pid control for electric vehicles subject to control and...

Research ArticlePID Control for Electric Vehicles Subject to Control andSpeed Signal Constraints

Amanda Danielle O da S Dantas 1

Andreacute Felipe O de A Dantas 2 Joatildeo Tiago L S Campos 2

Domingos L de Almeida Neto 2 and Carlos Eduardo T Doacuterea 1

1Department of Automation and Computing CT Federal University of Rio Grande do Norte 59078-970 Natal RN Brazil2Master of Engineering in Oil and Gas Potiguar University 59054-180 Natal Brazil

Correspondence should be addressed to Andre Felipe O de A Dantas andredantasunpbr

Received 1 June 2018 Accepted 19 July 2018 Published 1 August 2018

Academic Editor Darong Huang

Copyright copy 2018 Amanda Danielle O da S Dantas et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A PID control for electric vehicles subject to input armature voltage and angular velocity signal constraints is proposed A PIDcontroller for a vehicle DC motor with a separately excited field winding considering the field current constant was tuned usingcontrolled invariant set and multiparametric programming concepts to consider the physical motor constraints as angular velocityand input armature voltage Additionally the integral of the error derivative of the error constraints and 120582 were considered inthe proposed algorithm as tuning parameters to analyze the DC motor dynamic behaviors The results showed that the proposedalgorithm can be used to generate control actions taking into account the armature voltage and angular velocity limits Also resultsdemonstrate that a controller subject to constraints can improve the electric vehicle DC motor dynamic and at the same time itprotects the motor from overvoltage

1 Introduction

Some researchers state that electric vehicles can be one of therenewable solutions to energy and environmental problemscaused by oil based vehicles due to the various advantagesassociated with the use of electric energy such as low cost [1ndash5] In this scenario direct current (DC) motors are one of themost used actuators in the construction of electric vehicles[6] This type of actuator has numerous advantages suchas low cost high reliability easy maintenance and simplecontrol for both speed and position variables with PID beingone of the main used controllers [7 8]

The Proportional Integral Derivative Controller (PID)has been widely used for most industrial process due toits simplicity and effectiveness in control [9 10] This typeof controller is commonly used in level flow temperatureand vehicular systems as well as electric motors [10ndash12] Inaddition the design of the PID controller is considered easy

to implement since it is only necessary to tune three param-eters 119870119901 119870119894 and 119870119889 and tuning methods can be performedautomatically [13] Some ofmost used PID tuningmethods incontrol engineering literature are Ziegler and Nichols Cohenand Conn Relay method and Relatus Apparatus Thesemethods are effective and achieve excellent results when con-trolling unconstrained monovariable systems although someof these ones are also applicable for multivariate systems [9]

Despite all advantages of PID controllers most of tuningmethods do not consider the process constraintsThus manyresearches tried to consider these conditions in the controlloop using antireset windup control signal saturation andintegrator constraints These techniques aim to limit thecontrol action to suit the controller to constrained processes[14 15] However thesemethods still do not take into accountthe constraints while tuning the controller and thereforesuchmethods are not totally appropriate ie they do not leadto an optimal control signal for the constrained system

HindawiJournal of Control Science and EngineeringVolume 2018 Article ID 6259049 11 pageshttpsdoiorg10115520186259049

2 Journal of Control Science and Engineering

In order to solve the optimal constrained problem manycontrollers are being proposed One solution consists ofmaintaining the system trajectory within 120582-contractive con-trolled invariant polyhedron set defined in the state spaceThis set contains all states for which there is a state feedbackcontrol law that maintains the trajectory of the dynamicsystemwithinΩ [16 17]The state feedback control law can becalculated online from the solution of a linear programming(LP) problem or offline by solving a multiparametric linearprogramming problem (mp-LP) [17] This optimal solutionrepresents an explicit PWA (PieceWise Affine) state feedbackcontrol law defined under a set of polyhedral regions instate space [14] In a complementary way to feedback controlrecent research has shown that there is a state space formthat allows the tuning of PID controllers using the LinearQuadratic Regulator (LQR) [13 18] and this may allow us tocombine both strategies making a new tuning method thatconsiders constraints

Within this context in this paper a design of a new type ofgain-scheduling PID controller to control angular velocity ofelectric vehicle DC motors subject to constraints in angularvelocity and input voltage and PID states is proposed To thisend the formulations in the state space of the PID controllerare used as well as the concept of controlled invariant setstogetherwith the solution of amultiparametric programmingproblem [6 9 19 20] In this case we use the same techniquesapplied to obtain explicit controllers (which take into accountsystem constraints) to tune similar PID controllers (mp PID)to constrained systems

This work is organized as follows At first we willapproach the concept of the 120582-contractive controlled invari-ant set In sequence the problem of linear multiparametricprogramming will be described After that we will introducehow to tune PID controllers frommultiparametric linear pro-gramming technique An overview of electrical vehicle DCmotors will be discussed later Finally a set of simulations willbe carried out with the objective of proving the functionalityof the proposed algorithm and the concept of mp PID in thecontrol of electrical vehicle DC motors ie specified to workwith electric cars

2 Controlled Invariant Sets

The concept of controlled invariant sets has become impor-tant in the design of controllers for linear discrete-timesystems subject to constraints since it represents a fundamen-tal condition to maintain system stability ensuring that theconstraints are not violated [21]

Consider the linear time-invariant discrete-time systemdescribed by

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) (1)

where 119896 isin N is the sample time 119909 isin R119899 is the state of thesystem with 119909 isin Ω = 119909 119866119909 le 120588 (where 119866 isin R119892times119899 and120588 isin R119892) and 119906(119896) isin R119898 is the control input subject to theconstraints 119906(119896) isin U = 119906 119881119906 le 120593

A nonempty closed set Ω sub R119899 is controlled invariantwith respect to the system described in (1) if there existsa control signal 119906 such that 119909(119896 + 1) remains inside it for

every 119909(119896) belonging to the closed set Moreover if a givencontraction rate 0 lt 120582 lt 1 is considered a set Ω sub R119899 issaid to be 120582-contractive controlled invariant set with respectto system (1) if there exists a control signal 119906 such that 119909(119896 +1) belongs to the set 120582Ω for every 119909(119896) belonging to theclosed set [16 21] In general the set of constraintsΩ definedin state space is not a controlled invariant set ie there isnot necessarily a control law (119906(119896) isin U) which maintainsthe trajectory of the state vector completely contained inthe set of constraints However it is possible to computea controlled invariant set Ω119888 to be as large as possiblecontained within the set of constraints Ω [22] Thereforebefore starting the controller synthesis process it is necessaryto define a controlled invariant set and then to compute asuitable control law that is able to restrict the state vector to acontrolled invariant set forall119909 isin Ω119888

By defining the maximal contractive controlled invariantset (Ω119888 = 119866119888119909 le 120588119888) [16 22] a state feedback control law(119906(119896) isin U) capable of maintaining the system dynamics(1) contained in Ω119888 can be computed online by solving thelinear programming problem (LP) as described in [16] oroffline from the solution of the following multiparametricprogramming problem (mp-LP) [23]

minV

119888119879V + 119889119879119909subject to 119863V le 119882 + 119864119909 (2)

where

119888119879 = [0 1] 119863 = [119866119888119861 minus120588119888

119881 0 ]

