Appendices

FREQUENCY RESPONSE AND MEAN-SQUARED STABILITY

4.8 Appendices

Thus, we can write the following due to the binary nature of the index selections:

E h

DT𝑘 X DT𝑘

𝑖, 𝑗

i

=𝑋_{𝑖, 𝑗} P[𝑖 ∈ T𝑘, 𝑗 ∈ T𝑘]. (4.83) Regarding the probabilities in (4.83), we have the following for the non-diagonal (𝑖≠ 𝑗) entries:

P[𝑖 ∈ T𝑘, 𝑗 ∈ T𝑘] =P[𝑖 ∈ T𝑘] P[𝑗 ∈ T𝑘] = 𝑝_𝑖 𝑝_𝑗, (4.84) which follows from the fact that indices get updated independently from each other.

On the other hand, we have the following for the diagonal (𝑖 = 𝑗) entries:

P[𝑖 ∈ T𝑘, 𝑗 ∈ T𝑘] =P[𝑖 ∈ T𝑘] = 𝑝_𝑖, (4.85) which follows simply from the fact that𝑖 ∈ T𝑘 if and only if 𝑗 ∈ T𝑘 when𝑖 = 𝑗. Thus, we can write the following:

E

DT𝑘 X DT𝑘

𝑖, 𝑗

=











𝑝_𝑖 𝑝_𝑗 𝑋_{𝑖, 𝑗} 𝑖 ≠ 𝑗 , 𝑝_𝑖 𝑋_𝑖,𝑖, 𝑖 = 𝑗 ,

(4.86)

which is equivalent to the identity (4.81).

4.8.2 Proof of Theorem 4.1

Due to asynchronous updates described in (4.13), state vectorx𝑘 can be written as follows:

x𝑘 =(I−DT𝑘)x𝑘-1+DT𝑘 A x𝑘-1+

𝑅

Õ

𝑖=1

B𝑖 𝛼^𝑘-1

𝑖 +w𝑘-1

= I+DT𝑘 (A−I)

x𝑘-1+DT𝑘

𝑅

Õ

𝑖=1

B𝑖 𝛼^𝑘-1

𝑖 +w𝑘-1

!

. (4.87)

Taking expectation of (4.87) and using the facts that updated indices are selected independently, the input noise has zero mean, and the noise is uncorrelated with the index selections, we have the following:

E[x𝑘] =A¯ E[x𝑘-1] +

𝑅

Õ

𝑖=1

B¯𝑖𝛼^𝑘-1

𝑖 =A¯^𝑘E[x₀] +

𝑘−1

Õ

𝑛=0

A¯^𝑛

𝑅

Õ

𝑖=1

B¯𝑖𝛼^𝑘-1-𝑛

𝑖 (4.88)

=

𝑅

Õ

𝑖=1

(𝛼_𝑖I−A)¯ ^-1B¯𝑖𝛼^𝑘

𝑖 +A¯^𝑘

E[x₀] −

𝑅

Õ

𝑖=1

(𝛼_𝑖I−A)¯ ^-1B¯𝑖

,

=x^ss_𝑘 +A¯^𝑘

E[x₀] −x^ss₀

=x^ss_𝑘 +x^tr_𝑘 (4.89)

where ¯Aand ¯Bare as in (4.19).

4.8.3 Proof of Theorem 4.2

Using the definition of the error term in (4.29) and substituting x𝑘 =q𝑘+x^ss_𝑘 into (4.87), we have the following:

q𝑘+1+x^ss_𝑘

+1 = I+DT𝑘+1 (A−I)

(q𝑘+x^ss_𝑘) +DT𝑘+1 w𝑘+DT𝑘+1P^-1

𝑅

Õ

𝑖=1

B¯𝑖𝛼^𝑘

𝑖, (4.90) which can be written as follows by rearranging the terms:

q𝑘+1= I+DT𝑘+1 (A−I)

q𝑘+DT𝑘+1 w𝑘 + DT𝑘+1P^-1−I

δ𝑘, (4.91) where thedeterministicvectorδ𝑘 is defined as in (4.36).

