Modeling and Control of an Autonomous Linear DC Machine Powered Mine Locomotive

Modeling and Control of an Autonomous Linear DC Machine Powered Mine Locomotive

18th 18th IFAC IFAC Symposium Symposium on on Control, Control, Optimization Optimization and and Automation Automation in in Mining, Mineral and Proc...

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18th 18th IFAC IFAC Symposium Symposium on on Control, Control, Optimization Optimization and and Automation Automation in in Mining, Mineral and Processing 18th IFAC Symposium on Control, Optimization and Automation in Mining, Mineral and Metal Metal Processing Available online at www.sciencedirect.com Stellenbosch, South Africa, AugustOptimization 28-30, 2019 and Automation in 18th IFAC Symposium on Control, Mining, Mineral and Metal Processing Stellenbosch, South Africa, August 28-30, 2019 Mining, MineralSouth and Metal Processing Stellenbosch, Africa, August 28-30, 2019 Stellenbosch, South Africa, August 28-30, 2019

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IFAC PapersOnLine 52-14 (2019) 195–200

Modeling and Modeling and Modeling Linear and DC Modeling and Linear Linear DC DC Linear DC

Control of an Autonomous Control of an Autonomous Control of an Autonomous Machine Powered Mine Control ofPowered an Autonomous Machine Mine Machine Powered Mine Locomotive Machine Powered Mine Locomotive Locomotive Locomotive ∗∗ Mohlalakoma T. Ngwako ∗∗∗ Otis T. Nyandoro ∗∗ ∗∗

