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# Developing a fuzzy proportional-derivative controller optimization engine for engineering design optimization problems

## Abstract

In real-world engineering design problems, decisions for design modifications are often based on engineering heuristics and knowledge. However, when solving an engineering design optimization problem using a numerical optimization algorithm, the engineering problem is basically viewed as a pure mathematical problem. Design modifications in the iterative optimization process rely on numerical information. Engineering heuristics and knowledge are not utilized at all.

In this paper, the optimization process is analogous to a close loop control system, and a fuzzy proportional-derivative (PD) controller optimization engine is developed for engineering design optimization problems with monotonicity and implicit constraints. Monotonicity between design variables and the objective and constraint functions prevails in engineering design optimization problems. In this research, monotonicity of the design variables and activities of the constraints determined by the theory of Monotonicity Analysis are modeled in the fuzzy PD controller optimization engine using generic fuzzy rules. The designer only needs to define the initial values and move limits of the design variables to determine the parameters in the fuzzy PD controller optimization engine.

In the optimization process using the fuzzy PD controller optimization engine, the function value of each constraint is evaluated once in each iteration. No sensitivity information is required. The fuzzy PD controller optimization engine appears to be robust in the various design examples tested in this research.

Keywords: design optimization, monotonicity analysis, fuzzy control.

## 1.     Introduction

Engineering design optimization problems often have two characteristics. First, monotonicity between design variables and the objective and constraint functions prevails in engineering design optimization problems. The theory of Monotonicity Analysis (Papalambros and Wilde, 2000) was developed to analyze this type of problems to identify which constraints must be active at the optimum design point. However, numerical optimal solution cannot be obtained by Monotonicity Analysis alone.

Secondly, most real-world engineering design problems have complex phenomena, and the objective and constraint functions often cannot be expressed analytically in terms of design variables. These so-called “implicit functions” are evaluated by computer simulation or by physical experiments, which are usually the major cost of the optimization process. Moreover, the analytical forms of the first derivatives of the implicit functions are often not available. This fact hinders the use of formal numerical optimization techniques to solve these engineering design optimization problems.

An iterative optimization process can be analogous to a close-loop control system. Figure 1 shows the block diagram of a close-loop control system. The measured response of the system process being controlled is fed back and compared with a desired response. The control actions generated by the controller are determined in an attempt to fix the error between the system response and the desired response.

Figure 1. A close-loop control system

Figure 2 shows a block diagram for an optimization process. Initial parameters are input to the optimization algorithm, which in turn generates a trial design point according to its search rules. The optimization model is then evaluated at this trial design point, and the information such as objective and constraint function values and sensitivities are fed back to the termination test. If the termination test fails, the optimization algorithm is triggered again to generate the next design point, using the numerical information from previous iterations. Finally, while a control system attempts to achieve a stable, predefined output, the optimization process pursues a converging objective function value. Comparing Figure 1 and 2, an optimization model in an optimization model is analogous to the system process in a control system, while an optimization algorithm is analogous to the controllers.

Figure 2. Block diagram for an optimization process.

Traditional numerical optimization algorithms are analogous to direct digital controllers. The algorithms are usually ‘crisply’ designed for well defined mathematical models, and their numerical rules for generating the next design point are exact and definite. In real-world engineering design problems, decisions for design modifications are often based on engineering heuristics and knowledge. However, when solving an engineering design optimization problem using a numerical optimization algorithm, the engineering problem is basically viewed as a pure mathematical problem. Design modifications in the iterative optimization process rely on numerical information. Engineering heuristics and knowledge are not utilized at all.

This motivates the idea that, in addition to crisp numerical rules, the engineering heuristics and knowledge should also be modeled in an optimization algorithm using fuzzy rules. As suggested in Figure 2, the ‘controller’ in the optimization process may as well be a fuzzy controller!

In this research, a fuzzy proportional-derivative (PD) controller optimization engine is developed to deal with general engineering design optimization problems with monotonicity and implicit functions. In particular, monotonicity of the design variables and activities of the constraints determined by Monotonicity Analysis are modeled in the fuzzy PD controller optimization engine using generic fuzzy rules. The structure of an optimization algorithm is still maintained to guide the engineering decision process.

Section 2 of the paper further discusses the concept of the fuzzy PD controller optimization engine. Section 3 describes the process of using the fuzzy PD controller optimization engine, and Section 4 uses an example to illustrate the process. In Section 5, the fuzzy PD controller optimization engine is used to solve several engineering design optimization examples commonly seen in literature. Finally Section 6 concludes the paper.

## 2.     The concept of the fuzzy PD controller optimization engine

### Application of fuzzy theory in optimization problems

Fuzzy theory is primarily concerned with quantifying and reasoning using natural language in which many words have ambiguous meanings. Since the introduction of the basic theory of fuzzy sets by Zadeh (1965), fuzzy theory has been extended and applied to many different fields, including engineering design optimization.

