*4.2.1. Mathematical model of studied motor*

The induction motor is studied as an example for the explanation of the proposed sliding mode observer. Its mathematical model could be described in the rotating d-q frame [22] as follows.

Let *i <sup>d</sup>* and *i <sup>q</sup>*, *Vd* and*Vq*, *ωr*, *TL* , *ψrd* , *θ*, , *p*, and *J* stand for stator currents, stator voltages, rotor speed, load torque, direct-axis rotor flux, flux angle, pole pair number, and moment of inertia. *τ<sup>r</sup>* = *L <sup>r</sup>* / *Rr* represents the rotor time constant, and *σ* =1− *L <sup>m</sup>* <sup>2</sup> / *<sup>L</sup> <sup>s</sup><sup>L</sup> <sup>r</sup>* is the leakage magnetic coefficient. For analysis simplicity, set *m*=(*L <sup>r</sup>Rs* + *L <sup>m</sup>* 2 *Rr*) / (*σL <sup>s</sup>L <sup>r</sup>* 2 ), *γ* = *L <sup>m</sup>* / (*σL <sup>s</sup>L <sup>r</sup>*), *ς* =1 / (*σL <sup>s</sup>*), *κ* =1 / *τr*, *ρ* = *pL <sup>m</sup>* / *L <sup>r</sup>*, and the motor model can be written below:

$$\begin{cases} \frac{\text{d}I\_d}{\text{d}t} = -m\dot{\mathbf{i}}\_d + \kappa \gamma \rho \nu\_{rd} + p o \rho\_r \dot{\mathbf{i}}\_q + \kappa \frac{L\_m}{\rho \nu\_{nl}} \dot{\mathbf{i}}\_q^2 + \boldsymbol{\xi} \cdot \mathbf{V}\_d\\ \frac{\text{d} \dot{\mathbf{i}}\_q}{\text{d}t} = -m\dot{\mathbf{i}}\_q - \gamma p o \rho \nu\_{rd} - p o \rho\_r \dot{\mathbf{i}}\_d - \kappa \frac{L\_m}{\rho \nu\_{rd}} \dot{\mathbf{i}}\_d \dot{\mathbf{i}}\_q + \boldsymbol{\xi} \cdot \mathbf{V}\_q\\ \frac{\text{d} \frac{\partial}{\partial t} = p o \rho\_r + \kappa \frac{L\_m}{\rho \nu\_{rd}} \dot{\mathbf{i}}\_q\\ \frac{\text{d} \mathbf{v}\_{rd}}{\text{d}t} = -\kappa \eta \nu\_{rd} + \kappa L\_m \dot{\mathbf{i}}\_d\\ \frac{\text{d} \alpha \rho\_r}{\text{d}t} = \rho \nu\_{rd} \dot{\mathbf{i}}\_q - \frac{1}{J} T\_L \end{cases} \tag{13}$$

#### *4.2.2. Design of fixed-boundary-layer sliding mode observer*

In this study, the three-phase stator currents are the only required measures, and these are transformed from the three-phase reference frame to a diphasic reference frame, and then to the frame of the rotating field (d-q) as follows:

$$\begin{cases} \dot{\mathbf{i}}\_d = \sqrt{\frac{2}{3}} \left( \cos(\hat{\theta}) \mathbf{i}\_a + \cos(\hat{\theta} - \frac{2}{3}\pi) \mathbf{i}\_b + \cos(\hat{\theta} + \frac{2}{3}\pi) \mathbf{i}\_c \right) \\\\ \dot{\mathbf{i}}\_q = \sqrt{\frac{2}{3}} \left( -\sin(\hat{\theta}) \mathbf{i}\_a - \sin(\hat{\theta} - \frac{2}{3}\pi) \mathbf{i}\_b - \sin(\hat{\theta} + \frac{2}{3}\pi) \mathbf{i}\_c \right) \end{cases} \tag{14}$$

where *i a*, *i <sup>b</sup>*, and *i <sup>c</sup>* represent the three-phase stator currents, respectively, and *θ* ^ denotes the estimated flux angle. Relation between the estimated flux angle and rotor speed could be derived from system (13) as follows:

$$\frac{\mathbf{d}\,\hat{\theta}}{\mathbf{d}\,t} = p\hat{\phi}\_r + \kappa \frac{L\_w}{\hat{\boldsymbol{\nu}}\_{rd}} \mathbf{i}\_q \tag{15}$$

