## 3 Main Results
### Design of the Combined Estimator  
In speed regulation of PMSM, model simplification and operating environments may cause the parameter uncertainties and load disturbances. Hence, the dynamic and steady-state performance of PMSM will be reduced. In this paper, a compensational estimator is presented to reconstruct the modeled disturbance signal, which contains the influence of both parameter uncertainties and load disturbance. They can be used in controller design to improve the capability for anti-jamming. Based on the system (5), state observer is proposed as follows. \[ \begin{split}\dot{\hat{x}}&=A\hat{x}+Bu+E\hat{d}+L( \hat{y}-y)\\ \hat{y}&=C\hat{x}\end{split}  \tag{10}\]  
where \(\hat{x}\) and \(\hat{d}\) are the estimation of \(x\) and \(d\). \(L=\left[ \begin{array}{cc}L_{1}&L_{2}\\ \end{array} \right]^{T}\) represents the observer gain matrix. The compensational estimator is presented as Equation (11) if \(d\) is a slow time-varying signal. \[\dot{\hat{d}}=K(\hat{x}-x) \tag{11}\]  
in which, \(K\) is the estimator gain matrix, and the parameters in \(K\) are positive. Then it is followed by the stability analysis of the proposed method. **Lemma 1**. : _Based on system (9) and the estimator (11), one can get that the observer (10) and estimator (11) can achieve good performance if the following inequality holds [20]._  
\[(A-LC)\leq 0 \tag{12}\]  
_Hence, one can get that the proposed method can achieve stable convergence._  
**Proof 1**: The Lyapunov function is defined as  
\[V=\frac{E}{2K}\hat{d}^{2}+\frac{1}{2}\hat{x}^{2} \tag{13}\]  
where \(\hat{d}=d-\hat{d}\), \(\hat{x}=x-\hat{x}\). The derivative can be written as  
\[ \begin{split}\dot{V}&=\frac{\xi}{\hat{k}}\hat{d}+ \hat{x}\hat{x}\\ &=\hat{d}\Big{(}\hat{d}-\hat{d}\Big{)}+\hat{x}\Big{[}Ax+Bu+Ed- \Big{(}A\hat{x}+Bu+E\hat{d}+LC(\hat{x}-x)\Big{)}\Big{]}\\ &=\hat{d}\hat{d}-\hat{d}\hat{d}+\hat{x}\Big{[}A\hat{x}-E\hat{d}- LC\hat{x}\Big{]}\\ &=\hat{d}\hat{d}-E\hat{x}+A\hat{x}^{2}-E\hat{d}\hat{x}-LC\hat{x}^{ 2}\\ &=\hat{d}\hat{d}+(A-LC)\hat{x}^{2}\end{split}  \tag{14}\]  
Under the assumption that \(d\) is a slow time-varying signal, it can be obtained that \(\dot{\hat{d}}=0\). Hence, \(\dot{V}<0\) holds if and only if inequality (12) holds. In fact, disturbance signals always show the characteristic of non-stationary in many systems. To enhance the tracking performance, the proposed estimator in Equation (11) can be rewritten as  
\[\dot{\hat{d}}=K_{1}(\hat{x}-x)+K_{2}\Big{(}\dot{\hat{x}}-\dot{x}\Big{)}+\theta \tag{15}\]  
where \(K_{1}\), \(K_{2}\), and \(K_{3}\) are the gain matrices. It can be also obtained that the estimating signal can track actual signal well if Theorem 1 holds. The parameter \(\theta\) is a correction term of estimating results error due to the simplifications and constraint assumptions. \(\theta\) is determined by the experience and the bounds of disturbance signal. **Theorem 1**. : _Based on system (9) and the estimator (15), set that \(E-M[K_{1}+K_{2}(A-LC)]=0\) and assume that \(\dot{d}-K_{2}E\dot{d}\leq 0\), \(\dot{d}-\theta\leq\rho\dot{d}\), \(\rho\leq 0\), one can get that the observer (10) and estimator (15) can achieve good performance if the following inequality holds._  
\[\left\{ \begin{array}{l}(A-LC)\leq 0\\ K_{2}E\geq 0\\ K_{1}+K_{2}(A-LC)\geq 0\end{array} \right. \tag{16}\]  
_That is, the estimator (15) can be applied in some time-varying disturbance reconstructions if the derivative \(\dot{d}\) of disturbance signal is bounded._  
**Proof 2**: _Define that_  
\[V=\frac{E}{2K}\dot{d}^{2}+\frac{1}{2}\dot{x}^{2} \tag{17}\]  
_where \(V\) is the Lyapunov function, \(\ddot{d}=d-\dot{d}\), \(\dot{x}=x-\dot{x}\). Then one can get the derivative as follows._  
\[