Control assignment#2

4th
Year Biomedical Eng. Systems Engineering 2016 Assignment#2
Solve the following background problems:
1- Given the second order digital wave guide oscillator system shown in Fig.1
(a) Determine the state equation of this system.
(b) Find the eigenvalues of this system and show that they lie on the unit circle if |β| < 1
and the system will be unstable if |β| > 1.
(c) Determine the response of this system to an initial condition of [0.8 0.6] T
where β =
0.9. Use MATLAB in this part and notice that the response is periodic in the steady
state. Check the frequency of the periodic response and compare it with ω where the
eigenvalues λ1,2 = cos(ωT) ± j sin(ωT). The following code will guide you in finding
the response of this system to a given initial condition
G = [ as determined in part (a)
H = [0; 0]; C = [ 1 0; 0 1]; D = [ 0 ; 0]; Here we assume that y(k) = x(k)
Sysd = ss(G,H.C,D, 0.1) Discrete representation with T = 0.1 sec.
x0 = [ 0.8; 06]; This is the function that generates and plots the
response of a linear system “Sysd” to a given initial condition x0.
Initialplot(Sysd,x0,50) Output response corresponding to 50 samples
Look at the plot and deduce the frequency of the periodic response at steady state and
compare it with ω as calculated at part (c).
2- Given the third order digital lattice filter system shown in Fig.2
+
-
z-1
z
-1
ccc
+
+
+
Fig. 1: Block diagram of a second order waveguide oscillator
c
X1(k)
X2(k)
+

Z-1
Z-1
Z-1
C1 C2
C3 d y(k)
u(k)
α3
α2 α1
-α3
-α2
-α1
X3(k) X2(k) X1(k)
Fig. 2: Block diagram of a third order digital lattice filter & |αm| < 1, m = 1, 2, 3.
(a) Write the state equations of the shown third order digital lattice filter.
(b) Compute the unit sample response of this system corresponding to α1 = 0.5, α2 = 0.6, α3 =
-0.8, c1 = 2, c2 = 1, c3 = 1.5 and d = 1. Use MATLAB in computing the unit sample
response and calculate as many samples until the state response is close to zero.
(c) Using the result of part (b), show that the state variables x1(k), x2(k) and x3(k) are
orthogonal, i.e., ∑ xi(k)xj(k) = 0.∞
k=0 [ Note that irrespective of the values of α provided
that |α| < 1, the state response will be orthogonal]
The following MATLAB code can help you in calculating the unit sample response as
well as showing the orthogonality of the resulting state response:
>> syms a1 a2 a3 : where a1 stands for α1
>> RCO = [ a1 a2 a3]; RCN = [ 0.8 0.5 -0.6]; where RCO denote old reflection
coefficients as symbolic variables while RCN denote their assigned values.
>> G = [ x x x; x x x; x x x]; H = [ x; x; x]; you will replace x with the expression of
the resulting state equations of part (a).
>> C1 = eye(3,3); D1 = zeros(3,1); Define the C and D matrices that will be used in
the computation of the output response.
>> G1 = double(subs(G,RCO,RCN));
>> H1 = double(subs(H,RCO,RCN)); Evaluate the numeric values of the G and H
matrices denoted by G1 and H1 using the substitution function subs of MATLAB.
>> Sysd = ss(G1,H1,C1,D1, 0.1) Define the discrete system Sysd with sampling time =
0.1 sec.
>> [Yresp,t] = impulse(Sysd,10); compute the unit sample response from 0 to 10 sec.
>> plot(1, Yresp) Plot the output response Yresp against time.
>> Ydot = Yresp’ * Yresp Compute the dot product of the state variables and you will
see that the result is a diagonal matrix (Change the selected values of a1, a2 and a3 and
you will note that the dot product is still diagonal.
3- For the two cascaded systems shown in Fig. 3
H1 = double(subs(H,RCO,RCN));