V = [119906120576]

119882 = [0120593]

119864 = [1198661198600 ]

(3)

where 0 lt 120576 le 120582 is the contraction rate to be minimizedat each time step 119906(119896) is the control action to be computedand 119909(119896) is set of states contained inside Ω119888 The expression119866119888(119860119909(119896) + 119861119906(119896)) le 120576120588 represents a convex polyhedron inthe space R119899+119898 and U = 119906 119881119906 le 120593 is a convex polytopethat represents the constraints in the control variable

In the design of controllers under constraints the solutionof the mp-LP (problem (2)) results in a PWA state feedbackcontrol law over the polyhedral regions in the space ofparameters 119909(119896) as follows [24]

(1) The set Ω119888 (controlled invariant polyhedral) is parti-tioned into 119873119902 different polyhedral regions119877119895 = 119909 isin R

119899 | 119875119895119909 le 119887119895 119895 = 1 sdot sdot sdot 119873119902 (4)

Journal of Control Science and Engineering 3

K

s2 + 2dnds + (n

d)2

-

r(t) e(t)

1sX1(t)

X2(t)

s Kd

Kp

K

+

++

+

yu(t) (t)

X3(t)

Figure 1 System controlled by a PID

(2) The optimal solution 119906lowast(119909(119896)) Ω119888 997888rarr R119898 is a PWAfunction over 119877119895119906lowast (119909 (119896)) = 119865119895119909 (119896) + 119892119895 119891119900119903 119909 (119896) isin 119877119895 (5)

As the system is in the state space form it is possible tofind the largest 120582-contractive invariant set and in sequencethe parameters of the control law are computed whichmaintain the dynamics of the states within the 120582-contractiveinvariant set In order to associate with the PID controllerwe will call this ldquothe tuning steprdquo because we find thecontrollerrsquos parameters that guarantee positive invariance and120582-contractivity That is by using this process we will be ableto find a PID control law PWA that allows the controller tosynthesize control actions capable of controlling the processunder constraints

21 Tuning of Gain-Scheduling PID Control Design (mp PID)Based on formulation that allows the reorganization of asecond-order systems in state space form described in [20]whose states are the tracking error integral of the errorand derivative of the error we propose the tuning of a typeof gain-scheduling PID controller by using the PWA statefeedback control law computed by multiparametric linearprogramming described in (2)

211 PID Controllers Consider now the system presented in(1) is described by

119884 (119904)119880 (119904) = 1198701199042 + 2120585119900119897120596119900119897119899 119904 + (120596119900119897119899 )2 (6)

Because the external setpoint does not affect the con-troller design we assume 119903(119905) = 0 for the system of Figure 1[20]

The relation 119910(119905) = minus119890(119905) is valid for the standardregulation problem Thus it is possible to place the system infunction of the error and control signal as presented in [20]

[1199042 + 2120585119900119897120596119900119897119899 119904 + (120596119900119897119899 )2] 119864 (119904) = minus119870119880 (119904) (7)

997904rArr 119890 + 2120585119900119897120596119900119897119899 119890 + (120596119900119897119899 )2 119890 = minus119870119906 (8)

Use the following definitions

1199091 = int1199050119890 (119905) 119889119905

1199092 = 119890 (119905) 1199093 = 119889119890 (119905)119889119905

(9)

Equation (8) can be rewritten as

3 + 2120585119900119897120596119900119897119899 3 + (120596119900119897119899 )2 1199092 = minus119870119906 (10)

Considering the state space the formulation becomes[20]

[[[

123

]]]

= [[[[

0 1 00 0 10 minus (120596119900119897119899 )2 minus2120585119900119897120596119900119897119899

]]]][[[

119909111990921199093

]]]+ [[[

00minus119870

]]]119906 (11)

In this case we assume the reference signal equal tozero and because the system is regulatory it is organizedin a way that the states tend to zero and tend to eliminatethe disturbance When applied in electrical vehicle motorcontrol we intend to make changes in the reference so someconsiderations must be realized

(i) The external reference does not affect the controllerdesign

(ii) It is possible to work step-type references in twoways by using model illustrated in Figure 1 or byforcing changes in the operational point so the newsystemrsquos reference is forced to be at the origin likewiselinearizing a nonlinear system in the operationalpoint

Concerning the direct change of reference it is possibleto verify that as the system is stable in closed loop it tends toconverge to the referenceHowever in this case only the errorand derivative of the error states will converge to zero andthe integral of the error will become a value that maintainsthe necessary control action to force the output to zero in thesame way conventional PID does

The second way to control the system is using thelinearization idea and it can be observed in Figure 2


1 Convert the continuous system into a state space system where the states are the errors and the input is thecontrol action according to equation (11)

2 Compute the maximal 120582minus contractive controlled invariant set contained in the set of constraints which mustbe reexplained in terms of the new state representation

3 Solve the mp-LP problem described in equation (2)4 Compute the integral of error error and derivative of error5 Identify which polyhedral region the computed state 119909(119896) belongs to6 Use the affine control law to control the system with 119865119895 = [119870119894119895 119870119901119895 119870119889119895]119879 and 119892119895 being the mp PID parameters

corresponding to the j-th polyhedral region7 If the control routine is not interrupted return to step 4

Algorithm 1

ＨＯＧ(Ｍ)

＞Ｈ(Ｍ)

StepPIDMMPT

Step1 Gain1k

++ ++

Scope1 Scope2

Transfer Fcn Scopeminus minus


Figure 2 considers the reference equal to zero (step)and to make changes in reference we are changing theoperational point using step 1 The concept is very simple bychanging the reference (step 1) in this system the operationalpoint will change Thus the zero will be shifted ie theentered reference will be recognized as an instantaneouserror disturbance on the systems states The system wasdesigned to force the error integral of the error and thederivative of the error to the origin so the system will workto stabilize states in this ldquonew zerordquo which is the operationalpoint In this way the systems output will be forced towardthe reference using the same considerations as depicted inFigure 2 In this scheme when there is a reference change theinvariant set moves along with it ie if the error limit is equalto 1 for example and the reference is 5 the output will belimited between the values 6 and 4 However if the referenceis changed to 4 the output will be limited to the intervalbetween 3 and 5This is particularly useful when it is desirableto constrain the error around a variable reference Howeverit is needed to be careful not to vary the reference abovethe limits of the invariant set and not to exceed the physicallimitations of the system When the operator wants to limitthe output between 4 and 6 and use the operating points4 and 5 because of this the constraints on the magnitudeof the error would be within plusmn1 in the first example and inthe second one between 2 and 0 (119890119903119903119900119903 = minus119910) This meansthat for these examples it would be necessary to modify thesystems constraints by making a set of constraints for eachoperational point