Using (4.91), we can express the outer product q_𝑘

+1q^H_𝑘

+1 recursively in terms of q_𝑘q^H_𝑘 as follows:

q_𝑘

+1q^H_𝑘

+1 = I+DT𝑘+1 (A−I)

q𝑘q^H_𝑘 (A^H−I)DT𝑘+1 +I +DT𝑘+1w_𝑘 w^H_𝑘 DT𝑘+1

+ DT𝑘+1P^-1−I

δ𝑘 δ^H𝑘 P^-1DT𝑘+1 −I + I+DT𝑘+1 (A−I)

q𝑘 δ^H_𝑘 P^-1DT𝑘+1 −I + DT𝑘+1P^-1−I

δ^𝑘 q^H_𝑘 I+ (A^H−I)DT𝑘+1

, (4.92)

where the cross terms includingw𝑘 are left-out intentionally because these terms will disappear when we take the expectation since w𝑘 has a zero mean and it is uncorrelated with the index selections.

We now take the expectation of both sides of (4.92) and use the identities given by Lemma 4.6, and the independence assumption regarding the index selections, input noise and the initial condition. Then, we obtain the following:

Q𝑘+1 =𝜑(Q𝑘) +P𝚪P+𝚪 P−P²

+ δ𝑘 δ^H𝑘

P^-1−I +

(A¯ −I) x^tr_𝑘 δ^H_𝑘

P^-1−I +

δ𝑘 (x^tr_𝑘)^H(A¯^H−I)

P^-1−I

, (4.93) where the function 𝜑(·)is defined in (4.33). We also note thatE[q𝑘] =x^tr_𝑘 as given in (4.35).

AlthoughX+X^H ≠2<{X} in general, we note that the following equality holds true for anyX ∈C^𝑁×𝑁:

X+X^H

P^-1−I

=2<

X P^-1−I

, (4.94)

where<{·} denotes the real part of its argument. So, using the identity (4.94) in (4.93) gives the result in (4.35).

4.8.4 Proof of Corollary 4.2 We first define the following:

Z𝑘 =Q𝑘 −Q^r_𝑘−Qⁿ, (4.95)

where Q^r_𝑘 and Qⁿ are given as the solutions of (4.44) and (4.43), respectively.

Substituting (4.95) into the recursion (4.35), we get

Z𝑘+1+Q^r_𝑘+1+Qⁿ =𝜑(Z𝑘) +𝜑(Q^r_𝑘) +𝜑(Qⁿ) (4.96) +P𝚪P+𝚪 P−P²

+ <n δ𝑘δ𝑘^H

o

P⁻¹−I + <n

2(A¯ −I)x^tr_𝑘δ^H𝑘

o

P⁻¹−I ,

which can be simplified as follows due to the defining equations in (4.44) and (4.43):

Z𝑘+1 =𝜑(Z𝑘) + <n

2(A¯ −I)x^tr_𝑘 δ^H𝑘

o

P⁻¹−I

. (4.97)

Due to the stability assumption (4.41) and Lemma 4.3 we have 𝜌(A)¯ <1, so lim^𝑘→∞x^tr_𝑘 =0. As a result,

lim

𝑘→∞Z𝑘 =0, (4.98)

which gives the desired result.

Necessity of the condition (4.41) follows from (4.40). That is, when 𝜌(𝚽) ≥1 there exists a nonzero positive semi-definite matrixXthat cannot be reduced by the function𝜑(·).

4.8.5 Proof of Lemma 4.1

Assume that the stability condition (4.41) is met, and solution to (4.43) exists. Lete𝑖

denote the𝑖^{𝑡 ℎ} standard basis vector. It is readily verified that the following identity holds true for anyX∈C^𝑁×𝑁 and any index 1≤ 𝑖 ≤ 𝑁:

e𝑖^H 𝜑(X) e𝑖 =(1− 𝑝_𝑖) e^H𝑖 X e𝑖+𝑝_𝑖e^H𝑖 A X A^He𝑖. (4.99) Furthermore, we have the following:

e^H𝑖

P𝚪P+𝚪 P−P²

e𝑖 = 𝑝_𝑖 e^H𝑖 𝚪e𝑖. (4.100) So, by left and right multiplying (4.43) with e^H_𝑖 and e𝑖 respectively, we get the following:

e_𝑖^H Qⁿ−𝚪

e𝑖 𝑝_𝑖 = 𝑝_𝑖 e^H_𝑖 A QⁿA^He𝑖 ≥ 0, (4.101) where the inequality follows fromQⁿ 0, and the desired result follows from the fact that 𝑝_𝑖 > 0 for all𝑖.

