Power Control in Wireless Ad Hoc Networks: Stability and Convergence Under Uncertainties

Abstract

A successful distributed power control algorithm requires only local measurements for updating the power level of a transmitting node, so that eventually all transmitters meet their QoS requirements, i.e. the solution converges to the global optimum. There are numerous algorithm which claim to work under ideal conditions in which there exist no uncertainties and the model is identical to the real-world implementation. Nevertheless, the problem arises when real-world phenomena are introduced into the problem, such as uncertainties (such as changing environment and time delays) or the QoS requirements cannot be achieved for all the users in the network. In this chapter, we study some distributed power control algorithms for wireless ad hoc networks and discuss their robustness to real-world phenomena. Simulations illustrate the validity of the existing results and suggest directions for future research.

1 Appendix: Mathematical Preliminaries

Some notions that are used in this chapter are more thoroughly described in the appendix, just for completeness.

Lyapunov Stability

Consider a differential equation

$$\begin{array}{rcl} \dot{\mathbf{x}}(t) = f(\mathbf{x}(t)),\ \ \mathbf{x}(0) ={ \mathbf{x}}_{0}& &\end{array}$$

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that has a unique solution x _e , i.e. f(x _e ) = 0 has a unique solution given by $\mathbf{x} ={ \mathbf{x}}_{e}$.

Let $\mathbf{z}(t) = \mathbf{x}(t) -{\mathbf{x}}_{e}$. Then, the differential equation (1) becomes

$$\begin{array}{rcl} \dot{\mathbf{z}}(t) = g(\mathbf{z}(t)),\ \ \mathbf{z}(0) ={ \mathbf{z}}_{0} ={ \mathbf{x}}_{0} -{\mathbf{x}}_{e}& &\end{array}$$

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that has a unique solution ${\mathbf{z}}_{e} = \mathbf{0}$, i.e. g(0) = 0 has a unique solution given by $\mathbf{z} = \mathbf{0}$.

Theorem 5 ([51]). 

Consider a continuously differentiable function V ( z) such that

$$\begin{array}{rcl} V (\mathbf{z}) > 0,\ \ \forall \mathbf{z}\neq 0& &\end{array}$$

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and V (0) = 0. If $\dot{V }(\mathbf{z}) \leq 0$ $\forall \mathbf{z}$ , then the equilibrium point is stable. If in addition, $\dot{V }(\mathbf{z}) < 0$ $\forall \mathbf{z}\neq \mathbf{0}$ , then the equilibrium point is asymptotically stable. If in addition to these, V is radially unbounded, i.e., $V (\mathbf{z}) \rightarrow \infty$ when $\mathbf{z} \rightarrow \infty$ , then the equilibrium point is globally asymptotically stable.

D-Stability

The notion of D-stability was initially introduced in the field of mathematical economics, but its properties are very useful for the study of dynamic equilibria and many important classes of matrices are linked with D-stability. The following summary is adopted from [52].

Definition 4 ([52]). 

Matrix $A \in {C}^{n\times n}$ is D-stable if DA is stable for all diagonal matrix D with positive entries.

Remark 5.

If $A \in {C}^{n\times n}$ is D-stable, then:

• 1. AD is similar to DA ($DA = D(AD){D}^{-1}$), so it is irrelevant in defining D-stability whether D is multiplied by the left or the right side of A.

• 2. A is nonsingular.

• 3. A ^− 1 and A ^∗ are D-stable.

• 4. DAE is D-stable, D, E positive diagonal matrices.

• 5. P ^T AP is D-stable, where P is any permutation matrix.

The following are some of the sufficient conditions for D-stability:

• 1. There exists a diagonal matrix D with positive entries, such that $DA + {A}^{{_\ast}}D$ is positive definite.

• 2. $A \in {\mathbb{R}}^{n\times n}$ is an M-matrix.

