Basic information

Title: A Novel Sliding Mode Observer With Optimized Constant Rate Reaching Law for Sensorless Control of Induction Motor

Highlight authors: None

Publication: TIE, 2020

Contribution

• An optimized constant rate reaching law (OCRRL) is adopted to SMO
• Detaildely analysed the reaching speed for the proposed OCRRL, which behaves the same speed of CRRL

Motivation

• Compared with other estimators, SMO has a better characteristic of antimodeling uncertainty because of the inherent variable structure, which leads to a better robustness performance.
• In fact, how the state variables reach the sliding mode surface, i.e., the reaching way to the sliding mode surface, has also the significant influence on the performance of the observer, especially for the fast reaching transient performance and the chattering reduction.
• SMO + constant rate reaching law (CRRL) cannot balance the dilemma between the requirement of fast reaching transient and the chattering reduction on the sliding mode surface.

SMO DESIGN WITH CONSTANT RATE REACHING LAW FOR SENSORLESS IM DRIVES

Given the induction motor (IM)
[ d i s α d t d i s β d t ] = k 1 ( [ λ ω r − ω r λ ] [ ψ r α ψ r β ] − λ L m [ i s α i s β ] ) − k 2 [ i s α i s β ] + k 3 [ u s α u s β ] (sys1) \begin{aligned} \left[\begin{array}{c} \frac{d i_{s \alpha}}{d t} \\ \frac{d i_{s \beta}}{d t} \end{array}\right]= &k_{1} \left(\left[\begin{array}{cc} \lambda & \omega_{r} \\ -\omega_{r} & \lambda \end{array}\right]\left[\begin{array}{c} \psi_{r \alpha} \\ \psi_{r \beta} \end{array}\right]-\lambda L_{m}\left[\begin{array}{c} i_{s \alpha} \\ i_{s \beta} \end{array}\right]\right) \\ &-k_{2}\left[\begin{array}{c} i_{s \alpha} \\ i_{s \beta} \end{array}\right]+k_{3}\left[\begin{array}{c} u_{s \alpha} \\ u_{s \beta} \end{array}\right] \end{aligned}\tag{sys1} [dtdisα​​dtdisβ​​​]=​k1​([λ−ωr​​ωr​λ​][ψrα​ψrβ​​]−λLm​[isα​isβ​​])−k2​[isα​isβ​​]+k3​[usα​usβ​​]​(sys1)

[ d ψ r α d t d ψ r β d t ] = − ( [ λ ω r − ω r λ ] [ ψ r α ψ r β ] − λ L m [ i s α i s β ] ) (sys2) \left[\begin{array}{c} \frac{d \psi_{r \alpha}}{d t} \\ \frac{d \psi_{r \beta}}{d t} \end{array}\right]=-\left(\left[\begin{array}{cc} \lambda & \omega_{r} \\ -\omega_{r} & \lambda \end{array}\right]\left[\begin{array}{l} \psi_{r \alpha} \\ \psi_{r \beta} \end{array}\right]-\lambda L_{m}\left[\begin{array}{c} i_{s \alpha} \\ i_{s \beta} \end{array}\right]\right) \tag{sys2} [dtdψrα​​dtdψrβ​​​]=−([λ−ωr​​ωr​λ​][ψrα​ψrβ​​]−λLm​[isα​isβ​​])(sys2)

