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IFAC PapersOnLine 52-14 (2019) 201–206

A Condition for Singular Optimal Control A Condition for Singular Optimal Control A Condition for Singular Optimal Control Formulation for a Tandem Pair of Gravity A Condition for for aSingular Optimal Control Formulation Tandem Pair of Gravity Formulation for a Tandem Pair of Gravity Fed Linear for DCaMachine Mine Formulation Tandem Powered Pair of Gravity Fed Machine Powered Mine Fed Linear Linear DC DC Machine Powered Mine Locomotives Fed Linear DC Machine Powered Mine Locomotives Locomotives Locomotives ∗∗ Mohlalakoma T. Ngwako ∗∗∗ Otis T. Nyandoro ∗∗ ∗∗

Mohlalakoma T. Ngwako T. Nyandoro Mohlalakoma T. Otis T. ∗ Otis ∗∗∗ ∗∗∗∗∗∗ Mohlalakoma T. Ngwako Ngwako Otis T. Nyandoro Nyandoro ∗∗∗ ∗∗∗∗ John van Coller C. Madahana ∗∗∗ Milka ∗∗∗∗ John van Coller Milka C. Madahana John van Coller Milka ∗ C. Madahana ∗∗∗∗∗∗ ∗∗∗ Mohlalakoma T. Ngwako T. Nyandoro John van Coller MilkaOtis C. Madahana ∗∗∗ John ∗∗∗van Coller Milka C. Madahana ∗∗∗∗ (e-mail: [email protected]). (e-mail: [email protected]). (e-mail: [email protected]). ∗∗∗ (e-mail: ∗∗ (e-mail: [email protected]) ∗∗ (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]) ∗∗∗∗ (e-mail: [email protected]) (e-mail: [email protected]) ∗∗∗ ∗∗∗∗ (e-mail:[email protected]) [email protected]) (e-mail: ∗∗∗∗ School of Electrical and Information Engineering, University of School of Electrical and Information Engineering, University of (e-mail: [email protected]) School of Electrical and Information Engineering, University of School of Witwatersrand, Electrical and Information Engineering, University of Johannesburg, South Africa Witwatersrand, Johannesburg, South Africa Witwatersrand, Johannesburg, South Africa School of Witwatersrand, Electrical and Information Engineering, University of Johannesburg, South Africa Witwatersrand, Johannesburg, South Africa Abstract: The key focus in this study is on gravity fed linear DC machines used to drive Abstract: The The key key focus focus in in this this study study is is on on gravity gravity fed fed linear linear DC DC machines machines used used to to drive drive Abstract: Abstract: The key focus in this study is on gravity fed linear DC machines used to drive mine locomotives. A case for optimal control singularity analysis for this tandem pair of mine locomotives. locomotives. A A case case for for optimal optimal control control singularity singularity analysis analysis for for this this tandem tandem pair pair of of mine Abstract: The key in study is on gravity fed linear DC machines usedfeature topair drive mine locomotives. Afocus case forthis optimal control singularity analysis for this tandem of gravity fed linear DC machines on a slope is thus presented. The key operational is gravity fed linear DC machines on a slope is thus presented. The key operational feature is gravity fed linear DC machines on a slope is thus presented. The key operational feature is mine locomotives. A case for optimal control singularity analysis for this tandem pair of gravity fed linear DC machines on a slope is thus presented. The key operational feature is for a generator locomotive to drive a motor locomotive with gravity providing the machine for a generator locomotive to drive a motor locomotive with gravity providing the machine for a locomotive to motor with gravity providing the machine gravity fed DC and machines on aaa slope is locomotive thus presented. key operational feature is for a generator generator locomotive to drive drive motor locomotive with The gravity thethis machine disparity for generator motor action. Optimal control formulation is extended in study, disparity forlinear generator and motor action. Optimal control formulation is providing extended in this study, disparity for generator and motor action. Optimal control formulation is extended in this study, for a generator locomotive to drive a motor locomotive with gravity providing the machine disparity for generator and motor action. Optimal control formulation is extended in this study, particularly optimal optimal control control singularity singularity formulation. formulation. The The optimal optimal controller controller should should be be able able to to particularly particularly optimal control singularity formulation. The optimal controller should be to disparity for generator and motor action. Optimal control is extended in this study, particularly optimal control singularity formulation. The formulation optimal controller should be able able to minimize the relative displacement between the machines. Mine locomotive dynamics results minimize the relative displacement between the machines. Mine locomotive dynamics results minimize the relative displacement between the machines. Mine locomotive dynamics results particularly singularity formulation. Theand optimal controllerare be able to minimize theoptimal relative displacement thevelocity machines. Mine locomotive dynamics results are also presented to show that the differential displacement minimized. The are also also presented presented tocontrol show that the between differential velocity and displacement areshould minimized. The are to show that