## 4 Improved Control Method
### Vibration Suppression by the Feedback Controller  
To suppress the vibration, the velocity feedback PI controller is redesigned as follows,  
\[u_{c}^{*}(t)=K_{p}e^{*}(t)+K_{i}\int_{0}^{t}e^{*}(\tau)d\tau \tag{28}\]  
where \(e^{*}(t)=r_{v}(t)-\dot{\theta}_{e}(t)-\dot{\theta}_{b}^{*}(t)\) is the new angular velocity error. When compared with the PI controller described by (21), the modified controller uses the modified base angular velocity \(\dot{\theta}_{b}^{*}(t)\). The system from \(\dot{\theta}_{b}(t)\) to \(\dot{\theta}_{b}^{*}(t)\) is obtained via the following state space description,  
\[ \begin{cases}\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\dot{ \theta}_{b}(t)\\ \dot{\theta}_{b}^{*}(t)=\mathbf{C}\mathbf{x}(t)+\mathbf{D}\dot{\theta}_{b}(t) \end{cases}  \tag{29}\]  
where \(\mathbf{x}(t)\) is a state variable and the constant matrices are expressed in the controllable standard form. \[\mathbf{A}= \begin{bmatrix}0&1\\ a_{21}&a_{22}\end{bmatrix}  \tag{30}\]  
\[\mathbf{B}=[ \begin{array}{cc}0&1\end{array} ]^{T} \tag{31}\]  
\[\mathbf{C}=[ \begin{array}{cc}0&c_{12}\end{array} ] \tag{32}\]  
\[\mathbf{D}=1 \tag{33}\]  
where \(a_{21},a_{22},c_{12}\) are all constants in the state space equation, and they satisfy the following conditions, \(a_{21}~{}<~{}0,\,a_{22}~{}<~{}0,\,c_{12}=a+a_{22}<0\), and \(a>0\). Here, Equation (29) describes the conversion of the control signal from the base angular velocity \(\dot{\theta}_{b}(t)\) to the modified angular velocity \(\dot{\theta}_{b}^{*}(t)\). As this is a single-input single-output (SISO) system, the input signal is a sine wave with a frequency of \(\omega\) Hz, and the modified angular velocity \(\dot{\theta}_{b}^{*}(t)\) is represented as follows,  
\[\dot{\theta}_{b}^{*}(t)= k_{1}(\omega)\sin\omega t+k_{2}(\omega)\cos\omega t-k_{2}( \omega)e^{\frac{1}{2}a_{22}t}(\cos kt+k_{3}(\omega)\sin kt), \tag{34}\]  
where \(k\) is a constant \(\sqrt{\frac{1}{4}a_{22}^{2}+a_{21}}\). Further, the variable parameters are as follows,  
\[k_{1}(\omega)=\frac{\omega^{4}+2a_{21}\omega^{2}+a_{22}^{2}\omega^{2}-a_{22}c _{12}\omega^{2}+a_{21}^{2}}{\omega^{4}+2a_{21}\omega^{2}+a_{22}^{2}\omega^{2}+ a_{21}^{2}} \tag{35}\]  
\[k_{2}(\omega)=\frac{-c_{12}\omega^{3}-a_{21}c_{12}\omega}{\omega^{4}+2a_{21} \omega^{2}+a_{22}^{2}\omega^{2}+a_{21}^{2}} \tag{36}\]  
\[k_{3}(\omega)=\frac{1}{2}a_{22}-\frac{a_{21}a_{22}c_{12}\omega}{c_{12}\omega^{3 }+a_{21}c_{12}\omega}. \tag{37}\]  
By indicating that the parameter \(\sqrt{-a_{21}}\) is close to \(\omega\), the state space function can rapidly attenuate the input signal amplitude; however, it has minimal effect on the signal far from \(\omega\). The vibration frequency is limited by the above method to suppress the vibration. Furthermore, we introduce Lemmas 1 and 2 to prove the stability of the control signal conversion system (29) [39]. **Lemma 1**. : _Let us consider the following linear and time-invariant system,_  
\[ \begin{cases}\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{b} (t),&\mathbf{x}(0)=x_{0}\\ \mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)+\mathbf{D}\mathbf{b}(t)\end{cases}  \tag{38}\]_where the state \(\mathbf{x}(t)\in\mathbb{R}^{n}\), the input \(\mathbf{b}(t)\in\mathbb{R}^{p}\), and the output \(\mathbf{y}(t)\in\mathbb{R}^{q}\). If the system is internally stable or asymptotically stable in the sense of Lyapunov, it must be a bounded input bounded output (BIBO) system that is stable or externally stable._  
**Lemma 2**. : _For the \(n\)-dimensional linear and time-invariant system \(\dot{\mathbf{x}}(t)=\mathbf{Ax}(t)\), the necessary and sufficient conditions for the origin equilibrium state to be asymptotically stable is provided by the Lyapunov equation, which is given for any positive symmetric \(\mathbf{Q}\)._  
\[\mathbf{A}^{T}\mathbf{P}+\mathbf{P}\mathbf{A}=-\mathbf{Q} \tag{39}\]  
_has a unique positive symmetric matrix \(\mathbf{P}\)._  
**Theorem 1**.