Assume that system (13) has the outputs (*y*<sup>1</sup> *y*2)*<sup>T</sup>* =(*i <sup>d</sup>* i*q*)*<sup>T</sup>* . Unfortunately, it could be found that *ψrd* in system (13) is not observable. Reference [24] has proved that system (13) has a stable zero dynamics about *ψrd* with the output *y*<sup>1</sup> =*i <sup>d</sup>* and *κ* >0. Thus, it is possible to estimate *ψrd* by designing an estimator, and then *ω<sup>r</sup>* becomes observable. The sliding mode observer technol‐ ogy is employed to estimate *ωr*. The required voltage d-q frame in the observer is obtained from outputs of voltage feedforward compensation part. The description of the fixed-boun‐ dary-layer sliding mode (FLBSM) observer is given below.

#### Electric Drives in Alternative Fuel Vehicles — Some New Definitions and Methodologies http://dx.doi.org/10.5772/61645 17

$$\begin{cases} \frac{\mathbf{d}\hat{I}\_{d}}{\mathbf{d}t} = -m\mathbf{i}\_{d} + \mathbf{B}\_{1}\kappa\gamma\hat{\nu}\_{rd} + \mathbf{B}\_{1}\lambda\_{1}(\mathbf{i}\_{d} - \hat{\mathbf{i}}\_{d}) + \mathbf{B}\_{1}\kappa\frac{L\_{m}}{\mathcal{W}\_{rd}}\hat{\mathbf{i}}\_{q}^{2} + \boldsymbol{\xi}\boldsymbol{V}\_{d} \\ \frac{\mathbf{d}\,\hat{\nu}\_{rd}}{\mathbf{d}t} = -\kappa\hat{\nu}\_{rd} + \kappa L\_{m}\mathbf{i}\_{d} \\ \frac{\mathbf{d}\,\hat{\boldsymbol{\alpha}}\_{r}}{\mathbf{d}t} = \mathbf{B}\_{1}\rho\gamma\hat{\nu}\_{rd}\mathbf{i}\_{q} + \mathbf{B}\_{2}\lambda\_{2}\text{sat}(\{(\tilde{\boldsymbol{\alpha}}\_{r} - \hat{\boldsymbol{\alpha}}\_{r}), \boldsymbol{\phi}\_{l}\} \\ \hat{\boldsymbol{T}}\_{d} = -\mathbf{B}\_{3}\lambda\lambda\_{2}\text{sat}(\{(\tilde{\boldsymbol{\alpha}}\_{r} - \hat{\boldsymbol{\alpha}}\_{r}), \boldsymbol{\phi}\_{l}\} \end{cases} \tag{16}$$

where *ω*˜*<sup>r</sup>* <sup>=</sup> *B*2*λ*1sat((*i <sup>d</sup>* − *i* ^ *<sup>d</sup>* ), *ϕ*2) *piq* denotes the auxiliary state of the rotor speed positioned where the speed measurement originally stays for observer design, *i* ^ *<sup>d</sup>* is an intermediate variable required by the observer to obtain estimation of other parameters so that *i <sup>d</sup>* could be measured, and the function sat(*x*, *ϕ*) is defined below:

$$\text{satt}(\mathbf{x}, \phi) = \begin{cases} \mathbf{x} / \phi & \text{if } \begin{vmatrix} \mathbf{x} \end{vmatrix} < \phi \\ \text{sgn}(\mathbf{x}) & \text{otherwise} \end{vmatrix} \tag{17}$$

The coefficients *B*1, *B*2, and *B*3 in observer (16) are depicted as follows:

*τ<sup>r</sup>* = *L <sup>r</sup>* / *Rr* represents the rotor time constant, and *σ* =1− *L <sup>m</sup>*

coefficient. For analysis simplicity, set *m*=(*L <sup>r</sup>Rs* + *L <sup>m</sup>*

d 1

<sup>ï</sup> =- + <sup>ï</sup>

ï = -

*4.2.2. Design of fixed-boundary-layer sliding mode observer*

ry

*t J*

q

q

*<sup>L</sup> p i <sup>t</sup>*

ky

w k

d d d d d d d

ì

16 New Applications of Electric Drives

ï ï

ï ï

ï ï

ï

ï î d

the frame of the rotating field (d-q) as follows:

where *i*

*a*, *i*

*<sup>b</sup>*, and *i*

derived from system (13) as follows:

Assume that system (13) has the outputs (*y*<sup>1</sup> *y*2)*<sup>T</sup>* =(*i*

dary-layer sliding mode (FLBSM) observer is given below.

zero dynamics about *ψrd* with the output *y*<sup>1</sup> =*i*

*rd*

í = +

*r*

w

*t*

y

q

*ς* =1 / (*σL <sup>s</sup>*), *κ* =1 / *τr*, *ρ* = *pL <sup>m</sup>* / *L <sup>r</sup>*, and the motor model can be written below:

d <sup>2</sup>

*d m*

ï =- + + + +

kgy

g wy

*i L mi p i i V <sup>t</sup>*

 w

*<sup>i</sup> <sup>L</sup> mi p p i i i V <sup>t</sup>*

 w

In this study, the three-phase stator currents are the only required measures, and these are transformed from the three-phase reference frame to a diphasic reference frame, and then to

2 22 ˆˆ ˆ cos( ) cos( ) cos( ) 3 33

*da b c*

<sup>ì</sup> æ ö <sup>ï</sup> = +-++ ç ÷ <sup>ï</sup> è ø <sup>í</sup>

*ii i i*

 qp

*q abc*

*i iii*

ï æ ö <sup>ï</sup> =- - - - + ç ÷ <sup>î</sup> è ø

 qp

<sup>d</sup> <sup>ˆ</sup> <sup>ˆ</sup> <sup>d</sup> <sup>ˆ</sup>

w k

q

*<sup>L</sup> p i <sup>t</sup>*

2 22 ˆˆ ˆ sin( ) sin( ) sin( ) 3 33

*<sup>c</sup>* represent the three-phase stator currents, respectively, and *θ*

*m r q rd*

y

that *ψrd* in system (13) is not observable. Reference [24] has proved that system (13) has a stable

designing an estimator, and then *ω<sup>r</sup>* becomes observable. The sliding mode observer technol‐ ogy is employed to estimate *ωr*. The required voltage d-q frame in the observer is obtained from outputs of voltage feedforward compensation part. The description of the fixed-boun‐

estimated flux angle. Relation between the estimated flux angle and rotor speed could be

*q m*

<sup>ï</sup> =- - - - + <sup>ï</sup>

*m r q rd*

y

*rd m d*

 k

*L i*

*rd q L*

*i T*

*d rd r q q d*

*q r rd r d dq q*

 k

*rd*

y

 k

*rd*

 qp

 qp

= + (15)

*<sup>d</sup>* i*q*)*<sup>T</sup>* . Unfortunately, it could be found

*<sup>d</sup>* and *κ* >0. Thus, it is possible to estimate *ψrd* by

y

 V <sup>2</sup> / *<sup>L</sup> <sup>s</sup><sup>L</sup> <sup>r</sup>* is the leakage magnetic

), *γ* = *L <sup>m</sup>* / (*σL <sup>s</sup>L <sup>r</sup>*),

(13)

(14)

