## 4 Design of Controller
Combine Equations (17) and (19) to get:  
\[M_{\text{h}}\hat{\mathbf{r}}=-(\mathbf{K}_{\text{r}}+\mathbf{H}_{\text{h}}+\mathbf{D}_{\text{ Lh}})\mathbf{r}+\widetilde{\mathbf{\chi}}(\mathbf{x})-\tau_{\text{H}} \tag{20}\]  
where \(\widetilde{\mathbf{\chi}}=\mathbf{\chi}-\hat{\mathbf{\chi}}\). The primary reinforcement signal can be enhanced as follows:  
\[\mathbf{r}_{\text{s}}(t)=\mathbf{r}(t)+\mathbf{\Gamma}\mathbf{W}_{\text{C}}^{\text{*T}}(t)\bm {\psi}^{*}(t) \tag{21}\]  
where \(\mathbf{W}_{\text{C}}^{\text{*}}\) is the ideal weight matrix, \(\mathbf{\psi}^{*}\) is the ideal regression matrix, \(\mathbf{\Gamma}=\text{diag}(r_{\text{s1}},r_{\text{s2}},\cdot\cdot,r_{\text{s,n+1 }})\) and \(r_{\text{s}i}\) is the element in \(\mathbf{r}_{\text{s}}\). It is assumed that the enhancement signal can be approximated by ACN as follows:  
\[\hat{\mathbf{r}}_{\text{s}}(t)=\mathbf{r}(t)+\mathbf{\Gamma}\hat{\mathbf{W}}_{\text{C}}^{\text {T}}(t)\hat{\mathbf{\psi}}(t) \tag{22}\]  
\[\mathbf{e}=\mathbf{q}_{\rm d}-\mathbf{q}_{\rm h} \tag{14}\]0 The control block diagram based on fuzzy logic Reinforcement Learning. (**a**) The overall control block diagram; (**b**) The ACN and ASN unit. where \(\hat{W}_{\text{C}}=[\hat{w}_{\text{C1}},\hat{w}_{\text{C2}},\cdots,\hat{w}_{\text{ Cm}}]\) is the estimated value of weight matrix, and \(\hat{\mathbf{\psi}}\) is the estimated value of regression matrix. Expand \(\tilde{\mathbf{\psi}}=\mathbf{\psi}^{*}-\hat{\mathbf{\psi}}\) by [PERSON] as follows:  
\[\tilde{\mathbf{\psi}}=\left.\left[ \begin{array}{c}\left(\frac{\partial\hat{w}_{ 1}}{\partial\hat{w}}\right)^{\text{T}}\\ \vdots\\ \left(\frac{\partial\hat{w}_{1}}{\partial\hat{w}}\right)^{\text{T}}\end{array}  \right]\right|_{\mathbf{\omega}=\hat{\mathbf{\omega}}}\left.\left[ \begin{array}{c} \left(\frac{\partial\hat{w}_{1}}{\partial\hat{w}}\right)^{\text{T}}\\ \vdots\\ \left(\frac{\partial\hat{w}_{2}}{\partial\hat{w}}\right)^{\text{T}}\end{array}  \right]\right|_{t=\hat{1}}\bar{I}+\tilde{\mathbf{\xi}} \tag{23}\]  
where \(\tilde{\mathbf{\omega}}=\mathbf{\omega}^{*}-\hat{\mathbf{\omega}}\), \(\bar{I}=I^{*}-\hat{\mathbf{l}}\). The Equation (23) can be rewritten as:  
\[\tilde{\mathbf{\psi}}=\mathbf{\alpha}^{\text{T}}\tilde{\mathbf{\omega}}+\mathbf{\beta}^{ \text{T}}\bar{I}+\mathbf{\xi} \tag{24}\]  
**Assumption 2**. : _The ideal weight \(\mathbf{w}_{\text{Ck}}^{*}\) and \(\mathbf{\alpha}_{j}^{*}\), centre value \(l_{j}^{*}\), and width \(\mathbf{\alpha}_{j}^{*}\) of fuzzy wavelet neural network are bounded such that \(0<\left\|\mathbf{w}_{\text{Ck}}^{*}\right\|\leq b_{\text{nsck}}\), \(0<\left\|\mathbf{w}_{\text{Sk}}^{*}\right\|\leq b_{\text{nsck}}\), \(0<\left\|\mathbf{\alpha}_{j}^{*}\right\|\leq b_{\text{nsck}}\), \(0<\left\|l_{j}^{*}\right\|\leq b_{\text{lj}}\), \(k=1,2,\cdots,m\), \(j=1,2,\cdots,N\)._  
**Assumption 3**. : _The Taylor remainder \(\mathbf{\xi}\), \(\partial\psi_{j}/\partial\mathbf{\omega}\) and \(\partial\psi_{j}/\partial\mathbf{l}\) are bounded such that \(\left\|\mathbf{\xi}\right\|\leq b_{\xi}\), \(\partial\psi_{j}/\partial\mathbf{\omega}\leq b_{\eta j}\), \(\partial\psi_{j}/\partial\mathbf{\omega}\leq b_{\text{sj}}\)._  
Combined with Equation (24), the output of ACN can be expressed as follows:  
\[\hat{W}_{\text{C}}^{\text{T}}\hat{\mathbf{\psi}}=-\tilde{W}_{\text{C}}^{\text{T}} (\hat{\mathbf{\psi}}-\mathbf{\alpha}^{\text{T}}\hat{\mathbf{\omega}}-\mathbf{\beta}^{\text{T }}\bar{I})-\hat{W}_{\text{C}}^{\text{T}}(\mathbf{\alpha}^{\text{T}}\tilde{\mathbf{ \omega}}+\mathbf{\beta}^{\text{T}}\bar{I})-\mathbf{v}_{1} \tag{25}\]  
where \(\mathbf{v}_{1}=W_{\text{C}}^{\text{T}}(\mathbf{\psi}^{*}-\mathbf{\alpha}^{\text{T}}\mathbf{ \omega}^{*}-\mathbf{\beta}^{\text{T}}I^{*}-\mathbf{\xi})-\tilde{W}_{\text{C}}^{\text{T }}(\mathbf{\alpha}^{\text{T}}\mathbf{\omega}^{*}+\mathbf{\beta}^{\text{T}}I^{*})\), \(\tilde{W}_{\text{C}}=W_{\text{C}}^{*}-\hat{W}_{\text{C}}\).