The application of fuzzy theory in optimization problems can be categorized in two dimensions:

#### (1)   Using fuzzy theory to formulate the uncertainty of the optimization model

Bellman and Zadeh (1970) introduced the fuzzy set-based optimization on decision making in a fuzzy environment which includes the concept of fuzzy constraint, fuzzy objective and fuzzy decision. For fuzzy optimization, the objective and constraint functions are characterized by the membership functions in a fuzzy system. The decision can be viewed as the intersection of the fuzzy objective and constraint functions. Once the membership functions are known, the optimization problem can be viewed as a crisp optimization problem.

Different kinds of mathematical models have been proposed to solve fuzzy optimization problems in various engineering fields. Rao et al. (1992) and Xiong and Rao (2003, 2005) applied the concept of fuzzy optimization on the design optimization of mechanical and structural systems. Guan et al. (1995) presented the application of a fuzzy optimization technique to optimal power flow calculations. Sarma and Adeli (2000) presented a fuzzy discrete multi-objective optimization model for space steel structures design, subjected to constraints of commonly-used design codes. Inuiguchi and Ramik (2000) reviewed some fuzzy optimization methods and techniques from a practical point of view.

#### (2)   Implementing fuzzy theory into the optimization algorithms

In this research area, fuzzy theory is used to adjust or control the parameters of the numerical optimization algorithm, such that engineering knowledge and human supervision process is integrated into the optimization process.

Arakawa and Yamakawa (1990) demonstrated an optimization method using qualitative reasoning, which makes use of qualitative information to give an approximate direction of the optimum search. Trabia and Lu (2001) proposed a fuzzy adaptive simplex search optimization algorithm to minimize a function of n variables. This method uses fuzzy logic to decide the next move of the simplex.

Hsu et al. (1995a, b) proposed a fuzzy algorithm for determining the move limits in sequential linear programming algorithm. Arabshahi et al. (1996) pointed out that many optimization techniques involve parameters that are often adapted by the user through trial and error, experience, and other insight. Instead, they applied neural and fuzzy ideas to adaptively select these parameters. Mulkay and Rao (1998) proposed a modified sequential linear programming algorithm using fuzzy heuristics to control the optimization parameters.

Some researchers also tried to develop intelligent optimization algorithms based on fuzzy theory to solve optimization problems. Xiong and Rao [2003] combined fuzzy l-formulation with a hybrid genetic algorithm to solve the mixed-discrete design optimization problem. Mukuda et al. (2005) combined a fuzzy logic controller with a hybrid genetic algorithm and a local search technique to solve the multi-objective optimization problems.

The research presented in this paper falls to the second category. In implementing fuzzy theory into the optimization algorithms, most research focused on implementing optimization process knowledge into the numerical optimization algorithms using fuzzy theory. Few researches emphasize on implementing engineering knowledge to develop optimization algorithm specifically for solving certain type of engineering optimization problems.

### Fuzzy PD controller

Fuzzy control is similar to the classic closed-loop control approaches, but differs in that it substitutes imprecise, symbolic notions for precise numeric measures. The fuzzy controller takes input values from the real world. These crisp input values are mapped to the linguistic values through the membership functions in the fuzzification step. A set of rules that emulates the decision-making process of the human expert controlling the system is then applied using certain inference mechanism to determine the output. Finally the output is mapped into crisp control actions required in practical applications in the defuzzification step.

In the close-loop control system shown in Figure 1, the measured response of the system process being controlled is fed back to be compared with a desired response. The control actions generated by the controller are determined in part by the system response in an attempt to fix this error. The output of a proportional controller is a control signal u which is proportional to the error expressed in Equation (1), where Kp is the gain.

(1)

Derivative control is used to anticipate the future behavior of the error signal by using corrective actions based on the rate of change in the error signal. The output of a derivative controller is a control signal u which is proportional to the derivative of error expressed in Equation (2), where Kd is the gain.

(2)

The control action in a PD controller is in the form of Equation (3), which combines proportional and derivative control modes. The PD controller makes a control loop respond faster and with less overshoot, and is the most popular method of control by a great margin.

(3)

The fuzzy counterpart of the PD controller also has two inputs: system error e and error change . Fuzzy inference is used to compute the control signal u. Table 1 shows a typical rule base for a fuzzy PD controller with 25 rules. Five linguistic terms are used for each variable, NB (Negative Big), NS (Negative Small), ZE (Zero), PS (Positive Small), and PB (Positive Big). For example, the 5th rule in Table 1 (row 1, column 5) states that, ‘IF e is NB AND  is PB, THEN the control action is NS’.