the differential velocity and displacement are minimized. The minimize the relative displacement between the machines. Mine locomotive dynamics results are also presented to show that the differential velocity and displacement are minimized. The performance index and Lagrangian are also minimized. The optimal control formulation is performance index index and and Lagrangian Lagrangian are are also also minimized. The The optimal optimal control control formulation formulation is is performance are also presented toand show that the differential velocity and displacement minimized. The performance index Lagrangian are also minimized. minimized. The optimal formulation is demonstrated via switching switching function differential calculus. The solutioncontrol isare shown to improve improve demonstrated via function differential calculus. The solution is shown to demonstrated via switching function differential calculus. The solution is shown to improve performance index and performance. Lagrangian also minimized. formulation is demonstrated viaenergy switching functionare differential calculus.The Theoptimal solutioncontrol is shown to improve mine locomotive mine locomotive locomotive energy performance. mine performance. demonstrated viaenergy switching function differential calculus. The solution is shown to improve mine locomotive energy performance. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. mine locomotive energy performance. Keywords: Optimal control singularity analysis, gravity fed linear DC machine, mine Keywords: Optimal Optimal control control singularity singularity analysis, analysis, gravity gravity fed fed linear linear DC DC machine, machine, mine mine Keywords: Keywords: Optimal control singularity analysis, gravity fed linear DC machine, mine locomotives, slope. locomotives, slope. locomotives, slope. Keywords: Optimal locomotives, slope. control singularity analysis, gravity fed linear DC machine, mine locomotives, slope. 1. INTRODUCTION would also also reduce reduce the the intensity intensity of of physical physical labour labour in in the the 1. INTRODUCTION INTRODUCTION would 1. would also reduce reduce the the intensity intensity of of physical physical labour labour in in the the 1. INTRODUCTION would also mining sector. A key operational concept in the mining mining sector. A key operational concept in the mining miningalso sector. A key key operational concept in the mining mining 1. INTRODUCTION would reduce the operational intensity of concept physical labour in two the mining sector. A in the sector where the track has slope would be to use sector where where the track has aaa slope slope would be be to use use two sector the track has would to two Some of the approaches used to access mine deposits from mining sector. A key operational concept in the mining where the track has a slope would be to use two mine locomotives simultaneously to fully utilize all the Some of of the the approaches approaches used used to to access access mine mine deposits deposits from from sector Some mine locomotives locomotives simultaneously simultaneously to to fully fully utilize utilize all all the the mine Some of the approaches used to access mine deposits from sector underground include; adit, decline/ramp, incline shaft where the track has idea a slope would to use two mine locomotives simultaneously to fully aabe utilize all the energies in the system. The is to have set-up where underground include; adit, adit, decline/ramp, incline shaft shaft underground include; decline/ramp, incline energies in the system. The idea is to have set-up where energies in the system. The idea is to have a set-up where Some of the approaches used to access mine deposits from underground include; adit, decline/ramp, incline shaft and vertical vertical shafts shafts Rupprecht Rupprecht [2012]. [2012]. Adits Adits in in the the South South mine locomotives simultaneously fullywhilst the energies in the system. is to to have autilize set-up where one locomotive is at the top of the slope the other and and vertical shafts shafts Rupprecht [2012]. Adits in one locomotive locomotive is at at theThe topidea of the the slope whilst theall other one is the top of slope whilst the other underground include; adit, decline/ramp, incline shaft [2012]. Adits in the the South South and vertical Rupprecht African mining sector are often found in mountainous arenergies in the system. The idea is to have a set-up where one locomotive is at the top of the slope whilst the other locomotive is at the bottom of the slope. The locomoAfrican mining sector are often found in mountainous arAfrican mining sector are often often[2012]. found Adits in mountainous mountainous ar- locomotive locomotive is is at at the the bottom bottom of of the the slope. slope. The The locomolocomoand vertical Rupprecht in Rupprecht the South African mining sector are in areas where the deposits dip under the mountain one locomotive is at the top would ofofthe slope whilst locomotive is at the bottom thetake slope. Thethe locomotive at the top of the slope advantage of the eas where where theshafts deposits dip under underfound the mountain mountain Rupprecht eas the deposits dip the