Simultaneous Design of Structural and Control Systems using

Set-based Design Method

Haruo Ishikawa

and Naoko Sasaki

Graduate School of Informatics and Emgineering, The University of Electro-Communications, Chofu, Tokyo, Japan

Center for International Programs and Exchange, The University of Electro-Communications, Chofu, Tokyo, Japan

Keywords: Simultaneous Design, Structural System, Control System, Set-based Design.

Abstract: The design method capable of considering the influence factors related to control and structural design

systems under conditions that simultaneously satisfy multi-objective performances over both the systems is

investigated. The influence factors and performances have some kinds of uncertainty related to structural

design and control system design. The uncertainty may be expressed in terms of set interval. Set-based

design method is available as a design method that can take account of such uncertainty. In the method,

instead of optimization, the concept of satisficing is used. In the present study, the applicability of set-based

design method that has been studied in the field of structural design is investigated for the simultaneously

satisficing design of control and structural systems. For discussing the applicability, an example problem of

inverted pendulum with heteromorphic shape on cart is solved by optimal regulator method in modern

control theory. As a result, the set intervals of influence factors which simultaneously satisfy the set

intervals of multi-objective performances are obtained. From this result, the usefulness of set-based design

method for simultaneous design of control and structural systems can be confirmed.

1 INTRODUCTION

In the development of mechanical/structural system,

control function plays a great role in the system, and

optimal design of control system and structural

system that have deep interaction with each other is

considered to be of great importance. Therefore, in

product development, the simultaneity of optimum

design of both systems is indispensable from the

viewpoints of realization of better product

performances, short lead time and low cost for the

development of product (Mehra, R.K., 1976;

Kubrusly C.S. et al, 1985; Kabamba P.T. et al, 1983;

Soong, T.T., 1990). Main previous studies have

treated with mathematical optimization or coupled

computation by CAE which are based on point-

based calculation. The simultaneous optimization

problems have often been formulated for the

relatively simple cases, as the mathematical

minimization/maximization problem of evaluation

function (Kajiwara, I.,1994, Hara, F., 1992, Hatano,

H., 1993, Khot, N.S. et al, 1993, Nahm, Y. -E. and

Ishikawa, H., 2006). Generally speaking, in some

practical cases (Nakaminami, M., et al, 2007),

specified control target objects (structures) were

designed in advance, and in the other cases, in

addition to the characteristics of control system,

weight or damped oscillation characteristics that is a

factor governing the characteristics of structures is

considered later (Obinata, G., 1997). However, in

general, physical mechanisms of the mutual

interaction of both systems of structure and control

are still remained obscure. There are various issues

to be considered, such as types and number of

design variables, structural and control

characteristics of product, a large number of

restrictions for design, meta model relating design

variables and characteristics of product, complexity

and multi-peak solutions of evaluation function, and

dynamic characteristics governed by motion

equation with a large number of freedom that affects

the interaction and so on. Due to the complexity of

the problems, in the simultaneous design problem of

the structural and control systems, a large number of

design factors simultaneously satisfying multiple

conditions should be obtained. Conventionally,

practical design methodology for such situation has

been based on the point-based repetition design

procedure.

284

Ishikawa, H. and Sasaki, N.

Simultaneous Design of Structural and Control Systems using Set-based Design Method.

DOI: 10.5220/0006905502840289

In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 1, pages 284-289

ISBN: 978-989-758-321-6

On the other hand, it is general and essential that

the required performances and the influence factors

for them have some kinds of uncertainty.

Uncertainty especially in the initial stage of

structural design is based on imperfections of related

knowledge, influences from other design sectors or

customers and so on. In control design, there is

essential uncertainty due to the difference between

structural model and mathematical model for control.

In case of classical control design, as the typical

example of uncertainty of control design, it can be

said that feedback gains, like proportional gain,

integral gain and derivative gain, have uncertainty in

PID feedback control. The values of these gains can

be independently given for control object, and it is

necessary for each gain to determine an optimum

value that represents good performance regarding

stability of control. Namely, the gains are adjustment

factors. Also, in modern control theory, for

examples, turn over design method has the variation

in the position of folding line by which response

characteristics of control change and optimum

regulator control method has the variation in the

values of diagonal terms of weight matrix of

evaluation function to realize high stability of

control. Actually, the position of folding line in the

former example and the values of diagonal terms in

the latter example are decided by trial and error.

Then, these variations can be regarded as certain

kind of uncertainty.

In this research, for the expression of the

uncertainty, we use aggregate-based method rather

than point-based method. Also, as stated above, the

design system is often multi-objective, whether

structural system or control system. In the design of

structure (machine) system, there are multi-

objectives, like rigidity, strength, light weight,

compactness, cost, impact on environmental burden,

velocity, accurate positioning, Kansei performance

like comfort and so on. In case of control design

systems, generally, there are plural performances of

control stability, like rising time, maximum

overshoot rate, settling time and so on. Then, it is

important to investigate the simultaneously

satisficing multi-objective design problem in

consideration of uncertainty.

