A new robust speed-sensorless control strategy for high-performance brushless DC motor drives with reduced torque ripple

Control Engineering Practice 24 (2014) 42–54

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

A new robust speed-sensorless control strategy for high-performance brushless DC motor drives with reduced torque ripple S.A.KH. Mozaffari Niapour a,n, M. Tabarraie a, M.R. Feyzi b a b

Private Research Laboratory, 71557 Shiraz, Iran Faculty of Electrical and Computer Engineering, University of Tabriz, 51664 Tabriz, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 20 February 2013 Accepted 15 November 2013 Available online 10 December 2013

This paper presents an analysis, design, and strategy of a high-performance speed-sensorless control scheme for estimating the phase-to-phase trapezoidal back-EMF of BLDC motor drive by means of a novel stochastic deconvolution technique in the H1 setting, named robust stochastic H1 deconvolution filter. In the proposed method, unlike the conventional observer-based approaches, the back-EMF is considered as an unknown input, and no need is felt for the constancy assumption of the rotor position and speed of machine within a short period of the time in the modeling of the BLDC motor which leads to ignoring the back-EMF dynamic at high and variable speed. In addition, since high-speed operation is vital for the motor, an improved approach has also been proposed to reduce the commutation-torqueripple at high-speed for direct torque control (DTC) strategy of three-phase BLDC motor with 120° conduction mode in parallel with the proposed method due to the fact that drive performance intensely downgrades in this mode. & 2013 Elsevier Ltd. All rights reserved.

Keywords: H1 deconvolution filter Brushless dc (BLDC) motor Sensorless control High-performance drive Torque ripple

1. Introduction Brushless dc (BLDC) motors and their drives in vast range of high-performance applications have increasingly been taken into consideration due to their significant characteristics in different powers ranging from microwatts to megawatts. Realization of this important issue in these types of motors is, on the one hand, indebted to the ever-increasing progress of permanent-magnet technologies which has provided accessibility to high efficiency, power density, and torque for the motors, on the other hand, structure and special characteristics of these types of motors has created a basis for easier control, smaller size, and low maintenance in comparison with same-power motors. The BLDC motor drives according to their applications require position sensors such as Hall-effect, resolver, or absolute encoder for accurate implementation of current commutation in stator windings and/or empowerment of desired control. However, installation of these sensors in the motor for meeting the control needs poses several problems for the motor-drive system. Among the most important of these drawbacks are as follow: (1) these sensors increase the size and expense of the motor due to the fact that they require a particular mechanical arrangement for installation in such a way that this issue will be most effective where restricted-space is of utmost importance; (2) they can reduce reliability and system robustness

n

Corresponding author. Tel./fax: þ 98 711 735 38 39. E-mail address: [email protected] (S.A.KH. Mozaffari Niapour). 0967-0661/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2013.11.014

because of extra components and wiring; (3) sensitivity of these sensors to temperature, ambient light and contamination lead to the limitation of motor operation; (4) difficulty of installation and maintenance of these sensors in harsh working environment in which motor has to deal with high vibration and high temperature. Thus, considering the disadvantages mentioned above and powerful and economical accessibility of today's microprocessors, it is worthwhile to replace sensorless control methods of the BLDC motor drives by the drives with position-sensor. In the two recent decades, considerable efforts have been made for optimizing sensorless control techniques from the viewpoints of the BLDC motor drive (Chen & Cheng, 2007; Damodharan, Sandeep, & Vasudevan, 2008; Damodharan & Vasudevan, 2010; Fakham, Djemai, & Busawon, 2008; Iizuka, Uzuhashi, Kano, Endo, & Mohri, 1985; Kim & Ehsani, 2004; Shen & Tseng, 2003; Shao, Nolan, Teissier, & Swanson, 2003; Terzic & Jadric, 2001; Zhang & Wang, 2011). In reference Iizuka et al. (1985), the terminal voltage sensing method which is based on float phase voltage sensing with respect to virtual neutral point was originally proposed in order to detect zerocrossing point (ZCP) of the back-electromotive-force (back-EMF) waveform. However, when using techniques of chopping drive in this method, neutral point is no longer a standstill point and this point's potential varies between zero and dc-bus voltage. A compensation for the introduced phase delay of LPF in Shen and Tseng (2003) has been reported by using frequency-independent phase shifter which can shift ZCP of input signal by a known phase delay. In Shao et al. (2003) the direct back-EMF detection approach which is not in need of sensing or reconstructing motor neutral point and uses voltage difference of unexcited phase and power ground of

S.A.KH. Mozaffari Niapour et al. / Control Engineering Practice 24 (2014) 42–54

dc-link voltage for direct back-EMF information elicitation has been analyzed. In this method, sensing circuit can only operate during freewheeling period (off-time of PWM) with a minimum off-time 3 ms sampling which results in that the maximum duty cycle of PWM be lower than 100%. Another direct back-EMF detection approaches to extend duty cycle control from 5% to 95% has been proposed in Chen and Cheng (2007) by means of measurement of line voltages without considering the back-EMF. Under an ideal assumption that there exists no freewheeling current in nonconducted phase, recently a simple position-sensorless technique for detecting the back-EMF ZCPs in Damodharan and Vasudevan (2010) and starting of the motor in Damodharan, Sandeep, and Vasudevan (2008) are presented. This method emphasizes on the issue that by measuring difference of line voltages in the motor terminals, it will be possible to create amplified version of back-EMF in order to extend its ZCPs detection at lower speeds. Unfortunately, the considered assumptions in Damodharan and Vasudevan (2010) and Damodharan et al. (2008) methods can not always come true; in fact, using these methods, there may be a possibility that freewheeling currents in non-conducted phase exist both during normal commutation period and during un-commutated period in such a way that their amplitude, duration, and location of effectivity can differ according to the type of switching method. In Zhang and Wang (2011) a method based on proper PWM strategy (PWM-ONPWM) is offered in order that overcome the disadvantages in Damodharan and Vasudevan (2010) and Damodharan et al. (2008)). Although by using this method can realize good motor performance over a much speed range, there is no wonder it results in a tiny variety in application of BLDC motor drives. In Kim and Ehsani (2004) a speed-independent new physical concept has been proposed to detect commutation instants by utilizing speedindependent position function. However, since this function depends on calculations of current derivatives, this method, firstly, requires digital implementation, and, secondly, due to the extreme sensitivity of the method mentioned to measurement noises and machine parameters, this issue inevitably leads to a disorder in the determination of commutation points. Nevertheless, the strategies above-mentioned operate only in a bounded speed range and are considered to be among open-loop speed-sensorless methods, but observer-based methods are mainly considered to be among closed-loop speed-sensorless techniques which are more robust and are of high-accuracy with respect to uncertainty in parameters and disturbances. Therefore, observer-based drives for high-performance applications can be the best and safest choice. In Terzic and Jadric (2001), an extended Kalman filter (EKF) has been used for instantaneous estimation of system state variables and stator resistance by using line actual voltages and currents and utilizing complete model of the BLDC motor. Unfortunately, the most basic problem for EKF is that its robustness against parameter detuning is too weak. In addition, determining the values of noise covariance matrices is difficult in them, and as this method is based on having accurate knowledge of practical system noises, the parameters determined by simulation should still be adjusted in practical system which increases the inconveniences for EKF. In Fakham et al. (2008) a sliding-mode observer has been presented by means of the stator line voltages and currents and electrical motor model to estimate the phase-tophase back-EMF of the BLDC motor. In this respect, it should be pointed that a continuous approximation has been used for switching sign function by applying sliding-mode observer to drive system in order to reduce chattering effect in the method mentioned, which results in that, on the one side, it reduces the accuracy of observer in estimating state variables, and, on the other side, the applied approximation is no longer effective in the reduction of chattering effect when a high-level noise exists in the system output.

