Study of Energy Evaluation Control

Yasushi Yamamoto

, Shinya Hasegawa

, Satoru Iwamori

and Shigeru Yamaguchi

ISUZU Advanced Engineering Center Ltd., Fujisawa, Kanagawa-ken, Japan

Department of Mechanical Engineering, School of Science, Tokai University, Kanagawa-ken, Japan

Department of Physics, School of Science, Tokai University, Kanagawa-ken, Japan

Keywords: Lagrangian, Hamiltonian, Energy, Bang-bang Controller.

Abstract: One of the objectives in control theory is to ensure that a control system converges to a target state in the

shortest possible time. To achieve that objective, we studied a control method that combines the Lagrangian,

the Hamiltonian, and a bang-bang controller. Referred to as energy evaluation control (EEC), this method

evaluates the control state using the Lagrangian, and evolves the control output using the Hamiltonian. Here,

the Lagrangian and the Hamiltonian are defined for the deceleration field. The control result from the EEC

is fast and robust. Moreover, EEC has the same control strategy as the sliding-mode control, and hence can

be incorporated within it.

1 INTRODUCTION

The Lagrangian and Hamiltonian have been used in

control theory to establish robust control methods

(Bloch, et al., 2000), (Choi, et al., 1997). In addition,

the control logic by using these energys are

examined, and the application experiment to the

inverted pendulum is conducted (Fantoni et al.,

2000), (Ortega, and Spong, 2000),(Ortega, et al.,

2000).

Moreover, one objective in control execution is

to direct the control system towards target states in

the shortest possible time. A bang-bang controller is

one that is able to realize this objective. However, a

bang-bang controller has one drawback in that it

does not perform well when external forces are

changed. Although in examining the control rule,

which determines the feedback of a state quantity to

the change in output of the bang-bang controller

performed on the control object, the output is

changed in the second-half of the cycle and has a

slight complicated control structure (Vakilzadeh and

Keshavarz, 1982).

In this report, we studied a control method which

determines the control output of the bang-bang

controller using the Lagrangian and its convergence

control output using the Hamiltonian. We refer to it

as the “energy evaluation controller” (EEC) because

this controller evaluates the energy of the control

state. The features of EEC is that a formula can be

made simpler than the conventional energy method,

and that a control result becomes the shortest time

control because EEC based on the bang-bang

controller.

We begin by explaining the control rule of the

EEC for the simple control model. Next, we propose

the adjustment method for the external force of the

damper and friction. Finally, we explain that the

EEC as a kind of sliding-mode control (SMC).

2 SWITCHING OF CONTROL

OUTPUT USING THE

LAGRANGIAN

A bang-bang controller is a controller that using

maximum thrust enables abrupt changes in state

through acceleration and deceleration of a controlled

object. A simple model of the bang-bang controller

(Fig. 1) and its control cycles (Fig. 2) assumes that

the actuator can generate a fixed thrust ±F

max

. The

notation and significance of the variables are: x:

stroke, : velocity, X: target position, T: kinetic

energy and -F

max

(X-x): braking work. The switching

of the actuator thrust is the instant when the braking

energy and the kinetic energy are equal.

Next, the Lagrangian is calculated for the

deceleration field which is generated by the thrust

from the deceleration force of the actuator. Here, the

reference position of the deceleration field is set to a

target position X. The Lagrangian L is defined as the

550

Yamamoto Y., Hasegawa S., Iwamori S. and Yamaguchi S..

Study of Energy Evaluation Control.

DOI: 10.5220/0005538105500553

In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2015), pages 550-553

ISBN: 978-989-758-122-9

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

-F

max

Actuator

Target Position

Controller

Figure 1: Simple model of a bang-bang controller.

max

−F

max

Actuator

Force

Stroke

Velocity

max

Energy

Time



Figure 2: Operation cycle of the bang-bang Controller.

difference between kinetic energy T and potential

energy U:

L = T - U (1)

xmT



(2)

()

max



−=

dxFU

(3)

The potential energy of the decelerating field

corresponds to the possible work during braking.

Therefore the switching time to apply thrust from

the actuator is the instance when the value of the

Lagrangian takes value 0. It is necessary to consider

the direction of motion of the controlled object to

decide the control output. Hence the sign (plus or

minus) of the velocity is added to the kinetic energy,

and (1) is modified to:.

()

max



−−=

dxFxm



(4)

The sign of the Lagrangian L' then determines

the direction of the control output u:

max

u =

(5)

Here F

max

is the maximum thrust that can be

generated by the actuator. The control is simple and

seems to work fast. However, in actual control

systems, it does not work properly because the

estimation error for the Lagrangian is generated by

the external force that cannot be predetermined. To

address this issue, calculations of the Lagrangian

and the control output are done repeatedly at every

control cycle, in trying to move the controlled object

towards the target position despite the estimation

errors associated with the Lagrangian.

For the control target illustrated in Fig. 3, we

obtain the EEC control result presented in Fig. 4.