Given the main considerations about the state spacesystem it is necessary to tune and use the controller In thiscase the concept of invariant sets will be used to find anoptimum tuning for a PWA PID control law which we callmp PID Thus the 120582-contractive controlled invariant set iscomputed by using the algorithms proposed by [22] thenmp-LP problem (2) is solved to compute the parameters119865119895 and 119892119895 of the affine law 119906(119909) = 119865119895119909 + 119892119895 where 119865119894multiplies the state119883 = [1199091 1199092 1199093] corresponding to integralof the error the error and the derivative of the error Thismeans that the parameters of the proposed PID controller aredefined by 119865119895 = [119870119894119895 119870119901119895 119870119889119895] plus affine term 119892119895 whichis associated with a polyhedral region 119877119895 forming a gain-scheduling PID control (mp PID) Such approach can beobtained from the implementation of Algorithm 1

As described in Algorithm 1 the proposed PID controldesign is performed from a sequence of steps where at firstthe system must be rearranged in state space form so thatthe input is the control action and the states are the integralof the error error and derivative of the error according to(11) Then the new system is discretized with the desiredsampling period and then given the new state space extendedsystem the maximal 120582minuscontractive controlled invariant set iscomputed as well as themultiparametric linear programmingproblem (mp-LP) (2) The solution of the mp-LP problemrepresents a PWA control law over polyhedral regions 119877119895associated with a PID controller At the end of these steps thecontrol law is inserted into the control loop where the errortends to zero and the constraints are not violated as long as the


vi

R L

w TＧ

Ｃ

J

B

Figure 3 DC motor with separated winding field with constant current (119894119891(119905)) [25]

initial state is inside the 120582minuscontractive controlled invariantset

3 DC Motors Overview

After the definition of the PID tuningmethod the applicationmust be studied in order to have mp PID applied to it soin this section we present an overview of electric vehicle DCmotors

Recently electric vehicles are gaining popularity amongthe population resulting in more demand for these types ofcar Electric vehicles are efficient and need less maintenancethan fuel-based cars and they do not pollute the environment[15] However the electric car depends heavily on batterysystems that are finite and have fewer storage capabilitiesTherefore strategies to reduce and use more efficiently theenergy stored in the batteries are needed

The consuming of the electric vehicles depends heavily onthe used motor type and adopted control strategy ActuallyDC motors and induction motors are being proposed tobe used in the electric vehicle industry with DC motortype being a good candidate to be used in electric vehicleapplications [26 27] DCmotors are efficient presenting highreliability and easy maintenance They are easy to controlresulting in smooth acceleration and efficient battery usage[7 8] Although brushed DC motor suffers from the brushmaintenance the aforementioned advantages turn the DCmotor to be applicable for use in electric cars

DC motors have the advantage of being easy to controlresulting in several control techniques for velocity controlusing PID type control Techniques using metaheuristic PID[28 29] multivariate PID [30] genetic algorithm methodwith PID [12] and Adaptive PID Dynamic Fuzzy andNeurofuzzy Controller [19] were proposed to control DCmotors

DC motors have different configurations as compoundshunt series permanent magnetic DC and separated fieldwinding where each of them has advantages and disad-vantages DC motors in series configuration are the typeof motors that can be used in electric cars due to theirinstantaneous torque and smooth acceleration Howeverthese types ofmotors need aminimal load not to be damagedwhich is a condition that occurs in electric car applicationsOneway to limit the damage in theDCmotor series control is

to limit themaximumvelocity in the adopted control strategy[31] DC motors with separated field are the most versatilebecause they allow controlling torque and velocity separatelyusing the armature current and field current separately withmore options for control applications In this paper DCmotor with separated field was chosen to test the controlalgorithm because it allows more restrictions options to betested by the proposed control algorithm

In sequence we present the DC motor modelling withexcited separately field and afterwards we present the numer-ical examples controlling the motor and testing the perfor-mance of the tuned controllers

A DC motor controlled by current armature with inde-pendent field is depicted in Figure 3

The torque induced (119879119898) in the motor is given by

119879119898 (119905) = 119869119889119908 (119905)119889119905 + 119861119908 (119905) (12)

119879119898 (119905) = 119870119886119894 (119905) (13)

where 119894(119905) is the armature current 119869 is the inertia moment119908is the angular velocity 119861 is the friction coefficient and 119870119886 isthe torque constant

The armature induced voltage (119890119886(119905)) with constant fieldcurrent (119894119891) is given by

119890119886 (119905) = 119870119887119908 (119905) (14)

where 119870119887 is the velocity constantAs the DC motor has separated field the voltage applied

(V(119905)) in the armature windings is given by

V (119905) = 119871119889119894 (119905)119889119905 + 119877119894 (119905) + 119890119886 (119905) (15)

where 119871 is the armature inductance and 119877 is the armatureresistance

The following second-order transfer function resulted

119882(119904)119881 (119904) = 119870119886(119869119904 + 119861) (119871119904 + 119877) + 119870119886119870119887 (16)

4 Numerical Examples

(i) The parameters of a motor suitable for use in electriccars are determined


Table 1 Motor parameters

119881119899 48 VR 01 ΩL 0005 H119870119887 0004 119881119904119903119886119889119870119886 00036 119873119898119860119869 01 1198731198982119861 005 119873119898119904119903119886119889

(ii) The mp PID controllers are tuned(iii) The proposed tuning algorithm is performed using

different parameters and then the results are com-pared

(iv) The influence of each parameter on the final processperformance is analyzed

41 DC Motor Parameters At first in order to control anelectric vehicle DC motor we specify its parameters (themotor) in such a way that it is able to move the vehicleThe voltage armature resistance and inductance armaturesmoment of inertia velocity constant torque constant andfriction coefficient motor parameters presented in Table 1were used to test the proposed control algorithm perfor-mance

Replacing the motor parameters in (17)

119882(119904)119881 (119904) = 00036(119904 + 005) (0005119904 + 01) + 144119890minus5 (17)

then the simplified transfer function of the used motorbecomes

119866 (119904) = 721199042 + 205119904 + 1003 (18)

In the model of (18) the output is the angular velocity(119903119886119889119904) in the range of plusmn3456119903119886119889119904 and the input is thevoltage that will vary in the range of plusmn48119881 Note that as thistype of motor allows two-direction movement we will haveboth positive and negative voltages and positive and negativespeeds considered in the controller constraints

42 Tuning of the Controllers mp PID This section presentsthe general tuning process of the mp PID controllers Someimportant aspects to emphasize are described as follows

(i) The output constraints can be transformed into con-straints on the error especially when assuming theoperating point at zeroThe output constraints for theelectric vehicle DC motor case will be the maximumspeed allowed at a voltage of 48V

(ii) The control signal constraints must be limited toplusmn48119881 since we need to limit motor damage fromovercurrent as the armature current is directly depen-dent on the applied load and armature voltage so thealgorithm is able to optimize the system within thisset of constraints unlike other techniques which only

insert constraint on control action without taking inaccount the constraints at tuning phase resulting inlower dynamic performance of the vehicle DCmotor

(iii) The integral and derivative of the error values will beappropriately chosen ie theywill be used as a tuningparameter and the influence of these constraints willbe evaluated in order to emphasize how the finalperformance of the system is changed in aspects suchas overshooting and stabilization time

(iv) The main tuning parameter of the mp PID controlleris the value of 120582 It is related to the contractivity andwill also be an observed parameter

(v) In a specific case we observe the difference betweenvarying the operating point and varying the reference

43 Set of Tests Based on these considerations tests areperformed varying the constraints on the integral of the errorderivative of the error and the 120582 parameter After that acomparison between changing the reference and changingthe operating point will be made

