HW 3

Homework 3
Fall 2014 - EE 105, Feedback Control Systems (Prof. Khan)
Assigned: September 17, 2014
Due: September 24, 2014
Problem 1 (20 pts)
Consider the following matrix, A ∈ R2×2 ,
"
A=
1 α
0 2
#
.
(1)
Plot both its two-norm, kAk2 , defined as
kAk2 ,
p
λmax (AT A),
where λmax (·) is the maximum eigenvalue of the matrix in the argument; and spectral radius,
ρ(A), defined as
ρ(A) , max |λi (A)|.
i
as a function of α ∈ [−10, 10]. Comment on the dependence of α on kAk2 and ρ(A). Repeat
for
"
#
1 α
A=
.
(2)
α 2
and comment.
From the plot, provide a relationship between kAk2 and ρ(A).
Problem 2 (20 pts)
Write the state-space representation of the parallel RLC circuit in Fig. 1. From the state space,
find the following:
(a) internal, external and total response of the RLC circuit;
(b) write the eigenvalues in terms of R, L, and C;
(c) find the conditions under which the internal response is stable (goes to zero with out
oscillations), unstable, stable-oscillations, oscillatory.
(d) For each of the cases: stable (goes to zero with out oscillations), unstable, stable-oscillations,
oscillatory, use Matlab to plot the RLC internal response, i.e., the time evolution of vc (t)
and iL (t).
2
is(t)
Fig. 1.
R
L
C
Vc(t)
A parallel RLC circuit
Problem 3 (20 pts)
(a) For A ∈ R2×2 , let det(A) = −4 and let trace(A) = 3. Find the eigenvalues of A. Is A
positive-definite ?
(b) Is the eigenvalue information enough to conclude that A is symmetric ? If yes, justify, if
not, give a counter-example.
In other words, we know that a (real-valued) symmetric matrix has real eigenvalues, but if a
matrix has real eigenvalues, is it symmetric ?
Do not use MATLAB for Problem 4
Problem 4 (20 pts)
Consider the following continuous-time dynamical system:
 


1
−1 2
0
 


x(t)
˙
= 0 2
1  x(t) +  0  u(t).
0
0 0 0.1
{z
}
| {z }
|
Ac
(a)
(b)
(c)
(d)
(3)
Bc
Convert the above system into its discrete-time representation with a sampling interval, Ts .
Write the complete discrete-time dynamics for Ts = 100 , 10−1 seconds.
Find the eigenvalues for the continuous-time system and comment on stability.
Find the eigenvalues for the discrete-time system (for each Ts ) and comment on stability.
———–Problem 5, next page
September 22, 2014
DRAFT
3
Problem 5 (20 pts) (MATLAB) Consider the following continuous-time dynamical system:

 

−0.2 −0.1
0
1

 

x(t)
˙
=  1
(4)
−0.2 −0.1  x(t) +  0  u(t)
1
0
−0.1
0
{z
}
|
| {z }
Ac

1

y(t) =  0
0
|
Bc

0 0

1 0  x(t).
0 1
{z
}
(5)
Cc
In MATLAB, we can use the following to create the CT state-space model.
A = [-0.2 -0.1 0;1 -0.2 -0.1;1 0 -0.1]; B = [1;0;0]; C = eye(3);
sys_c = ss(A,B,C,0);
We now find the discretization using the following:
T1=1; Ad1 = expm(A*T1); Bd1 = inv(A)*(Ad1 - eye(3))*B;
sys_d1 = ss(Ad1,Bd1,C,0, T1);
The following plots the step response for the continuous and discrete-time systems: (Tf is the
final time for the step-response output.)
Tf = 50;
step(sys_c,Tf), hold on, step(sys_d1,Tf)
(a) Repeat the above for the following sampling times: Ts = 0.1, 1, 2 seconds. Plot the stepresponse for the CT and all of the DT models on the same figure (using hold on). Comment
on the discretization.
(b) Plot the step response of the CT model and DT models (Ts = 1 seconds) using Ad , Bd with
no approximation and Ad , Bd with first-order approximation on x(t)
˙
(all on the same figure).
Comment.
Remarks:
•
Recall zero-order hold from DSP ? Note that the discretized model is still LTI (of course,
with a fixed Ts ). How about a first-order hold ?
September 22, 2014
DRAFT
4
A PPENDIX : F IRST O RDER A PPROXIMATION OF DISCRETIZED LTI SS
Recall from the lecture:
x(k + 1) = Ad x(k) + Bd u(k),
Ad , eATs ,
Bd , A−1 (eATs − I)B.
The first-order approximation of the matrix exponential is
eATs = I + ATs +
A2 Ts2 A3 Ts3
+
+ ...,
2!
3!
' I + ATs ,
when Ts is small enough such that the higher order terms are close to 0. Now the system
ed and B
ed , after this approximation become
matrices, A
ed , I + ATs ,
A
ed = A−1 (I + ATs − I)B,
B
= Ts B.
Finally, the first-order approximation of the discretized LTI SS is
ed x(k) + B
ed u(k).
x(k + 1) = A
˙
Alternatively, recall the definition of x:
x(t + Ts ) − x(t)
.
Ts →0
Ts
x˙ = lim
˙
Now, choose t = kTs and replace x˙ with the above (with very small Ts ) in x(t)
= Ax(t)+Bu(t):
x(kTs + Ts ) − x(kTs )
= Ax(kTs ) + Bu(kTs ),
Ts
leading to
x(kTs + Ts ) − x(kTs ) = Ts Ax(kTs ) + Ts Bu(kTs ),
⇒ x((k + 1)Ts ) = x(kTs ) + Ts Ax(kTs ) + Ts Bu(kTs ),
= (I + Ts A)x(kTs ) + Ts Bu(kTs ),
which results into the same approximated system matrices as above. This alternate process is
also sometimes called the Euler method of approximating derivatives and follows the first-order
approximation of the matrix exponential in the case of LTI SS.
September 22, 2014
DRAFT

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