43

In this paper, a novel robust stochastic H1 deconvolution filter has been proposed for sensorless BLDC motor drives with the aim of improving the robustness and dynamic performance in a vast speed range of the conventional aforementioned methods. This innovative deconvolution approach is the first known study that is utilized for linear stochastic systems which contain a known input as well as state-, input- and exogenous disturbance-dependent noise. The proposed deconvolution filter is also extended to the case in which the deterministic part of state space model matrices and the covariance matrices of multiplicative noises are uncertain but reside in a given polytope. To be more precise, the proposed observer is used to estimate phase-to-phase trapezoidal back-EMF of the BLDC motor by utilizing actual line voltages and currents in such a way that the rotor speed and position can easily be obtained by the backEMF estimation. On the other hand, in the proposed filter the backEMF dynamic has been considered for gaining access to better high and variable speed performance in contrast with the conventional observer-based approaches which have assumed that back-EMF variations are very slow in the modeling of the BLDC motor. Owing to the fact that high-speed operation is one of the most important situations that a motor should successfully undergo in order to overcome the big commutation-torque-ripple created under this condition, an improved approach has also been proposed to reduce the commutation-torque-ripple at high-speed for direct torque control (DTC) strategy of three-phase BLDC motor with 1201 conduction mode in parallel with the proposed method. Utilizing this improved approach, the electromagnetic torque remains constant without feeling any need to an accurate calculation of the duration of the applying effective complementary controller of the varying input voltage, and the commutation-torque-ripple, which derives from the uncontrollable conduction of freewheeling diodes at highspeed, is minimized. Likewise, in this paper, basic principles of designing the proposed method has analytically been studied, and following that, modeling of the BLDC motor and the proposed overall drive system configuration have been presented in detail complete with principles of its operation. At the end, the proposed system has been simulated under different operating conditions of the BLDC motor by computer simulation, and the performance of the proposed control strategy have been evaluated via simulation results from four perspectives: steady-state accuracy, dynamic performance, parameter and noise sensitivity, and low-speed-operation performance. Simulation results verify that the proposed system has very good robustness and dynamic performance, high estimation accuracy, and low commutation-torque-ripple under different operating conditions in spite of the existence of measurement noise and electric parameter uncertainty. Therefore, the proposed strategy with its strong robustness and reduced commutation-torque-ripple makes it possible for the drive to enable the motor to undergo a stable tensionless operation without confronting any problem at high- and low-speeds and braking regime.

2. Modeling of BLDC motor The BLDC motor model is described in the stationary reference frame abc by the following equations (see Mozaffari Niapour., Tabarraie, & Feyzi, 2012): 30 2 2 3 21 3 32 0 0 ia  ib ia  ib R 0 0 L 6 7B 6 d6 7 6 7 7 1 7 4 ib  ic 5 ¼ 6 4 0 L 0 5@  4 0 R 0 54 ib  ic 5 dt ic  ia ic  ia 0 0 R 0 0 1L 2

3 2 31 va vb ea  eb 6 e  e 7 6 v  v 7C 4 b c 5þ4 b c 5A ec  ea vc  va

ð1Þ

44

S.A.KH. Mozaffari Niapour et al. / Control Engineering Practice 24 (2014) 42–54

where va , vb , vc , ia , ib , ic , ea , eb , ec , R and L are the phase voltages, currents, back-EMFs, resistance, and inductance respectively. As regards to the fact that equations related to phases ða  bÞ, ðb  cÞ, and ðc  aÞ are similar, the deconvolution filter is designed for phase ða  bÞ and then utilized for phase ðb  cÞ in a similar way for facility purposes. It should be noted that since the system has been considered in a balanced way and for achieving back-EMF between the two phases c and a ðeca Þ, we can readily utilize the equation eca ¼  ðeab þ ebc Þ. The phase ða  bÞ equations in the state space are written as follows by considering process and measurement noises: R 1 1 x_ ¼  x  ω þ u1 þ ζ 1 L L L y ¼ x þ ζ2

ð2Þ

where x ¼ ia ib and u1 ¼ va  vb are the state and input variables respectively, y is the actual output, and ω ¼ ea  eb is the unknown input signal. ζ 1 and ζ 2 are the uncorrelated zero-mean white noises that satisfy Efζ 1 ðtÞζ 1 ðt  τÞg ¼ Q δðτÞ Efζ 2 ðtÞζ 2 ðt  τÞg ¼ RδðτÞ where Q and R are the covariances of noises ζ 1 and ζ 2 respectively. 3. Robust stochastic H1 deconvolution filter The aim of deconvolution filter is the estimation of the unknown input signal of a system by means of actual outputs and known inputs. Deconvolution problem has comprehensive use in environments such as equalization, image restoration, speech processing, fault detection, see Peng and Chen (1997), Yaesh and Shaked (2000) and You, Wang, and Guan (2011) and the references therein. H1 deconvolution filtering has received a widespread attention in two decades ago. One of their significant advantages is the fact that it requires no statistical knowledge about the exogenous disturbances, and the disturbances should only have bounded energy. Likewise, these filters are robust against the noises and parameter uncertainties. Two types of uncertainties have been considered in the literature. The first type is the deterministic uncertainties which are commonly posed in two forms: norm-bounded uncertainty and convex-polytopic uncertainty. Polytopic uncertainty is utilized exhaustively in robust control and estimation of uncertain systems. In these types of uncertainties, deterministic parameters of these systems are not known thoroughly, and it is assumed that they lie in a given polytope (Mozaffari Niapour et al., 2012). The second type is the stochastic uncertainties which have been considered to be multiplicative noise or Markov jump perturbations. In the case of stochastic systems with multiplicative noise, the parameter uncertainties are modeled as white noise processes (see Gershon, Limebeer, Shaked, & Robust, 2001; Ugrinovskii & Petersen, 1999). Markov jump systems are efficiently used to model the systems which sudden variations occur in their structures (Dragan & Morozan, 2004). Stochastic uncertainty in the system under study can be considered according to inductance model. Motions of magnetic materials which are close to each other can induce rapid changes in the inductance value. To be more precise, inductance Lsto ¼ Lsto ðtÞ can be modeled in the form 1 of Lsto ¼ L  1 þ Lr 1 η_ ðtÞ (Ugrinovskii & Petersen, 1999) where η_ ðtÞ is a zero-mean Gaussian white noise with unity covariance. Furthermore, L is the phase inductance and the value of Lr is obtained by estimation of reciprocal inductance covariance. The model recalled for the inductance creates state-, input- and exogenous disturbance-dependent noise in state space model.

The stochastic bounded real lemma (BRL) (Hinriechsen & Pritchard, 1998) is of significant use in designing H1 filtering for the linear stochastic systems. Based on this important lemma, a proper filter has been proposed for the reduced-order H1 estimating in Xu and Chen (2002) which estimates a linear combination of state and exogenous input signals and in Wei, Wang, Shu, and Fang (2007) an H1 deconvolution filter has also been designed for the linear stochastic systems with state-multiplicative noise and deterministic interval uncertainties. In this part of the paper, by using the stochastic BRL (Hinriechsen & Pritchard, 1998) and linear matrix inequalities (LMIs) techniques, we design a novel H1 deconvolution filtering approach for linear stochastic system with known input signal as well as state-, input- and exogenous disturbance-dependent noise. In this filter, the worst-case energy gain from the exogenous input signals to the estimation error is bounded by a prescribed level. This filter is also extended to the case in which the deterministic component of state space model matrices and the covariance matrices of multiplicative noises are uncertain but reside in a convex-bounded polytopic domain. Notation: the superscript T shows matrix transposition. ℜn determines the n-dimensional Euclidean space, and ‖ U ‖ is the Euclidean vector norm, and ℜnm is a set of all the n  m real matrices. The notation P 4 0 for P nn means that P is symmetric and positive definite. Ef U g stands for expectation. The symbol n is used for the symmetric terms in a symmetric matrix. By L2 ðΩ; ℜk Þ we denote the space of square-integrable ℜk -valued functions on the probability space ðΩ; ϑ; Ψ Þ, where Ω is the sample space, ϑ is a s-algebra of subsets of the sample space, and Ψ is a probability measure on ϑ. By ðϑt Þt 4 0 we denote an increasing family of 2 s-algebras ϑt  ϑ. Likewise, let L~ ð½0; 1Þ; ℜk Þ denote the space of non-anticipative stochastic process f ð UÞ ¼ ðf ðtÞÞt A ½0;1Þ in ℜk with R1 respect to ðϑt Þt A ½0;1Þ which satisfies ‖f ‖2L~ ¼ Ef 0 ‖f ðtÞ‖2 dtg ¼ R1 2 2 0 Ef‖f ðtÞ‖ gdt o1. It should be mentioned that stochastic differential equations are of Itô type. 3.1. Bounded real lemma for linear stochastic systems with multiplicative white noise We consider the following linear stochastic system with stateand exogenous disturbance-dependent noise: dxðtÞ ¼ ½AxðtÞ þB1 ωðtÞdt þ ½G1 xðtÞ þ G2 ωðtÞdβ ðtÞ; xð0Þ ¼ x0 zðtÞ ¼ C 1 xðtÞ þ D11 ω

