## 3 Compound Controller Design
### Design of the Novel DTSMC  
In this subsection, the nonlinear disturbance term \(d(k)\) is considered, which describes the lumped effect of unmodeled hysteresis nonlinearity, inverse compensation errors, residual dynamics, parameter uncertainties and external disturbances. In order to design the proposed controller, Equation (11) is rewritten as follows:  
\[y(k)=-\sum_{i=1}^{n_{a}}a_{i}y(k-i)+\sum_{i=1}^{n_{b}}b_{i}u(k-i)+d(k) \tag{14}\]  
**Assumption 1**. : _The nonlinear disturbance term \(d(k)\) is assumed to be unknown but bounded, which can be expressed as follows:_  
\[|d(k)|\leq\delta_{0} \tag{15}\]  
_where \(\delta_{0}\) is a known positive constant._  
**Assumption 2**. : _The change rate of the nonlinear disturbance term \(d(k)\) is assumed to be unknown but bounded, which can be expressed as follows:_  
\[\frac{|d(k)-d(k-1)|}{T}\leq\delta \tag{16}\]  
_where \(T\) is the sampling period, \(k\in N\), and \(\delta\) is a known positive constant._  
As reported in [32], the nonlinear disturbance term \(d(k)\) possesses the following property. **Property 1**. : \(d(k)=O(T)\)_, \(d(k)-d(k-1)=O(T^{2})\), and \(d(k)-2d(k-1)+d(k-1)=O(T^{3})\)._  
In order to complete the controller design and its stability analysis, a controllable canonical representation can be obtained according to the discrete-time plant model (14)  
\[\left\{ \begin{array}{l}\mathbf{X}(k+1)=\mathbf{AX}(k)+\mathbf{R}u(k)+ \mathbf{BU}(k)+\mathbf{D}(k)\\ y(k)=\mathbf{CX}(k)\end{array} \right. \tag{17}\]  
in which  
\[\mathbf{U}(k)=\big{[}u(k-n_{b}+1),u(k-n_{b}),\cdots,u(k-1)\big{]}^{T}\in\Re^{( n_{b}-1)\times 1},\]  
\[\mathbf{X}(k)=\big{[}x(k-n_{a}+1),\cdots,x(k-1),x(k)\big{]}^{T}\in\Re^{n_{b} \times 1},\]  
Figure 7: The structure of the compound controller. \[\mathbf{A}= \begin{bmatrix}0&1&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1\\ -a_{n_{a}}&-a_{n_{a}-1}&\cdots&-a_{1}\end{bmatrix} \in\Re^{n_{a}\times n_{a}}, \mathbf{B}= \begin{bmatrix}0&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&0\\ b_{n_{b}}&b_{n_{b}-1}&\cdots&b_{2}\end{bmatrix} \in\Re^{n_{a}\times(n_{b}-1)},\]  
\[\mathbf{C}=[0,\cdots,0,1]\in\Re^{1\times n_{a}},\mathbf{R}=[0,\cdots,0,b_{1}] ^{T}\in\Re^{n_{a}\times 1},\mathbf{D}(k)= \begin{bmatrix}0,\cdots,0,d(k)\end{bmatrix} ^{T }\in\Re^{n_{a}\times 1}.\]  
The control objective is to make the system output \(y(k)\) track a desired displacement in the presence of the nonlinear disturbance \(\mathbf{D}(k)\). To achieve this objective, a novel DTSMC is presented in this subsection. First, the nonlinear disturbance \(\mathbf{D}(k)\) is estimated. Then, the DTSMC is designed based on a new discrete-time terminal sliding function and a terminal-sliding-model-type reaching law. **Remark 1**. : _In order to obtain the nonlinear disturbance term \(\mathbf{D}(k)\), the technique of perturbation estimation [33] is employed under the Assumption 1, the nonlinear disturbance term is estimated by its one-step delayed value, which can be expressed as_  
\[\hat{\mathbf{D}}(k)=\mathbf{D}(k-1)=\mathbf{X}(k)-\mathbf{A}\mathbf{X}(k-1)- \mathbf{R}u(k-1)-\mathbf{B}\mathbf{U}(k-1) \tag{18}\]  
The state-space model (17) can be rewritten as follows:  
\[\left\{ \begin{array}{l}\mathbf{X}(k+1)=\mathbf{A}\mathbf{X}(k)+\mathbf{R}u(k )+\mathbf{B}\mathbf{U}(k)+\hat{\mathbf{D}}(k)-\tilde{\mathbf{D}}(k)\\ y(k)=\mathbf{C}\mathbf{X}(k)\end{array} \right. \tag{19}\]  
where \(\tilde{\mathbf{D}}(k)=\tilde{\mathbf{D}}(k)-\mathbf{D}(k)\) is the nonlinear disturbance estimation error, it can be further expressed as  
\[\hat{\mathbf{D}}(k)=\mathbf{A}\big{[}\mathbf{X}(k)-\mathbf{X}(k-1)\big{]}+ \mathbf{R}\big{[}u(k)-u(k-1)\big{]}+\mathbf{B}\big{[}\mathbf{U}(k)-\mathbf{U} (k-1)\big{]}-\big{[}\mathbf{X}(k+1)-\mathbf{X}(k)\big{]} \tag{20}\]  
Define  
\[e(k)=y(k)-y^{*}(k) \tag{21}\]  
as the trajectory tracking error of the DEAP actuator, where \(y^{*}(k)\) is the desired displacement, \(y(k)\) is the actual displacement.