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Technical Communique

Controller Design Using Fuzzy Logic A Case Study* KARL HEINZ KIENITZt Key Words--Control systems; optimization; fuzzy logic.

controller parameters to be determined. Section 4 is devoted to the selection of preference functions which truly incorporate the control specifications. In Section 5 the preference function optimization problem over the controller parameter space is treated. Preference function optimization yields adequate parameter values which will satisfy the specifications. Results for the case study problem and final comments are found in Sections 6 and 7, respectively.

Abstract--Controller design is considered for system specifications which are not handled naturally by analytical methods. Using fuzzy sets and related theory, system specifications are translated into preference functions which are readily combined with search methods to determine adequate controller parameters. This contribution integrates the discussion of the theory and its step-by-step application to aircraft control during the flare-out phase of landing.

1. Introduction

2. Aircraft landing problem definition

IN MANYENGINEERING applications controller design, tuning and adjustment have been necessary to guarantee the attainment of a variety of application specific performance and stability requirements. The control engineer, in general, is able to formulate control specifications informally, using some sentences and, eventually, some sketches. Nevertheless a methodic approach to the design, adjustment or tuning problem itself may be intricate, depending on these specifications. As long as quadratic performance indices or eigenvalue specifications are used, for example, controller design is almost straightforward. Often, however, such specifications do not arise naturally and have to be "designed" by themselves to be hopefully equivalent to the original specifications. In this context it is highly desirable to have a tool for generating a preference function which translates the engineers' "informal" specifications into mathematical language. Such preference function should allow for quantitative control system quality evaluation and be readily combinable with search methods to generate an adequately tuned or adjusted controller. The main concern of this contribution is to show how fuzzy sets and the related theory may be used to translate relatively informal controller design specifications into a preference function, which is then used to determine adequate controller parameter values using an optimization algorithm such as that due to Nelder-Mead (see references and comments by Himmelblau, 1972). Thus fuzzy sets and the related theory are used as a design decision tool. In the fuzzy set literature several decision applications were reported, such as those by Zimmermann et al. (1984). However, a specific application to control engineering is not known to this author. For better comprehension and illustration of the simplicity and potential of the proposal, this contribution integrates the discussion of the theory and its step-by-step application to a well-known, non-trivial aircraft landing system design problem (Ellert and Merriam, 1963). The contribution is set up as follows. In Section 2 the case study problem is described. Section 3 presents the chosen controller structure. This choice defines the number of

This section closely follows Ellert and Merriam (1963) and Tou (1964). This case study is concerned with the final phase of aircraft landing, also called the flare-out phase. In this phase the aircraft must be guided along the desired flare-path from a given altitude until it touches the runway. It is assumed that the aircraft is guided to the proper initial conditions for flare-out begin, which in this example range from 80 to 120 ft altitude, and 16-24ftsec -1 descent rate. Variation of descent rate is zero at flare-out begin. The aircraft is waved off for values outside these ranges. At flare-out begin in any case the angle of attack value is supposed to be 12.6 °, which is 70% of its stall value. Through adequate control (not discussed herein) aircraft velocity v is maintained constant equal to 256ftsec -1 during flare-out. Only longitudinal motion is considered. The linearized state equations for the initial angle of attack, adopted henceforth as the nominal value, read:

dx/dt:Ax+Bu,

y=Cx,

idh/dt/,

x

L

tl

d

where x is the state vector, y is the output and measurement vector, dO/dt is the pitch rate (rad sec-~), A0 is variation in pitch relatively to the nominal value (rad), dh/dt is rate of ascent (ft sec -1) and h is the altitude (ft). System input u is the elevator deflection (rad). System matrices are given as: i -0"61 - 0 . 7 6 0 2 " 9 6 8 7 ~5 x 170 - 30 A=

0 0

102.4 0

-0.4 1

ca0001 o,

'

'

0 0

0 0

o

o

10 -3 0

0 10 -3

"

In addition to the state variables, the angle of attack tr is of primary importance since for tr = 18° the aircraft reaches stall conditions. Its geometric relation to the pitch angle and the rate of ascent is well known (see Etkin, 1982):

* Received 25 August 1991; recommended for publication by Editor W. S. Levine. t Department of Control, Instituto Tecnol6gico de Aeromiutica--ITA, Centro T6cnico Aeroespacial----CTA, 12228-900 Silo Jos6 dos Campos SP, Brazil.

t r = 0 - s i n -I dh/dt U

549

550

Technical Communique

The following specifications and control value limitations are given for the flare-out phase of landing. (a) For reasons of safety and passenger comfort, the desired altitude hd(t ) of the aircraft during flare-out is

~ lOOe -t/5 hd(t) = L 20 - t

0<_.t___15 15 <- t -< 20'

Thus the desired duration of the flare-out is 20 sec, including 5 sec over the runway. (b) As a consequence of (a) the desired rate of ascent is the time derivative of hd(t ). (c) The aircraft must touch the runway with a slightly negative rate of ascent to ensure proper landing. (d) During landing the angle of attack must remain below the stall value. The aircraft enters the flare-out phase in equilibrium with an angle of attack 0.7 times this value. (e) The pitch angle O(T) at real touchdown time T (not desired touchdown time!) must lie between 0° and 15° to prevent either the nose or the aircraft tail from touching the runway first. (f) The elevator, which is the only actuator for this problem, has its mechanical stops at - 3 5 ° and 15° .

