Speed-sensorless control of SR motors based on GPI observers

Control Engineering Practice 46 (2016) 115–128

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Speed-sensorless control of SR motors based on GPI observers A. De La Guerra n, Marco A. Arteaga-Pérez, Alejandro Gutiérrez-Giles, P. Maya-Ortiz Universidad Nacional Autónoma de México (UNAM), Engineering Faculty, C.P. 04510 México D.F., Mexico

art ic l e i nf o

a b s t r a c t

Article history: Received 26 January 2015 Received in revised form 8 September 2015 Accepted 17 October 2015

This paper presents a robust speed-sensorless controller based on Generalized Proportional Integral (GPI) Observers for Switched Reluctance Motors. It compensates the unknown load torque while naturally estimates the output phase variables, making unnecessary to directly measure the angular velocity. An experimental comparison of the proposed controller with two other well-known similar schemes, carried out on a three phase 12/8 SR motor, shows the good performance of the proposed scheme. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Switched reluctance motors Speed control Robust control Disturbance rejection Observers design

1. Introduction The Switched Reluctance Motor (SR motor) represents a good option for direct-drive variable-speed applications (Krishnan, 2001). Moreover, it can deliver high torque at low speed (Miller, 2001). Additionally, it has windings only on the stator, resulting in a smaller motor for a given rating and frame size. These characteristics, coupled with its low cost and high reliability, make it suitable to applications such as wind power like in Nassereddine, Rizk, and Nagrial (2011) or electric-hybrid vehicles as in Takano et al. (2010). However, its main limitation is its strong nonlinear electromagnetic behavior which makes it difficult to control (Toliyat & Kliman, 2012). Velocity tracking over a wide range of speeds, despite changes in the electrical/mechanical dynamics, has led to the implementation of a variety of control techniques. Some classical results are the exact linearization control presented in Ilic'-Spong, Marino, Peresada, and Taylor (1987), the variable structure control in Buja, Menis, and Valla (1993), a PWM based control in Husain and Ehsani (1996), passivity based control in Espinosa-Pérez, Maya-Ortiz, Velasco-Villa, and Sira-Ramírez (2004), backstepping control in Alrifai, Chow, and Torrey (2000), among others. However, these schemes are not designed to cope with unknown parameters or load torque. Current research on speed control of the SR motor is focused on the robustness properties with respect n

Corresponding author. E-mail addresses: [email protected] (A. De La Guerra), [email protected] (M.A. Arteaga-Pérez), [email protected] (A. Gutiérrez-Giles), paulm@dctrl.fi-b.unam.mx (P. Maya-Ortiz). http://dx.doi.org/10.1016/j.conengprac.2015.10.010 0967-0661/& 2015 Elsevier Ltd. All rights reserved.

to disturbances (e.g., changes in the load torque, parametric uncertainties, noise measurement, etc.). For instance, the gain-scheduling PI control presented in Hannoun, Hilairet, and Marchand (2011) or the sliding mode control proposed in Sahoo, Panda, and Xu (2005). Also, there are control schemes where the emphasis relies on a model-free compensation of the load torque and the parametric uncertainties, such as the self-tuning fuzzy control designed in Chwan-Lu, Shun-Yuan, Shao-Chuan, and Chang (2012) and the fuzzy-PI controller in Paramasivam and Arumugam (2005). Another problem of interest is the lack of speed measurement because, when the speed is not available, a common practice is to use numerical differentiation of the angular position, which may lead to noise amplification. To solve this drawback, a reliable alternative is the use of an state observer. Regarding this issue, it is easy to show that the SR motor is globally observable if the stator current and the rotor position are measured (but not the angular velocity). This fact has led to the design of the so-called speedsensorless controllers like the PI2D presented in Loria, EspinosaPerez, and Chumacero (2012, 2014). Notice that all the mentioned schemes still rely on the position measurements. The full-sensorless problem, where neither the angular position nor the velocity are measured, is much more involved and it is not treated here. In short, the objective of this work is to design a robust speed control scheme, which does not need to measure the angular velocity of the rotor shaft by using a simplified model of the SR motor. For this, a Generalized Proportional Integral (GPI) observer, first presented in Sira-Ram (2010), is designed to estimate in an approximate way the unknown mechanical dynamics and the angular velocity of the SR motor, while simultaneously, providing some good robustness characteristics as well. The GPI observer is a

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linear high-gain Extended State Observer (ESO) (Radke & Gao, 2006). This kind of observers relies on a simplified model of the plant and an approximate time-dependent model of the disturbance (in this work, a polynomial approximation), resulting in a simplified control design. This approach has been already applied to control the Induction Motor speed in the presence of unknown torque load (Sira-Ramírez, Gonzalez-Montanez, Cortés-Romero, & Luviano-Juárez, 2013). In this work, a cascade scheme is presented, where the external loop achieves the tracking control of the angular velocity of the rotor based on a GPI observer, which helps to compensate the unknown load torque and friction terms while it naturally estimates the position error phase variables. Meanwhile the inner loop calculates the stator voltages using the passivity based control law reported in Espinosa-Pérez et al. (2004). The commutation of the stator phase currents is based on the torque sharing approach (Miller, 2001), in order to minimize the speed ripple and control the instantaneous torque. The paper is organized as follows: Section 2 describes the SR motor and its mathematical model, Section 3 is devoted to the main result of this paper while experimental results are presented in Section 4. Some conclusion are given in Section 5.

θ̇ = ω

1 T d 1 i C (θ ) i − ω − τ L, 2J J J

ω̇ =

m

⎡ x1⎤ ⎡ i ⎤ x = ⎢ x2 ⎥ = ⎢ θ ⎥, ⎢ ⎥ ⎢ ⎥ ⎣ x3 ⎦ ⎣ ω ⎦

D (x2 ) x1̇ = u − x3 C (x2 ) x1 − Rx1

(4a)

x2̇ = x3

(4b)

1 d 1 τ − x3 − τ L, J J J

The SR motor has salient poles in both rotor and stator; while the windings are concentrated in the stator, the rotor is made up only of laminate steel. The torque in this electric machine is produced by the tendency of the rotor to move to the maximum inductance position. Therefore, the stator phases must be sequentially switched to generate a rotor continuous movement. Additionally, to control the angular velocity of the SR motor is necessary to measure the angular position in order to properly switch the stator phases, for a detailed description see Miller (2001).

where

(i) The mutual inductances are negligible, i.e., each phase is electrically independent, which means that the stator phases are decoupled. (ii) The phase inductance is defined as

⎛ 2π ⎞ L j (θ ) = l0 − l1 cos ⎜ Nr θ − (j − 1) ⎟, ⎝ m⎠

(1)

where j = 1, 2, 3, … , m, with m the number of stator phases, θ ∈  is the angular position, Nr represents the total number of rotor poles, and l0 > l1 > 0 are the coefficients of static winding inductance. (iii) The phase flux linkage ψj is defined as

ψj = L j (θ ) i j , where i j is the stator phase current, which is assumed lower than the saturation current. Therefore, the mathematical model of a m-phases SR motor is

D (θ )

di = u − ωC (θ ) i − Ri dt

(2a)

