c}\right\rceil,\left\lceil\frac{M(1-\alpha)}{(1-\alpha)c-pg(\alpha)}\right\rceil\Bigg{\\}}$.
We define the events

$A_{t_{1},t_{2}}:\displaystyle\sum_{l=t_{1}}^{t2}X_{l}<\alpha\displaystyle\sum_{l=t_{1}}^{t2}Z_{l}+\displaystyle\sum_{l=t_{1}}^{t2}X_{l}S_{l}-\alpha
M,$
$A_{\tau}=\displaystyle\bigcup_{t_{1}=1}^{\tau}A_{t_{1},\tau},A=\displaystyle\bigcup_{\tau=t-\lambda\widetilde{M}}^{t-1}A_{\tau},$

$B_{t_{1},t_{2}}:\displaystyle\sum_{l=t_{1}}^{t2}S_{l}X_{l}\geq(1-\alpha)\displaystyle\sum_{l=t_{1}}^{t2}Z_{l}+(1-\alpha)M,$
$B_{\tau}=\displaystyle\bigcup_{t_{1}=1}^{\tau}B_{t_{1},\tau},B=\displaystyle\bigcup_{\tau=t-\lambda\widetilde{M}}^{t-1}B_{\tau},$

$E:\displaystyle\sum_{l=t-\lambda\widetilde{M}}^{t-1}(S_{l}X_{l}-(1-\alpha)Z_{l})+(1-\alpha)M<0,$
$F:\displaystyle\sum_{l=t-\lambda\widetilde{M}}^{t-1}((1-S_{l})X_{l}-\alpha
Z_{l})\geq\alpha M.$

Using Hoeffding’s inequality,

$\displaystyle\mathbb{P}(A_{t_{1},t_{2}})$
$\displaystyle\leq\exp\left(\frac{-2(p(1-g(\alpha))-\alpha
c)(t_{2}-t_{1}+1)+\alpha M)^{2}}{(t_{2}-t_{1}+1)(1+\alpha
c_{\text{max}}-\alpha
c_{\text{min}})^{2}}\right)\leq\delta_{A}\exp\left(\frac{-2(p(1-g(\alpha))-\alpha
c))^{2}(t_{2}-t_{1}+1)}{(1+\alpha c_{\text{max}}-\alpha
c_{\text{min}})^{2}}\right),$

where $\delta_{A}=\exp\left(\frac{-4(p(1-g(\alpha))-\alpha c))\alpha
M}{(1+\alpha c_{\text{max}}-\alpha c_{\text{min}})^{2}}\right)$. Note that
since $p(1-g(\alpha))>\alpha c$, the quantity $\delta_{A}<1$ and it decreases
exponentially with increase in $M$.

Using Union bound,

$\displaystyle\mathbb{P}(A_{\tau})$
$\displaystyle\leq\displaystyle\sum_{t_{1}=1}^{\tau}\mathbb{P}(A_{t_{1},\tau})\leq\displaystyle\sum_{t_{1}=1}^{\tau-\frac{M}{c_{\text{max}}}}\mathbb{P}(A_{t_{1},\tau})\leq\delta_{A}\displaystyle\sum_{l=0}^{\infty}\exp\left(\frac{-2(p(1-g(\alpha))-\alpha
c))^{2}(\frac{M}{c_{\text{max}}}+1+l)}{(1+\alpha c_{\text{max}}-\alpha
c_{\text{min}})^{2}}\right)$
$\displaystyle\leq\delta_{A}\frac{\exp\left(-2\frac{M+c_{\text{max}}}{c_{\text{max}}}\frac{(p(1-g(\alpha))-\alpha
c)^{2}}{(1+\alpha c_{\text{max}}-\alpha
c_{\text{min}})^{2}}\right)}{1-\exp\left(-2\frac{(p(1-g(\alpha))-\alpha
c)^{2}}{(1+\alpha c_{\text{max}}-\alpha c_{\text{min}})^{2}}\right)}.$ (14)
$\displaystyle\mathbb{P}(A)\leq\lambda\widetilde{M}\mathbb{P}(A_{\tau})\leq\lambda\widetilde{M}\frac{\delta_{A}\exp\left(-2\frac{M+c_{\text{max}}}{c_{\text{max}}}\frac{(p(1-g(\alpha))-\alpha
c)^{2}}{(1+\alpha c_{\text{max}}-\alpha
c_{\text{min}})^{2}}\right)}{1-\exp\left(-2\frac{(p(1-g(\alpha))-\alpha
c)^{2}}{(1+\alpha c_{\text{max}}-\alpha c_{\text{min}})^{2}}\right)}.$

Along similar lines we prove the following bounds.

(15) $\displaystyle\mathbb{P}(B)$
$\displaystyle\leq\lambda\widetilde{M}\mathbb{P}(B_{\tau})\leq\lambda\widetilde{M}\frac{\delta_{B}\exp\left(-2(\frac{(1-\alpha)M}{1-(1-\alpha)c_{\text{min}}}+1)\frac{((1-\alpha)c-pg(\alpha))^{2}}{(1+(1-\alpha)(c_{\text{max}}-c_{\text{min}}))^{2}}\right)}{1-\exp\left(-2\frac{((1-\alpha)c-pg(\alpha))^{2}}{(1+(1-\alpha)(c_{\text{max}}-c_{\text{min}}))^{2}}\right)},$

where
$\delta_{B}=\exp\left(\frac{-4((1-\alpha)c-pg(\alpha)))(1-\alpha)M}{(1+(1-\alpha)(c_{\text{max}}-c_{\text{min}}))^{2}}\right)$
which is less than one and decreases exponentially with $M$.

$\displaystyle\mathbb{P}(E^{c})$
$\displaystyle\leq\exp\left(\frac{-2(\lambda\widetilde{M}((1-\alpha)c-pg(\alpha))-(1-\alpha)M)^{2}}{\lambda\widetilde{M}(1+(1-\alpha)(c_{\text{max}}-c_{\text{min}}))^{2}}\right)\leq\exp\left(\frac{-2(\lambda(1-\alpha)M-(1-\alpha)M)^{2}}{\lambda\widetilde{M}(1+(1-\alpha)(c_{\text{max}}-c_{\text{min}}))^{2}}\right)$
(16)
$\displaystyle\leq\exp\left(\frac{-2(\lambda-1)^{2}M^{2}(1-\alpha)^{2}}{\lambda\widetilde{M}(1+(1-\alpha)(c_{\text{max}}-c_{\text{min}}))^{2}}\right).$

Similarly,

(17)
$\displaystyle\mathbb{P}(F^{c})\leq\exp\left(\frac{-2(\lambda-1)^{2}M^{2}\alpha^{2}}{\lambda\widetilde{M}(1+\alpha(c_{\text{max}}-c_{\text{min}}))^{2}}\right).$

Note that the right hand side of inequalities (16), (17) diminishes
exponentially increase in $M$.

  * –

If $\alpha$ fraction of service is at the edge at time
$t-\lambda\widetilde{M}$ then $A^{c}$ implies
$\text{totalCost}(R_{0}^{n},I_{m})>\text{totalCost}(R_{\alpha}^{n},I_{m})$,
for any $n\in[1,t-1]$ and $m\in[n,t-1]$. In the same case, $B^{c}$ implies
$\text{totalCost}(R_{1}^{n},I_{m})>\text{totalCost}(R_{\alpha}^{n},I_{m})$,
for any $n\in[1,t-1]$ and $m\in[n,t-1]$. Thus $A^{c}\cap B^{c}$ ensures
$\alpha$ fraction of service will be at the edge at time-slot $t$.

  * –

If no fraction of service is at the edge at time-slot
$t-\lambda\widetilde{M}$, and $\alpha$ fraction of service is fetched during
$t-\lambda\widetilde{M}+1\leq\tilde{\tau}\leq t-2$ then $A^{c}$ implies
$\text{totalCost}(R_{0}^{n},I_{m})>\text{totalCost}(R_{\alpha}^{n},I_{m})$,
for any $n\in[t-\lambda\widetilde{M}+1,t-1]$ and $m\in[n,t-1]$. In the same
case, $B^{c}$ implies
$\text{totalCost}(R_{1}^{n},I_{m})>\text{totalCost}(R_{\alpha}^{n},I_{m})$
,for any $n\in[t-\lambda\widetilde{M}+1,t-1]$ and $m\in[n,t-1]$. Thus
$A^{c}\cap B^{c}$ ensures $\alpha$ fraction of service will be at the edge at
time-slot $t$.

  * –

If full service is at the edge at time-slot $t-\lambda\widetilde{M}$ and only
$1-\alpha$ fraction of service is evicted during
$t-\lambda\widetilde{M}+1\leq\tilde{\tau}\leq t-2$ $A^{c}$ implies
$\text{totalCost}(R_{0}^{n},I_{m})>\text{totalCost}(R_{\alpha}^{n},I_{m})$ for
any $n\in[t-\lambda\widetilde{M}+1,t-1]$ and $m\in[n,t-1]$. In the same case,
$B^{c}$ implies
$\text{totalCost}(R_{1}^{n},I_{m})>\text{totalCost}(R_{\alpha}^{n},I_{m})$,
for any $n\in[t-\lambda\widetilde{M}+1,t-1]$ and $m\in[n,t-1]$. Thus
$A^{c}\cap B^{c}$ ensures $\alpha$ fraction of service will be at the edge at
time-slot $t$.