Rupprecht tive at the top of the slope would take advantage ofother the tive at the top of the slope would take advantage of the African mining sector are often found in mountainous areas where the deposits dip under the mountain Rupprecht [2012]. Incline shafts offer quick access to the ore body locomotive is at the bottom of the slope. The locomotive at the top of the slope would take advantage of the gravitational force applied to it to drive the locomotive at [2012]. Incline Incline shafts shafts offer offer quick quick access access to to the the ore ore body body gravitational [2012]. gravitational force applied to it to drive the locomotive at force applied to it to drive the locomotive at eas where the deposits dip under the mountain Rupprecht [2012]. Incline shafts offer quick access to the ore body which reduces cost significantly. However, they are labour tive at the top of the slope would take advantage of the gravitational force applied to it to drive the locomotive at the bottom of the slope. Therefore, the locomotive at which reduces cost significantly. However, they are labour which reduces cost significantly. However, they are labour the bottom of the slope. Therefore, the locomotive at the the bottom bottom of offorce the slope. slope. Therefore, the locomotive locomotive at the [2012]. Incline shafts offer access to theare orelabour body which reduces cost significantly. However, they intensive and they have low operational efficiencies comapplied to it to the locomotive at the the Therefore, at top of the slope would behave as aathe generator whilst the intensive and they they have lowquick operational efficiencies com- gravitational intensive and have low operational efficiencies comtop of of the the slope slope would behave as drive generator whilst the top would behave as a generator whilst the which reduces cost significantly. However, they are labour intensive and they have low operational efficiencies compared to vertical shafts Rupprecht [2012], Holl and Fairon the bottom of the slope. Therefore, the locomotive at the of the slope would behave as a generator whilst locomotive at bottom of the slope would behave as pared to to vertical vertical shafts shafts Rupprecht Rupprecht [2012], [2012], Holl Holl and and Fairon Fairon top pared locomotive at at the the bottom bottom of of the the slope slope would would behave behave as as locomotive intensive and they have lowshaft operational efficiencies com- top pared shafts Rupprecht [2012], Holl and Fairon [1973]. In the 1960s, incline power consumption costs of the as lift a generator whilst locomotive at thewould bottom of the slope would behavethe as motor. This approach is similar to with counter-weight [1973].to Invertical the 1960s, 1960s, incline shaft power consumption costs [1973]. In the incline shaft power consumption costs motor. Thisslope approach is behave similar to lift with counter-weight motor. This approach is similar to lift with counter-weight pared to vertical shafts Rupprecht [2012], Holl and Fairon [1973]. In the 1960s, incline shaft power consumption costs and maintenance costs were estimated to amount to 3.5 at theOptimal bottom of the slope would behave as motor. approach is similar toalgorithms lift with counter-weight control problem. control based on the and maintenance maintenance costs costs were were estimated estimated to to amount amount to to 3.5 3.5 locomotive and control This problem. Optimal control algorithms based on the the control problem. Optimal control algorithms based on [1973]. In tonne the 1960s, incline shaft power consumption and maintenance costs were estimated to Holl amount tocosts 3.5 motor. cents per compared to vertical shafts and Fairon This approach is similar to lift with counter-weight control problem. Optimal control algorithms based on the work by Tsiotras [2000] could be applied for optimal speed cents per tonne compared to vertical shafts Holl and Fairon centsmaintenance per tonne tonne compared to vertical vertical shafts Holl and Fairon Fairon work by by Tsiotras Tsiotras [2000] [2000] could could be be applied applied for for optimal optimal speed speed work and costsare were estimated to Holl amount to 3.5 control cents to and [1973]. Vertical shafts efficient in transporting mining control algorithms based on the work byproblem. Tsiotras Optimal [2000] could be applied for optimal speed and displacement. displacement. [1973].per Vertical compared shafts are efficient inshafts transporting mining [1973]. Vertical shafts are efficient in transporting mining and and displacement. cents per tonne compared to vertical shafts Holl and Fairon [1973]. Vertical shafts are efficient in transporting mining equipment. However, they experience difficulty in transwork by Tsiotras [2000] could be applied for optimal speed and displacement. equipment. However, However, they they experience experience difficulty difficulty in in transtransequipment. Singular optimal control for gravity fed pair of mine [1973]. shafts they are efficient inshafts transporting equipment. However, experience difficulty inmining transporting large equipment. Vertical are often used Singular optimal control control for for aa gravity gravity fed fed pair