Near Optimal Closed-Loop Motion Control Minimising frictional Energy Loss

7/26/2019 Near Optimal Closed-Loop Motion Control Minimising frictional Energy Loss

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NEAR OPTIMAL CLOSED-

LOOP MOTION CONTROL

MINIMISING FRICTIONAL

ENERGY LOSS

Xavier Matieni


2/36

Presentation OverviewMotivation

Main contributions

A wider view of the energy issues

A Glimpse of the global CO2 emissions by region for 1973 and 2010

Estimated USA energy use in 2013

The ConsequencesElectric drive systems energy consumption in industry

Electric drive systems use and losses by industry

Typical motion control system

Energy conversion and losses in a drive for motoring and regenerative braking modes

Electric drive modelling

Simplification of Drive Model

Model of Combined Electric drive and Mechanical Load

Linear model with kinematic integratorNonlinear model with kinematic integrator

The general optimal control problem with saturation constraints

Method1: Bellmans Dynamic Programming

Method2: Pontryagins maximum principle

Method3: Koppels method applied to frictional energy minimisation using linear plant model

The minimum friction energy position control by Pontryagins method

Feedback Controller Designs and Results Traditional Controller Results

Near Optimal Reference Input And Dynamic Lag Pre-compensation

Near Optimal controller Results

The sliding mode based near optimal controller

Sliding Mode Based Near Optimal Controller

Results

Frictional loss comparison

Robustness comparisons by simulationOverall Conclusions and recommendations for further research


3/36

Motivation

The current need to minimise energy wastage to reduce the impact ofindustrial development on the environment has promoted the author to re-visit optimal control theory with a view to deriving new practicable closedloop optimal laws that could save terawatts of electrical power by

replacement of classical controllers throughout industry.


4/36

Main contributions

Practicable rest-to-rest position control strategyfor a single degree of

freedom mechanism achieving frictional energy loss minimisation.

A theoretical contribution is the proof of optimality using Pontryagins

method.

Two forms of Implementation:

Nonlinear sliding mode control law

Linear control law with dynamic lag pre-compensator together with

minimum friction loss position reference input generator


5/36

From 2000 to 2010, worldwide energy consumption increased by 28%.

This will probably increase by 50% from 191 quadrillion (x1015) Btu in2008 to 288 quadrillion Btu in 2035. Adriana (2012)

Growing energy demand entails proportional energy losses unless

measures are taken to increase efficiency

The future developments in engines and powertrain systems will

necessarily be sympathetic to nature. Rahnejat (2010) There is a need to educate industrial developers of energy consuming

applications to adopt new technologies and methodologies

Manufacturers especially need to develop an energy management

culture.

We should learn from nature in our inventions, including engines and

powertrains, conserve energy and strive to maintain peace and

tranquillity. The affinity to nature translates to adherence to the principle of

parsimony and a quest for the principle for least action (optimal

conservation of energy). Rahnejat (2010)

A Wider View of the Energy Issues


6/36

Africa

Asia

Bunkers

China

Middle-East

OECD Europe

And Eurasia

Non OECD Americas

OCDE

Mt of CO210 20 30 40 50 60 70

OCDE: countries and those EU

countries that are not members

of the OECD (i.e. Bulgaria,

Cyprus, Latvia, Lithuania, Malta

and Romania).

Mt: million tonnes

Mtoe: million tonnes

of oil equivalent

1973

66.1%

2.6%

16.2%

0.8%

5.9%

3.6%

3.0%

1.8%Africa

Asia

Bunkers

China

Middle-East

OECD Europe

And Eurasia

Non OECD Americas

OCDE

Mt of CO210 20 30 40 50 60 70

2010

41%

3.5 %

8.6%

5.1%

24.1%

3.6%

11%

3.1%

A Glimpse of the global CO2 emissions

by region for 1973 and 2010


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Estimated USA energy use in 2013


8/36

The USA has 4.3% of the worlds population, but generates 23% of

carbon dioxide emissions

Singapore carbon dioxide emission rose from 1 to 22 metric tons of CO2

per capita (person) in about three decades

The economic surge of china is remarkable but unfortunately this coulddestroy the planet in a long run (dirty coal)

The USA generates 18 times as much carbon dioxide as India and 100

times as much as most of Africa.

