# An Improved Hybrid Control Scheme of a Switched Reluctance Motor for Torque Ripple Reduction
## 4 Speed Controller Based on Modified LADRC  
The input of the speed controller is the given speed and feedback speed, and the output is the given torque. LADRC was chosen for lower complexity in terms of model and parameter tuning, which consists of a linear tracking differentiator, a linear extended-state observer, and a linear state-error feedback. The speed loop controller is designed as a first-order LADRC based on the first-order system mathematical model. As the key component, the linear extended-state observer is used to observe and compensate for model uncertainties and external disturbances, with good adaptability and robustness. Based on (2), the state equation of the controlled object is as follows:  
\[\dot{\omega}=\frac{\mathrm{d}\omega}{\mathrm{d}t}=\frac{T^{*}-D\cdot\omega-T_{L}} {J} \tag{14}\]  
Then, (14) can be rewritten as [42]  
\[\dot{\omega}=bT^{*}+f_{t} \tag{15}\]  
\[f_{t}=\frac{-D\cdot\omega-T_{L}}{J} \tag{16}\]  
where \(f_{t}\) is the total disturbance, and \(bT^{*}\) is the control quantity. According to the design principle of the linear extended-state observer, (16) can be expressed as  
\[\left\{ \begin{array}{l}\hat{\dot{\omega}}=\hat{f}_{t}+bT^{*}+\beta_{1}( \omega-\hat{\omega})\\ \hat{\dot{f}}_{t}=\beta_{2}(\omega-\hat{\omega})\end{array} \right. \tag{17}\]  
where \(\omega\), \(\hat{\omega}\), and \(\hat{f}_{t}\) are the feedback value of speed, the estimated value of speed, and the estimated value of disturbance, respectively, and \(\beta_{1}\) and \(\beta_{2}\) are tunable gain parameters. To improve the observation speed of the linear extended-state observer in response to disturbance, a proportional gain of the observation speed error derivative can be added in the disturbance observation channel. Therefore, (17) can be further expressed as  
\[\left\{ \begin{array}{l}\hat{\dot{\omega}}=\hat{f}_{t}+bT^{*}+\beta_{1}( \omega-\hat{\omega})\\ \hat{\dot{f}}_{t}=\beta_{2}(\omega-\hat{\omega})+\beta_{3}\big{(}\dot{\omega} -\dot{\hat{\omega}}\big{)}\end{array} \right. \tag{18}\]  
where \(\beta_{3}\) is the tunable gain parameter. The control law adopts the form of a linear combination for fast response and easy implementation. The linear state-error feedback is mainly applied to offset the disturbance, and \(bT^{*}=u_{0}-\hat{f}_{t}\) is selected, where \(u_{0}\) is the linear state-error feedback. The controlled object can be equivalent to a pure integral link, and (13) can be derived as  
\[\dot{\omega}=u_{0}-\hat{f}_{t}+f_{t}\approx u_{0} \tag{19}\]  
Next, the proportional linear control law can be used to achieve tracking of the given error. Therefore, the linear state-error feedback can be expressed as  
\[u_{0}=k_{p}(\omega^{*}-\hat{\omega}) \tag{20}\]  
where \(\omega^{*}\) is the reference speed, and \(k_{p}\) is the proportional gain. A structural diagram of LADRC is shown in Figure 6.  
Figure 6: Structural diagram of LADRC. Based on the introduction of the bandwidth concept [43], the value of tunable parameters in LADRC can be expressed as  
\[\left\{ \begin{array}{l}\beta_{1}=\beta_{3}=2\omega_{0}\\ \beta_{2}=\omega_{0}^{2}\\ k_{p}=\omega_{cv}=(0.1\sim 0.2)\omega_{0}\end{array} \right. \tag{21}\]  
where \(\omega_{0}\) and \(\omega_{cv}\) are the bandwidth of the linear extended-state observer and the speed loop, respectively. However, for constant bandwidth gain, stronger anti-disturbance ability is associated with a more prominent overshoot phenomenon. It is difficult to ensure no overshoot and strong anti-interference performance simultaneously. Low gain parameters can ensure no overshoot and the controllability of the system, whereas anti-interference ability and speed tracking effect are reduced. Conversely, high gain parameter behaves well in terms of anti-disturbance ability and tracking performance, whereas overshoot under varied conditions will be obvious.