Mohlalakoma T. Ngwako ∗ Otis T. Nyandoro ∗∗∗ ∗∗∗∗∗∗ Mohlalakoma T. Ngwako Otis T. Nyandoro ∗∗∗ ∗∗∗∗ John C. ∗∗∗ Milka ∗∗∗∗ John van van Coller Coller Milka C. Madahana Madahana ∗ ∗∗ ∗∗∗ Mohlalakoma T. Ngwako Otis T. Nyandoro John van Coller Milka C. Madahana ∗∗∗∗ ∗∗∗ ∗∗∗∗ John∗∗∗ van Coller Milka C. Madahana (e-mail: (e-mail: [email protected]). [email protected]). ∗∗∗ (e-mail: [email protected]). ∗∗ (e-mail: [email protected]) ∗∗ (e-mail: [email protected]) ∗ ∗∗ ∗∗∗ (e-mail: [email protected]). (e-mail: [email protected]) ∗∗∗ (e-mail: [email protected]) ∗∗∗ ∗∗ (e-mail: [email protected]) ∗∗∗ ∗∗∗∗ (e-mail: [email protected]) (e-mail: [email protected]) ∗∗∗∗ [email protected]) ∗∗∗∗ (e-mail: (e-mail: [email protected]) ∗∗∗ ∗∗∗∗ (e-mail:[email protected]) [email protected]) (e-mail: ∗∗∗∗ School and Information (e-mail: School of of Electrical Electrical [email protected]s.wits.ac.za) Information Engineering, Engineering, University University of of School of Witwatersrand, Electrical and Information Engineering, University of Johannesburg, South Africa Witwatersrand, Johannesburg, South Africa School of Witwatersrand, Electrical and Information Engineering, University of Johannesburg, South Africa Witwatersrand, Johannesburg, South Africa Abstract: An An autonomous autonomous linear linear DC DC machine machine based based mine mine locomotive locomotive is is presented. presented. The The linear linear Abstract: Abstract: linear DC machine based presented. linear DC is modeled an field based Lagrange analysis. The Lagrange DC machine machineAn is autonomous modeled using using an electromagnetic electromagnetic fieldmine basedlocomotive Lagrange is analysis. TheThe Lagrange Abstract: An linear DC machine based mine is presented. The linear DC machine is autonomous modeled using an is electromagnetic field basedlocomotive Lagrange analysis. The Lagrange based linear DC machine model found to have the same result as the result that exists in based linear DC machineusing model is found to have field the same result as the result The that Lagrange exists in DC machine is modeled an electromagnetic based Lagrange analysis. based linear DC machinebased model is found to have control the same result as the result thatthe exists in literature. State feedback on pole-placement algorithm is used to control speed literature. State based onispole-placement control algorithm is used to control the speed based DCfeedback machine model found to have the same as thewould result that exists ina literature. feedback on The pole-placement control is used to control the speed of the the linear mineState locomotive tobased 3 m/s. m/s. success criteria of algorithm theresult controller be; achieving of mine locomotive to 3 The success criteria of the controller would be; achieving literature. State feedback based onofThe pole-placement control algorithm is used to control the speedaa of the mine locomotive to 3 m/s. success criteria of the controller would be; achieving settling time of 0.6 s and a speed 3 m/s. The results presented show the desired performance settling time of 0.6 s and a speed of 3 m/s. The results presented show the desired performance of the mine m/s.and success criteria of the controller would be;performance achieving a settling timelocomotive of 0.6s ssettling andtoa 3speed ofThe 3 3m/s. The results presented the desired specification (0.6 time m/s speed). Therefore, theshow Lagrange modeling and the the specification s ssettling time and m/s speed). Therefore, the Lagrange modeling and settling time (0.6 ofof0.6 and alocomotive speed of 3 3 m/s. The results presented the desired performance specification (0.6 s settling time and 3is m/s speed). Therefore, theshow Lagrange modeling and the speed control the mine effective. speed control of the mine locomotive effective. specification s settling time and 3is speed). Therefore, the Lagrange modeling and the speed control(0.6 of the mine locomotive is m/s effective. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. speed control of the mine locomotive is effective. Keywords: Autonomous Autonomous linear linear DC DC machine machine powered powered locomotive, locomotive, Lagrange, Lagrange, state state feedback feedback Keywords: Keywords: Autonomous linear DC machine powered locomotive, Lagrange, state feedback based on placement, speed control based on placement, speed control Keywords: Autonomous linear DC machine powered locomotive, Lagrange, state feedback based on placement, speed control based on placement, speed control 1. INTRODUCTION INTRODUCTION the mine mine transport transport productivity productivity would would improve improve although although 1. the 1. INTRODUCTION the statistics mine transport productivity would improve although would differ. the statistics would differ. 1. INTRODUCTION the mine transport would improve although would productivity differ. In the the mining mining sector, sector, underground underground mine mine locomotives locomotives are are Thisstatistics paper advocates for modelling modelling and and control control of of aa In the statistics would differ. This paper advocates for In the mining sector, underground locomotives are This a of for ore also paper advocates fortomodelling and control of a linear DC machine used power a mine locomotive. a form form of transportation transportation for coal, coal, mine ore and and also materimaterilinear paper DC machine used tomodelling power a and mine locomotive. In the mining sector, underground locomotives are aalsform of transportation for coal,et mine ore and also materiin underground underground mines Polnik al. [2014]. [2014]. There are This advocates for a linear DC machine used minecontrol