Table 1. A typical rule base for a fuzzy PD controller (25 rules)

 e PB PS ZE NS NB PB PB PB ZE NS NS PS PB PS ZE NS NS ZE PS PS ZE NS NB NS PS PS ZE NS NB NB PS PS ZE NB NB

### The concept of the fuzzy PD controller optimization engine

As discussed in the previous section, the iterative optimization process can be analogous to a close-loop control system. In the authors’ previous work (Hsu et. al, 2005), the fuzzy PD controller was used as the optimization engine in an optimization process to handle the specific type of objective function in Equation (4),

min.                                                                      (4)

that is, to minimize the differences between system outputs yi and the target values Yi. As shown in Figure 3, initial errors e and change of errors De are input to the fuzzy PD controller, which generates the changes of design variables Dx for the next iteration. Then, the system inputs are updated (xq+1 = xq + Dxq) and the new system process outputs yq+1 are fed back to compare with the set point Y again.

System outputs y are functions of design variables x. In engineering design optimization problems, the monotonicity of design variables in system outputs are often known empirically. This monotonicity is modeled in the fuzzy PD controller optimization engine in order to correctly update the design variables in the next iteration. The optimization process terminates when e and De approach zero, that is, when no change in design variables Dx will be generated by the fuzzy PD controller optimization engine.

Figure 3. Using the fuzzy PD controller in an optimization process

The objective function in Equation (4) is rather restricted. In the next section, the application of the fuzzy PD controller optimization engine is extended to more general engineering optimization problems with monotonicity and implicit functions.

## 3.     Using the fuzzy PD controller optimization engine for engineering design optimization problems

As discussed earlier, monotonicity between design variables and the objective and constraint functions prevails in engineering design optimization problems. Monotonicity is often known through engineering knowledge or experience without knowing the explicit mathematical form of the optimization model. The idea of analyzing the monotonicity of objective function and constraints was first introduced by Wilde (1975) for checking the model boundedness. Papalambros (1979) then developed Monotonicity Analysis as a generalized systematic methodology. Monotonicity Analysis seeks to identify rigorously, in advance to any extensive numerical computation, in which combinations of inequality constraints can be active. The active constraints are to be satisfied with strict equality at the optimum.

There are two Monotonicity Principles (Papalambros and Wilde, 2000). The First Monotonicity Principle (MP1) states that, ‘in a well-constrained minimization problem, every (strictly) increasing (decreasing) variable is bounded below (above) by at least one active constraint’. The Second Monotonicity Principle (MP2) deals with variables which are not in the objective function: ‘every monotonic nonobjective variable in a well-bounded problem is either (a) irrelevant and can be deleted from the problem together with all constraints in which it occurs, or (b) relevant and bounded by two active constraints, one from above and one from below’.

Figure 4 shows the conceptual structure of how to use the output of Monotonicity Analysis to construct the fuzzy PD controller optimization engine in this research. In general, Monotonicity Analysis can be carried out by explicit algebraic substitution for the constraints that are identified active. In this research, Monotonicity Analysis is done by a computer program MONO developed by Hsu [1993], which implements Monotonicity Analysis into an automated computerized process.

The input of MONO is a monotonicity table which only contains the monotonicity signs of the design variables with respect to the objective function and constraints. The explicit mathematical form of the optimization model is not required. In MONO, the Monotonicity Principles are applied to the design variables one by one. When a constraint is identified as a critical constraint by the Monotonicity Principles, MONO uses “implicit elimination” to eliminate the critical constraint using only the monotonicity signs in the monotonicity table. The monotonicity table is then updated according to the implicit elimination rules, and the size of the table is reduced. The new table is then passed to the next analysis cycle.

In some cases, the monotonicity of a design variable with respect to the objective function or certain constraint cannot be determined after implicit elimination. The monotonicity sign of the variable in the monotonicity table is marked ‘i’ for ‘indeterminate’. MONO terminates when all design variables have been analyzed, or the monotonicity table has only ‘i’ in the objective function row and cannot be analyzed further. Details of implicit elimination are described in Hsu (1993).

Figure 4. The conceptual structure of using the output of Monotonicity Analysis to construct the fuzzy PD controller optimization engine

The complete output from MONO includes rigorous Monotonicity Analysis steps, the reduced monotonicity tables after each step, and the monotonicity analysis results. After Monotonicity Analysis, some of the constraints are identified as ‘critical constraints’ by MP1. These constraints have to be satisfied with strict equality at the optimum. If more than one constraint can be critical for a design variable x by MP1, these constraints form a ‘conditionally critical set’ (Papalambros and Wilde, 2000). That is, at least one of the constraints in the set must be critical at the optimal design point. Some constraints are identified as ‘uncritical’ by MP1, while other constraints are identified as ‘irrelevant’ by MP2. These constraints can be temporarily eliminated from the optimization model. Finally, some constraints may remain undecided by either MP1 or MP2.

As shown in Figure 4, the inactive (uncritical or irrelevant) constraints are temporarily eliminated from the optimization model. It is necessary to check the inactive constraints again after the optimal design point is obtained. The critical, conditionally critical and undecided constraints, as well as the monotonicity signs of the design variables in the original objective function are implemented into the fuzzy PD controller optimization engine.

