## 4 Main Contribution of the Control Design
### The Control Challenge  
The problem considered in this paper consists of solving the design of a controller that ensures convergence between the variables considered in (1) and (3) and the states of the reference model \(\Phi^{*}\) inside of an invariant zone with a size proportional to the power of the non-matched perturbations of the arm. This statement can be reformulated to design an output feedback controller such that \(\|\Delta_{\Phi\ t}\parallel\leq\beta\) as long as \(t\) tends to infinity. By design, the reference trajectory \(\Phi^{*}\) can be obtained by implementing the inverse kinematics (a twice differentiable time-dependent function) of the robotic arm. The same option can be used to drive the position dynamics of the piezoelectric actuator:  
\[\frac{d^{2}}{dt^{2}}\Phi_{t}^{*}=h\left(\Phi_{t}^{*},\frac{d}{dt}\Phi_{t}^{*}\right) \tag{6}\]where the the reference trajectory is represented by \(\Phi^{*}\). In (6), the initial conditions \(\Phi^{*}(0)\) and \(\dfrac{d}{dt}\[PERSON]^{*}(0)\) are given vectors with bounded components. The function \(h\) is a local Lipschitz function, which ensures the existence and uniqueness of the reference trajectories.  
The dynamics of the reference trajectories presented in (6) have a stable equilibrium point and, by the converse Lyapunov theorem, it can be ensured that the system in the new coordinates \([z^{*}]^{\top}=\left[(\[PERSON]^{*})^{\top}\quad\left(\dfrac{d}{dt}\[PERSON]^{*}\right)^ {\top}\right]\) satisfies  
\[\left\|z^{*}(t)\right\|^{2}\leq Z_{+}^{*},\quad Z_{+}^{*}\in\mathbb{R}^{+}, \quad\forall t\geq 0 \tag{7}\]  
Due to the assumption that the inequality presented in (7) is considered valid and taking into consideration that both vector fields \(f\) and \(h\) are continuous, the following inequality can be straightforwardly verified. There is a positive constant \(g^{+}\) such that the following inequality is valid:  
\[\left\|f\left(\Phi_{t}^{*},\dfrac{d}{dt}\Phi_{t}^{*}\right)-h\left(\Phi_{t}^{* },\dfrac{d}{dt}\Phi_{t}^{*}\right)\right\|\leq g^{+} \tag{8}\]  
Considering the inclusion of a pair of auxiliary state variables, \(\Delta_{a}=\Delta_{\Phi}\) and \(\Delta_{b}=\dfrac{d}{dt}\Delta_{\Phi}\), for the tracking error, its dynamics then satisfy  
\[\dfrac{d}{dt}\Delta_{a,t}=\Delta_{b,t} \tag{9}\] \[\dfrac{d}{dt}\Delta_{b,t}=h\left(\Phi_{t}^{*},\dfrac{d}{dt}\Phi_{ t}^{*}\right)-F\left(\Phi_{t},\dfrac{d}{dt}\Phi_{t}\right)-G(\Phi_{t})\tau_{t}- \xi\left(\Phi_{t},\dfrac{d}{dt}\Phi_{t},t\right)\]  
where the function \(F\left(\Phi_{t},\dfrac{d}{dt}\Phi_{t}\right)\) can be derived straightforwardly from (1) and (3). The extended state-dependent function related to the extended controller is provided in \(G(\Phi_{t})\), which is left invertible according to the nature of the mechanical system and the piezoelectric device. The controller proposal presented in this study satisfies the following structure, which coincides with a state feedback structure having adaptive gains depending on the tracking error and its derivative:  
\[\tau=\Gamma(\Phi_{t})h\left(\Phi_{t}^{*},\dfrac{d}{dt}\Phi_{t}^{*}\right)- \Gamma(\Phi_{t})\left[F\left(\Phi_{t},\dfrac{d}{dt}\Phi_{t}\right)+K_{t} \Delta_{t}\right] \tag{10}\]  
\[\Gamma(\[PERSON]_{t})=G^{\top}(\[PERSON]_{t})\left(G(\Phi_{t})G^{\top}(\Phi_{t})\right) ^{-1}\]  
where \(K_{t}=\left[k_{p}\ k_{d}\right]\), with \(k_{p}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{n\times n}\) and \(k_{d}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{n\times n}\) being the time varying matrix gains for the sections of the controller proportional to the tracking error and derivative, respectively.