locomotive. The suspension suspension system of to thepower mine alocomotive locomotive willofnot not alsform in mines Polnik et ore al. There are The system of the mine will a of transportation for coal, and also materials in underground mines Polnik et al. [2014]. There are The several types of underground locomotives such as battery linear DC machine used to power a mine locomotive. suspension system of the mine locomotive will not be modelled since the linear DC machine for locomotion several types of underground locomotives such as battery be modelled since the linear DC machine for locomotion als in underground mines Polnik et al. [2014]. There are The several types diesel of underground locomotives suchlocomotives as battery locomotives, locomotives and suspension system of the locomotive will not be modelled since the linear DCmine machine for use locomotion is the the area of of interest.The objective is to to energylocomotives, diesel locomotives and electric electric locomotives is area interest.The objective is use energyseveral types of underground locomotives such as Battery battery be locomotives, diesel locomotives andGreen electric locomotives Polnik et al. [2014], Eaton [1914], [1956]. modelled since the linear DC machine for locomotion the modelling area of interest.The objective to use and energybased to model model aa linear linear DC ismachine machine also Polnik et al. diesel [2014],locomotives Eaton [1914], Green [1956]. Battery is based modelling to DC and also locomotives, andGreen electric locomotives Polnik et al.have [2014], [1956]. Battery locomotives beenEaton found[1914], to have have an advantage of lack lack is the modelling area of result interest.The objective to use energyto model ausing linear DC ismachine based and also control it. The found energy-based techniques, locomotives have been found to an advantage of control it. The result found using energy-based techniques, Polnik et al.of [2014], Battery locomotives have beenEaton found to haveGreen an advantage of lack based of emission emission harmful gases[1914], compared to [1956]. diesel locomolocomomodelling to model linear DC machine and control it. Thethe result found ausing energy-based techniques, particularly Lagrange should be the the same as also the of of harmful gases compared to diesel particularly the Lagrange should be same as the locomotives have been found to have an mine advantage of lack particularly of emission of harmful gases compared to diesel locomotives Polnik et al. [2014]. Nonetheless, locomotives it. The result foundusing using energy-based techniques, the Lagrange should be the same as the result one would obtain physics laws as discussed tives Polnik et al. [2014]. Nonetheless, mine locomotives control result one would obtain using physics laws as discussed of emission ofet harmful gases compared mine to rope diesel locomotives Polnik al. advantage [2014]. Nonetheless, locomotives generally offer an compared to haul and particularly the Lagrange should the as of the result one would obtain using physics laws same as discussed by Chiasson Chiasson [2005]. It is is desirable desirable tobeachieve achieve speed generally offer an advantage compared to rope haul and by [2005]. It to aa speed of 33 tives Polnik et al. [2014]. Nonetheless, mine locomotives generally offerbelt an Polnik advantage compared to rope to haul and result the conveyor et al. [2014]. According Sarata one would obtain using physics laws as discussed by Chiasson [2005]. It is desirable to achieve a speed of m/s for a loaded locomotive, this is the speed for practical the conveyor belt Polnik et al.compared [2014]. According to Sarata m/s for a loaded locomotive, this is the speed for practical3 generally offer an Polnik advantage to rope haul and by the conveyor belt etworking al. [2014]. According to Sarata et al. [1986], underground conditions include hard Chiasson [2005]. It islocomotive desirable achieve aforspeed of 3 m/s for aAlso, loaded this isto the speed practical systems. thelocomotive, mine should be autonomous. autonomous. et al.conveyor [1986], underground working conditions include hard systems. Also, the mine locomotive should be the belt Polnikthe etworking al. [2014]. According to Sarata et al. [1986], conditions include hard m/s physical work and conditions underground autoforfeature aAlso, loaded this ison the speed for practical systems. thelocomotive, mine locomotive should be autonomous. A key that could be added the mine locomotive physical workunderground and also, also, the conditions underground autoA key feature that couldlocomotive be added on the mine locomotive et al. [1986], underground working include hard systems. physical work andmine also, workers the conditions underground automatically expose to conditions various health hazards the mine should besurface autonomous. A key feature that could added on thethe mine locomotive would beAlso, a sensor, sensor, that is isbeused used to track track of the the matically expose mine workers to various health hazards would be a that to the surface of physical work and also, the conditions underground automatically expose mineconditions workers toSarata various hazards due to to environmental environmental et health al. [1986]. [1986]. The A key feature that could beused added on the mine locomotive would be a track sensor, that is to track the surface of the the mining so that the necessary control is performed due conditions Sarata et al. The the mining track so that the necessary control is performed matically expose mineconditions workers toSarata various hazards due to environmental et health al.mine [1986]. The would environmental conditions in an an underground underground