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

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Low-Cost Implementation of Passivity-Based Control and Estimation of Load Torque for a Luo Converter with Dynamic Load electronics 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 2 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, 2 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 (!∗) under no-load and load conditions. The desired speed profile is obtained in an offline fashion using bezier polynomials. For this desired speed profile (!∗), 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∗ 2 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∗ 2 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 2 − @x∗ The value of obtained from Equation (9) is substituted in Equation (13), then the error dynamics 2 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 2 !2 !T33 . @H(e) 6 @J(u) 6 @H(x∗) 77 = [ ( ) ] + 6 + 6 77 e J u R 46b 46 5757eu (15) − @e @u @x∗ !T !T . @H(e) @H(e) e = J(u) R + beˇ u (16) @e − @e where 2 !2 !T33 6 @J(u) 6 @H(x∗) 77 ˇ = 6 + 6 77 b 46b 46 5757 (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].
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