  * –

If no service is at the edge at time-slot $t-\lambda\widetilde{M}$ and no
amount of service is fetched $t-\lambda\widetilde{M}+1\leq\tilde{\tau}\leq
t-2$, then the event $F$ implies
$\text{totalCost}(R_{0}^{t-\lambda\widetilde{M}},I_{t-1})>\text{totalCost}(R_{\alpha}^{t-\lambda\widetilde{M}},I_{t-1})$,
that is $\alpha$ fraction of service will be at the edge at time-slot $t$.

  * –

If service is not hosted at time-slot $t-\lambda\widetilde{M}$ and full
service is fetched during $t-\lambda\widetilde{M}+1\leq\tilde{\tau}\leq t-2$
then the event $E$ implies
$\text{totalCost}(R_{1}^{t-\lambda\widetilde{M}},I_{t-1})>\text{totalCost}(R_{\alpha}^{t-\lambda\widetilde{M}},I_{t-1})$,
that is $\alpha$ fraction of service will be at the edge at time-slot $t$.

  * –

If full service is at the edge at time-slot $t-\lambda\widetilde{M}$ and not
evicted till $t-2$ then the event $E$ implies
$\text{totalCost}(R_{1}^{t-\lambda\widetilde{M}},I_{t-1})>\text{totalCost}(R_{\alpha}^{t-\lambda\widetilde{M}},I_{t-1})$,
that is $\alpha$ fraction of service will be at the edge at time-slot $t$.

$\displaystyle\mathbb{P}(A^{c}\cap B^{c}\cap E\cap F)\geq
1-\mathbb{P}(A)-\mathbb{P}(B)-\mathbb{P}(E^{c})-\mathbb{P}(F^{c}).$

From the inequalities (14), (15), (16) and (17), we see that
$\mathbb{P}(A^{c}\cap B^{c}\cap E\cap F)$ approaches unity as the value of $M$
increases.

Conditioned on $G=A^{c}\cap B^{c}\cap E\cap F$, the $\alpha$ fraction of
service is hosted at the edge during time-slot $t$. The expected cost incurred
by the $\alpha-$RR policy is
$\mathbb{E}[C_{t}^{\alpha-\text{RR}}]=\mathbb{E}[C_{t}^{\alpha-\text{RR}}|G]\mathbb{P}(G)+\mathbb{E}[C_{t}^{\alpha-\text{RR}}|G^{c}]\times\mathbb{P}(G^{c}).$
Note that,
$\mathbb{E}[C_{t}^{\alpha-\text{RR}}|G]=\mathbb{E}[C_{t}^{\alpha-\text{OPT-
ON}}],\ \mathbb{E}[C_{t}^{\alpha-\text{RR}}|G^{c}]\leq\max\\{M+p,M+c\\}.$
Therefore,

$\displaystyle\mathbb{E}[C_{t}^{\alpha-\text{RR}}]-\mathbb{E}[C_{t}^{\alpha-\text{OPT-
ON}}]=$
$\displaystyle\mathbb{E}[C_{t}^{\alpha-\text{RR}}|G](\mathbb{P}(G)-1)\max\\{M+p,M+c\\}\mathbb{P}(G^{c})\leq\max\\{M+p,M+c\\}\mathbb{P}(G^{c}).$
$\displaystyle\frac{\mathbb{E}[C_{t}^{\alpha-\text{RR}}]}{\mathbb{E}[C_{t}^{\alpha-\text{OPT-
ON}}]}\leq$ $\displaystyle 1+\max\\{M+p,M+c\\}\frac{\mathbb{P}(G^{c})}{\alpha
c+g(\alpha)p}$

Case 2: When $p>\max\\{c,\frac{(1-\alpha)c}{g(\alpha)}\\}$
Let
$\widetilde{M}=\max\Bigg{\\{}\frac{M}{p-c},\left\lceil\frac{M(1-\alpha)}{pg(\alpha)-(1-\alpha)c}\right\rceil\Bigg{\\}}$.
We define the events

$A_{t_{1},t_{2}}:\displaystyle\sum_{l=t_{1}}^{t2}X_{l}+M<\displaystyle\sum_{l=t_{1}}^{t2}Z_{l},$
$A_{\tau}=\displaystyle\bigcup_{t_{1}=1}^{\tau}A_{t_{1},\tau},A=\displaystyle\bigcup_{\tau=t-\lambda\widetilde{M}}^{t-1}A_{\tau},$
$B_{t_{1},t_{2}}:\displaystyle\sum_{l=t_{1}}^{t2}S_{l}X_{l}+(1-\alpha)M<(1-\alpha)\displaystyle\sum_{l=t_{1}}^{t2}Z_{l},$
$B_{\tau}=\displaystyle\bigcup_{t_{1}=1}^{\tau}B_{t_{1},\tau},B=\displaystyle\bigcup_{\tau=t-\lambda\widetilde{M}}^{t-1}B_{\tau},$
$D:\displaystyle\sum_{l=t-\lambda\widetilde{M}}^{t-1}X_{l}\geq\displaystyle\sum_{l=t-\lambda\widetilde{M}}^{t-1}Z_{l}+M,$
$E:\displaystyle\sum_{l=t-\lambda\widetilde{M}}^{t-1}S_{l}X_{l}\geq(1-\alpha)\displaystyle\sum_{l=t-\lambda\widetilde{M}}^{t-1}Z_{l}+(1-\alpha)M.$

Using Hoeffding’s inequality,

$\displaystyle\mathbb{P}(A_{t_{1},t_{2}})$
$\displaystyle\leq\exp\left(\frac{-2((p-c)(t_{2}-t_{1}+1)+M)^{2}}{(t_{2}-t_{1})(1+c_{\text{max}}-c_{\text{min}})^{2}}\right)\leq\delta_{A}\exp\left(\frac{-2(p-c)^{2}(t_{2}-t_{1}+1)}{(1+c_{\text{max}}-c_{\text{min}})^{2}}\right),$

where $\delta_{A}=\exp\left(\frac{-4(p-c)\alpha
M}{(1+c_{\text{max}}-c_{\text{min}})^{2}}\right)$. Note that since $p>c$, the
quantity $\delta_{A}<1$ and it decreases exponentially with increase in $M$.
Using Union bound,

$\displaystyle\mathbb{P}(A_{\tau})$
$\displaystyle\leq\displaystyle\sum_{t_{1}=1}^{\tau}\mathbb{P}(A_{t_{1},\tau})\leq\displaystyle\sum_{t_{1}=1}^{\tau-\frac{M}{c_{\text{max}}}}\mathbb{P}(A_{t_{1},\tau})\leq\delta_{A}\displaystyle\sum_{l=0}^{\infty}\exp\left(\frac{-2(p-c))^{2}(\frac{M}{c_{\text{max}}}+1+l)}{(1+c_{\text{max}}-c_{\text{min}})^{2}}\right)$
$\displaystyle\leq\delta_{A}\frac{\exp\left(-2\frac{M+c_{\text{max}}}{c_{\text{max}}}\frac{(p-c)^{2}}{(1+c_{\text{max}}-\alpha
c_{\text{min}})^{2}}\right)}{1-\exp\left(-2\frac{(p-c)^{2}}{(1+c_{\text{max}}-c_{\text{min}})^{2}}\right)}.$
(19)
$\displaystyle\mathbb{P}(A)\leq\lambda\widetilde{M}\mathbb{P}(A_{\tau})\leq\lambda\widetilde{M}\delta_{A}\frac{\exp\left(-2\frac{M+c_{\text{max}}}{c_{\text{max}}}\frac{(p-c)^{2}}{(1+c_{\text{max}}-\alpha
c_{\text{min}})^{2}}\right)}{1-\exp\left(-2\frac{(p-c)^{2}}{(1+c_{\text{max}}-c_{\text{min}})^{2}}\right)}.$

Along similar lines we prove the following bounds.

(20) $\displaystyle\mathbb{P}(B)$
$\displaystyle\leq\lambda\widetilde{M}\mathbb{P}(B_{\tau})\leq\lambda\widetilde{M}\frac{\delta_{B}\exp\left(-2\frac{M+c_{\text{max}}}{c_{\text{max}}}\frac{(pg(\alpha)-(1-\alpha)c)^{2}}{(1+(1-\alpha)(c_{\text{max}}-c_{\text{min}}))^{2}}\right)}{1-\exp\left(-2\frac{(pg(\alpha)-(1-\alpha)c)^{2}}{(1+(1-\alpha)(c_{\text{max}}-c_{\text{min}}))^{2}}\right)},$

where
$\delta_{B}=\exp\left(\frac{-4(pg(\alpha)-(1-\alpha)c))(1-\alpha)M}{(1+(1-\alpha)(c_{\text{max}}-c_{\text{min}}))^{2}}\right)$
which is less than one and decreases exponentially with $M$.

$\displaystyle\mathbb{P}(E^{c})$
$\displaystyle\leq\exp\left(\frac{-2(\lambda\widetilde{M}(pg(\alpha)-(1-\alpha)c)-(1-\alpha)M)^{2}}{\lambda\widetilde{M}(1+(1-\alpha)(c_{\text{max}}-c_{\text{min}}))^{2}}\right)\leq\exp\left(\frac{-2(\lambda(1-\alpha)M-(1-\alpha)M)^{2}}{\lambda\widetilde{M}(1+(1-\alpha)(c_{\text{max}}-c_{\text{min}}))^{2}}\right)$
(21)
$\displaystyle\leq\exp\left(\frac{-2(\lambda-1)^{2}M^{2}(1-\alpha)^{2}}{\lambda\widetilde{M}(1+(1-\alpha)(c_{\text{max}}-c_{\text{min}}))^{2}}\right).$

Similarly,

(22)
$\displaystyle\mathbb{P}(D^{c})\leq\exp\left(\frac{-2(\lambda-1)^{2}M^{2}\alpha^{2}}{\lambda\widetilde{M}(1+\alpha(c_{\text{max}}-c_{\text{min}}))^{2}}\right).$

Note that the right hand side of inequalities (21), (22) diminishes
exponentially increase in $M$.