pair of of mine mine displacement. Singular optimal portingVertical large equipment. Vertical shafts are often often used and porting large equipment. Vertical shafts are used Singular optimal controlNkomo for aa et gravity fed Zhang pair of mine locomotives is presented presented Nkomo et al. [2017], [2017], Zhang [2019]. equipment. However, they experience difficulty in transporting large equipment. Vertical shafts are often used where the ore body is 250 500 m deep Rupprecht [2012]. locomotives is al. [2019]. locomotives is presented Nkomo et al. [2017], Zhang [2019]. where the ore body is 250 500 m deep Rupprecht [2012]. where the the ore body body is 250 250 --Vertical 500 m m deep deep Rupprecht [2012]. Singular optimal controlNkomo for a et gravity fed Zhang pair of[2019]. mine locomotives is presented al. [2017], porting large equipment. shafts are used where ore is 500 Rupprecht Decline shafts are often used in Australia and the ore is Decline shafts shafts are often often used in Australia Australia andoften the[2012]. ore is locomotives The rest of the journal paper is structured as follows; in Decline are used in and the ore is The rest of the journal paper is structured as follows; in is presented Nkomo et al. [2017], Zhang [2019]. The rest of the journal paper is structured as follows; in where the ore body is 250 500 m deep Rupprecht [2012]. Decline shafts often used or inconveyor Australiabelts and Rupprecht the ore is The transported by truck haulage rest22 of the modeling journal paper is structured as gravity follows;fed in transported by are truck haulage or conveyor belts Rupprecht section , the the modeling of aa tandem tandem pair of of gravity fed transported by truck haulage or conveyor belts Rupprecht section , of pair section 2 , the modeling of a tandem pair of gravity fed Decline shafts are often used in Australia and the ore is transported by truck haulage or conveyor belts Rupprecht [2012]. The form of transportation used in mining includes the modeling journal paper is structured as gravity follows; in section 2 of , machines the offor a tandem pair are of fed [2012]. The The form form of of transportation transportation used used in in mining mining includes includes The linear DC used locomotion presented, [2012]. linearrest DC machines used for locomotion are presented, linear DC machines used for locomotion are presented, transported by truck haulagebelts or conveyor Rupprecht [2012]. The form ofconveyor transportation usedrope in belts mining includes mine locomotives, and hauling Polnik section 2 , the modeling of a tandem pair of gravity fed linear DC machines used for locomotion are presented, mine locomotives, conveyor belts and rope hauling Polnik the Hamiltonian based optimal control formulation is mine locomotives, locomotives, conveyor belts and and rope haulingincludes Polnik the the Hamiltonian Hamiltonian based based optimal optimal control control formulation formulation is is [2012]. The form ofconveyor transportation usedrope in mining mine belts hauling et al. [2014]. Now introducing autonomous mine equipDC machines used for locomotion are presented, the Hamiltonian based optimal control formulation is et al. al. [2014]. [2014]. Now introducing autonomous mine Polnik equip- linear presented in section 3 where the simulation results are et Now introducing autonomous mine equippresented in section 3 where the simulation results are presented in section 3 where the simulation results are mine locomotives, conveyor belts and rope hauling Polnik et al. in [2014]. Now introducing mine equipment, in particular, autonomousautonomous mine locomotives locomotives would the Hamiltonian based formulation is presented in section section 3 where the control simulation results are ment, particular, autonomous mine would 4. Aoptimal conclusion is found found in section section ment, in particular, autonomous mine locomotives would presented in 4. A conclusion is in presented in section 4. 4. A conclusion is found results in section et al. in [2014]. Now efficiency, introducing autonomous mine equipment, particular, autonomous mine locomotives would improve transport productivity and mine safety presented 3 where the simulation are in section A conclusion is found in section improve transport efficiency, productivity and mine safety 5. improve transport efficiency, productivity and mine mine would safety 5. 5. ment, in transport particular, autonomous mineThe locomotives improve efficiency, and safety Kononov [1995], Jiang et al. [2017]. aforementioned presented in section 4. A conclusion is found in section 5. Kononov [1995], Jiang Jiang et al. al.productivity [2017]. The aforementioned Kononov [1995], et [2017]. The aforementioned improve transport efficiency, productivity and mine safety Kononov [1995], Jiang et al. [2017]. The aforementioned 5. Kononov Jiang et al. [2017]. The of aforementioned 2405-8963 © ©[1995], 2019, IFAC IFAC (International Federation Automatic Control) Copyright 2019 205 Hosting by Elsevier Ltd. All rights reserved. 205 Copyright © 2019 205 Copyright © under 2019 IFAC IFAC 205 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 205 10.1016/j.ifacol.2019.09.188 Copyright © 2019 IFAC 205