The estimation of the matrix G G G can be provided by the sliding mode function as follows:
[ f α f β ] = G ^ = [ G ^ α G ^ β ] = [ λ ω ^ r − ω ^ r λ ] [ ψ ^ r α ψ ^ r β ] [ − λ L m 7 ] [ i ^ s α i ^ s β ] \left[\begin{array}{c} f_{\alpha} \\ f_{\beta} \end{array}\right]=\hat{G}=\left[\begin{array}{c} \hat{G}_{\alpha} \\ \hat{G}_{\beta} \end{array}\right]=\left[\begin{array}{cc} \lambda & \hat{\omega}_{r} \\ -\hat{\omega}_{r} & \lambda \end{array}\right]\left[\begin{array}{c} \hat{\psi}_{r \alpha} \\ \hat{\psi}_{r \beta} \end{array}\right]\left[\begin{array}{c} -\lambda L_{m} \\ 7 \end{array}\right]\left[\begin{array}{l} \hat{i}_{s \alpha} \\ \hat{i}_{s \beta} \end{array}\right] [fα​fβ​​]=G^=[G^α​G^β​​]=[λ−ω^r​​ω^r​λ​][ψ^​rα​ψ^​rβ​​][−λLm​7​][i^sα​i^sβ​​]

SMO structure

Based on ( s y s 1 ) (sys1) (sys1), the SMO for current observation is transformed into:
[ d i ^ s α d t d i ^ s β d t ] = k 1 [ f α f β ] − k 2 [ i ^ s α i ^ s β ] + k 3 [ u s α u s β ] \left[\begin{array}{c} \frac{d \hat{i}_{s \alpha}}{d t} \\ \frac{d \hat{i}_{s \beta}}{d t} \end{array}\right]=k_{1}\left[\begin{array}{c} f_{\alpha} \\ f_{\beta} \end{array}\right]-k_{2}\left[\begin{array}{c} \hat{i}_{s \alpha} \\ \hat{i}_{s \beta} \end{array}\right]+k_{3}\left[\begin{array}{l} u_{s \alpha} \\ u_{s \beta} \end{array}\right] [dtdi^sα​​dtdi^sβ​​​]=k1​[fα​fβ​​]−k2​[i^sα​i^sβ​​]+k3​[usα​usβ​​]
Based on ( s y s 2 ) (sys2) (sys2), the SMO for flux-linkage observation is as follows:
[ d ψ ^ r α d t d ψ ^ r β d t ] = − [ f α f β ] \left[\begin{array}{c} \frac{d \hat{\psi}_{r \alpha}}{d t} \\ \frac{d \hat{\psi}_{r \beta}}{d t} \end{array}\right]=-\left[\begin{array}{l} f_{\alpha} \\ f_{\beta} \end{array}\right] [dtdψ^​rα​​dtdψ^​rβ​​​]=−[fα​fβ​​]
where
[ f α f β ] = − [ λ 0 sign ⁡ ( S α ) λ 0 sign ⁡ ( S β ) ] \left[\begin{array}{l} f_{\alpha} \\ f_{\beta} \end{array}\right]=-\left[\begin{array}{l} \lambda_{0} \operatorname{sign}\left(S_{\alpha}\right) \\ \lambda_{0} \operatorname{sign}\left(S_{\beta}\right) \end{array}\right] [fα​fβ​​]=−[λ0​sign(Sα​)λ0​sign(Sβ​)​]

With λ 0 > 0 \lambda_0 > 0 λ0​>0 being the gain of the sliding mode function.

Sliding surface design

Conventionally, the sliding surface of this paper is merely defined as the error
S = [ S α S β ] = [ i ~ s α i ~ s β ] = [ i ^ s α − i s α i ^ s β − i s β ] S = \left[\begin{array}{c} S_{\alpha} \\ S_{\beta} \end{array}\right]=\left[\begin{array}{l} \tilde{i}_{s \alpha} \\ \tilde{i}_{s \beta} \end{array}\right]=\left[\begin{array}{l} \hat{i}_{s \alpha}-i_{s \alpha} \\ \hat{i}_{s \beta}-i_{s \beta} \end{array}\right] S=[Sα​Sβ​​]=[i~sα​i~sβ​​]=[i^sα​−isα​i^sβ​−isβ​​]
In this paper, the sliding mode surface is defined by considering the estimation error of stator currents and the rate of the estimation error simultaneously (merely the PI-type SS):
S I = [ S I α S I β ] = [ p 1 i ~ s α + p 2 ∫ i ~ s α d t p 1 i ~ s β + p 2 ∫ i ~ s β d t ] S_{I}=\left[\begin{array}{c} S_{I \alpha} \\ S_{I \beta} \end{array}\right]=\left[\begin{array}{l} p_{1} \tilde{i}_{s \alpha}+p_{2} \int \tilde{i}_{s \alpha} d t \\ p_{1} \tilde{i}_{s \beta}+p_{2} \int \tilde{i}_{s \beta} d t \end{array}\right] SI​=[SIα​SIβ​​]=[p1​i~sα​+p2​∫i~sα​dtp1​i~sβ​+p2​∫i~sβ​dt​]
The parameter selection criteria one might refer to PID method.