431 Test 1 Integral of the Error Constraints Change EffectIn this first test the objective is to observe the influencein systemrsquos performance of integral of the error constraintschange Despite the direct relationship between error andthe output of the motor (angular velocity) the integral ofthe error is an internal state of the system in state spaceand influences the motor dynamic performance but notnecessarily forces physically the system outside its limitsTherefore by hypothesis we assume that we can freelymodifythese parameters In this way the test conditions presentedin Table 2 were chosen In these test conditions the integralof the error limits was varied and the derivative of the errorlimits and 120582 were maintained constant

The systemrsquos outputs and the control actions for testcondition 1 are depicted in Figures 5 and 6 Also Figure 4presents the polyhedron formed by the first test case (test 1)

As seen Figure 4 presents the polyhedron formed by thefirst test case (test 1) where the default value is 3456 so119872119886119909119868119890 = 1 means 1 times 3456 and 119872119886119909119863119890 = 5 means5 times 3456 and this concept is applicable to all the othersfigures Because this polyhedron changes its shape for eachtest in all tests we present only the quantity of sets generatedwithin the invariant set that in this case are 10 8 and 10The number of regions within the invariant set reveals thecomputational cost difference (in Algorithm 1 we need tosearch the region to find which PID controller parameterswill be used to compute the control signal) In this case asobserved the quantity is very similar for a reference theremay be cases in which the amount of computed regions maybe greater than 400 which in this case points out that thecontrol signal computation will be inexpensive

Figure 5 shows the output ie the angular velocity in the3 performed tests We observe that with lower constraintvalues in the integral of the error signal the system tendsto converge faster on the other hand with more relaxedrestriction in the integral of the error signal the system tendsto stabilize slower Another important factor to be analyzed is


Table 2 Table with the 3 tests varying integral of error

Simulations 120582 Integral of the Error Limits Derivative of the Error Limits1 0999 plusmn3456 plusmn172802 0999 plusmn17280 plusmn172803 0999 plusmn1728 plusmn17280

150

100

50

0

minus50

minus100

minus150

3(D

eriv

ativ

e of t

he E

rror

)

20200

0minus20 minus20

2 (Error)1 (Integral of the Error)

0999 MaxIe 1 MaxDe 5

Figure 4 Polyhedron test 1 for test condition 1

0999 MaxIe 1 MaxDe 5 0999 MaxIe 5 MaxDe 5 0999 MaxIe 50 MaxDe 5

2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

Velo

city

(rad

s)

Figure 5 Systemrsquos outputs for test condition 1

that the lower the restriction value in the integral of the erroris the greater the overshooting system tends to present

Figure 6 corresponds to the behavior of the control signalHere the similarity in behavior with respect to the outputsignal is evident ie the smaller the restriction value in theintegral of the error is the closer to the limits the signal willbe which in this case is plusmn48


minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)

2 4 6 8 10 12 14 16 18 200Time (s)

Figure 6 Control actions for test condition 1

432 Test 2 Derivative of the Error Constraints ParametersChange Effect Similarly to the first test in this second testour objective is to observe the influence in the performanceof the system when the derivative of the error constraintsis changed This also happened to the integral of the errorconstraint the derivative of the error is an internal state of thesystem in state space and influences the performance but not


Table 3 Table with the 3 tests varying derivative of error


Table 4 Table with the 3 tests varying 120582Simulations 120582 Integral of Error Limits Derivative of Error Limits1 099 plusmn17280 plusmn172802 090 plusmn17280 plusmn172803 060 plusmn17280 plusmn17280


2 4 6 8 10 12 14 16 18 200Time (s)

minus10

minus5

0

5

10

15

20

25

30

35

Velo

city

(rad

s)


necessarily physically forces the system outside its limits (weare not considering variation effects in this case although insome cases they are included in problems with restrictions)Therefore by hypothesis we assume that we can freelymodifythese parameters In this way the test conditions presented inTable 3 were chosen In these test conditions the derivative ofthe error limits was varied and the integral of the error limitsand 120582 were maintained constant

The systemrsquos outputs and the control signals for testcondition 2 are depicted in Figures 7 and 8

Figure 5 shows the output ie the angular velocity inthe 3 performed tests We observe that as the derivativeof the error limits increases the system begins to presentnonminimum phase and it is more difficult to stabilize it insmaller times this means that in the case in which the limitsare relaxed the integral of the error values can be larger andconsequently the system tends to have a better transient

In these tests the quantity of sets generated within thecontrolled invariant set is 8 8 and 12 Similarly to theprevious test we emphasize that it will be very inexpensiveto compute the control action


2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)


The figure of this second test (Figure 6) corresponds tothe behavior of the control signal Here the similarity inbehavior with respect to the output action is evident ie thesmaller the constraint value in the derivative of the error isthe closer the control signal will be to its limit A secondimportant factor to be highlighted here is that in the casewhere the system tends to exceed the control action it isevident that there is a saturation and because the controllerwas obtained from a constrained optimization problem itsperformance will be optimal

433 Test 3 Effect of Changing 120582 Unlike the first 2 tests thisone will not present the effect in constraints but in the tuningparameter (120582) of the optimization problem What should behighlighted here is that this parameter is associated with thecontraction rate which makes the invariant set smaller ateach iteration and it is related to the systemrsquos convergencespeed In this way the test conditions presented in Table 4were chosen In these test conditions 120582 was varied and theintegral and derivative of the error limits were maintainedconstant



0 4 6 8 10 12 14 16 18 202Time (s)

0

5

10

15

20

25Ve

loci

ty (r

ads

)

Figure 9 Systemrsquos output for test condition 3


2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)


The systemrsquos outputs and the control actions for the testcondition 3 are depicted in Figures 9 and 10

This third test resulted as the quantities of sets within theinvariant sets 8 14 and 12 Also Figure 9 shows the outputie the angular velocity in the 3 performed tests We observethat the lower the 120582 value is the more aggressive the systemsbehavior is forcing the stabilization to happen before butreflecting in greater overshooting This also means that byincreasing the value of this parameter we have less aggressivesystems but with slower responses

The last figure of this third test (Figure 10) corresponds tothe behavior of the control signal Here the similarity to theoutput signal is evident ie the smaller the value of 120582 is thecloser to its limits it is

099 MaxIe 5 MaxDe 5Regulatory control - 099 MaxIe 5 MaxDe 5

2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

Velo

city

(rad

s)

Figure 11 Systemrsquos output


minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50Vo

ltage

(V)

2 4 6 8 10 12 14 16 18 200Time (s)

Figure 12 Control action

434 Test 4 Reference Changes versus Setpoint Changes Thefirst three tests aimed to show the results related to thecontrollers tuning when the systems are subject to referencevariations (step) This means that the controlled invariantset remained unchanged as well as the operating point ofthe system In this fourth test the objective is to present thechanges related to the operating point Note that there is apossibility of differences in behaviors because although it hasthe same amount of variation this test directly affects theactivated regions in the control process which consequentlychanges the response of the system Thus the test conditionis to change the setpoint by 20 units and compare it with thechanged operating point (setpoint 1) by 20