The simulation settings are actuator thrust:

()

N100± , mass: 1(kg), initial position:-1(m), target

position: 0(m). Coefficient of damping C(Ns/m) and

frictional force F

(N) is set as follows: Case A

(C=10, F

=10), Case B (C=20, F

=0), Case C (C=0,

=70),. Furthermore, switching of the control

output is assumed to occur abruptly. The simulation

result shows that the EEC can control the object to

the target and it's result seems robust. At the same

time, this simulation result shows that the control out

put is not shape of ideal bang-bang controller's

output.

Figure 3: System of control target with dumper.

3 CORRECTION OF THE

POTENTIAL ENERGY

EEC can converge the control object to the target

position even if the external force act on the control

ovject. However, hunting of control output occurs,

as shown in Fig. 4. This hunting is caused by the

estimation error associated with the potential energy

of the deceleration field which generated by the

damping force and frictional force. On the other

hand the braking work done by damping force W

calculated from the equation (6) and the braking

work done by frictional force W

is given by (7).

Here the Coulomb's friction model is adopted and

the coefficient of friction F

is assumed as constant.

Therefor the potential energy is corrected to (8).

Figure 5 shows the simulation result of using

equation (8). The simulation result shows that the

bang-bang control is almost realised by this

correction.

StudyofEnergyEvaluationControl

551

Force (N)

0.4

0.2

Stroke (m)

Time

(

)

Case A

C=10(Ns/m)

=10(N)

Case B

C=20(Ns/m)

=0(N)

Case C

C=0(Ns/m)

=70(N)

Case B

Case C

100

-100

100

-100

Figure 4: Control result of the EEC Simulation conditions

are changed over 3 cases.

ddc

xCxW



(6)

dff

xFW =

(7)

WWdxFU ++−=



)(

max

(8)

C : coefficient of damper, F

: frictional force.

4 CONVERGENCE OF

CONTROL OUTPUT USING

THE HAMILTONIAN

Figure 5 also shows the actuator output after the

controlled object has reached its target state. To

reduce energy consumption, it is desirable to stop

the control output after control finishes (velocity is

zero and positional deviation is zero). Here, the

Hamiltonian H for the deceleration field (9) is

introduced into the EEC procedure. As the

Hamiltonian represents the total energy of the

deceleration field, takes the value zero when the

object velocity and positional deviation become

zero. Therefore, to stop the actuator thrust at the end

of the control period, the convergence rate of the

actuator thrust must be determined; this is achieved

by multiplying the Hamiltonian by the actuator

thrust F

max

and the gain K

. To take into account the

upper and lower limits of the control output, HK

restricted to values between 0 and 1. Therefore, the

formula expressing the thrust of the actuator is

modified to (10). Because of this, the Lagrangian is

Figure 5: Control result of the EEC Effect of the potential

energy correction potential energy.

also corrected to (11). Finally, the control output is

modified to give (12)

()

WWdxFxmH ++−+=



max



(9)

maxmax

' FHKF

= )10( ≤⋅≤

(10)













++−=



WWdxFxm

max



(11)

max

u =

(12)

Because F

max

is changed in the next control step,

the Hamiltonian also changes:.

WWdxFxmH ++−+=



max



(13)

-1

Stroke (m)

Force (N)

100

-100

Control output is converged

0.4

0.2

Time (s)

100

Energy (J)

Hamiltonian

Lagrangian

L=0

Accel. field

Deceleration. field

H=0

Figure 6: Using the Hamiltonian to control output

convergence.

ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

552

Convergence of the control output using the

Hamiltonian (Fig. 6) shows that the control output

has converged to zero when the control converges.

On the other hand the Lagrangian is restrained to 0.

It is means that the control object reached to the

target position according to the deceleration field.

5 CONTROL STRATEGY OF ECC

Figure 7 shows the phase plane, obtained using (7),

that determines the control output in EEC. For

comparison, a switching phase plane for SMC is sho

wn in Fig. 8. SMC has a switching phase plane of

control output that is similar to EEC (

Efe, et al., 2000),

and hence these two methods have similar control

structures. Usually, the displacement and velocity

are associated with the axes of the phase plane of

SMC. However, other axes can be used because

SMC can be extended to any dimension. This

implies that the EEC is included in SMC.

EEC has good characteristics in that the gradient

of the output switching line is always set to −1,

because the kinetic energy and braking energy have

the same units (J). Moreover, the Lagrangian is

restricted to zero within the SMC context. This

behavior indicates that the controlled object will

move naturally. Thereby EEC can simplify the

problem of control, as mentioned above.

Figure 7: Phase plane for EEC and conventional SMC.

6 CONCLUSIONS

EEC was proposed that switches the control output

using the Lagrangian and uses the Hamiltonian to

converge the control output. And we proposed the

correction method for the damping force and

frictional force. Finaliy the control output of EEC

becams almost same as the bang-bang controller's

output. Therefor the EEC can control the controlled

object in shortest time.

EEC was found to have the same phase space

structure as SMC and hence is included to the SMC.

However, the EEC has practical advantages in that

the phase plane can be simplified by choosing

energy as one of the axis variables in the phase plane

for the control output. In consequence, the output

switching line for EEC is the diagonal of gradient −1.

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