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2019 IFAC MMM 202 Mohlalakoma T. Ngwako et al. / IFAC PapersOnLine 52-14 (2019) 201–206 Stellenbosch, South Africa, August 28-30, 2019

2. MODELLING 2.1 Physical model

Fig. 1. Free body diagram for a tandem pair of linear DC machines The free body diagram of a tandem pair of gravity fed linear DC machines (LDCMs) is shown in fig.1. The LDCMs are subject to a gravitational force Fg , an applied force, a normal force Fnormal , a Lorentz force FL and a friction force Ff r . The arrangement of the LDCM locomotives allows for the gravitational force to be applied on either machines when positioned at the top of the slope. All this is under the assumption that the machines are at steady state. When the gravitationally fed LDCM starts moving, a reverse current is generated. The aforementioned implies that the DC machine is operating as a generator. The goal is to achieve a set-up where one LDCM drives the other LDCM. That would mean that the desired set-up would have a generator driving a motor. The LDCM at the top of the slope would be the generator whilst the LDCM at the bottom of the slope would be the motor. Remark: From the free body diagram, it should be noted that the block represents the LDCM powered locomotive. Zooming into the block, the linear DC machine circuit shown in fig.2 is used for the analysis. This is the same circuit used by Chiasson [2005].

current flowing along the circuit. This force as used by Chiasson [2005], is linearly proportional to the current flowing in the circuit. The Lorentz force is described by: (1) F Ln = in ln × B n Where n identifies the circuit of interest and F is the Lorentz force, i is the current flowing along the circuit, l is the length of the bar and B is the magnetic field strength. For the LDCM shown in fig.2, the resultant Lorentz force for a single LDCM is expressed by: F L = il(−ˆ y ) × B(−ˆ z) (2) = ilB x ˆ As it can be seen from Eq.1 - 2, the direction of the Lorentz force is found using the cross product of the length of the bar and the magnetic field. The Lorentz force is found to be in the x ˆ direction. Notation: It should be noted that bold symbols represent vectors throughout the paper. The linear DC machine is in a 3D plane where x ˆ, yˆ, zˆ are unit vectors representing the direction. By Newtons law, the translation motion of the LDCM locomotives on a slope is described by Chiasson [2005], Chapman [2012]: m1 v˙ 1 − i1 l1 B1 = Fapp1 (3) and m2 v˙ 2 − i1 l2 B2 = Fapp2 (4) Where m1 and m2 represent the mass of the locomotive, v˙ is the acceleration of locomotive, and Fapp is the applied force. It should also be known that Fapp is the gravitational force applied to the mine locomotive. Therefore, the applied force is dependent on the mass of the locomotive. It should be noted that the applied force is also the gravitational force. The applied force is expressed by: Fapp = mg

(5)

Where m is the mass of the locomotive and g is the gravitational acceleration. Using Kirchoff’s law, the current flowing through the independent pair of LDCMs is described by: 1 i = (V − l1 B1 v1 − l2 B2 v2 ) (6) R The term (lBv)from equation Eq.6 represents the induced emf as used by Chiasson [2005]. The induced emf is governed by the Faraday’s law and the Lentz law. R = R1 +R2 is the equivalent resistance of the electrically series connected LDCMs and V is the voltage. Remark: In practice, there is friction force due to the bar of the LDCM sliding along the rail. However, in this study the friction force is assumed to be negligible. This assumption is made primarily to simplify optimal control formulation, in particular, singularity analysis, which will be demonstrated in the sections that follow.

Fig. 2. Linear DC machines 2.2 Mathematical model The mathematical model of a tandem pair of gravitationally fed LDCM powered locomotives is based on Chiasson’s model for a single LDCM Chiasson [2005]. The LDCM locomotive model is formulated using physics laws i.e. Faraday’s law, Kirchoff’s and Newton’s laws. The Lorentz force is experienced by the pair of bars as a result of a 206

2.3 State equations From the equations found using physics laws in sec.2.2, let: x1 = v1 , x˙1 = v˙1 , x2 = v2 , x˙2 = v˙2 , and x˙3 = i. The following state equations are obtained:

2019 IFAC MMM Mohlalakoma T. Ngwako et al. / IFAC PapersOnLine 52-14 (2019) 201–206 Stellenbosch, South Africa, August 28-30, 2019

Fapp1 1 + (V − l1 B1 x1 − l2 B2 x2 )l1 B1 m1 Rm1 Fapp2 1 x˙2 = − + (V − l1 B1 x1 − l2 B2 x2 )l2 B2 m2 Rm2 V − l1 B1 x1 − l2 B2 x2 x˙3 = R x˙1 =

(7)

For sake of compactness of the state equations; let α1 = Fapp1 Fapp2 V m 1 , α2 = m2 , f (x) = −l1 B1 x1 − l2 B2 x2 , u = R , ρ1 = Rl11Bm11 , ρ2 = Rl22Bm22 , γ = R1 . R = R1 + R2 . This step is necessary in that it allows for a neater treatment of the switching function which will be formally introduced in optimal control analysis. The state equation is now described by: u m1 u x˙2 =α2 + ρ2 f (x) + m2 x˙3 =γf (x) + u

x˙1 =α1 + ρ1 f (x) +

(8)

i.e. x˙ = g(x, u, t)

203

L = x1 + x2 , where x1 and x2 are the velocities of the locomotives. Computing the Hamiltonian associated with the optimal control problem follows; the Hamiltonian for a general case is described using the Legendre transformation and this is shown by Tsiotras [2000], Boscain and Piccoli [2016]: H = L + λT (g(x, u, t))