ð3Þ

n

where x A ℜ is the system state vector, and x0 represents the 2 initial state. ωðtÞ A L~ ð½0; 1Þ; ℜq Þ is the exogenous input vector, and m z A ℜ is the objective vector. A, B1 , G1 , G2 , C 1 , and D11 are the constant matrices with appropriate dimensions. β ðtÞ is a zeromean real scalar Wiener process which satisfies EfdβðtÞg ¼ 0; Efdβ ðtÞ2 g ¼ dt

ð4Þ

In fact, G1 β_ and G2 β_ can be interpreted as white noise parameter perturbations in the matrices A and B1 respectively by adopting the fact that white noise signals are formally the derivatives of Wiener processes. The following performance index is considered: J E :¼ ‖z‖2L~  γ 2 ‖ω‖2L~ 2

ð5Þ

2

which γ 4 0 is a given scalar. In this part of the paper, we use the following definition: Definition 1. The system (3)  with ωðtÞ ¼ 0 is called asymptotically mean-square stable if lim E ‖xðtÞ‖2 ¼ 0, for all x0 A ℜn . t-1

S.A.KH. Mozaffari Niapour et al. / Control Engineering Practice 24 (2014) 42–54

45

pþq ω~ ðtÞ A L~ ð½0; 1Þ; Þ.iTTaking into account Eqs. (9) and (10) and h ℜ denoting ξ ¼ xT x^ T , the following augmented system, which 2

Now, we give the following important lemma which is celebrated to the stochastic bounded real lemma.

shows the filtering error dynamic, will be obtained:

Lemma 1. (Hinriechsen & Pritchard, 1998) The system (3) is asymptotically mean-square stable, and J E of (5) is negative for all 2 nonzero ωðtÞ A L~ ð½0; 1Þ; ℜq Þ under zero initial condition if and only if there exists Q 4 0, which satisfies " T # A Q þ Q A þ G1 T Q G1 þ C T1 C 1 Q B1 þ G1 T Q G2 þ C T1 D11 o0 ð6Þ B1 T Q þGT2 Q G1 þ DT11 C 1 GT2 Q G2 þDT11 D11  γ 2 I

~ dt þ ½G~ 1 ξ þ G~ 2 ω ~ dβ; dξ ¼ A~ ξdt þ B~ ω

~ω ~ z~ ¼ C~ ξ þ D

ð13Þ

where " A ~ A¼ B1f C 2

3.2. Design of stochastic H1 deconvolution filter

# " # " # 0 B B1 B2 ~ ; B¼ ¼ Af B1f D21 B2f Bf " #     G1 0 G2 G3 G2 ; ; G~ 2 ¼ ¼ G~ 1 ¼ 0 0 0 0 0 h i h i C~ ¼ C 1  Df C 2  C f ; D~ ¼ D11  Df D21 0

Now we consider the following asymptotically mean-square stable system with stochastic uncertainties and a known input signal:

Theorem 1. Consider the system (7a)–(7c) and the deconvolution filter (8). For γ 40 the following results hold:

dxðtÞ ¼ ½AxðtÞ þ B1 ωðtÞ þ B2 rðtÞdt

i) the system (13) is asymptotically mean-square stable, and J S is 2 ~ ðtÞ A L~ ð½0; 1Þ; ℜp þ q Þ under zero initial negative for all nonzero ω condition, if and only if there exist R ¼ RT A ℜnn , W ¼ W T A ℜnn , Z A ℜnr , Z A ℜnp , S A ℜnn , T A ℜmn , and Df A ℜmr such that

þ ½G1 xðtÞ þ G2 ωðtÞ þ G3 rðtÞdβðtÞ; xð0Þ ¼ x0 yðtÞ ¼ C 2 xðtÞ þ D21 ωðtÞ zðtÞ ¼ C 1 xðtÞ þ D11 ωðtÞ

ð7a–cÞ

∑ðR; W; Z; Z; S; T; Df Þ o0

whose description is similar to that of system (3), in addition, r A ℜp is the known deterministic input signal and y A ℜr is the 2 6 6 6 6 6 6 ∑ :¼ 6 6 6 6 6 6 4

AT W þ C T2 Z T þST

RB1

RB2

GT1 R

GT1 W

n

 S  ST

WB2 þZ 0

0 GT2 R

0

TT

GT2 W

DT11  DT21 DTf

n

n

n

n

n

 γ 2 Ip

GT3 R

GT3 W

0

n

n

n

n

R

0

0

n

n

n

n

n

W

0

n

n

n

n

n

n

 Im

~ dt þ ðG1 x þ G2 ω ~ Þdβ dx ¼ ½Ax þ Bω

ð9Þ

where B2 ; G2 ¼ ½ G2

G3 

In the same way, in (8) by substituting dy of (7b) we obtain ~ dt dx^ ¼ ½Ax^ þ B1f C 2 xdt þ Bf ω

ð10Þ

where Bf ¼ ½ B1f D21

7 7 7 7 7 7 7 7 7 7 7 7 5

Af ¼  W  1 S;

B1f ¼  W  1 Z;

B2f ¼  W  1 Z;

Cf ¼ T

ð16Þ

needless to say, Df is obtained of (15). Proof. This theorem can be proved through a trend similar to those used in Gershon et al, (2001) and Yaesh and Shaked (2000). i) According to Lemma 1, necessary and sufficient condition for system (13) to be asymptotically mean-square stable, and J S to be 2 ~ ðtÞ A L~ ð½0; 1Þ; ℜp þ q Þ, is that there exist negative for all nonzero ω Q 4 0 that satisfies the following inequality: 2 T 3 T T T T A~ Q þ Q A~ þ G~ 1 Q G~ 1 þ C~ C~ Q B~ þ G~ 1 Q G~ 2 þ C~ D~ 4 5o0 ð17Þ T T T T ~  γ 2 Ip þ q ~ TD B~ Q þ G~ 2 Q G~ 1 þ D~ C~ G~ 2 Q G~ 2 þ D Applying Schur complement, we obtain 3 T T T A~ Q þ Q A~ Q B~ G~ 1 Q C~ 6 7 T 6 ~T 7 6 7 n  γ 2 I p þ q G~ 2 Q D 6 7o0 6 n n Q 0 7 4 5 n n n  Im 2

B2f 

Denoting z~ ðtÞ ¼ zðtÞ  z^ ðtÞ

ð11Þ

and for a given scalar γ 4 0, the following cost function is defined: ~ ‖2~ J S :¼ ‖z~ ‖2L~  γ 2 ‖ω L 2

3

ii) If (15) is satisfied, the deconvolution filter parameters can be extracted using the following equations:

ð8Þ

where x^ A ℜn and z^ A ℜm . in (7a) we substitute the exogenous input vector ωðtÞ with the augmented exogenous input vector ω~ ðtÞ ¼ ½ ωðtÞT rðtÞT T ; then have

B ¼ ½ B1

C T1  C T2 DTf  T T

WB1 þ ZD21  γ 2 Iq

dx^ ¼ Af x^ dt þ B1f ydt þ B2f rdt; x^ ð0Þ ¼ 0; z^ ¼ C f x^ þ Df y

ð15Þ

where

RA þ AT R

measurement vector. B2 , G3 , C 2 and D21 are the constant matrices with appropriate dimensions. Now we consider the following deconvolution filter to estimate zðtÞ:

ð14Þ

2

ð12Þ

The aim of stochastic H 1 deconvolution filter is to seek for estimation z^ ðtÞ from the zðtÞ over the infinite time horizon ½0; 1Þ in such a way that J S of (12) is negative for all nonzero

Q and Q  1 are partitioned in the form of Q :¼   Y N , where we require that X 4 Y  1 . Q  1 :¼ T V N

ð18Þ



X MT

M U

 and

46

S.A.KH. Mozaffari Niapour et al. / Control Engineering Practice 24 (2014) 42–54

 Defining J :¼

Y NT

In 0



  and J~ :¼ diag J; I p þ q ; J; I m , (18) is pre-

In order to design the stochastic H 1 deconvolution filter, which estimates eab as the unknown input, the state space model (2) including state-, input- and exogenous disturbance-dependent 1 noise terms resulted from inductance model ðLsto ¼ L  1 þ Lr 1 η_ ðtÞÞ can be written in the form of (7a)–(7c) as the following:

and post-multiplied by J~ and J~ . Carrying out some multiplications and through the substitution of T

Z :¼ MB1f ;

Z~ :¼ C f N T ;

Z :¼ MB2f ;

Z^ :¼ MAf N T

ð19Þ

^ þ B2 u1 dt þ ðG1 x þ G2 ω ^ þG3 u1 Þdη dx ¼ ½Ax þ B1 ω

we obtain 2

AY þYAT

6 T 6 A þ XAY þ ZC 2 Y þ Z^ 6 6 6 BT1 6 6 6 BT2 6 6 G1 Y 6 6 6 XG 1Y 4 C 1 Y  Df C 2 Y  Z~

n

n

n

n

n

n

XA þ AT X þ C T2 Z T þZC 2

n

n

n

n

n

BT1 X þ DT21 Z T

 γ 2 Iq

n

n

n

n

T BT2 X þZ

0

 γ 2 Ip

n

n

n

G1

G2

G3

Y

n

n

XG1

XG2

XG3

I

X

n

C 1  Df C 2

D11 Df D21

0

0

0

 Im

Defining ϒ as below (" # " R 0 R ϒ :¼ diag ; Iq ; Ip ;  R In R

0 In

#

T :¼ Z~ R;

R :¼ Y  1 ;

) ; Im

ð21Þ

W ¼ X R

ð22Þ

with pre- and post-multiplying (20) by ϒ and ϒ , respectively, (15) is achieved. If there exists a solution to (15), from (19) we obtain that T

^ T; Af ¼ M  1 ZN B2f ¼ M  1 Z;

B1f ¼ M  1 Z;

C f ¼ Z~ N  T

ð23Þ

Applying (23) in the transfer function matrix of the deconvolution filter, which is obtained of (8), we find that " # " #   B1f H z^ y ðsÞ Df ¼ C f ðsI  Af Þ  1 þ H deconvolver ðsÞ ¼ B2f H z^ r ðsÞ 0     D Z f T  1 ^ þ ¼ Z~ ðsMN  ZÞ Z 0     Df 1 Z ~ ^ ¼ Z ðsðI n  XYÞ  ZÞ þ ð24Þ Z 0 now owing to (22), H deconvolver is obtained as the following:     Df Z þ H deconvolver ðsÞ ¼ TðsðR  XÞ  SÞ  1 Z 0 " #   1 ðR  XÞ Z Df 1 1 ¼ TðsI  ðR  XÞ SÞ þ 0 ðR  XÞ  1 Z

ð25Þ

considering the relation above, (16) is obtained. Due to the fact that LMIs are affine in the system parameters, Theorem 1 can be extended for the case which these parameters are uncertain. Assume that A, B1 , B2 , C 2 , D21 , G1 , G2, and G3 reside in the polytope as follows: s

Ω :¼ ðA; B1 ; B2 ; C 2 ; D21 ; G1 ; G2 ; G3 Þ ¼ ∑ μi Ωi ; μi Z 0; i¼1

7 7 7 7 7 7 7 7 o0 7 7 7 7 7 5

ð20Þ

^ y ¼ C 2 x þD21 ω ^ z ¼ C 1 x þD11 ω

and substituting ^ S :¼ ZR;

3

s

∑ μi ¼ 1

ð26Þ

i¼1

where Ωi :¼ ðAi ; B1i ; B2i ; C 2i ; D21;i ; G1i ; G2i ; G3i Þ; i ¼ 1; …; s are the polytope vertices. Corollary 1. Consider the system (7a)–(7c) and the deconvolution 2 ~ ðtÞ A L~ ð½0; 1Þ; ℜp þ q Þ filter (8). For a given γ 4 0 and for all nonzero ω and for all ðA; B1 ; B2 ; C 2 ; D21 ; G1 ; G2 ; G3 Þ A Ω, J S is negative if (15) is satisfied by a single set of ðR; W; Z; Z; S; T; λÞ for all the polytope vertices. In the latter case, filter matrices are obtained via (16).

ð27Þ

where h i pffiffiffiffi R 1 Q 0 ; B2 ¼ ; A ¼  ; B1 ¼  1L L L h i R 1 1 G1 ¼  ; G2 ¼  Lr 0 0 ; G3 ¼ ; L Lr h r pffiffiffi i D21 ¼ 0 0 R ; D11 ¼ ½ 1 0 0  h ^ ¼ ω and ω

ζ1 ζ2

iT

with Ef½ ζ 1 ðtÞ

C 1 ¼ 0;

C 2 ¼ 1;

ζ 2 ðtÞ T ½ ζ 1 ðt  τÞ ζ 2 ðt 

τÞg ¼ I 2 δðτÞ, namely, we have embedded the covariance matrices of noises in B1 and D21 . In addition, η_ ðtÞ is not correlated with the other noise signals. Since the covariance matrices of the additive noises, i.e., Q and R are not available in practice, the values of the covariance matrix elements are used as tuning parameters. For simplicity and avoiding computational complexity, the covariance matrices are chosen diagonal and constant, and they are tuned by trial and error (Mozaffari Niapour et al., 2012). 4. Proposed commutation-torque-ripple minimization method In this section an improved approach has been proposed for reducing commutation-torque-ripple at high-speeds for DTC strategy of the three-phase BLDC motor with 1201 elec. conduction mode, and also the commutation current and torque ripple have been studied in similar way to the method presented in Nam, Lee, Lee, and Hong (2006), and only its results have been provided. The proposed method, in fact, makes use of a combination of the methods presented in Liu, Zhu, and Howe (2007) and Nam et al. (2006) in such a way that the hybridization of these two methods leads to that; first, there is no need to an accurate knowledge of the duration of precise applying of the controller during the commutation interval, second, the complimentary controller used in the commutation interval is different from controllable threephase switching mode, and an effective method of varying input voltage has been used instead. Fig. 1 illustrated the current flow paths during the commutation between phases c and a, i.e., when the switching states change from (000110) to (100100), in such a way that these logical values describe the switching states (“0” represents off-switch and “1” represents on-switch) of the upper and lower switches for phases a, b, and c respectively. In order to analyze the varying input voltage method, the current commutation interval is divided

S.A.KH. Mozaffari Niapour et al. / Control Engineering Practice 24 (2014) 42–54

47

Fig. 1. Switching states and current flow of the BLDC motor. (a) Before current commutation. (b) During current commutation with freewheeling region. (c) After current commutation with build-up region.

into freewheeling and build-up regions. While the commutation is occurring from phase c to phase a, current ripple will not be generated in case current of phase a is constant. In the other words, the commutation-torque-ripple can be reduced by keeping the un-commutated phase current constant in freewheeling region and applying of varying direct current voltage instead of dc voltage with fixed amplitude. Supposing the commutation occurring from phase c to phase a, as it can be understood from Fig. 1, all the three phases are conducting the current during the commutation interval as long as the phase current, whose switch has been off, does not decay to zero. In this interval, a getting-off phase of the motor lies parallel to the getting-on phase, that is why the off-phase freewheeling diode current drops faster than on-phase current, the issue which causes the generation of the commutation-torque-ripple. As the off-phase current is reaching zero, the build-up region starts in the two on-phases, in this case, if the electrical period of the BLDC motor rotation is shorter than the time constant of the on-phase current, the current can not reach its steady-state value bounded by resistance, and this region will also be a factor for generation of the current ripple and torque. In this method (Nam et al., 2006) the instantaneous currents and voltages can be calculated by circuit analysis in the Laplace domain as well as the applying of Kirchhoff's law in each of the equivalent circuits. Finally, the current ripple and commutation torque could be reduced by the applying of the freewheeling varying input voltage during the freewheeling interval which have been expressed in Eqs. (28) and (29) respectively. V dc  f ree ¼ 3RI 0 þ 3E þ E′  3RI 0 þ 4E