4.8.6 Proof of Lemma 4.2

Enumerate all possible index update sets. Namely, letT𝑗 denote the 𝑗^{𝑡 ℎ} update set, and let 𝛾_𝑗 denote the probability of selecting the set T𝑗 for 1≤ 𝑗 ≤ 2^𝑁. So, the system is equivalent to the following state transition matrix:

A𝑗 =I+DT𝑗 (A−I) (4.102)

which now depends on the index 𝑗. This has probability𝛾_𝑗 in model (4.50), where the probability is given as follows (due to the stochastic model in (4.15)):

𝛾_𝑗 = Ö

𝑖∈T𝑗

𝑝_𝑖

! Ö

𝑖∉T𝑗

(1− 𝑝_𝑖)

!

. (4.103)

Then, for a givenX∈C^𝑁×𝑁 we have the following:

2^𝑁

Õ

𝑗=1

𝛾_𝑗 A𝑗 X A^H𝑗 =

2^𝑁

Õ

𝑗=1

𝛾_𝑗

I+DT𝑗 (A−I)

X

I+ (A^H−I)DT𝑗

=E h

I+DT𝑗 (A−I) X

I+ (A^H−I)DT𝑗

i

=X+P(A−I)X+X(A^H−I)P+E h

DT𝑗 (A−I)X (A^H−I)DT𝑗

i

(4.104)

=X+P(A−I)X+X(A^H−I)P+P(A−I) X (A^H−I)P +

(A−I) X (A^H−I)

(P−P²)

=

I+P(A−I) X

I+ (A^H−I)P +

(A−I) X (A^H−I)

(P−P²)

=𝜑(X),

where we use the identity (4.81) in (4.104). Thus, the statement of the corollary follows directly from [42, Corollary 3.26].

4.8.7 Proof of Corollary 4.3

Using the recursive definition of the error correlation matrix in (4.35), we have the following regarding the trace of the error correlation matrix:

tr(Q𝑘+1)=tr(𝜑(Q𝑘)) +tr(P𝚪) +tr

δ_𝑘δ^H_𝑘

P⁻¹−I +tr

<

n

2(A¯ −I)x^tr_𝑘δ^H𝑘

o

P⁻¹−I

, (4.105)

where we use the fact that tr(XD) =tr(XD)holds true for any diagonal matrixD. Next, we will provide bounds for individual elements on the right-hand-side of (4.105). We first get the following trace equality by summing (4.99) over all indices:

tr 𝜑(X)

=tr

X I+A^HP A−P

. (4.106)

Now defineΨas follows:

Ψ =𝜆

max I+A^HP A−P

. (4.107)

Then we have the following:

ΨI I+A^HP A−P=A¯^HA¯ + (A^H−I) (P−P²) (A−I) A¯^HA¯, (4.108) which implies the following:

kA¯k₂ ≤Ψ^1/2 and tr 𝜑(X)

≤ Ψ tr(X). (4.109) Using the bound in (4.53), we can write the following:

tr

δ^𝑘δ^H𝑘

P⁻¹−I

=δ^H𝑘 P⁻¹−I

δ^𝑘 ≤ 𝚫²

P⁻¹−I

2. (4.110) For the last term on the right-hand-side of (4.105), we can write the following inequalities:

tr

<n

2(A¯ −I)x^tr_𝑘δ^H𝑘

o

P⁻¹−I

=<n

2δ^H𝑘 (P^-1−I) (A¯ −I)x^tr_𝑘 o

≤

2δ^H𝑘 (I−P) (A−I)A¯^𝑘 (E[x₀] −x^ss

0)

≤ 𝑐Ψ^𝑘/2, (4.111) where (4.111) follows from the definition of x^tr_𝑘 in (4.18), and the constant 𝑐 in (4.111) is given as follows:

𝑐 =2𝚫

(I−P) (A−I) 2

E[x₀] −x^ss

0

2, (4.112)

where we use the bounds from (4.53) and (4.109).

