Energy Efficient Distributed Volterra Modeling Approach with ADMM-Based Sparse Signal Recovery

Abstract

Modeling the behavior of nonlinear systems in a distributed fashion is of paramount in many industrial applications. Distributed means no sensor node in the wireless network has complete information about the data. Also, there is no centralized unit, and the communication can take place only with the single-hop neighborhood. The prodigious amount of data exchange between nodes limits the life of any distributed network that results in an inefficient ad-hoc network deployment. The current article develops a compressed-sensing (CS) based distributed Volterra–Laguerre modeling approach. It can remarkably decrease the communication load among the nodes that benchmarks the performance of full information exchange configuration. Further reduction in communication is achieved by capitalizing the spatial sparsity of the localized phenomena. The latter approach utilizes the inactive node strategy along with CS where a fraction of nodes are turned off. It conserves more energy than CS-based approach. ADMM-based sparse signal recovery method is then proposed to encompass the NP-hard limitation of $l_0$-norm minimization. The proposed recovery method is employed at each node to recover the uncompressed estimates of the neighboring nodes compressed data. Simulation results on a 2nd-order nonlinear system are obtained, expressing the potential performance of the proposed approaches.

Appendix 1

In ADMM, the first step is to find the updates of Lagrangian multipliers using gradient ascent iterations

$$\begin{aligned} {\bar{h}}_j^{j'}(P) &= {\bar{h}}_j^{j'}(P - 1) + c\left[ {{{\bar{\mathcal {C}}} _j}(P) - {\bar{Z}}_j^{j'}(P)} \right] \end{aligned}$$

(24)

$$\begin{aligned} {\bar{\mu }} _j^{j'}(P) &= {\bar{\mu }} _j^{j'}(P-1) + c\left[ {{{\bar{\mathcal {C}}} _j}(P) - {\bar{Z}}_{j'}^j(P)} \right] . \end{aligned}$$

(25)

Second step involves the minimization ofaugmented Lagrangian function (21) w.r.t. ${{\bar{\mathcal {C}}} _j}$ holding all other variables fixed to their most recent updated values. Then, the variables ${{\bar{\mathcal {C}}} _j}$ can be recursively updated as

$$\begin{aligned} {{{\bar{\mathcal {C}}}}_j}\left( {P + 1} \right) &= {\left[ \begin{array}{l} 2\sum \limits _{m = 0}^{P+1} {{\lambda ^{P +1 - m}}\left[ {{{\varvec{B}}_{jj}}\left( m \right) {{{{\bar{\phi }}} }_j}(m){{\bar{\phi }}} _j^T(m)} \right] } \nonumber \\ \quad +2{P^{ - 1}}{\lambda ^{P+1}}{{\varvec{B}}_{jj}}\left( m \right) {{\varvec{\Psi }}_0} + 2c\left| {{N_j}} \right| \\ \end{array} \right] ^{ - 1}} \nonumber \\&\quad \times \, \left[ \begin{array}{l} {\text {2}}\sum \limits _{m = 0}^{P+1} {{\lambda ^{P +1 - m}}\left[ {{{\varvec{B}}_{jj}}\left( m \right) {{{{\bar{\phi }}} }_j}(m){x _j}(m)} \right] } \\ \quad - {\sum _{j' \in {{\mathcal {N}}_j}}}\left( {{\bar{h}}_j^{j'}(P) + {\bar{\mu }} _j^{j'}(P)} \right) \\ \quad + c{\sum _{j' \in {{\mathcal {N}}_j}}}\left( {{\bar{Z}}_j^{j'}\left( {P} \right) + {\bar{Z}}_{j'}^j\left( {P} \right) } \right) \\ \end{array} \right] \end{aligned}$$

(26)

The third step of ADMM minimizes the augmented Lagrangian function (21) w.r.t. ${{\bar{Z}}_j^{j'}}$ with all other variables kept to their warm-start values. Then ${{\bar{Z}}_j^{j'}}$ can be expressed as

$$\begin{aligned} {\bar{Z}}_j^{j'}\left( {P + 1} \right) = 0.5\left[ {{{\bar{\mathcal {C}}} _j}\left( {P + 1} \right) + {{\bar{\mathcal {C}}} _{j'}}\left( {P + 1} \right) } \right] + 0.5{c^{ - 1}}\left[ {{\bar{h}}_j^{j'}\left( {P } \right) + {\bar{\mu }} _{j'}^j\left( {P} \right) } \right] . \end{aligned}$$

(27)

Using (27) into (24) and (25), and if Lagrange multipliers are initialized as ${\bar{h}}_j^{j'}(0) = - {\bar{\mu }} _{j'}^j(0)$, it follows that ${\bar{h}}_j^{j'}(P) = - {\bar{\mu }} _{j'}^j(P),\,\forall \,t$, which leads to the expression (22)

$$\begin{aligned} {\bar{h}}_j^{j'}(P) ={\bar{h}}_j^{j'}(P- 1) + 0.5c\left[ {{{\bar{\mathcal {C}}} _j}(P) - {{\bar{\mathcal {C}}} _{j'}}(P)} \right] . \end{aligned}$$

(28)

Now, substituting (27) into (26) and following ${\bar{h}}_j^{j'}(P) = - {\bar{\mu }} _{j'}^j(P),\,\forall \,t$, (26) can be simplified as (23).