# EE363 homework

1. Lyapunov condition for passivity. The system described by

x(0) = 0, with u(t), y(t) ∈ R^{m}, is said to be passive if

holds for all trajectories of the system, and for all t.

Here we interpret u and y as power-conjugate quantities (i.e., quantities
whose product

gives power) such as voltage and current or force and velocity. The inequality

above states that at all times, the total energy delivered to the system since t
= 0 is

nonnegative, i.e., it is impossible to extract any energy from a passive system.

(a) Establish the following Lyapunov condition for passivity: If there exists
a function

V such that V (z) ≥ 0 for all z, V (0) = 0, and
for all w and z,

then the system is passive.

(b) Now suppose the system is
,
and consider the quadratic

Lyapunov function V (z) = z^{T}Pz. Express the conditions found in part (a) as a

matrix inequality involving A, B, C, and P.

Remark: you will not be surprised to learn that for a linear system, the
condition

you found in this problem is not only sufficient but also necessary for the
system to

be passive. This result is called the Kalman-Yakubovich-Popov (KYP) or positive

real (PR) lemma.

(c) Now consider the specific case with

Use an LMI solver to find a matrix P for which the Lyapunov function V (z) =

z^{T}Pz establishes passivity of the system.

2. Finding a discrete-time diagonal Lyapunov function. Recall problem 4 from
the last

homework, which concerned the stability of a digital filter with saturation. In
that

problem you proved that the system x_{t+1} = sat(Ax_{t}) is globally asymptotically
stable

if there exists a nonsingular diagonal D such that ||DAD^{-1}||< 1.

(a) Show how to find a nonsingular diagonal D that satisfies
||DAD^{-1}||
< 1, or

determine that no such D exists, using LMIs.

Hints:

• A matrix Z satisfies |Z| < 1 if and only if Z^{T}Z < I.

• You might find it easier to search for E = D^2, which is positive and
diagonal.

(b) Use an LMI solver to find such a D for the specific case

3. Finding a stabilizing state feedback via LMIs. We consider the time-varying LDS

with x(t) ∈ R^{n} and u(t) ∈ R^{m}, where A(t) ∈ {A_{1}, . . . ,A_{M}}.
Thus, the dynamics

matrix A(t) can take any of M values, at any time. We seek a linear state
feedback

gain matrix K ∈ R^{m×n} for which the closed-loop system

is globally asymptotically stable. But even if you’re given a specific state
feedack gain

matrix K, this is very hard to determine. So we’ll require the existence of a
quadratic

Lyapunov function that establishes exponential stability of the closed-loop
system, i.e.,

a matrix P = P^{T} > 0 for which

for all z, and for any possible value of A(t). (The parameter β > 0 is given, and sets

a minimum decay rate for the closed-loop trajectories.)

So roughly speaking we seek

• a stabilizing state feedback gain, and

• a quadratic Lyapunov function that certifies the closed-loop performance.

In this problem, you will use LMIs to find both K and P, simultaneously.

(a) Pose the problem of finding P and K as an LMI problem.

Hint: Starting from the inequality above, you won’t get an LMI in the variables

P and K (although you’ll have a set of matrix inequalities that are affine in K,

for fixed P, and linear in P, for fixed K). Use the new variables X = P^{-1}
and

Y = KP^{-1}. Be sure to explain why you can change variables.

(b) Carry out your method for the specific problem instance

B = (1, 0, 0), and β = 1. (Thus, we require a closed-loop decay at least as
fast as
e^{−t/2}.)

4. Stability of a switching system. We consider the nonlinear dynamical system

where

You can assume that c > 0. Roughly speaking, the system switches between two
linear

dynamical systems, depending on the sign of a quadratic function of the state.
The

function f can be discontinuous, but don’t let it worry you.

We seek a positive definite quadratic Lyapunov function V (x) = x^{T}Px
for which

for all x. (The parameter β is given.)

(a) Explain how to find such a P, or determine that no such P exists, by
formulating the problem as an LMI.

(b) Use an LMI solver to find such a P for the specific case

5. Perron-Frobenius theorem for nonnegative but not regular matrices. Suppose
A ∈ R^{n×n}

and is nonnegative, with Perron-Frobenius eigenvalue λ_{pf}
. Show by examples that the

following can occur:

(a) The multiplicity of λ_{pf}
can exceed one.

(b) The eigenvalue λ_{pf}
is associated with a Jordan block of size larger than 1 × 1.

(c) There are multiple PF eigenvectors, i.e., there are nonzero nonnegative
vectors v

and
,
not multiples of each others, such that

(None of these can occur if A is regular, i.e., A^{k} > 0 for some
k.)

6. A bound on the Perron-Frobenius eigenvalue. Let A ≥ 0, with PF eigenvalue λ_{pf}
. Show

that

i.e., λ_{pf}
lies between the minimum and maximum of all row sums of A. Show that the

same holds for the column sums.

7. Some relations between a matrix and its absolute value. In this problem, A
∈ R^{n×n},

and |A| denotes the matrix with entries |A|_{ij} = |A_{ij}|.

For each of the following statements, give a proof of the statement or
provide a specific

counterexample.

(a) If all eigenvalues of |A| have magnitude less than one, then all
eigenvalues of A

have magnitude less than one.

(b) If all eigenvalues of A have magnitude less than one, then all eigenvalues
of |A|

have magnitude less than one.

(c) If
then

(d) If
then

8. A weighted maximum Lyapunov function. Suppose A is nonnegative, regular,
and

stable, and let v be the PF eigenvector of A, and λ_{pf}
the PF eigenvalue. Consider the

Lyapunov function

and the system
.
Show that along trajectories of this system, V decreases

at each step by at least the factor λ_{pf}
.

9. Iterative power control with receiver noise. We consider the power control
problem

described in the lecture, with one modification: we include a receiver noise
term, so

the signal to interference plus noise ratio (SINR) is

Note that by increasing the powers of all transmitters, we can make the
effects of the

noise on the SINR small, so we can achieve a minimum SINR as close as we like
(but

not equal) to the optimal SIR when there is no noise.

Now suppose the following iterative power control scheme is used to set the
powers:

at each step of the iteration, the power P_{i} is adjusted to that the SINR of
receiver i

would equal γ, provided the other powers are not changed.

Show that this scheme works, provided
, where λ_{pf}
is the PF eigenvalue of

(‘Works’ means the powers converge to a power
allocation for which each SINR is

equal to γ). You can assume that G_{ij} > 0, and that N_{i} > 0.