(3)

so that system (2) can be rewritten as

x3̇ =

The standard assumptions of the non-saturated SR motor model are (Krishnan, 2001):

(2c) m

where i ∈ is the vector of stator currents, u ∈ is the vector of voltage inputs, ω ∈  is the angular velocity, R ∈ m × m is a diagonal matrix accounting for the winding resistances, D (θ ) ∈ m × m is a diagonal matrix of winding inductances, τ L ∈  is the load torque, J ∈  is the rotor inertia and d ∈  the viscous friction ∂D (θ ) coefficient. The matrix C (θ ) ∈ m × m is given by C (θ ) = ∂θ . Define the vector state

2. Mathematical model of the switched reluctance motor

2.1. The non-saturated SR motor model

(2b)

τ=

1 T x1 C (x2 ) x1. 2

(4c)

(5)

System (4) has the following property. Property 1. D (x2 ) is a symmetric positive definite matrix and it satisfies λm ∥ x ∥2 ≤ xTD (x2 ) x ≤ λM ∥ x ∥2 ∀ x ∈ n , where λm and λM are the minimum and the maximum eigenvalue of D (x2 ), respectively. □ Remark 1. Model (4) is valid only at low currents, i.e., for the nonsaturated operation (Miller, 2001). Nevertheless, although it is a very simplified representation, it preserves the essential characteristics of the machine dynamics what makes it useful for control design. In the case of observer design this model is valid only for currents lower than the saturation current. □

3. Main result In Espinosa-Pérez et al. (2004) a cascade speed control strategy is proposed where, given the desired speed x3d , an external (mechanical) control loop calculates the corresponding torque τd, which in turn is used to define the desired currents x1d via a system inversion. Based on the desired currents, an internal (electrical) control loop determines the stator phase voltage u using an output feedback controller based on the passivity properties of the SR motor. In this work, we preserve the cascade structure of EspinosaPérez et al. (2004), but replacing the mechanical loop with a GPI observer control. The GPI observer permits us to reconstruct the unknown mechanical dynamics i.e., load torque τ L and matched perturbations, by means of a time-dependent polynomial approximation. At the same time, the phase variables of the position error are estimated, what makes unnecessary to measure the rotor velocity. Also, the electrical loop has been changed to avoid the dependence on the velocity measurement.

A. De La Guerra et al. / Control Engineering Practice 46 (2016) 115–128

3.1. Mechanical control loop For the external control loop to guarantee the speed tracking objective x3 → x3d , the mechanical errors are defined as

117

̇ e^2 = e^3 + λ p + 1e˜2

(14a)

e2 ≜ x2 − x2d

(6)

1 ̇ e^3 = τ + z^1 + λ p e˜2 J

(14b)

e3 ≜ x3 − x3d ,

(7)

̇ z^1 = z^2 + λ p − 1e˜2

(14c)

̇ z^2 = z^3 + λ p − 2 e˜2

(14d)

⋮ ̇ z^p − 1 = z^p + λ1e˜2

(14e)

̇ z^p = λ 0 e˜2 ,

(14f)

where

x2d ≜

∫0

t

x3d (ϑ) dϑ,

x2d (0) = x2 (0).

(8)

Substituting these definitions in the mechanical dynamics (4b) and (4c) yields

e2̇ = e3

e3̇ =

(9)

1 τ + z1, J

(10)

where z1 is defined as

z1 ≜ −

d 1 ̇ . x3 − τ L − x3d J J

s p + 2 + λ p + 1s p + 1 + ⋯ + λ1s + λ 0 = 0. (11)

To make use of this approximation as an internal model for the controller-observer design, we make the following assumptions (Sira-Ram, 2010). Assumption 1. The time signal z1 (t ) can be written as a fixeddegree Taylor polynomial p−1

z1 (t ) =

∑ ai t i + r (t ), i=0

where e˜ 2 ≜ e2 − e^2, e˜3 ≜ e3 − e^3 and λ 0, … , λ p + 1 are the coefficients of the characteristic polynomial

(12)

Finally, using the estimates of observer (14), the following mechanical control law is proposed:

τd = J

( −Λ

m2 e2

)

− Λm3 e^3 − z^1 ,

e3̇ =

1 (τ − τd ) − Λm2 e2 − Λm3 e3 + Λm3 e˜3 + z˜1. J

To simplify the representation it is defined the vector

Assumption 2. At least the first p derivatives of the residual term r(t) exist, i.e., up to r (p) (t ). □

⎡ e2 ⎤ e 23 = ⎢ ⎥. ⎣ e3 ⎦

Remark 2. Assumption 1 theoretically reduce the type of velocity references that can be used (they must have bounded derivatives). However, in practice the controllers based on GPI observers work properly with references with abrupt changes, such as step references. Also, these observers have compensated physical phenomena such as Coulomb friction in robot control (Arteaga-Pérez & Gutiérrez-Giles, 2014). This is shown by means of experiments in Section 4. □

Then, the mechanical error dynamics can be rewritten as

After Assumption 1, an internal model of zi(t), with i = 1, … , p, can be written as

⎡ 0⎤ B m = ⎢ ⎥, ⎣ 1⎦

(13a)

z2̇ = z3

(13b)

⋮ z ṗ − 1 = z p

z ṗ = r (p) (t ).

(16)

where Λm2 > 0 and Λm3 > 0 are constant gains. Thus, the mechanical error dynamics can be obtained by substituting (16) into (10)

where i = 1, … , p, and ai are constant coefficients. □

z1̇ = z2

(15)

(17)

(18)

e ̇23 = Am e 23 + B m r¯1,

(19a)

where

⎡ 0 1 ⎤ Am = ⎢ ⎥ ⎣ − Λm2 − Λm3⎦

(19b)

and

r¯1 =

1 (τ − τd ) + Λm3 e˜3 + z˜1. J

(19c)

Similarly, the mechanical system (9) and (10) and (13) in closed loop with observer (14) deliver the estimation error dynamics

e˜2̇ = e˜3 − λ p + 1e˜2

(20a)

e˜3̇ = z˜1 − λ p e˜2

(20b)

z˜1̇ = z˜2 − λ p − 1e˜2

(20c)

z˜2̇ = z˜3 − λ p − 2 e˜2

(20d)

(13c)

(13d)

To estimate the phase variables of the position error and the internal model of zi(t), the following linear observer is proposed:

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A. De La Guerra et al. / Control Engineering Practice 46 (2016) 115–128

⋮

⎧ a¯ x^ + b if C (x ) ≠ 0 ̇ j j 2 x^1dj ≜ ⎨ j 3 , ⎩0 otherwise ⎪

z˜ ṗ − 1 = z˜ p − λ1e˜2

(20e)

z˜ ṗ = r (p) − λ 0 e˜2 ,

(20f)

where z˜i ≜ zi − z^i , i = 1, … , p. By substituting (20a) into (20b), it can be obtained

e˜¨2 + λ p + 1e˜2̇ + λ p e˜2 = z˜1,

(22)

z˜1̇ = e˜2(3) + λ p + 1e˜¨2 + λ p e˜2̇ .