  * –

If $\alpha$ full service is at the edge at time $t-\lambda\widetilde{M}$ then
$A^{c}$ implies
$\text{totalCost}(R_{0}^{n},I_{m})>\text{totalCost}(R_{1}^{n},I_{m})$, for any
$n\in[1,t-1]$ and $m\in[n,t-1]$. In the same case, $B^{c}$ implies
$\text{totalCost}(R_{\alpha}^{n},I_{m})>\text{totalCost}(R_{1}^{n},I_{m})$,
for any $n\in[1,t-1]$ and $m\in[n,t-1]$. Thus,$A^{c}\cap B^{c}$ ensures that
full service will be at the edge at time-slot $t$.

  * –

If full service or $\alpha$ fraction of service is at the edge at time-slot
$t-\lambda\widetilde{M}$, and is evicted during
$t-\lambda\widetilde{M}+1\leq\tilde{\tau}\leq t-2$ then the event $D$ implies
$\text{totalCost}(R_{0}^{t-\lambda\widetilde{M}+1},I_{t-2})>\text{totalCost}(R_{1}^{t-\lambda\widetilde{M}+1},I_{t-2})$,
that is full service will be at the edge at time-slot $t$.

  * –

If $\alpha$ fraction of service is at the edge at time-slot
$t-\lambda\widetilde{M}$, and no service is fetched during
$t-\lambda\widetilde{M}+1\leq\tilde{\tau}\leq t-2$ then $E$ ensures that
$\text{totalCost}(R_{\alpha}^{t-\lambda\widetilde{M}+1},I_{t-2})>\text{totalCost}(R_{1}^{t-\lambda\widetilde{M}+1},I_{t-2})$,
that is full service will be at the edge at time-slot $t$.

  * –

If service is not at the edge at time-slot $t-\lambda\widetilde{M}$, and
$\alpha$ fraction of service is fetched during
$t-\lambda\widetilde{M}+1\leq\tilde{\tau}\leq t-2$, then $D\cap E$ ensures
that
$\text{totalCost}(R_{\alpha}^{t-\lambda\widetilde{M}+1},I_{t-2})>\text{totalCost}(R_{1}^{t-\lambda\widetilde{M}+1},I_{t-2})$,
that is full service will be at the edge at time-slot $t$.

  * –

If service is not at the edge at time-slot $t-\lambda\widetilde{M}$, and no
fraction of service is fetched during
$t-\lambda\widetilde{M}+1\leq\tilde{\tau}\leq t-2$, then $D\cap E$ ensures
that
$\text{totalCost}(R_{\alpha}^{t-\lambda\widetilde{M}+1},I_{t-2})>\text{totalCost}(R_{1}^{t-\lambda\widetilde{M}+1},I_{t-2})$,
that is full service will be at the edge at time-slot $t$.

Using union bound and Hoeffding inequality,

$\displaystyle\mathbb{P}(A^{c}\cap B^{c}\cap D\cap E)\geq
1-\mathbb{P}(A)-\mathbb{P}(B)-\mathbb{P}(D^{c})-\mathbb{P}(E^{c})$

From the inequalities (19), (20), (22) and (21), we see that
$\mathbb{P}(A^{c}\cap B^{c}\cap D\cap E)$ approaches unity as the value of $M$
increases.

Conditioned on $G=A^{c}\cap B^{c}\cap D\cap E$, the full service is hosted at
the edge during time-slot $t$. The expected cost incurred by the $\alpha-$RR
policy is
$\mathbb{E}[C_{t}^{\alpha-\text{RR}}]=\mathbb{E}[C_{t}^{\alpha-\text{RR}}|G]\mathbb{P}(G)+\mathbb{E}[C_{t}^{\alpha-\text{RR}}|G^{c}]\times\mathbb{P}(G^{c}).$
Note that,
$\mathbb{E}[C_{t}^{\alpha-\text{RR}}|G]=\mathbb{E}[C_{t}^{\alpha-\text{OPT-
ON}}],\ \mathbb{E}[C_{t}^{\alpha-\text{RR}}|G^{c}]\leq\max\\{\alpha M+\alpha
c+g(\alpha)p,M+p\\}.$ Therefore,

$\displaystyle\mathbb{E}[C_{t}^{\alpha-\text{RR}}]-\mathbb{E}[C_{t}^{\alpha-\text{OPT-
ON}}]=$
$\displaystyle\mathbb{E}[C_{t}^{\alpha-\text{RR}}|G](\mathbb{P}(G)-1)+\max\\{\alpha
M+\alpha c+g(\alpha)p,M+p\\}\mathbb{P}(G^{c})$ $\displaystyle\leq$
$\displaystyle\max\\{\alpha M+\alpha c+g(\alpha)p,M+p\\}\mathbb{P}(G^{c}).$
$\displaystyle\frac{\mathbb{E}[C_{t}^{\alpha-\text{RR}}]}{\mathbb{E}[C_{t}^{\alpha-\text{OPT-
ON}}]}\leq$ $\displaystyle 1+\max\\{\alpha M+\alpha
c+g(\alpha)p,M+p\\}\frac{\mathbb{P}(G^{c})}{c}$

Case 3: When $p<\min\\{c,\frac{\alpha c}{1-g(\alpha)}\\}$
Let
$\widetilde{M}=\max\Bigg{\\{}\frac{M}{c-p},\left\lceil\frac{M\alpha}{\alpha
c-p(1-g(\alpha))}\right\rceil\Bigg{\\}}$. We define the events

$A_{t_{1},t_{2}}:\displaystyle\sum_{l=t_{1}}^{t_{2}}X_{l}\geq\displaystyle\sum_{l=t_{1}}^{t_{2}}Z_{l}+M,$
$A^{\tau}=\displaystyle\bigcup_{t_{1}=1}^{\tau}A_{t_{1},\tau},A_{t}=\displaystyle\bigcup_{\tau=t-\lambda\widetilde{M}}^{t}A^{\tau},$
$B_{t_{1},t_{2}}:\displaystyle\sum_{l=t_{1}}^{t_{2}}(1-S_{l})X_{l}\geq\alpha\displaystyle\sum_{l=t_{1}}^{t_{2}}Z_{l}+\alpha
M,$
$B^{\tau}=\displaystyle\bigcup_{t_{1}=1}^{\tau}B_{t_{1},\tau},B_{t}=\displaystyle\bigcup_{\tau=t-\lambda\widetilde{M}}^{t}B^{\tau},$
$D:\displaystyle\sum_{l=t-\lambda\widetilde{M}}^{t-1}X_{l}+M<\displaystyle\sum_{l=t-\lambda\widetilde{M}}^{t-1}Z_{l},$
$E:\displaystyle\sum_{l=t-\lambda\widetilde{M}}^{t-1}(1-S_{l})X_{l}+\alpha
M<\alpha\displaystyle\sum_{l=t-\lambda\widetilde{M}}^{t-1}Z_{l}.$

$\displaystyle\mathbb{P}(A_{t_{1},t_{2}})$
$\displaystyle\leq\exp\left(\frac{-2((c-p)(t_{2}-t_{1}+1)+M)^{2}}{(t_{2}-t_{1})(1+c_{\text{max}}-c_{\text{min}})^{2}}\right)\leq\delta_{A}\exp\left(\frac{-2(c-p)^{2}(t_{2}-t_{1}+1)}{(1+c_{\text{max}}-c_{\text{min}})^{2}}\right),$

where $\delta_{A}=\exp\left(\frac{-4(c-p)\alpha
M}{(1+c_{\text{max}}-c_{\text{min}})^{2}}\right)$. Note that since $c>p$, the
quantity $\delta_{A}<1$ and it decreases exponentially with increase in $M$.
Using Union bound,

$\displaystyle\mathbb{P}(A^{\tau})$
$\displaystyle\leq\displaystyle\sum_{t_{1}=1}^{\tau}\mathbb{P}(A_{t_{1},\tau})\leq\displaystyle\sum_{t_{1}=1}^{\tau-\frac{M}{1-c_{\text{min}}}}\mathbb{P}(A_{t_{1},\tau})\leq\delta_{A}\displaystyle\sum_{l=0}^{\infty}\exp\left(\frac{-2(c-p)^{2}(\frac{M}{1-c_{\text{min}}}+1+l)}{(1+c_{\text{max}}-c_{\text{min}})^{2}}\right)$
$\displaystyle\leq\delta_{A}\frac{\exp\left(-2(\frac{M}{1-c_{\text{min}}}+1)\frac{(c-p)^{2}}{(1+c_{\text{max}}-\alpha
c_{\text{min}})^{2}}\right)}{1-\exp\left(-2\frac{(c-p)^{2}}{(1+c_{\text{max}}-c_{\text{min}})^{2}}\right)}.$
(24)
$\displaystyle\mathbb{P}(A_{t})\leq\lambda\widetilde{M}\mathbb{P}(A_{\tau})\leq\lambda\widetilde{M}\frac{\delta_{A}\exp\left(-2(\frac{M}{1-c_{\text{min}}}+1)\frac{(c-p)^{2}}{(1+c_{\text{max}}-\alpha
c_{\text{min}})^{2}}\right)}{1-\exp\left(-2\frac{(c-p)^{2}}{(1+c_{\text{max}}-c_{\text{min}})^{2}}\right)}.$

Along similar lines we prove the following bounds.