include be athe sensor, is to track surface ofThe the track so that that theused necessary control is performed the mining to ensure ensure speed of the the locomotive isthe maintained. environmental conditions in mine include to the speed of locomotive is maintained. The due to environmental conditions Sarata ettoal.the [1986]. The environmental conditions in an underground mine include humid, dusty and noisy conditions due nature of mining track so that thelocomotive necessary control is performed to ensure the speed of also the is maintained. The mine locomotive could have a positioning system used humid, dusty and noisy conditions - due to the nature of the mine locomotive could also have a positioning system used environmental conditions in an underground mine include humid, and noisy conditions - due to the nature of to the work workdusty which involves drilling through rocks. Introducthe speed of also the locomotive is maintained. The mine locomotive could have a so positioning used to ensure track the distance travelled that the system locomotive the which involves drilling through rocks. Introducto track the distance travelled so that the locomotive humid, dusty and noisy conditions due to the nature of the work which involves drilling through rocks. Introducing autonomous machinery in mining is of great benefit mine locomotive could also have a positioning system used to track the distance travelled so that the locomotive knows the location where it is supposed to stop. Also, ing work autonomous machinery in mining is rocks. of great benefit knows the location where it is supposed to stop. Also, the drilling through Introducing machinery in but mining great benefit to not autonomous only which to the theinvolves mine worker worker but alsoisto toof the the company. distance travelled sodetect that obstacles the locomotive knows thethe location it is supposed to stop. Also, the track locomotive could where autonomously (mine not only to mine also company. the locomotive could autonomously detect obstacles (mine ing autonomous machinery in but mining great benefit not onlymine to the mineautonomous worker alsoistoof the company. For worker machinery would mean the location iteven is supposed to stop. Also, the locomotive could where autonomously detect obstacles (mine walls, other locomotives or rocks on the pathway) For the the mine worker autonomous machinery would mean knows walls, other locomotives or even rocks onobstacles the pathway) not onlymine to work the mine worker but also to the company. For the worker autonomous machinery would mean the less physical and it would also improve safety signifilocomotive could autonomously detect (mine walls, locomotives rocks on the pathway) Zein et etother al. [2018], [2018], Man et et or al. even [2018]. less physical work andautonomous it would alsomachinery improve safety signifiZein al. Man al. [2018]. For theInmine worker would mean less physical work it would also improve safety cantly. fact, fataland accidents would by 80%signifiwhile walls, locomotives rocks on the pathway) Zein etother al. [2018], Man et or al. even [2018]. cantly. In fact, fatal accidents would improve by 80% while less work it would also would improve safety significantly. In fact, fataland accidents would by 80% while fromphysical these statistics, fatal accidents improve by 50% Zein et al. [2018], Man et al. [2018]. 2. CHIASSON ELECTROMAGNETIC LOCOMOTIVE from these statistics, fatal accidents would improve by 50% 2. CHIASSON ELECTROMAGNETIC LOCOMOTIVE cantly. In fact, fatal accidents wouldwould improve by 80% from these statistics, improve bywhile 50% Kononov [1995]. For the mining autonomous 2. CHIASSON ELECTROMAGNETIC LOCOMOTIVE DRIVE ANALYSIS Kononov [1995]. Forfatal the accidents mining company, company, autonomous DRIVE ANALYSIS from these[1995]. statistics, fatal accidents would improve by 50% 2. CHIASSON ELECTROMAGNETIC Kononov thetransport mining company, autonomous machinery would For mean productivity, and the DRIVE ANALYSIS LOCOMOTIVE machinery would mean transport productivity, and the Kononov [1995]. thetransport mining company, autonomous DRIVE ANALYSIS productivity, and the machinery would For mean study in Kononov [1995], estimated an improvement of The following analysis summarizes the linear linear DC DC machine machine study in Kononov [1995],transport estimated an improvement of The following analysis summarizes the machinery would mean productivity, and the study in Kononov [1995], estimated an improvement of The following analysis summarizes linearThe DCChiasson machine 6% -- 17%. 17%. Although Although the the study study in in Kononov Kononov [1995] [1995] was was done done modelling modelling and and analysis analysis by by Chiasson Chiassonthe [2005]. The Chiasson 6% [2005]. study in Kononov [1995], estimated improvement of The following linear DCChiasson machine 6% - 17%. Although the study in [1995]safety was done modelling andanalysis analysis by Chiasson [2005]. years ago, its still remains remains that theKononov mineanworkers workers and electromagnetic circuitsummarizes analysis is the based onThe physics laws years ago, its still that the mine safety and electromagnetic circuit analysis is based on physics laws 6% - 17%. Although the study [1995]safety was done and analysis Chiasson [2005]. Chiasson years ago, its still remains that in theKononov mine workers and modelling electromagnetic circuit by analysis is based onThe physics laws years ago,©its stillIFAC remains that theFederation mine workers safetyControl) and electromagnetic circuit analysis is based on physics laws 2405-8963 2019, (International of Automatic Hosting by Elsevier Ltd. All rights reserved.