(23)

Accordingly, using (20a)–(20f) in an iterative process it can be found

(24)

Then, by defining the vector

(

∂mj (x 2 ) ∂x 2

Cj (x2 ) − mj (x2 )

(

τ^ḋ = J

( −Λ

(31)

(32)

)

^ − Λ e^ ̇ − z^ ̇ , m3 3 1

m2 e3

(33)

∂Dj (θ )

where Cj (x2 ) ≜ ∂θ and mj (x2 ) are the torque sharing functions (Taylor, 1992), which define the stator phase switching and allows to minimize the torque/speed ripple. Note that x1d cannot be chosen freely but it is a function of τd in (16). Also, note the slightly ̇ abuse of notation in the definition of x^1dj in (30) and τ^ḋ . Using the electrical error definition, the control law (27), and the electric dynamics given by (4a), it follows

(25)

where

aj = a¯ j − (26a)

1 mj Cj (x2 ) JΛm2 . x1dj

e1̇ = A e e1 + Be.

(26b)

T Bo = ⎡⎣ 0 0 ⋯ 1⎤⎦ ,

(26c)

(35)

(36a)

where

with r¯2 = r (p), and Ao ∈ p × p , Bo ∈ p × 1. Also, {λ 0, … , λ p + 1} must be chosen in such a manner that the polynomial (15) of degree p + 2 is Hurwitz. 3.2. Electrical control loop To achieve the electrical control objective, i.e., x1 → x1d , the control law introduced in Espinosa-Pérez et al. (2004) was employed, but instead of the measured angular velocity an estimated value x^3 is used, i.e.,

(27)

where

(28)

m × m

is a matrix of gains, e1 ≜ x1 − x1d is the electrical error Kv ∈ and the desired per phase currents and their derivatives are given by

x1dj

⎟ ⎠

)

⎡ 0 1 … 0 ⎤ ⎥ ⎢ 0 0 1 ⎥ … Ao = ⎢ ⋮ ⋮ ⎥ ⎢ ⋮ ⎢⎣ − λ 0 − λ1 … − λ p − 1⎥⎦

⎧ 2m (x ) τ j 2 d ⎪ if Cj (x2 ) ≠ 0 , ≜⎨ Cj (x2 ) ⎪ ⎩0 otherwise

) ⎞⎟⎟,

1 mj (x2 ) Cj (x2 ) τ^ḋ , x1dj

Eq. (34) can be rewritten as

x^3 ≜ e^3 + x3d ,

∂x 2

C2j (x2 )

where

̇ u = D (x2 ) x^1d + x^3 C (x2 ) x1d + Rx1d − Kv e1,

∂C j (x 2 )

e1̇ = − D (x2 )−1(R + Kv + x3 C (x2 )) e1 − e˜3 D (x2 )−1C (x2 ) x1d + e˜3 aj , (34)

Eq. (24) can be rewritten as

e˜ ȯ = A o e˜ o + Bo r¯2,

a¯ j =

bj =

z˜1̇ = z˜2 − λ p − 1e˜2

⎡ e˜2 ⎤ ⎥ ⎢ ⎢ e˜2̇ ⎥ e˜ o = ⎢ , ⋮ ⎥ ⎢ (p + 1) ⎥ ⎥⎦ ⎢⎣ e˜2

⎛ τ 1 ⎜ d ⎜ x1dj ⎜ ⎝

(30)

(21)

and from (20c) and (21) it is

e˜2(p + 2) + λ p + 1e˜2(p + 1) + ⋯ + λ1e˜2̇ + λ 0 e˜2 = r (p) (t ).

⎪

A e = − D (x2 )−1(R + Kv + x3 C (x2 ))

(36b)

Be = − e˜3 D (x2 )−1C (x2 ) x1d + e˜3 aj .

(36c)

The complete closed loop dynamics is described by Eqs. (19), (26) and (36), for which the corresponding state vector is defined as

⎡ e1 ⎤ y = ⎢⎢ e 23⎥⎥. ⎣ e˜ o ⎦

(37)

Thus, the dynamics of y can be written as

ẏ = Ay + B,

(38a)

where

⎡ Ae O O ⎤ ⎢ ⎥ A = ⎢ O Am O ⎥ , ⎢⎣ O O A o ⎥⎦

⎡ Be ⎤ ⎥ ⎢ B = ⎢ B m r¯1⎥. ⎢⎣ BO r¯2 ⎥⎦

The main result of the paper is presented in the following Proposition 1. Consider the SR motor model (4) in closed loop with the control law (27), where the error e^3 is estimated by the observer (14) and x1d per phase is obtained using (29) with τd given by (16). Define a region D ∈ m × 2 × p + 1 as

D = {y ∈ m × 2 × p + 1| ∥ y ∥ ≤ ymax } (29)

(38b)

(39)

where ymax is a positive constant arbitrarily large. Suppose that the desired trajectory x3d and the load torque τ L are bounded with at least

A. De La Guerra et al. / Control Engineering Practice 46 (2016) 115–128

119

Fig. 1. Block diagram of the control system.

its first p time derivatives bounded. Then, a combination of roots of (15), gains Λm2 and Λm3 in (16), and Kv in (27), can be always found to make the tracking errors (e1, e2, e3 ) and the observation errors ( e˜ 2, e˜3, z˜1, … , z˜p ) bounded, as long as the initial condition y (t0 ) lies in a subset of D chosen small enough to prevent the state y (t ) leaving D during the transient response. Furthermore, if the roots of (15) are chosen far in the left-half of the complex plane, then y (t ) is ultimately bounded with ultimate bound arbitrary small. Moreover, the estimate z^1 becomes an arbitrarily close approximation of the actual value of z1. □ The proof of Proposition 1 is given in Appendix A. Remark 3. As in Arteaga-Pérez and Gutiérrez-Giles (2014) the boundedness of r (p) (t ) is not assumed a priori but formally proven. □ Remark 4. Given τd in (16), the scheme can be modified to track the angular position instead of the angular velocity just by changing the reference definition in (8) for

̇ . x3d ≜ x2d

(40)

□

Fig. 2. Dc motor (left) coupled with the SR motor (right). Table 1 SR motor parameters.

4. Experimental analysis of speed-sensorless controllers 4.1. Experimental setup An experimental analysis and comparison of three control schemes, including the proposed in this work, have been carried out using the setup described in Fig. 1. For the evaluation the controller was implemented in Matlab Simulink Real-Time with a sampling period of T = 0.5 ms.1 The DSP dSPACE DS1104 R&D controller board serves as the interface to the current and position sensors and implements the PWM signals, with a frequency of 20 kHz for the electronic converter. The electronic converter is an asymmetric bridge with a DC-bus voltage equal to 45 V . The position sensor is an encoder with a resolution of 1024 pulses/rev . A DC motor with a torque constant Kτ = 1.5556 Nm/A , is used as a variable torque load for the SR motor shown in Fig. 2. The SR motor nominal parameters are shown in Table 1. 4.2. Experimental results The GPI approach was compared with the PI2D controller (PI2D) in Loria et al. (2014) and with the passivity based controller (PBC) 1 Since our objective is the robust control at low speed, the sample time used for the experiments is adecuated. However, for speed control at high speeds it is mandatory to use a faster sample time in the electrical control loop, as explained in Rain, Hilairet, and Arias (2014).

