Page 1

ICIC Express Letters

Volume 3, Number 1, March 2009

ICIC International c °2009 ISSN 1881-803X

pp. 41—46

OPTIMAL INTEGRATED DESIGN FOR MECHANICAL STRUCTURE

AND CONTROLLER USING BARGAINING GAME THEORY

Yasuo Konishi, Nozomu Araki, Yoji Iwai and Hiroyuki Ishigaki

Graduate School of Engineering

University of Hyogo

2167 Shosha, Himeji-shi, Hyogo, 671-2201, Japan

konishi@eng.u-hyogo.ac.jp

Received September 2008; accepted November 2008

Abstract. This paper proposes the optimal integrated design method using the Nash

bargaining model and Krstic Extremum-Seeking method. The bargaining game theory is

often introduced to explain situations in economic activity. However, it is also applicable

to engineering problem. Applying the Nash bargaining model to integrated design prob-

lem, sevral design goals are mapped into utility functions defined by design parameters,

and the utility function for integrated design is easily constructed by the utility functions

of each design goals. Using this method, the integrated design problem is replaced as

the optimization problem that maximize the utility function within feasible parameter re-

gion. However, it is difficult to find the maximum of the utility function analytically.

To solve this difficulty, we use extremum seeking proposed by Krstic to find the maxi-

mum of the utility function. The effectiveness of our proposed method has been clarified

through integrated design simulation of mass-damper-spring system with proportional-

integral controller.

Keywords: Nash bargaining model, Game theory, Extremum seeking

1. Introduction. When a design includes several purposes, it is often difficult to find

the optimum compromise point rationally. There are several ways to set an utility func-

tion for integrated design. To generate this utility function, it is generally calculated a

weighted sum of multiple (two or more) indexes. However, this method requires the weight

constants to be determined by the designer’s experience and a trial and error process.

To solve this difficulty, we apply the Nash bargaining model for integrated design prob-

lem of mechanical structure and controller. Using the Nash bargaining model, the design

problem corresponds the n-person bargaining game, and it is easy to set an utility func-

tion for integrated design which is set by multiplying by the cost functions for all design

purposes [1]. However, this utility function often becomes a high-dimensional function, it

is difficult to find a design solution analytically. Consequently, we use Krstic Extremum

seeking (KES) [2] to find the maximum of the utility function. This method can be ef-

fectively used to find the maximum solution of a certain function which cannot be found

analytically. To study the effectiveness of our integrated design method, we design mass-

damper-spring system with proportional-integral controller imposed three design purposes

and verify the design result.

2. Bargaining Game Theory and Nash Bargaining Solution. Bargaining game

theory is a mathematics theory which studies how decision-making entity decides ratio-

nally and distributes equal profit to all members rationally on the ground that there are

some conflicts among the members. In Bargaining game theory, ”player” means a mem-

ber who participates in the bargaining, and ”payoff” means benefit barometer to a player

when he selects a strategy. Payoff function which represents a player gets the payoff when

41

Page 2

42Y. KONISHI, N. ARAKI, Y. IWAI AND H. ISHIGAKI

he selects one strategy is defined. The insist of all players in the bargaining game theory

is that one preferentially selects the best strategy that brings him the most benefit.

In order to obtain the good solution that can prove all player’s payoff in a bargaining

problem, we use the Nash bargaining model. This model can give an optimum point for

Pareto problem and it gives an unique solution. This characteristic is useful for solving

engineering problems. In the Nash bargaining model, the Nash solution is defined that

multiply all player’s profit-loss. The profit-loss means the difference between the payoff

which a player obtain if the bargaining is settled and the loss which a player lose if the

bargaining broke down. This solution gives an optimum compromise point by ”mixed

strategy”. Mixed strategy means a linear coupling constructed with several multiplying

a strategy and its possibility.

