ICIC Express Letters
Volume 3, Number 1, March 2009
ICIC International c °2009 ISSN 1881-803X
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
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-
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 . 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)  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
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).
, subject to
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
Thus, decision of the optimum design parameters for our integrated design means finding
the maximum of the utility function expressed by Eq.(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
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
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
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.
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
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 < λ(μ) < 0
λ(μ) ≤ −1
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
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.
1Gmax(ν) < 1.2
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:
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
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
From optimal parameters shown in Table 2 and Eq.(3), the transfer function that is
obtained by proposed integrated design method given as
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
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.