ð28Þ



3V dc  f ree þ 2E′ L T f ree ¼  ln R 9RI 0 þ 3V dc  f ree þ 2E′

ð29Þ

V dc  f ree and T f ree are the applying voltage in the freewheeling region and the time of the freewheeling region respectively. I 0 is 0 the initial current value, E is the back-EMF value at the commutation starting for ongoing phase, and E is the peak value of the back-EMF waveform. As it can be inferred from (29), T f ree , on the one hand, is sensitive to the motor parameters, and, on the other hand, has resulted under an ideal assumption that no back-EMF current exists in the silent phase; therefore, its precise determination seems impossible, and its calculation is difficult. To cope with this problem, the time during which the controller of the varying input voltage should be utilized by the criterion of the torque error ΔT, which has been presented in Liu et al. (2007), is determined in the proposed method. In this method during the commutation, when ΔT r ΔT n , the varying input voltage controller is not applied; rather, when ΔT 4 ΔT n the controller mentioned starts operating in parallel with the proposed sensorless method for reducing the commutation-torque-ripple. Therefore, using this improved approach, the electromagnetic torque will be constant without feeling any need to calculate the duration of the effective

complementary controller of the varying input voltage, and the commutation-torque-ripple, which derives from the conduction of uncontrollable freewheeling diodes at high-speeds, minimizes.

5. Description of the overall drives system configuration The key issue in the DTC method in the constant torque region for driving a BLDC motor is correct torque estimation. For a BLDC motor whose back-EMF waveforms are non-sinusoidal (trapezoidal), the electromagnetic torque can be calculated by means of stationary reference frame as the following form (Ozturk, Alexander, & Toliyat, 2010): " # dψ r β 3 p dψ r α 3p 1 T em ¼ isα þ isβ ¼ ½eα isα þeβ isβ  ð30Þ 2 2 dθ e 2 2 ωe dθ e where eα ; eβ , isα ; isβ are the motor back-EMFs and stator currents in the stationary reference frame (αβ-axis) respectively. Since the balanced systems in αβ reference frame require no zero sequence term, the torque equation (30), which includes ab  bc reference frame variables, can be obtained as the following by means of the transformation of the proposed αβ  ðab  caÞ in Ozturk and Toliyat (2011) and Eq. (30). Transformation of αβ  ðab  bcÞ can be written as the following matrix form: " # 0 2 1 1" # xα xab 3 3 pffiffi A ¼@ ð31Þ xβ xbc 0 33 where xα and xβ are the stationary reference frame components, xab ¼ xa  xb and xbc ¼ xb  xc are ab bc reference frame components, and x in (31) represents the current and back-EMF in (30). Thus, using transformation (31), substituting its parameters mentioned in (30), and some algebraic manipulations the electromagnetic torque equation (30) can be rewritten as the following form in ab  bc reference frame:   1 p ½2eab þ ebc  ½2eca þ ebc  T em ¼ ð32Þ iab þ ibc 32 ωe ωe The overall block diagram of the proposed speed-sensorless control drive strategy has been sketched in Fig. 2. In this controlling strategy, the five effective key blocks in order to provide the proposed sensorless control scheme are: speed controller, estimators of the back-EMF, speed, position, and electromagnetic torque of the motor, torque comparator, varying input voltage complementary controller for reducing the commutation-torque-ripple at high-speed, and switching logic that is determinant of the inverter switching status. The block outputs of the proposed deconvolution filter provide an estimation of the phase-to-phase back-EMFs for detecting rotor position and speed estimation in such a way that B1f and B2f values as well as Af , Df , C f in this block are obtained via solving matrix inequality according to (15) and (16) respectively. In order to detect commutation points and 1201 electrical conduction

48

S.A.KH. Mozaffari Niapour et al. / Control Engineering Practice 24 (2014) 42–54

Fig. 2. Overall block diagram of the proposed speed-sensorless control drive strategy.

Table 1 Selection of the switching status for the DTC method in the constant torque region: torque increase ðTIÞ, torque decrease ðTDÞ, and no-change in flux ðFÞ. Sector Tst

F

TI TD

θ1

θ2

θ3

θ4

θ5

θ6

V2(001001) V5(000110)

V3(011000) V6(100100)

V4(010010) V1(100001)

V5(000110) V2(001001)

V6(100100) V3(011000)

V1(100001) V4(010010)

mode signals for each active phase the following trend could be followed. If the desired phase is considered x1 , and the motor nþ1 nþ2 sequence cycle direction x1 ¼ a; b; c - x2 ¼ b; c; a - x3 ¼ c; a; b is focused; then it can be claimed that the conditions exn xn þ 1 40 and exn þ 2 xn o 0 mean the positive conduction of the desired phase, and the opposite of the condition mentioned should be for the negative conduction of the relevant phase. Instantaneous speed of the motor can easily obtained out of a mathematical relationship of speed and back-EMF which itself can be achieved out of the proportion between the maximum absolute value of the phase-tophase back-EMF, each 601elec. of which results from eab ; eca ; ebc and rotor position, and double back-EMF constant. At the same time, speed feedback derived from the proposed observer is compared with speed reference, and then torque command is generated through PI speed controller. The torque estimator is responsible for the generation or estimation of the generated torque feedback signal, and the torque comparator, which includes hysteresis controller block, is utilized for comparing torque command with its values. Finally, the appropriate command for the electromagnetic torque is obtained by comparing it with its corresponding demanded values via hysteresis controller. While the motor is undergoing the high-speed operation, the key, shown in Fig. 2, changes its situation from status 1 to 2 for the parallel operation of the proposed complementary controller to reduce commutation-torque-ripple with the proposed strategy. As it can be drawn from Table 1, the inverter switching pattern is determined at any instant of time according to the torque status, which is obtained by the regulator outputs shown in Fig. 2 as well as by sectors determined by estimated electrical position of the rotor. In each sector, regarding that it is assumed that the value of the actual stator flux-linkage is identical with the value of the reference stator flux-linkage, since only two phases of the BLDC

Table 2 Specifications and parameters of the BLDC motor. Parameter

Value

Unit

Number of poles DC link voltage Rated speed Phase resistance Phase inductance Load torque Moment of inertia Torque constant Damping constant

2 300 1500 0.4 13 3 0.004 0.4 0.002

pole V rpm Ω mH Nm kg m2 V/(rad/s) N m/(rad/s)

130 110

Estimated

80

Back-EMF [V]

Fst

Actual

50 20 -10 -40 -70 -100 -130 2

2.015

2.03

2.045

2.06

2.075

2.09

2.105

2.12

2.135

2.15

Time [sec] Fig. 3. Waveforms of estimated (dashed line) and actual (solid line) steady-state phase-to-phase back-EMF for the proposed method under full-load at rated-speed.

motor drive is excited and controlled during each 60° elec., a nonzero voltage space vector is employed for torque increase and decrease as demonstrated in Fig. 2.

