1.
Sampled-Data Systems :

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

3.
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

4.
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

5.
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.

6.
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)
12-May-16

7.
.
12-May-16
Gh0(s)
1
s(s+5)
u(t)
-
+
T=1s
u(k)
ZOH
plant
Q(t)
u+(t)

8.
• 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
12-May-16
Example :

9.
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

10.
Obtain Z-domain TF using MATLAB
• .
12-May-16
ZOH 1
s(s+2)
R(s) E(s) E*(s)
T=1s
C(s)
_
+

11.
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
12-May-16

12.
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

13.
P = tf(1,[1, 1e-5,0]);
T = 1.0/20;
C = ss([-1.5 T/4; -2/T -.5],[ .5 2;1/T 1/T],...
[-1/T^2 -1.5/T], [1/T^2 0],T);
Pd = c2d(P,T,'zoh');
systemnames = 'Pd C';
inputvar = '[ref]';
outputvar = '[Pd]';
input_to_Pd = '[C]';
input_to_C = '[ref ; Pd]';
sysoutname = 'dclp';
cleanupsysic = 'yes';
sysic;
[yd,td] = step(dclp,20*T);
M = [0,1;1,0;0,1]*blkdiag(1,P);
t = [0:.01:1]';
u = ones(size(t));
y1 = sdlsim(M,C,u,t);
plot(td,yd,'r*',y1{:},'b-')
axis([0,1,0,1.5])
xlabel('Time: seconds')
title('Step response: discrete (*) and continuous')
y2 = sdlsim(M,C,u,t,1,0,[0.25;0]);
plot(td,yd,'r*',y1{:},'b-',y2{:},'g--')
axis([0,1,0,1.5])
xlabel('Time: seconds')
title('Step response: nonzero initial condition')
M2 = [0,1,1;1,0,0;0,1,1]*blkdiag(1,1,P);
t = [0:.001:1]';
dist = 0.1*sin(41*t);
u = ones(size(t));
[y3,meas,act] = sdlsim(M2,C,[u dist],t,1);
plot(y3{:},'-',t,dist,'b--',t,u,'g-.')
xlabel('Time: seconds')
title('Step response: disturbance (dashed) and output (solid)')