State-variable representation-Two systems with transfer functions H_1(z) = 0.2/1
ID: 2084258 • Letter: S
Question
State-variable representation-Two systems with transfer functions H_1(z) = 0.2/1 + 0.5z^-1, H_2(z) = 0.8 - 0.2z^-1/1 - z^-1 + 0.5z^-2 are connected in parallel. (a) Use MATLAB to determine the transfer function H(z) of the overall system. (b) use the function tf2ss to obtain state-variable representations for H_1(z) and H_2(z), and the use ss2tf to verify these transfer functions are the transfer functions obtained from the models. (c) obtain a state-variable representation for H(z) and compare it with the one you would obtain from the state variable models for H_1(z) and H_2(z). Answers: H(z) H_1(z) H_2(z) = 1/(1 - 0.5z^-1 0.25z^-3).Explanation / Answer
type the following commands in MATLAB
num1= 0.2;
den1= [1 0.5];
num2= [0.8 -0.2];
den2= [1 -1 0.5];
H1 = tf(num1,den1,0.01,'variable','z^-1');
H2 = tf(num2,den2,0.01,'variable','z^-1');
You will get,
num1 =
0.2000
den1 =
1.0000 0.5000
num2 =
0.8000 -0.2000
den2 =
1.0000 -1.0000 0.5000
Transfer function: H1
0.2
------------
1 + 0.5 z^-1
Transfer function:H2
0.8 - 0.2 z^-1
-------------------
1 - z^-1 + 0.5 z^-2
Sampling time (seconds): 0.01
How put H! and H2 in parallel using following command
H = parallel(H1,H2);
Transfer function: H
1
------------------------
1 - 0.5 z^-1 + 0.25 z^-3
Sampling time (seconds): 0.01
(b)
Check tf2ss and ss2tf functions in MATLAB as below.
H1(z):
>> [A1,B1,C1,D1]=tf2ss(num1,den1) % convert from transfer function model to state space.
A1 =
-0.5000
B1 =
1
C1 =
0.2000
D1 =
0
>> [test_num1,test_den1]=ss2tf(A1,B1,C1,D1) % get back numerator and denominator from the matrices A1,B1,C1,D1
test_num1 =
0 0.2000
test_den1 =
1.0000 0.5000
We got the system with numerator and denominator.
H2(z):
>> [A2,B2,C2,D2]=tf2ss(num2,den2) % convert from transfer function model to state space.
A2 =
1.0000 -0.5000
1.0000 0
B2 =
1
0
C2 =
0.8000 -0.2000
D2 =
0
>> [test_num2,test_den2]=ss2tf(A2,B2,C2,D2) % get back numerator and denominator from the matrices A2,B2,C2,D2
test_num2 =
0 0.8000 -0.2000
test_den2 =
1.0000 -1.0000 0.5000
(C)
type the following commands in MATLAB command window for state space representatio for H(z)
>> ss(H)
a =
x1 x2 x3
x1 0.5 0 -0.5
x2 0.5 0 0
x3 0 1 0
b =
u1
x1 1
x2 0
x3 0
c =
x1 x2 x3
y1 0.5 0 -0.5
d =
u1
y1 1
Sampling time (seconds): 0.01
It is evident that H(z) is a third order system, meaning that H(z) has three state variables as given above. Now lets compare it with the H1(z) and H2(z).
H1(z) is a first order system and H2(z) is a second order system. Hence the overall system (H) will be of third order.
Type the following command in matlab to generate state space equivalant of (H1+H2)
>> ss(H1+H2)
a =
x1 x2 x3
x1 0.5 0 -0.5
x2 0.5 0 0
x3 0 1 0
b =
u1
x1 1
x2 0
x3 0
c =
x1 x2 x3
y1 0.5 0 -0.5
d =
u1
y1 1
Sampling time (seconds): 0.01
Discrete-time model.
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