Geometric Number of Number of Number of

parameters rotor poles stator poles phases

Electric parameters Maximum voltage Maximum current Winding resistance

8 12 3

Vmax = 120 V imax = 4 A r = 1.6 Ω

Coefficients of winding inductance

l0 = 0.05 mH l1 = 0.008 mH Mechanical parameters Rotor inertia

J = 0.0016 kg m2

reported in Espinosa-Pérez et al. (2004). These controllers share the same cascade structure i.e., an external control loop for speed control and an inner control loop for the stator current control. However, it is important to point out that the desired currents are calculated via Eq. (29) for the PBC and GPI approaches while in the PI2D scheme they depend on the rotor inertia parameter. Also, the stator phase commutation is the same for the three controllers, and it is based on the Instantaneous Torque Control (Miller, 2001) approach, a commutation scheme that profiles the phase currents to produce the demanded torque by coordinating the torque produced by the individual phases. This is made using the torque-sharing functions (Taylor, 1992), which are formed with

120

A. De La Guerra et al. / Control Engineering Practice 46 (2016) 115–128

polynomial functions of fifth order defined as

90

⎛ h ⎞5 ⎛ h ⎞4 ⎛ h ⎞3 pr (h) = 10 ⎜ ⎟, ⎟ + 6⎜ ⎟ − 15 ⎜ ⎝ θm ⎠ ⎝ θm ⎠ ⎝ θm ⎠ where h = |x2 − θm |, θm =

π 3Nr

80

(41)

70

and pf (h) = 1 − pr (h). The torque60

(when

∂Lj (x2 ) ∂x2

> 0) and polynomial pf(h) (when

∂Lj (x2 ) ∂x2

< 0) following

equations:

⎫ ⎧ ∂L j (x2 ) Θj+ = ⎨ x2 : > 0⎬ ⎭ ⎩ ∂x2 ⎧ ⎫ L x ∂ ( ) j 2 Θj− = ⎨ x2 : < 0⎬ , ⎩ ⎭ ∂x2

x [rad/s]

sharing functions mj (x2 ) are formed using the polynomial pr(h)

20 10

∀ x2 ∈ Θj+;

mj (x2 ) > 0

∀ x2 ∈ Θj−;

∑ mj (x2 )+ = 1 ∑ mj (x2 )− = 1

0

2

4

Remark 5. The syntonization of the GPI scheme is rather straightforward following the next empirical approach:

 The electrical control loop is a proportional gain, therefore the gain must be increased to reduce the current tracking error. For the mechanical control loop the gains Λm2, Λm3 corresponds to a PI controller which is tuned to obtain a stable behavior. The higher the degree of the polynomial p in (12) the better the approximation of the uncertain dynamics. However, in practice, up to a certain degree there is no advantage in increasing p, while the computational burden grows. As a matter of fact, in general p ¼3 turns out to be a good choice and therefore it is the value employed in this work. Finally, the observer poles should be chosen as far on the left in the complex plane as the bandwidth of the system allows it. □

In the next subsection, three experiments are presented, designed to show the performance of the controllers in different operating conditions. 4.2.1. Set point variation The first experiment considers a variable velocity reference, with no load torque, i.e., τ L = 0, in the range of 0 − 80 rad/s, corresponding with the low velocity interval of operation of the SR motor (the base velocity is 100 rad/s). It is worth noting, that this is the interval of velocities where the non-saturated model and the observer are valid (Miller, 2001). For the speed tracking control loop, Fig. 3 shows the desired and actual velocities for the three control laws, where it can be seen that the performance of the GPI and the PI2D controllers is pretty similar.

6

8

10

12

t[s]

∀ x2 .

These functions allow to write the generated torque as the sum of the torque delivered by each stator phase. Note that the conduction angles ( θon to switch on the phase, θoff to switch off the phase), necessary for the commutation of the SR motor (Miller, 2001), are implicitly calculated via the torque-sharing functions, which in turn obey the behavior of the inductances. The gains for the PBC electric control law are Kv = diag {30, 30, 30} and for the mechanical control law a ¼36, b¼40. For the GPI control, the gains for the electric control law are Kv = diag {30, 30, 30}, for the mechanical control law Λm2 = 40, Λm3 = 100 and for the observer (14) λ 0 = 750, λ1 = 2.25 × 105, (i.e., the λ2 = 3.375 × 107 , λ3 = 2.5313 × 109 , λ4 = 7.5938 × 1010 poles have been set as p1 = p2 = p3 = p4 = p5 = − 150). Meanwhile for the PI2D scheme, the gains used for the mechanical law are kd = 150, kp = 300, ki ¼ 0.1, a ¼320, b¼ 661, and for the electric law Kpx = diag {60, 60, 60}.



0

∀ x2

Fig. 3. Experiment 1, velocity tracking, ). x3PI 2D (

x3d (- - -),

x3GPI (

),

x3PBC (

),

20 15 10 5

e3

mj (x2 ) > 0



40 30

such as



50

0 −5 −10 −15 −20

0

2

4

6

8

10

12

t[s]

Fig. 4. Experiment 1, velocity tracking error e3, GPI(

), PBC(

), PI2D(

).

Note that the differences between the set point changes from 40 to 60 rad/s and from 60 to 80 rad/s are minimal for the GPI and the PI2D, being the later slightly fast for the first change and the former also slightly faster in the second change. Meanwhile, the PBC control has larger transient responses and overshoot always. Fig. 4 depicts the velocity error over all the experiment, with a zoom in the maximum velocity interval, where the velocity error for the proposed scheme and for the PI2D are smaller than for the PBC. Fig. 5 shows (a) the desire torque, τd , (b) the stator phase-1 current behavior and (c) the stator phase-1 torque sharing function. This figure presents the relationship between the torque sharing functions, the desired current for the phase-1 and the desired torque. Where it can be seen, how function m1 defines the conduction angles for the stator phase current one. Meanwhile, Fig. 6, shows the torque sharing functions for the three stator phases in the same interval. Fig. 7 shows the commutation for stator phase-1, where the ̇ phase current x1d1 and its derivative x^1d1 are different from zero when the inductance derivative C1 (x2 ) is positive, to obtain forward ̇ movement (the magnitude of C1 (x2 ) and x^1d1 has been multiplied by 50 and 0.001 respectively, for comparison). The torque-sharing

A. De La Guerra et al. / Control Engineering Practice 46 (2016) 115–128

121

τd [Nm]

0.4 0.2 0 −0.2 7.45

7.46

7.47

7.48

7.49

7.5

7.51

7.52

7.53

7.54

7.55

7.51

7.52

7.53

7.54

7.55

7.51

7.52

7.53

7.54

7.55

t[s]

x1 [A]

4 3 2 1 0 7.45

7.46

7.47

7.48

7.49

7.5

t[s]

m

1

1

0.5

0 7.45

7.46

7.47

7.48

7.49

7.5

t[s] Fig. 5. Experiment 1, (a) desired torque, (b)x1 (

), x1d (- - -), (c) torque sharing function m1.