Using the bounds in (4.109), (4.110), and (4.111) in the equality (4.105), we get the following:

tr(Q𝑘+1) ≤Ψtr(Q𝑘) +tr(P𝚪) +𝚫²kP⁻¹−Ik₂+𝑐Ψ^𝑘/2. (4.113) Using the inequality (4.113) recursively, we get the following:

tr(Q𝑘) ≤ Ψ^𝑘tr(Q₀) +𝑐 Ψ^𝑘/2−Ψ^𝑘

/ Ψ^1/2−Ψ + 1−Ψ^𝑘

1−Ψ

tr(P𝚪) +𝚫²kP⁻¹−Ik₂

. (4.114)

In the final step we use the assumptionA^HP A≺ Pin (4.51), and conclude from (4.107) thatΨ < 1. So, the upper bound in (4.114) gives the following result:

lim sup

𝑘→∞ tr(Q𝑘) ≤ tr(P𝚪) +𝚫²kP⁻¹−Ik₂

1−Ψ , (4.115)

which is equivalent to (4.52) since 1−Ψ =𝜆

min(P−A^HP A).

4.8.8 Proof of Lemma 4.5

Assume thatA∈C^𝑁×𝑁 is a triangular matrix with 𝐴_𝑖,𝑖 being the𝑖^{𝑡 ℎ} diagonal entry.

Then, we first note that both ¯Aand𝚽are triangular matrices, which follows simply from the fact that P and J are diagonal matrices, and the Kronecker product of triangular matrices is a triangular matrix. In particular, we note that the𝑖^{𝑡 ℎ}diagonal entry of ¯Ais as follows:

¯ 𝐴_𝑖,𝑖 =

1+ 𝑝_𝑖(𝐴_𝑖,𝑖−1)

. (4.116)

Then, it is straightforward to verify that the𝑘^{𝑡 ℎ}diagonal entry of𝚽is as follows for 𝑘 =𝑖+ (𝑗 −1)𝑁with 1 ≤ 𝑖, 𝑗 ≤ 𝑁:

𝑆_{𝑘 , 𝑘} = 𝐴¯^∗

𝑗 , 𝑗 𝐴¯_𝑖,𝑖+𝛿_{𝑖, 𝑗} (𝑝_𝑖− 𝑝²

𝑖) |𝐴_𝑖,𝑖−1|², (4.117) where𝛿_{𝑖, 𝑗} =1 if and only if𝑖= 𝑗.

Since𝚽is triangular, the diagonal entries of𝚽correspond to the eigenvalues of𝚽. Thus,

𝜌(𝚽)= max

1≤𝑘≤𝑁²

|𝑆_{𝑘 , 𝑘}|= max

1≤𝑖, 𝑗≤𝑁

𝐴¯^∗

𝑗 , 𝑗 𝐴¯_𝑖,𝑖+𝛿_{𝑖, 𝑗} (𝑝_𝑖− 𝑝²

𝑖) |𝐴_𝑖,𝑖−1|²

(4.118)

= max

1≤𝑖≤𝑁|𝐴¯_𝑖,𝑖|²+ (𝑝_𝑖− 𝑝²

𝑖) |𝐴_𝑖,𝑖−1|² = max

1≤𝑖≤𝑁1+ 𝑝_𝑖 |𝐴_𝑖,𝑖|²−1

. (4.119) We now prove the equivalence in (4.64). Assume thatAis a stable matrix. SinceA is a triangular matrix, its diagonal entries are the eigenvalues. Thus,

𝜌(A) < 1 =⇒ |𝐴_𝑖,𝑖| <1 ∀𝑖 =⇒ 𝜌(𝚽) < 1, (4.120) where the last implication follows from (4.119).

For the converse direction,

𝜌(A) ≥ 1 =⇒ ∃𝑖 s.t. |𝐴_𝑖,𝑖| ≥ 1 =⇒ 𝜌(𝚽) ≥1, (4.121) where the last implication follows from (4.119).

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