(25) $\displaystyle\mathbb{P}(B_{t})$
$\displaystyle\leq\lambda\widetilde{M}\mathbb{P}(B_{\tau})\leq\lambda\widetilde{M}\frac{\delta_{B}\exp\left(-2(\frac{\alpha
M}{1-g(\alpha)-\alpha
c_{\text{min}}}+1)\frac{(c\alpha-p(1-g(\alpha)))^{2}}{(1+\alpha(c_{\text{max}}-c_{\text{min}}))^{2}}\right)}{1-\exp\left(-2\frac{(c\alpha-p(1-g(\alpha)))^{2}}{(1+\alpha(c_{\text{max}}-c_{\text{min}}))^{2}}\right)},$

where $\delta_{B}=\exp\left(\frac{-4(\alpha c-p(1-g(\alpha))\alpha
M}{(1+\alpha(c_{\text{max}}-c_{\text{min}}))^{2}}\right)$ which is less than
one and decreases exponentially with $M$.

(26) $\displaystyle\mathbb{P}(E^{c})$
$\displaystyle\leq\exp\left(\frac{-2(\lambda\widetilde{M}(\alpha
c-p(1-g(\alpha))-\alpha
M)^{2}}{\lambda\widetilde{M}(1+\alpha(c_{\text{max}}-c_{\text{min}}))^{2}}\right)\leq\exp\left(\frac{-2(\lambda\alpha
M-\alpha
M)^{2}}{\lambda\widetilde{M}(1+\alpha(c_{\text{max}}-c_{\text{min}}))^{2}}\right)\leq\exp\left(\frac{-2(\lambda-1)^{2}M^{2}\alpha^{2}}{\lambda\widetilde{M}(1+\alpha(c_{\text{max}}-c_{\text{min}}))^{2}}\right).$

Similarly,

(27)
$\displaystyle\mathbb{P}(D^{c})\leq\exp\left(\frac{-2(\lambda-1)^{2}M^{2}}{\lambda\widetilde{M}(1+(c_{\text{max}}-c_{\text{min}}))^{2}}\right).$

Note that the right hand side of inequalities (26), (27) diminishes
exponentially increase in $M$.

  * –

If full service or $\alpha$ fraction of service is at the edge at time-slot
$t-\lambda\widetilde{M}$, and is not evicted during
$t-\lambda\widetilde{M}+1\leq\tilde{\tau}\leq t-2$ then $A^{c}_{t}\cap
A^{c}_{t-1}\cap B^{c}_{t}\cap B^{c}_{t-1}\cap D\cap E$ implies
$\text{totalCost}(R_{0}^{n},I_{m})<\text{totalCost}(R_{1}^{n},I_{m})$ and
$\text{totalCost}(R_{0}^{n},I_{m})<\text{totalCost}(R_{\alpha}^{n},I_{m})$ ,
for any $n\in[1,t-1]$ and $m\in[n,t-1]$. Thus no service will be at the edge
at time-slot $t$.

  * –

If full service or $\alpha$ fraction of service is at the edge at time-slot
$t-\lambda\widetilde{M}$, and $1-\alpha$ fraction of service is evicted during
$t-\lambda\widetilde{M}+1\leq\tilde{\tau}\leq t-2$ then the event $E$ implies
that
$\text{totalCost}(R_{0}^{t-\lambda\widetilde{M}+1},I_{t-1})<\text{totalCost}(R_{\alpha}^{t-\lambda\widetilde{M}+1},I_{t-1})$,that
is no service will be at the edge at time-slot $t$.

  * –

If no service is at the edge at time-slot $t-\lambda\widetilde{M}$, and full
or $\alpha$ fraction of service is fetched during
$t-\lambda\widetilde{M}+1\leq\tilde{\tau}\leq t-2$ then $D\cap E$ ensures that
$\text{totalCost}(R_{0}^{t-\lambda\widetilde{M}+1},I_{t-1})<\text{totalCost}(R_{\alpha}^{t-\lambda\widetilde{M}+1},I_{t-1})$
and
$\text{totalCost}(R_{0}^{t-\lambda\widetilde{M}+1},I_{t-1})<\text{totalCost}(R_{1}^{t-\lambda\widetilde{M}+1},I_{t-1})$.
Thus no service will be at the edge at time-slot $t$.

Using union bound and Hoeffding inequality,

$\displaystyle\mathbb{P}(A^{c}_{t}\cap A^{c}_{t-1}\cap B^{c}_{t}\cap
B^{c}_{t-1}\cap D\cap E)\geq
1-\mathbb{P}(A_{t})-\mathbb{P}(A_{t-1})-\mathbb{P}(B_{t})-\mathbb{P}(B_{t-1})-\mathbb{P}(B)-\mathbb{P}(D).$

Conditioned on $G=A^{c}_{t}\cap A^{c}_{t-1}\cap B^{c}_{t}\cap B^{c}_{t-1}\cap
D\cap E$, the full service is hosted at the edge during time-slot $t$. The
expected cost incurred by the $\alpha-$RR policy is
$\mathbb{E}[C_{t}^{\alpha-\text{RR}}]=\mathbb{E}[C_{t}^{\alpha-\text{RR}}|G]\mathbb{P}(G)+\mathbb{E}[C_{t}^{\alpha-\text{RR}}|G^{c}]\times\mathbb{P}(G^{c}).$
Note that,
$\mathbb{E}[C_{t}^{\alpha-\text{RR}}|G]=\mathbb{E}[C_{t}^{\alpha-\text{OPT-
ON}}],\ \mathbb{E}[C_{t}^{\alpha-\text{RR}}|G^{c}]\leq\max\\{\alpha M+\alpha
c+g(\alpha)p,M+c\\}.$ Therefore,

$\displaystyle\mathbb{E}[C_{t}^{\alpha-\text{RR}}]-\mathbb{E}[C_{t}^{\alpha-\text{OPT-
ON}}]=$
$\displaystyle\mathbb{E}[C_{t}^{\alpha-\text{RR}}|G](\mathbb{P}(G)-1)+\max\\{\alpha
M+\alpha c+g(\alpha)p,M+c\\}\mathbb{P}(G^{c})$ $\displaystyle\leq$
$\displaystyle\max\\{\alpha M+\alpha c+g(\alpha)p,M+c\\}\mathbb{P}(G^{c}).$
$\displaystyle\frac{\mathbb{E}[C_{t}^{\alpha-\text{RR}}]}{\mathbb{E}[C_{t}^{\alpha-\text{OPT-
ON}}]}\leq$ $\displaystyle 1+\max\\{\alpha M+\alpha
c+g(\alpha)p,M+c\\}\frac{\mathbb{P}(G^{c})}{p}$

∎


# Unsupervised deep learning framework for temperature-compensated damage
assessment using ultrasonic guided waves on edge device

Pankhi Kashyap  Department of Electrical Engineering, Indian Institute of
Technology Bombay, Mumbai 400076, MH India  Kajal Shivgan  Department of
Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400076,
MH India  Sheetal Patil  Department of Electrical Engineering, Indian
Institute of Technology Bombay, Mumbai 400076, MH India  Ramana Raja B
Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai
400076, MH India  Sagar Mahajan  Department of Electrical Engineering, Indian
Institute of Technology Bombay, Mumbai 400076, MH India  Sauvik Banerjee
Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai
400076, MH India  Siddharth Tallur<EMAIL_ADDRESS>Department of
Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400076,
MH India

###### Abstract

Fueled by the rapid development of machine learning (ML) and greater access to
cloud computing and graphics processing units (GPUs), various deep learning
based models have been proposed for improving performance of ultrasonic guided
wave structural health monitoring (GW-SHM) systems, especially to counter
complexity and heterogeneity in data due to varying environmental factors
(e.g., temperature) and types of damages. Such models typically comprise of
millions of trainable parameters, and therefore add to cost of deployment due
to requirements of cloud connectivity and processing, thus limiting the scale
of deployment of GW-SHM. In this work, we propose an alternative solution that
leverages TinyML framework for development of light-weight ML models that
could be directly deployed on embedded edge devices. The utility of our
solution is illustrated by presenting an unsupervised learning framework for
damage detection in honeycomb composite sandwich structure (HCSS) with disbond
and delamination type of damages, validated using data generated by finite
element (FE) simulations and experiments performed at various temperatures in
the range 090. We demonstrate a fully-integrated solution using a Xilinx
Artix®-7 FPGA for data acquisition and control, and edge-inference of damage.

Unsupervised learning, damage detection, light-weight neural network,
temperature variations, TinyML, structural health monitoring

††preprint: APS/123-QED

##  I Introduction

Ultrasonic guided waves (GW) have the ability to propagate long distances
without significant attenuation. GW based structural health monitoring (SHM)
systems are highly sensitive to damages, and require minimal instrumentation
of sensors on the structure to be monitored. Over the years, many studies has
been reported on such systems, and a vast body of literature is available on
GW SHM [1]. However, unlike vibration condition based monitoring of rotating
machinery [2], the industrial acceptance towards replacement of periodic
nondestructive testing (NDT) with such permanently deployed SHM systems has
been quite limited [3]. The primary hurdle in this regard is the impact of
variations in environmental and operating conditions (EOCs) on GW propagation.
The amplitude and group velocity of different GW modes have distinct
temperature-dependent coefficients, and therefore assessing damages in
presence of time-varying EOCs imposes a significant challenge [4].

