Adaptive sliding mode control based on a combined state/disturbance observer for the disturbance rejection control of PMSM

Abstract

An improved adaptive sliding mode controller (ASMC) based on a combined state/disturbance observer (CSO) is proposed for the high-performance control of permanent magnet synchronous motor (PMSM). In order to estimate the unknown state and the disturbance including the parameter uncertainty and the external disturbance, the CSO is proposed. Different from the extended state observer (ESO) or generalized ESO (GESO), the CSO uses the linear combination of the extended high-order states to construct the new estimation. The CSO resolves the contradiction between the estimation accuracy and the noise insensitivity, so it is more applicable to time-varying disturbances. Then, the CSO and the ASMC are integrated in the PMSM speed controller by the feed-forward compensation to enhance the system robustness. A simple nonlinear adaptive law is presented to solve the unknown upper bound of the estimation error and minimize the chattering. The theoretical analysis shows that the global stability of the closed-loop system is strictly guaranteed. The simulation and experimental results are presented to demonstrate the effectiveness of the proposed control method.

Appendices

Appendix A

Proof

Define the following state variables: $x_{0} = \int_{0}^{t} {x_{1} (t)} {\text{d}}t$, $x_{ - 1} = \int_{0}^{t} {x_{0} (t){\text{d}}t}$, …, and $x_{{1 - r_{1} }} = \int_{0}^{t} {x_{{1 - (r_{1} - 1)}} (t){\text{d}}t}$, the system (6) can be rewritten as

$$ \left\{ {\begin{array}{*{20}l} {\dot{x}_{{1 - r_{1} }} = x_{{1 - (r_{1} - 1)}} } \hfill \\ \vdots \hfill \\ {\dot{x}_{1} = x_{2} } \hfill \\ {\dot{x}_{2} = u + x_{3} } \hfill \\ {\dot{x}_{3} = \dot{D}} \hfill \\ \vdots \hfill \\ {\dot{x}_{{n + 2 - r_{1} }} = D^{{(n - r_{1} )}} } \hfill \\ \end{array} } \right. $$

(A1)

For system (A1), a GESO is designed as

$$ \left\{ {\begin{array}{*{20}l} {\dot{\eta }_{{1 - r_{1} }} = \eta_{{1 - (r_{1} - 1)}} - \omega_{0} k_{1} \xi_{1} } \hfill \\ \vdots \hfill \\ {\dot{\eta }_{1} = \eta_{2} - \omega_{0}^{{r_{1} + 1}} k_{{r_{1} + 1}} \xi_{1} } \hfill \\ {\dot{\eta }_{2} = u + \eta_{3} - \omega_{0}^{{r_{1} + 2}} k_{{r_{1} + 2}} \xi_{1} } \hfill \\ {\dot{\eta }_{3} = \eta_{4} - \omega_{0}^{{r_{1} + 3}} k_{{r_{1} + 3}} \xi_{1} } \hfill \\ \vdots \hfill \\ {\dot{\eta }_{{n + 2 - r_{1} }} = - \omega_{0}^{n + 2} k_{n + 2} \xi_{1} } \hfill \\ \end{array} } \right. $$

(A2)

where $\xi_{1} = \eta_{{1 - r_{1} }} - x_{{1 - r_{1} }}$, $k_{i} = \left( {n + 2} \right)!/i! \times (n + 2 - i)!$, and $- \omega_{0}$ is the desired pole. Supposing $\xi = [\xi_{1} , \ldots ,\xi_{n + 2} ] = [\eta_{{1 - r_{1} }} - x_{{1 - r_{1} }} ,\eta_{{1 - (r_{1} - 1)}} - x_{{1 - (r_{1} - 1)}} , \ldots ,\eta_{{n + 2 - r_{1} }} - x_{{n + 2 - r_{1} }} ]$, and subtracting (A1) from (A2) we have

$$ \left\{ {\begin{array}{*{20}l} {\dot{\xi }_{1} = \xi_{2} - \omega_{0} k_{1} \xi_{1} } \hfill \\ {\dot{\xi }_{2} = \xi_{3} - \omega_{0}^{2} k_{2} \xi_{1} } \hfill \\ \vdots \hfill \\ {\dot{\xi }_{n + 2} = - \omega_{0}^{n + 2} k_{n + 2} \xi_{1} - D^{{(n - r_{1} )}} } \hfill \\ \end{array} } \right. $$

(A3)

