The MEMS Handbook Introduction & Fundamentals (2nd Ed) - M. Gad el Hak Part 16 potx

If the pendulum angle is positive and large and the angular velocity is zero, then the control force should be positive and large... If the pendulum angle is positive and small and the p

16.3.2 Heat Exchanger Application

This SC technique is applied to the heat exchanger described before The optimization problem here is to

find the best correlation that fits experimental data A set of N  214 experimental runs provided the database In each case, the heat rate Q . is found as a function of the two flow rates m w and m aas well as

the two inlet fluid temperatures Iin

a and I win Details are in Pacheco-Vega et al (1998)

There are two resistances to the flow of heat by convection: on the inside with water and on the out-side with air The conventional way of handling data is determining correlations for the inner and outer

heat transfer coefficients For example, power-law relations of the form Nu  aRe nbetween the Nusselt

and Reynolds numbers, Nu and Re, respectively, on both sides of the tube wall are often assumed There are then four constants to determine: a1, a2, n1, and n2 One possible procedure is to minimize the root

mean square (rms) error S U (a1, a2, n1, n2) in total thermal resistance to heat transfer between prediction and data in the least-square sense The total resistance is the sum of the air-side and water-side resistances This procedure leads to a large number of local minima due to the nonlinearity of the function to be

minimized Figure 16.15 shows a pair of such minima In the figure, a section of the error surface S U (a1, a2,

n1, n2) that passes through two local minima A and B is shown The coordinate z is a linear combination

of a1, a2, n1, and n2such that it is zero at A and unity at B, and the ordinate is the rms error The values

S U of the two correlations obtained at A and at B are very similar, and the heat rate predictions for the result-ing correlations are also almost equally accurate However, a1, a2, n1, n2, and the predictions of the ther-mal resistances on either side are very different This shows the importance of using global minimization

techniques for nonlinear regression analysis If the GA is used to find the global minimum, the point A is

the global minimum The correlation (not shown) found as a result of the global search is the best that fits the assumed power laws and is closest to the experimental data

16.3.3 Other Applications

Many other applications of GAs to optimization and control problems include optimization of a control scheme by Seywald et al (1995), Michalewicz et al (1992), Perhinschi (1998), and Tang et al (1996b) Reis

0 0.5 1 1.5 2 2.5 3

z

2 K/W

2 )

× 10−5

FIGURE 16.15 Global vs local minima in optimization problem

et al (1997) and Kao (1999) have used the GA to find the optimal location of control valves in a piping net-work Gaudenzi et al (1998) optimized the control of a beam using the technique Several workers have applied the method to the motion of robots [Nakashima et al., 1998; Nordin et al., 1998] Katisikas et al (1995) and Tang et al (1996a) used the genetic algorithm for active noise control Nagaya and Ryu (1996) controlled the shape of a flexible beam using a shape memory alloy, and Keane (1995) optimized the geometry of structures for vibration control Dimeo and Lee (1995) controlled a boiler and turbine using the genetic algorithm Sharatchandra et al (1998) used the GA for shape optimization of a micropump Kaboudan (1999) used genetic algorithms for time-series prediction Luk et al (1999) developed a GA-based fuzzy logic control of a solar power plant using distributed collector fields Additional applications of GAs combined with other SC techniques have been used for optimization of the control process [Matsuura et al., 1995; Trebi-Ollennu and White, 1997; Rahmoun and Benmohamed, 1998; Ranganath et al., 1999; Lin and Lee, 1999]

16.3.4 Final Remarks

There are two main advantages when using a genetic or evolutionary approach to optimization One is that the methods seek the global optimum The other advantage is that they can be used in discrete systems,

in which derivatives do not exist or are meaningless Examples of this are piping networks and position-ing of electronic components As with all tools, the reader must evaluate the advantages and disadvantages

in terms of specific applications

16.4.1 Introduction

Fuzzy sets and fuzzy logic date back to Lotfi Zadeh’s [Zadeh, 1965, 1968a, 1968b, 1971] work concerning complex systems Fuzzy sets and fuzzy logic have been present in controls applications since the late 1970s [Mamdani, 1974; Mamdani and Assilian, 1975; Mamdani and Baaklini, 1975] Fuzzy logic and its appli-cation to feedback control is comprised of two components First, fuzzy logic is not model based so it can

be applied to systems for which developing analytical models, either from first principles or from some identification techniques, is impractical or expensive Second, it provides a convenient mechanism for application to feedback control of human (or expert) intuition regarding how a system should be con-trolled This section outlines basic fuzzy set definitions, fuzzy logic concepts, and their primary applica-tion to control systems First, an illustrative controls applicaapplica-tion of fuzzy logic is presented in complete detail The example is followed by a more complete exposition of the mathematics of fuzzy logic intended

to provide the reader with a complete set of tools with which to approach a fuzzy control problem

