# A Novel Robust Control System Design and Its Application to Servo Motor Drive
## 3 Sliding Mode Control Design for a Class of Linear System Models with Perturbations  
The following will add the sliding mode control structure to the linear model-represented system. As shown in (13), the integral state feedback control system has the control law \(u=-\mathbf{k}_{1}^{\mathsf{T}}\mathbf{z}\). The system with the linear control law can eliminate the steady-state error from the reference input and external disturbance input. However, transient performance defined in (14) cannot be guaranteed if the system has parameter variation, i.e., the resulting eigenvalues of the closed-loop system will not be the designed ones. A system with external disturbance and parameter variations included is represented as follows:  
\[\dot{\mathbf{z}}=(\mathbf{A}_{1}+\Delta\mathbf{A}_{1})\mathbf{z}+(\mathbf{b}_ {1}+\Delta\mathbf{b}_{1})u+\mathbf{w} \tag{19}\]  
where \(\Delta\mathbf{A}_{1}\) and \(\Delta\mathbf{b}_{1}\) are variations in \(\mathbf{A}_{1}\) and \(\mathbf{b}_{1}\) and \((\mathbf{b}_{1}+\Delta\mathbf{b}_{1})\) does not change the sign of \(\mathbf{b}_{1}\) to maintain controllability. The variations and disturbances in (19) can be lumped and expressed as:  
\[\mathbf{p}=\Delta\mathbf{A}_{1}\mathbf{z}+\Delta\mathbf{b}_{1}u+\mathbf{w} \tag{20}\]  
When the system is expressed in a controllable canonical form, the lumped perturbation can be in the following format:  
\[\mathbf{p}=\begin{bmatrix}0&0&\cdots&\widetilde{p}\end{bmatrix}^{\mathsf{T}} \tag{21}\]  
The SMC control is added to recover the system with the desired transient performance as defined by the equivalent system state, \(\mathbf{A}_{c1}\). First, with a constant vector \(\mathbf{g}\), where \(\mathbf{g}^{\mathsf{T}}\mathbf{b}_{1}\
eq 0\)[11; 29], a sliding surface  
\[\sigma(t,\mathbf{z})=\mathbf{g}^{\mathsf{T}}(\mathbf{z}-\mathbf{z_{0}})- \mathbf{g}^{\mathsf{T}}\mathbf{A}_{\mathsf{c1}}\int\mathbf{z}(t)\mathbf{dt} \tag{22}\]  
is used to determine an extra control law, \(u_{smc}\). For the closed-loop system (3) under nominal conditions (without the uncertainty, perturbation, and load), with an initial value \(\mathbf{x}_{0},\sigma(t,\mathbf{x})=0\) for \(t\geq 0\), i.e., the system states are initially on the sliding surface and will stay on it for the entirety of the next proceeding time. However, this is not for a perturbed system (19). Extra control effort is needed to make and keep the states on the sliding surface as a nominal system does.  
The extra control law is defined as:  
\[u_{smc}=-q\text{sgn}(\sigma) \tag{23}\]  
where \(\text{sgn}(\cdot)\) is the sign function and \(q\) is a positive scalar constant defined as:  
\[\frac{q}{b}>|\widetilde{p}|_{\max} \tag{24}\]  
where \(|\widetilde{p}|_{\max}\) is the upper bound of the total perturbation. Then, the total control law for the augmented system is:  
\[u=u_{L}+u_{smc}=-\mathbf{k}_{1}^{\mathsf{T}}\mathbf{z}-q\text{sgn}(\sigma) \tag{25}\]  
where \(u_{L}\) is the linear state feedback control. The second part of (25) in the control structure mainly guarantees transient performance since the integral action can ensure steady-state performance. Furthermore, the chattering phenomenon due to the switching control law of \(u_{smc}=-q\text{sgn}(\sigma)\) must be solved to let the steady-state performance fit the requirement as a linear control system does.