3
------------
(z – 0.6)
5(z -1)
------------
(z + 0.8)
2
------------
(z – 1)
U(k) X3(k) X2(k) X1(k)
3
------------
(z – 0.6)
2
------------
(z - 1)
5(z - 1)
------------
(z + 0.8)
U(k) X3(k) X2(k) X1(k)
Y(k)
Y(k)
(a) Write the state equations of the two systems shown in Fig 3.
(b) Check the controllability and observability of those systems and comment on your
results.
4- Consider a causal LTI system with input u(k) and pulse-transfer function specified as
H(z) = H1(z) H2(z)
where H1(z) =
1
1+
1
4
𝑧−1−
1
8
𝑧−2
and H2(z) = 1 −
7
4
𝑧−1
−
1
2
𝑧−2
A block diagram corresponding to H(z) may be obtained as a cascade connection of a
block diagram for H1(z) followed by a block diagram for H2(z) . The result is shown in
Fig. 4, in which we have also labeled the intermediate signals x1(k), x2(k), v1(k), and
v2(k) .
Fig. 3

-7/4
-1/2
Z-1
Z-1
Z-1
Z-1
-1/4
1/8
x1(k)
x2(k)
v1(k)
v2(k)
+
+
++
+
+
+
+
+
+
y(k)
Fig. 4
U(k)
a) How is x1(k) related to v1(k) ?
b) How is x2(k) related to v2(k) ?
c) Using your answers to the previous two parts as a guide, construct a direct form block
diagram for H(z) that contains only two delay elements.
d) Write the state space equations of this direct form.
e) Draw a cascade-form block diagram representation of H(z) based on the observation
that 𝐻( 𝑧) = (
1+
1
4
𝑧−1
1+
1
2
𝑧−1
) (
1 −2 𝑧−1
1−
1
4
𝑧−1
)
f) Draw a parallel form block diagram representation based on the observation that
𝐻(𝑧) = 4 +
5/3
1 +
1
2
𝑧−1
−
14/3
1 −
1
4
𝑧−1
5 − Consider the following two pulse transfer functions
𝐻1(𝑧) =
1
(1 − 𝑧−1 +
1
4
𝑧−2) (1 −
2
3
𝑧−1 +
1
9
𝑧−2)
𝐻2(𝑧) =
1
(1 − 𝑧−1 +
1
2
𝑧−2) (1 −
1
2
𝑧−1 + 𝑧−2)

a) For each pulse transfer function, draw a block diagram that corresponds to the cascade
connection of two second-order block diagrams. Each second order block should be in
direct form.
b) For each pulse transfer function, draw a direct form block diagram.
6- Consider the single input - single output pulse transfer function which is given by
𝑌(𝑧)
𝑈(𝑧)
=
5 ( 𝑧 + 2)(𝑧 + 1)
(𝑧 − 1)(𝑧 − 0.6)(𝑧 − 0.8)
a) Obtain a state space representation in the standard controllable form.
b) Obtain a state space representation in the standard observable form.
c) Obtain a state space representation in the diagonal form.
d) Sketch the block diagram representations of a), b) and c).
7- Repeat question No. 6, for the discrete pulse transfer function given by
𝑌(𝑧)
𝑈(𝑧)
=
2 ( 𝑧 + 2)(𝑧2
+ 0.4 𝑧 − 0.6)
(𝑧 − 1)(𝑧 − 0.6)(𝑧 − 0.8)
8- For the block diagram shown in Fig. 5
a) Find two different state space representations for this block diagram.
b) Find a nonsingular matrix P that relates these two representations.
c) Determine the pulse transfer function of this system.
d) Obtain the standard controllable and the standard observable form representations for
this system.
Z-1
-2
Z-1
-1
Z-1
0.5
2
3
-3
-1.5
y(k)
u(k)
+
+
++
+ +
+
+
+
+
Fig. 5
+
+
9- Given the third order discrete system

),(
01
10
11
)(
210
101
010
)1( kukxkx























  )(123)( kxky 
a) Find the state space representation in terms of the new state x^
(k), where
x1
^
(k) = 0.5 ( x1(k) + x2(k) )
x2
^
(k) = 0.5 ( x1(k) - x2(k) )
x3
^
(k) = x1(k) + x2(k) + x3(k)

Control assignment#2

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Control assignment#2