S.A.KH. Mozaffari Niapour et al. / Control Engineering Practice 24 (2014) 42–54

Erorr [rpm]

Speed [rpm]

Speed [rpm]

In order to reach an insight distinct from the whole system performance and emphasize on the advantages of the proposed sensorless control scheme, the motor operation needs to be evaluated under different conditions so that a successful assessment of the motor operation can be performed. The proposed control scheme has been simulated under different operating conditions of the motor. To set the gating signals of the power switches easily and represent the real conditions in simulation as close as possible the electrical model of the BLDC motor with R–L elements and the inverter with power semiconductor switches considering the snubber circuit, the simulation model has been designed in Matlab/Simulink using the SimPower System toolbox. Moreover, the dead-time of the inverter and non-ideal effects of the BLDC motor are neglected in the simulation model. In the simulations, sampling interval of 25 μs and the magnitude of the torque hysteresis band 0.1 N m have been taken into account and all simulations have been made in the discrete time implementation. Simulation parameters of a standard BLDC motor for testing the proposed sensorless drive technique performance are given in Table 2. In this part, the effectiveness of the proposed strategy will be analyzed from four perspectives including: steady-state accuracy, dynamic performance, parameter and noise sensitivity, and lowspeed-operation performance. The controlling parameters that have been selected for all the simulations accomplished are as follows: Q ¼ 1  10  2 , R ¼ 1  10  6 , ηðtÞ ¼ 0, and γ ¼ 2:7.

0.5

1

1.5

2

2.5

3

3.5

4

Time [sec] Fig. 4. Waveforms of estimated speed (upper trace: 0.2 rpm/div), actual speed (middle trace: 0.2 rpm/div), and speed estimation error (lower trace: 0.2 rpm/div) for the proposed method at rated-speed.

Electromagnetic Torque [N.m]

3.5 3.4 3.3 3.2 3.1 3

0.5

1

1.5

2

2.5

3

3.5

4

Time [sec]

Estimated Speed [rpm]

Fig. 5. Waveform of estimated electromagnetic torque for the proposed method at rated-speed.

Actual Speed [rpm]

6. Simulation results

3.6

Erorr [rpm]

When the estimated torque value is smaller than its command one (i.e., TI is the torque error Tst indicator), a non-zero voltage space vector is utilized for torque increase as well as when the estimated torque value is bigger than its command one (i.e., TD is the indicator of Tst), another non-zero voltage space vector is used for torque decrease. Processing torque output status, which are functions of optimal switching logic for selecting appropriate stator voltage vector, is performed through voltage selector for satisfying torque output. In fact, there exist only six non-zero stator voltage vectors for the BLDC motor drive in this method, the non-zero stator voltage sectors that the voltage inverter can produce. While the motor is undergoing the high-speed operation, the key, shown in Fig. 2, changes its situation from status 1 to 2 to accompany the operation of the proposed complementary controller with the proposed strategy for reducing the commutationtorque-ripple. In this way, with harmonic and simultaneous operation of each of these controllers, the BLDC motor can experience a drive with successful performance.

49

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time [sec]

Fig. 6. Dynamic responses of the proposed method when load torque and speed reference change. From top to bottom: estimated speed (400 rpm/div), actual speed (400 rpm/div), and speed estimation error (5 rpm/div).

6.1. Steady-state accuracy analysis The first category of simulations under study in this part has been brought to emphasize the behavioral effects of the proposed controlling drive according to the steady-state accuracy analysis of the motor in nominal operating conditions. Fig. 3 shows the estimated and actual phase-to-phase back-EMF ðeab Þ under fullload and rated-speed for the proposed method. As the figure clearly illustrates, accuracy of the estimated back-EMF is very high with the proposed method such that the distinction between the estimated back-EMF error and the actual one is extremely difficult, and also the estimation back-EMF error includes oscillations of small amplitude with the maximum peak 7 1:8 V imposed on top and bottom of zero. Fig. 4 depicts the estimated and actual speeds as well as speed estimation error in the nominal operating conditions of the motor by employing the proposed method, in which the white dashed lines indicate the reference values of each one of the quantities. This figure replicates a very good speed estimation of the actual value with a very small ripple. In this state, as Fig. 4(lower trace) delineates, the speed estimation error is variable and its maximum value, which occurs during current commutation instants, when there are sharp changes in phase currents, reach less than 7 0:3 r/min, i.e., 0.04% of rated-speed. Generally, the speed estimation error and the phase shift created between the estimated and actual back-EMF values, which tacitly mean a phase difference between quasi-square current waveforms and trapezoidal back-EMF of the motor, bears a direct relationship with the torque ripple produced by the motor. The estimated

S.A.KH. Mozaffari Niapour et al. / Control Engineering Practice 24 (2014) 42–54

electromagnetic torque waveform for the proposed method has been represented in Fig. 5. Noting the 9% torque ripple, first it can be inferred that the proposed method does not suffer from any phase-shift (see Fig. 3) and second guarantees a very good estimation for the motor. 6.2. Dynamic performance analysis The second category of the simulations under investigation in this section has been brought to emphasize on the effects of the proposed sensorless control drive behavior based on dynamic performance analysis of the motor according to torque and speed profiles applied to it is as follows.

 Speed profile: during interval 0  1:5 s. the motor starts up in



nominal operating conditions at 1500 r/min; t ¼ 1:5  3 s: acceleration and following that high-speed operation ðV dc o4EÞ at 2500 r/min; t ¼ 3  4:5 s: deceleration and lowspeed operation ðV dc 4 4EÞ at 500 r/min. Load torque profile: during interval 0 1:5 s the motor starts to operate under half of the full-load (1.5 N m); t ¼ 0:75  4 s: increase of the applied load to the motor and following that full-load operation (3 N m); t ¼ 4  4:5 s: decrease of the applied load to the motor and on track of it half of the fullload operation.

Fig. 6 depicts the dynamic responses of the actual speed, estimated speed, and speed estimation error for the proposed method under the above conditions. Three significant points can be concluded from the speed estimation error viewpoint out of this figure. First, the error observed in the transient instants

Erorr [rpm]

Speed [rpm]

Actual Estimated

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time [sec] Fig. 7. Dynamic responses of the proposed method when load torque and speed reference change without using LPF. From top to bottom: estimated and actual speed (400 rpm/div), and speed estimation error (2 rpm/div). 140 Actual

105

Estimated

Back-EMF [V]

70 35 0 -35 -70 -105 -140

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Time [sec] Fig. 8. Waveforms of estimated (dashed line) and actual (solid line) transient behavior phase-to-phase back-EMF for the proposed method during start-up under full-load at rated-speed.

360 Actual

Rotor Position [degree]

50

300

Estimated

240 180 120 60 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time [sec] Fig. 9. Waveforms of estimated (dashed line) and actual (solid line) rotor position (in electrical degree) for the proposed method during start-up under full-load at rated-speed.

operation, acceleration and deceleration, has been produced due to the phase delay created by LPF, which is used in the proposed sensorless drive scheme strategy for smoothing the estimated speed that has been utilized as feedback signal in speed and torque controllers. To verify such a claim, another similar test has been carried out for the proposed method by establishing the conditions mentioned in this part without using any LPF (see Fig. 7). Thus, as it can be understood from the figure, the LPF can be eliminated where speed transient state is of high importance. As it can be seen, in such state there is no phase delay between the estimated and actual speeds, of course, in return of addition of some increase in estimation error in the steady state. The origin of the high-frequency ripples observed in this figure are the same as the commutation notches whose frequency is six times as much as the electric frequency of the motor. The frequency of these commutation notches results from the multiplication of the number of motor poles by one-twentieth of the motor speed in rpm, and the relative width and depth of these notches will increase as the speed increases. Furthermore, Figs. 8 and 9 verify a good transient response for the proposed method in the backEMF estimation and electric position of the rotor in the operating transient instants of the motor. As the figures demonstrate, the estimated back-EMF and rotor position fully match with their values, and this issue is another verification for the claim made above. Second, although a load torque step-change has been applied to the motor at 0.75 and 4 s for testing the sensorless drive response, and the motor experience an undershoot and overshoot less than 7 14 r/min approximately within a short period of time, according to Fig. 6, it should be evident that no error of the steady state is found in the motor speeds after passing the undershoot and overshoot. This issue means that the proposed sensorless drive strategy has a good stability against un-modeled mechanical disturbances of the motor. Another point is that a few overshoots, which emanate from a big rapid transient response in the back-EMF estimation, are observed in the transient states (see Fig. 6) in response to speed profile. In explanation for that, it should be pointed that the observers used in the BLDC motor drives require a high enough gain for convergence, which cause it to peak to big values before the transient response rapidly decays towards zero. This impulsive-like behavior is recognized as the peaking phenomenon (Sussmann & Kokotovic, 1991). As it was expected, the proposed method has acted much successfully in response to high-speed operation, and the motor can experience a stable operation, as Fig. 6 reveals it. In such state, the speed estimation error is restricted between 7 0:9 r/min limits for proposed method. Since commutation currents form in an inharmonic and distorted manner in high-speed operation mode, this speed ripple might be justified as the motor be affected by a current noise at high-speed, and the proposed method can easily

ib

ic

-1500 rpm

Estimated Speed [rpm] 1500 rpm

Commutation Torque Ripple

3.4

3.405

3.41

3.415

3.42

3.425

-1500 rpm

0

1

2

3

Time [sec]

5

6

Fig. 12. Estimated (upper trace) and actual (lower trace) speed response using the proposed method, when the demanded speed is varied from 1500 to  1500 r/min (500 rpm/div).

