Algorithm 1: PID Wavenet Controller

1:Read the wavenet parameters: number of neurons $L_{\sigma }$, learning rates (${\mu }_{W_{\sigma }},{\mu }_{a_{\sigma }},{\mu }_{b_{\sigma }}$), synaptic weight values $W_{\sigma }$ and the wavelet ${\psi }_{\sigma }$. 2:Read the IIR filter parameters: number of feed-backs and feed-forward coefficients, $M_{\sigma }$ and $N_{\sigma }$, respectively; learning rates (${\mu }_{C_{\sigma }},{\mu }_{D_{\sigma }}$), the IIR coefficient values ($C_{\sigma }{D}_{\sigma }$) and the persistent signal $v_{\sigma }\left(k\right)$. 3:Read the PID controller parameters: gain values ($K_{p_{\sigma }},K_{i_{\sigma }},K_{d_{\sigma }}$) and its update rates (${\mu }_{p_{\sigma }},{\mu }_{i_{\sigma }},{\mu }_{d_{\sigma }}$). 4:Read the operation parameters: number of epochs $epks$. 5:Initialize the internal parameters and $k=0$. 6:while it is working do 7:    for $epk=0;epk\le epks;i++$ do 8:        Get the control signal $u_{\sigma }\left(k\right)$ and the output from plant $y_{\sigma }\left(k\right)$, Equations (39) and (9). 9:        Compute the vector ${\tau }_{L_{\sigma }\left(k\right)}$ using (14). 10:        Evaluate the mother’s wavelet ${\psi }_L\left({\tau }_{L_{\sigma }}\left(k\right)\right)$ with (13). 11:        Compute the wavelet output $z_{\sigma }\left(k\right)$ with (23). 12:        Compute estimated ${\widehat{y}}_{\sigma }\left(k\right)$ using (24). 13:        Compute the estimated error using (28). 14:        Compute the energy function $E_{\sigma }\left(k\right)$ with (27). 15:        Compute the parameter gradients: $\Delta W_{\sigma }\left(k\right)$, $\Delta a_{\sigma }\left(k\right)$, $\Delta b_{\sigma }\left(k\right)$, $\Delta C_{\sigma }\left(k\right)$, $\Delta D_{\sigma }\left(k\right)$, Equations (29)–(33). 16:        Update the parameter values using its learning ratios: $\Delta W_{\sigma }(k+1)$, $\Delta a_{\sigma }(k+1)$, $\Delta b_{\sigma }(k+1)$, $\Delta C_{\sigma }(k+1)$, $\Delta D_{\sigma }(k+1)$; Equations (34)–(38). 17:    end for 18:    Get the parameter ${\widehat{\Gamma }}_{\sigma }\left(k\right)$ and the tracking error ${\epsilon }_{\sigma }\left(k\right)$, Equations (25) and (40). 19:    Tune controller gains: $K_{p_{\sigma }}\left(k\right),K_{i_{\sigma }}\left(k\right),K_{d_{\sigma }}\left(k\right)$: Equations (41)–(43). 20:    Calculate the new control signal $u(k+1)$ using (39). 21:    Reassign the new values. 22:    Increase the value of while loop operator, $k=k+1$. 23: end while