There are 4 possible situations of implementing the analysis results from MONO into the Fuzzy PD controller optimization engine:

#### (1)   For design variables with only one critical constraint

For a design variable x with only one critical constraint gk, the change of the design variable Δx in the next iteration is decided by the current value of gk. Since the target value for gk is zero, e and  of constraint gk can be calculated, and are converted into fuzzy membership in the fuzzyfication step using a ‘quantization table’ defined by the user. The definition of the quantization table will be described in details in the following section. From the two inputs, the membership of control action contributed by constraint gk can be determined using a rule base similar to Table 1. Finally, Δx in the next iteration is calculated by:

Δx = defuzzification(mgk(x)),                                                                      (6)

where mgk is the membership of control action contributed by the critical constraint gk, and ‘defuzzification( )’ defuzzifies the membership function into crisp value of Δx, again using a quantization table.

#### (2)   For design variables with a conditionally critical set

For a design variables x with a conditionally critical set, at least one of the constraints in the set must be critical at the optimal design point. In this situation, the change of the design variable Δx in the next iteration is decided by the current values of these constraints. At the current design point, if constraints g1, g2, …, gi in the conditionally critical set are violated, Δx is decided by

Δx =defuzzication                                      (7)

The ‘max’ and union operators are used to pick the strongest membership among all violated constraints in order to force the design point back to the feasible domain. On the other hand, if all constraints are satisfied, Δx is decided by the monotonicity sign of x in the original objective function. In this case, Δx will be equal to the move limit defined by the user in the proper direction which reduces the value of the objective function.

#### (3)   For design variables whose monotonicity signs are ‘indeterminate’

For a design variable x whose monotonicity sign is ‘indeterminate’ in the objective function of the final monotonicity table output by MONO, all constraints in which the variable appears can be active. Thus, the change of the design variable Δx in the next iteration is decided by the current values of these constraints in the same fashion described in (2).

#### (4)   For design variables that do not appear in the objective function

If a design variable does not appear in the objective function, and is proved to be ‘relevant’ by MP2, all constraints in which the design variable appears can be active according to MP2. Thus, if some constraints are violated, Δx is decided by the current values of the violated constraints in the same fashion described in Equation (7). On the other hand, if all constraints in which the non-objective variable appears are satisfied, the amount of change of the design variable Δx for the next iteration is zero to maintain the current value of the design variable.

In the following section, an air tank design optimization example is used to describe the complete process of applying the fuzzy PD controller optimization engine on general engineering optimization problems with monotonicity and implicit constraints.

## 4.     An air tank design optimization problem

### Monotonicity Analysis using MONO

Figure 5 and Equation (8) show the optimal design problem for a cylindrical air tank (Papalambros and Wilde, 2000, Unklesbay et al., 1972). The objective is to minimize the quantity of material used, which depends on the inner radius r, the shell thickness s, the shell length l, and the head thickness h. The volume of the tank has to be larger than the specified volume (constraint g1); the thicknesses of the head and the wall have to satisfy the ASME code (g2, g3), and there are constraints on the size of the tank (g4, g5, g6).

Figure 5. The air tank design problem

(8)

Table 2 is the corresponding monotonicity table for Equation (8) which is the only input required for the computer program MONO. In the air tank design optimization example, MONO concludes that constraints g1, g3, and g2 are critical to design variables r, s, and h, respectively. The monotonicity sign of design variable l is ‘indeterminate’ in the objective function of the final monotonicity table in the output from MONO. Note that this analysis result is obtained using only the monotonicity table in Table 2. Explicit formulations of the objective function and the constraints are not required. The complete Monotonicity Analysis of this example using explicit substitution of the active constraints can be found in (Papalambros and Wilde, 2000).

Table 2. Monotonicity table of the air tank example

 r s l h F ＋ ＋ ＋ ＋ g1 － . － . g2 ＋ . . －. g3 ＋ － . . g4 . . － . g5 ＋ ＋ . . g6 . . ＋ .

### Preparing the fuzzy PD controller

Four simple generic rules are used to describe how to adjust the values of the monotonic design variables in order to find the design point that satisfies the critical (active) constraints with strict equality. For monotonically increasing variables,

l          Rule I: ‘IF the constraint function value is positive, THEN the variable must be decreased’.

l          Rule II: ‘IF the constraint function value is negative, THEN the variable must be increased’.

Similarly, for monotonically decreasing variables,

l          Rule III: ‘IF the constraint function value is positive, THEN the variable must be increased’.

l          Rule IV: ‘IF the constraint function value is negative, THEN the variable must be decreased’.

For example, constraint g1 is critical for design variables r, which is monotonically decreasing with respect to g1. At the initial design point, if  is positive, design variable r must be increased by Rule III in the next iteration so that g1 will be pushed to zero.