8

Vdc

Actual

6

ib

ia

Electromagnetic Torque [N.m]

Currents [A] & Dc Link Voltage [V]

4

Time [sec]

Fig. 10. Voltage waveform of dc bus (upper dashed line trace: 50 V/div), phase currents (middle trace: 2 A/div), electromagnetic torque (lower trace: 0.2 N m/div): without using proposed complementary control method at 2500 rpm.

Torque [N.m]

51

1500 rpm

Estimated Speed [rpm]

ia

Actual Speed [rpm]

Torque [N.m] Currents [A] & DC Link Voltage [V]

S.A.KH. Mozaffari Niapour et al. / Control Engineering Practice 24 (2014) 42–54

ic

Commutation Torque Ripple

Estimated

4 2 0 -2 -4 -6

3.4

3.405

3.41

3.415

3.42

3.425

Time [sec]

-8

0.5

1

1.5

2

2.5

3

3.5

4

Time [sec]

Fig. 11. Voltage waveform of dc bus (upper dashed line trace: 65 V/div), phase currents (middle trace: 1 A/div), electromagnetic torque (lower trace: 0.2 N m/div): using proposed complementary control method at 2500 rpm.

Fig. 13. Dynamic torque response using the proposed method, when the demanded torque varied from 3 to  3 N m in every 0.5 s.

confront it. It is not surprising that due to such situation of the commutation currents and speed estimation error, the torque ripple at 2500 r/min increases to 15.5% value in comparison with rated-speed for the proposed method. In order to overcome the aforementioned commutation-torque-ripple or torque dips at high-speed, the proposed complementary controller starts operating in parallel with the proposed sensorless method for reducing the commutation-torque-ripple. Figs. 10 and 11 show the simulation results for waveforms of dc-bus voltage, phase currents, and electromagnetic torque without and with utilizing the proposed method to reduce commutation-torque-ripple at 2500 r/min respectively. As it is clear from Fig. 10, the controller has not been able to cope with high-speed commutation current ripple of the motor during commutation interval without applying the proposed complementary control even by utilizing the DTC method such that this issue has lead to the formation of an undesired commutation-torque-ripple (15.5%) for the motor. This exists whereas, as shown in Fig. 11, current ripple and consequently commutation-torque-ripple of the motor have intensely decreased to 9.3% value of the load average torque by applying the proposed complementary approach in the commutation interval which has been marked by dashed line in this figure. Hence, the torque ripple

can almost decrease to its half value by utilizing the proposed drive strategy. Another test has been allocated in order to evaluate the motor and braking operations of the proposed strategy at 71500 r/min, as shown in Fig. 12. In this test, the machine is operated for 3 s at 1500 r/min, and then its speed reference value is reversed to  1500 r/min, while the load torque has been kept constant (braking mode). Fig. 12 shows the estimated and actual speeds, again as evident, the estimated speed has a good accordance with its actual value in both regimes. The speed estimation error remains between 7 0:3 r/min limits during the braking operation mode, and this issue indicates that the proposed filter is by no means affected by the reverse state and is not sensitive to the speed transient states. Fig. 13 illustrates the dynamic response of the estimated and actual torques by using the proposed method when the torque command changes from 3 to  3 N m in each 0.5 s, and the speed is kept constant at 1500 r/min. It is seen from Fig. 13 that the actual torque is almost controlled within the hysteresis band and rapidly follows its reference value (less than 0.04 s) while compared with the conventional current control methods. The torque estimation error maximizes to 7 0:1 N m value for the transient

S.A.KH. Mozaffari Niapour et al. / Control Engineering Practice 24 (2014) 42–54

Erorr [rpm]

Erorr [rpm] Speed [rpm]

Erorr [rpm]

Speed [rpm]

52

0.5

1

1.5

2

2.5

3

3.5

4

Time [sec] Fig. 14. Waveforms of speed estimation error under full-load sensorless operation at rated-speed for resistance detuning. From top to bottom: proposed method with þ 40% deviation (0.5 rpm/div), proposed method with  40% deviation (0.5 rpm/div).

0.5

1

1.5

2

2.5

3

3.5

4

Time [sec]

1

1.5

2

2.5

3

3.5

4

Time [sec] Fig. 15. Waveforms of speed estimation error under full-load sensorless operation at rated-speed for inductance detuning. From top to bottom: proposed method with þ 15% deviation (2 rpm/div), proposed method with  30% deviation (3 rpm/div).

Speed [rpm]

0.5

Erorr [rpm]

Erorr [rpm]

Speed [rpm]

Erorr [rpm]

Fig. 16. Waveforms of speed estimation error under full-load sensorless operation at rated-speed for 40% noisy current measurement. From top to bottom: estimated speed (5 rpm/div), actual speed (5 rpm/div), and speed estimation error (5 rpm/div).

0.25

states, whereas this value oscillates between 7 0:025 N m for the steady state. It is worth to note that the torque overshoots observed in Fig. 13 are translated from the speed to the torque due to the peaking phenomenon. 6.3. Parameter and noise sensitivity analysis Since parameter detuning, modeling error, and measurement noise are among the most important disturbances which affect the observer's accuracy, in this section, another test based on sensitivity to parameters and noise has been carried out on the system for analyzing the robustness of the proposed system against the disturbances. In sensorless drives based on the observer in the BLDC motors, stator phase resistance R and phase inductance L are used as constant parameters of the model in estimating state variables. Nevertheless, the stator resistance can be deviated a lot from their nominal values due to skin-effect, temperature variation, and the inductance due to flux-saturation, demagnetization effect, and other disturbances. Fig. 14 demonstrates the waveforms of the speed estimation error in full-load and rated-speed for resistance increase and decrease ( 7 40% R) for the proposed method. As it is self-evident from the figure, the proposed method is not sensitive to resistance detuning, and under such a test, maximum speed error oscillates between 7 0:5 r/min and 7 0:75 r/min values for resistance increase and decrease respectively. It seems necessary to point that the rest of the tests accomplished in this part have been performed for similar condition, i.e., under full-load and rated-speed. Similarly, Fig. 15 reports the results of the sensorless drive sensitivity tests in contrast with inductance uncertainty for the proposed method. As it can be seen, the proposed method is robust against the overestimation (15%) of L and underestimation (  15%) of L, and these mentioned parameter variations create

2.25

4.25

6.25

8.25

10.25

12.25

14.25

16.25

18.25

20

Time [sec] Fig. 17. Full-load sensorless operation at 15 r/min with the proposed method. From top to bottom: estimated speed (0.05 rpm/div), actual speed (0.05 rpm/div), and speed estimation error (0.05 rpm/div).

very small disturbances in speed estimation such that the speed estimation error reaches 7 1:9 r/min and 73:4 r/min respectively in this state. According to the results obtained in this part it can be claimed that the proposed method could create a very satisfactory compromise between the existence of uncertainty in electric parameters and speed accuracy required for the system in such a way that it does not require an online estimation of electric parameters for gaining access to a good speed accuracy in contrast to the conventional observer methods. This issue will inevitably decrease the complexity of the calculation power of the controllers. It is worth mentioning that the variations considered in this category may not be realistic, yet they have been used only for studying the estimation performance. The second category of tests accomplished in this section has been brought forth for analyzing the capability of the proposed method under the applying conditions of current measurement noise. In order to realize such a test, a uniform random noise with zero-mean and intensity of 0.4 have been added to ia and ib currents separately, and the results of the test under full-load and rated-speed have been exhibited in Fig. 16. According to this figure, as expected, the proposed method shows to be of strong robustness in face of measurement noises such that the speed estimation error created in this state varies between 7 6:5 r/min values, and this can be another emphasis on the features of the proposed method. Regarding these relatively small ripples generated for the proposed method under aforementioned test, there is no wonder to expect an acceptable behavior of the motor drive.