The worlds economies could either spend 1% of their GDP on green

energy sources and environment technologies now or be confrontedwith having to spend 20 times that much in the near future. Stern report

(2006).

Business as usual will derail the growth

The Consequences


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1990 2000 2011

0

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Unclassified

Other industries

Construction

Paper, printing, publishing

Textiles, leather, clothing

Food, drink, tobacco

Electrical & instrument engineering

Mechanical engineering & metal products

Chemicals

Mineral products

Non-ferrous metals

Iron & steel

Vehicles

Energy losses in

drive systems

represent for 12 %of

total energy end

used inmanufacturing and

mining sub-sector

45% only is used in

processes

55% is lost due to

inefficiencies inequipment and

distribution system

Electric drive systems:

energy consumption in industry


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Mining

chemicals

Forest products

Food and beverage

Iron and steel

Petroleum refining

100 200 300 400 500

To processes

Conversion losses

Distribution losses

482 TBtu

183 TBtu

429 TBtu

121 TBtu

185 TBtu

142 TBtu

TBtu =Trillion Btu

300

250

200

150

100

50

0MiningChemicals. Forest

products

Food

&

beverage

Iron

&

steel

Petrol.

Ref.

350

63

6

88

8

80

5

202

165

18

89

Distribution losses

Conversion losses

301

In view of the

foregoing, the new

optimal control

technique resulting

from this researchcould save between

35% and 40% of

electric energy

throughout industry

in global scale. It

would therefore be a

very significantcontributor to the

overall mission of

reducing energy

wastage and

reducing the carbon

footprint

For six largest

energy

consuming

industries

which

represent the

bulk of the total

electric drives

systems losses

Energyconversion and

distribution

losses

Electric drive systems use and losses by industry


11/36

Typical motion control system


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Reverse

energy flow

Electrical

energy

Voltage and

frequency

change in the

converter

Electric to

magnetic

energy

conversion

Magnetic to

mechanical

energy

conversion

Mechanical

load

Harmonic

losses due to

converter

Commutation

and conducting

losses in the

converter

Core losses

(hysteresis

& eddy

current )

Mechanical

losses

Forward

energy flowELWESW

MLW

Electric Motor

MRW

MaW

MaRWCpW

CpRWrCW

rRCW

ERcW

Co

CoR

W

W

ERSW

Copper

losses

2I RMechanical

losses

FLossW

FLoss ML MR

EL ERc

W W W

W W

100%EL FLoss

EL

W W

W

The efficiency is defined in terms of thefrictional energy loss in the mechanical

load, which is dominant:

Energy conversion and losses in a drive for

motoring and regenerative braking modes

The frictional energy loss:


13/36

dd r q d

q

r d q r q

rd q L

diAi B i Fv

dt

diC i Di E Gvdt

dH Ki i M

dt

The standard PMSM drive model consists of the following

differential equations Quang & Dittrich (2008):

Here, id, iqand vd, vqare the stator current vector and voltage vector

components in the rotor-fixed frame, is the rotor angular velocity, Lis

the net load torque and the constants,A,B,,M, are dependent on the

stator resistance and inductance and the permanent magnet flux.