The function value of the critical constraint (such as g1) is the errors e in the fuzzy PD controller optimization engine. The error change Δe (the change in the constraint function values) reflects the trend of the constraint function. Considering the error change, the rules can be extended as follows:

For monotonically increasing variables,

l          Rule Ia: ‘IF the constraint function value is positive and is increasing, THEN decrease the variable strongly’.

l          Rule Ib: ‘IF the constraint function value is positive and is decreasing, THEN decrease the variable softly’.

l          Rule IIa: ‘IF the constraint function value is negative and is increasing, THEN increase the variable softly’.

l          Rule IIb: ‘IF the constraint function value is negative and is decreasing, THEN increase the variable strongly’.

For monotonically decreasing variables,

l          Rule IIIa: ‘IF the constraint function value is positive and is increasing, THEN increase the variable strongly’.

l          Rule IIIb: ‘IF the constraint function value is positive and is decreasing, THEN increase the variable softly’.

l          Rule IVa: ‘IF the constraint function value is negative and is increasing, THEN decrease the variable softly’.

l          Rule IVb: ‘IF the constraint function value is negative and is decreasing, THEN decrease the variable strongly’.

Similar to the typical rule base for a fuzzy PD controller in Table 1, the rules discussed above are further interpreted into the fuzzy rule base in Table 3 and 4 using 5 linguistic terms: negative big (NB), negative small (NS), Zero (ZE), positive small (PS), and positive big (PB).

Table 3. Rule base for the monotonically increasing variables

 e Δe PB PS ZE NS NB PB NB NB ZE PS PS PS NB NS ZE PS PS ZE NB NS ZE PS PB NS NS NS ZE PS PB NB NS NS ZE PB PB

Table 4. Rule base for the monotonically decreasing variables

 e Δe PB PS ZE NS NB PB PB PB ZE NS NS PS PB PS ZE NS NS ZE PS PS ZE NS NB NS PS PS ZE NS NB NB PS PS ZE NB NB

In the air tank design optimization example, design variables r, s, h have only one critical constraint (g1, g3, and g2, respectively), which is the situation (1) in the 4 possible situations discussed in Section 3. Therefore, the change of design variable r, s, h in the next iteration can be defined as

Δr = defuzzification (mg1(r)),

Δs = defuzzification (mg3(s)),

Δh = defuzzification (mg2(h)),                                                                     (9)

The monotonicity sign of design variable l is ‘indeterminate’ in the objective function of the final monotonicity table output from MONO, which is the situation (3) discussed in Section 3. The change of design variable l in the next iteration is decided by constraints (g1, g4 and g6) in which l appears using Equation (7), and can be written as

Δl=defuzzification,                                                                                                                                            (10)

### Defining the initial values and move limits of the design variables

The inputs of the fuzzy PD controller optimization engine are the constraint function values (e) and the change of the constraint function values (Δe). Table 5 gives the quantitative definitions for the error inputs (e: g1, g2, g3, g4 and g6), and Table 6 gives the quantitative definitions for the error change inputs (Δe: Δg1, Δg2, Δg3, Δg4 and Δg6). As discussed in the previous section, these quantization tables are used in the fuzzification step to convert e and Δe into fuzzy membership for the rule base in Table 3 and 4.

Table 5. The quantization table of error inputs of the air tank example

 Quantized level g1 g2 g3 g4 g6 2 || || || || || 1 ||/2 ||/2 ||/2 ||/2 ||/2 0 0 0 0 0 0 -1 -||/2 -||/2 -||/2 -||/2 -||/2 -2 -|| -|| -|| -|| -||

Table 6. The quantization level of error change inputs of the air tank example

 Quantized level Δg1 Δg2 Δg3 Δg4 Δg6 2 Δg1(x0)max Δg2(x0)max Δg3(x0)max Δg4(x0)max Δg6(x0)max 1 Δg1(x0)max/2 Δg2(x0)max/2 Δg3(x0)max/2 Δg4(x0)max/2 Δg6(x0)max/2 0 0 0 0 0 0 -1 -Δg1(x0)max/2 -Δg2(x0)max/2 -Δg3(x0)max/2 -Δg4(x0)max/2 -Δg6(x0)max/2 -2 -Δg1(x0)max -Δg2(x0)max -Δg3(x0)max -Δg4(x0)max -Δg6(x0)max

As shown in Table 5, the values of the error inputs at different quantized levels are determined by the initial values of the constraint functions. In this example, the initial values of the variables are:

h = 10cm,        l = 800cm,       r = 150cm,       s = 3cm.                         (11)

The constraint function values can be evaluated at the initial design point, which give:

= -0.63,                = 0.95,         = -0.52,                                        = -0.99,                = 0.31.                                                 (12)

In Table 6, the quantization level value of error change (Δe: Δg1, Δg2, Δg3, Δg4 and Δg6) are determined by

(13)

where x0 is a vector of initial values of the design variable, and Δxmax is the vector of ‘move limits’ of the variables. The ‘±’ sign depends on the monotonicity of the variable to ensure that Δgi(x0)max is always positive. If the variable is monotonically increasing in the constraint function, the sign will be ‘+’, and vice versa.

