This document discusses state descriptions of digital processors, sampled-data systems, and systems with dead time. It provides an overview of state variables and state space, and describes how to represent digital processors, sampled continuous-time plants, and systems with dead time using state-space models. It also discusses converting between transfer functions and state-space representations, and applications of state-space analysis such as sampled-data controller synthesis and analyzing time responses of sampled-data feedback systems.

1. State description –
Digital Processors,
Sampled-data systems,
Systems with dead time
Manish A Tadvi
M.E Student
New Roll No.: 167
(A.C.R)
M. S. University 17 Sep 2012
India

2. Why State Description?
Limitation of Transfer Function
technique.
1.Highly cumbersome
2. It reveals only the system output for
a given input
 Advantage Of State Description
o Provides a feedback proportional to
the internal variables of a system

3. CONTROLLABLE (FIRST) CANONICAL FORM:
Given a transfer function.
The coefficients can now be inserted directly into the state-
space model by the following approach:
This state-space realization is called controllable canonical
form.
.

4. OBSERVABLE (SECOND) CANONICAL FORM:
. This state-space realization is called observable canonical form.

5. State Descriptions of Digital Processors
 SISO DTS:
State variables x1(k),x2(k),…,xn(k)
Input u(k), Output y(k)
Assumption- I/p switched on to the system at k=0
i.e. u(k)=0 for k<0,
Initial state is given by:
x(0)=x0 (n*1)vector

6.  LTI system:
X(k+1)=F x(k) + g u(k); (state equation)…(1)
Y(k)=c x(k) + d u(k); (output equation)…(2)
where,
 u(k)=system input, d=scalar, (direct coupling between i/p & o/p.
 Y(k)=defined output
x1(k ) f 11 f 12 ... ... f 1n g1
x 2(k ) f 21 f 22 ... .. f 2n g2
F . .. .. .. .. g ...
xk ...
.. .. . .. .. ...
xn 1(k ) fn1 fn .. fnn
2 .. gn
xn (k )
c c1 c2 c3 c4

7. State Variables:- The smallest set of variables
which determine the state of a dynamic system
is called state variable.
State variable describes the future response of a
system, given the present state, the excitation
i/p, and the eqn describing the dynamics.
State space:- The n dimensional state variables are
elements of n dimensional space is called state
space.

8. Basic Structure Of DCS :
Digital Controlled o/p
set pt Digital computer D/A Plant
Sensor
A/D
20-Sep-12

9. The State-Space block implements a system whose behavior is defined
by :
X(k+1)=F x(k) + G u(k) n = number of states.
m= number of inputs.
Y(k)=C x(k) + D u(k) r= number of outputs
where x is the state vector,
u is the input vector,
and y is the output vector.
F must be an n-by-n matrix,
G must be an n-by-m matrix,
C must be an r-by-n matrix,
D must be an r-by-m matrix.
n m
n F G
r C D

10. Conversion of state variable to TF
X(k+1)=F x(k) + g u(k)
Y(k)=c x(k) + d u(k)
zX(z) – z x0 =F x(z) + g u(z) % z transform
(zI - F) X(z) = zx0 + g u(z) % I is n x n identity matrix
X(z) = (zI-F)-1 * z x0 + (zI-F)-1 * g u(z)
Y(z) = c x(z) + d u(z) % z transform
Y(z) = c * (zI-F)-1 * z x0 + (zI-F)-1 * g u(z)*c + d u(z)
Y(z) = c * (zI-F)-1 * z x0 + [ c * (zI-F)-1 * g + d ] u(z)
Y(z) = G(z) = c * (zI-F)-1 * g + d % In case of Initial condition x0 = 0
U(z)
Y(z) = G(z) = c * adj(zI-F) * g + d
U(z) | zI-F |

