Low-Cost Implementation of Passivity-Based Control and Estimation of Load Torque for a Luo Converter with Dynamic Load

Article Low-Cost Implementation of Passivity-Based Control and Estimation of Load Torque for a Luo Converter with Dynamic Load

Ganesh Kumar Srinivasan 1,* , Hosimin Thilagar Srinivasan 1 and Marco Rivera 2

1 Department of Electrical and Electronics Engineering, Anna University, Chennai 600025, Tamil Nadu, India; [email protected] 2 Department of Electrical Engineering, Centro Tecnologico de Conversión de Energía, Faculty of Engineering, Universidad de Talca, Curico 3465548, Chile; [email protected] * Correspondence: [email protected]; Tel.: +91-9791-071-498

Received: 9 October 2020; Accepted: 1 November 2020; Published: 13 November 2020

Abstract: In this paper, passivity-based control (PBC) of a Luo converter-fed DC motor is implemented and presented. In PBC, both exact tracking error dynamics passive output feedback control (ETEDPOF) and energy shaping and damping injection methods do not require a speed sensor. As ETEDPOF does not depend upon state computation, it is preferred in the proposed work for the speed control of a DC motor under no-load and loaded conditions. Under loaded conditions, the online algebraic approach in sensorless mode (SAA) is used for estimating diﬀerent load torques applied on the DC motor such as: constant, frictional, fan-type, propeller-type and unknown load torques. Performance of SAA is tested with the reduced order observer in sensorless mode (SROO) approach and analyzed, and the results are presented to validate the low-cost implementation of PBC for a DC drive without a speed and torque sensor.

Keywords: DC motor; Luo converter; passivity-based control; load torque estimation

1. Introduction Electrical motors are the workforce in industrial applications. They are generally needed to work under variable speeds and for unknown load torque scenarios. DC and AC motors are used for this purpose. Though an induction motor is robust, cost-effective and cheaper in maintenance, a DC motor is preferred due to its good dynamic response in comparison to an AC motor, simple controlling methods and wide speed control. Due to these reasons, the DC motor is extensively used in rolling mills, paper machines and unwinding and rewinding machines. The feedback regulation of a DC motor is achieved through the pulse width modulation (PWM) of the DC-DC converter. The typical way of controlling DC-DC converters relies on an averaged, small-signal model, facilitating the application of linear control theory [1] which yields satisfactory results in many applications. In some situations, this approach offers limited performance or may even fail. Such situations may require tight tracking of the reference voltage during sudden load variations. Passivity-based control (PBC) will alleviate this kind of problem [1]. It is based on the energy calculations in the system. If the model of any system is converted to its energy perspectives, PBC can be implemented easily. PBC is developed for achieving stabilized output trajectories based on the required reference trajectories. In PBC, all the systems are modeled in terms of conservative forces, dissipative forces and the energy acquisition part. The control function of PBC for any system is derived in such a way that an error in the energy satisfies Lyapunov’s stability along with the dissipation matching condition.

Electronics 2020, 9, 1914; doi:10.3390/electronics9111914 www.mdpi.com/journal/electronics Electronics 2020, 9, 1914 2 of 25

Robustness, a stability feature of the PBC approach in various applications, confirms the selection of PBC [2–7] in the present work. Further, PBC outperforms the proportional–integral controller (PIC) in the control of the power flow [4]. Tzann-Shin Lee proved a better performance of PBC + PIC in power converter circuits. Tofighi et al. achieved a good tracking response with PBC in comparison with PIC [5]. The authors demonstrated the robustness of PBC in a photovoltaic power management system [5]. In synchronous reluctance motor drive systems [6] and in DC drive systems, PBC outperforms PIC. From the above, it is concluded that PBC performs better than other controllers. Due to this, PBC finds applications in microgrids [8,9], inverters [10], wind power systems [11], HVDC [12], power converter-based applications [13–22], robotics [23–25], MEMS [26], ball and beam dynamical systems [27] and photovoltaic systems [28]. In PBC, the exact tracking error dynamics passive output feedback (ETEDPOF) method is preferred for the present work in comparison with energy shaping and damping injection (ESDI) due to the absence of controller state computation in ETEDPOF [29]. Linares-Flores J et al. [30] implemented ETEDPOF for a DC motor under loaded conditions with two load torque estimation schemes, namely: the online algebraic approach and the reduced order observer in sensorless mode approach with a speed sensor [30]. In continuation of that, Ganesh Kumar S and Hosimin Thilagar attempted the same approaches for a DC motor without any speed sensor which is a fourth-order flat system [31]. The schemes are termed as online algebraic approach in sensorless mode (SAA) and reduced order observer approach in sensorless configuration (SROO). A system is called flat when the order of the system equals to the relative degree and the corresponding reference profiles can be generated easily. Hence, it is decided to propose the ETEDPOF technique with SAA and SROO schemes for a sixth-order partially flat system, namely: a Luo converter with dynamic load without any speed as well as torque sensor. This paper is organized as follows: controller development and estimation of applied load torque for the Luo converter system are presented in Section2. Results and discussions are made in Section3. The conclusions are given in Section4.

2. Development of Controller and Estimation of Torque with Reduced Number of Sensors The ETEDPOF method [31] is developed from the energy managing structure of the error dynamics, which can be placed in generalized Hamiltonian form, identifying the passive output associated with this stabilization of the error dynamics. Based on these error dynamics, a simple linear, time-invariant feedback controller may be synthesized which will achieve the desired equilibrium point into a semi-globally asymptotically stable equilibrium for the closed-loop system, provided a structural dissipation matching condition would be satisﬁed [2]. This chapter is organized as follows: the general procedure for controller development is presented in Section 2.1. ETEDPOF and load torque schemes for the Luo converter with dynamic load are presented in Section 2.2.

2.1. General Procedure for ETEDPOF Development The Hamiltonian of any system represents an energy function (H), and it can be written in terms of state vector “x”, which is given in Equation (1):

1 H = xTMx (1) 2 where M is a positive-deﬁnite and constant matrix. On taking the partial derivative of Equation (1) with respect to “x”, Equation (2) is obtained.

!T ∂H = Mx (2) ∂x Electronics 2020, 9, 1914 3 of 25

Generally, any averaged state model can be represented in Equation (1)

!T . ∂H(x) x(t) = ( J(u) R ) + bu+ (3) − ∂x ∈

Matrix J is skew-symmetric in nature which will not aﬀect the system stability as

xTJ(u)x = xJ(u)xT = 0 (4)

The term “bu” is the energy acquisition term. External disturbances like load torque are introduced in “ ”. R is symmetric and positive-semi-deﬁnite, ∈ i.e., RT = R 0 (5) ≥ The skew symmetric matrix J(u) can be written as

J(u) = J0 + J1u (6)

The main objective of the present work is to regulate the speed of a DC motor for a given desired speed profile (ω∗) under no-load and load conditions. The desired speed profile is obtained in an offline fashion using bezier polynomials. For this desired speed profile (ω∗), it is assumed that a state reference trajectory x*(t) satisfies the desired open-loop dynamics and it is given by

!T . ∂H(x∗) x∗(t) = (J(u∗) R) + bu∗+ ∗ (7) − ∂x∗ ∈

The matrix J(u) is skew-symmetric which is related to an average control input “u” and it satisﬁes the following expansion property:

∂J(u) J(u) = J(u∗) + (u u∗) (8) − ∂u

∂J(u) where ∂u is a skew symmetric constant matrix. Under steady-state conditions, state trajectory “x” will reach x* and control input “u” becomes u*. Now, Equation (7) is modiﬁed to

!T ∂H(x∗) 0 = [J(u∗) R] + bu∗ + (9) − ∂x∗ ∈

In order to implement closed-loop operation, errors in the system’s state trajectories should be identiﬁed to modify the control input. These errors in the state trajectory (e) and control input (eu) are calculated as given below: !T ∂H(e) e = x x∗ = (10) − ∂e

eu = u u∗ (11) − On diﬀerentiating Equation (10) with respect to time “t”,

. . e = x (12) Electronics 2020, 9, 1914 4 of 25

On adding and subtracting the required steady-state values to Equation (12), this yields

