辅导SINR、讲解Matlab、辅导Matlab语言程序、讲解linear systems

Homework 6

1. Analysis of a power control algorithm. In this problem we consider again the power

control method described in homework 1 problem 1. Please refer to this problem for the setup

and background. In that problem, you expressed the power control method as a discrete-time

linear dynamical system, and simulated it for a specific set of parameters, with several values

of initial power levels, and two target SINRs. You found that for the target SINR value γ = 3,

the powers converged to values for which each SINR exceeded γ, no matter what the initial

power was, whereas for the larger target SINR value γ = 5, the powers appeared to diverge,

and the SINRs did not appear to converge. You are going to analyze this, now that you know

alot more about linear systems.

a) Explain the simulations. Explain your simulation results from the problem 1(b) for the

given values of G, α, σ, and the two SINR threshold levels γ = 3 and γ = 5.

b) Critical SINR threshold level. Let us consider fixed values of G, α, and σ. It turns out

that the power control algorithm works provided the SINR threshold γ is less than some

critical value γcrit (which might depend on G, α, σ), and doesn’t work for γ > γcrit.

(‘Works’ means that no matter what the initial powers are, they converge to values for

which each SINR exceeds γ.) Find an expression for γcrit in terms of G ∈ R

n×n

, α, and

σ. Give the simplest expression you can. Of course you must explain how you came up

with your expression.

2. Real modal form. We learned about the modal form of a system in class. Show that when

some of eigenvalues of the dynamics matrix A are complex, the system can be put in real

modal form (Assuming the eigenvectors of A are independent):

S

1AS = diag (Λr, Mr+1, Mr+3, . . . , Mn?1)

where Λr = diag(λ1, . . . , λr) are the real eigenvalues, and

Mj =



σj ωj

ωj σj



, λj = σj + iωj , j = r + 1, r + 3, . . . , n ? 1

where λj are the complex eigenvalues (one from each conjugate pair). Clearly explain what

the matrix S is.

Generate a matrix A in R

10×10 using A=randn(10). (The entries of A will be drawn from

a unit normal distribution.) Find the eigenvalues of A. If by chance they are all real, generate

a new instance of A. Find the real modal form of A, i.e., a matrix S such that S

1AS has

the real modal form. Your solution should include the source code that you use to find S, and

some code that checks the results (i.e., computes S

1AS to verify it has the required form).

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3. Output response envelope for linear system with uncertain initial condition. We

consider the autonomous linear dynamical system x˙ = Ax, y(t) = Cx(t), where x(t) ∈ R

n and

y(t) ∈ R. We do not know the initial condition exactly; we only know that it lies in a ball of

radius r centered at the point x0:

kx(0) x0k ≤ r.

We call x0 the nominal initial condition, and the resulting output, ynom(t) = CetAx0, the

nominal output. We define the maximum output or upper output envelope as

y(t) = max{y(t) | kx(0) x0k ≤ r},

i.e., the maximum possible value of the output at time t, over all possible initial conditions.

(Here you can choose a different initial condition for each t; you are not required to find a

single initial condition.) In a similar way, we define the minimum output or lower output

envelope as

y(t) = min{y(t) | kx(0) x0k ≤ r},

i.e., the minimum possible value of the output at time t, over all possible initial conditions.

a) Explain how to find y(t) and y(t), given the problem data A, C, x0, and r.

b) Carry out your method on the problem data in uie_data.m. On the same axes, plot

ynom, y, and y, versus t, over the range 0 ≤ t ≤ 10.

4. Spectral mapping theorem. Suppose f : R → R is analytic, i.e., given by a power series

expansion

f(u) = a0 + a1u + a2u

2 + · · ·

(where ai = f

(i)

(0)/(i!)). (You can assume that we only consider values of u for which this

series converges.) For A ∈ R

n×n

, we define f(A) as

f(A) = a0I + a1A + a2A

2 + · · ·

(again, we’ll just assume that this converges).

Suppose that Av = λv, where v 6= 0, and λ ∈ C. Show that f(A)v = f(λ)v (ignoring the

issue of convergence of series). We conclude that if λ is an eigenvalue of A, then f(λ) is an

eigenvalue of f(A). This is called the spectral mapping theorem.

To illustrate this with an example, generate a random 3 × 3 matrix, for example using

A=randn(3). Find the eigenvalues of (I + A)(I ? A)

1 by first computing this matrix, then

finding its eigenvalues, and also by using the spectral mapping theorem. (You should get very

close agreement; any difference is due to numerical round-off errors in the various compuations.)

5. Interconnection of linear systems. Often a linear system is described in terms of a

block diagram showing the interconnections between components or subsystems, which are

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themselves linear systems. In this problem you consider the specific interconnection shown

below:

S T

u v y

w1

w2

Here, there are two subsystems S and T. Subsystem S is characterized by

x˙ = Ax + B1u + B2w1, w2 = Cx + D1u + D2w1,

and subsystem T is characterized by

z˙ = F z + G1v + G2w2, w1 = H1z, y = H2z + Jw2.