1

m1, m2, m3

0.8

0.6

0.4

0.2

0

7.45

7.46

7.47

7.48

7.49

7.5

7.51

7.52

7.53

7.54

7.55

t[s] Fig. 6. Experiment 1, Torque sharing functions for stator phase-1, m1, phase 2, m2, phase 3, m3.

functions help to commutate every phase only when Cj (x2 ) and x1dj are different from zero in their corresponding cycle of operation, i.e. for every jth phase, as shown in Fig. 5. On the other hand, for the current control loop, Fig. 8 presents the current tracking of one stator phase for the higher speed interval in the experiment, where the error of the three controllers is comparable, (a) shows the actual and desired currents for the GPI approach while (b) presents the actual and desired currents for the PBC controller and (c) shows the actual and desired currents for the PI2D controller.

4.2.2. Constant load torque The second experiment considers a velocity reference defined by

x3d = 40 ( 1 + tanh (5 (t − 0.8)) ) [rad/s], with a constant load torque τ L = 0.85 Nm applied at time 0.8 s. A constant torque load is typical when fixed volumes are being handled. The velocity tracking is shown in Fig. 9, while the velocity tracking error is presented in Fig. 10, where it is clear that the GPI

122

A. De La Guerra et al. / Control Engineering Practice 46 (2016) 115–128

40

20

35

15

30

25

5

x [rad/s]

Conmutation

10

0

15

−5

10

−10

5

−15

−20 0.91

20

0

0.92

0.93

0.94

0.95

0.96

0.97

0

0.98

1

2

3

4

̇ Fig. 7. Experiment 1, Phase-1 Current x1d (—), Phase-1 Current derivative x^1d ( ). C1 (x2 )(

5

6

),

and PI2D controllers can compensate the load torque in contrast to the PBC approach that has a non-zero error throughout the whole experiment. For the inner control loop, Fig. 11 shows the current tracking for the three controllers: (a) shows the actual and desired currents for the GPI approach while (b) presents the actual and desired currents for the PBC controller and (c) shows the actual and desired currents for the PI2D controller. Just as before, the current error seems to be similar for the three controllers. 4.2.3. Variable load torque The third experiment has the same velocity reference than the former one, but the load torque, shown in Fig. 15, is variable. This load torque was designed to test the robustness of the controllers

Fig. 9. Experiment 2, velocity tracking, ). x3PI 2D (

x3d (- - -),

x1[A]

2 1 6.43

6.44

6.45

6.46

6.47

6.48

6.49

6.5

6.46

6.47

6.48

6.49

6.5

6.46

6.47

6.48

6.49

6.5

t[s] 5

1

x [A]

4 3 2 1 0 6.4

6.41

6.42

6.43

6.44

6.45

t[s] 5

1

x [A]

4 3 2 1 0 6.4

6.41

6.42

6.43

6.44

10

x3GPI (

),

x3PBC (

),

In order to have a better insight of the performance of the schemes presented above, the RMSE index given by (Arteaga-

3

6.42

9

4.3. Discussion

4

6.41

8

and therefore is not related to any practical scenario. The speed tracking is shown in Fig. 12, where it can be appreciated that the performance for the GPI and PI2D control laws is similar. This can be better appreciated in Fig. 13, that shows the velocity error, where it is clear that the GPI and PI2D controllers can compensate the effect of the load torque while the PBC controller has a constant error along the experiment. The current regulation for this experiment is presented in Fig. 14, where (a) corresponds to the GPI current tracking, (b) shows the PBC current tracking and (c) is the PI2D current tracking.

5

0 6.4

7

t[s]

t[s]

6.45

t[s] Fig. 8. Experiment 1, phase current, (a) GPI, (b) PBC, (c) PI2D, x1 (

), x1d (- - -).

A. De La Guerra et al. / Control Engineering Practice 46 (2016) 115–128 0.5

123

40

35

0 30

25

e

x [rad/s]

−0.5

−1

20

15

10

−1.5 5

−2

0

0

1

2

3

4

5

6

7

8

9

0

10

5

10

15

Fig. 10. Experiment 2, velocity tracking error e3, GPI(

), PBC(

), PI2D(

).

Pérez, Gutiérrez-Giles, & Weist, 2015)

RMSE =

1 n

20

25

30

t[s]

t[s]

Fig. 12. Experiment 3, velocity tracking, ). x3PI 2D (

x3d (- - -),

x3GPI (

),

x3PBC (

),

has gains that depends directly on the stability proof and on the model parameters.

n

∑ ei2 ,

(42)

i=1

has been calculated for the tracking velocity and tracking current error of stator phase one, where i is the current sample number, ei is the corresponding error and n is the total number of samples. Table 2 shows that the GPI controller is better in the first two cases. Only for the variable load torque the PBC is better for current tracking and the PI2D for velocity tracking. Still, the GPI is, in both cases, the second best and not for much, so altogether, it can be concluded it has a better performance. Also, it is worth noting that it requires less model parameters knowledge and tuning it, is more straightforward than for example the PI2D approach, which

5. Conclusions This paper presents a speed-sensorless control for the SR motor based on a GPI observer, designed to compensate variable load torque. The GPI observer can estimate the unknown perturbation input, in an arbitrarily close manner, thanks to the internal polynomial model of this input. The advantages of the proposed scheme are:

 It estimates the phase variables of the error position, what makes unnecessary to measure angular velocities.

x1[A]

6 4 2 0

9

9.02

9.04

9.06

9.08

9.1

9.12

9.14

9.16

9.18

9.2

9.12

9.14

9.16

9.18

9.2

9.12

9.14

9.16

9.18

9.2

t[s]

x1[A]

6 4 2 0

9

9.02

9.04

9.06

9.08

9.1

t[s]

x1[A]

4 2 0

9

9.02

9.04

9.06

9.08

9.1

t[s] Fig. 11. Experiment 2, phase current, (a) GPI, (b) PBC, (c) PI2D, x1 (

), x1d (- - -).

124

A. De La Guerra et al. / Control Engineering Practice 46 (2016) 115–128

 It does not needs the exact knowledge of the model parameters 

1.4

except for the rotor inertia. It compensates the unknown varying load torque.

1.2

The performance of the proposed scheme was compared with a similar scheme presented in Loria et al. (2014) by means of an experimental evaluation performed with a three phase 12/8 SR motor. The results of this experimental validation show that the GPI controller has a better performance for velocity tracking, with no load, in the range of 0–80 rad/s, and at operation with constant load torque at low velocity. At variable load torque operation the GPI approach shows to be competitive against the PI2D scheme which has a little better SRME index for velocity tracking.

Load torque [Nm]

1

0.8

0.6

0.4

0.2

1

0

0

5

10

15

20

25

30

t[s] 0.5

Fig. 15. Experiment 3, load torque.