11. Conversion of state variable to TF using MATLAB
Matlab Simulation : Example :
A=[0 1 0;-5 -2 -1;0 0 3]
B=[0;1;1]
c=[4 1 0]
D=0
[num,den]=ss2tf(A,B,C,D)
sys=tf(num,den)
O/P : -
A=
0 1 0
-5 -2 -1
0 0 3
B=
0
1
1
20-Sep-12

12. c=
4 1 0
D=
0
num =
0 1.0000 0.0000 -16.0000
den =
1 -1 -1 -15
Transfer function:
s^2 - 16
------------------
s^3 - s^2 - s - 15

13. csys= canon(sys,'companion')
a=
x1 x2 x3
x1 0 0 15
x2 1 0 1
x3 0 1 1
b=
u1
x1 1
x2 0
x3 0
c=
x1 x2 x3
y1 1 1 -14
d=
u1
y1 0
Continuous-time model.
>>

14. Conversion of Transfer Function to Canonical State
Variable Model :
First Companion form : % Direct Form : 1
Second Companion form : % Direct Form : 2
Jordan Canonical Form : % Parallel Form
zn+ 1zn-1+….+ n-1z+ n
=
Transfer Function : G(z) zn+ 1zn-1+…+ n-1z+ n
1, 2,… n as feedback element
1, 2,… n as feed forward element

15. Direct form : 1
+ + + y(k)
+ + +
b0 b1 bn-1 bn
+ x2(k) x1(k)
u(k) Xn(k)
_
a1 an-1 an
+ +
20-Sep-12
+ +

16. Direct Form : 1
x(k+1)=Fx(k)+gu(k)
y(k)=cx(k)+du(k)
0 1 0 ... 0 0
0 0 1 .. 0 0
F . .. .. .. .. g ...
0 0 0 .. 1 ..
a a .. a 1
1
n n 1 ..
c [ n n 0, n -1 - n -1 , ....,
0 1 - 1 ]
0
d =β 0
This is called first companion form.

17. Direct form :- 2
u(k)
bn bn-1 b1 b0
+ + +
+ + y(k)
+ +
xn-1(k) _ xn(k)
x1(k) _
_
an an-1 a1
20-Sep-12

18. Direct Form : 2
x(k+1)=Fx(k)+gu(k)
y(k)=cx(k)+du(k)
0 0 .. ... n n n 0
1 0 .. .. n 1 n 1 n 1 0
F 0 1 .. .. a n 2 g ...
.. .. .. .. .. ..
0 0 .. .. a 1 1 1 0
c [0 0 ... 0 1]
d = β0
This is called Second companion form.

19. • F, g and c matrices of one companion form
correspond to the transpose of F, c and g matrices,
respectively, of the other.
• play an important role in pole-placement design
through state feedback.

20. Case 1: Parallel Form :
b0 1
+
u(k) + x1(k) + y(k)
r1
+
+
l1
+ xn(k)
rn
+
ln 20-Sep-12

21. Case 1:
If the transfer function involves distinct poles only as shown
below :
zn+ 1zn-1+….+ n-1z+ n
G(z) =
Transfer Function : zn+ 1zn-1+…+ n-1z+ n
r1 r2 rn
=
(z- 1) (z- ) ……………….. (z- n)

22. Case 1:
x(k+1)=Fx(k)+gu(k)
y(k)=cx(k)+du(k)
1 0 .. ... 0 1
0 2 .. .. 0 1
.. .. .. .. .. g ...
.. .. .. .. .. ..
1
0 0 .. .. n
c [r1 r2 ... rn]
d= β 0

23. Case 2:
If the transfer function involves multiple poles as shown
below :
zn+ 1zn-1+….+ n-1z+ n
G(z) =
Transfer Function : zn+ 1zn-1+…+ n-1z+ n
= ’1zn-1+ ’2zn-2+… ’n
(z- 1)m(z- m+1)…(z- n)
G(z) = H1(z)+Hm+1(z)+….+Hn(z)
Hm+1(z) = rm+1 ,…., Hn(z) = rn
z- m+1 z- n
And,
r11 r12 … r1m
H1(z) =
(z- 1)m (z- )m-1 (z- )
The realization of H1(z) is shown Here