By using Laplace transform of (A2) and (A3), the following equation can be obtained

$$ \begin{aligned} \eta_{2} (s) & = \frac{{l_{{r_{1} + 2}} s^{{n - r_{1} }} + l_{{r_{1} + 3}} s^{{n - r_{1} - 1}} + \cdots + l_{n + 2} }}{{s^{n + 2} + l_{1} s^{n + 1} + l_{2} s^{n} + \cdots + l_{n + 2} }}x_{2} (s) \\ & \quad + \frac{{s^{n + 1} + l_{1} s^{n} + \cdots + l_{{r_{1} + 1}} s^{{n - r_{1} }} }}{{s^{n + 2} + l_{1} s^{n + 1} + l_{2} s^{n} + \cdots + l_{n + 2} }}u(s) \\ \end{aligned} $$

(A4)

Comparing (A4) with (17), it can be seen that $\hat{x}_{2} (s) = \eta_{2} (s)$, so $\hat{x}_{2} (t) = \eta_{2} (t)$. Then using the theorem in [27], for GESO (A2), there is $\lim_{t \to \infty } \left| {\eta_{2} - x_{2} } \right| \le O(\varepsilon^{{n + 1 - r_{1} }} )$, where ε = 1/ω₀. Thus $\lim_{t \to \infty } \left| {\hat{x}_{2} - x_{2} } \right| \le O(\varepsilon^{{n + 1 - r_{1} }} )$ is satisfied. Similarly, $\lim_{t \to \infty } \left| {\hat{D} - D} \right| \le O(\varepsilon^{{n - r_{2} }} )$ can be proved. Thus, Theorem 1 is proved.

Here, the existing results of GESO are used in the proof of Theorem 1. And it can be seen that the GESO (A2) achieves the same estimation as the CSO. However, (A2) cannot be implemented because the “integral saturation” problem will appear when calculating x₀, $x_{ - 1}$, …, $x_{{1 - r_{1} }}$. The CSO uses the linear combination of the extended high-order states to generate the unknown state and disturbance estimations, so this problem is eliminated.

Appendix B

Proof

When x = 0, it can be easy known f(0) = 0. When x > 0, if k < k_d, defining $g(x) = f^{'} (x) = - 2k_{s} x - (k - k_{d} ) + \frac{1}{\lambda }(k - k_{d} )k_{m} px^{p - 1}$, then $g^{'} (x) = - 2k_{s} + \frac{1}{\lambda }(k - k_{d} )k_{m} p(p - 1)x^{p - 2}$ can be obtained. Supposing $g^{'} (x_{0} ) = 0$, we have

$$ x_{0} = \left(\frac{{2k_{s} \lambda }}{{(k - k_{d} )k_{m} p(p - 1)}}\right)^{{\frac{1}{p - 2}}} $$

(B1)

It can be known that x₀ > 0. Noting $g^{''} (x_{0} ) = (k - k_{d} )k_{m} {{p(p - 1)(p - 2)x_{0}^{p - 3} } \mathord{\left/ {\vphantom {{p(p - 1)(p - 2)x_{0}^{p - 3} } \lambda }} \right. \kern-0pt} \lambda }$, and considering k < k_d, 0 < p < 1 and x₀ > 0, there is $g^{''} (x_{0} ) < 0$. Thus the maximum value of g(x) is g(x₀), i.e., g(x) ≤ g(x₀). Selecting $\lambda = {{(k - k_{d} )k_{m} p(p - 1)} \mathord{\left/ {\vphantom {{(k - k_{d} )k_{m} p(p - 1)} {2k_{s} }}} \right. \kern-0pt} {2k_{s} }} \cdot \left[ { - \frac{1}{{2k_{s} }}(k - k_{d} )} \right]^{p - 2} > 0$, and substituting it into (B1), we have $x_{0} = - \frac{1}{{2k_{s} }}(k - k_{d} )$. Substituting x₀ into g(x), it yields

$$ g(x_{0} ) = {{(k - k_{d} )k_{m} px_{0}^{p - 1} } \mathord{\left/ {\vphantom {{(k - k_{d} )k_{m} px_{0}^{p - 1} } \lambda }} \right. \kern-0pt} \lambda } $$

(B2)

Because x₀ > 0, there is g(x₀) < 0, and thus $f^{'} (x) = g(x) \le g(x_{0} ) < 0$ for all x > 0.

If k = k_d, there is $f(x) = - k_{s} x^{2}$, it is obvious that $f^{'} (x) = - 2k_{s} x < 0$ for all $x > 0$. Therefore, there exists a positive real number λ such that $f^{'} (x) < 0$ for all x > 0. And noting that f(0) = 0 and f(x) is continuous, it can be known that $f(x) \le 0$ for all $x \ge 0$. Lemma 2 is proved.