Figure 11 shows the output ie the angular velocity inthe 3 performed tests We observe that the systemrsquos response


which had its setpoint changedwas less aggressive (observingthe overshooting) Despite the presented output it is notenough to emphasize that in any and every change ofoperating point the response will behave accordingly sincethis difference is related to the fact that different regions of theinvariant set are activated in both cases Another importantfact to note is that depending on the stabilization criteria weobserve the operating point change case stabilizing before theresponse to the setpoint change case this is mainly due to afaster reaction of this response

Figure 12 of the fourth test corresponds to the behaviorof the control signal Here the similarity in behavior withrespect to the output signal is evident We observe that theinitial value of control action in operating point change case isgreater than that of the reference change case this impacts onthe initial response and the possibility of faster stabilizationNote that additionally to the performed tests other possibili-tieswere investigated ie the change of the objective functionto minimize the integral of the error and the derivative of theerror (as alternatives to changing constraints) However theresults were not satisfactory impacting directly on the sizeof the set of constraints and affecting the performance of theprocess

5 Conclusion

This paper presented a PID controller for electric vehicle DCmotors based on proving the functionality of the proposedalgorithm from a set of simulations The algorithm provedthat it is possible to generate control actions that considerthe 48V limits of the motor as well as forcing the outputspeed of the motors to be limited to their specified valuesIn addition it was verified that the integral and derivativeof the error constraints make tuning parameters ldquoconstrainrdquothe performance of the process output In other words wefound a direct relationship between error constraints and theDC motor dynamics Another important point to note is thatchanges in 120582 also interfere with systemrsquos performance and theoutput behavior also depends on how the error is changedIn this second case we verify the behavior of the system toreference and setpoint variations

In conclusion control of electric vehicle DC motorsunder constraints and tuning controllers adjusting the behav-ior of the systems output is possible by using the proposedalgorithm In this sense important improvements can beproposed The first one would be to use a methodology toautomatically tune parameters since there are no tuning rulesfor these constraints (in the integral of the error and in thederivative of the error) A second improvement would be toextend the tests by verifying that the behavior presented to theDC motors ie verified can be extended to other processes

6 Future Works

Although DC motor with excited separately field can be usedin an electric vehicle new types of motors as permanentmagnet synchronous motor (PMSM) are being used recently[11 32 33] The usage of the proposed algorithm in this typeof motors will be investigated

Data Availability

The [algorithms] data used to support the findings ofthis study have not been made available because scientificresearch is still being carried out using the presented algo-rithms

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to mention the institutions thatcontributed to the development of this project UFRN (Fed-eral University of Rio Grande do Norte) DCA (Depart-ment of Computation and Automation) CNPQ (NationalCouncil for Scientific and Technological Development) UnP(Potiguar University) and LAUT (Laboratory of Automationin Petroleum)

References

[1] S Chu and A Majumdar ldquoOpportunities and challenges for asustainable energy futurerdquo Nature vol 488 no 7411 pp 294ndash303 2012

[2] X Zhu H Zhang D Cao and Z Fang ldquoRobust control of inte-grated motor-transmission powertrain system over controllerarea network for automotive applicationsrdquo Mechanical Systemsand Signal Processing vol 58 pp 15ndash28 2014

[3] Z Shuai H Zhang J Wang J Li and M Ouyang ldquoCombinedAFS andDYC control of four-wheel-independent-drive electricvehicles over CAN Network with time-varying delaysrdquo IEEETransactions onVehicular Technology vol 63 no 2 pp 591ndash6022014

[4] X Wang W Kong D Zhang and L Shen ldquoActive disturbancerejection controller for small fixed-wing UAVs with modeluncertaintyrdquo in Proceedings of the 2015 IEEE International Con-ference on Information and Automation ICIA 2015 - In conjunc-tionwith 2015 IEEE International Conference onAutomation andLogistics pp 2299ndash2304 China August 2015

[5] Y Luo T Chen and K Li ldquoMulti-objective decoupling algo-rithm for active distance control of intelligent hybrid electricvehiclerdquo Mechanical Systems and Signal Processing vol 64-65pp 29ndash45 2015

[6] Z Has A H Muslim and N A Mardiyah ldquoAdaptive-fuzzy-PID controller based disturbance observer for DC motor speedcontrolrdquo in Proceedings of the 2017 4th International Conferenceon Electrical Engineering Computer Science and Informatics(EECSI) pp 1ndash6 Yogyakarta September 2017

[7] N Matsui ldquoSensorless PM brushless DC motor drivesrdquo IEEETransactions on Industrial Electronics vol 43 no 2 pp 300ndash308 1996

[8] J S Ko J H Lee S K Chung and M J Youn ldquoA robust digitalposition control of brushless DC motor with dead beat loadtorque observerrdquo IEEE Transactions on Industrial Electronicsvol 40 no 5 pp 512ndash520 1993

[9] H Jeongheon and R Skelton ldquoAn LMI optimization approachfor structured linear controllersrdquo in Proceedings of the 42ndIEEE International Conference on Decision and Control pp5143ndash5148 Maui HI USA 2003


[10] A Kalangadan N Priya and T K S Kumar ldquoPI PID controllerdesign for interval systems using frequency response modelmatching techniquerdquo in Proceedings of the International Confer-ence on Control Communication and Computing India ICCC2015 pp 119ndash124 India November 2015

[11] S Hu Z Liang W Zhang and X He ldquoResearch on theintegration of hybrid energy storage system and dual three-phase PMSM Drive in EVrdquo IEEE Transactions on IndustrialElectronics vol 65 no 8 pp 6602ndash6611 2018

[12] M Jaiswal M Phadnis and H O D Ex ldquoSpeed controlof DC motor using genetic algorithm based PID controllerrdquoInternational Journal of Advanced Research in Computer Scienceand Software Engineering vol 3 no 7 pp 2277-128 2013

[13] E Vinodh Kumar and J Jerome ldquoLQR based optimal tuningof PID controller for trajectory tracking of magnetic levitationsystemrdquo in Proceedings of the 2013 International Conference onDesign and Manufacturing IConDM 2013 pp 254ndash264 IndiaJuly 2013

[14] M Kvasnica P Grieder M Baotic and M Morari ldquoMulti-parametric toolbox (MPT)rdquo Lecture Notes in Computer Science(including subseries Lecture Notes in Artificial Intelligence andLecture Notes in Bioinformatics) Preface vol 2993 pp 448ndash4622004

[15] S-M Lu ldquoA review of high-efficiencymotors specification pol-icy and technologyrdquo Renewable amp Sustainable Energy Reviewsvol 59 pp 1ndash12 2016

[16] F Blanchini ldquoUltimate boundedness control for uncertaindiscrete-time systems via set-induced Lyapunov functionsrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 39 no 2 pp 428ndash433 1994

[17] A D O d S Dantas C E T Dorea and A F O dA Dantas ldquoProjeto de controladores para sistemas linearessujeitos a restricoes utilizando analise de agrupamento dedadosrdquo Simposio Brasileiro de Automacao Inteligente - SBAI2015

[18] S Das I Pan K Halder S Das and A Gupta ldquoLQR basedimproved discrete PID controller design via optimum selectionof weighting matrices using fractional order integral perfor-mance indexrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol37 no 6 pp 4253ndash4268 2013