!T !T . ∂H(e) ∂H(x∗) e = [ J(u) R] + beu+ +[ J(u) R] + bu∗ (13) − ∂e ∈ − ∂x∗

The value of obtained from Equation (9) is substituted in Equation (13), then the error dynamics ∈ will become !T !T . ∂H(e) ∂H(x∗) e = [ J(u) R] + beu + [J(u) J(u∗)] (14) − ∂e − ∂x∗ On using Equation (7) in Equation (14), the error dynamics of the system are modiﬁed into

!T  ! !T . ∂H(e)  ∂J(u)  ∂H(x∗)  = [ ( ) ] +  +   e J u R b  eu (15) − ∂e ∂u ∂x∗

!T !T . ∂H(e) ∂H(e) e = J(u) R + beˇ u (16) ∂e − ∂e where  ! !T  ∂J(u)  ∂H(x∗)  ˇ =  +   b b   (17) ∂u ∂x∗ T ( ) ∂H(e) In Equation (13), “J u ∂e ” is a conservative term which will not aﬀect the stability property of the system, ! !T ∂H(e) ∂H(e) i.e., J(u) = 0 (18) ∂e ∂e Other remaining terms exactly coincide with the tangent linearization part of the dynamics [2]. The passive output tracking error is given by

!T T ∂H(e) ey = y y∗= bˇ (19) − ∂e

A linear time-varying average incremental passive output feedback controller is simply given by

!T T ∂H(e) eu = γe = γbˇ (20) − y − ∂e

On substituting Equation (20) in Equation (16), the error dynamics of the system will become

!T !T !T . ∂H(e) ∂H(e) T ∂H(e) e = J(u) R bˇγbˇ (21) ∂e − ∂e − ∂e

!T !T . ∂H(e) T ∂H(e) e = J(u) R + bˇγbˇ (22) ∂e − ∂e . Now, with the skew symmetric property of J(u), H(e) is given by

! !T . ∂H(e) ∂H(e) H(e) = eR < 0 (23) − ∂e ∂e

Equation (23) is negative-deﬁnite if the dissipation matching condition (24) is satisﬁed.

T eR = R + bˇγbˇ (24) Electronics 2020, 9, 1914 5 of 25

In general, satisfaction of the dissipation matching condition will be in two forms and both forms are explained below: Case I: eR > 0 This indicates that the dissipation matching condition is strictly satisfied and Equation (18) will become negative-definite. Case II: eR 0 ≥ Here, the dissipation matching condition is not strictly satisfied. For these cases, LaSalle’s theorem [4] is used to establish the global asymptotic stability of the origin of the tracking error space. LaSalle’s theorem can be stated as follows: suppose that there exists a positive-definite function, . H: Rn R, whose derivative satisfies the inequality H 0 and if, moreover, the largest invariant set → . ≤ contained in the set {x: H = 0 } is equal to {0}, then the system is globally asymptotically stable. Due to the bounded nature of the control input [0 & 1] in power electronic converters, the system can be semi-globally asymptotically stable. Now, the dissipation matching condition and the average control input are given by

 ! !T  ! !TT T  ∂J(u) ∂H(x∗)   ∂J(u) ∂H(x∗)  + ˇγˇ = e = +  + γ +  R b b R R b  b  (25) ∂u ∂x∗ ∂u ∂x∗

!T T ∂H(e) T u = u∗ γbˇ = u∗ γbˇ Me (26) − ∂e − where “γ” represents the damping injection coeﬃcient whose value can be considered of low value to avoid noise ampliﬁcation. Thus, the control input for regulating the speed of the DC motor is obtained. The control function (u) in Equation (26) clearly indicates the absence of the derivative term which makes the controller simpler.

2.2. ETEDPOF and Estimation of Torque in Sensorless Mode for the Luo Converter System Earlier differentially flat and non-flat systems of order four were considered for sensorless load torque estimation and ETEDPOF implementation [22,31]. In this paper, a sixth-order non-flat system is proposed for analysis. One example of a non-flat sixth-order system is a Luo converter-fed DC motor. Out of many pump circuits, the fundamental Luo converter is taken for analysis. The relative degree of the Luo converter-fed DC motor is three which is less than the order of the system. This confirms the unstable internal dynamics of the Luo converter inductor current and capacitor voltage with respect to the speed. Hence, reference trajectories are developed in an indirect manner. Figure1 represents the low-cost load torque estimation of the Luo converter with dynamic load using the feedback signal’s armature voltage (v2) and armature current (iam), whereas ETEDPOF can be implemented with inductor currents (i1 and i2) and capacitor voltage (v1). With the available wide variety of pump circuits, the fundamental positive output Luo converter is taken for the present work. Using Kirchhoff’s laws and Newton’s laws, a linear averaged model for a Luo converter-fed DC motor can be derived. Due to the selection of the armature control method for speed control, field circuit equations are omitted. The derived linear averaged model is given by

di1 uE (1 u) = − v1 (27) dt L1 − L1

di uE uv v 2 = + 1 2 (28) dt L2 L2 − L2 dv (1 u)i ui 1 = − 1 2 (29) dt C1 − C1 ElectronicsElectronics2020 2020, ,9 9, 1914, x FOR PEER REVIEW 65 of of 25 26

Now, the dissipation matching condition and the average control input are given by dv2 i2 iam = (30) ∂J(u)dt C∂H2 −(x∗C)2 ∂J(u) ∂H(x∗) (25) R + bγb =R=R+b+ ∗ γ b + ∗ diam ∂uv2 Rm∂x k ∂u ∂x = iam ω (31) dt Lm − Lm − Lm ( ) ∗dω k∂H e B ∗TL u=u − γ=b iam ω=u − γb Me (26)(32) dt J ∂e− J − J where:where "γ” represents the damping injection coefficient whose value can be considered of low value to avoid noise amplification. -i 1—Inductor (L1) current (Ampere) Thus, the control input for regulating the speed of the DC motor is obtained. The control -i 2—Inductor (L2) current (Ampere) function (u) in Equation (26) clearly indicates the absence of the derivative term which makes the -v —Capacitor (C ) voltage (Volt) controller1 simpler. 1 -v 2—Capacitor (C2) voltage (Volt) -i am—Motor armature current (Ampere) 2.2. ETEDPOF and Estimation of Torque in Sensorless Mode for the Luo Converter System - ω—Angular velocity 2πN (rad/seconds) Earlier differentially flat60 and non-flat systems of order four were considered for sensorless load -torque k—Torque estimation constant and (N.ETEDPOF m/A) implementation [22,31]. In this paper, a sixth-order non-flat -systemR m —Motoris proposed armature for analysis. resistance One (Ohm) example of a non-flat sixth-order system is a Luo converter- -fedL DCm —Motormotor. Out armature of many inductance pump circuits, (Henry) the fundamental Luo converter is taken for analysis. The -relative u—Control degree inputof the Luo converter-fed DC motor is three which is less than the order of the -system. N—Speed This confirms (RPM) the unstable internal dynamics of the Luo converter inductor current and -capacitor E—Supply voltage voltage with (Volt)respect to the speed. Hence, reference trajectories are developed in an

-indirectR fm —Motormanner. ﬁeld resistance (Ohm) Figure 1 represents the low-cost load torque estimation of the Luo converter with dynamic -L fm—Motor ﬁeld inductance (Henry) -load J—Moment using the offeedback inertia (Kgm signal’s2) armature voltage (v2) and armature current (iam), whereas ETEDPOF can be implemented with inductor currents (i1 and i2) and capacitor voltage (v1).

L S C RmL

i2 iam iag

v1 R

v2 M G Lf Ef i1 C Ef Lf E L1 D

v1 v i i i1 2 a

u CONTROLL ω* ER

FigureFigure 1.1. Non-flatNon-ﬂat Luo Luo converter converter with with dynamic dynamic load load for low-cost for low-cost passivity-based passivity-based control control (PBC) (PBC)implementation. implementation.