^ denotes the

2

2

 V

*Rr*) / (*σL <sup>s</sup>L <sup>r</sup>*

$$B\_1 = \begin{cases} 0, & \text{if } \left| \hat{\boldsymbol{\psi}}\_{rd} - \boldsymbol{\psi}\_{rd} \right| > \varepsilon \\ 1, & \text{otherwise} \end{cases} \tag{18}$$

$$B\_2 = \begin{cases} 0, & \text{if } \hat{\mathbf{i}}\_d \neq \mathbf{i}\_d \\ 1, & \text{otherwise} \end{cases} \tag{19}$$

$$B\_3 = \begin{cases} 0, & \text{if } \hat{\alpha}\_r \neq \tilde{\alpha}\_r \\ 1, & \text{otherwise} \end{cases} \tag{20}$$

Then, the theorem that the sliding mode observer can exponentially converge will be proved. Assume that the observation error of *ψrd* , *e*<sup>1</sup> =*ψrd* −*ψ* ^ *rd* , exceeds *ε* (i.e., *B*<sup>1</sup> =0), and d*e*<sup>1</sup> / d*t* = −*κe*<sup>1</sup> where *<sup>κ</sup>* >0 can be obtained. A Lyapunov function *Ve*<sup>1</sup> =0.5*e*<sup>1</sup> <sup>2</sup> is designed. Then d*Ve*<sup>1</sup> / d*t* = −*κe*<sup>1</sup> <sup>2</sup> < −*κε* <sup>2</sup> <0 can be obtained. The exponential convergence of *ψ* ^ *rd* to *ψrd* is proved. Therefore, a certain instant *t*<sup>1</sup> can always be found such that when *t*>*t*1, we have *e*<sup>1</sup> ≤*ε* given a sufficiently small real*ε* >0. At this moment, *B*<sup>1</sup> =1. Next *e*<sup>2</sup> =*i <sup>d</sup>* − *i* ^ *<sup>d</sup>* is defined. When *t*>*t*1, the following equation can be derived as:

#### 18 New Applications of Electric Drives

$$\frac{\mathbf{d}e\_2}{\mathbf{d}t} = p\rho\_r \mathbf{i}\_q + \left(\kappa\gamma - \frac{\kappa L\_m \mathbf{i}\_q^2}{\nu\_{rd}\hat{\nu}\_{rd}}\right) e\_1 - \lambda\_\mathbf{i} \text{sat}(e\_2, \phi\_2) \tag{21}$$

To prove the convergence of *i* ^ *<sup>d</sup>* to *i <sup>d</sup>* , a Lyapunov function *Ve*<sup>2</sup> = 1 2 *e*2 2 is used. We set

$$\mathcal{L}\_1 = \max \left\{ \left( \kappa \gamma - \frac{\kappa L\_m i\_q^2}{\nu\_{rd} \hat{\rho}\_{rd}} \right) e\_1 + p o\_r i\_q \right\} + \mathcal{\xi}\_2 \tag{22}$$

and the differential of the Lyapunov function can be delineated as:

$$\frac{\mathbf{d}V\_{e\_2}}{\mathbf{d}t} = \left( \left( \kappa \gamma - \frac{\kappa L\_m \mathbf{i}\_q^2}{\nu\_{\text{ref}} \hat{\nu}\_{\text{ref}}} \right) e\_1 + p \alpha\_r \mathbf{i}\_q - \lambda\_1 \text{sat}(e\_2, \phi\_2) \right) e\_2 \le -\xi\_2 \left| e\_2 \right| = -\sqrt{2V\_{e\_2}} \xi\_2 \tag{23}$$

Consequently, *i* ^ *<sup>d</sup>* converges to *i <sup>d</sup>* in a finite time *t*2. When *t*>*t*2, *B*<sup>1</sup> =1 and d*e*<sup>2</sup> <sup>d</sup>*<sup>t</sup>* <sup>=</sup>*e*<sup>2</sup> =0 are obvious. So according to (21), the new relation establishes below:

$$
\rho \rho \rho\_r \mathbf{i}\_q + \left(\kappa \gamma - \frac{\kappa L\_m \mathbf{i}\_q^2}{\nu\_{nl} \hat{\nu}\_{rd}}\right) \mathbf{e}\_1 - \lambda\_1 \text{sat}(\mathbf{e}\_2, \phi\_2) = \mathbf{0} \tag{24}$$