[19] A K Hassan M S Saraya M S Elksasy and F F AreedldquoBrushless DC motor speed control using PID controller fuzzycontroller and neuro fuzzy controllerrdquo International Journal ofComputer Applications vol 180 no 30 pp 47ndash52 2018

[20] J-B He Q-G Wang and T-H Lee ldquoPIPID controller tuningvia LQR approachrdquoChemical Engineering Science vol 55 no 13pp 2429ndash2439 2000

[21] F Blanchini and S Miani Set-theoretic methods in controlSystems amp Control Foundations amp Applications BirkhauserBoston Inc Boston MA USA 2008

[22] C E Dorea and J C Hennet ldquo(a b)-invariant polyhedral setsof linear discrete-time systemsrdquo Journal of OptimizationTheoryand Applications vol 35 pp 521ndash542 1999

[23] G P Weinkeller J L Salles and T F B Filho ldquoControle pred-itivo via programacao multiparametrica aplicado no modelocinematico de uma cadeira de rodas roboticardquo in Proceedingsof the DINCON 2013mdashConferencia Brasileira de DinamicaControle e Aplicacoes 2008

[24] A Bemporad F Borrelli and M Morari ldquoModel predictivecontrol based on linear programmingmdashthe explicit solutionrdquo

IEEE Transactions on Automatic Control vol 47 no 12 pp1974ndash1985 2002

[25] A A R Coelho and L S Coelho ldquoIdentificacao de sistemasdinamicos linearesrdquo Didactica (Editora da UFSC) 2004

[26] S Lixin ldquoElectric vehicle development the past present ampfuturerdquo in Proceedings of the 2009 3rd International Conferenceon Power Electronics Systems and Applications PESA 2009China May 2009

[27] C Chan ldquoThe past present and future of electric vehicledevelopmentrdquo in Proceedings of the IEEE 1999 InternationalConference on Power Electronics and Drive Systems PEDSrsquo99(Cat No99TH8475) pp 11ndash13 vol1 Hong Kong July 1999

[28] D Puangdownreong ldquoOptimal PID controller design for DCmotor speed control system with tracking and regulating con-strained optimization via cuckoo searchrdquo Journal of ElectricalEngineering amp Technology vol 13 no 1 pp 460ndash467 2018

[29] P Payakkawan K Klomkarn and P Sooraksa ldquoDual-line PIDcontroller based on PSO for speed control of DC motorsrdquoin Proceedings of the 2009 9th International Symposium onCommunications and Information Technology ISCIT 2009 pp134ndash139 Republic of Korea September 2009

[30] B Gasbaoui and A Nasri ldquoA novel multi-drive electric vehiclesystem control based on multi-input multi-output PID con-trollerrdquo Serbian Journal of Electrical Engineering vol 9 no 2pp 279ndash291 2012

[31] Z Bitar S Al Jabi and I Khamis ldquoModeling and simulationof series DC motors in electric carrdquo in Proceedings of theInternational Conference on Technologies and Materials forRenewable Energy Environment and Sustainability TMREES2014 - EUMISD pp 460ndash470 Lebanon April 2014

[32] R Xiao Z Wang H Zhang J Shen and Z Chen ldquoA NovelAdaptive Control of PMSM for Electric Vehiclerdquo in Proceedingsof the 2017 IEEE Vehicle Power and Propulsion Conference(VPPC) pp 1ndash8 Belfort France December 2017

[33] C Shi Y Tang and A Khaligh ldquoA three-phase integratedonboard charger for plug-in electric vehiclesrdquo IEEE Transac-tions on Power Electronics vol 33 no 6 pp 4716ndash4725 2018

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018


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VLSI Design



Shock and Vibration


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Journal of

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Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

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Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

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wwwhindawicom Volume 2018

Advances in

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Submit your manuscripts atwwwhindawicom


In order to solve the optimal constrained problem manycontrollers are being proposed One solution consists ofmaintaining the system trajectory within 120582-contractive con-trolled invariant polyhedron set defined in the state spaceThis set contains all states for which there is a state feedbackcontrol law that maintains the trajectory of the dynamicsystemwithinΩ [16 17]The state feedback control law can becalculated online from the solution of a linear programming(LP) problem or offline by solving a multiparametric linearprogramming problem (mp-LP) [17] This optimal solutionrepresents an explicit PWA (PieceWise Affine) state feedbackcontrol law defined under a set of polyhedral regions instate space [14] In a complementary way to feedback controlrecent research has shown that there is a state space formthat allows the tuning of PID controllers using the LinearQuadratic Regulator (LQR) [13 18] and this may allow us tocombine both strategies making a new tuning method thatconsiders constraints

Within this context in this paper a design of a new type ofgain-scheduling PID controller to control angular velocity ofelectric vehicle DC motors subject to constraints in angularvelocity and input voltage and PID states is proposed To thisend the formulations in the state space of the PID controllerare used as well as the concept of controlled invariant setstogetherwith the solution of amultiparametric programmingproblem [6 9 19 20] In this case we use the same techniquesapplied to obtain explicit controllers (which take into accountsystem constraints) to tune similar PID controllers (mp PID)to constrained systems

This work is organized as follows At first we willapproach the concept of the 120582-contractive controlled invari-ant set In sequence the problem of linear multiparametricprogramming will be described After that we will introducehow to tune PID controllers frommultiparametric linear pro-gramming technique An overview of electrical vehicle DCmotors will be discussed later Finally a set of simulations willbe carried out with the objective of proving the functionalityof the proposed algorithm and the concept of mp PID in thecontrol of electrical vehicle DC motors ie specified to workwith electric cars

2 Controlled Invariant Sets

The concept of controlled invariant sets has become impor-tant in the design of controllers for linear discrete-timesystems subject to constraints since it represents a fundamen-tal condition to maintain system stability ensuring that theconstraints are not violated [21]

Consider the linear time-invariant discrete-time systemdescribed by

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) (1)

where 119896 isin N is the sample time 119909 isin R119899 is the state of thesystem with 119909 isin Ω = 119909 119866119909 le 120588 (where 119866 isin R119892times119899 and120588 isin R119892) and 119906(119896) isin R119898 is the control input subject to theconstraints 119906(119896) isin U = 119906 119881119906 le 120593

A nonempty closed set Ω sub R119899 is controlled invariantwith respect to the system described in (1) if there existsa control signal 119906 such that 119909(119896 + 1) remains inside it for

every 119909(119896) belonging to the closed set Moreover if a givencontraction rate 0 lt 120582 lt 1 is considered a set Ω sub R119899 issaid to be 120582-contractive controlled invariant set with respectto system (1) if there exists a control signal 119906 such that 119909(119896 +1) belongs to the set 120582Ω for every 119909(119896) belonging to theclosed set [16 21] In general the set of constraintsΩ definedin state space is not a controlled invariant set ie there isnot necessarily a control law (119906(119896) isin U) which maintainsthe trajectory of the state vector completely contained inthe set of constraints However it is possible to computea controlled invariant set Ω119888 to be as large as possiblecontained within the set of constraints Ω [22] Thereforebefore starting the controller synthesis process it is necessaryto define a controlled invariant set and then to compute asuitable control law that is able to restrict the state vector to acontrolled invariant set forall119909 isin Ω119888