UsingWith matrixthe available notation, wide the statevariety vector of ispump shown ci inrcuits, Equation the (33)fundamental positive output Luo converter is taken for the present work. Using Kirchhoff’s laws and Newton’s laws, a linear T averaged model for a Luo converter-fedx (t) = ( iDC1 , i 2motor, v1 , vcan2 , ibeam ,derived.ω) Due to the selection of(33) the Electronics 2020, 9, 1914 7 of 25 and the matrices b, ,J(u), R are given from Equation (34) to Equation (37): ∈ T b = [E/L1 , E/L2 , 0 , 0 , 0, 0] (34) " # T T= 0 , 0 , 0 , 0 , 0, − L (35) ∈ J

 (1 u)   0 0 − − 0 0 0   L1C1   u 1   0 0 − 0 0   L2C1 L2C2   (1 u) u   − − 0 0 0 0  J(u) =  L1C1 L2C1  (36)  1 1   0 0 0 − 0   L2C2 LmC2   1 k   0 0 0 L C 0 JL−   m 2 m   0 0 0 0 k 0  JLm   0 0 0 0 0 0      0 0 0 0 0 0     0 0 0 0 0 0    R =   (37)  0 0 0 0 0 0     0 0 0 0 Rm 0   L2   m   0 0 0 0 0 B  J2 Matrix J is skew-symmetric in nature. R is symmetric and positive-semi-deﬁnite,

i.e., RT = R 0 (38) ≥ The skew symmetric matrix J(u) can be written as

J(u) = J0 + J1u (39) where J0 and J1 are skew symmetric constant matrices which are given in Equations (40) and (41):

 1  0 0 − 0 0 0  L1C1   1   0 0 0 − 0 0   L2C2     1 0 0 0 0 0   L1C1  J0 =  1 1  (40)  0 0 0 − 0   L2C2 LmC2   1 k   0 0 0 0 −   LmC2 JLm     0 0 0 0 k 0  JLm

 1   0 0 L C 0 0 0   1 1   0 0 1 0 0 0   L2C1   1 1   − − 0 0 0 0  J =  L1C1 L2C1  (41) 1    0 0 0 0 0 0     0 0 0 0 0 0     0 0 0 0 0 0  The total stored energy of the system is given as

1 H(x) = xTM x (42) 2 Electronics 2020, 9, 1914 8 of 25 where the matrix “M” is given by   L 0 0 0 0 0  1     0 L2 0 0 0 0     0 0 C 0 0 0  =  1  M    0 0 0 C2 0 0  (43)    0 0 0 0 Lm 0     0 0 0 0 0 J  which is positive-definite and constant. Thus, the average model of the Luo converter-fed DC motor is modified. The ETEDPOF control law can be derived by substituting the necessary matrices in Equation (26) and it is given by h i u = u∗ + γ v i∗ v∗ i v∗ i + v i∗ E i i∗ E i i∗ (44) 1 1 − 1 1 − 1 2 1 2 − 1 − 1 − 2 − 2 In the present case, the dissipation matching condition is not strictly satisfied, and the final dissipation matching matrix is given by

 2 2  E+x E+x γ(E+x ∗ )(x ∗ +x )  γ 3∗ γ 3∗ 3 1 2∗ 0 0 0   L1 L1L2 L1C1   −   + 2 + 2 γ( + )( +   E x3∗ E x3∗ E x 3∗ x 1∗ x2∗ )   γ γ 0 0 0   L1L2 L2 − L2C1   2   γ(E+x ∗ )(x ∗ +x ) γ(E+x ∗ )(x ∗ +x ) x +x  ∼ =  − 3 1 2∗ 3 1 2∗ γ 1∗ 2∗  R  L C L C C 0 0 0  0 (45)  1 1 − 2 1 1  ≥    0 0 0 0 0 0     0 0 0 0 Rm 0   L2   m   0 0 0 0 0 B  J2

Since eR is positive-semi-deﬁnite whenever t 0, the control law (44) makes the origin of the error ≥ space an asymptotically stable equilibrium point by virtue of LaSalle’s theorem.

2.2.1. Stability Proof The error dynamics in the Luo converter and the ﬁrst derivative of energy can be written as

!T  ! !T . ∂H(e)  ∂J(u)  ∂H(x∗)  = [ ( ) ] +  +   e J u R b  eu (46) − ∂e ∂u ∂x∗

! !T . ∂H(e) T ∂H(e) H(e) = R + bˇγbˇ (47) − ∂e ∂e and they are calculated as

. h i2 h i2 h i h i2 H(e) = γ E + x* e + γ E + x* e e 2γ E + x* x + x e e + γ E + x e e − 3 1 3 1 2 − 3 1∗ 2∗ 1 3 3∗ 1 2 h i2 h i h i2 (48) +γ E + x∗ e2 γ E + x∗ x∗ + x∗ e2e3 + γ x∗ + x∗ e3 + γRme5 3 − 3 1 2 1 2 +γBe6

. where H(e) = 0 and e = e = e = e = e = 0 e = 0. This indicates that LaSalle’s theorem is 1 2 3 5 6 → 4 established, and the origin of the error space is globally asymptotically stable. Therefore, due to the bounded nature of the control input between 0 and 1, the origin of the error space is semi-globally asymptotically stable. Electronics 2020, 9, 1914 9 of 25

In terms of converter inductor currents i1 and i2 and voltage v1, the feedback control law (44) governs the speed of the Luo converter-fed DC motor to stabilize the reference trajectory ω∗. The value of 0 γ 0 can be taken as 0.1. The control law (44) indicates the necessity of reference trajectories in terms of speed, and the load torque estimated will be presented in the next section.

2.2.2. Generation of Reference Trajectories In continuation of the derived feedback law (44), it is necessary to generate voltage and current references for the Luo converter circuit, i.e., v1*(t), i2*(t) and i1*(t). In order to realize a smooth starter, reference trajectories in terms of speed and load torque have to be calculated. Reference trajectories can be calculated easily for the differentially flat system. As the relative degree of the system is three, the Luo converter-fed DC motor is a non-flat system. Relative degree can be calculated by differentiating the output variable continuously till the control input is reached [31]. In the Luo converter with dynamic load, the control input is reached in the third step and hence the relative degree is three. Thus, the relative degree is not equal to the system of order six. Due to this reason, the system is considered as a non-flat system. As the system is non-flat, reference trajectories for v1*(t), u* and i1*(t) are generated from the fundamental working principle of the Luo converter which is explained below: Figures2 and3 depict the operation of the Luo converter in “on” and “o ff” conditions. Due to the high-frequency operation, variations in the inductor currents i1 and i2 are negligible. Inductor current profile i1 is obtained by equating the amount of charge in the capacitor when the switch is “on” (Figure2) and “o ff” (Figure3). The charge balance equation is presented in Equation (33).

+ (Charge during switch oﬀ conditions)Q = Q−(Charge during switch on conditions) (49)

(1u)*T*i1 = u*T*i2 (50) where “T” represents the time duration for one switching. From (34), the reference proﬁle for i1 is given by u∗ Electronics 2020, 9, x FOR PEER REVIEW i∗ = i∗ 10 (51)of 26 1 1 u 2 − ∗

S (ON) + v1 C1 L2

i2 iam

iD + M i1 E L1 C2 v2

Figure 2. Converter status when the switch is on (ﬁeld circuit of motor is omitted).

Figure 2. Converter status when the switch is on (field circuit of motor is omitted). As capacitor C2 performs as a low-pass ﬁlter, the load current iload = i2. Hence,

S (OFF) C1 L2 u∗ i∗ = i∗ (52) 1 1 u load − ∗ i2 + v1 iam

iD + M i1 E L1 C2 v2

Figure 3. Converter status when the switch is off (motor field circuit is omitted).

As capacitor C2 performs as a low-pass filter, the load current iload = i2. Hence, u∗ i∗ = i∗ (52) 1−u∗ Source current is given by isource = i1 + i2 when the switch is on and it is zero when the switch is off. Thus, the average source current (isourcea) is presented in Equation (53):

isourcea = isource=i + i =i (53) Hence, the load current is given by iam = i (54) and the output voltage is v2 = E (55)

when the switch is on, inductor current i increases and is supplied by E. Further, current i decreases when the switch is off. Therefore,

uTE = (1-u)Tv1 (56)

u∗ v∗ = E=v∗ (57) 1−u∗ Hence, variations in capacitor voltages are equal, i.e., Electronics 2020, 9, x FOR PEER REVIEW 10 of 26

S (ON) + v1 C1 L2

i2 iam

iD + M i1 E L1 C2 v2

Electronics 2020, 9, 1914 10 of 25 Figure 2. Converter status when the switch is on (field circuit of motor is omitted).