Electric drive modelling

r


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( )rddem qdem L

dH i i M

dt

In vector control of electric drives, however, setting iddem= 0 keeps the stator

current vector perpendicular to the magnetic flux vector of the rotor carrying

permanent magnet, similar to the situation in a DC motor, which maximises the

torque produced for a given stator current magnitude. This further simplifies the

model:

r

qdem L

dHi M

dt

Simplification of Drive Model

In the Unidrive from Control Technique Dynamics Ltd, two current loops tightly

control idand iqto follow the demanded currents, iddemand iqdem, according to Vittekand Dodds (2003). Then it may be assumed that id= iddemand iq= iqdem, which

become the control variables. Thus


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Let the motor torque be denoted and define where

the model becomes

Lerb U s U s

ss a

The corresponding transfer function relationship is

is the external load torque, and is the dynamic load torque,

3

2

PM

m qdem

pi

rLd L rd

J F

dt

Le

the first term of which is the is the inertial torque. The second term is the all important

friction torque which, in general, is nonlinear. The model then becomes

For the design of a linear drive controller, which is relevant to the pre-compensator

method, a linear model of the combined drive and mechanical load is obtained by

replacing by a linear viscous friction term, . . rF rB

Let . Then

r L

Ba

J J

r Le rd

b u u adt

L Le Ld

1 1

r r r

m Le L r m Le r

r r L

d d dJ F F

dt J dt dt J J

PM

r L r L

3 3, , and2 2

rPM

qdem Le Le r Fp pi u b u f

J J J J

r Le rd

b u u f dt

Then if ,

Model of Combined Electric drive and Mechanical Load


16/36

The additional equation needed for a model suitable for the design of a position

controller is , where is the rotor angle. The kinematic integrator is that in

b

s a

1

s

1X s 2X s

Y s Y s

U s

LeU s

1

s

1

s

ab

b

LeU s

U s

LdU s

+

-

+

+

1X s 2X s

Y s Y s

b) in the control canonical forma) directly from the transfer function relationship

LU s-

+

r r r

2

Le

r

b U s U ss

s as

, which yields the transfer function relationship, r r1

r rdt s s

s

, from which the following block diagrams are derived:

This leads to the state space model:

1 2

2 Le 2 Le 2

1

x x

ax b u u x b u u ax

b

y x

Linear model with kinematic integrator


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1 2

2 Le 2 Le 2

1

x x

a

x b u u x b u u f xby x

2ax 2f x

Nonlinear model with kinematic integratorAs previously, the more general model is obtained by replacing the linear

friction term,

, by the nonlinear term, , which agrees with Athans & Falb (1982):


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, ,

,

t t t t

t t t

x f x u

y h x

0

, , , .

T

t

J G t t t t dt x u r

max max maxu u t u uu t

The general optimal control problem with saturation constraints

a) The plant is represented by a state space model:

where xis the plant state, uis the set of control

inputs, and yis the set of measured outputs.

b) There are magnitude limits on u(t)and x(t)imposed

by the hardware. Thus .c) A reference signal, r(t), is provided that y(t)is intended to follow.

d) A performance criterion or cost function has to be minimised:

where t0 is the initial time, T is the end time of the action, andG(.), is the loss function or the measure of the instantaneouschange in the performance.


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max, , , , ,t f t t t t h t t u t u x x u y x

0

0 0, , , , , .

fT

t

J t t G t t t t dt u x x u r

00

, , , , 0o o

J Jf x t u t G x t u t r t t

t x

maxsgno

o Ju t u B

x

Method : Bellmans Dynamic Programming

Dynamic programming is the optimisation method originated by Bellman in the

USA in the 1950s.

The plant:

The cost function:

The scalar Hamilton-Jacobi equation has to be satisfied when J has the requiredminimum value, Jo. This yields the following nonlinear partial differential equation

to be solved for Jo :

The solution is then used to calculate the optimal control:

This can only be solved numerically except in very simple cases.


20/36

max, , , , ,t f t t t t h t t u t u x x u y x

0

0 0, , , , , .

fT

t

J t t G t t t t dt u x x u r

1

, , ,n

i i

i

H Gpx t

x u r

, 1,2, ,i

i

H tp t i nx t

Method2: Pontryagin Minimum PrinciplePontryagin originated this method in Russia in the 1950s.

The plant:

The cost function:

The optimal control input, uo(t), that minimises Jwill maximise the scalar Hamiltonian

function

wherepi(t)are the co-states obeying the ordinaryadjoint system differential equation:

The optimal control is then obtained by maximising the Hamiltonian with respect to u(t).