Copyright © 2019 199 Copyright © under 2019 IFAC IFAC 199 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 199 10.1016/j.ifacol.2019.09.187 Copyright © 2019 IFAC 199

199 199 199 199

2019 IFAC MMM 196 Mohlalakoma T. Ngwako et al. / IFAC PapersOnLine 52-14 (2019) 195–200 Stellenbosch, South Africa, August 28-30, 2019

3. LAGRANGE ELECTROMAGNETIC LINEAR DC MACHINE ANALYSIS

Fig. 1. Chiasson linear DC machine Chiasson [2005] in that, the Lorentz law, Faraday’s law, Newton’s law and Kirchoff’s law are applied on an elementary machine, the linear DC machine Chiasson [2005]. For this analysis, consider a linear DC machine as shown in figure 1. The machine comprises a voltage source V, a resistor R due to copper losses, and a sliding bar placed on a pair of rails. It should be noted that bar can only move in the x ˆ direction. Also, the electrical circuit described is placed in a constant magnetic field B in the zˆ direction where the magnetic field is associated with the flux density and the magnetic field takes the direction of the magnetic lines. Magnetic field notation is such that ’x’ points into the page and ’o’ points out of the page. The current flowing along the circuit is denoted by ’i’ and the length l of the bar is in the yˆ direction. Notation: If a vector is represented in bold face, zˆ represents the unit vector in the direction of z. The paper positive F represents the magnitude of F . When a current flows in the circuit, the bar experiences a Lorentz force which is described by: F = il × B (1) Where F is the Lorentz force. For the linear DC machine shown in figure 1, the resultant force is expressed by: F = il(−ˆ y ) × B(−ˆ z) (2) = ilB x ˆ Using Kirchoff’s law, the current flowing along the circuit is found and it is described by; V − Blx˙ i= (3) R Blx˙ is the back emf as used by Chiasson and the back emf is a result obtained by Chiasson after applying Faraday’s law Chiasson [2005]. Where x˙ is the velocity of the bar. The current is dynamic in that it changes as the bar accelerates. The motion of the bar is found using Newton’s law where including an applied as used by Chapman [2012] yields: ml x ¨ − ilB = Fapp − bx˙ (4) where ml is the mass of the locomotive which is the mass of the bar added to the mass of the locomotive trolley (ml = mbar + mtrolley ), Fapp is an external force applied to the locomotive and b is the coefficient of the rail friction of the sliding bar. This brief summary is the modelling and analysis of a linear DC as modelled by Chiasson [2005]. 200

The Lagrange is applied on an electromagnetic circuit where the key contribution is the magnetic field based Lagrange analysis. The Lagrange analysis is performed on a linear DC machine under the same assumptions as Chiasson’s linear DC machine which are typically ideal electromagnetic conditions such as eddy current losses, no saturation, perfectly constant B. The Lagrange is used because the Lagrange is a very simple way of solving machine modelling problems in that, the technique is very systematic and structured. The objective of this paper is to obtain Chiasson’s model to demonstrate the correctness of the field based Lagrange analysis. The Lagrange analysis is commonly known for energy based modeling of RLC circuits, AC machines Combes et al. [2016, 2014]. It is desirable that models must satisfy reciprocity conditions to be physically acceptable hence, the Lagrange, energy based approach Combes et al. [2014] is one way of satisfying these conditions. Therefore, the Lagrange analysis is preferred due to its merits. The Lagrange analysis steps can be summarized as follows: • • • • • •

Identify the kinetic energy Identify the potential energy Identify the dissipation power Compute the Lagrange equation Evaluate the Euler-Lagrange Formulate the state space model

For the Chiasson linear DC machine, the kinetic energy is due to the motion of the bar, the dissipation power is due to copper losses and friction losses whilst the potential energy is due to the magnetic field Civeleka and Bechteler [2008]. The potential energy is due to the magnetic field because the magnetic field is the primary field in that it actuates the electromagnetic circuit without the help of the voltage source. 3.1 Identifying energies The Lagrange standard notation used in Wach [2011], Spiegel [1967] is utilized for the purpose of explaining the Lagrange formulation. Hence, the following is a summary of the theoretical description provided in Wach [2011], Spiegel [1967]. The kinetic energy and potential energy are denoted by T and U respectively. Where both types of energies are a function of generalized coordinates and generalized velocities. The generalized coordinates are described by; r = (r1 , r2 , ..., rn ) (5) The generalized coordinates describe the position of a point. As such, the Euclidean space is described by polar coordinates thus the generalized coordinates in a Euclidean space are (x,y,z) Tatum [2018]. For a cylinder and a sphere, the generalized coordinates are described by (ρ,φ,z) and (r, θ, φ), respectively. The time derivative of the generalized coordinates is known as the generalized velocity and it is denoted by; (6) r˙ = (r˙1 , r˙2 , ..., r˙n ) It should be noted that each generalized coordinate has a force associated to it. The degree of freedom for an un-

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constrained system is the sum of generalized coordinates. However, for a system with R generalized coordinates and N particles the degree of freedom is RN. In practice, systems are constrained to movement. This means that typically, a system either has a rotational motion or a translation motion. A constraint imposed on a system associated with the position is known as a holonomic constraint and it is described by a holonomic equation. When an unconstrained system with a degree of freedom of RN is constrained by k constraints the resulting degree of freedom is RN − k.