S (OFF) C1 L2

i2 + v1 iam

iD + M i1 E L1 C2 v2

Figure 3. Converter status when the switch is oﬀ (motor ﬁeld circuit is omitted). Figure 3. Converter status when the switch is off (motor field circuit is omitted).

Source current is given by isource = i1 + i2 when the switch is on and it is zero when the switch is As capacitor C2 performs as a low-pass filter, the load current iload = i2. Hence, off. Thus, the average source current (i ) is presented in Equation (53): sourcea u∗ i∗ = i∗ (52) 1−u∗ u isourcea = isource = i1 + i1 = i1 (53) Source current is given by isource = i1 + i2 when the switch1 u is on and it is zero when the switch is − off. Thus, the average source current (isourcea) is presented in Equation (53): Hence, the load current is given by isourcea = isource=i + i =i (53) u i = i (54) am 1 u 1 Hence, the load current is given by − and the output voltage is iam = i (54) u v2 = E (55) and the output voltage is 1 u − when the switch is on, inductor current i1 increases and is supplied by E. Further, current i1 decreases v2 = E (55) when the switch is off. Therefore, when the switch is on, inductor currentuTE i =increases(1 u)Tv and1 is supplied by E. Further, current(56) i − decreases when the switch is off. Therefore, u∗ v∗ = E = v∗ (57) 1 1 u 2 uTE =− (1-u)Tv∗ 1 (56) Hence, variations in capacitor voltages are equal, i.e., ∗ ∗ u ∗ v = ∗ E=v (57) v1−u1∗ = v2∗ (58) Hence, variations in capacitor voltages are equal, i.e., From this, it can be concluded that, for the desired voltage profile v2∗ , v1∗ will follow v2∗ . Hence, the control input, inductor current i1 and other profiles are given by

v∗ u = 2 (59) ∗ + E v2∗

u∗ i∗ = i∗ (60) 1 1 u 2 − ∗ 2 LmJ .. (BLm + RmJ ) . BRm + k RmTˆL v ∗ = ω∗ + ω∗ + ω∗ + (61) 2 k k k k 2 LmC2J ... (BLmC2 + RmJC2) .. BRmC2 + k C2 + J . B TˆL i ∗ = ω∗ + ω∗ + ω∗ + ω∗ + (62) 2 k k k k k

v1∗ = v2∗ (63)

For the generation of reference trajectories of the Luo converter voltage v2∗ and inductor current i2∗, differential parameterizations in terms of the desired angular velocity and the estimated load torque Electronics 2020, 9, 1914 11 of 25 are performed and the equations are given in (61) and (62). In order to define the trajectory, the Bezier polynomial of the fifth order is used. Thus, the trajectories of the desired speed profile and its corresponding voltage and current profiles have been achieved in this section. These reference values are used for sensorless operation under no-load and loading conditions of a Luo converter-fed DC motor. In order to update the value of load torque under closed-loop operation, estimation of load torque is essential, and the method is discussed in the next section. Load torque estimation is essential in the speed control of DC motors under loaded conditions. The space constraint restricts the use of a torque sensor and thus the observer scheme and algebraic approach [30].

2.2.3. Reduced Order Observer in Sensorless Conﬁguration for Estimating Load Torque Linares-Flores et al. [30] proposed a reduced order observer (ROO) approach for the load torque estimation of a DC motor. The load torque is obtained from the mechanical model of the DC motor using the measured speed and armature current as

dξ = λξ + λkiam(t) + λ(λJ B)ω (64) dt − − Setting ξ = τˆ + λωJ,τˆ = ξ λωJ. L L − Clearly, for λ 0, the observation error eτ , where eτ = τ τˆ , exponentially converges to L L L − L zero. This estimated torque Tˆ L is used in the ETEDPOF implementation along with other feedback signals received from the power converter. To restrict the use of the speed sensor, ROO is implemented without a speed sensor and the scheme is known as the reduced order observer approach in sensorless mode (SROO). SROO-based estimation of load torque is found by replacing speed “ω” by Equation (65) in Equation (64). The resultant expression is shown in Equation (66). diam(t) v(t) Rmiam(t) Lm − − dt ω = (65) k diam(t) λ(λJ B) v(t) Rmiam(t) Lm dξ − − − dt = λξ + λkiam(t) + (66) dt − k Based on the value of λ, the convergence rate will vary. Moreover, for a high value of λ, estimation will become unstable. To avoid this problem, the algebraic approach is employed [30].

2.2.4. Online Algebraic Approach in Sensorless Mode (SAA) Online algebraic approaches of estimating load torque with a speed sensor [2] and in sensorless mode are given in (67) and (70). As the estimated torque is getting updated for every o.o3 s (δ), estimation becomes quick. " Z Z Z # 2 t t t τ = J ω(τ)dτ J(t t )ω(t) + k (t t ) i (τ)dτ B (t t )ω(τ)dτ (67) L 2 i i am i (t t ) t − − t − − t − − i i i i Load torque values are decided based on the constraints mentioned below:

n(t) τˆ = τˆ (t −) for t [t , t + δ]; for t > t + δ (68) L L i i i d(t) i Electronics 2020, 9, 1914 12 of 25

With

h R t R t R t i 2 n(t) = 2 J ω(τ)dτ J(t ti)ω(t) + k (t ti)iam(τ)dτ B (t ti)ω(τ)dτ ; d(t) = (t ti) ti − − ti − − ti − − (69) And t = ksT, ks = 0, 1, 2, ...... , T δ i where δ —reset time. 2 R t 1 diam(t) 1 diam(t) τL = 2 J v(t) Rmiam(t) Lm dτ J(t ti) v(t) Rmiam(t) Lm (t t ) ti k dt k dt − i − − − − − − (70) R t R t 1 diam(t) +k (t ti)iam(τ)dτ B (t ti) v(t) Rmiam(t) Lm dτ ti − − ti − k − − dt

3. Results To validate the features, an ETEDPOF controller is implemented for the Luo converter-fed DC motor under no-load condition and loaded conditions.

3.1. Flow Diagram Figure4 represents the ﬂow diagram for the implementation of ETEDPOF in the Luo converter. Further, Figure4 indicates the path for obtaining the control input for speed regulation. Two types of load torque estimation methods, namely SAA and SROO, are applied and the performances are compared. From the desired speed proﬁle and estimated load torque, the desired armature voltage v2* and desired inductor current i2* are obtained using Equations (61) and (62), respectively. With the help of v2*, the desired control function u* is obtained using Equation (59). The desired capacitor * * * voltage v1* is obtained using Equation (63). i2 and u are used for obtaining i1 by using Equation (60). From the feedback signals v1, i1 and i2, the control input is calculated using Equation (44) along with the knowledge of i *, i *, v * and u*. This control input is given as input to the switch of the Luo converter. Electronics 2020, 91, x FOR2 PEER1 REVIEW 13 of 26

i1* (61)

⍵* v2* u* (62) (60)

u From v1* DC (64) (44) motor To Luo converter SAA (71) V Or SROO i2* iam (67) (63)

From Luo converter v1 i1 i2

Figure 4. Flow diagram for the ETEDPOF implementation.

Sensorless load torqueFigure estimation4. Flow diagram with for the the ETEDPOF ETEDPOF implementation implementation. is completed for 12 s in the hardware and simulation. Simulation is performed using MATLAB SIMULINK.

Figure 5. Hardware setup for the Luo converter-fed DC motor.

The Luo converter and controller are used for controlling the DC motor speed under the armature control method. The specifications for the system of interest and the base values are given in Table 1. Hardware waveforms are stored in comma separated value (CSV) format and then these values are exported to MATLAB along with the simulated values for the comparison of both hardware and simulation results.