By defining the maximal contractive controlled invariantset (Ω119888 = 119866119888119909 le 120588119888) [16 22] a state feedback control law(119906(119896) isin U) capable of maintaining the system dynamics(1) contained in Ω119888 can be computed online by solving thelinear programming problem (LP) as described in [16] oroffline from the solution of the following multiparametricprogramming problem (mp-LP) [23]

minV

119888119879V + 119889119879119909subject to 119863V le 119882 + 119864119909 (2)

where

119888119879 = [0 1] 119863 = [119866119888119861 minus120588119888

119881 0 ]

V = [119906120576]

119882 = [0120593]

119864 = [1198661198600 ]

(3)

where 0 lt 120576 le 120582 is the contraction rate to be minimizedat each time step 119906(119896) is the control action to be computedand 119909(119896) is set of states contained inside Ω119888 The expression119866119888(119860119909(119896) + 119861119906(119896)) le 120576120588 represents a convex polyhedron inthe space R119899+119898 and U = 119906 119881119906 le 120593 is a convex polytopethat represents the constraints in the control variable

In the design of controllers under constraints the solutionof the mp-LP (problem (2)) results in a PWA state feedbackcontrol law over the polyhedral regions in the space ofparameters 119909(119896) as follows [24]

(1) The set Ω119888 (controlled invariant polyhedral) is parti-tioned into 119873119902 different polyhedral regions119877119895 = 119909 isin R

119899 | 119875119895119909 le 119887119895 119895 = 1 sdot sdot sdot 119873119902 (4)


K

s2 + 2dnds + (n

d)2

-

r(t) e(t)

1sX1(t)

X2(t)

s Kd

Kp

K

+

++

+

yu(t) (t)

X3(t)






119884 (119904)119880 (119904) = 1198701199042 + 2120585119900119897120596119900119897119899 119904 + (120596119900119897119899 )2 (6)



[1199042 + 2120585119900119897120596119900119897119899 119904 + (120596119900119897119899 )2] 119864 (119904) = minus119870119880 (119904) (7)

997904rArr 119890 + 2120585119900119897120596119900119897119899 119890 + (120596119900119897119899 )2 119890 = minus119870119906 (8)


1199091 = int1199050119890 (119905) 119889119905

1199092 = 119890 (119905) 1199093 = 119889119890 (119905)119889119905

(9)


3 + 2120585119900119897120596119900119897119899 3 + (120596119900119897119899 )2 1199092 = minus119870119906 (10)


[[[

123

]]]

= [[[[

0 1 00 0 10 minus (120596119900119897119899 )2 minus2120585119900119897120596119900119897119899

]]]][[[

119909111990921199093

]]]+ [[[

00minus119870

]]]119906 (11)











Algorithm 1

ＨＯＧ(Ｍ)

＞Ｈ(Ｍ)

StepPIDMMPT

Step1 Gain1k

++ ++

Scope1 Scope2







vi

R L

w TＧ

Ｃ

J

B













119879119898 (119905) = 119869119889119908 (119905)119889119905 + 119861119908 (119905) (12)

119879119898 (119905) = 119870119886119894 (119905) (13)



119890119886 (119905) = 119870119887119908 (119905) (14)



V (119905) = 119871119889119894 (119905)119889119905 + 119877119894 (119905) + 119890119886 (119905) (15)



119882(119904)119881 (119904) = 119870119886(119869119904 + 119861) (119871119904 + 119877) + 119870119886119870119887 (16)





119881119899 48 VR 01 ΩL 0005 H119870119887 0004 119881119904119903119886119889119870119886 00036 119873119898119860119869 01 1198731198982119861 005 119873119898119904119903119886119889






119882(119904)119881 (119904) = 00036(119904 + 005) (0005119904 + 01) + 144119890minus5 (17)


119866 (119904) = 721199042 + 205119904 + 1003 (18)

















150

100

50

0

minus50

minus100

minus150

3(D

eriv

ativ

e of t

he E

rror

)

20200

0minus20 minus20





2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

Velo

city

(rad

s)





minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)

2 4 6 8 10 12 14 16 18 200Time (s)








2 4 6 8 10 12 14 16 18 200Time (s)

minus10

minus5

0

5

10

15

20

25

30

35

Velo

city

(rad

s)







2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)






0 4 6 8 10 12 14 16 18 202Time (s)

0

5

10

15

20

25Ve

loci

ty (r

ads

)



2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)






2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

Velo

city

(rad

s)



minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50Vo

ltage

(V)

2 4 6 8 10 12 14 16 18 200Time (s)








5 Conclusion



6 Future Works


Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



K

s2 + 2dnds + (n

d)2

-

r(t) e(t)

1sX1(t)

X2(t)

s Kd

Kp

K

+

++

+

yu(t) (t)

X3(t)






119884 (119904)119880 (119904) = 1198701199042 + 2120585119900119897120596119900119897119899 119904 + (120596119900119897119899 )2 (6)



[1199042 + 2120585119900119897120596119900119897119899 119904 + (120596119900119897119899 )2] 119864 (119904) = minus119870119880 (119904) (7)

997904rArr 119890 + 2120585119900119897120596119900119897119899 119890 + (120596119900119897119899 )2 119890 = minus119870119906 (8)


1199091 = int1199050119890 (119905) 119889119905

1199092 = 119890 (119905) 1199093 = 119889119890 (119905)119889119905

(9)


3 + 2120585119900119897120596119900119897119899 3 + (120596119900119897119899 )2 1199092 = minus119870119906 (10)


[[[

123

]]]

= [[[[

0 1 00 0 10 minus (120596119900119897119899 )2 minus2120585119900119897120596119900119897119899

]]]][[[

119909111990921199093

]]]+ [[[

00minus119870

]]]119906 (11)











Algorithm 1

ＨＯＧ(Ｍ)

＞Ｈ(Ｍ)

StepPIDMMPT

Step1 Gain1k

++ ++

Scope1 Scope2







vi

R L

w TＧ

Ｃ

J

B













119879119898 (119905) = 119869119889119908 (119905)119889119905 + 119861119908 (119905) (12)

119879119898 (119905) = 119870119886119894 (119905) (13)



119890119886 (119905) = 119870119887119908 (119905) (14)



V (119905) = 119871119889119894 (119905)119889119905 + 119877119894 (119905) + 119890119886 (119905) (15)



119882(119904)119881 (119904) = 119870119886(119869119904 + 119861) (119871119904 + 119877) + 119870119886119870119887 (16)





119881119899 48 VR 01 ΩL 0005 H119870119887 0004 119881119904119903119886119889119870119886 00036 119873119898119860119869 01 1198731198982119861 005 119873119898119904119903119886119889






119882(119904)119881 (119904) = 00036(119904 + 005) (0005119904 + 01) + 144119890minus5 (17)


119866 (119904) = 721199042 + 205119904 + 1003 (18)

















150

100

50

0

minus50

minus100

minus150

3(D

eriv

ativ

e of t

he E

rror

)

20200

0minus20 minus20





2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

Velo

city

(rad

s)





minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)

2 4 6 8 10 12 14 16 18 200Time (s)