We assume a bargaining problem in which n players participate with the Nash bar-

gaining model. This bargaining model is that player i (i = 1,...,n) selects strategy si

(si∈ U, U : the class of all realizable strategies) at the possibility αi, and obtains a payoff

pi(si). Player i preferentially selects strategy siwhich maximizes his payoff pi(si). The

Nash bargaining solution is expressed by Eq.(1) using an expectation of payoff αipi(si).

Ã

j=1

max

n

Y

i=1

m

X

αjpi(sj)

!

, subject to

m

X

j=1

αj= 1, sj∈ U, αj≥ 0(1)

Note that the solution of Eq.(1) is a mixed strategy which uses possibility and cannot

be used for engineering problems in practice. Hence, we assume that all αiof siselect-

ing a strategy to be homogeneity, yielding an approximate Nash bargaining solution as

expressed Eq.(2).

n

Y

Thus, decision of the optimum design parameters for our integrated design means finding

the maximum of the utility function expressed by Eq.(2).

max

s∈U

i=1

pi(s)(2)

3. Krstic Extremum Seeking Method. We use KES to find the maximum of the

utility function constructed by the Nash bargaining model. The extremum seeking means

a method of finding input u∗that fills extreme value f∗of function f(u). Figure 1 shows

the block diagram of KES.

Figure 1. Block line chart of krstic extremum seeking

The feature of KES is that it calculates the inclination of function f(u) using the search

signal asin(ωt). The input u added to the search signal is substituted for the function

f(u). Then, the search signal changes the output y peturbatively. The Washout-Filter

detects the variation of output y, and calculates the inclination of f(u). In the loop of

KES, input u is automatically changed with reference to the inclination information to

Page 3

ICIC EXPRESS LETTERS, VOL.3, NO.1, 2009 43

approximate the extremum. Thus, using KES for a fixed time, input u and output y are

converged to u∗and f∗respectively.

4. Integrated Design Simulation.

4.1. Simulation setup and design goals. We introduce the integrated design simula-

tion used by the bargaining game theory. The design object is mass-damper-spring system

shown in Figure 2.

Figure 2. Integrated design object

The objective of this system is to control the position of mass y to the desired position yd

using PI controller with propotional gain Cpand integrator gain Ci. The design parame-

ters and range of each parameters considered in this paper are shown in Table 1.

Table 1. Design parameters

Design Parameter

Proportional Gain

Integrator Gain

Truck mass

Damper coefficient

Spring constant

Symbol

Cp

Ci

m

d

k

Range

0.1≤ Cp≤10.0

0.01≤ Ci≤2.0

0.6≤ m ≤2.0

0.3≤ d ≤ 3.0

0.5≤ k ≤ 1.5

Using design parameters shown in Table 1, the transfer function of this system is given

as follows

Cps + Ci

ms3+ ds2+ (k + Cp)s + Ci

Moreover, the design goals that designer wants to achieve are considered as follows.

(A): To make the settling time short. Therefore, the real part of all characteristic

roots are set -1 or less.

(B): The peak gain of G(jω) is assumed to be less than 1.2.

(C): The PI controller gain Cpand Ciare set as small as possible.

Above goals (A) and (B) are terms of the performance of the control system. On the

other hand, (C) is a goal that conflicts with the goal (A). In general, the convergence

performance of PI controller tend to improve as the gain Cpand Ciincrease. Therefore,

the goal (C) has the purpose to reduce the input force.

G(s) =

(3)

4.2. The utility function for integrated design. In order to do the integrated design

and consider all design goals, we set the utility functions used with design parameters

which represent each design purposes. In order to deal all design goals as equal, the range

of the utility functions is from 0 to 1.

First, we set the utility function representing the design goal (A). If a set of design

parameters ν = [Cp Ci m d k]>is decided, we can calculate the characteristic roots

of transfer function G(s) in Eq.(3). Here, the maximum value of the real part of the

Page 4

44Y. KONISHI, N. ARAKI, Y. IWAI AND H. ISHIGAKI

characteristic roots about design parameters ν is assumed to be λ(ν). To achieve (A), the

utility function has to set that the evaluated value of the utility function decrease when

λ(ν) > −1. Therefore, we define the utility function P(ν) for (A) as follows:

(

1

P(ν) =

|λ(ν)|10

−1 < λ(μ) < 0

λ(μ) ≤ −1

. (4)

Second, we set the utility function representing the design goal (B). Using the transfer

function G(s) in Eq.(3), the peak gain Gmaxcan be expressed as

Gmax= max

ω∈[0,∞)|G(jω)|.