Electronics 2020, 9, x FOR PEER REVIEW 13 of 26

i1* (61)

⍵* v2* u* (62) (60)

u From v1* DC (64) (44) motor To Luo converter SAA (71) V Or SROO i2* iam Electronics 2020, 9, 1914(67) (63) 13 of 25

3.2. Experimental Setup From Luo The experimental setup for the ETEDPOF implementationconverter under load and no-load conditions is v1 i1 shown in Figure5. The setup includes a Luo converter, a DC motor coupled with a DCi motor2 which will act as a generator, a controller and appropriate sensors with signal conditioning circuits. Speed can be measured using a phototransistor (MOC 7811) and encoder disc. The load torque in the DC motor can be modiﬁed by varyingFigure 4. the Flow load diagram in the DC for generator. the ETEDPOF implementation.

FigureFigure 5. 5. HardwareHardware setup setup for for the the Luo Luo converter-fed converter-fed DC DC motor. motor.

TheThe Luo converterconverter and and controller controller are are used used for controllingfor controlling the DC the motor DC speedmotor under speed the under armature the armaturecontrol method. control Themethod. speciﬁcations The specifications for the system for the of system interest of and interest the base and values the base are values given inare Table given1. inHardware Table 1. Hardware waveforms waveforms are stored are in comma stored separatedin comma valueseparated (CSV) value format (CSV) and format then these and then values these are valuesexported are to exported MATLAB to along MATLAB with thealong simulated with the values simulated for the values comparison for the of comparison both hardware of both and hardwaresimulation and results. simulation results.

Table 1. Speciﬁcations for ETEDPOF control of a Luo converter with dynamic load.

DC Motor Luo Converter Armature Side S.No Field Side Symbol Value Symbol Value Symbol Value

1. L1 18 mH Po 1 HP Rfm 696.1 Ω 2. C1 200 µFEa 180 Volts Lfm 25.023 H 3. L2 20.769 mH Iam 5.1 A Ef 180 V 4. C2 440.1 µF N 1500 RPM Base Values 1500 5. Rated Current 7 A L 111.6 mH Speed m RPM R 6.1 Ω Current 5.1 A 6. DC Supply Voltage 220 V m Laf 3.44 H Voltage 220 V 7. Switching Frequency 32 kHz B 2.7 10 3 Nm/rad -- × − 8. - - J 3.4 10 3 kgm2 -- × −

The total operation is divided into two broad categories, namely servo control and regulatory control operations, based on the free acceleration, change of speed reference and step change in load torque conditions. Electronics 2020, 9, x FOR PEER REVIEW 14 of 26

Table 1. Specifications for ETEDPOF control of a Luo converter with dynamic load.

DC Motor Luo Converter Armature Side S.No Field Side Symbol Value Symbol Value Symbol Value 1. L1 18 mH Po 1 HP Rfm 696.1 Ω 2. C1 200 µF Ea 180 Volts Lfm 25.023 H 3. L2 20.769 mH Iam 5.1 A Ef 180 V 4. C2 440.1 µF N 1500 RPM Base Values 5. Rated Current 7 A Lm 111.6 mH Speed 1500 RPM Rm 6.1 Ω Current 5.1 A 6. DC Supply Voltage 220 V Laf 3.44 H Voltage 220 V 7. Switching Frequency 32 kHz B 2.7 × 10−3 Nm/rad - - 8. - - J 3.4 × 10−3 kgm2 - -

ElectronicsThe2020 ,total9, 1914 operation is divided into two broad categories, namely servo control and regulatory14 of 25 control operations, based on the free acceleration, change of speed reference and step change in load torque conditions. InIn servo servo control control mode, mode, speed speed control control of theof the Luo Luo converter-fed converter-fed DC DC motor motor is performedis performed under under freefree acceleration acceleration and and at at di ﬀdifferenterent constant constant load load torques torques of of value value 0.5, 0.5, 1.0 1.0 and and 0.25 0.25 p. p. u.u. (Figure 66).). In In regulatory controlcontrol mode,mode, speedspeed control control is is achieved achieved along along with with two two load load torque torque estimation estimation schemes, schemes, namelynamely the the algebraic algebraic approach approach in sensorless in sensorless mode (SAA) mode and (SAA) the reducedand the order reduced observer order in sensorlessobserver in modesensorless approach mode (SROO), approach and both (SROO), schemes and are both compared schemes without are compared any speed without sensor. Inany SROO, speed based sensor. on In theSROO, value ofbased the tuningon the constant,value of itthe is tuning classiﬁed constant, into SROO1 it is classified and SROO2 into for SROO1 which and the λSROO2value isfor equal which to 5 andλ 10, respectively. the value is equal to 5 and 10, respectively.

Applied load torque (p. u.) vs. time (seconds) APPLIED LOAD TORQUE vs TIME

0.8

0.6

0.4 Applied Applied load torque (p. u)

0.2 Applied load torque (p. u.) (p. u.) torque load Applied 0 0 2 4 6 8 10 12 Time (seconds) Time (seconds)

FigureFigure 6. Applied 6. Applied load load torque. torque. 3.3. Discussion 3.3. Discussion Simulation and hardware results are shown from Figures6–11. In the simulation, ideal components Simulation and hardware results are shown from Figures 6–11. In the simulation, ideal are used. Hence, the simulation performance is slightly better than hardware implementation which components are used. Hence, the simulation performance is slightly better than hardware is clearly exhibited in Figures6–11. As v 1 = v2 and iam = i2, inductor current and armature voltage implementation which is clearly exhibited in Figures 6–11. As v1 = v2 and iam = i2, inductor current waveforms alone are presented. Simulation and hardware results are explained below. Electronicsand armature 2020, 9, x voltageFOR PEER waveforms REVIEW alone are presented. Simulation and hardware results15 ofare 26 explained below. Reference Speed (p. u.) vs. time (seconds) REFERENCE SPEED vs TIME 1.4

1.2

0.8

0.6 Speed reference (p. u) Speed (p. u.) 0.4

0.2

0 0 2 4 6 8 10 12 Time (seconds) Time (seconds)

FigureFigure 7. Speed7. Speed reference reference proﬁle. profile.

Inductor current (p. u.) vs. time (seconds) INDUCTOR CURRENT vs TIME 1.6

1.4

1.2

0.8

0.6

Inductor current (p. u,) 0.4

0.2 Inductor current (p. u.) u.) (p. Inductor current

0 SROO1-SIM SROO2-SIM SAA-SIM SROO1-HW SROO2-HW SAA-HW -0.2 0 2 4 6 8 10 12 TIme (seconds) Time (seconds) Figure 8. Inductor current of the Luo converter. Electronics 2020, 9, x FOR PEER REVIEW 15 of 26

Reference Speed (p. u.) vs. time (seconds) REFERENCE SPEED vs TIME 1.4

1.2

0.8

0.6 Speed reference (p. u) Speed (p. u.) 0.4

0.2

0 0 2 4 6 8 10 12 Time (seconds) Time (seconds)

Electronics 2020, 9, 1914 Figure 7. Speed reference profile. 15 of 25

Inductor current (p. u.) vs. time (seconds) INDUCTOR CURRENT vs TIME 1.6

1.4

1.2

0.8

0.6

Inductor current (p. u,) 0.4

0.2 Inductor current (p. u.) u.) (p. Inductor current

0 SROO1-SIM SROO2-SIM SAA-SIM SROO1-HW SROO2-HW SAA-HW -0.2 0 2 4 6 8 10 12 TIme (seconds) Time (seconds) Electronics 2020, 9, x FOR PEER REVIEW 16 of 26 FigureFigure 8. Inductor 8. Inductor current current of the of Luo the Luoconverter. converter.

Capacitor voltageCAPACITOR (p. VOLTAGE u.) vs. vs TIMEtime (seconds) 1.2

Servo control 1 SERVO CONTROL

0.8

0.6

0.4

Capacitor voltage (p. u) (p. voltage Capacitor RegulatoryREGULATORY control CONTROL 0.2 Capacitor voltage (p. u.)

0 SROO1 SIM SROO2 SIM SAA SIM SROO1 HW SROO2 HW SAA HW SROO1-SIM SROO2-SIM SAA-SIM SROO1-HW SROO2-HW SAA-HW

-0.2 0 2 4 6 8 10 12 TimeTime (seconds) FigureFigure 9. 9.Armature Armature voltage voltage of of the the Luo Luo converter-fed converter-fed DC DC motor. motor.