2 4 6 8 10 12 14 16 18 200Time (s)

minus10

minus5

0

5

10

15

20

25

30

35

Velo

city

(rad

s)







2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)






0 4 6 8 10 12 14 16 18 202Time (s)

0

5

10

15

20

25Ve

loci

ty (r

ads

)



2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)






2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

Velo

city

(rad

s)



minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50Vo

ltage

(V)

2 4 6 8 10 12 14 16 18 200Time (s)








5 Conclusion



6 Future Works


Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







Algorithm 1

ＨＯＧ(Ｍ)

＞Ｈ(Ｍ)

StepPIDMMPT

Step1 Gain1k

++ ++

Scope1 Scope2







vi

R L

w TＧ

Ｃ

J

B













119879119898 (119905) = 119869119889119908 (119905)119889119905 + 119861119908 (119905) (12)

119879119898 (119905) = 119870119886119894 (119905) (13)



119890119886 (119905) = 119870119887119908 (119905) (14)



V (119905) = 119871119889119894 (119905)119889119905 + 119877119894 (119905) + 119890119886 (119905) (15)



119882(119904)119881 (119904) = 119870119886(119869119904 + 119861) (119871119904 + 119877) + 119870119886119870119887 (16)





119881119899 48 VR 01 ΩL 0005 H119870119887 0004 119881119904119903119886119889119870119886 00036 119873119898119860119869 01 1198731198982119861 005 119873119898119904119903119886119889






119882(119904)119881 (119904) = 00036(119904 + 005) (0005119904 + 01) + 144119890minus5 (17)


119866 (119904) = 721199042 + 205119904 + 1003 (18)

















150

100

50

0

minus50

minus100

minus150

3(D

eriv

ativ

e of t

he E

rror

)

20200

0minus20 minus20





2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

Velo

city

(rad

s)





minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)

2 4 6 8 10 12 14 16 18 200Time (s)








2 4 6 8 10 12 14 16 18 200Time (s)

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Velo

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s)







2 4 6 8 10 12 14 16 18 200Time (s)

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0

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Volta

ge (V

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0 4 6 8 10 12 14 16 18 202Time (s)

0

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ty (r

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2 4 6 8 10 12 14 16 18 200Time (s)

minus50

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Volta

ge (V

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2 4 6 8 10 12 14 16 18 200Time (s)

0

5

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Velo

city

(rad

s)



minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50Vo

ltage

(V)

2 4 6 8 10 12 14 16 18 200Time (s)








5 Conclusion



6 Future Works


Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



vi

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119879119898 (119905) = 119869119889119908 (119905)119889119905 + 119861119908 (119905) (12)

119879119898 (119905) = 119870119886119894 (119905) (13)



119890119886 (119905) = 119870119887119908 (119905) (14)



V (119905) = 119871119889119894 (119905)119889119905 + 119877119894 (119905) + 119890119886 (119905) (15)



119882(119904)119881 (119904) = 119870119886(119869119904 + 119861) (119871119904 + 119877) + 119870119886119870119887 (16)





119881119899 48 VR 01 ΩL 0005 H119870119887 0004 119881119904119903119886119889119870119886 00036 119873119898119860119869 01 1198731198982119861 005 119873119898119904119903119886119889






119882(119904)119881 (119904) = 00036(119904 + 005) (0005119904 + 01) + 144119890minus5 (17)


119866 (119904) = 721199042 + 205119904 + 1003 (18)

















150

100

50

0

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minus100

minus150

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eriv

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he E

rror

)

20200

0minus20 minus20





2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

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city

(rad

s)





minus50

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Volta

ge (V

)

2 4 6 8 10 12 14 16 18 200Time (s)








2 4 6 8 10 12 14 16 18 200Time (s)

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minus5

0

5

10

15

20

25

30

35

Velo

city

(rad

s)







2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

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Volta

ge (V

)






0 4 6 8 10 12 14 16 18 202Time (s)

0

5

10

15

20

25Ve

loci

ty (r

ads

)



2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

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Volta

ge (V

)






2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

Velo

city

(rad

s)



minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50Vo

ltage

(V)

2 4 6 8 10 12 14 16 18 200Time (s)








5 Conclusion



6 Future Works


Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




119881119899 48 VR 01 ΩL 0005 H119870119887 0004 119881119904119903119886119889119870119886 00036 119873119898119860119869 01 1198731198982119861 005 119873119898119904119903119886119889






119882(119904)119881 (119904) = 00036(119904 + 005) (0005119904 + 01) + 144119890minus5 (17)


119866 (119904) = 721199042 + 205119904 + 1003 (18)

















150

100

50

0

minus50

minus100

minus150

3(D

eriv

ativ

e of t

he E

rror

)

20200

0minus20 minus20





2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

Velo

city

(rad

s)





minus50

minus40

minus30

minus20

minus10

0

10

20

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Volta

ge (V

)

2 4 6 8 10 12 14 16 18 200Time (s)








2 4 6 8 10 12 14 16 18 200Time (s)

minus10

minus5

0

5

10

15

20

25

30

35

Velo

city

(rad

s)







2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

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Volta

ge (V

)






0 4 6 8 10 12 14 16 18 202Time (s)

0

5

10

15

20

25Ve

loci

ty (r

ads

)



2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

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Volta

ge (V

)






2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

Velo

city

(rad

s)



minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50Vo

ltage

(V)

2 4 6 8 10 12 14 16 18 200Time (s)








5 Conclusion



6 Future Works


Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





150

100

50

0

minus50

minus100

minus150

3(D

eriv

ativ

e of t

he E

rror

)

20200

0minus20 minus20





2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

Velo

city

(rad

s)





minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)

2 4 6 8 10 12 14 16 18 200Time (s)








2 4 6 8 10 12 14 16 18 200Time (s)

minus10

minus5

0

5

10

15

20

25

30

35

Velo

city

(rad

s)







2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)






0 4 6 8 10 12 14 16 18 202Time (s)

0

5

10

15

20

25Ve

loci

ty (r

ads

)



2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)






2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

Velo

city

(rad

s)



minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50Vo

ltage

(V)

2 4 6 8 10 12 14 16 18 200Time (s)








5 Conclusion



6 Future Works


Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







2 4 6 8 10 12 14 16 18 200Time (s)

minus10

minus5

0

5

10

15

20

25

30

35

Velo

city

(rad

s)







2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)






0 4 6 8 10 12 14 16 18 202Time (s)

0

5

10

15

20

25Ve

loci

ty (r

ads

)



2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)






2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

Velo

city

(rad

s)



minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50Vo

ltage

(V)

2 4 6 8 10 12 14 16 18 200Time (s)








5 Conclusion



6 Future Works


Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




0 4 6 8 10 12 14 16 18 202Time (s)

0

5

10

15

20

25Ve

loci

ty (r

ads

)



2 4 6 8 10 12 14 16 18 200Time (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Volta

ge (V

)






2 4 6 8 10 12 14 16 18 200Time (s)

0

5

10

15

20

25

Velo

city

(rad

s)



minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50Vo

ltage

(V)

2 4 6 8 10 12 14 16 18 200Time (s)








5 Conclusion



6 Future Works


Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





5 Conclusion



6 Future Works


Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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