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1.5 Actual

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0.5 0 -0.5 -1 -1.5

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17

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Time [sec] Fig. 18. Waveforms of estimated (dashed line) and actual (solid line) behavior of the phase-to-phase back-EMF for the proposed method under low-speedoperation.

360

Rotor Position [degree]

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Time [sec] Fig. 19. Waveforms of estimated (dashed line) and actual (solid line) behavior of the rotor position for the proposed method under low-speed-operation.

6.4. Low-speed-operation performance analysis The last category of the simulations under study in this section has been allocated to low-speed-operation performance since it is one of the most challenging tests and the most critical situations for a sensorless BLDC motor drive. One of the most important reasons that make the estimation of the state variables problematic in very low speeds where the excitation amplitude is low, is that in these conditions the observer's sensitivity to the parameter variations especially the resistance increases. However, resistive voltage drop is smaller than stator voltage in high-speed operation range; consequently, the back-EMF and speed estimation can be performed with high accuracy. In contrast, in low-speed regime, the observer faces the serious problem of considerable effect of the resistive voltage drop against low stator voltage or increase in the proportion of noise to actual signals. Thus, in order to achieve an accurate estimation in very low-speed-operation mode for the sensorless scheme under analysis in this paper, it is self-evident that the observer that is of a high robustness against sensitivity to electrical parameters will act successfully. Fig. 17 depicts the fullload sensorless operation with the proposed method at 15 r/min (1% of rated-speed) for the actual and estimated speeds and speed estimation error. As the figure clearly and expectedly shows, the estimated speed replicates the speed by a tiny ripple such that the oscillation ripple of the speed estimation error restricts in 70:12 r/min values, i.e., 2.4% of the speed under study. This small speed ripple, on the one hand, indicates a very good estimation of the back-EMF and, on the other hand, verifies the statements concerning the observer's success with more robustness in face of the electric parameter uncertainty. The small ripples observed in Fig. 17 recall the worthwhile point that the proposed filter does

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not confront with severe oscillations in zero-crossing instants; in the other words, the proposed method acts very precisely in the accurate estimation of the back-EMF during commutation instants. Interesting to point that the ripple of the system decreases as the load applied to the motor decreases. Figs. 18 and 19 depict the estimated and actual responses of the phase-to-phase back-EMF, and rotor position under low-speed-operation, respectively. Figs. 18 and 19 depict the estimated and actual responses of the phase-to-phase back-EMF, and rotor position under low-speedoperation, respectively. The figures verify a good response for the proposed method in the back-EMF estimation at low-speedoperation of the motor operation, and the estimated parts are well in accordance with the actual ones in such a way that the distinction between them is very difficult. This proves that the introduced results for the sensorless operation at very low speeds, due to the excellent robustness and high estimation accuracy can provide excellent performance in another stressful condition for the motor by presenting a very low back-EMF, and position rotor ripples. Since the sensorless schemes are not self-starting, speedsensorless methods can not be applied well when the motor is at standstill. Thus, a starting procedure is needed to start the motor from standstill. Most of these starting strategies are based on arbitrarily energizing the two or three windings and expecting the rotor to align to a certain definite position (Asaei and Rostami, 2009; Damodharan et al., 2008; Iizuka et al., 1985; Jang, Park, & Chang, 2002; Lee & Sul, 2006; Moreira, 1996). Consequently, for the motor starting, one of the already known procedures would have to be applied. Among the simplest of them, for instance two phases can be excited to result in the rotor to rotate and lock into position. If the rotor is not in the desired position, the forcing torque from the excited phases causes it to rotate and stop at the desired position. After prepositioning, the next commutation signal advancing the switching pattern by 60 electrical degrees is applied. Then instantaneously the proposed speed-sensorless scheme can be taken over to detect the next commutation instant. With regard to the reasons mentioned about the proposed method in the section, the 60 electrical degrees rotor movement is enough to detect the commutation instants and speed of the motor. After the first detection of the commutation instant, both current and speed control is possible using the estimated speed. 6.5. Comments on computational complexity In this section the proposed filter has been compared with two conventional sensorless methods in BLDC motor drives, is that, Kalman filter and sliding-mode observers from computational complexity point of view. For the Kalman filter and sliding-mode observers, there are three and two parameters that should be regulated respectively, and for proposed method the number of these parameters is three. Alternatively, there are only sum and product for sliding-mode while computation process also consists of matrix product and inverse for Kalman filter method. However, for proposed method the condition will exacerbate in comparison with Kalman filter method because in addition to multiplying matrix by its inverse it needs to be solved by LMI. The LMI can be solved rapidly and user-friendly manner by using efficient numerical algorithm and software package such as MATLAB. It should be noted that the calculation time for proposed method can dramatically be decreased by optimization of LMI's algorithm and programmer's ability which is easy to obtain with nowadays microcontrollers of DSP systems. One more thing that deserves to be mentioned is that the use of DSP for similar on-line complicated mathematical computations has been reported in different references such as Lain, Chiang, and Tu (2007) and Zhu, Kaddouri, Dessaint, and Akhrif (2001). Nonetheless, what can have

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more effect on computation time is on-line structure of estimator. In this respect, the sliding-mode and proposed methods suffer from using saturation function and extra matrix product, respectively with respect to Kalman filter method. On the other hand, the sliding-mode method has superiority over the Kalman filter and proposed methods from computation complexity parameter tuning point of view, while the Kalman filter method is very beneficial in computation time perspective in comparison with sliding-mode and proposed methods. In conclusion, the proposed control scheme enhances robustness and accuracy of the drive system at the expense of more computational complexity and adjusting some more additional parameters compared to Kalman filter and sliding-mode methods. 7. Conclusion In this paper a strategy has been presented for designing a BLDC motor speed sensorless control based on a novel robust stochastic H 1 deconvolution filter and a complementary controller for the purpose of torque ripple reduction and drive performance promotion. The innovative deconvolution filter is derived using the stochastic bounded real lemma for the linear stochastic system with state- and input- and exogenous disturbancedependent noise. The proposed deconvolver has also been applied to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are uncertain and reside in a given polytope. Based on the simulation results, the proposed deconvolution filter and generally the proposed strategy proves to have a high robustness, very low sensitivity, excellent estimation, and reduced torque-ripple in response to all scenario considered for evaluating the performance of the motor. Thus, regarding the successful operation results, the proposed sensorless drive strategy with its high robustness and elimination of any limitation in the back-EMF dynamic can make it possible for the drive to conduct the motor toward a stable tensionless operation in various high-performance applications in a vast speed range. Acknowledgements The first and second authors would like to thank their parents separately for all their spiritual and financial supports. Furthermore, they would like to express their sincere and ultimate gratitude to Prof. M. R. Feyzi for his patient guidance, continued support, encouragement, and advice he has provided in all stages of preparing the paper. References Asaei, B., & Rostami, A. (2009). A novel starting method for BLDC motors without the position sensor. Energy Conversion and Management, 50(2), 337–343. Chen, C.-H., & Cheng, M.-Y. (2007). New cost effective sensorless commutation method for brushless dc motors without phase shift circuit and neutral voltage. IEEE Transactions on Power Electronics, 22(2), 644–653. Damodharan, P., Sandeep, R., & Vasudevan, K. (2008). Simple position sensorless starting method for brushless DC motor. IET Electric Power Applications, 2(1), 49–55. Damodharan, P., & Vasudevan, K. (2010). Sensorless brushless DC motor drive based on the zero-crossing detection of back electromotive force (EMF) from the line voltage difference. IEEE Transactions on Energy Conversion, 25(3), 661–668.

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