Speed (p. u.)SPEED vs. vstime TIME (seconds) 1.4 Servo control 1.2

0.8

0.6

Speed (p.u.) Regulatory control 0.4

0.2

Speed (p. u.) u.) (p. Speed 0 SROO1 SIMSROO1-SIM SROO2SROO2-SIM SIM SAASAA-SIM SIM SROO1SROO1-HW HW SROO2-HW SROO2 HW SAA-HW SAA HW

-0.2 0 2 4 6 8 10 12 Time (seconds) Time (seconds)

Figure 10. Measured speed of the DC motor fed from the Luo converter. Electronics 2020, 9, x FOR PEER REVIEW 16 of 26

Capacitor voltageCAPACITOR (p. VOLTAGE u.) vs. vs TIMEtime (seconds) 1.2

Servo control 1 SERVO CONTROL

0.8

0.6

0.4

Capacitor voltage (p. u) (p. voltage Capacitor RegulatoryREGULATORY control CONTROL 0.2 Capacitor voltage (p. u.)

0 SROO1 SIM SROO2 SIM SAA SIM SROO1 HW SROO2 HW SAA HW SROO1-SIM SROO2-SIM SAA-SIM SROO1-HW SROO2-HW SAA-HW

-0.2 0 2 4 6 8 10 12 TimeTime (seconds) Electronics 2020, 9, 1914 16 of 25 Figure 9. Armature voltage of the Luo converter-fed DC motor.

Speed (p. u.)SPEED vs. vstime TIME (seconds) 1.4 Servo control 1.2

0.8

0.6

Speed (p.u.) Regulatory control 0.4

0.2

Speed (p. u.) u.) (p. Speed 0 SROO1 SIMSROO1-SIM SROO2SROO2-SIM SIM SAASAA-SIM SIM SROO1SROO1-HW HW SROO2-HW SROO2 HW SAA-HW SAA HW

-0.2 0 2 4 6 8 10 12 Time (seconds) Time (seconds) Electronics 2020, 9, x FOR PEER REVIEW 17 of 26 FigureFigure 10. 10. MeasuredMeasured speed speed of of the the DC DC moto motorr fed fed from from the the Luo Luo converter. converter.

Estimated TorqueESTIMATED (p. TORQUE u.) vs. vs TIME time (seconds) 1.2

0.8

0.6

0.4 Estimated torque (p.u.) torque Estimated 0.2 Estimated Torque (p. u.) u.) (p. Torque Estimated

SROO1 SIMSROO1-SIM SROO2SROO2-SIM SIM SAASAA-SIM SIM SROO1-HW SROO1 HW SROO2-HW SROO2 HWSAA-HW SAA HW -0.2 0 2 4 6 8 10 12 TIme (seconds) Time (seconds)

Figure 11. Estimated torque. Figure 11. Estimated torque. Inductor current variations of the Luo converter for the given speed reference (Figure6) and load torqueInductor applied current (Figure 7variations) are shown of inthe Figure Luo 8converter. These variations for the given are obtained speed forreference regulatory (Figure and 6) servo and loadcontrol torque operations. applied Under (Figure regulatory 7) are shown control in operations, Figure 8. These SAA settles variations the current are obtained faster when for regulatory compared andwith servo SROO. control This is operations. due to the fastUnder estimation regulatory of load control torque operations, using SAA. SAA settles the current faster whenUnder compared regulatory with SROO. control This operations, is due to SAAthe fast and estimation SROO methods of load are torque implemented using SAA. to modify the referenceUnder proﬁles regulatory for ETEDPOF control operations, control. Due SAA to and the fastSROO estimation methods nature are implemented of SAA, armature to modify voltage the reference profiles for ETEDPOF control. Due to the fast estimation nature of SAA, armature voltage settles faster than in the case of SAA when compared with SROO. As the armature voltage is equal to the capacitor voltage (v1), the armature voltage alone is presented in Figure 9. Speed variation under servo and regulatory modes is shown in Figure 10. During starting, the speed is varied smoothly using ETEDPOF. Under loading conditions, the speed is regulated using ETEDPOF with load torque estimation. Table 2 compares the performances of SAA and SROO in terms of speed settling time. From the above analysis and Table 2, it can be concluded that SAA performs better than SROO with less overshoot/undershoot in speed responses. Thus, the speed responses for the DC motor with ETEDPOF, SAA and SROO are analyzed.

Electronics 2020, 9, 1914 17 of 25 settles faster than in the case of SAA when compared with SROO. As the armature voltage is equal to the capacitor voltage (v1), the armature voltage alone is presented in Figure9. Speed variation under servo and regulatory modes is shown in Figure 10. During starting, the speed is varied smoothly using ETEDPOF. Under loading conditions, the speed is regulated using ETEDPOF with load torque estimation. Table2 compares the performances of SAA and SROO in terms of speed settling time. From the above analysis and Table2, it can be concluded that SAA performs better than SROO with less overshootElectronics /2020undershoot, 9, x FOR PEER in speed REVIEW responses. Thus, the speed responses for the DC motor with ETEDPOF,18 of 26 SAA and SROO are analyzed. Table 2. Comparison between SAA and SROO: constant load torque/Luo converter with dynamic Tableload. 2. Comparison between SAA and SROO: constant load torque/Luo converter with dynamic load.

SpeedSpeed Settling Time Time (Seconds) (seconds) in in Regulatory Control Operation * Regulatory Control Operation S. No Time (Seconds) ω* (p. u.) TL (p. u.) S. No Time (seconds) ω (p. u.) TL (p. u.) SROO1SROO1 SROO2SROO2 SAA SAA SIMSIM HW SIMSIM HWHW SIMSIM HWHW 1. 2.0 2.0 0.6 0.6 0.50 0.50 1.0 1.0 1.4 0.50.5 0.60.6 0.20.2 0.250.25 2. 6.0 6.0 1.0 1.0 1.00 1.00 1.0 1.0 1.4 0.50.5 0.60.6 0.20.2 0.250.25 3. 10.0 10.0 0.6 0.6 0.25 0.25 1.0 1.0 1.4 0.50.5 0.60.6 0.20.2 0.250.25 LoadLoad Torque Torque EstimationEstimation Time Time (Seconds) (seconds) in in Regulatory Control Operation ω* T (p. u.) Regulatory Control Operation S.S. No TimeTime (seconds) (Seconds) ω* (p.(p. u.) u.) TL L(p. u.) SROO1SROO1 SROO2SROO2 SAA SAA SIMSIM HW SIMSIM HWHW SIMSIM HWHW 1. 2.0 2.0 0.6 0.6 0.50 0.50 1.0 1.0 1.5 0.50.5 0.60.6 0.20.2 0.250.25 2. 6.0 6.0 1.0 1.0 1.00 1.00 1.0 1.0 1.5 0.50.5 0.60.6 0.20.2 0.250.25 3. 10.0 10.0 0.6 0.6 0.25 0.25 1.0 1.0 1.5 0.50.5 0.60.6 0.20.2 0.250.25 SIM—simulation;SIM—simulation; HW—hardware. HW—hardware.

FigureFigure 11 11 shows shows the the estimated estimated torque torque for for the the DC DC motor motor using using the the SAA SAA and and SROO SROO methods; methods; bothboth methods methods are are tested tested in thein simulationthe simulation and hardwareand hardware setup setup for the for step the change step change in load torquein load values torque ofvalues 0.50, 1.00 of 0.50, and 1.00 0.25 and p. u. 0.25 p. u. ComparisonComparison between between SAA SAA and and SROO SROO estimation estimation times times are are tabulated tabulated in in Table Table2. 2. From From the the table, table, itit can can be be concluded concluded that that the the performance performance of of SAA SAA is is better better than than SROO SROO for for all all load load conditions conditions with with a a lowerlower integral integral square square error error (ISE) (ISE) (Figure (Figure 12 12).).

ISE vs.ISE time vs TIME (seconds) 0.12

SROO1 SROO2 0.1 SAA

0.08

ISE 0.06 ISE

0.04

0.02

0 0 2 4 6 8 10 12 Time (seconds) Time (seconds)

FigureFigure 12. 12.Integral Integral square square error error (ISE) (ISE) vs. vs. time—Luo time—Luo converter-fed converter-fed DC DC motor. motor.