On the other hand, set-based design method has

been studied as the simultaneously satisficing

method for multi-objective design considering the

uncertainty in structural (mechanical) design field

(Nahm, Y. -E. and Ishikawa, H., 2005, Nahm, Y. -E.

and Ishikawa, H., 2006, Nahm, Y. -E. and Ishikawa,

H., et al, 2006,). In the present study, PSD

(Preference Set-Based Design) method that is one of

the set-based design methods is used. The concept of

preference that is given by designer is used to find

subsets that show higher satisfaction and robustness

of solutions. In the present study, the applicability or

potential of PSD method is studied by using an

example problem which is a virtual inverted

pendulum with non-uniform shape on cart. This

problem is solved by optimal regulator formulation

in modern control theory. The inverted pendulum

with non-uniform shape (the size is change) and

holes in the inside is adopted to emphasize the

structural factors of the design. Depending on these

factors, the position of the moment of inertia of

pendulum which is one of the control factors varies.

In other words, it can be said that the simultaneous

design problem like this time implies the design of

structural system (mechanical system) that realizes

the performances of control system. In this problem,

five design variables related to structural and control

systems are adopted. Three of them are structural

variables and two are control related variables. Two

performance objectives such as transient

characteristic of control (settling time) and

lightweight of structure are adopted.

In this research, MATLAB is used for simulation

of control system design (MATLAB is registered

trade mark of MathWorks, USA).

2 PREFERENCE SET-BASED

DESIGN (PSD METHOD)

“Preference” is attached to the name of the method

because “preference” includes one of the basic

concepts of the method. The outline of PSD method

is shown below.

Figure 1: Concept of preference set-based design method.

Simultaneous Design of Structural and Control Systems using Set-based Design Method

285

The PSD method consists of (1) set representation,

(2) set propagation and (3) set narrowing. These

concepts of the PSD method are represented as the

different layers in Figure 1, respectively. Also, the

procedure of the method is illustrated in Figure 2. In

this study, the solution space for structural

requirements and control design performances are

aligned and a final narrowed interval set of design

solution is identified. In the process, instead of the

elimination of those uncertainties by repeatedly

correcting point values of adjustment parameters, in

the present study, preference set-based design

method is used. That is, set solution candidates are

evaluated by the concept of satisfaction and

robustness defined by set interval with preference

and the subsace of low evaluation is eliminated.

Each process of PSD method, shown in Figure 2, is

as follows;

(1) The set representation is to express uncertainty

and variability of performances and design

variables in terms of set concept.

(2) The set propagation means the set mapping of

design variables to performance set, based on

the relationship (equation of theory, numerical

calculation or experiment) between design

variables and performances.

(3) In the set narrowing, first, overlap region

between the mapped result and the initially set

of performance is obtained. Next, the common

set of the overlap region for each performance

is found. Finally, this approach narrows the

common set of design variables by eliminating

infeasible design subspaces by using the

concept of satisfaction and robustness. A

detailed description of the PSD method can be

found in references (Nahm, Y. -E. and

Ishikawa, H., 2005; Nahm, Y. -E. and Ishikawa,

H., 2006).

3 SIMULTANEOUS DESIGN OF

STRUCTURAL AND CONTROL

SYSTEMS

An example problem of PSD method for the

simultaneous design of structural and control

systems is virtual problem of inverted pendulum on

cart, shown in Figure 3. The pendulum has non-

uniform (step type) outline shape with internal holes

as structural factors. That is, the outer shape and

dimensions of the pendulum, and the number of

holes are design variables. In other words, it can also

be said that the design problem of the shape and

Figure 2: Procedure of preference set-based design method.

Figure 3: A virtual example problem of inverted pendulum

on cart.

dimensions of pendulum is a problem of obtaining

structure design solution that realizes the control

performance of pendulum on cart. Although this

problem is certainly not a realistic issue, it is suitable

for examining the usefulness of the method. There

has been no trial of application of concept and

approach of set-based design method to control

system design itself and simultaneous design of

control and structural systems, the present study is

the first one.

By the linearization of the equations of motion

of pendulum and cart around a certain value of

rotation (θ=0) of pendulum, the equations are given

as follows,

(1)

where M, x and f are mass, position and driven force

of cart, respectively, and m, l, J, θ and g are mass,

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length from rotation fulcrum to centroid of

pendulum, rotation angle of pendulum and

gravitational acceleration, respectively.

Mathematical model given by equation (1) is

expressed as state equations (2) and (3).