Analytical solutions, obtained by eliminating the co-states to obtain a state feedback

control law are only possible in a few particular cases. Fortunately, this can be done for

the minimum friction energy problem.


21/36

1 2

2 2

x x

x ax bu

0

2

2

fT

t

J x dt

0

2 2 21 2 0, 0

fT

tJ u Px x Su dt P S

0

2 2 2

3 1 2

fT

t

x Px x Su dt 1 2

2 2

2 2 2

3 1 2

x x

x ax bu

x px x Su

The Hamiltonian is then 2 2 2

1 2 2 2 3 1 2H p x p ax bu p Px x Su x,u,p

. The correct cost functional is but another restriction

, is introduced, giving the augmented plant model,

of Koppels method is that the cost functional contains a control weighting term. Thus

The method was applied with with the intention of obtaining meaningful results

by reducing in steps and predicting the control behaviour. An additional state variable,0P

S

Method 3: Koppels method applied to frictional

energy minimisation using linear plant model

Koppels method is a variant of Pontryagins method in which the adjoint system

variables are eliminated but it is subject to restrictions. First the plant model has to

be linear, so only viscous friction is considered and the plant model is


22/36

1 3 1

1

2 1 2 3 2

2

3

3

2

2

0

Hp p Pxx

Hp p p a p x

x

Hp

x

2

2

2 0

1

2

o o

Hp b Su

uu t bp t

S

2op t

2

2 2 0

HS

u

* maxsatu t u u max

max max

,sat sgn ,

u u uu u u u u

2max2

bp tu

Su

2

**1

*2* 22

2

**11

*

* * *21 2 2

2

2

2

dxx

dt

p tdxax b

dt S

dpPx

dt

dpp p a x

dt

1 0 10

2 0

1

2

0

f

f

x t x

x t

p T free

p T free

1 0p t 2 0p t 1 f 2 f 0x T x T

The co-state differential

Equations are as follows:

The optimal control then satisfies with

and are found that yield .

where is

to be determined.

If is unconstrained, , indicating a minimum, as required.u

If is constrained, , whereu

Then which is a normalised control candidate.

Applying this method revealed that could not be reduced sufficiently. So this method

was abandoned in favour of pursuing Pontryaginsmethod, particularly as this allowsnonlinear friction, which is often significant.

Method 3: Koppels method applied to frictional energy

minimisation using linear plant model (continued)

Two-point Boundary Value

Problem


23/36

dt dt

2f x

b

Leu t

u t

du t

+

-

+

+

1x t 2x t

y t y t

Lu t

Iu t

1 2

2 2 L max,

x x

x f x b u u u u

m

2 2

0

T

J x f x dt

1 2 2 2 L 2 2H p x p f x b u u x f x

The adjoint system differential equations are

mT

ou t

2 20 0H u p b p

1 1 0p H x 2 2 1 2 2 2 2 2p H x p p f x f x x f x and

Since the manoeuvre time, , is fixed, it is

postulated that the optimal control,

contains at least one continuous segment

2 2 2 1 2. .f x x f x p const x const

max maxu u u

So far, it may be concluded that there exists a portion of the optimal

control trajectory that has a constant velocity with .

for which .

Since ,2 0p 2 1 2 2 20 0p p f x x f x Hence

The minimum friction energy position control by

Pontryagin s method

The plant model block diagram with the nonlinear friction, the corresponding

state space model and the optimal control formulation are as follows:


24/36

2coastx

1rx

0t

mT0

1 0x

t

2

x t

2 0x

u t t t b

1x t

2coast

x

1rx

0t0

1 0x

t

2

x t 1x t

maxu

maxu

u t 2 0

xu t t t

b

a) hypothetical case withoutcontrol saturation

b) practical case with controlsaturation

u t

0 adt T

mTm adT T

u t

u t

A practicable optimal control imparts the maximum possible acceleration magnitude to reach a constant

velocity and the maximum possible deceleration magnitude close to the end of the manoeuvre to bring the

velocity to zero and the position to the correct value in the specified time, .mT

acc max 1r

dec max 1r

sgn

sgn

u u bx

u u bx

1r 1acc 1decacc dec m

2coast

x x xT T T

x

1r 1ad

m ad

2coast

22

x xT T

x

2 2 2

max m max m max 1r

acc dec admax

4

2

bu T b u T bu x

T T T bu

m1r 20T

x x t dt 2 ( )x t

2 ( )x t

Since is the area

under the graph of against t,

and Tmis fixed, then any attempt to

reduce below the constant

optimal value, , in an attempt to

reduce the frictional energy loss would

result in a catch up peak value of

larger than and incur a

considerable frictional energy penalty

since the transfer characteristic of

frictional power loss againstgenerally has a slope that increases

with . Hence a constant velocity

of is the extreme of the

velocity for the optimal control.

*2coastx

2x

*

2coast

x

2 ( )x t*2 coastx

and

2x

The minimum friction energy position

control by Pontryagin s method

For a double integrator plant,


25/36

First, traditional controllers will be designed to attempt to minimise the

frictional energy loss within the constraints of these linear control systems.

The performances of these traditional controllers will be compared with two

different implementations of the optimal control strategy to assess the

improvement in energy savings that could be brought about by theiradoption.

Approach:

FEEDBACK CONTROLLER DESIGNS

AND RESULTS


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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0

0.2

0.4

0.6

0.8

1

Ts

Steady state value

Time [sec]

Desiredclosedloop

stepresponse[-]

+/- 5%

c s

1.6 1.51 nn ns s

T T

s c1.6 1.5T n T 1,2, , c1ns T cT

The desired closed loop

characteristic polynomial for the

pole placement with x=2 is

Example for x=5:

TRADITIONAL FEEDBACK CONTROLLER DESIGNS

Since non-overshooting step responses are preferred to minimise excessive velocities with the

attendant penalties in frictional energy wastage, and the orders of these control loops exceed three,

the 2% settling time formula of Dodds (2008) will be applied, as follows.

The settling time for the x% criterion is defined as the time taken for the step response to reach and

stay within a +/-x% band of the steady state value, centred on the steady state value.

If the closed loop poles are placed coincidently at , where is the time

constant of the multiple closed loop pole, the settling time for x=2 is given by


27/36

2ss

2ss const

const

axbx u b

a u

2 ss

a const

xb

T u

According to the elementary theory of linear first order systems,

reaches 63% of its steady state value in a period equal to one time

constant, which will be denoted .

aT 2ss0.63x

1aT a a

It is clear that may be measured experimentally by detecting the crossing of

Since, , the plant parameter , may be

determined from the equation, 1

a

aT

It is also clear that , may

be measured experimentally.

Then is determined asfollows,

2 ssx

b

Substituting for

yields,

a

a

a

ss

a

1.83

10.546

8868.71.83 0.7

T

aT

b T u

0 1 2 3 4

0

a

1

T

ss

ss0.63

[s]t

Plant Parameter Determination

2x t

aT


28/36

b

s a1

s

1

sIK

PK

DK s

den s

U s 2X s 1X s

1rX s

b

s a

1

s

1

sIK

PK

DK s

den s

U s 2X s 1X s

1rX s

_

b

s a1

s

1k 2k

s

den s

U s 2X s 1X s 1rX s

_

2X s

r

2

b

s as

1

s2

2 1 0

1

f s f s f

U s 1X s

1rX s

_ dU s

U s

22 1 0h s h s h

Polynomial Controller

r

PID

IPD

LSF

Polynomial

0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

50

60

70

80

Experimental

Simulated

PIDcontollerpositionresponses:

simulatedandexperimental[rad]

t[s]

0 0.5 1 1.5 2 2.5 3 3.5 4-5

0

5

1015

20

25

30

35

40

45

50Simulated

Experimental

IPD

position

responses:simulated

withex

perimental[rad]

t[s]