Hence, it can be concluded that SAA estimates the unknown load torque quickly when compared with SROO with the lowest integral square error (ISE), as shown in Figure 12.

3.4. Estimation of Different Load Torque Profiles The following time-varying load torques are used for studying the capability of SAA with the speed reference, as shown in Figure 13: a. Load torque proportional to speed (Figure 14). b. Load torque proportional to square of the speed (Figure 15). c. Load torque proportional to cube of the speed (Figure 16). Electronics 2020, 9, 1914 18 of 25

Hence, it can be concluded that SAA estimates the unknown load torque quickly when compared Electronics 2020, 9, x FOR PEER REVIEW 19 of 26 withElectronics SROO 2020 with, 9, thex FOR lowest PEER REVIEW integral square error (ISE), as shown in Figure 12. 19 of 26

3.4.d.d. Estimation UnpredictableUnpredictable of Diﬀerent load load Load torque torque Torque (Figure (Figure Proﬁles 17). 17). TheTheseThese following casecase studiesstudies time-varying areare usedused load toto testtest torques thethe abilityability are used ofof SAASAA for studying alongalong withwith the thethe capability assumptionassumption of SAA ofof a witha boundedbounded the speedvaluevalue reference, inin loadload torque. astorque. shown MATLABMATLAB in Figure Simulink Simulink13: isis usedused foforr studyingstudying thethe robustnessrobustness ofof SAA.SAA. InIn thethe LuoLuo converter,converter, asas capacitorcapacitor voltagesvoltages areare equal,equal, thethe armaturearmature voltagevoltage alonealone isis considered.considered. InIn additionaddition toto a. Load torque proportional to speed (Figure 14). 2 this,this, thethe armaturearmature currentcurrent isis eqequalual toto thethe inductorinductor currentcurrent (i(i2)) andand hencehence thethe armaturearmature currentcurrent alonealone b. Load torque proportional to square of the speed (Figure 15). isis consideredconsidered here.here. SpecificationsSpecifications forfor thethe systemsystem ofof interestinterest areare takentaken fromfrom TableTable 1.1. FromFrom thethe c. simulationsimulationLoad torque resultsresults proportional shownshown fromfrom to cubeFiguresFigures of the 18–23,18–23, speed itit is (Figureis confirmedconfirmed 16). thatthat SAASAA outperformsoutperforms SROOSROO forfor allall d.types typesUnpredictable of of load load torques. torques. load torque (Figure 17).

SpeedSPEEDSpeedSPEED reference reference REFERENCEREFERENCE (p. (p. u.) u.) vs. vs. t timeime vs(seconds) vs(seconds) TIMETIME

0.80.8

0.60.6 Speed referenceSpeed Speed reference (p. u.) u.) (p. Speed reference Speed referenceSpeed Speed reference (p. u.) u.) (p. Speed reference 0.40.4

0.20.2

00 00 11 22 33 44 55 66 77 88 99 1010 TimeTime (seconds)(seconds) Time Time (seconds) (seconds) Figure 13. Speed reference profile: Load torque proportional to speed TL α ω. FigureFigure 13. 13.Speed Speed reference reference proﬁle: profile: Load Load torque torque proportional proportional to speedto speed TL αTLω α. ω.

AppliedAppliedAPPLIEDAPPLIED torque torque (p. (p. TORQUETORQUE u. u.)) vs. vs. time time (seconds) (seconds) vsvs TIMETIME

0.80.8

0.60.6

0.4

Applied torque (p.u.) 0.4 Applied torque (p.u.) Applied torque (p. u.) u.) (p. torque Applied Applied torque (p. u.) u.) (p. torque Applied 0.20.2

0 0 00 22 44 66 88 1010 1212 TimeTime (seconds)(seconds) Time Time (seconds) (seconds)

L 2 2. FigureFigureFigure 14. 14. 14.Applied Applied Applied torque: torque: torque: TL T αTLα ω α ω ω. 2. ElectronicsElectronics2020, 9 2020, 1914, 9, x FOR PEER REVIEW 19 of 2520 of 26 Electronics 2020, 9, x FOR PEER REVIEW 20 of 26 APPLIEDApplied torque TORQUE (p. u.) vs. vstime TIME (seconds) 1.2 APPLIEDApplied torque TORQUE (p. u.) vs. vstime TIME (seconds) 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4

Applied torque (p.u.) 0.2 Applied torque (p.u.) 0.2

Applied torque (p. u.) u.) (p. torque Applied 0

Applied torque (p. u.) u.) (p. torque Applied 0 -0.2 0 2 4 6 8 10 12 -0.2 Time (seconds) 0 2 4 6 8 10 12 Time Time(seconds) (seconds) Time (seconds) 3. Figure 15. Applied torque: T3L α ω Figure 15. Applied torque: TL α ω . Figure 15. Applied torque: TL α ω3. AppliedAPPLIED torque (p.TORQUE u.) vs. time vs(seconds) TIME AppliedAPPLIED torque (p.TORQUE u.) vs. time vs(seconds) TIME

1 1

0.8 0.8

0.6 0.6

0.4 Applied torque (p.u.) 0.4 Applied torque (p.u.)

Applied torque (p. u.) u.) (p. torque Applied 0.2

0 0 1 2 3 4 5 6 7 8 9 10 0 0 1 2 3 4 Time5 (seconds)6 7 8 9 10 Time Time(seconds) (seconds) Time (seconds) Figure 16. Applied load torque. Figure 16. Applied load torque. These case studies are used toFigure test the 16. ability Applied of load SAA torque. along with the assumption of a bounded value in load torque. MATLAB Simulink is used for studying the robustness of SAA. In the Luo converter, as capacitor voltages are equal, the armature voltage alone is considered. In addition to this, the armature current is equal to the inductor current (i2) and hence the armature current alone is considered here. Speciﬁcations for the system of interest are taken from Table1. From the simulation results shown from Figures 18–23, it is conﬁrmed that SAA outperforms SROO for all types of load torques. ElectronicsElectronics2020 2020, 9, ,9 1914, x FOR PEER REVIEW 2021 of of 25 26 Electronics 2020, 9, x FOR PEER REVIEW 21 of 26 AppliedAPPLIED torque (p. TORQUE u.) vs. time (seconds) vs TIME 1.2 AppliedAPPLIED torque (p. TORQUE u.) vs. time (seconds) vs TIME 1.2

1 1

0.8 0.8

0.6 0.6

0.4 0.4 Applied torque (p.u.)

Applied torque (p.u.) 0.2

Applied torque (p. u.) u.) (p. torque Applied 0.2 Applied torque (p. u.) u.) (p. torque Applied 0 0

-0.2 -0.20 2 4 6 8 10 12 0 2 4 Time (seconds)6 8 10 12 TimeTime (seconds) (seconds) Time (seconds) Figure 17. Torque reference: TL α ω. FigureFigure 17. 17.Torque Torque reference: reference: TL TαL α ω ω. . Speed (p.SPEED u.) vs. time vs TIME (seconds) Speed (p. u.) vs. time (seconds) 1.2 SPEED vs TIME 1.2 SROO2 SROO1SROO2 1 SAASROO1 1 SAA

0.8 0.8

0.6 0.6

0.4

Speed Speed (p.u.) 0.4 Speed Speed (p.u.)

