l2=-5
elt=g(l)=b0+b1l
eAt=b0+b1A= 1 1/5(1-e-5T)
0 e-5T
13-May-16
Example :

10. g= 0.2(T-0.2+0.2e-5T)
0.2(1-e-5T)
for T=0.1 sec
F = 1 0.0787
0 0.6065
g = 0.043
0.0787
bdT
eAg  0

11. Obtain Z-domain TF using MATLAB
• .
13-May-16
ZOH 1
s(s+2)
R(s) E(s) E*(s)
T=1s
C(s)
_
+

12. Matlab Simulation :
num=1
den=[1 2 0]
T=1
[numz,denz]=c2dm(num,den,T,'ZOH')
printsys(numz,denz,'z')
sys = tf(numDz,denDz,-1)
axis([-1 1 -1 1])
zgrid
13-May-16

13. num =
1
den =
1 2 0
T =
1
numz =
0 0.2838 0.1485
denz =
1.0000 -1.1353 0.1353
num/den =
0.28383 z + 0.1485
------------------------
z^2 - 1.1353 z + 0.13534
numDz =
1
denDz =
1.0000 -0.3000 0.5000
Transfer function:
1
-----------------
z^2 - 0.3 z + 0.5
Sampling time: unspecified