(2)

(3)

where A and B are matrix of 4 row by 4 column and

matrix of 4 row by 1 column, respectively. In the

present problem, mathematical model given by

equation (2) has controllability and observability.

Then, the concept of state feedback control and its

formulation of optimal regulator are applicable to

the inverted pendulum problem on cart which is

inherently unstable control system. Conceptual

diagram of optimum regulator system is shown in

Figure 4. In Figure 4, K is state feedback gain matrix.

According to the optimal regulator method, optimal

control input that minimizes evaluation function can

be obtained. The evaluation function is given by

(4)

In equation (4), Q and R are weighting factor

matrices which are given by the equations,

(5)

Here, diagonal terms of diagonal matrix, Q, are

assumed to be equal.

Weighting factor matrices, Q and R, are

currently determined by trial and error, because the

relationship between the weighting factors, Q and R,

and the response of control has not been elucidated.

That means they cannot be definitely determined as

point values at the beginning of control design. That

is, they have each fluctuation range, because they

are independent each other.

By solving Riccati algebraic equation under the

provisional values of Q and R, it is possible to

obtain the response of closed loop system under

initially given position value of cart.

Figure 4: Optimum regulator system.

4 APPLICATION RESULTS

The preference set-based design was carried out

using the initial sets of five design variables and two

kinds of performances (settling time on stability of

pendulum and lightweight of structure). The settling

time is defined as the time when the swing angle of

pendulum falls within ±0.01 (rad). Among the five

design valuables, three are structural variables, the

length l

and width h

of the upper part of pendulum

and the discrete number n of holes in the pendulum,

shown in Figure 3. The diameter of hole is fixed as

φ=0.04 (M). The remaining two are control system

variables that are the weighting factors of evaluation

function (diagonal matrix), Q and R in equation (4).

Thus, their diagonal terms, q1 and r1 in equation (5)

are also design variables. The length l

and width h

of the lower part of pendulum are fixed as l

=0.2

(M) and h

=0.06 (M).

Using the initial interval sets of the design

variables and required performances, shown in Table

1, the procedure of PSD method in Figure 2 is

applied. Table 2 shows that all the design variables

are narrowed so as to realize the set range of two

performances which simultaneously satisfy the

narrowed performances. This can be understood

from the algorithm of PSD method. As an example

of concrete results, the interval set of the discrete

number of holes, shown in the first row of Table 2,

is narrowed from the interval set, [2, 9], to the

interval set, [7, 9].

Table 1: The initial interval sets of design variables and

required performances.

Design variables

n [2, 9]

[0.5, 0.8] (M)

[0.06, 0.2] (M)

Q(q1) [1, 9]

R(r1) [0.1, 1.5]

Required performances

settling time, T

≦3.5 (sec)

mass, m

≦0.12 (kg)

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287

Table 2: Narrowed set of design variables and

performance variables.

Design variables

n [7, 9]

[0.650, 0.725] (M)

[0.130, 0.165] (M)

Q(q1) [7, 9]

R(r1) [1.15, 1.5]

Required performances

settling time, T

[3.485, 3.50] (sec)

mass, m [0.073, 0.111] (kg)

5 CONCLUSION

Various kinds of variables related to structural

design and control design have uncertainty, such as

lack of knowledge, influence from other sector or

customer, certain weight factor to be subjectively

determined and so on. In the present study, the

uncertainty is expressed by set interval. Then, the

uncertainty of design can also be considered as

adjustment factor. In the present study, the set-based

design method capable of considering the

uncertainties related to the state feedback controller

in control system design and the shape/size of

controlled object in structural system design at the

same time and satisfying the multiple design objects

as well is investigated. The design method is

different from the traditional point-based design

method. In PSD (Preference Set-based Design)

method which is one of the set-based design method,

the design variables and performance variables are

represented by interval set and solution sets of multi-

objective performances can be obtained considering

subspaces with higher satisfaction and robustness of

required performances.

In order to consider the applicability of PSD

method, a virtual inverted pendulum problem on cart

controlled by optimum regulator method in modern

control theory is studied. In the example problem,

five design variables (three and two design variables

for structural system and control system,

respectively), and two required performances (one is

performance for structural system and another for

control system) are selected. As a result of the

application, the effectiveness of PSD method could

be confirmed.

6 FUTURE WORK

Future study based on set-based design method can

be expected from various viewpoints including the

application of other control theory. For example,

turn over design method in modern control theory

has the variation in the position of folding line. By

using the position as a design variable, it is possible

to appropriately select pole placement that desires

transient characteristics. Based on these control

theories, simultaneous design of control and

structural systems will be discussed. Although not

discussed in this paper, we will consider the

influence of fluctuation of the preference in the PSD

method on the results of interval set solutions.

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