16.4.2 Example Implementation of Fuzzy Control

This section first introduces a typical structure of fuzzy controllers by presenting an example of a common fuzzy control application — namely, to stabilize the inverted pendulum system illustrated in Figure 16.16

where the control input is a force of magnitude u In this problem, only the pendulum angle is stabilized.

This is accomplished via linguistic variables and fuzzy if–then rules such as:

1 If the pendulum angle is zero and the angular velocity is zero, then the control force should be zero

2 If the pendulum angle is positive and small and the angular velocity is zero, then the control force should be positive and small

3 If the pendulum angle is positive and large and the angular velocity is zero, then the control force should be positive and large

4 If the pendulum angle is positive and small and the pendulum angular velocity is negative and small, then the control force should be zero

The linguistic variables are the angle error and the angular velocity These rules are better expressed in tabular form in Table 16.3 The first enumerated rule is expressed in the third column and third row of the table The second rule is in the third column and fourth row The third rule is in the third column and fifth row The fourth rule is in the second column and fourth row These rules were determined by intu-ition For example, whether the second column and second row should be “negative small” or “negative large” is determined by experience, guesswork, or tuning

The next basic element of the fuzzy controller is the fuzzy set, which basically encapsulates the notion

of to what degree the angle is “zero,”“negative small,” etc.Figure 16.17 illustrates the fuzzy sets that define the fuzzy state of the angle of the pendulum system In the figure, if the pendulum angle is
7.5°, then the degree of membership in the “negative small” fuzzy set is 0.5, and the degree of membership in the “zero” fuzzy set is also 0.5 The degree of membership in the other fuzzy sets is 0.Figures 16.18 and 16.19 illus-trate similar fuzzy sets that are defined for the angular velocity and the control force, respectively Figure 16.20 illustrates the overall control structure First, a sensor measures the state (θ,θ

) Second,

the state is “fuzzified” by computing the degree of membership of the state in each of the fuzzy sets, A i, used in the if–then rules Third, the if–then rules in the rule base are evaluated in parallel, and the output

of each rule is the fuzzy set (control force), which has the shape of the fuzzy set associated with the output

of the if–then rule but is “capped” or “cut off ” at the degree of membership of the state in the associated

x

y

m

l

u

M

FIGURE 16.16 Pendulum system

TABLE 16.3 Fuzzy Logic Rules to Determine Control Force

Angular Velocity Negative Large Negative Small Zero Positive Small Positive Large

(1) Negative large Negative large Negative large Negative large Negative small Zero

(2) Negative small Negative large Negative large Negative small Zero Positive small (3) Zero Negative large Negative small Zero Positive small Positive large (4) Positive small Negative small Zero Positive small Positive large Positive large (5) Positive large Zero Positive small Positive large Positive large Positive large

fuzzy set If there is a logical operation, such as “and” in the antecedent (the “if ” part) of the rule, then the minimum of the degree of membership in each of the fuzzy sets is used

As a concrete example of this “fuzzy inference,” consider the case where the pendulum angle is
20° and the angular velocity is 22.5°/s The fuzzy state of the angle of the system is determined according to

positive small

positive large

0

1

Pendulum angle (degrees)

large

negative negative

FIGURE 16.17 Pendulum angle fuzzy set

positive small

positive large

0

1

large

negative negative

Pendulum angular velocity (degrees/sec)

FIGURE 16.18 Pendulum velocity fuzzy set

positive small

positive large

0

1

large

negative negative

Control force (N)

FIGURE 16.19 Pendulum force fuzzy set

Figure 16.21, where the state of the system is represented by a 0.25 degree of membership in the “negative large” fuzzy set, and a 0.75 degree of membership in the “negative small” fuzzy set Figure 16.22 shows the velocity is characterized by a 0.5 degree of membership in both the “positive large” fuzzy set and the “pos-itive small” fuzzy set

Now, the output of each rule will be the corresponding force fuzzy set, but modified so that its maxi-mum value is capped to be the minimaxi-mum degree of membership of the two elements of the antecedent