We set *E* =(*κγ* − *κL <sup>m</sup>i q* 2 *ψrdψ* ^ *rd* )*e*<sup>1</sup> / *pi <sup>q</sup>*, so the auxiliary state of rotor speed is given by:

$$
\rho \tilde{o}\_r = o\_r + \left(\kappa \gamma - \frac{\kappa L\_m \dot{i}\_q^2}{\nu\_{rd} \hat{\nu}\_{rd}}\right) e\_1 / \dot{p} i\_q = o\_r + E \tag{25}
$$

*<sup>E</sup>* and d*<sup>E</sup>* <sup>d</sup>*<sup>t</sup>* exponentially converge to zero due to the convergence of the rotor flux. Thus, the fact that *ω*˜*r* exponentially converges to *ωr* is proved. Then we have:

$$\frac{\mathbf{d}\,\tilde{\rho}\_r}{\mathbf{d}t} = \frac{\mathbf{d}\,\rho\_r}{\mathbf{d}t} + \frac{\mathbf{d}E}{\mathbf{d}t} = \rho\psi\_{rd}i\_q - \frac{1}{J}T\_\perp + \frac{\mathbf{d}E}{\mathbf{d}t} \tag{26}$$

A Lyapunov function *Ve*<sup>3</sup> = 1 2 *e*3 2 is used for proof of convergence of *ω* ^ *<sup>r</sup>* to *ω*˜*<sup>r</sup>* assuming *<sup>e</sup>*<sup>3</sup> <sup>=</sup>*<sup>ω</sup>*˜*<sup>r</sup>* <sup>−</sup>*<sup>ω</sup>* ^ *<sup>r</sup>*. Let

Electric Drives in Alternative Fuel Vehicles — Some New Definitions and Methodologies http://dx.doi.org/10.5772/61645 19

$$\mathcal{A}\_2 = \max \left\{ \rho i\_q e\_1 - \frac{1}{J} T\_L + \frac{\mathbf{d}E}{\mathbf{d}t} \right\} + \xi\_3 \tag{27}$$

given *ξ*<sup>3</sup> >0. Then, the following equation can be obtained:

2

*<sup>d</sup>* , a Lyapunov function *Ve*<sup>2</sup>

<sup>d</sup> sat( , ) <sup>d</sup> <sup>ˆ</sup> *m q*

k

æ ö = +- - ç ÷ ç ÷ è ø

y y

*p i e e*

*rd rd*

2 <sup>1</sup> max 1 2 <sup>ˆ</sup> *m q*

*rd rd*

*e pi e e e V <sup>t</sup>*

=- +- ç ÷ ç ÷ £- =- ç ÷

2

+- - = ç ÷

2 <sup>1</sup> / <sup>ˆ</sup> *m q r r q r rd rd*

*L i*

d d d 1d d dd d

*t tt J t*

= += - + ry

k

æ ö =+ - =+ ç ÷ ç ÷ è ø

y y

fact that *ω*˜*r* exponentially converges to *ωr* is proved. Then we have:

*r r*

 w

w

= 1 2 *e*3 2 *m q*

*L i p i e e* k

*rd rd*

y y

ç ÷ è ø

æ ö

wl

*L i*

ì ü æ ö ï ï = - ++ í ý ç ÷ ç ÷ ï ï î þ è ø

k

y y 1 1 22

*r q*

 x

w

1 1 2 2 2 22 2

 f

*<sup>d</sup>* in a finite time *t*2. When *t*>*t*2, *B*<sup>1</sup> =1 and

1 1 22 sat( , ) 0 <sup>ˆ</sup>

*<sup>q</sup>*, so the auxiliary state of rotor speed is given by:

*e pi E*

<sup>d</sup>*<sup>t</sup>* exponentially converge to zero due to the convergence of the rotor flux. Thus, the