The kinetic and potential energy are generally described by Lyshevski [2008]; T (t, r1 , r2 , ..., rn , r˙1 , r˙2 , ..., r˙n ) (7) U (t, r1 , r2 , ..., rn ) (8) Where T is the kinetic energy and U is the potential energy. Both energies are a function of generalized coordinates. The above Lagrange formulation is the summary of the Lagrange formulation as presented in Wach [2011]. 3.2 Computing the Lagrange equation

The Lagrange equation is the kinetic energy less the potential energy. This is shown in the equation Eq.9. L = T (t, r1 , r2 , ..., rn , r˙1 , r˙2 , ..., r˙n )−U (t, r1 , r2 , ..., rn ) (9) Where L is the Lagrange equation. It can be seen that the Lagrange equation is also a function of generalized coordinates and generalized velocities. 3.3 Evaluate the Euler-Lagrange The Euler-Lagrange is further evaluated. For a loss-less system with a zero generalized force, the Euler-Lagrange is given by; d ∂L ∂L − =0 (10) dt ∂ r˙ ∂r However, the Euler-Lagrange is expressed as a function of n independent generalized coordinates and generalized velocities. Therefore, there are n Euler-Lagrange equations for a system with n generalized coordinates. For cases where a generalized force is applied to the system, the Euler-Lagrange has the form; d ∂L ∂L − =Q (11) dt ∂ r˙ ∂r The generalized force Q is the input applied to the system and it can be a voltage in electric circuits or even a torque in mechanical system. Now, consider a system with losses, such a system is described by the equation that follows; ∂P d ∂L ∂L − =Q− (12) dt ∂ r˙ ∂r ∂ r˙ It follows that P is the dissipation power function. The dissipation function may result in either of the three types of systems; under-damped system, critically damped system or the over-damped system. The dissipation function is the force that acts in the opposite direction to the motion of a system. The dissipation function is also nonconservative because it is dependent on the path of an object. It is common that a system oscillates when a disturbance is applied to the system. The dissipation force influences the behaviour of system in that the dissipation force attempts to stop the oscillations. In cases where there 201

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is no dissipation force, by law of conservation of energy, the system oscillates infinitely. For low losses, the system oscillates. However, the dissipation force acting on the system will result in a system that decays exponentially so that the system can be forced back to equilibrium. Now, considering a system with a high dissipation function, the system will not oscillate. However, this does not mean that the system reaches its equilibrium fast. A system of this nature is over-damped. A system with not too low or high losses does not oscillate and it reaches the equilibrium fast. This means that a critically-damped system is achieved.

3.4 Lagrange on linear DC machine The kinetic energy of the bar is: 1 (13) Tr = ml x˙ 2 2 The potential energy of the bar in the presence of the magnetic field is given by Jain [2009]:  x UM = −qlB ˙ x ˆ · dl (14) 0

The potential energy is the path integral of the Lorentz force where q˙ is the current and UM is the potential energy due to the magnetic field. The dissipation power is described by Wach [2011], Ortegs et al. [1998], Stutts [2017] : 1 P1 = Rq˙2 (15) 2 1 2 bx˙ (16) 2 The total dissipative power is given by; 1 1 P = Rq˙2 + bx˙ 2 (17) 2 2 The Lagrange equation is given by; 1 L(x, x, ˙ q, q) ˙ = mx˙ 2 + qlBx ˙ (18) 2 Where x and q are the generalized coordinates and x˙ and q˙ are the generalized velocities. As described in Eq.9 12 after obtaining Eq. 18. The key step of the Lagrange analysis is to perform the Euler-Lagrange. P2 =

Evaluating the Euler-Lagrange with respect to x: d ∂L ∂L ∂ 1 2 1 2 − =U− ( Rq˙ + bx˙ ) (19) dt ∂ x˙ ∂x ∂ x˙ 2 2 yields: ml x ¨ − qlB ˙ = F − bx˙ (20) Whilst evaluating the Euler-Lagrange with respect to the generalized coordinate q: ∂ 1 2 1 2 d ∂L ∂L − =V − ( Rq˙ + bx˙ ) (21) dt ∂ q˙ ∂q ∂ q˙ 2 2 yields; 1 ˙ (22) q˙ = (V − lB x) R Since q˙ is the current, the dynamic model of the linear DC can be represented as: ml x ¨ − ilB = F − bx˙

(23)

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1 ˙ i = (V − lB x) R

(24)