• 3. There exists a positive diagonal matrix D, such that $DA = B =\{ {b}_{ij}\}$ satisfies

  $$\begin{array}{rcl} \mathfrak{R}({b}_{ii}) >\sum\limits_{j=1,j\neq }^{n}\vert {b}_{ ij}\vert,\ \forall i = 1,2,\ldots,n.& & \\ \end{array}$$

• 4. $A =\{ {a}_{ij}\}$ is triangular and $\mathfrak{R}({a}_{ii}) > 0$, i = 1, 2, …, n.

• 5. For each $0\neq x \in {C}^{n\times n}$, there is a diagonal matrix D with positive entries such that $\mathfrak{R}({x}^{{_\ast}}DAx) > 0$.

• 6. $A \in {\mathbb{R}}^{n\times n}$ is oscillatory.

Fixed-Points, Contractions and Para-Contractions

We consider iterative algorithms on the form

$$\begin{array}{rcl} \mathbf{x}(n + 1)& =& T{\bigl (\mathbf{x}(n)\bigr )},\;\;\;n = 0,1,2,\ldots,\end{array}$$

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where T is a mapping from a subset X of ${\mathbb{R}}^{K}$ into itself. A vector x ^⋆ is called a fixed point of T if $T({\mathbf{x}}^{\star }) ={ \mathbf{x}}^{\star }$. If T is continuous at ${\mathbf{x}}^{\star }$ and the sequence {x(n)} converges to ${\mathbf{x}}^{\star }$, then ${\mathbf{x}}^{\star }$ is a fixed point of T [53, Chapter 3]. Therefore, the iteration (4) can be viewed as an algorithm for finding such a fixed point. T is called a contraction mapping, if it has the following property

$$\|T(\mathbf{x}) - T(\mathbf{y})\| \leq c\;\|\mathbf{x} -\mathbf{y}\|\;,\forall x,y \in X,$$

where $\|\cdot \|$ is some norm on X, and c ∈ [0, 1). The following proposition shows that contraction mappings have unique fixed points and linear convergence rates.

Proposition 2 (Convergence of Contracting Iterations [53, Chapter 3]). 

If $T : X \rightarrow X$ is a contraction mapping and that X is a closed subset of ${\mathbb{R}}^{K}$ , then:

• (Existence and uniqueness of fixed points) The mapping T has a unique fixed point x ^⋆ ∈ X.

• (Linear convergence) For every initial vector x(0) ∈ X, the sequence { x(n)} generated by $\mathbf{x}(n + 1) = T{\bigl (\mathbf{x}(n)\bigr )}$ converges to x ^⋆ linearly. In particular,

  $$\|\mathbf{x}(n) -{\mathbf{x}}^{\star }\| \leq {c}^{n}\|\mathbf{x}(0) -{\mathbf{x}}^{\star }\|\;.$$

An operator T on X is called para-contraction if

$$\begin{array}{rcl} \|T(\mathbf{x}) - T(\mathbf{y})\| <\| \mathbf{x} -\mathbf{y}\|\;,\;\;\mbox{ for all}\;\;\mathbf{x}\neq \mathbf{y}\;.& & \\ \end{array}$$

Para-contractions have at most one fixed point and, in contrast to contractions, may not have a fixed point. As an example, consider the para-contracting function $T(x) = x + {e}^{-x}$ in [0, ∞). It is easily seen that T has no fixed point. The following theorem summarises properties of para-contractions.

Proposition 3 ([54]). 

If $T : X \rightarrow X$ is a para-contraction, then:

• If T has a fixed point x ^⋆, then that fixed point is unique; moreover

• If X is a finite-dimensional space, for every initial vector x(0) ∈ X, the sequence { x(n)} generated by $\mathbf{x}(n + 1) = T{\bigl (\mathbf{x}(n)\bigr )}$ converges to x ^⋆.

As can be seen from Proposition 3, para-contractivity does not yield any estimate of the rate of convergence to the fixed point.