Table 2 RMSE: root mean square error

e

0

Controller

RMSE for e1

RMSE for e3

Set point change GPI PI2D PBC

0.7125 0.9959 0.9531

3.8159 17.7647 14.4264

Constant load torque GPI PI2D PBC

0.5946 1.4673 0.9814

13.6498 15.1473 14.4434

Variable load torque GPI PI2D PBC

0.6356 0.7203 0.6308

0.2161 0.1568 0.2929

−0.5

−1

−1. 5

−2 0

5

10

15

20

25

30

t[s]

Fig. 13. Experiment 3, velocity tracking error e3, GPI(

), PBC(

), PI2D(

).

5

1

x [A]

4 3 2 1 0

9

9.02

9.04

9.06

9.08

9.1

9.12

9.14

9.16

9.18

9.2

9.12

9.14

9.16

9.18

9.2

9.12

9.14

9.16

9.18

9.2

t[s] 5

1

x [A]

4 3 2 1 0

9

9.02

9.04

9.06

9.08

9.1

t[s] 5

1

x [A]

4 3 2 1 0

9

9.02

9.04

9.06

9.08

9.1

t[s] Fig. 14. Experiment 3, phase current, (a) GPI, (b) PBC, (c) PI2D, x1 (

), x1d (- - -).

A. De La Guerra et al. / Control Engineering Practice 46 (2016) 115–128

Acknowledgement

e¨2 + Λm3 e2̇ + Λm2 e2 =

This work was supported by CONACYT México, as scholarship for the corresponding author, CVU. 332351.

Appendix A. Proof of Proposition 1 In this appendix Proposition 1 is proven following the next three steps, (Gutiérrez-Giles & Arteaga-Pérez, 2014). 1. A local stability analysis is carried out, which is only valid in the region D (39). Since it cannot be guaranteed a priori that every signal of interest is bounded whenever y ∈ D, the first step is to check it out. First of all, after (21) it is clear that z˜1 is bounded, given that (e˜ 2, e˜ 2̇ , … , e˜2(p + 1) ) are bounded in D. Actually, one can also calculate its derivatives as

z˜1̇ = e˜2(3) + λ p + 1e˜¨2 + λ p e˜2̇

⋮

(A.1)

⋮

z˜1(p − 1)

(p ) = e˜2(p + 1) + λ p + 1e˜¨ + λ p e˜2(p − 1) ,

(A.2)

which means that (z˜1, … , z˜1(p − 1) ) are bounded, (but not z˜1(p) nor

z˜1(p + 1)

up to this point!). On the other hand, from Eqs. (20c) to (20e) it can be shown that (z˜2, … , z˜p, z˜2̇ , … , z˜ṗ − 1) are bounded too. Moreover, from (20a) e˜3 is bounded and from (20b) e˜3̇ is bounded. Whenever y is bounded it is clear that (e1, e2, e3 ) are bounded too. From (7), since x3d is bounded by assumption, then x3 must be bounded, and z1 in (11) must be bounded because the load torque τ L is assumed to be bounded. Now, if (16) is rewritten as

it can be seen that τd is bounded. Again, from the definition of error z˜1, one can write z^1 = z1 − z˜1, thus z^1 is bounded. In the same fashion, from the definition of τ given by (5) and from the fact that the expression for the desired current comes from a system's inversion, it is

τ − τd =

1 T 1 T x1 C (x2 ) x1 − x1d C (x2 ) x1d. 2J 2J

(A.3)

As in Espinosa-Pérez et al. (2004), by taking norms and from the triangle inequality, (A.3) can be shown to satisfy

∥ τ − τd ∥ ≤ α1 ∥ e1 ∥2 + α2 ∥ x1d ∥∥ e1 ∥, where α1 > 0 and α2 > 0 are constants. Given that e1 is bounded, one only needs to show that x1d is bounded. The desired current given by (29), is a function of τd and x2, i.e.,

x1d = f1 (τd, f (x2 )),

3

By writing (17) as

e¨2 + Λm3 e2̇ + Λm2 e2 =

τ = f2 (x1, f (x2 )),

1 (τ − τd ) + Λm3 e˜3 + z˜1, J

and by combining (20a) and (A.5) one gets

(A.6)

(A.7)

which means that

τ ̇ = f2̇ (x1, x1̇ , f (x2 ), f ̇ (x2 ), x3 ),

(A.8)

where f ̇ (x2 ) is the time derivative of f (x2 ). To show that τ ̇ is bounded one needs to show that x1̇ is bounded. From the defini̇ , it is necessary to show that tion of the current error, x1̇ = e1̇ + x1d ̇ and e1̇ are bounded. For the later, given that the right hand side x1d of Eq. (34) is bounded, so that e1̇ is bounded. Then, from (A.4) the desired current is a function of (τd, x2 ) and from (30) it is

̇ = f1̇ (τd, τḋ , f (x2 ), f ̇ (x2 ), x3 ), x1d

(A.9)

̇ is computed here only for stability analysis but is not Note that x1d ̇ used for implementation. Instead, x^1d in (30) is employed. Then to ̇ is bounded, it remains to show that the first derishow that x1d vative of τd is bounded. From (16) it is

τḋ = J

( −Λ

m2

)

̇ ̇ e2̇ − Λm3 e^3 − z^1 .

(A.10)

From the definition of z1 in (11), one can write

d 1 x3̇ − τ L̇ − x¨3d , J J

(A.11)

where given that x¨2 = x3̇ , τ L̇ (t ) and x¨3d are bounded, then z1̇ (t ) is ̇ bounded. Given that z˜1̇ is bounded then from z^1 = z1̇ − z˜1̇ it can be said ̇ ̇ is bounded that z^1 is bounded too. Hence τḋ is bounded, whence x1d and consequently x1̇ is also bounded; recalling that e1̇ is bounded. ̇ can be written as Finally, τ ̇ is bounded too. Additionally, x1d

̇ = a¯ j x3 + bj (τ^ḋ ), x1d

(A.12)

and from (30) and (35) it follows

̇ ̇ − x^1d x1d = a¯ j x3 + bj (τḋ ) − a¯ j x^3 + bj (τ^ḋ ),

(A.13)

which is a function of only bounded arguments. ̇ On the other hand, if z^1 is bounded then from (14c) z^2 is also bounded and from z2 = z˜2 + z^2 , z2 is bounded. Actually, from (A.6) one has

(A.4)

where f (x2 ) ≜ C (x2 ). Because x2 is only argument of bounded trigonometric functions and τd is bounded, then x1d is bounded. Thus, it can be said that τ − τd is bounded. Moreover, from the current error definition it is x1 = e1 + x1d , meaning that x1 is bounded. In addition, from (17), e3̇ is bounded. If e3̇ and z1 are bounded, from (10), τ is bounded too. Also, given that the right hand side of (4c) is bounded, then x3̇ is bounded. Again from the definition of e˜3, it is ̇ e^3 = e3 − e˜3, thus e^3 is bounded. From (14a) it can be seen that e^2 is ̇ bounded. Furthermore, from (14b) it is clear that e^ is also bounded.