(12)

Where H is the Hamiltonian, λ is a vector of co-state variables (also known as a vector of Lagrange multipliers), this vector has the same dimensions as the state vector x. Evaluating the Hamiltonian yields: u ) + λ2 (α2 + ρ2 f (x) H = x1 + x2 + λ1 (α1 + ρ1 f (x) + m1 u ) + λ3 (γf (x) + u) + m2 (13) It follows that the time derivative of the co-state variables is equal to the Jacobian of the Hamiltonian. The relationship between the Hamiltonian and co-states variables (co-state equation) deduced by a first order optimality condition is necessary for determining the optimal control input which will be shown later in the paper:

(9)

Another thing to note is that, throughout this document, g(x, u, t) will be used to represent the state equation. In this section, electrically series connected LDCM powered locomotives are modeled and presented. The model provided is based on Faraday’s law, Lorentz law, Kirchoff’s law and Newtons law. An applied force due to gravity is also incorporated when modeling the locomotives. The friction force is omitted. Future work in this model will incorporate a detailed friction model of the locomotives. This friction model will be an extension of Pacejka’s friction model and the LuGre dynamic model Tsiotras [2000]. 3. HAMILTONIAN BASED OPTIMAL CONTROL FORMULATION The Hamiltonian based optimal control from Tsiotras [2000] is used to formulate the optimal control in this study. Formulating the optimal control demands that the control problem is well defined. A key step that follows is the Pontryagin Maximum Principle (PMP). In this step a first order condition necessary for optimality is used to determine an explicit solution for the optimal control problem Hale et al. [2016], Nkomo et al. [2017]. The optimal control problem would be to minimize the differential displacement of the of the locomotives over a finite time and a space of admissible controls Hale et al. [2016]. The performance index Hu et al. [2019]: tf J= Ldt (10) 0

is subjected to the state space equation: x˙ = g(x, u, t) (11) Where J is the cost function (performance index), and L is the running cost (Lagrangian) Boscain and Piccoli [2016]. Now consider: 207

∂H λ˙ = − ∂x Evaluating Eq.14 yields the following equations: ∂H ∂f (x) = −1 + (λ1 ρ1 + λ2 ρ2 + λ3 γ) λ˙1 = ∂x1 ∂x1 ∂H ∂f (x) = −1 + (λ1 ρ1 + λ2 ρ2 + λ3 γ) λ˙2 = ∂x2 ∂x2 λ˙3 = 0

(14)

(15)

(16) (17)

Now imposing the transversality condition for finite time: (18) −λT |tf dx(tf ) + H|tf dtf = 0 so; H(tf ) = 0 (19) therefore: (20) H(t) = 0 ∀t ∈ [ 0, tf ] and it is desired that: λ(tf ) = 0 (21) The optimal control input is also given by: u∗ = argminH(x, λ, t) (22) The admissible control input is described by: umin ≤ u ≤ umax (23) Where u is the input voltage normalized with the resistance and it ranges between u ∈ [ 0, 20/R ] Stationary condition: ∂H =0 (24) ∂u 3.1 Pontryagin Singular control H∗ ≤ H

(25)

2019 IFAC MMM 204 Mohlalakoma T. Ngwako et al. / IFAC PapersOnLine 52-14 (2019) 201–206 Stellenbosch, South Africa, August 28-30, 2019

L∗ + λT (g((x), u, t))∗ ≤ L + λT (g((x), u, t)) Which yields λT ∗ u ∗ ≤ λ T u

(26)

(27)

The switching function is known as the coefficient of the control input deduced from the Hamiltonian represented by Eq.28. Thus: (28) H1 = ρ1 λ1 + ρ2 λ2 + λ3 The control strategy is found using Eq.23: umin H1 ≥ 0 u∗ = umax H1 ≤ 0 using H1 ≡ 0 At the singular arc, H1 ≡ 0, which implies that: ρ1 λ1 + ρ2 λ2 + λ3 = 0

(30)

Now, taking the time derivative of the switching function: (31) H˙ 1 = ρ1 λ˙1 + ρ2 λ˙2 + λ˙3 substituting Eq.15 - Eq.17 into Eq.31 yields: ∂f (x) ∂f (x) + ρ2 ) = ρ1 + ρ2 (λ1 ρ1 + λ2 ρ2 + λ3 γ)(ρ1 ∂x1 ∂x2 Since H˙ 1 does not result in an explicit solution for the optimal control problem, now evaluate: ∂f (x) ∂f (x) + ρ2 ) H¨1 = (λ˙1 ρ1 + λ˙2 ρ2 + λ˙3 γ)(ρ1 ∂x1 ∂x2 d ∂f (x) d ∂f (x) + (λ1 ρ1 + λ2 ρ2 + λ3 γ)(ρ1 + ρ2 ) dt ∂x1 dt ∂x2 (32) Now evaluating: d ∂f (x) ∂ 2 f (x) = x˙1 + dt ∂x1 ∂x21 and d ∂f (x) ∂ 2 f (x) = x˙2 + dt ∂x1 ∂x22