If A1 then B1

If A2 then B2

If An then Bn

x

B'1

B'2

B'n

FIGURE 16.20 Fuzzy control structure

positive small

positive large

0

1

Pendulum angle (degrees)

large

negative negative

0.25 0.75

FIGURE 16.21 Fuzzification of pendulum angle

positive small

positive large

0

1

large

negative negative

Pendulum angular velocity (degrees/sec) 0.5

FIGURE 16.22 Fuzzification of pendulum angular velocity

part of each rule In particular, only four of the rules listed in the table will evaluate to nonzero values — namely, the top two rows in the last two columns ofTable 16.3 Considering the “negative large” position and

“positive small” velocity first, the “negative small” force output will be capped at 0.25, which is the degree of membership in the “negative large” position fuzzy set which is less than the 0.5 membership of the angular velocity in the “positive small” fuzzy set In the “negative large” position and “positive large” velocity, the out-put will again be capped at 0.25, as similarly, it is less than the 0.5 membership of the angular velocity in the

“positive large” fuzzy set In the cases of “negative small” position and “positive small” velocity, as well as “neg-ative small” position and “positive large” velocity, the output of the “zero” and “positive small” output force fuzzy sets will both be capped at 0.5 Once the outputs from each if–then rule are computed, they are aggre-gated into one large fuzzy set In this aggregation, if two of the fuzzy outputs overlap, then (opposite to the

“and” combination for the fuzzy rules) the maximum of the two sets is taken Returning to the example, Figure 16.23 illustrates the aggregation of the four rules for the angle of
20° and angular velocity

of 22.5°/s “Defuzzification” is necessary to have a crisp output force, and Figure 16.24 demonstrates a com-mon technique to compute the value of the crisp output as the centroid of the aggregated fuzzy output set Simulating such a system is straightforward using Matlab If the pendulum mass is 0.1 kg, the cart mass 2.0 kg, the length of the pendulum 0.5 m, and the values of the membership functions are as illustrated

inFigure 16.25, the response of the cart and pendulum system is illustrated in Figures 16.26 and 16.27 Figure 16.26 illustrates the response of the pendulum angle, and Figure 16.27 illustrates the velocity of the pendulum.Figure 16.28 illustrates the control effort Because the cart position was not controlled, its steady-state response is actually a constant, nonzero velocity.Figure 16.29 illustrates the “response surface”

positive small

positive large

0

1

large

negative negative

0.25 0.5

Control force (N)

FIGURE 16.23 Aggregation of fuzzy output sets

Crisp output force

0

1

Control force (N)

0.25 0.5

FIGURE 16.24 Defuzzification of output by computing centroid

60 40 20 0 20 40 60 0

0.5

1

Angle error (deg)

0

0.5

1

Angular velocity error (deg/s)

0

0.5

1

Cart force

FIGURE 16.25 Membership functions for cart and pendulum simulation

0 1 2 3 4 5 6 7 8 9 10

Time (sec)

FIGURE 16.26 Pendulum position

(i.e., the plot of the function defining the control force computed by the fuzzy controller as a function of the two input variables)

The remainder of this section outlines the mathematical foundations of fuzzy logic which allow the reader to adapt this example for a particular application Note that in the pendulum example, the “and” conjunction, the aggregation of the outputs, and the means to defuzzify the output were all implemented

in certain, specific ways These are not necessarily the only or best implementations The mathematical outline will consider in more general terms fuzzy statements such as, “If A and B, then C” or “If A or B, then C,” which will lead to a list of possible alternative implementations of such a fuzzy inference system Which type of implementation is best may be application dependent, although the previous procedure is the predominant approach to fuzzy control

− 6

− 5

− 4

− 3

− 2

− 1 0

Time (sec)

FIGURE 16.27 Pendulum velocity

− 14

− 12

− 10

− 8

− 6

− 4

− 2 0

Time (sec)