1 (τ − τd ) + Λm3 e˜2̇ + Λm3 λ p + 1e˜2 + z˜1, J

which means that e¨ 2 is bounded. Thus, from the definition of e2, ̇ one can write x¨2 = e¨ 2 + x¨2d , so that x¨2 is bounded, because x¨2d = x3d is bounded by assumption. Furthermore, from Eq. (5) it can be seen that τ is a function of (x1, x2 ), i.e.,

z1̇ = −

τd = J ( − Λm2 e2 + Λm3 e˜3 − Λm3 e3 − z1 + z˜1),

125

e2(3) = − Λm3 e¨ 2 − Λm2 e 2̇ +

then

e2(3) x2(3)

1 (τ ̇ − τḋ ) + Λm3 e˜¨2 + Λm3 λp + 1e˜ 2̇ + z˜1̇ , J

is bounded and given that

(3) x2d

(A.14)

= x¨3d is assumed bounded

then is bounded too. From (A.11) and (4b) one can write

z¨1 = −

d 1 (3) x¨2 − τ¨ L − x3d . J J

(A.15)

Given that τ¨L is bounded by assumption, then z¨1 = z2̇ = z3 are bounded and consequently from the definition of z˜2, one can write ̇ ̇ z^2 = z˜2̇ + z2̇ thus z^2 = z^3 are bounded. From (A.10) τ¨d can be written as

(A.5) τ¨d = J

( −Λ

m2

)

¨ ¨ e¨2 − Λm3 e^3 − z^1 ,

(A.16)

126

A. De La Guerra et al. / Control Engineering Practice 46 (2016) 115–128

¨ since from (14b) it can be shown that e^3 is bounded, then τ¨d is also bounded. Again from (A.9) one has

x¨1d = f¨1 (τd, τḋ , τ¨d, f (x2 ), f ̇ (x2 ), f¨ (x2 ), x3, x3̇ ),

(A.17)

with f¨ (x2 ) the second time derivative of f (x2 ), so x¨1d is bounded since it is a function of bounded signals. Moreover, from (34) one has

̇ , f (x2 ), x1d, e˜2 , e˜2̇ ), e1̇ = f3̇ (e1, e2̇ , x2d

(A.18)

then one can write ̇ , e˜ 2, e˜ 2̇ , e˜¨2 ), ̇ , x¨2d, x3, f (x2 ), f ̇ (x2 ), x1d, x1d e¨ 1 = f¨3 (e 1, e 1̇ , e 2̇ , e¨ 2, x2d

(A.19)

thereupon e¨1 is bounded. As well, from the definition of the current error it can be shown that x¨1 is also bounded. Correspondingly, from (A.8) one has

τ¨ = f¨2 (x1, x1̇ , x¨1, f (x2 ), f ̇ (x2 ), f¨ (x2 ), x3, x3̇ ),

(A.20)

which is bounded since it is function of bounded signals. From (A.14) one has

e2(4) = − Λm3 e2(3) − Λm2 e¨2 +

1 (τ¨ − τ¨d ) + Λm3 e˜2(3) J

+ Λm3 λ p + 1e˜¨2 + z˜¨1,

(A.21)

which implies that e2(4) is bounded and consequently x2(4) is bounded too. Hence, by following an iterative procedure, it can be shown after (A.14) and (A.15) that

z1(p)

d 1 (p + 1) = − x2(p + 1) − τL(p) − x2d , J J

(A.22)

hence z1(p) is bounded. Therefore, from (10)–(13d), zṗ is bounded which in turn implies that r (p) (t ) is also bounded. Accordingly, from (24), e˜2(p + 2) is bounded. This concludes the first part of the proof. 2. Once it has been shown that all the signals of interest are bounded in the region D. It must be proven that the state y is ultimately bounded provided the initial condition y (t0 ) is small enough. As can be seen in Fig. 16, the initial condition y (t0 ) must begin in a subregion of D, say Bμ , small enough to guarantee that y will never leaves D. As time increases, y will enter and stay in the region Br , which can be made arbitrarily small. Consider system (26), where the matrix Ao in (26c) has been chosen to have all its eigenvalues with negative real part and different. Since this is a time invariant system it must hold (Arteaga-Pérez & Gutiérrez-Giles, 2014)

∥ e˜ o(t − t 0 ) Ao ∥ ≤ ke−γ (t − t 0 ) ,

k ∥ Bo ∥ sup ∥ r¯2 (ϑ)∥, γ t 0 ≤ϑ≤ t

As well, consider the Lyapunov equation given by

A oT P1 + P1A o = − Q 1

k ∥ Bo ∥ r2M, γ

which can be made arbitrarily small since large.

(A.24)

γ can be set arbitrarily

Remark 6. Note that in fact the observation errors e˜o can be made arbitrarily small independently of the rest of the state vectors e1, e 2, e3 and that the initial condition e˜o (t0 ) could be arbitrarily

(A.25)

where P1 and Q1 are positive definite matrices since the system is stable. Therefore the following function is positive definite:

V1 = e˜ oT P1e˜ o,

(A.26)

where it can be shown that its derivative along (26a) is given by

V1̇ = − e˜ oT Q 1e˜ o + 2e˜ oT P1Bo r¯2.

(A.27)

Thus it holds

V1̇ ≤ − λm1 (Q 1)∥ e˜ o ∥2 + 2λM1 (P1)∥ e˜ o ∥∥ Bo ∥ r2M = − ∥ e˜ o ∥ ( λm1 (Q 1)∥ e˜ o ∥ − 2λM1 (P1)∥ Bo ∥ r2M ).

(A.28)

Then, V1̇ ≤ 0 if

∥ e˜ o ∥ ≥

2λM1 (P1) ∥ Bo ∥ r2M, λm1 (Q 1)

(A.29)

with ultimate bound given by

∥ e˜ o ∥ ≤

where it has been shown that r¯2 = r (p) (t ) is bounded in D, so that supt0≤ϑ≤ t ∥ r¯2 (ϑ)∥ can be replaced by a constant value, say r2M . Therefore the ultimate bound for ∥ e˜o ∥ is given by

∥ e˜ o ∥ ≤

large. However, to achieve the velocity tracking objective it is necessary to have a small enough initial condition for e˜o .

(A.23)

with γ = |λ1| = λM1 (Ao ) and k = p + 1. Note that, |λM1 (Ao )| is the smaller magnitude of all of them. In addition, the state is bounded by

∥ e˜ o ∥ ≤ ke−γ (t − t 0 ) ∥ e˜ o (t0 )∥ +

Fig. 16. Ultimate boundedness of the state y .

2λM1 (P1) λM1 (P1) ∥ Bo ∥ r2M, λm1 (Q 1) λm1 (P1)

(A.30)

this implies that the state is ultimately bounded as in GutiérrezGiles and Arteaga-Pérez (2014). Since λM1 (P1) /λm1 (P1) ≥ 1 and ∥ Bo ∥ r2M is fixed in D, it means that the term

2λM1 (P1) , λm1 (Q 1)

(A.31)

can be made arbitrarily small since from (A.24) it is known that the ultimate bound can be made arbitrarily small2 2 The pair ( P1, Q 1) is not unique. However, there must exist a pair such that Eqs. (A.25) and (A.30) are equivalent. We assume that this is the case.