0 0.5 1 1. 5 2 2.5 3 3.5 40

5

10

15

20

25

30

35

40

45

t[s]

Simulated

Experimental

LSFpositionresponses:simulated

with

experimental[rad]

0 0.5 1 1.5 2 2.5 3 3.5 4

0

5

10

15

20

25

30

35

40

45 Simulated

Experimental

t[s]

Polynomialcontrollerpositionresponses:

simulatedwithexperimental[rad]

0 0.5 1 1.5 2 2.5 3 3.5 4-5

0

5

10

15

20

25

30

35

40

45

t[s]Polynomialcontrollervelocityresponses:

simulatedwithexperimental[rad/s]

Simulated

Experimental

0 0.5 1 1.5 2 2.5 3 3.5 4-0.5

0

0.5

1

1.5

2

2.5

3

t[s]

Simulated

Experimental

Polynomia

lcontrollercontrolvariableresponses:

sim

ulatedwithexperimental[Volts]

Simulated

Experimental

t[s]Poly

nomialcontrollerfrictionalenergyloss

responses:simulatedwithexperimental

[Joules]

0 0.5 1 1.5 2 2.5 3 3.5 40

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5 3 3.5 4-10

0

10

20

30

40

50

60Simulated

Experimental

LSFvelocityresponses:simulated

withex

perimental[rad/s]

0 0.5 1 1.5 2 2.5 3 3.5 4-1

0

1

2

3

4

5

6

7

t[s]

LSFcontro

lvariableresponses:

simulatedw

ithexperimental[Volts]

Simulated

Experimental

0 0.5 1 1.5 2 2.5 3 3.5 4-200

0

200

400

600

800

1000

1200

1400Simulated

Experimented

LSFfrictionalenergyloss

respo

nses:simulatedwith

exp

erimented[Joules]

t[s]

0 0.5 1 1.5 2 2.5 3 3.5 4-5

0

5

1015

20

25

30

35

40

45

t[s]

IPD

velocityresponses:simulated

withexp

erimental[rad/s]

Simulated

Experimental

0 0.5 1 1.5 2 2.5 3 3.5 4-0.5

0

0.5

1

1.5

2

2.5

3

3.5

t[s]

IPD

contro

lvariableresponses:

simulatedwithexperimental[Volts]

Simulated

Experimental

0 0.5 1 1.5 2 2.5 3 3.5 4-100

0

100

200

300

400

500

600

700

800

900

IPD

frictionalen

ergylossresponses

simulatedwithe

xperimented[Joules]

t[s]

Simulated

Experimented

0 0.5 1 1.5 2 2.5 3 3.5 4-20

0

20

40

60

80

100

120

140

t[s]

PID

velocityresponses:

simulated

withexperimental[rad/s]

Experimental

Simulated

0 0.5 1 1.5 2 2.5 3 3.5 4-4

-2

0

2

4

6

8

10

12 Simulated

Experimental

PID

controlvariableresponses:

simulatedwithexperimenta

l[Volts]

t[s] 0 0.5 1 1.5 2 2.5 3 3.5 4-500

0

500

1000

1500

2000

2500

3000

3500

4000

4500

t[s]

PID

controllerfrictionalene

rgyloss

responses:simulatedwithexperimented

[Joules]

Simulated

Experimented

Position Velocity Control Energy Loss

Traditional Controller Results


29/36

Optimal

Position

Reference

InputGenerator

Dynamic

lag pre-

compensator

Position

controller

Plant

(controlled

mechanism)

ry t 'ry t

u t

y t

Closed loop control

system

2 3

1 2 3

1

1 a s a s a s

rY s 1rX s

Y s1a

2a

1rsX s

2 1rs X s

Dynamic lagpre compensator

3a 3 1rs X s

1rx tdtdt

1rx t

1rx t

1rx t

1r t 3r tdt

2r t r t

r t

t0

accT

mT

R

R pT

pT

pT

pT

3r t

0 t

maxbu

maxbu

2r t

0 t

2coastx

accT

1r t

0 t

1cx

accT

m accT T mT

1ax

1c 1ax x

Bang-zero-bang triple integrator input function and

state variables of reference input generator for 1c 0x

NEAR OPTIMAL REFERENCE INPUT AND DYNAMIC LAG PRE-COMPENSATION


30/36

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

35

4045

t[s]