FIGURE 16.28 Control effort required to stabilize inverted pendulum

16.4.3 Fuzzy Sets and Fuzzy Logic

16.4.3.1 Introduction

This section introduces fuzzy sets, fuzzy logic, and their mathematical foundations First, this section con-siders the concept of a membership function, and more specifically, whether an element belongs to a set

or whether membership in a set is a matter of degree Instead of either belonging or not belonging to a crisp set, an element can partially belong to a “fuzzy” set Several examples of fuzzy sets are provided, and the properties of traditional crisp sets are compared with the analogous properties of fuzzy sets There is

a “crisp” aspect to the normal definition of fuzzy sets because the membership function returns a crisp value Fuzzy sets can be generalized to have fuzzy-valued membership functions After defining fuzzy sets and outlining their properties, operations on fuzzy sets such as the complement, intersection, etc are defined and contrasted with the analogous operations on crisp sets Finally, fuzzy arithmetic and fuzzy logic are introduced as well as the notion of an additive fuzzy system, which is the basic framework used

in most fuzzy controls (in fact, the pendulum example above used this type of inference system)

16.4.3.2 Fuzzy vs Crisp Sets

The traditional notion of a set is called a crisp set Examples of crisp sets include:

1 The set of integers {…,
2,
1, 0, 1, 2, …}

3 Closed or open intervals of real numbers between a and b: [a, b], (a, b), respectively

4 A set defined by explicitly listing its elements, such as the set containing the letters a, b, and c: {a, b, c}.

Unless otherwise indicated, crisp sets are not considered ordered Crisp sets can be distinguished from fuzzy sets because in crisp sets an element either is a member of the set or is not a member of the set

− 60

− 40

− 20

0

20

40

60

− 40

− 20 0

20

40

− 40

− 30

− 20

− 10

0

10

20

30

40

Angle err

or

FIGURE 16.29 Response surface for pendulum fuzzy controller

Mathematically, one can define a membership function m which maps from a universal set U which is the set of all possible elements, to the set {0, 1}, where for set A and element x ∈ U:

That is, the membership function returns a 1 if x is a member of A, and returns 0 if x is not a member of A.

Crisp sets have a list of standard properties related to concepts in classical logic In particular, if the fol-lowing operations are defined:

1 Complement: A –  U
A  {x 僆 U|x 僆 A}

2 Union: A 艛 B  {x 僆 U|x 僆 A or x 僆 B}

3 Intersection: A 艚 B  {x 僆 U|x 僆 A and x 僆 B}

then verifying the following partial list of fundamental properties of crisp sets is straightforward:

1 Involution: A –  A

2 Contradiction: A 艚 A – φ

3 Excluded middle: A 艛 A –  U

Having defined the membership function as a mapping from the universal set to the set containing zero and one, it is natural to consider a generalization of the mapping Instead of considering the membership function as a binary mapping, the membership function for a fuzzy set is a mapping to the interval [0, 1]:

Now the mapping returns a value anywhere in the range between and including zero and one which encap-sulates the notion that membership can be a matter of degree This notion of degree enables fuzzy sets to express transitions between membership in sets where the transition is gradual (as opposed to crisp)

A prototypical example is temperature and whether the temperature on any given day is hot or cold There

is the set of hot days and the set of cold days If these sets were crisp, they would require sharp boundaries For example, if the temperature is above 80°F, it is hot; otherwise, it is not hot Similarly, if the temperature is below 45°F, it is cold; otherwise, it is not cold Such a rigid mathematical treatment of the notions of hot and cold is not appealing because humans are inclined to treat the transition to and from the set of hot and cold temperatures as gradual A more appealing notion is that a given temperature may have a degree of mem-bership in the set of hot days having a value of zero, one, or some value between zero and one These values

in between zero and one represent the transition from a day being not hot to the day being hot

Membership functions have been described only as a mapping from the universal set to the interval from zero to one.Figure 16.30 illustrates several examples of typical membership functions The membership function illustrated in the upper left figure is an example of a membership function that may model cold

where the variable x represents temperature For low temperatures, the value of the membership function

is one, illustrating that the temperature is cold High temperatures do not belong to the set of cold days, hence the value of the membership function is zero Between the two extremes is a transition period where the temperature only partially belongs to the set of cold days The figure in the upper right-hand corner is the analogous membership function for the set of hot days Other fuzzy sets may require that only values within

a certain range have a significant degree of membership in the fuzzy set Possible examples of such mem-bership functions are illustrated in the bottom two figures, which could represent warm days

An interesting feature of all the examples of fuzzy sets presented above is that the membership

func-tions are crisp values; that is, m(x) is a crisp number Depending on the application, requiring m to return

a crisp value may be overly precise Fuzzy sets can be generalized by defining membership functions to return a range of values instead of a crisp value In particular,

where I represents the family of all closed intervals of real numbers in [0, 1] that the shaded portion in Figure

16.31 illustrates Note that further generalization is possible because interval valued membership functions