The move limit of a design variable is the maximum amount of change of the variable allowed in one iteration. In this example, the move limits are:

h)max = 2,      (Δl)max = 160,   r)max = 30,    s)max = 0.6.                   (14)

Therefore, from Equation (14)

Δg1(r, l)max = 0.36,           Δg2(r, h)max = 0.98,          Δg3(r, s)max = 0.24,                          Δg4(l)max = 3.1×10-3,  Δg6(l)max = 0.26.                                               (15)

The outputs from the fuzzy PD optimization engine are the changes of the design variables in the next iteration. Table 7 shows the quantization level of the outputs, which are also determined by the move limits defined by the designer. As discussed in the previous section, this quantization table is used to ‘defuzzify’ the outputs into crisp values of Δh, Δl, Δr, and Δs for the next iteration (Equation (9) and (10)).

Table 7. The quantization level of fuzzy outputs in the air tank example

 Quantized level Δh Δl Δr Δs 2 (Δh)max (Δl)max (Δr)max (Δs)max 1 (Δh)max/2 (Δl)max/2 (Δr)max/2 (Δs)max/2 0 0 0 0 0 -1 -(Δh)max/2 -(Δl)max/2 -(Δr)max/2 -(Δs)max/2 -2 -(Δh)max -(Δl)max -(Δr)max -(Δs)max

### The optimization results

Using the initial design point and move limits described above, the fuzzy PD controller optimization engine terminates after 49 iterations, when the change in objective function value in consecutive iterations is less than 0.01%, and all constraints are satisfied within a tolerance of 0.01% of the initial values of the constraints. The numerical results h =13.67cm, l = 610.00cm, r = 105.18cm, s = 1.01cm are identical to the analytical solution. At this optimal design point, the values of the active constraints are g1 = 4.02×10-10, g2 = 9.17×10-5, g3 = 2.20×10-10, and g6 = 7.83×10-10. The values of other constraints are g4 = -0.98 and g5 = -0.29. The value of the objective function is 1.38×106.

Note that in this optimization process, the function value of each constraint is evaluated 51 times, including two evaluations to construct Table 5 and Table 6. No sensitivity information is required. The designers only need to define the initial values and the move limits of the design variables in the optimization process.

Figure 6 shows the iteration history of the objective function of the air tank example. The objective function value drops rapidly in the first several iterations, and shows stable converging trend afterward. Different initial values and different move limits are also tested in this example (Tables 8 and 9). As expected, the number of iterations increases when the starting design point is farther from the optimal design point. In some tests, larger move limits help to reduce the number of iterations, while in other tests, larger move limits cause overshoot and the number of iterations increases. In all 19 tests shown in Table 8 and 9, the fuzzy PD controller optimization engine appears to be robust. Same numerical solutions are obtained though the number of iterations required varies. The iteration histories of all 19 tests also show converging trend similar to that in Figure 6. Table 8 also compares the number of iterations required for different termination criteria. Note that in these tests, 1/3 to 1/2 of the iterations are used to drive the optimal solution from a 0.1% tolerance to a 0.01% tolerance.

Figure 6. The iteration history of the objective function of the air tank example

Table 8. Using different initial values in the air tank example

 r (cm) s (cm) h (cm) l (cm) Iteration (0.01%) Iteration (0.1%) Original 150 3 10 800 49 34 Test 1 200 3 10 800 48 21 Test 2 50 3 10 800 51 38 Test 3 150 5 10 800 49 34 Test 4 150 0.1 10 800 138 81 Test 5 150 3 15 800 17 13 Test 6 150 3 5 800 154 104 Test 7 150 3 10 1000 63 45 Test 8 200 5 15 1000 44 29 Test 9 50 0.1 5 100 214 135 Optimal 105.0 1.0 13.6 610.0

Table 9. Using different move limits in the air tank example

 Variables r (cm) s (cm) h (cm) l (cm) Iteration 150 3 10 500 49 Move limits Δr (cm) Δs (cm) Δh(cm) Δl (cm) Original 30 0.6 2 160 49 Test 1 15 0.6 2 160 34 Test 2 45 0.6 2 160 52 Test 3 30 0.3 2 160 49 Test 4 30 0.9 2 160 49 Test 5 30 0.6 1 160 107 Test 6 30 0.6 3 160 22 Test 7 30 0.6 2 80 56 Test 8 30 0.6 2 200 49 Test 9 15 0.3 1 80 102 Test 10 45 0.9 3 200 32

The fuzzy PD controller optimization engine has been tested in various engineering optimization problems with implicit functions and monotonicity. Three representative design examples are described in the following section to demonstrate the practicality of this approach.