*rd q L E E i T*

is used for proof of convergence of *ω*

w

% (25)

% (26)

 f

l

*r q e*

sat( , ) 2

 x

*e pi*

 f

> = 1 2 *e*2 2

is used. We set

2

d*e*<sup>2</sup>

^

*<sup>r</sup>* to *ω*˜*<sup>r</sup>* assuming

<sup>d</sup>*<sup>t</sup>* <sup>=</sup>*e*<sup>2</sup> =0 are obvious.

x

(21)

(22)

(23)

(24)

l

2

*t*

l

To prove the convergence of *i*

18 New Applications of Electric Drives

2

^

*κL <sup>m</sup>i q* 2

*ψrdψ* ^ *rd* )*e*<sup>1</sup> / *pi*

d ˆ *e m q*

*V L i*

kg

d

Consequently, *i*

We set *E* =(*κγ* −

*<sup>E</sup>* and d*<sup>E</sup>*

*<sup>e</sup>*<sup>3</sup> <sup>=</sup>*<sup>ω</sup>*˜*<sup>r</sup>* <sup>−</sup>*<sup>ω</sup>* ^ *<sup>r</sup>*. Let

A Lyapunov function *Ve*<sup>3</sup>

*r q*

w

^ *<sup>d</sup>* to *i*

*e L i*

 kg

 kg

and the differential of the Lyapunov function can be delineated as:

æ ö æ ö

è ø è ø

2

*rd rd*

So according to (21), the new relation establishes below:

*r q*

 kg

w

w w kg

k

y y

*<sup>d</sup>* converges to *i*

$$\frac{\mathrm{d}V\_{\epsilon\_3}}{\mathrm{d}t} = \left(\rho i\_q e\_1 - \frac{1}{J} T\_L + \frac{\mathrm{d}E}{\mathrm{d}t}\right) e\_3 \le -\xi\_3 \left|e\_3\right| = -\sqrt{2V\_{\epsilon\_3}} \xi\_3 \tag{28}$$

Therefore, the convergence of *ω* ^ *<sup>r</sup>* to *ω*˜*r* in a finite time *t*3 is proved. When *t*>*t*3, *B*<sup>3</sup> =1and d*e*<sup>3</sup> <sup>d</sup>*<sup>t</sup>* <sup>=</sup>*e*<sup>3</sup> =0 are obvious. Thus,

$$
\hat{T}\_{\perp} = T\_{\perp} - J\rho i\_q e\_1 - J\frac{\mathbf{d}E}{\mathbf{d}t} \tag{29}
$$

As mentioned above, both *ω*˜*r* and *<sup>ψ</sup>* ^ *rd* exponentially converge, so *e*1, *E*, and d*E* / d*t* exponentially converge to zero. Therefore, the convergence of *T* ^ *<sup>L</sup>* to *TL* is finally proved.

The reference, experimental, and estimated values of rotor speed of the electric motor are compared in Figure 6. To test the robustness of the observer, +100% resistance change is considered and the -50% load torque and +100% stator leakage inductance changes as well. We can see that the actual speed tracks the reference trajectory well using the proposed controller, and no distinct speed differences between estimation and measurement occurred. Because of small inaccuracy due to the current harmonics, differences between the estimated and measured speed at low speeds seem a little bit bigger than those at high speeds. However, the situation is also acceptable in real operation.

**Figure 6.** Comparison between references and estimated speeds using FBLSM observer

Rotor flux comparison is also studied, given the flux reference. Unlike the rotor speed results, estimated rotor flux is not precisely approaching the reference (see Figure 7), which is however close to actual results. The "actual value" shown in Figure 7 for comparison is obtained from normal flux calculation method through measured voltages and currents and could be considered as actual flux due to proof of incomputable applications. It is clear that the estimated flux with +200% resistance, 50% load torque, or +200% stator leakage inductance is not influenced by these disturbances dramatically.

**Figure 7.** Comparison between references and estimated rotor flux using FBLSM observer