Recently, data-driven damage assessment methods using machine learning (ML)
techniques have been proposed to overcome this limitation, by identifying
patterns hidden in the features computed from GW recordings [5]. Such reports
include methods based on artificial neural networks (ANNs) and support vector
machines (SVMs) [6], principal component analysis (PCA) [7], singular value
decomposition (SVD) [8, 9], Gaussian mixture models (GMMs) [10, 11] etc. On
the other hand, deep learning techniques for GW based SHM eliminate the
dependencies on domain knowledge for computation of features, by utilizing the
deep learning models themselves for feature computation and classification
[12, 13, 14]. Recently, [15] demonstrated a method which utilizes supervised
convolutional neural networks (CNNs) to learn the features of GW signals
acquired from healthy and damage structures for varying temperature
conditions, combined with GMMs for damage index computation and localization.
Unlike such supervised methods that require labeled data collected for healthy
as well as damaged structures, unsupervised learning approaches offer greater
utility, as most practical applications of SHM systems will result in
generation of significantly higher amount of data for healthy operating
conditions as compared to damaged structure. Recently reported unsupervised
learning demonstrations for GW-SHM include convolutional denoising autoencoder
(AE) for temperature compensation [16] (albeit, without damage assessment) and
deep AEs operating on time-domain GW data obtained on composite panels [17,
18]. Typical deep learning architectures containing convolutional layers
result in millions of trainable parameters and require large amount of
computational resources and take long time for training. Cloud based data
storage and inference may be the most suitable option for meeting these
requirements; however, the added cost of data transmission and storage,
latency, and security and privacy aspects potentially limit their
applications. Hence, there is a need for development of data driven
unsupervised algorithms for damage assessment, that could be implemented in
edge devices such as microcontrollers and field programmable gate arrays
(FPGAs) used for data acquisition.

TinyML has emerged as a promising solution for seamless development of light-
weight ML models on a variety of microcontrollers, resulting in a vast body of
work on memory-efficient and low-power edge-implementation of ML applications
[19, 20]. The ML models are typically trained on a workstation, and deployed
on edge devices using device-specific libraries and open-source frameworks
such as TensorFlow Lite Micro (TFLM) [21, 22]. While this enables deployment
of models on low-cost microcontrollers with limited resources (e.g. single-
board platforms such as Arduino), ML models typically require to be retrained
in the field to accommodate drift in the sensor recordings, and therefore more
powerful edge devices such as Raspberry Pi single board computer or FPGA
platforms may be more suitable for practical applications [23, 24]. FPGAs
offer design flexibility in GW-SHM applications, by allowing seamless
reconfiguration of the ML model architecture and hardware implementation and
acceleration of the multiply-accumulate (MAC) operations involved in data
processing, without hardware redesign. The high speed clock offered (tens of
MHz and higher) by most FPGAs also makes them suitable for interfacing with
high speed digital to analog converters (DACs) and analog to digital
converters (ADCs) for the GW transmitters and receivers,respectively.
Aranguren et al. recently demonstrated an FPGA-based GW-SHM system for
detection of progressive damage in aircraft structures [25]. Malviya et al.
demonstrated an FPGA-based implementation of light-weight convolutional AE
model for anomaly detection in vibration-based monitoring, validated using the
Airbus SAS helicopter accelerometer dataset [24]. Thus, while embedded systems
with edge-learning have been developed for a variety of applications, there
have been no such reports for GW SHM to counter EOC variations. Moreover,
models that leverage convolutional layers to learn features from vast amounts
of training data are impractical due to low memory availability on such
embedded systems.

Figure 1: Overview of the proposed methodology, described in section II.1.

In this paper, we demonstrate a TinyML-enabled unsupervised learning framework
for GW-SHM, implemented on the Xilinx Artix®-7 FPGA, to realize a fully-
integrated SHM system for data acquisition and damage assessment
(identification and localization). From a network of 8 PZT sensors covering an
undamaged and two damaged portions of a panel made of honeycomb composite
sandwich structure (HCSS), GW data are recorded at different temperatures from
090. A light-weight neural network is trained on features obtained from data
collected on an undamaged panel, and is then tested on independently collected
data from undamaged as well as damaged portions of the panel. The features
comprise of various parameters obtained from time-domain GW signal recordings.
The light-weight model is deployed using TensorFlow Lite framework on the
MicroBlaze® reduced instruction set computer (RISC) processor core in the
Xilinx Artix®-7 FPGA for damage assessment at the edge. The efficacy of this
method is demonstrated using GW data from a network of PZT sensors on an HCSS
panel with teflon release film (TRF) and lack of film adhesive (LFA) damages,
data collected by experimental measures at various temperatures (from 090). A
parametric study is also done using data generated by finite element (FE)
simulations to see the model performance for detection of damages with
different sizes. The mean square error (MSE) of the reconstructed signal is
used as signal difference coefficient (SDC) for damage identification. The
system presented in this work is a promising development towards fully-
integrated data-driven GW-SHM solution.

##  II Methodology

###  II.1 Overview of proposed method

The proposed damage assessment method broadly comprises of three steps:(i)
feature extraction, (ii) training of unsupervised learning model, and (iii)
edge implementation of trained model. Figure 1 shows the overview of our
proposed methodology. First, time-series data were collected from a structure
instrumented with PZT sensors, for healthy (undamaged) and damaged conditions.
Data were generated using FE simulations and experimental setup for
temperatures varying from 090. Next, features are computed from the signals,
based on the sensitivity to damage, computational cost and memory requirement
associated with computing the features for edge implementation. Note that the
features can be computed from time and frequency domain representations of the
signals. In this work, we have only focused on the use of time-domain
features, as they are simpler to compute and do not introduce any significant
computation overhead on the embedded system. To study the performance of the
model in presence of large errors, white noise and pink noise were added to
the data to obtain signal to noise ratio (SNR) of
$20\text{\,}\mathrm{d}\mathrm{B}$. Next, an artificial neural network (ANN)
was trained on the manually computed set of features, obtained using data
collected for undamaged (baseline) structure under various temperature
conditions. When the trained model is tested on data obtained from damaged
panels, the paths containing damage located between the transmitter and
receiver positions produce higher reconstruction error (mean squared error,
MSE) for the model prediction, which in turn is used to identify damage. The
trained model is deployed on a Xilinx Artix®-7 FPGA for damage inference using
TensorFlow Lite framework operating in the MicroBlaze® RISC processor in the
FPGA. The FPGA is also used for data acquisition and feature computation, and
thus serves as an end-to-end smart GW-SHM solution.

###  II.2 Experimental data acquisition

In this work, we have evaluated the proposed unsupervised learning method for
damage assessment of a HCSS panel, which is an advanced composite structure
widely used in aerospace, automotive and marine industries. The panel of
lateral dimensions 1x1.2m was manufactured by sandwiching two composite face
sheets with a $12.7\text{\,}\mathrm{m}\mathrm{m}$ thick aluminum honeycomb
core between the face sheets. The face sheets comprised of six
$0.125\text{\,}\mathrm{m}\mathrm{m}$ thick unidirectional layers with a layer-
wise orientation of $[0,+60,-60,-60,+60,0]$ and were separately cured and
bonded to the core. Disbond was artificially created between the core and face
sheet by removing the film adhesive in a region of dimensions 30x30mm, and
delamination was created between the second and third layer of the face sheet
adjacent to the core by inserting a thin layer of teflon release film of
dimensions 30x30mm. The effect of temperature variation on the characteristics
of GW signals for disbond (LFA) and delamination (TRF) damages in this
structure was extensively characterized and reported in our previous work
[26]. The amplitude and group velocity of the fundamental anti-symmetric
(A$0$) mode were found to increase in presence of disbond and decrease in
presence of face sheet delamination. However, the amplitude of A$0$ mode
reduced linearly with increasing temperature for both healthy and damaged
cases. The variation in amplitude of the A$0$ mode due to temperature makes it
difficult to use this property to evaluate the presence of damage, especially
in presence of large noise.

Ultrasonic GW data from the structure were recorded by attaching a network of
eight PZT-5H transducers on one of the face sheets of the structure (Figure
2(a)). Each transducer is of dimension 30x30x0.4. The transducers were placed
in such a manner so that the distance between transmitter and receiver remains
$180\text{\,}\mathrm{m}\mathrm{m}$ for any horizontal or vertical path in a
unit cell. A multicore cable of length $3\text{\,}\mathrm{m}$ with
electromagnetic shielding was used to connect the PZTs with the FPGA board.
Electromagnetic shielding prevents, or at the very least, reduces the coupling
of undesired radiated electromagnetic energy in systems and cables. This is
necessary in our experiment because the FPGA board is kept outside an
environmental chamber (Arcade Scientific Instruments Pvt. Ltd., Figure 2(b))
used to record ultrasonic GW signals on the HCSS panel, and thus requires
analog voltage signals from the PZTs to be connected with a long cable, that
is susceptible to electromagnetic interference. A closer view of the region of
interest on the HCSS panel (Figure 2(c)) shows that the TRF damaged portion
lies at the center of unit cell with PZTs numbered $2$, $7$ and $4$, $5$, and
LFA damaged portion lies at the center of unit cell with PZTs numbered $3$,
$8$ and $5$, $6$. Similarly, the region at the intersection of horizontal and
vertical paths in unit cell with PZTs numbered $1$, $5$ and $2$, $3$ does not
have any damage and is considered as baseline region. We collected time-series
data from the above mentioned vertical and horizontal paths. Ultrasonic GW
data were recorded for these paths by placing the HCSS panel in the
environmental chamber and varying temperature from 090, in intervals of
$5\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

Figure 2: Overview of the experimental setup. a) PZT network on HCSS structure
connected to FPGA board using a shielded multicore cable. Inset: $8$ pairs of
shielded cables in the multicore cable. (b) Environmental chamber used for
temperature control. (c) HCSS panel showing positions of PZT transducers and
damages (TRF and LFA) and undamaged (baseline) locations. (d) FPGA enabled
embedded system for data acquisition and damage assessment.