## 5.     Design Examples

### A tension/compression spring design optimization problem

Figure 7 and Equation (16) show an example in which the weight of a tension/compression spring is to be minimized, subject to constraints on minimum deflection (g1), shear (g2), surge frequency (g3) and limits on outside diameter (g4) (Coello and Monts, 2002; Arora, 1989; Belegundu and Arora, 1985). The design variables are the mean coil diameter D, the wire diameter d and the number of active coils N.

Figure 7. The tension/compression spring design optimization problem

,

,

,

.                                                                        (16)

Table 10 shows the monotonicity table of the spring design example. After Monotonicity Analysis by the computer program MONO, it is concluded that design variable D has only one critical constraint g1. One of the constraints in the conditionally critical set g2 and g3 must be critical for design variable d. The monotonicity sign of design variable N is ‘indeterminate’ in the objective function at the end of Monotonicity Analysis.

Table 10. Monotonicity table of the spring design example

 d D N f + + + g1 + － － g2 － + . g3 － + + g4 + + .

In this example, the initial values of the design variables are:

d = 0.1,                    D = 0.25,                 N = 13,                                    (17)

The move limits are:

d)max = 0.001,       (ΔD)max = 0.01,        N)max = 0.1.                          (18)

Similar to Tables 5-7, these user-defined values are used to determine the quantization levels of the error inputs, change of error inputs, and outputs in this example.

The fuzzy PD controller optimization engine terminates after 225 iterations, when the change in objective function value in consecutive iterations is less than 0.01%, and all constraints are satisfied within a tolerance of 0.01% of the initial values of the constraints. Table 11 shows the comparison of the optimization results by the fuzzy PD controller optimization engine with those obtained in the literature. Note that only constraints g1 and g2 are active at the optimum design point, though there are 3 design variables. Figure 8 shows the iteration histories of the objective function. In this process, the function value of each constraint is evaluated 227 times, and no sensitivity information is required.

The purpose of the comparison in Table 11 is to confirm the quality of the solution obtained by the fuzzy PD controller optimization engine. The information and computational cost required in one iteration are different for the 4 algorithms listed in Table 11. Therefore the number of iterations of the algorithms in literature cannot be compared directly.

Table 11. Comparison of the results for the spring design example

 Fuzzy PD Coello and Monts (2002) Arora (1989) Belegundu (1982) Objective function value 0.0126503 0.0126810 0.0127303 0.0128334 Iteration 225 - - - d 0.052362 0.051989 0.053396 0.050000 D 0.373153 0.363965 0.399180 0.315900 N 10.36486 10.89052 9.185400 14.25000 g1 0.001987 -0.000013 0.000019 -0.000014 g2 0.000084 -0.000021 -0.000018 -0.003782 g3 -0.803753 -4.061338 -4.123832 -3.938302 g4 -0.716237 -0.722698 -0.698283 -0.756067

Figure 8. The iteration history of objective function of the spring design example

### Tubular column design optimization problem

Figure 9 shows an example for designing a uniform column of tubular section to carry a compressive load P = 2500 kgf for minimum cost (Rao, 1996). Equation (19) shows the optimization model of this problem. The column is made up of a material that has a yield stress (sy) of 500 kgf/cm2, modulus of elasticity (E) of 0.85×106 kgf/cm2, and density (r) of 0.0025 kgf/cm3. The length of the column (L) is 250 cm. The stress included in the column should be less than the bucking stress (constraint g1) and the yield stress (constraint g2). The mean diameter of the column is restricted between 2 and 14 cm (constraint g3 and g4), and columns with thickness outside the range 0.2 to 0.8 cm are not available in the market (constraint g5 and g6). The cost of the column includes material and construction costs and can be taken as 9.82dt+2d, where d is the mean diameter of the column in centimeters, and t is tube thickness.

Figure 9. Tubular column under compression design problem

(19)

After Monotonicity Analysis by the computer program MONO, it is concluded that there are two conditionally critical sets. One of the constraints in the conditionally critical set {g1, g2, g3} must be critical for design variable d, and one of the constraints in the other conditionally critical set {g1, g2, g5} must be critical for design variable t.

In this example, the initial values of the design variables are:

d = 14.0 cm,                            t = 0.8 cm                                                (20)

The values of the move limits are:

d)max = 1.0,                           t)max = 0.1.                                           (21)

Using the same termination criteria of the previous example, the fuzzy PD controller optimization engine terminates after 53 iterations. Table 12 compares the results obtained from the fuzzy PD controller optimization engine with that from the literature. Figure 10 shows the iteration histories of the objective function.