Simulated

Experimental

IPD+PC

positionresponses:

simulatedwithexperimental[rad]

0 0.5 1 1.5 2 2.5 3-5

0

5

10

15

20

25

43.5

Simulated

Experimented

IPD+PC

velocityresponses:

simulatedwithexperimental[rad/s]

t[s] 0 0.5 1 1.5 2 2.5 3 3.5 40

100

200

300

400

500

600

700

800

900

Simulated

Experimental

IPD+PC

frictionalenergyresponses:

simulatedwithexperimental[Joules]

t[s]

0 0.5 1 1.5 2 2.5 3 3.5 4-10

-8

-6

-4

-2

0

2

4

6

8

10Simulated

Experimental

t[s]Poly+PC

controlvar

iableresponses:

simulatedwithexpe

rimental[Volts]

0 0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

510

15

20

25

30

t[s]

SimulatedExperimental

Polynomial+PC

velo

cityresponses:

simulatedwithexpe

rimental[rad/s]

0 0.5 1 1.5 2 2.5 3 3.5 4-5

0

5

10

15

20

25

30

35

40

45

t[s]

Poly+PC

Positio

nresponses

simulatedwithexp

erimental[rad]

Simulated

Experimental

0 0.5 1 1.5 2 2.5 3 3.5 40

100

200

300

400

500

600

700

800

900

t[s]

Poly+PC

frictiona

lenergyloss

responses:simulatedwith

experimented

[Joules]

Simulated

Experimented

Near

optimal

controller Results

IPD controller with dynamic lag pre-compensator and reference input generator

Position Velocity Control Energy loss

Polynomial controller with dynamic lag pre-compensator and reference input generator


31/36

1e 2, 0S x x

2coastx

2coastx

c 2coastT x

1e 1 1r x x x

0S

0S

2x

0S

0S

c 2coastT x

1e 1 1r x x x max 1e 2sgn ,u u S x x

2 1e 1e c

c1e 2

2

2coast

2coa 1e 1e cst 2coastsgn

1 for

,

for

x x x TTS x x

x x x T

x

x x

1ex

2x

2coastx

2coastx

maxu u

maxu u

1ex

2x

2coastx

2coastx

'n' saturation boundary

maxu u

maxu u

'p' saturation boundary

original switching boundary

statetrajectory boundary layer

2boundary layer width, x

max

c 2

uK

T x

1e 2 max maxsat . , , ,u K S x x u u

1e 2 2 1e 2coast 2coastc

1, sat , ,S x x x x Sx Sx

T

2 1e 2coast 2coast

c

1sat , ,x x Sx Sx

T

Boundary layer Closed loop phase portraitSwitching boundary

Basic bang-bang control law:

or

Equivalent saturating control lawwith switching boundary:



32/36

Demanded

position

change 3

x2smc

2

x1smc

1

usmc

Unity slope,

Saturation lim its

+/-S.x2coast

1/Tc

Slope

magnitude

of central

switching boundary

segment

Saturation

limits:

+/-umax

1

s

b

s+a

PLANT

-K

K=umax/(Tc*Dx2) for

boundary layer width of Dx2

in x2 direction of phase plane plane

1

Yc

0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

50

t[s]SMC

pos

ition

responses:

simulated

withexperimental

[rad]

Simulated

Experimental

0 0.5 1 1.5 2 2.5 3 3.5 4-5

0

5

10

15

20

25

30

35

40

Simulated

Experimental

SMC

velo

cityresponses:simulated

with

experimental[rad/s]

t[s] 0 0.5 1 1.5 2 2.5 3 3.5 4

-10

-5

0

5

10 Simulated

Experimental

t[s]