Figure 2(d) shows the portable embedded system built for GW ultrasonic signal
transduction, data acquisition, and processing. The system is based on a low-
cost FPGA (Xilinx Artix®-7 XC7A15T-1CPG236C, Digilent Cmod A7-15 T module).
The FPGA is interfaced with a $12$ bit, parallel input, multiplying digital to
analog converter (DAC) — Texas Instruments DAC7821, with a transimpedance
amplifier (Texas Instruments TL072) to generate the $5-$cycle Hanning pulse
actuation voltage centered at $75\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ with
peak-to-peak amplitude of $10\text{\,}\mathrm{V}$. The receiver PZT output is
amplified using Texas Instruments INA128 instrumentation amplifier and
digitized using Maxim Integrated MAX-1426 analog to digital converter (ADC)
with $10$ bit resolution and
$10\text{\,}\mathrm{M}\mathrm{s}\mathrm{p}\mathrm{s}$ sampling rate. The
printed circuit board (PCB) also consists of an on-board power supply module
with voltage regulators for generating necessary supply rails for the
amplifier and data converter ICs, so that the entire board can be powered with
a USB type-C charger. Voltage controlled switches implemented with Texas
Instruments CD4051 allow digital selection of the PZTs to be connected as
transmitter and receiver to the signal chain. The system is capable of storing
$4096$ samples obtained at
$10\text{\,}\mathrm{M}\mathrm{s}\mathrm{p}\mathrm{s}$ i.e.,
$409.6\text{\,}\mathrm{\SIUnitSymbolMicro s}$ for each channel in the memory
(Block RAM) in the FPGA. A graphical user interface (GUI) created using Python
is used for system control and configuration, and operating the experiment
(selection of PZT channels, visualizing GW data and saving data to file). To
incorporate uncertainties caused by variation in other environmental and
operating conditions that one may expect in the field, we also added white
noise and pink noise to the data to obtain SNR of
$20\text{\,}\mathrm{d}\mathrm{B}$ [15] (note that the SNR of the data recorded
with the system described above is $>$$50\text{\,}\mathrm{d}\mathrm{B}$).
Fifty noise-augmented copies were generated from each data recording.

Figure 3: (a) Noise-augmented data series recorded from 090, for all
horizontally and vertically oriented paths passing through baseline, TRF and
LFA regions. Lighter shades of color represents lower temperature and darker
shades represent progressively higher temperatures. (b) Variation in amplitude
of A$0$ mode due to temperature, shown as distribution for baseline, TRF and
LFA damages.

Figure 3(a) shows a plot of representative noise-augmented data gathered at
all temperatures for vertical and horizontal paths in all unit cells up to
$200\text{\,}\mathrm{\SIUnitSymbolMicro s}$. Data acquired at higher
temperatures are shown using deeper color shades, whereas data obtained at
lower temperatures are indicated by lighter color shades. The variation in
amplitude of the A$0$ mode due to temperature for vertical and horizontal
paths in these noise-augmented data are more clearly seen in Figure 3(b). In
addition to change in amplitude, the group velocity of the A$0$ mode also
changes due to temperature, making it difficult to rely solely on analytical
methods to assess damage.

###  II.3 Data generation through finite element method (FEM) simulations

The experimental sample used in this study contained only a few damages, one
each of TRF and LFA damage. Therefore, to study the sensitivity of the method
proposed in this work for damage assessment for various sizes of defects,
additional data were generated through FEM simulations. Building a 3D model of
the entire panel (lateral dimensions 1x1.2m) is computationally expensive and
therefore impractical. Since the GW propagation paths used in our study are
straight lines, we employed a 2D model to simplify this exercise. GW
propagation in a 1200x14.2 HCSS with surface mounted PZTs of dimensions
20x0.7mm was analyzed using FE simulations (COMSOL Multiphysics $5.5$). The
simulation model was set up in a manner similar to the method described in our
earlier work [26]. A brief overview of the method is presented here. A
stationary study was first carried out where the temperature of the HCSS panel
was gradually increased from initial room temperature to the desired set-point
using Thermal Expansion subnode in material properties, and temperature-
dependent material properties for the HCSS were derived. Next, time-dependent
study was carried out to simulate GW propagation in HCSS at elevated
temperature, using Structural Mechanics module coupled with Piezoelectric
interaction physics analyzed using time-implicit study. Mapped quadrilateral
mesh elements were used in the FE model, with maximum and minimum element
sizes of $1\text{\,}\mathrm{m}\mathrm{m}$ and
$0.01\text{\,}\mathrm{m}\mathrm{m}$, respectively. At
$75\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ actuation frequency, the group
velocity of the A$0$ mode (the mode employed in this work) is
$1.061\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$, and the wavelength
is calculated to be $14.44\text{\,}\mathrm{m}\mathrm{m}$. Accurate simulation
of Lamb wave propagation in implicit analysis requires at least ten mesh
elements per wavelength, and this criterion is thus satisfied. Low reflecting
boundary condition was specified at the outer edges of HCSS to reduce
reflections from edges, and Rayleigh damping was added for the composite face
sheet and the core to account for damping of the GWs during propagation.

The FE model contained $81\,040\text{\,}$ elements with $498\,604\text{\,}$
degrees of freedom. The simulation time step was chosen to be
$25\text{\,}\mathrm{ns}$ for $5-$cycle Hanning pulse actuation centered at
$75\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, for total simulation time of
$400\text{\,}\mathrm{\SIUnitSymbolMicro s}$. The computation time for each
simulation on $64-$bit Intel® Core™-i$7-10$ $750$ H
$2.60\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ CPU workstation with
$16\text{\,}\mathrm{G}\mathrm{B}$ RAM was $4.15\text{\,}\mathrm{h}$.
Simulations for TRF and LFA damages were conducted with the simulated damage
with conditions similar to the experiment. Data were generated for various
sizes of damage: $5\text{\,}\mathrm{mm}$, $10\text{\,}\mathrm{mm}$,
$15\text{\,}\mathrm{mm}$, and $20\text{\,}\mathrm{mm}$ and at various
temperatures in the range of 3090 in intervals of
$10\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The data were added with white and
pink noise, in a manner similar to the noise-augmentation performed for
experimental data to obtain SNR of $20\text{\,}\mathrm{d}\mathrm{B}$.

###  II.4 Feature extraction

In any damage detection approach, choosing the right features from GW signals
is a crucial step since feature selection greatly affects accuracy of the
damage assessment algorithm. Automated feature extraction using CNNs require
training large networks that may not be feasible for edge implementation.
While frequency-domain methods such as frequency decomposition and time-
frequency methods such as spectrogram (obtained using short time Fourier
transform) and scalogram (obtained using wavelet transform) are best suited to
capture the non-stationary nature of GW signals [27, 28], they require
substantial computation resources and are not feasible for embedded system
implementation for edge-inference [29]. Features computed from time-domain
signals acquired using the embedded system are therefore best suited for
implementing ML models for damage inference at the edge. The utility of easy-
to-compute statistical metrics obtained from time-domain signals for machine
fault diagnosis was shown by Bandyopadhyay et al. [30]. For the damage
assessment model presented in this work, we utilized $16$ features computed
from the time-domain signals, summarized in Table 1. Along with widely used
statistical features such as mean absolute deviation (MAD), variance, standard
deviation, root mean square energy (RMS), root mean square deviation (RMSD),
kurtosis, skew, crest factor, impulse factor, shape factor and difference in
peak to peak amplitude of signals, more advanced features based on signal
energy [31] were included to capture the impact of damage on amplitude and
phase of the GW signals. MAD, variance, and standard deviation measure the
deviation of the guided wave signal relative to the mean. However, these
features only use one signal at a time, which does not allow comparison
between two signals obtained for different states of the structure (e.g.
healthy and damaged). Therefore, the remaining features were formulated by
comparing a reference signal, usually derived from data for healthy structure
taken at room temperature, to the signal obtained using the system in presence
of damage or temperature variations.