We don’t specify the dimensions of the signals (which can be vectors) or matrices here. You can

assume all the matrices are the correct (i.e., compatible) dimensions. Note that the subscripts

in the matrices above, as in B1 and B2, refer to different matrices. Now the problem. Express

the overall system as a single linear dynamical system with input, state, and output given by

respectively. Be sure to explicitly give the input, dynamics, output, and feedthrough matrices

of the overall system. If you need to make any assumptions about the rank or invertibility

of any matrix you encounter in your derivations, go ahead. But be sure to let us know what

assumptions you are making.

6. Analysis of investment allocation strategies. Each year or period (denoted t = 0, 1, . . .)

an investor buys certain amounts of one-, two-, and three-year certificates of deposit (CDs)

with interest rates 5%, 6%, and 7%, respectively. (We ignore minimum purchase requirements,

and assume they can be bought in any amount.)

B1(t) denotes the amount of one-year CDs bought at period t.

B2(t) denotes the amount of two-year CDs bought at period t.

B3(t) denotes the amount of three-year CDs bought at period t.

We assume that B1(0) + B2(0) + B3(0) = 1, i.e., a total of 1 is to be invested at t = 0. (You

can take Bj (t) to be zero for t < 0.) The total payout to the investor, p(t), at period t is a

sum of six terms:

1.05B1(t1), i.e., principle plus 5% interest on the amount of one-year CDs bought one

year ago.

1.06B2(t2), i.e., principle plus 6% interest on the amount of two-year CDs bought two

years ago.

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1.07B3(t  3), i.e., principle plus 7% interest on the amount of three-year CDs bought

three years ago.

0.06B2(t  1), i.e., 6% interest on the amount of two-year CDs bought one year ago.

0.07B3(t 1), i.e., 7% interest on the amount of three-year CDs bought one year ago.

0.07B3(t 2), i.e., 7% interest on the amount of three-year CDs bought two years ago.

The total wealth held by the investor at period t is given by

w(t) = B1(t) + B2(t) + B2(t 1) + B3(t) + B3(t1) + B3(t2).

Two re-investment allocation strategies are suggested.

The 35-35-30 strategy. The total payout is re-invested 35% in one-year CDs, 35% in twoyear

CDs, and 30% in three-year CDs. The initial investment allocation is the same:

B1(0) = 0.35, B2(0) = 0.35, and B3(0) = 0.30.

The 60-20-20 strategy. The total payout is re-invested 60% in one-year CDs, 20% in twoyear

CDs, and 20% in three-year CDs. The initial investment allocation is B1(0) = 0.60,

B2(0) = 0.20, and B3(0) = 0.20.

a) Describe the investments over time as a linear dynamical system x(t + 1) = Ax(t),

y(t) = Cx(t) with y(t) equal to the total wealth at time t. Be very clear about what the

state x(t) is, and what the matrices A and C are. You will have two such linear systems:

one for the 35-35-30 strategy and one for the 60-20-20 strategy.

b) Asymptotic wealth growth rate. For each of the two strategies described above, determine

the asymptotic growth rate, defined as limt→∞ w(t+ 1)/w(t). (If this limit doesn’t exist,

say so.) Note: simple numerical simulation of the strategies (e.g., plotting w(t+ 1)/w(t)

versus t to guess its limit) is not acceptable. (You can, of course, simulate the strategies

to check your answer.)

c) Asymptotic liquidity. The total wealth at time t can be divided into three parts:

B1(t) + B2(t1) + B3(t2) is the amount that matures in one year (i.e., the

amount of principle we could get back next year)

B2(t) + B3(t1) is the amount that matures in two years

B3(t) is the amount that matures in three years (i.e., is least liquid)

We define liquidity ratios as the ratio of these amounts to the total wealth:

L1(t) = (B1(t) + B2(t1) + B3(t2))/w(t),

L2(t) = (B2(t) + B3(t1))/w(t),

L3(t) = B3(t)/w(t).

For the two strategies above, do the liquidity ratios converge as t → ∞ If so, to what

values? Note: as above, simple numerical simulation alone is not acceptable.

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d) Suppose you could change the initial investment allocation for the 35-35-30 strategy,

i.e., choose some other nonnegative values for B1(0), B2(0), and B3(0) that satisfy

B1(0) + B2(0) + B3(0) = 1. What allocation would you pick, and how would it be

better than the (0.35, 0.35, 0.30) initial allocation? (For example, would the asymptotic

growth rate be larger?) How much better is your choice of initial investment allocations?

Hint for part d: think very carefully about this one. Hint for whole problem: watch

out for nondiagonalizable, or nearly nondiagonalizable, matrices. Don’t just blindly

type in matlab commands; check to make sure you’re computing what you think you’re

computing.