Note: Future work will discuss evaluating the EulerLagrange without using the general coordinate as is done in the case particularly in Eq.15 to Eq.24. This will form the basis of model order reduction based on Lagrange analysis. 3.5 Equivalent of electromagnetic modelling At this juncture it is worthwhile highlighting the key results. The same model i.e. Eq.3 - 4 obtained by Chiasson in modelling the linear DC machine has been obtained by Lagrange analysis i.e. Eq.23 - 24. With the same assumptions as Chiasson [2005], the Lagrange analysis result is the same as Chiasson’s results. The result is Newtonian and Kirchoff formulations. This highlights the correctness of the Lagrange analysis.

black lines. From figure 2 and 3, it is deduced that the current decreases gradually with an increase in velocity. This is due to electromagnetic induction Chiasson [2005]. Now consider a case where a force is applied in the direction of motion of the bar. This case happens at t = 12s to t = 15s. The results for this case are such that when the velocity increases from steady state current (t = 12s to t = 15s) a reverse current is obtained. The machine is operating as a generator . When the force is applied in a direction opposing the direction of motion of the bar when the bar is at steady-state, the velocity decreases resulting in the current increasing. This case happens between t = 5s to t = 10s. The machine is behaving as a motor .

3.6 Linear DC machine performance Lagrange physical equations typically allow for state space model equations to be obtained as below. Further state space trajectory can be performed to accomplish system performance analysis. When we let x1 = x, ˙ and x˙2 = q, ˙ V = v and F = u, the state space model is found to be;

x˙1 = −

u 1 B 2 l2 lB + b)x1 + v+ ( ml R ml R ml

1 x˙2 = (v − lBx1 ) R

Fig. 2. Linear DC machine current dynamics

(25)

4. MINE LOCOMOTIVE DRIVE RESULTS Using the linear DC machine parameters found in table 2, the results of the linear DC machine are plotted and discussed.

Table 1. Linear DC parameters Fig. 3. Linear DC machine velocity dynamics locomotive mass (ml ) magnetic field (B) rail friction coefficient (b) rail separation length (l) resistor (R) input voltage (v) applied force (u)

0.25 kg 0.5 T 0.05 0.6 m 100 mΩ 1 V, 3 V, 5 V 1 N, 3 N, 5 N

From the results, it is deduced that when there is no current flowing in the electromagnetic circuit the bar is not moving. This is shown at t = 0 to t = 1s. When current starts flowing, it starts at its maximum current because electromagnetic induction does not happen instantaneously. As the current continues flowing the bar accelerates until the the steady-state velocity is reached where the current is also at steady state. This shown by the 202

Fig. 4. Power curve

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The power curve for an input voltage of 1 V, 3 V and 5 V, respectively, is also shown in fig.4. The power as shown on the aforementioned figure starts from 0 W and increases until it reaches a peak power of 2 W, 22 W, and 62 W, for the voltage of 1 V, 3 V and 5 V, respectively. Thereafter, the power decreases gradually. When an external force is applied in the direction opposing the bar of the linear DC machine at steady state, the power increases. This is shown by the pronounced region between 5 s - 10 s. When an external force is applied in the direction of the bar of the linear DC machine at steady state, the linear DC machine supplies power. The aforementioned is shown by a pronounced region between 12 s - 15 s. 5. SPEED CONTROL FOR THE MINE LOCOMOTIVE DRIVE

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are found to have full rank. Therefore, the control design can proceed. For control design purposes, a second order system is assumed because the nature of a second order equation allows for the settling time, rise time and damping ratio to be determined. The aforementioned parameters are necessary to be able to realize the desired linear DC machine performance. The performance specifications are also chosen, and presented in table 2. Table 2. Linear DC performance specifications

Settling time (ts ) Percentage overshoot (PO) ζ Reference velocity ωn Settling time tolerance band

0.6s 5% 0.69 3m/s 9.4rad/s ±2%

Where for a chosen PO, the damping ratio is found using Burns [2001]:    ln( P O 2 100 ζ= (30) O2 2 π + P100 For a settling time tolerance band of ±2%, and a chosen settling time of 0.6 s, the natural frequency is Burns [2001]:

Fig. 5. Controlled linear DC machine

ωn =

The state-feedback using pole placement control algorithm is used to improve the performance of the linear DC machine. By performance, this refers to improving the settling time, and overshoot of the linear DC machine. The control algorithm also allows that the linear DC machine operates at the desired speed. The control design can only proceed after a controllability and observability test are performed and the system is both controllable and observable. According to reference Lathi [2005], Burns [2001] a system is controllable if a state x(t0 ) can be transferred to x(tf ) and a system is observable if at time t0 , the state x(t0 ) can be determined from the output. The observability and controllability test demand the following Burns [2001]:

2 −ln( 100 ) ζts

(31)

the desired closed-loop poles are determined using a second order equation and pole placement method Burns [2001]: s2 + 2ζωn + ωn2 Where the poles, p, are found to be;

(32) T

p = [−6.5200 + 6.8375i −6.5200 − 6.8375i] (33) Where K is the gain matrix, and poles. The resulting gain matrix is; K = [−74.2282 89.2622] (34) 6. RESULTS AND ANALYSIS

• a controllability and observability matrix • the controllability and observability matrices must be of full rank For a system with the state space equation x˙ = Ax + Bf y(t) = Cx + Df

(26)

and the control law is described f = r − Kx (27) Where K is the state-feedback gain matrix, r is the reference input, u is the control law The controllability and observability matrix are described by Quevedo [2018]:   C = B AB ... An−1 B (28)

T (29) O = C CA ...CAn−1 Where C is the controllability matrix and O is the observability matrix. for the linear DC machine both matrices 

203

Fig. 6. Linear DC machine velocity dynamics From the results it can be seen that when the speed is controlled using a negative state-feedback controller the steady state velocity is 3 m/s. Also, the controlled linear DC machine has a settling time of 0.6 s while the uncontrolled linear DC machine has a settling time of 1s. It

2019 IFAC MMM 200 Mohlalakoma T. Ngwako et al. / IFAC PapersOnLine 52-14 (2019) 195–200 Stellenbosch, South Africa, August 28-30, 2019

is worthwhile to pay attention to the steady state velocity of the controlled and uncontrolled linear DC machine. The controlled linear DC machine has a steady state velocity of 3 m/s whilst the uncontrolled linear DC machine has a state state velocity of 16m/s for an input voltage of 5 V. This shows that the controlled linear DC machine preforms better that the uncontrolled machine, also, the controlled linear DC machine is able to meet the desired performance specifications as summarized in table 2. 7. AUTONOMOUS POSITION TRACKING AND COLLISION DETECTION The locomotive presented in this journal paper is a modern mine locomotive. The locomotive moves autonomously. The mine locomotive is equipped with an autonomous position tracking system and an autonomous obstacle detection system. The tracking and detection algorithms are implemented on a low cost and low power microcontroller, the pic16F690 microcontroller (11 µA, 32 kHz, 2 V). The programming language used is the assembly language. A real-time camera is positioned at the front of the locomotive to detect curves and obstructions. The camera allows for the microcontroller to calculate the distance which the locomotive is required to move before reaching a curve. Similarly to the autonomous tracking system in Zein et al. [2018], the locomotive’s reference position data points are stored in a look-up table. When the locomotive moves, the current distance of the locomotive is compared with a reference distance. An 8-bit software analogue-to-digital converter (ADC) reads a voltage (Vin /Vref × 256) from a position sensor. Every position sensor voltage corresponds to a distance. The ADC allows the microcontroller to read the sensor voltage which allows for the current distance of the locomotive to be calculated. The camera measures the distance before a curve/turning point is detected. The distance measured by the camera is used to adjust the angle of the locomotive. When the locomotive reaches the curved area, an interrupt occurs and the next reference distance is selected as the new reference distance from the look-up table. The algorithm repeats until the last reference distance from the look-up table is equal to the current locomotive distance. The aforementioned would mean that the locomotive has reached its destination. If there is an obstacle in front of the locomotive such as a pile of rocks/coal/ore or potholes. The camera measures the length and height of the obstruction in front of the locomotive. The size of the obstruction is then calculated. If the obstruction is big and in front of the locomotive, the locomotive stops moving. When there are potholes in the rail, the locomotive would also stop moving automatically. 8. CONCLUSION The Lagrange analysis has been shown to generate the same physical model equations as is obtained by electromagnetic analysis of the same ideal linear machine as modelled by Chiasson. Therefore Lagrange analysis is a viable and verified modelling approach for physical modelling of an ideal linear DC machine. Consequently linear DC machine performance analysis is demonstrated by extending the Lagrange physical model to a state space model and performance analysis through state space trajectory formulation. 204

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