Table 1: Features utilized for damage assessment. Glossary: $f(i)$: data samples in the signal, $i=1,2,\dotsc,n$, $n$: length of time series, $X$: ordered list of values in the series, $\mu$:mean value of samples in time-series $f$,$\sigma$: standard deviation of samples in time-series $f$, $f_{b}$: baseline signal, $A$: peak to peak amplitude of signal $f$, $A_{b}$: peak to peak amplitude of baseline signal $f_{b}$ Feature |  Expression
---|---
Mean |  $\frac{1}{n}\sum_{i=1}^{i=n}f(i)$
Median |  $\begin{cases}X[\frac{n+1}{2}],&n\text{ is odd}\\\ \frac{{X[\frac{n}{2}]+X[\frac{n}{2}+1]}}{2},&n\text{ is even}\end{cases}$
Mean absolute deviation |  $\frac{1}{n}\sum_{i=1}^{i=n}|f(i)-\mu|$
Variance |  $\frac{\sum_{i=1}^{i=n}{(f(i)-\mu)^{2}}}{n}$
Standard deviation |  $\sqrt{\frac{\sum_{i=1}^{i=n}{(f(i)-\mu)^{2}}}{n}}$
Root mean square (RMS) |  $\sqrt{(\frac{1}{n})\int^{n}[f(i)]^{2}dt}$
Root mean square deviation (RMSD) |  $\sqrt{\frac{\int^{n}[f(t)-f_{b}(t)]^{2}dt}{\int^{n}[f_{b}(t)]^{2}dt}}$
Kurtosis |  $\frac{1}{n}\sum_{i=1}^{i=n}\left(\frac{x_{i}-\mu}{\sigma}\right)^{4}$
Skew |  $\frac{3(\mu-\mu_{1/2})}{\sigma}$
Crest factor |  $\text{max}(|x_{i}|)\div\sqrt{\frac{\sum_{i=1}^{i=n}x_{i}^{2}}{n}}$
Impulse factor |  $\text{max}(|x_{i}|)\div{\frac{\sum_{i=1}^{i=n}|x_{i}|}{n}}$
Shape factor |  $\sqrt{\frac{\sum_{i=1}^{i=n}x_{i}^{2}}{n}}\div\frac{\sum_{i=1}^{i=n}|x_{i}|}{n}$
Peak to peak |  $A-A_{b}$
Ratio of signal energy |  ${\frac{\int^{T}{|f(t)|^{2}}dt}{\int^{T}{|f_{b}(t)|^{2}}dt}}$
Damage Index |  ${\frac{\int^{T}{|f(t)-f_{b}(t)|^{2}}dt}{\int^{T}{|f_{b}(t)|^{2}}dt}}$
Normalized difference of signal energy |  ${\frac{\int^{n}{|f(t)|^{2}}dt-\int^{n}{|f_{b}(t)|^{2}}dt}{\int^{n}{|f_{b}(t)|^{2}}dt}}$

###  II.5 Unsupervised learning

Figure 4: t-SNE plot of experimental data.

Variations in the experimental data were studied using T-distributed
stochastic neighbor embedding (t-SNE) technique in two dimensions, as shown in
Figure 4. The figure shows clusters belonging to different classes, with each
data recording represented using $16$ features listed in Table 1. The features
of data obtained from damaged and undamaged paths are not linearly separable,
and therefore an ML model is suitable for distinguishing damaged and healthy
operating conditions. Generating extensive labeled data to capture all
possible damage scenarios under all possible varying environmental and
operating conditions is impractical, and thus significantly limits the
applications of supervised learning algorithms for GW-SHM. Therefore, we have
explored the use of unsupervised learning methods to overcome these
limitations. One such approach is anomaly detection, which involves finding
outliers in data that were not represented in the training data, and mark them
as possible damage scenarios. This involves a neural network that is trained
for an approximate reproduction or reconstruction of the input signal by
passing it through a series of neural layers. This method is different from
supervised learning, because there are no separate discrete (classification)
or continuous (regression) labels associated with input data. Training an
unsupervised learning model using only data obtained for baseline operation
(healthy or undamaged condition) at various temperatures would enable the
model to learn variations due to temperature. This would in turn be used to
distinguish the same from variations due to damages, which would be identified
as outliers, since data from damaged conditions would not be part of the
training exercise. Anomalies in the data due to damage will result in higher
error in reconstruction, and therefore be identified as data belonging to
damaged condition. The reconstruction error is quantified by the mean square
error (MSE) between reconstructed and input data:

$MSE={\frac{1}{n}}\times{\sum_{j=1}^{j=n}{\left(a_{j}-\bar{a_{j}}\right)^{2}}}$
(1)

where $n$ denotes number of elements in the input data, ${a_{j}}$ is the
$j^{th}$ element in the input data and ${\bar{a_{j}}}$ is $j^{th}$ element in
the reconstructed output data. The accuracy of the model is calculated by
setting the reconstruction error threshold for determining anomaly as
$\mu+\sigma$, where $\mu$ and $\sigma$ are the mean and standard deviation
respectively of the distribution of reconstruction error obtained from
training with healthy data.

##  III Results and discussion

Instead of setting up a separate model for learning features from vertical and
horizontal paths, a single model was trained on combined data from horizontal
and vertical paths, to reduce the computational burden on the embedded edge
platform. The performance of the model was evaluated for experimental and
simulation datasets. To calculate the $16$ features from time-series data,
segments corresponding to the initial $200\text{\,}\mathrm{\SIUnitSymbolMicro
s}$ portion of the time-series were utilized. Note that the A$0$ mode is
expected to be most sensitive to damage. However, we have also included
portions corresponding to other GW modes, so that a generic model may be
developed, that would not require prior knowledge of material properties of
the structure under test, for determination of group velocity and thereby
arrival time of a desired GW mode.

###  III.1 Analysis of experimental data

Experimental data collected on undamaged panel were used for training the
model, which was then tested on data collected on panel with TRF and LFA
damage. A total of $50\times 19\times 6=5700$ time-series recordings were
available for each of baseline, TRF and LFA conditions: $50$ noise augmented
copies generated at $19$ temperatures varying from 090 for $6$ paths. For
damage assessment, we implemented a neural network with only fully-connected
layers, in order to reduce the number of trainable parameters, and therefore
resource utilization on the FPGA device. All data were normalized such that
all values lie in the range of $[-1,1]$, and the normalized data were used for
noise augmentation and feature extraction. The model architecture is
summarized in Table 2. The model was developed in Python using TensorFlow
library and Keras environment. The network was trained using rectified linear
unit (ReLU) activation function and Adam optimizer. Keeping the number of
neurons fixed, the batch size, learning rate and number of epochs were
optimized using RandomSearch algorithm executed for $10$ iterations. The best
set of parameters was then selected based on maximum accuracy obtained with
$5-$fold cross-validation score. The range of hyper-parameter values passed to
the algorithm and optimal values thus obtained are listed in Table 3. Data
samples corresponding to undamaged (i.e., baseline) condition were randomly
split into $50$% for training , $20$% for validation, and the remaining $30$%
for testing the model. After training, we tested the model performance on
another independently collected dataset consisting of data from undamaged, as
well as damaged portions. Figure 5 shows distribution of reconstruction error
for model predictions for these experimentally generated data for test data of
the baseline set used for training, and the independently recorded dataset for
testing the model, comprising of baseline, TRF and LFA conditions. The paths
that do not contain damage in-line (i.e., baseline and test baseline) are
correctly identified, as the reconstruction error is lesser than the
$\mu+\sigma$ threshold. Similarly, damaged paths TRF and LFA are also
identified correctly, as the reconstruction error is greater than the
$\mu+\sigma$ threshold. Accuracy and F1 score for these results are shown in
Table 4. These results suggest that the chosen set of input features are
suitable for capturing the variation in data due to temperature, while
simultaneously enabling the model to distinguish changes due to presence of
damage.

Table 2: Summary of model architecture Layer (# filters) | # Parameters
---|---
Dense ($16$) | $272$
Dense ($32$) | $544$
Dense ($64$) | $2112$
Dense ($64$) | $0$
Dense ($64$) | $4160$
Dense ($32$) | $2080$
Dense ($16$) | $528$
Total trainable parameters | $9696$
Table 3: Hyperparameter tuning using RandomSearch Hyperparameter | Range | Optimal value
---|---|---
Learning rate | [$0.001$, $0.01$, $0.1$] | $0.01$
Batch size | [$16$, $28$, $32$, $64$] | $32$
No. of epochs | [$50$, $100$, $150$, $200$] | $150$
Figure 5: Reconstruction error for experimentally generated data Table 4: Accuracy and F1 score for experimental data Case | Accuracy (%) | F1 score (%)
---|---|---
Test Baseline | $99$ | $99$
TRF | $95$ | $86$
LFA | $95$ | $83$

###  III.2 Analysis of simulation data

To study the performance of the damage identification for temperature-affected
and noise-augmented data corresponding to various damage sizes, FE simulation
data were generated using COMSOL Multiphysics (with noise augmentation
performed using MATLAB). A total of $50\times 7\times 3=1050$ datasets were
available for training and validation of the model: $50$ noise augmented
copies were generated for each of the $7$ temperatures varying from 3090 for
$3$ paths. The same model architecture as used for evaluating experimental
data was used for analyzing these simulation generated data, with the only
exception being the batch size, which was set as $28$, since we observed the
model to over-fit to the data for batch size of $32$.

Figure 6: Reconstruction error for TRF data generated by FE simulations Figure
7: Reconstruction error for LFA data generated by FE simulations

Figures 6 and 7 show distributions of reconstruction error for model
predictions for defects of various sizes corresponding to TRF and LFA,
respectively. The corresponding accuracy and F1 score metrics are reported in
Table 5. Since defects of larger sizes produce larger changes in signal
amplitude, the model is able to easily discern TRF and LFA damages of sizes
$10\text{\,}\mathrm{m}\mathrm{m}$ and larger. The increasing trend of
reconstruction error with defect size indicates that the absolute value of
reconstruction error could possibly be used as an indicator of defect size
(i.e., damage intensity). Defects of size $5\text{\,}\mathrm{m}\mathrm{m}$ are
not as accurately classified as damaged. Note that the sensitivity of the
model to smaller damages could potentially be improved by exploring
transduction at higher frequencies. However, the PZT transducers used in this
study were found to have degradation in transduction efficiency at frequencies
higher than $150\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, and therefore only
$75\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ actuation was explored in this
work.

