1.1 Calculate and plot the Fourier transform of
the sampled signal f(t)=Cos(wt) when w=2 and the sampling frequencies w_s=6,
4.5 and 3 respectively. Verify your answers by MATLAB, see matlab file.
1.2 Let f(t)=Sin(t). Plot
sum_(k=-N)^(N) f(kh) Sinc[(t-kh)/h]
for N=5 & 10, and h=pi & pi/4 respectively. Verify your answers by MATLAB, see
matlab file.
1.3
Let f(t)=Cos(5t)+0.5Cos(20t) be the input to a low-pass filter with the cutoff
frquency 10(rad/sec). Plot f(t) and the output of the filter.
Verify your answers by MATLAB,
see simulink simulation in text file or
in graph file.
1.4
If f(t) has a Fourier transform
F(w) with F(w) =0 for |w| larger than or equal to w_0 and let f*(t) be
the sampled signal of f(t), then the Sampling Theorem tells us that
f(t) can be recovered from f*(t) as long as the sampling frequency
w_s is larger than or equal to 2w_0. Now consider two signals f_1(t)=0
and f_2(t)=sin(t). Let the sampling frequency w_s=2 for the
both signals and we have f*_1(kh)=f*_2(kh)=0 for all k. This
seems to contradict the sampling theorem. Explain.
2.1 Solve the difference equations described
in Textbook 2.3 (a) and (c) using the Z-transform.
Assuming zero initial conditions and unit step input.
2.2 Textbook 2.17.
2.3 Find the discrete transfer functions for
the continuous time system in Textbook 2.5 with h=1.
3.1 Verify your answer to HW 2.3
by using MATLAB, see e.g., c2d, tf, and ss.
Attach your MATLAB code.
3.2
Find exp(At) for A=[1 4 10; 0 2 0; 0 0 3].
3.3
Find exp(At) for A=[1 0; 0 2].
3.4
Find exp(At) for the matrix A in Textbook 2.2 (a) on page 68.
3.5 Verify your answers to HW 3.2-3.4
by using MATLAB expm(A*t) for t=1,2,10, see expm. Attach your MATLAB code.
3.6 Textbook 2.1.
3.7 Textbook 2.2 (a). Verify the result for h=1(second)
using MATLAB code sys_dis=c2d(sys_con,h), where
sys_con=ss(A,B,C,0).
4.1 Textbook 2.9, assuming distinct
eigenvalues.
4.2 Using simulink to simulate the
continuous time system in Textbook 2.10 and its sampled systems
at h=0.3, 0.6, and 0.8(sec)
respectively with the input u(t)=2000sin(t), see
simulink picture.
4.3
Let input u(k) be the delta (impulse) function and x(0)=(0,0)'.
Calculate x(k), y(k), the impulse response h(k) and the transfer function
H(z) of the state space equation in Textbook 2.8.
Verify the numerical results of y(k), k=1,2,...,10
by using MATLAB, see dlsim(Phi,Gamma,C,D,U,x0).
4.4 Let input u(k) be the unit step function to
the systme in Textbook 2.3 (b). Calculate x(k), y(k) in terms of x(0)=(0,0)'
and u(k). Verify the results by using MATLAB.
4.5 Find the (discrete) transfer function for Textbook
Problem 3.5.
5.1 Textbook 3.18 (a) and (b).
5.2 Check the asymptotic and BIBO stabilities
of the system in Textbook 3.6, verified by MATLAB, see eig, roots...
5.3 Check the asymptotic and BIBO stabilities
of the system in Textbook 3.17.
5.4 Textbook 3.1.
6.1 Textbook 3.2.
6.2 Textbook 3.3. For both parts (a) and (b), change
q to z and only consider the case when tau=0, i.e., no delay. Note in the problem,
H(z) is the sampled system of G(s)=1/s with ZOH and H_c(z)=k, which is NOT
NECESSARILY AN INTEGER but any real number.
6.3 Textbook 3.8.
6.4 Textbook 3.16, where u(k)=e(k)=u_c(k)-y(k)
as shown in Figure 3.5 on page 85.
7.1 Textbook 3.6.
7.2 Textbook 3.7.
7.3 Textbook 3.8 (again, now you can explain.).
7.4 Textbook 3.17.
7.5 Textbook 3.19, (a) only.
8.1 Textbook 4.1 (no verification of Example 4.5) .
8.2 Textbook 4.2
8.3 Textbook 4.4 (a) & (c).
8.4 Textbook 4.8, (b). (Hint: The transfer function from
disturbance to the output is zero at frequency 0...).
9.1 Textbook 4.12,(a) and (b).
9.2 Design a state feedback using the state observer
u(k)=-L\hat{x}(k)+v(k)
for the textbook problem 11.5 on page 442 (change c=(0,1) to c=(1,0)) so that the closed loop eigenvalues are
0.5 and -0.5 and the eigenvalues of the observer are -0.25 and 0.25. (no parts (a), (b) and (c).)
9.3 Repeat 9.2 for the textbook problem
11.20 on page 445 (stationary = constant L).