SMC

controlvariableresponses:

simulatedwithexperimental[Volts]

0 0.5 1 1.5 2 2.5 3 3.5 40

400

600

800

1000

1200

200

t[s]SMC

frictiona

lenergyloss:simulated

withexp

erimental[Joules]

Simulated

Experimental

Position Velocity Control Energy Loss

SMC block diagram:


Results


33/36

Conventional Controllers

PID IPD LSF POLY

4250 850 1300 1040

0 0.5 1 1.5 2 2.5 3 3.5 40

200

400

600

800

1000

1200

SMCPOLYnoIPDno

t[s]Frictionalenergylosscomparisonsfor

thenearoptimalcontrolsystems[Joules]

0 0.5 1 1.5 2 2.5 3 3.5 40

1000

2000

3000

4000

5000

6000 PIDLSF

POLY

IPD

t[s]

Jrconv[Joules]

4250

8501300

1040

852.2119

810.2108

985.25PID

IPD

LSF

POLY

IPD+PC

POLY+PC

SMC

Near Optimal Controllers

IPD+PC POLY+PC SMC

810.2108 852.2119 985.252

PID is worst performer due to overshoot induced

by controller zeros.

The sliding mode controller performance is not sogood as that of the reference input generator based

systems due to the exponential settling in the sliding

mode on the sloping boundary segment approaching

the origin of the phase plane. The SMC, however,

is the most robust.

Frictional loss comparison


34/36

max min

pu

max min

1

2

J JJ

J J

10%a

-10%a

10%b

-10%b

puJ b

puJ a

CONTROLLER

SMC IPD+PC Polynomial +PC

Plantparametervariations

985.201 810.2403 852.2216

Nominal 985.225 810.2108 852.2119

985.1994

810.1904

852.2052

985.2502 810.698 852.4945

Nominal 985.2125 810.293 852.2205

985.1505

810.095

852.105

Perunit

loss

deviation

0.000078 0.00074 0.00046

0.0000012 0.00062 0.000019

Robustness comparisons by simulation

Conclusions and Recommendations


35/36

An optimal control strategy proven using Pontryagins method has been establishedfor minimising frictional energy loss in motion control systems. It imparts themaximum possible acceleration magnitude to reach a constant optimal velocity andthe maximum possible deceleration magnitude close to the end of the manoeuvre toreach the required position in the specified manoeuvre time. Three practicable

feedback control systems have been developed that approximate this optimal control,one based on a sliding mode controller and the other two based on traditional linear(PID, IPD and LSF) and polynomial controllers used with a dynamic lag pre-compensator to follow a pre-planned near optimal state trajectory.

The new contribution of the sliding mode control system presented in this thesis isthe piecewise linear switching boundary replacing the conventional linear switchingboundary to yield the near optimal frictional energy minimisation.

The new contribution of the IPD and polynomial controller based systems is the nearoptimal reference input generator used in conjunction with a zero dynamic lag pre-compensator, to achieve the near optimal frictional energy loss minimisation. It isenvisaged to be attractive to industrial users in view of its use of linear and thereforerelatively easily understood controllers.

It is recommended to focus on minimum energy optimal position control of a widerange of mechanisms throughout industry including those with multiple degrees offreedom in order to make a real impact regarding environmental protection.

Robust pole placement that emulates the behaviour of a sliding mode control systemwith a boundary layer should be investigated, to improve the robustness of thereference input based near optimal controllers.

For appropriate applications, it would be advantageous to research into an adaptivecontrol system based on dynamic updating of the demanded constant coasting

velocity using real-time measurements of the acceleration and deceleration times, toachieve a much closer approach to the ideal performance.

Conclusions and Recommendations


36/36

Thank you for your attention.

Please feel free to ask any questions.

Date post:	13-Apr-2018
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Near Optimal Closed-Loop Motion Control Minimising frictional Energy Loss

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