Table 12. Comparison of the result for the tubular column design problem

 Fuzzy PD Rao (1996) Objective Function 25.5316 26.5323 Iteration 53 d 5.4507 5.4400 t 0.2920 0.2930 g1 -7.8×10-5 -0.8579 g2 -7.8×10-5 -0.9785 g3 -0.6331 -0.8571 g4 -0.6107 0.0000 g5 -0.3151 -0.7500 g6 -0.6350 0.0000

Figure 10. The iteration histories of the objective function of the tubular column example

### The speed reducer design optimization problem

The design of the speed reducer (Golinski, 1973), shown in Figure 11, is considered with the face width (b), module of teeth (m), number of teeth on pinion (z), length of shaft 1 between bearings (l1), length of shaft 2 between bearings (l2), diameter of shaft 1 (d1), and diameter of shaft 2 (d2). The objective is to minimize the total weight of the speed reducer. The constraints include limitations on the bending stress of gear teeth, surface stress, transverse deflections of shafts 1 and 2 due to transmitted force, and stresses in shafts 1 and 2. The design optimization model can be summarized in Equation (22).

Figure 11. Speed reducer design problem

,

,

,

,

,

,

,

,

,

,

,

,

,

.                                                               (22)

There are 7 design variables and 25 constraints in this model. After Monotonicity Analysis by the computer program MONO, it is concluded that there are 7 conditionally critical sets for the 7 design variables in this problem.

In this example, the initial values of the design variables are:

b = 3.6,            m = 0.8,           z = 28,             l1 = 8.3,                                                    l2 = 8.3,            b1 = 3.9,           b2, = 5.5.                                                  (23)

The values of the move limits are:

b)max = 0.1,           m)max = 0.01,        z)max = 1.0,           (Δl1)max = 0.1,                  (Δl2)max = 0.01,        (Δb1)max = 0.01,      (Δb2)max = 0.01.                       (24)

Following the same process, the fuzzy PD controller optimization engine terminates after 302 iterations. Table 13 compares the results obtained from the fuzzy PD controller optimization engine with that from the literature. Figure 12 shows the iteration history of the objective function.

Table 13. Comparison of the results for the speed reducer example

 Fuzzy PD Golinski (1973) Objective function value 3007.8 2985.2 Iteration 302 - x1 3.5197 3.5 x2 0.7039 0.7 x3 17.3831 17.0 x4 7.3000 7.3 x5 7.7152 7.3 x6 3.3498 3.35 x7 5.2866 5.29 g1 -0.1095 -0.0739 g2 -0.2458 -0.1980 g3 -0.5127 -0.4990 g4 -0.9073 -0.9194 g5 0.0000 0.0001 g6 0.0000 -0.0020 g7 -0.6941 -0.7025 g8 0.0000 0.0000 g9 0.5833 -0.5833 g10 -0.2613 -0.2571 g11 -0.0223 -0.0278 g12 -0.0056 0.0000 g13 -0.1201 -0.1250 g14 -0.0220 0.0000 g15 -0.3792 -0.3929 g16 0.0000 0.0000 g17 -0.1205 -0.1205 g18 -0.0538 0.0000 g19 -0.0705 -0.1205 g20 -0.1343 -0.1343 g21 -0.1411 -0.1410 g22 -0.0542 -0.0548 g23 -0.0388 -0.0382 g24 -0.0514 -0.0514 g25 0.0000 0.0574

Figure 13. The iteration history of objective function of the speed reducer example

## 6.     Conclusions and discussion

In real-world engineering design problems, engineering heuristics often relate to the monotonic behavior of certain design variables. Indeed, monotonicity between the design variables and the objective and constraint functions prevails in engineering design optimization problems. Monotonicity Analysis can generate valuable qualitative information for this type of optimization models, such as the activity of the constraints at the optimum design point. However, numerical optimal solution cannot be obtained by Monotonicity Analysis alone.

The fuzzy PD controller optimization engine developed in this research utilizes the monotonicity of the design variables and the activity of the constraints concluded from Monotonicity Analysis in the optimization process to obtain the numerical optimum solution. The basic structure of this optimization process is actually very similar to the traditional line search algorithms. In a numerical line search algorithm, the search direction and step length are decided by numerical information, such as the gradients of the objective function and constraints. In the optimization process presented here, the search direction and step length are generated by the fuzzy PD controller optimization engine using the information on the monotonicity of the design variables and the activity of the constraints, as well as the initial values and move limits of the design variables defined by the user. In the optimization process, the objective and constraint functions only need to be evaluated once in each iteration. No sensitivity information is required. This characteristic makes the optimization algorithm especially suitable for engineering optimization problems with implicit constraints. The fuzzy PD controller optimization engine appears to be robust in the various design examples tested in this research.

In this research, the optimization process is analogous to a close loop control system, and the fuzzy PD controller optimization engine was developed following the general concept of developing a fuzzy PD controller for a close loop control system. In the future development, concepts and methods commonly used for improving fuzzy PD controllers, for example, how to speed up the response, reduce the overshoot, and improve the transient response, etc., can also be tested to further improve the performance of the fuzzy PD controller optimization engine. Moreover, many design variables are discrete in nature in engineering optimization problems. This should also be considered in the future development of the fuzzy PD controller optimization engine.

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