Here, we assume that the peak gain of transfer function G(s) for the design parameter ν

is given by Gmax(ν). Using Gmax(ν), the utility function Q(ν) for the design goal (B) is

defined as following equation.

(

exp(1.2 − Gmax(ν))

Finally, we define the utility function for the design goal (C). (C) gives the condition

that the PI controller gain Cpand Ciare set as small as possible. Therefore, we simply

define following monotonically decreasing functions as the utility functions for design

parameters Cpand Ci.

Q(ν) =

1Gmax(ν) < 1.2

otherwise

(5)

R(ν) = 1 − 0.04Cp= 1 − [0.04 0 0 0 0] · ν

S(ν) = 1 − 0.2Ci= 1 − [0 0.2 0 0 0] · ν

From Eqs.(4)-(7), we define the utility function for the integrated design using the

bargaining game theory. In the game theory, each utility function P(ν), Q(ν), R(ν) and

S(ν) for our design goals are corresponded to the payoff pi(s) in Eq.(2). Therefore, using

the Nash bargaining model to this design problem, the utility function J(ν) for integrated

design can be written as follows:

(6)

(7)

J(ν) = P(ν) · Q(ν) · R(ν) · S(ν)(8)

Then, our optimal solution of integrated design can be obtained to find the parameters

maximized Eq.(8) within the range given by Table 1.

In this paper, we applied KES to calculate the optimal parameters. The KES block

diagram used to solve the integrated design problem is shown in Figure 3.

Figure 3. The KES block diagram used to solve the integrated design problem

Page 5

ICIC EXPRESS LETTERS, VOL.3, NO.1, 2009 45

Using the KES shown in Figure 3, the output y is convergence to the extremum value of

utility function J(ν). Therefore, the parameter ν at steady state condition of the system

shown in Figure 3 becomes our optimal parameter.

4.3. Simulation result. We carried out the integrated design simulation using proposed

mathod. Table 2 shows calculation result of optimal parameters obtained from KES.

Table 2. Optimal parameters obtained from KES

Parameter

Proportional Gain

Integrator Gain

Mass

Damping Coefficient

Spring Constant

Symbol Result

Cp

Ci

m

d

k

1.08

1.008

0.6

2.172

1.5

From optimal parameters shown in Table 2 and Eq.(3), the transfer function that is

obtained by proposed integrated design method given as

G(s) =

1.08s + 1.008

0.6s3+ 2.172s2+ 2.58s + 1.008.(9)

And the characteristic roots of above system is [−1.5 −1.12 −1.0]>. Therefore, it is

obvious that the design goal (A) is achieved. Moreover, Figure 4 shows the gain charac-

teristic curve of Eq.(9).

Figure 4. The gain characteristic curve of Eq.(9)

where, the peak gain Gmaxof this system is 1.008. Thus, the parameters shown in Table

2 also achieved the design goal (B).

According to the design goal (C), the value of controller gain Cpand Ciare 1.08 and

1.008, respectively. Especially, the feasible region of proportional gain Cpis 0.1 ≤ Cp≤ 10,

it is sufficiently small. To verify this proportional gain Cp, we carried out the control

simulation of the system shown in Figure 2. In this simulation, we assumed that initial

value is y0= 0 and desired value is yd= 1.0. Furthermore, we employed the proportional

gain Cp= 0.5,2.0 for comparison. The time history of control system output is shown in

Figure 5.

As shown in Figure 5, if Cp is smaller than the design result, the output response

occurs overshoot. Oppositely, if Cp is greater than the design result, the convergence

of the output response to the desired value becomes slow. This fact indicate that our

proposed design method can select the parameters suitable for human sense.