∂ 2 f (x) x˙2 ∂x1 ∂x2

(33)

∂ 2 f (x) x˙1 ∂x1 ∂x2

(34)

Recall from Eq. 8 that; x˙1 =α1 + ρ1 f (x) + ρ1 u x˙2 =α2 + ρ2 f (x) + ρ2 u Now Eq.32 can be represented as: using ζ(x) + β(x) = 0 (35) Where using is the control input at the singular arc. Therefore, rearranging Eq.35 results in: ζ(x) (36) using = − β(x) Solving Eq.36 yields: using = −f (x) (37) Remark: An interesting observation is made when performing an analysis on singular arc. It is found that the control input for a singularity case is independent of the mass of the locomotives. The control input is a function of f (x) where only the length of the bar and the magnetic field act on the f (x). 208

Now applying the Legendre-Clebesch second optimality condition to verify the correctness of the optimal control input on the singular arc, the following equation is used: ∂ d2k H1 ≥0 (38) ∂u dt k is the order of the optimal control input at the singular arc. The order for this system is 1. Therefore the LegendreClebesch condition is satisfied, making the control input on the singular arc valid. A Hamiltonian based optimal control formulation is presented and discussed. For an optimal control formulation where a tandem pair of gravity fed LDCM locomotives is evaluated, the optimal control input at the singularity arc is found using Pontryagin’s maximum principle. The Legendre-Clebesch second optimality condition is further applied to verify the existence of the control input on the singular arc. Remark: While this approach is similar to Tsiotras [2000] yet a maximizing friction is not needed as is done for Tsiotras [2000] approach. In this case, the state function solves the singularity problem. 4. SIMULATION RESULTS The current and velocity relations are observed on fig. 3 and fig. 4, respectively. As it can be seen, the pair of linear DC machines have the same current as expected. This same current is owing to the fact that the linear DC machines are connected in series. The maximum current observed is 2.2A whilst the steady state current appears to be 0A. As it had been mentioned prior, the friction effects are not taken into consideration hence, a steady state current of 0A. An illustration of the velocity of the pair of gravitationally fed linear DC machines is found on fig. 4. Again, the velocities are found to be same for both the linear DC machines. This means that the locomotives have the same velocities. Hence, the same displacement. These illustrations are for a case where there is no external force applied to the locomotives. Results that will follow will show; the effects of changing the slope on which the locomotive move along, effects of applying a force (gravity force) on the locomotives, optimal control results where the performance index, displacements and velocities of the linear DC machines will be shown.

Fig. 3. Current dynamics for a system of linear DC machines

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Fig. 4. Velocity dynamics for a system of mine locomotives Now, consider a case whereby the one locomotive is at the top of the slope whilst the other is at the bottom of the slope. The locomotives are at a steady state. When the locomotive at the top of the slope experiences an external force due to gravity, the DC machine starts operating as a generator . The generator at the top of the slope results in a reverse current in the system of LCDMs. The reverse current is shown in fig. 6. The velocities of the locomotives are shown on fig. 7. When this gravity force is applied to the locomotive at the top of the slope, the locomotives behaves as a generator. Hence, the negative current.When this happens, the locomotive at the bottom behaves as a motor . This conclusion is made based on the fact that a motor would result in a positive but equal current to the generator since a force with the same magnitude is applied to the linear DC machine at the bottom of the slope. On a general case; when a force is applied to the system of locomotives, the locomotives electric circuit either behave as a motor or generator . The aforementioned depend on whether the force is in the direction towards or opposite the motion locomotive. It is of paramount importance to mention that the force is applied at two seconds. But since there is a generator and motor , the negative and positive current due to these different modes of operation cancel each other completely. For this reason, the current will remain at steady-state as can be seen on fig. 5 (exponential graph). Also, to illustrate that a generator behaviour is obtained when a force is applied to the system of locomotives, refer to fig. 6. The explanation provided is for an uncontrolled pair of linear DC machines. When an optimal control algorithm is applied on the system of linear DC machines with the same conditions as the locomotive explained. That would be for a case where gravitational force is applied to the locomotives. The applied force on the locomotives is the equal in magnitude but opposite in direction. LDCM for the locomotive at the top of the slope is behaving as a generator whilst the LDCM at the bottom of the slope is behaving as a motor. After applying the co-state initial conditions that result in optimality by trial and error (transversality condition), the results are represented by the black line on fig. 5. A steady-state current is maintained for a controlled system. The velocity is also observed and it can be seen that for an uncontrolled system of linear DC machines, the velocity starts at 0 until reaching a steady state at two seconds. At exactly two seconds, a force is applied to the system of locomotives. The locomotives accelerate in opposite directions. This is because the one locomotive is sliding down the slope while the other locomotive is going up the slope. The aforementioned is supported by fig. 7 where the locomotives have velocities in opposite directions. For a 209