SMO with reaching law

SMO+CRRL

Conventional constant rate reaching law (CRRL):
S ˙ I = − k sign ⁡ ( S I ) = − k [ sign ⁡ ( S I α ) sign ⁡ ( S I β ) ] \dot{S}_{I}=-k \operatorname{sign}\left(S_{I}\right)=-k\left[\begin{array}{l} \operatorname{sign}\left(S_{I \alpha}\right) \\ \operatorname{sign}\left(S_{I \beta}\right) \end{array}\right] S˙I​=−ksign(SI​)=−k[sign(SIα​)sign(SIβ​)​]

SMO+Optimized CRRL

{ S ˙ I = − g ( i ~ s ) ⋅ sign ⁡ ( S I ) g ( i ~ s ) = k ′ ε + ( 1 + 1 / ∣ i ~ s ∣ − ε ) e − η ∣ δ ∣ (OCRRL) \left\{\begin{array}{l} \dot{S}_{I}=-g\left(\tilde{i}_{s}\right) \cdot \operatorname{sign}\left(S_{I}\right) \\ g\left(\tilde{i}_{s}\right)=\frac{k^{\prime}}{\varepsilon+\left(1+1 /\left|\tilde{i}_{s}\right|-\varepsilon\right) e^{-\eta|\delta|}} \end{array}\right.\tag{OCRRL} {S˙I​=−g(i~s​)⋅sign(SI​)g(i~s​)=ε+(1+1/∣i~s​∣−ε)e−η∣δ∣k′​​(OCRRL)

where k ′ > 0 , η > 0 , 0 < ε < 1 , k^{\prime}>0, \eta>0,0<\varepsilon<1, k′>0,η>0,0<ε<1, and δ = i ^ s α i s β − i s α i ^ s β \delta=\hat{i}_{s \alpha} i_{s \beta}-i_{s \alpha} \hat{i}_{s \beta} δ=i^sα​isβ​−isα​i^sβ​

Analysis of Dynamic Performance for SMO+OCRRL

Recalling ( O C R R L ) (OCRRL) (OCRRL), the reaching time t 1 t_1 t1​ can be given as:
S ˙ I [ ε + ( 1 + 1 / ∣ g ( i ~ s ) ∣ − ε ) e − η ∣ δ ∣ ] = − k ′ ⋅ sign ⁡ ( S I ) (SSM) \dot{S}_{I}\left[\varepsilon+\left(1+1 /\left|g\left(\tilde{i}_{s}\right)\right|-\varepsilon\right) e^{-\eta|\delta|}\right]=-k^{\prime} \cdot \operatorname{sign}\left(S_{I}\right) \tag{SSM} S˙I​[ε+(1+1/∣∣​g(i~s​)∣∣​−ε)e−η∣δ∣]=−k′⋅sign(SI​)(SSM)
The reaching phase details can be given as:

1. If the state variables are far away from the sliding mode surface, which means that when ∣ δ ∣ |δ| ∣δ∣ increases, then e − η ∣ δ ∣ e^{−η |\delta|} e−η∣δ∣ converges to 0 0 0, and g ( i ~ s ) g(\tilde{i}_s ) g(i~s​) will converge to k ′ / ε k'/ε k′/ε. Since 0 < ε < 1 , k ′ / ε > k ′ 0<ε<1, k'/ε>k' 0<ε<1,k′/ε>k′, which means that the gain of the CRRL is large, and the fast reaching speed is obtained at this time.
2. If the state variables are close to the sliding mode surface, which means that when ∣ δ ∣ |δ| ∣δ∣ decreases, e − η ∣ δ ∣ e^{−η |δ |} e−η∣δ∣ converges to 1 1 1, and g ( i ~ s ) g(\tilde{i}_s ) g(i~s​) will converge to k ′ ∣ i ~ s ∣ / ( 1 + ∣ i ~ s ∣ ) k^{\prime}\left|\tilde{i}_{s}\right| /\left(1+\left|\tilde{i}_{s}\right|\right) k′∣∣​i~s​∣∣​/(1+∣∣​i~s​∣∣​). Since the state variables are close to the sliding mode surface, is converges to zero, which means that the gain of the CRRL is small, and the chattering reduction is achieved at this time.

Considering time span [ t 0 , t 1 ] [t_0,t_1] [t0​,t1​] with final variable S I ( t 1 ) = 0 S_I(t_1) = 0 SI​(t1​)=0, the reaching time can be calculated by integrating ( S S M ) (SSM) (SSM):
t 1 = 1 k ′ [ ε ∣ S I ( 0 ) ∣ + 1 + 1 / ∣ i ~ s ∣ − ε η ( 1 − e − η S I ( 0 ) ) ] (Tr1) t_{1}=\frac{1}{k^{\prime}}\left[\varepsilon\left|S_{I}(0)\right|+\frac{1+1 /\left|\tilde{i}_{s}\right|-\varepsilon}{\eta}\left(1-e^{-\eta S_{I}(0)}\right)\right]\tag{Tr1} t1​=k′1​[ε∣SI​(0)∣+η1+1/∣∣​i~s​∣∣​−ε​(1−e−ηSI​(0))](Tr1)
Since 1 − e − η S I ( 0 ) < 1 1 - e^{-\eta S_I(0)} < 1 1−e−ηSI​(0)<1, ( T r 1 ) (Tr1) (Tr1) can be derived as:
t 1 < 1 k ′ [ ε ∣ S I ( 0 ) ∣ + 1 + 1 / ∣ i ~ s ∣ − ε η ] (Tr2) t_{1}<\frac{1}{k^{\prime}}\left[\varepsilon\left|S_{I}(0)\right|+\frac{1+1 /\left|\tilde{i}_{s}\right|-\varepsilon}{\eta}\right]\tag{Tr2} t1​<k′1​[ε∣SI​(0)∣+η1+1/∣∣​i~s​∣∣​−ε​](Tr2)
In this paper, η \eta η is chosen such that η ≫ ( 1 + 1 / ∣ i ~ s ∣ − ε ) / ( ε ∣ S I ( 0 ) ∣ ) \eta \gg\left(1+1 /\left|\tilde{i}_{s}\right|-\varepsilon\right) /\left(\varepsilon\left|S_{I}(0)\right|\right) η≫(1+1/∣∣​i~s​∣∣​−ε)/(ε∣SI​(0)∣), in which η \eta η is a positiv efactor which is used to adjust the rate of g ( i ~ s ) g(\tilde{i}_s) g(i~s​). Simply setting η = 3000 \eta = 3000 η=3000, ( T r 2 ) (Tr2) (Tr2) can be given as:

t 1 < ε ∣ S I ( 0 ) ∣ k ′ t_{1}<\frac{\varepsilon\left|S_{I}(0)\right|}{k^{\prime}} t1​<k′ε∣SI​(0)∣​

In the original context of the paper, the author further illustrated that the proposed OCRRL has the same reaching rate with the CRRL, which is omitted here.

Simulation results

Observation performance with (a) SMO+CRRL (b) SMO+OCRRL

My finding

1. Merely adopted a new reaching law (which is studied by the existing literature)
2. The chattering phenomenon is well alleviated