## 3 Cell Manipulation Control
### Cell Manipulation Control System  
NSOFCMNN has strong learning and generalization ability for nonlinear mapping. It is used as the main controller of a control system, a robust compensator is added for eliminating approximation errors, and a higher order sliding surface is used to further weaken the influence of the external disturbances and the chattering on the system. Thus, the basic structure of the optical trap position manipulation control system is built. Firstly, \(\varepsilon(t)\) is defined as an error:  
\[\varepsilon(t)=\mathbf{q}^{d}(t)-\mathbf{q}(t)=\left[\varepsilon_{x}(t),\varepsilon_{ y}(t)\right]^{T} \tag{35}\]  
where \(\mathbf{q}^{d}\) is the expected position of the cell and \(\mathbf{q}\) is the actual position of the cell. \(\varepsilon_{x}(t)\) is the error on the \(X\)-axis, \(\varepsilon_{y}(t)\) is the error on the \(Y\)-axis, and the sliding mode plane is introduced to represent another error function. \[\mathbf{s}(\varepsilon,t)=\varepsilon(t)^{\prime}+\mathbf{K}\varepsilon(t) \tag{36}\]  
where \(\mathbf{K}\) is a matrix of constants and \(\mathbf{s}(\varepsilon,t)\) is a sliding mode vector. In the practical application of cell manipulation control, some uncontrollable unknown factors will inevitably appear in the process of control, which is unavoidable in the control system, that is, the uncertainty in the cell dynamics. If the uncertainty term \(\mathbf{L}\) is known, the ideal neural network controller is defined as:  
\[\mathbf{u}_{\text{NSOFCMNN}}^{*}(t)=\frac{\delta}{k}*\left(\mathbf{q}(t-1)-\dot{\mathbf{q} }^{d}(t)\right)+\mathbf{q}(t-1)-\mathbf{d} \tag{37}\]  
\[\mathbf{d}=\mathbf{d}_{\text{NSOFCMNN}}\left(X,\ \alpha_{ijk},\ \beta_{ijk},\ \omega_{jkc},\ \psi_{jk}\right)+\mathbf{\sigma} \tag{38}\]where \(\sigma=[\sigma_{1},\sigma_{2}]^{T}\) is the error between the ideal neural network controller and the actual manipulation controller. This approximation error is assumed to be bounded and satisfy \(0\leq\|\sigma_{1}\|\leq N_{1}\), \(0\leq\|\sigma_{2}\|\leq N_{2}\), \(\mathbf{N}=[N_{1},N_{2}]^{T}\), \(\|\cdot\|\) represents the norm. An approximate error bound \(\mathbf{N}\) is assumed; however, it is difficult to measure in practical applications. Therefore, a bound estimate is set to estimate this bound error. The estimated error for the bounds is defined as follows:  
\[\widetilde{\mathbf{N}}=\mathbf{N}-\hat{\mathbf{N}} \tag{39}\]  
where \(\hat{\mathbf{N}}\) is an estimate of \(\mathbf{N}\), and the robust controller aims to compensate for the effects of approximation errors, expressed as:  
\[\mathbf{u_{r}}=\hat{\mathbf{N}}\text{tanh}(\mathbf{s}) \tag{40}\]  
Combining the NSOFCMNN controller with a robust compensator, an optical trap position manipulation control system is finally built. \[\mathbf{u}=u_{\text{NSOFCMNN}}+\mathbf{u_{r}} \tag{41}\]  
The structure of the cell manipulation control system is shown in Figure 4:  
After the error \(\varepsilon(t)\) between the expected position \(\mathbf{q}^{d}\) and the actual position \(\mathbf{q}\) of the cell passes through the sliding mode surface, \(\mathbf{s}(t)\) is used as the input data of NSOFCMNN, and its output plus the output of the robust compensator constitutes the total output of the control system. The holographic optical tweezers system finally realizes the manipulation of biological cells by generating several optical traps at specific expected positions at each time point according to the number of cells to be controlled and the total output of the control system. In the practical application of holographic optical tweezers, the maximum moving distance of the optical trap needs to be limited. When \(\mathbf{u}\leq\left(\mathbf{q}\pm[r_{0},r_{0}]^{T}\right)\) and \(\|\mathbf{l}-\mathbf{q}\|\leq r_{0}\) are satisfied at the same time, the position movement range of the output optical trap is  
Figure 4: Structure of the cell manipulation control system. limited.