Speed (p. u.) u.) (p. Speed 0.2 Speed (p. u.) u.) (p. Speed 0.2

0 0

-0.2 -0.20 2 4 6 8 10 12 0 2 4 Time (seconds)6 8 10 12 Time (seconds) Figure 18. Measured speed of the DC motor: TL α ω. Figure 18. Measured speed of the DC motor: TL α ω. Figure 18. Measured speed of the DC motor: TL α ω. Electronics 2020, 9, 1914 21 of 25 Electronics 2020, 9, x FOR PEER REVIEW 22 of 26

EstimatedESTIMATED torque (p. TORQUE u.) vs. time (seconds) vs TIME 1.2 SROO2 SROO1 1 SAA

0.8

0.6

0.4 Estimated torque (p.u.) torque Estimated Estimated torque (p.u.) torque Estimated 0.2 Estimated torque (p. u.) u.) (p. torque Estimated Estimated torque (p. u.) u.) (p. torque Estimated 0

-0.2 0 2 4 6 8 10 12 Time (seconds) Time (seconds)

2 2. FigureFigure 19. 19.Estimated Estimated torque: torque: TL TαL ωα ω. 2. Speed (p. u.) vs. time (seconds) Speed (p.SPEED u.) vs. time vs (seconds)TIME 1.2 SROO2 SROO1 1 SAA

0.8

0.6

0.4 Speed (p.u.) Speed (p.u.)

0.2 Speed (p. u.) u.) (p. Speed Speed (p. u.) u.) (p. Speed 0

-0.2 0 2 4 6 8 10 12 Time (seconds) Time (seconds)

L 3 3. FigureFigure 20. 20.Measured Measured speed: speed: TL TαL ωα ω. Electronics 2020, 9, x FOR PEER REVIEW 23 of 26

Electronics 2020, 9, 1914 22 of 25 Estimated torque (p. u.) vs. time (seconds) Electronics 2020, 9, x FOR PEER REVIEW ESTIMATED TORQUE vs TIME 23 of 26 1.2 EstimatedESTIMATED torque (p. u.) vs. TORQUE time (seconds) vs TIME SROO2 1.2 1 SROO1 SAA SROO2 10.8 SROO1 SAA

0.80.6

0.60.4

0.4Estimated torque (p.u.) 0.2 Estimated torque (p. u.) u.) (p. torque Estimated

Estimated torque (p.u.) 0.2 0 Estimated torque (p. u.) u.) (p. torque Estimated

0-0.2 0 2 4 6 8 10 12 Time (seconds) Time (seconds) -0.2 0 2 4 6 8 10 12 Figure 21.Time Estimated (seconds) torque: TL α ω3. Time (seconds)

3 3. FigureFigure 21. 21.Estimated Estimated torque: torque: TL TαL α ω ω. Speed (p. u.) vs. time (seconds) SPEED vs TIME

Speed (p. SPEEDu.) vs. time vs (seconds) TIME 1

10.8

0.80.6

0.60.4 Speed Speed (p.u.)

0.40.2 Speed Speed (p.u.) SROO2 Speed (p. u.) u.) (p. Speed SROO1 0.2 0 SAA SROO2 Speed (p. u.) u.) (p. Speed SROO1 0-0.2 0 1 2 3 4 5 6 7 8 SAA9 10 Time (seconds) Time (seconds) -0.2 0 1 2 3 4 5 6 7 8 9 10 Figure 22. Armature speed of the DC motor: unknown load torque. Figure 22. Armature Timespeed of(seconds) the DC motor: unknown load torque. Time (seconds)

Figure 22. Armature speed of the DC motor: unknown load torque. Electronics 2020, 9, 1914 23 of 25 Electronics 2020, 9, x FOR PEER REVIEW 24 of 26

EstimatedESTIMATED torque (p. u.)TORQUE vs. time (seconds) vs TIME 1.2 SROO2 SROO1 1 SAA

0.8

0.6

0.4

Estimated torque (p.u.) 0.2 Estimated torque (p. u.) u.) (p. torque Estimated

-0.2 0 1 2 3 4 5 6 7 8 9 10 Time (seconds) Time (seconds) Figure 23. Estimated torque: unknown load torque. Figure 23. Estimated torque: unknown load torque. 4. Conclusions 4. Conclusions A methodology for sensorless speed control of a DC motor-driven system with a faster and stableA trajectorymethodology tracking for approachsensorless is sp developed.eed control Considering of a DC motor-driven the robustness, system stability with and a highfaster speed and stableof trajectory trajectory tracking tracking offered approach by the is passivity-based developed. Considering control law, the it hasrobustness, been chosen stability to implement and high speedthe speed of trajectory control of tracking the DC motoroffered where by the two pass methodologiesivity-based control of PBC, law, viz., it ETEDPOFhas been chosen and ESDI, to implementare investigated. the speed A further control two of algorithms, the DC motor viz., SAA where and two SROO, methodologies were considered of PBC, for theviz., estimation ETEDPOF of andload ESDI, torque are which, investigated. in turn, areA further used for two implementing algorithms, viz., the sensorlessSAA and SROO, speed controlwere considered of the DC for motor the estimationdrive system of load under torque variable which, load in torque turn, are conditions. used for implementing All the investigations the sensorless are performed speed control on the of theprototype DC motor experimental drive system setup ofunder a DC motorvariable drive load system torque fed conditions. by the converter All the topologies, investigations viz., a Luoare performedconverter and on athe sixth-order prototype non-di experimentalfferentially setup flat system.of a DC motor drive system fed by the converter topologies,Here, aviz., sixth-order a Luo converter non-diff anderentially a sixth-order flat system non-different Luo converter-fedially flat system. DC motor is considered in whichHere, Lyapunov’s a sixth-order stability non-differenti condition isally not flat strictly system satisfied. Luo converter Therefore,-fed LaSalle’s DC motor theorem is considered is used forin whichestablishing Lyapunov’s asymptotic stability stability. condition is not strictly satisfied. Therefore, LaSalle’s theorem is used for establishingThe relative asymptotic degree of stability. the Luo converter-fed DC motor is three and it is less than the order. ThisThe confirms relative the degree non-di ffoferential the Luo flatness converter-fed nature of DC the motor Luo converter. is three and So, referenceit is less than profiles the fororder. the Thiscapacitor confirms voltage the (vnon-diffe1), controlrential input flatness (u) and nature inductor of the current Luo conv (i1) areerter. obtained So, reference from the profiles fundamental for the capacitorworking principlevoltage (v of1), thecontrol Luo input converter. (u) and Using inductor these current references (i1) are and obtained the capacitor from the voltage fundamental (v2) and workinginductor currentprinciple (i 2of) reference the Luo profilesconverter. in termsUsing of these speed references and load torque,and the the capacitor ETEDPOF voltage control (v2 law) and is inductorimplemented current successfully (i2) reference in theprofiles Luo converter-fedin terms of speed DC motorand load tested torque, successfully the ETEDPOF for various control speed law isreferences implemented and at successfully different load in conditions the Luo converter as servo and-fed regulatory DC motor control, tested respectively.successfully for various speedHence, references from and the at above, different it can load be concludedconditions that,as servo irrespective and regulatory of the flatness control, nature respectively. and order of the system,Hence, thefrom ETEDPOF the above, control it can law be concluded is developed that, and irrespective implemented of the successfully. flatness nature This clearly and order shows of thethe faster,system, stabler the ETEDPOF trajectory controlcontrol withlaw lessis develope overshootd /undershoots.and implemented successfully. This clearly showsFurther, the faster, a comparative stabler trajectory studyof co thentrol SAA with and less SROO overshoot/undershoots. methods confirms that SAA is capable of any typeFurther, of load torque. a comparative Due to thestudy fast of estimation the SAA ofand load SROO torque methods using SAA,confirms the controlthat SAA input, is capable armature of anyvoltage type and of load armature torque. currents Due to are the changing fast estimati at aon faster of load rate intorque comparison using SAA, with the SROO control where input, the armature currentvoltage andand inductorarmature current currents exhibit are somechanging overshoot at a faster which rate is duein comparison to dω . with SROO dt where the armature current and inductor current exhibit some overshoot which is due to . Therefore, the highlighted work in this paper may be resumed as follows: Therefore, the highlighted work in this paper may be resumed as follows: Electronics 2020, 9, 1914 24 of 25

(a) Development and implementation of a sensorless passivity based-control scheme, that is, ETEDPOF—SAA for the servo and regulatory control of buck/boost converter-fed DC motor drive systems with a good dynamic response. (b) Under variable load torque types, viz., constant-type, frictional-type, fan-type, propeller-type and unpredictable torque-type, a good dynamic control, that is, faster and stabler control of DC motor drive systems, is achieved.

Author Contributions: Conceptualization, methodology and writing, G.K.S.; editing and supervision, H.T.S. and M.R. All authors have read and agreed to the published version of the manuscript. Funding: The authors thank the support of DEEE, Anna University, Chennai, India for funding through NamPET Lab Scheme and the authors thank the support of Fondecyt Regular 1191028 research project and SERC Chile. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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