Fig. 5. Optimal and uncontrolled current for a pair of linear DC machines

Fig. 6. Current dynamics for a system of linear DC machines controlled system of locomotives, the velocities are shown in fig. 8. The velocities are equal but opposite in direction for a controlled system. For an uncontrolled system, there is an offset equal to the steady state velocity. Thus, if the steady state was zero, the velocities would be exactly the same in magnitude. This is because it is deduced that the velocities of the locomotives is the same gradient in magnitude but opposite in direction.

Fig. 7. Velocity dynamics for a system of mine locomotives

Fig. 8. Optimal velocity

2019 IFAC MMM 206 Mohlalakoma T. Ngwako et al. / IFAC PapersOnLine 52-14 (2019) 201–206 Stellenbosch, South Africa, August 28-30, 2019

The displacements of the locomotives are also shown. It is clear from the simulations that the locomotives are moving in the opposite direction hence, the displacement is in the opposite direction. For a case where the optimal controller is applied, the displacements are equal but opposite in direction. More details on the displacement will be shown in the performance index plot where it will be verified that the optimal displacement is achieved. Therefore, this will mean that the an optimal control solution for the optimal control problem is found and illustrated. The performance index is shown on fig. 11. The performance index is also described by Eq.10. It can be seen from this figure that the displacement of the locomotives is minimized. The performance index before applying the optimal control algorithm is also illustrated on the same axis, fig. 11. The illustration mentioned shows that before an optimal control is applied, the performance index increases in a linear fashion until the locomotive has travelled the whole length along the rail. If an infinite path is considered, the performance index will also approach infinity.

Fig. 9. Displacement dynamics for mine locomotives

Fig. 10. Optimal displacement for a pair of mine locomotives

Fig. 11. Performance index for a pair of linear DC machines

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5. CONCLUSION A condition for singular optimal control for a tandem pair of gravity fed linear DC machines is presented. A Chiasson mathematical model of the linear DC machine is used. These linear DC machines are connected in series hence, share the same current. Thereafter, an optimal control is formulated targeting; the singularity analysis and also, aiming for the one of the linear DC machines operating as a generator to drive the DC machine operating as motor. The results for a pair of gravity fed linear DC machines are also shown and it can be deduced from the results that a solution for the optimal control is found. REFERENCES Boscain, U. and Piccoli, B. (2016). An introduction to optimal control. 9966. Semantic Scholar. Chapman, S. (2012). Electric machinery fundamentals. 191–264. Mcgrawhill. Chiasson, J. (2005). Modeling and high performance control of electric machines. 1–400. IEEE Power engineering. Hale, M., Wardi, Y., Jaleel, H., and Egerstedt, M. (2016). Hamiltonian-based algorithm for optimal control. 18. Elsevier. Holl, G. and Fairon, E. (1973). A review of some aspects of shaft design. The Southern African Institute of Mining and Metallurgy. Hu, Y., Huang, P., Meng, Z., Zhang, Y., and Wang, D. (2019). Optimal control of approaching target for tethered space robot based on non-singular terminal sliding mode method. volume 63, 3848–3862. Elsevier, Advances in research. Jiang, Y., Li, Z., Yang, G., Zhang, Y., and Zhang, X. (2017). Recent progress on smart mining in china: Unmanned electric locomotive. volume 9, 1–10. SAGE, Advances in Mechanical Engineering. Kononov, V. (1995). Telecontrol for mining machinery. IFAC Automation in mining, mineral and metal processing. Nkomo, L., Dove, A., Ngwako, M., and Nyandoro, O. (2017). Heaviside based optimal control for ride comfort and actuation energy optimization in half-car suspension systems. volume 50, 259–264. Elsevier. Polnik, B., Budzynski, Z., and Miedzinski, B. (2014). Effective control of a battery supplied mine locomotive unit. volume 20, 39–43. ELEKTRONIKA IR ELEKTROTECHNIKA. Rupprecht, R. (2012). Mine development access to deposit. The Southern African Institute of Mining and Metallurgy Platinum. Tsiotras, P. (2000). On the optimal braking of wheeled vehicles. 569–573. Proceedings of the American Control Conference. Zhang, L. (2019). Singular optimal controls of stochastic recursive systems and hamiltonjacobibellman inequality. volume 266, 6383–6425. Elsevier, Journal of differential equations.