A. De La Guerra et al. / Control Engineering Practice 46 (2016) 115–128

127

In a similar way, for the mechanical errors consider the Lyapunov equation

∥ e1 ∥ ≤

T Am P2

here, by similar arguments as the pointed above, the term,

+ P 2 Am = − Q 2

(A.32)

where P2 and Q 2 are positive definite matrices since the gains Λm2 and Λm3 are chosen in order for the system in (19) to be stable. The function T V2 = e23 P 2 e 23,

(A.33)

is positive definite. Thus it follows that the derivative of (A.33) along (19a) is T T V2̇ = − e23 Q 2 e 23 − 2e23 P 2 B m r¯1,

(A.35)

where supt0≤ϑ≤ t ∥ r¯1 (ϑ)∥ = r1M, given that from (19c) r¯1 is the sum of bounded signals in D. Then, V2̇ ≤ 0 if

∥ e˜ 23 ∥ ≥

2λM2 (P 2 ) ∥ B m ∥ rmax, λm2 (Q 2 )

(A.36)

its ultimate bound is given by

∥ e˜ 23 ∥ ≤

2λM2 (P 2 ) λM2 (P 2 ) ∥ B m ∥ rmax, λm2 (Q 2 ) λm2 (P 2 )

can be made arbitrarily small. 3. By adding V1 in (A.26), V2 in (A.33) and Ve in (A.39), a positive definite function for system (38) is given by T V (y ) = V1 + V2 + Ve = e˜ oT P1e˜ o + e23 P 2 e 23 +

λm ∥ y ∥2 ≤ V (y ) ≤ λM ∥ y ∥2 .

1 Ve = e1T D (x2 ) e1, 2

(A.39)

whose derivative along system (34) is

⎛ ⎞ λ (P ) V̇ ≤ − λm1 (Q 1)∥ e˜ o ∥ ⎜ ∥ e˜ o ∥ − 2 M1 1 ∥ Bo ∥ r2M ⎟ Q λ ( ) ⎝ ⎠ m1 1 ⎛ ⎞ λ (P ) − λm2 (Q 2 )∥ e 23 ∥ ⎜ ∥ e 23 ∥ − 2 M2 2 ∥ B m ∥ r1M ⎟ λm2 (Q 2 ) ⎝ ⎠

where |e˜3 | ≤ rf and ∥ x1d ∥ ≤ x1M , ∥ aj ∥ ≤ x1Mp in D. By choosing the eigenvalues of Ao and Am with real part negative and large enough, and by setting the value of the gain k v large enough as well, the terms

2

λM1 (P1) , λm1 (Q 1)

λM2 (P 2 ) , λm2 (Q 2 )

1 , kv

(A.48)

if

∥ y ∥ ≥ μ,

(A.49)

⎧ 2λ (P )∥ Bo ∥ r2M 2λM2 (P 2 )∥ B m ∥ rmax μ = max ⎨ M1 1 , λm λm ⎩

⎛ ⎞ x Vė = − e1T ⎜ R + Kv1 + 3 C (x2 ) ⎟ e1 − e1T Kv2 e1 + e˜3 e1T C (x2 ) x1d ⎝ ⎠ 2 aj,

2

can be made arbitrarily small. In consequence, it holds

1

+

(A.47)

where

It is defined Kv ≜ Kv1 + Kv2 where Kv1 = Kv2 ≜ 2 k v I , with I the m × m identity matrix. Then, (A.40) can be written as

e1T D (x2 ) e˜3

kv k ∥ e1 ∥2 − v ∥ e1 ∥ ( ∥ e1 ∥ − (c1rf x1 M + d1x1Mp ) ). 4 4

V (y ) ≤ 0

x Vė = e1T D (x2 ) e1̇ − 3 e1T C (x2 ) e1 2 ⎞ x T⎛ = e1 ⎜ − (R + Kv ) e1 + 3 C (x2 ) e1 − e˜3 C (x2 ) x1d + e˜3 aj D (x2 ) ⎟. ⎝ ⎠ 2

(A.46)

By taking into account Eqs. (A.29), (A.36) and (A.43), the derivative of function (A.45) satisfies

−

can be made arbitrarily small. On the other hand, for the electrical errors, consider the following positive definite function, (Espinosa-Pérez et al., 2004):

(A.45)

Note that there must exist two positive constants λm and λM such that

where, by similar arguments as the pointed out above,

(A.38)

1 T e1 D (x2 ) e1 2

⎡ P1 O O ⎤ ⎢ ⎥ = yT ⎢ O P 2 O ⎥ y ≜ yT M (x2 ) y. ⎢⎣ O O D (x2 )⎥⎦

(A.37)

2λM2 (P 2 ) λm2 (Q 2 )

(A.43)

(A.44)

and it holds

= − ∥ e 23 ∥ ( λm2 (Q 2 )∥ e 23 ∥ − 2λM2 (P 2 )∥ B m ∥ r1M ),

λM (D (x2 )) ( c1|e˜3 |∥ x1d ∥ + d1|e˜3 |∥ aj ∥), λm2 (D (x2 ))

4 , kv

(A.34)

V2̇ ≤ − λm2 (Q 2 )∥ e 23 ∥2 + 2λM2 (P 2 )∥ e 23 ∥∥ B m ∥ r1M

4 kv

,

4 (c1rf x1M + d1x1Mp ) ⎫ ⎬, λm ⎭

with

λm = min {λm1 (Q 1), λm2 (Q 2 ), k v } . (A.40)

(A.50)

Therefore μ can be made arbitrarily small. After (A.46) it follows that y must be ultimately bounded by

where the quadratic term −e1T (R + Kv1) e1 can be used to comx pensate 23 e1T C (x2 ) e1 as in Espinosa-Pérez et al. (2004). Thus, (A.40) can be shown to satisfy

∥y∥≤

Vė ≤ − e1T Kv2 e 1 − e˜ 3 e1T C (x2 ) x1d ⎛k ⎞ k ≤ − v ∥ e 1 ∥2 − ∥ e 1 ∥ ⎜ v ∥ e 1 ∥ − |e˜ 3 |(c1 ∥ x1d ∥ − d1 ∥ a j ∥) ⎟ . ⎝ 4 ⎠ 4

Additionally it is necessary to have ∥ y ∥ ≤ ymax in order for the state to stay in D for all time. This can be achieved if the initial condition satisfies

(A.41)

Then, Vė ≤ 0 if

⎛ c1|e˜3 |∥ x1d ∥ + d1|e˜3 |∥ aj ∥ ⎞ ∥ e1 ∥ ≥ 4 ⎜ ⎟, kv ⎝ ⎠

λM μ ≜ b. λm

∥ y (t0 )∥ < (A.42)

by Property 1, the ultimate bound for ∥ e1 ∥ is then given by

λm y . λM max

(A.51)

(A.52)

Accordingly, the trajectories of system (38) are uniformly ultimately bounded with ultimate bound defined by (A.51). This means that, the elements of y can be made arbitrarily small which

128

A. De La Guerra et al. / Control Engineering Practice 46 (2016) 115–128

implies that the estimation and tracking errors can be made approximately zero and after the discussion on the first step, all the signals of interest are bounded. This concludes the proof. □

Appendix B. Supplementary data Supplementary data associated with this paper can be found in the online version at http://dx.doi.org/10.1016/j.conengprac.2015. 10.010.

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