Table 5: Accuracy and F1 score for simulation generated data Damage | TRF | LFA
---|---|---
size | Accuracy | F1 score | Accuracy | F1 score
$5\text{\,}\mathrm{m}\mathrm{m}$ | $50$ | $35$ | $50$ | $35$
$10\text{\,}\mathrm{m}\mathrm{m}$ | $90$ | $73$ | $96$ | $88$
$15\text{\,}\mathrm{m}\mathrm{m}$ | $90$ | $73$ | $96$ | $88$
$20\text{\,}\mathrm{m}\mathrm{m}$ | $96$ | $88$ | $96$ | $88$
Table 6: Resource utilization for model implementation on FPGA device for experimental data Resource | Utilized | Available | Utilization (%)
---|---|---|---
LUT | $5463$ | $20\,800\text{\,}$ | $26.26$
FF | $4352$ | $41\,600\text{\,}$ | $10.46$
BRAM | $49$ | $50$ | $98$
DSP | $5$ | $90$ | $5.56$
Figure 8: Reconstruction error for experimental data for edge implementation of model Table 7: Comparison of this work with other unsupervised learning algorithms for GW-SHM reported in literature Reference |  Application |  Input |  Architecture |  Model size and edge-compatibility
---|---|---|---|---
Lee et al. [17] |  Fatigue damage detection and classification in composite structures |  Time-domain signals |  1D deep AE |  Model size not reported; model not edge-compatible
Sawant et al. [18] |  Damage identification and localization in OGW dataset |  Time-domain signals |  1D CAE with transfer learning |  $1.1$M parameters; model not edge-compatible
Eybpoosh et al. [32] |  Online damage detection of pipelines under varying environmental and operational conditions |  Energy-based sparse representation of GW signals |  K-means, SVM and Kullback–Leibler divergence (KL) |  Model sizes not reported; models not edge-compatible
Zhang et al. [33] |  Automated damage severity and orientation detection |  Frequency and time domain features in aluminum beam |  SVM |  Model size not mentioned; accuracy drop on prediction of noise augmented data; model not deployed on FPGA
Rai et al. [34] |  GW-based semi-supervised damage diagnosis in composite laminates using ResNet autoencoder with transfer learning (ResNet CAE is not edge-compatible) |  Time-domain signals |  1D CAE |  $\approx 41$k trainable parameters in 1D CAE with residual layer and $\approx 260$k trainable parameters with transfer learning
Rautela et al. [35] |  Anomaly (delamination) detection in composite panels |  Time-frequency representation as 2D images (continuous wavelet transform) |  2D CAE |  $\approx 1.2$M parameters in encoder and $\approx 1$M trainable parameters in decoder
Yang et al. [36] |  Assessment of damage under regular and irregular variations of EOCs in aluminum plate |  Time-domain signals |  AE |  Method incompatible with edge implementation
Sikdar et al. [37] |  Assessment of breathing-like debonds in lightweight stiffened composite panels |  Time-frequency representation (continuous wavelet transform) |  Deep CNN |  Model not edge-compatible
This work |  TinyML enabled temperature-compensated damage detection in HCSS panel implemented on edge device |  Features extracted from time-domain and frequency domain data |  ANN |  $9696$ parameters; model deployed on Cmod A7-15T FPGA module for data acquisition and inference

###  III.3 Edge implementation of trained model

The trained model was deployed using TensorFlow Lite framework on the
MicroBlaze® RISC processor core in the Xilinx Artix®-7 device (Digilent Cmod
A7-35T module). The challenge in implementing machine learning model in such a
device is the absence of a processor core which is required to perform complex
control tasks and run TFLite. The MicroBlaze® CPU is a family of highly
configurable, drop-in, modifiable preset $32-$bit Harvard RISC microprocessor
architecture optimised for implementation in Xilinx FPGAs. MicroBlaze® is
fully supported by Eclipse based integrated development environments (IDEs)
Xilinx SDK/Vitis, which allow direct creation of an application project for
the Cmod A7-15T module and auto-generation of all board support package (BSP)
files. The corresponding TFLite kernels were also modified to invoke the
custom IPs for deploying the trained model on MicroBlaze®. The hardware for
FPGA is configured and bitstrean is generated using Vivado $2019.1$. Once the
bitstream is generated, BSP is created to provide all low level drivers for
the hardware using SDK $2019.1$. The TensorFlow Lite model header file
generated using Python application programming interface (API) TensorFlow Lite
converter is imported in the SDK project, and stored as C byte array in read-
only program memory on the FPGA device. For more details on the
implementation, please see source code available at [38], which describes the
use of this framework for handwritten digit classification in MNIST database
and [39], which describes the use of this framework for implementing a
reinforcement learning model. The model architecture described in Table 2 was
deployed on the device and the performance was evaluated on the same test
cases used for evaluating model performance on experimental data reported in
section III.1. We observed similar accuracy as that reported in Table 4, with
average inference time approximately $900\text{\,}\mathrm{\SIUnitSymbolMicro
s}$ for $100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ clock frequency. The
resource utilization in the FPGA is summarized in Table 6, and a plot of
reconstruction error for edge implementation is shown in Figure 8.

###  III.4 Discussion

Limited memory and computational resources available on embedded edge devices
make it challenging to design and deploy machine learning tools for GW-SHM, as
arbitrarily large models with significant amount of signal pre-processing
typically presented in literature are rendered impractical. This is especially
true for permanently deployed SHM systems, as compared to solutions for
traditional periodic inspection and nondestructive testing and evaluation
(NDT&E). The application of TinyML for GW-SHM presented in this work
illustrates the potential for further exploration towards implementation of
autonomous damage inspection strategy on resource constrained hardware. No
knowledge of material properties of the structure or impact of damage on
guided wave propagation is necessary to apply this method, and therefore it is
also potentially scalable to various types of damages in other structures.
However, this would require careful examination of damage-sensitive features
that must be calculated from the time-series data and provided as input to the
model. Generating the $16$ hand-crafted features from experimentally acquired
data reported in this work, and computing reconstruction error takes only a
few milliseconds. While the performance can be optimized further using a more
capable albeit expensive FPGA device, the Cmod A7-15T module offers an
excellent trade-off between performance and price ($<$USD $\$100$). A
comparison of our method to other unsupervised learning based damage
assessment methods in GW-SHM reported in literature is presented in Table 7.
Compared to many of these implementations, our model is extremely light-weight
($<$$10\,000\text{\,}$ trainable parameters) and can therefore be effectively
stored and executed on the edge device used for configuration and transduction
of GW signals. The model inference time on the edge device is
$<$$1\text{\,}\mathrm{m}\mathrm{s}$, thereby enabling efficient duty-cycling
of such an SHM solution, helping conserve power and battery life in the field.
Choosing suitable features allows such a light-weight model to distinguish
changes in data due to damage from those due to temperature variation. Thus,
our work presents an important breakthrough of relevance to the GW-SHM
community for realizing practically relevant smart-sensor nodes that may be
permanently deployed on infrastructure.

##  IV Conclusion

In summary, we present a lightweight unsupervised learning algorithm framework
that is deployed on an FPGA device for end-to-end implementation of a GW-SHM
embedded system for data acquisition, storage, feature extraction and damage
assessment. The variance in peak signal amplitude, variation in signal energy,
and variation in signal from baseline are all captured by the $16$ hand-
crafted features used to represent the time-domain data. These features are
adaptable to a wide range of GW-SHM applications because they capture the
typical impact of damages on guided wave propagation in structures. The method
was tested using data generated from FE simulations on HCSS panels equipped
with PZT-5H transducers and an experimental setup for inducing temperature
change in the range of 090. The unsupervised machine learning model used for
damage assessment consists of only $9696$ parameters, and shows high accuracy
when trained and tested with noise augmented data with
$20\text{\,}\mathrm{d}\mathrm{B}$ SNR. The trained model was also successfully
deployed on the MicroBlaze® RISC processor core in the Xilinx Artix®-7 FPGA
device. Despite the limited number of features and small model size employed
for edge implementation, the models evaluated in this work showed reasonably
high accuracy for disbond and delamination defects. We aim to investigate
feasibility of the method proposed in this work for different types of
structures, damages and severity. In future work, we shall also explore
strategies to improve the sensitivity of this framework to defects of smaller
sizes and incorporate online training on the edge device to realize completely
autonomous self-adaptable GW-SHM systems.

## Acknowledgments

This work was supported through grants from Science and Engineering Research
Board (SERB), Government of India [grant no. CRG/2021/001959] and Indian Space
Research Organization (ISRO) [grant no. RD/0118-ISROC00-006]. The authors
acknowledge support from staff and access to facilities at the Wadhwani
Electronics Lab (WEL), Department of Electrical Engineering, IIT Bombay for
carrying out experiments reported in this work. The authors thank Mr. Aryan
Lall for initial assistance with deploying TFLite model on Artix®-7 FPGA.

## Declaration of competing interests

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Data availability

Data available upon reasonable request.

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