24. Case : 2 + + y1(k)
+ +
r1m r12 r11
u(k) +
+ + x1(k)
z-1 z-1 z-1
xm(k) x2(k)
+
+ +
λ1 λ1 λ1
Realization of H1(z)

25. Case 2:
1 0 .. ... 0 0
0
0 m 1 .. .. 0
:
.. .. .. .. ..
:
.. .. 1
1
.. 0 ..
.. ...
..
0 1 .. 0 g 1
0 1
0 ..
..
1
.. ..
.. .. .. n
.. .. .. .. ..
0 0 .. .. 1
1
:
:
1
c [r11 r12 ... r1m | rm 1 . r]
.. n
d= β 0

26. Application :
State space analysis of DCS applicable for LTI as well
as LTV system.
LTI systems are SISO.
State variable describes the future response of a
system, given the present state, the excitation
i/p, and the eqn describing the dynamics.

27. Sampled-Data
Systems :

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

29. A model of D/A converter
f(k) f+(t)
ZOH
f+(t)=f(k); kT <= t< (k+1)T
t
0 1 2 3 k

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

31. x(k+1)=Fx(k)+gu(k)
y(k)=cx(k)+du(k)
F = eAt
Find eigen values By equation | λI – A | = 0
Get λ1, λ2 …
e t = g( ) = 0 + 1
e t = g( ) = 0 + 1
e t = g( ) = 0 + 1 Thus we find λ1, λ2.
eAt = 0+ 1A
g
T Aθbdθ
0 e

32. 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)
20-Sep-12

33. .
u(t) + u(k) u+(t)
1 Q(t)
- T=1s Gh0(s) s(s+5)
plant
ZOH
20-Sep-12

34.  A= 0 1 b= 0 c=[1 0]
0 5 1
Example :
Find eigen values, 1 =0, 2=-5
e t=g( )= 0+ 1
eAt= 0+ 1A= 1 1/5(1-e-5T)
0 e-5T
20-Sep-12

35. g T A bd
0e
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

36.  .
+ E*(s) C(s)
R(s) E(s)
ZOH 1
_ T=1s s(s+2)
Obtain Z-domain TF using MATLAB
20-Sep-12

37. 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
20-Sep-12

38. 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

39. Applications :
Sampled-data H controller synthesis
L2 norm of sampled-data system
Time response of sampled-data
feedback system

40. State Description of Systems
with Dead-Time:

41. What is Dead time?
•Appears in many processes in Industry and in other
fields like Economical and Biological Systems
They are Caused By Following Phenomena:
Transport Time
Accumulation of Time Lags
The required Processing time For Sensors
Effect of Dead time In System
Introduces additional lag in System Phase

42. A Heated Tank with A Long Pipe
The control input is the power W at the resistor.
The plant output is the temperature T at the end of the
pipe.

44. d X(t)/dt=Ax(t)+bu+(t- D)
t A( t
x(t) e A( t t 0 ) x (t 0) to e )b u ( t D) d
kT T A[ kT T
x((kT T) e AT x ( kT ) kT e )bu ( D ]d
X(kT+T)=Fx(kT)+g1u(kT-NT-T)+g2u(kT-NT)
T
g1 mT e A bd
mT A
g2 0 e bd

45. We can specify a first-order transfer function with dead time
.
Matlab Simulation :
• >> num = 5;
•den = [1 1];
•P = tf(num,den,'InputDelay',3.4)
•
•Transfer function:
• 5
•exp(-3.4*s) * -----
• s+1
•
•>> P0 = tf(num,den);
•step(P0,'b',P,'r')
•>>

46. Step Response
5
4.5
4
3.5
3
Amplitude
2.5
2
1.5
1
0.5
0
0 1 2 3 4 5 6 7 8 9 10
Time (sec)

47. Reference :
• Digital Control and State Variable Methods
- by M. Gopal
.Wikipedia
.Matlab