Kalman Filtering Applied to Induction Motor State Estimation

*Yassine Zahraoui and Mohamed Akherraz*

### **Abstract**

This chapter presents a full definition and explanation of Kalman filtering theory, precisely the filter stochastic algorithm. After the definition, a concrete example of application is explained. The simulated example concerns an extended Kalman filter applied to machine state and speed estimation. A full observation of an induction motor state variables and mechanical speed will be presented and discussed in details. A comparison between extended Kalman filtering and adaptive Luenberger state observation will be highlighted and discussed in detail with many figures. In conclusion, the chapter is ended by listing the Kalman filtering main advantages and recent advances in the scientific literature.

**Keywords:** Kalman filtering, stochastic algorithm, non-linear discrete system, state variables estimation, standard Kalman filter, extended Kalman filter

### **1. Introduction**

Kalman filtering is an algorithm that employs a series of observations over time, containing noise and other inaccuracies, and generates approximations of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe, as in [1].

The Kalman filter is a state observer which detects the presence of measurement noises as well as uncertainties about an unknown dynamic state system, this system is generally assimilated to state noise by stochastic algorithms tending to minimise the variance of the estimation error, as described in [2].

The Kalman filter is suitable for recursive linear filtering of discrete data. It provides an estimation of a state vector or a parameter and its error covariance and variance matrix that contain information about the accuracy of its state variables, as in [3]. The natural presence of noise when an induction machine is driven by an inverter represents a strong argument for the choice of this kind of observers. Its characteristics will relate to the observation of the speed and the components of the rotor fluxes. The only needed measurements are the stator currents. Some state variables will be provided directly by the control law. Thus, the stator voltages will be considered as inputs for the filter. **Table 1** shows a technical comparison between the adaptive observer and the stochastic filter.


representations. These model classes depend directly on the control objectives (torque, speed, position), the nature of the power source of the work repository and

stator fixed reference frame (*α*, *β*) (stationary frame) by assuming the stator

*Y* ¼ **C***:X*

; **B** ¼

;*<sup>K</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>σ</sup> σ:Lm*

Where *J* is the motor inertia,*Tem* is the electromagnetic torque,*TL* is the load

**Figure 2** shows the state space mathematical model of an induction motor.

In this chapter, the process to be observed is an induction motor. Its state is composed of stator currents and rotor fluxes in *α*-*β* reference frame, the motor model and its components are shown in **Figure 3**. The motor model is defined by a

In this chapter, the mathematical model of the machine in use is described in the

*<sup>X</sup>*\_ <sup>¼</sup> **<sup>A</sup>***:<sup>X</sup>* <sup>þ</sup> **<sup>B</sup>***:<sup>U</sup>*

Where *X*, *U* and *Y* are the state vector, the input vector and the output vector,

; *U* ¼ *us<sup>α</sup> us<sup>β</sup>* � �*<sup>t</sup>*

1 *σLs*

0

; *<sup>σ</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>L</sup>*<sup>2</sup>

*m Ls:Lr*

*dt* <sup>¼</sup> *Tem* � *TL* � *<sup>f</sup>:*Ω*<sup>r</sup>* (5)

; *Tr* <sup>¼</sup> *Lr Rr*

0

; *Y* ¼ *is<sup>α</sup> is<sup>β</sup>*

� �*<sup>t</sup>* (2)

<sup>0100</sup> � �

*:* (4)

; **<sup>C</sup>** <sup>¼</sup> <sup>1000</sup>

(1)

(3)

the choice of state vector components (flux or currents, stator or rotor).

(

currents and the rotor fluxes as state variables:

*DOI: http://dx.doi.org/10.5772/intechopen.92871*

*Kalman Filtering Applied to Induction Motor State Estimation*

*X* ¼ *is<sup>α</sup> is<sup>β</sup> ϕr<sup>α</sup> ϕr<sup>β</sup>* � �*<sup>t</sup>*

*Kω<sup>r</sup>*

*K Tr*

�*ω<sup>r</sup>*

*Tr*

*J: d*Ω*<sup>r</sup>*

*<sup>ω</sup><sup>r</sup>* � <sup>1</sup>

*K Tr*

respectively:

**A** ¼

With:

�*λ* 0

*Lm Tr*

0

0 �*λ* �*Kω<sup>r</sup>*

*Lm Tr*

<sup>0</sup> � <sup>1</sup> *Tr*

*<sup>λ</sup>* <sup>¼</sup> *Rs σ:Ls* þ 1 � *σ σ:Tr*

The rotor motion is expressed by:

torque, and *f* is the friction coefficient.

**3. Standard Kalman filter**

*Induction motor state space mathematical model.*

**Figure 2.**

**63**

#### **Table 1.**

*Comparison between ALO and EKF.*