3. Selecting the control structure To satisfy requisites (a)-(e) under restriction (f), all the foregoing section, a linear state feedback controller structure is adopted. The choice of the controller structure depends on control engineering considerations which primarily involve realization simplicity and measure availability. In the case of this aircraft landing problem, measures of all state variables are available, which suggests the use of static state feedback. The use of two mutually reinforcing references hd(t ) and dho/dt is natural, since the controlled system is expected to track them. Thus the control function becomes: a = - K 1 x I - K2x 2 - Ka(x3 - dhd/dt) - K4(x4 - hd), where K1, K2, K 3 and g 4 are the controller parameters. In the light of these considerations, the controller design process reduces to finding proper values for these four parameters. At this point it should be noticed that the design specifications will never be satisfied exactly because of the most likely existence of non-nominal permissible initial conditions of altitude and rate of ascent. Hence the formulation of the problem is uncertain in nature and fuzzy set theory will provide the best means of formulating mathematically which controller performs "best" in the most adequate sense.

4. Generating a preference function Design specifications were formulated in a quite complete fashion in Section 2. In this section the specifications are translated into a preference function, a function that allows for the ranking of sets of parameter values as well as for the determination of the acceptability or not of a parameter value set. For the sake of generality it is assumed that the desired properties of a successfully controlled system are formulated linguistically in terms of n linguistic variables X I . . . . . X , , (see Kandel, 1986) as follows [(X l i s A l l ) A N D ( X 2isA2t ) A N D . . .

AND (X, is A , t ) l O R

[(Xt is A 12) AND (X 2 is A22) A N D . . . AND

unique):

(Pitch AND AND AND AND AND AND

angle at touchdown is positive less than 15°) (Maximum angle o f attack is below stall value) (Error in h(t) is low) (Error in d h / d t is low) (Time at touchdown is about 20 sec) (Rate o f ascent at touchdown is slightly negative) (Maximum rate o f ascent is negative or about zero)

How was this sentence generated from specifications (a)-(e)? As a general characterization of an acceptable landing performance in the light of the specifications. "Pitch angle at touchdown is positive less than 15°'' and "'Maximum angle o f attack is below stall value" are specifications (d) and (e), respectively. The terms "Error in h(t) is low" and "'Error in dh/dt is low" account for the requirements that altitude and rate of ascent should remain reasonably close to those specified in (a) and (b). "Time at touchdown is about 20 seconds" enforces that touchdown should really occur near the time specified in (a). This is necessary to hinder the aircraft either from touching the ground before reaching the runway or from overrunning it. "Rate o f ascent at touchdown is slightly negative" is specification (c). " M a x i m u m rate o f ascent is negative or about zero" simply points out that, in order to correct initial conditions and follow the flare-path, the aircraft should not ascend in a perceptible manner. The reason for this is passenger comfort, which was already implicitly stated in the formulation of the desired flare-path hd(t). Restriction (f) on elevator excursion is not included in the description of landing behavior since it is supposed that the elevator deflection commanded by the controller will not reach saturation values. This supposition is confirmed later in Section 6. In the next step the linguistic formulation (1) of the desirable situation has to be translated into a preference function. As mentioned before, a preference function allows for the ranking of different sets of controller parameters; the set which produces system behavior with the highest preference function value being regarded as the most satisfactory one. Thus the membership function of the set of desirable (i.e. satisfactory) controlled systems would lend itself as preference function. To derive this membership function, recall that in the specific case of control systems, the linguistic values taken by X i in (1) generally will be singletons, which means that only one point x~ of the support set will have membership 1 whereas at all other points membership will assume its minimum value. Hence the membership of a certain controlled system in the set of desirable (i.e. satisfactory) systems is the membership function induced by (1), which, according to general definitions of fuzzy logic (Kandel, 1986), is: ~t(x, . . . . .

x.) = max m/in

Time o f touchdown = T such that h ( T ) = 0 ha(t ) - h ( t ) l

t~l(I, TI

Error in d h / d t = max I d h o / d t - dh/dtl te[O. T]

[(X I is A 1,,,) A N D (X 2 is Az,. ) A N D . . . AND

Descent rate at touchdown = ( dh / dt ) ( T ) (An is Anm)]. (1)

The Aij are linguistic values characterized as fuzzy sets on the support set of Xi. For the sample problem a description of desirable landing behavior could read (the choice is not

(2)

where x i is the actual scalar value on the support set of X i. There obviously remains the choice of the fuzzy sets which characterize the Aij. For computational efficiency the membership functions of these sets should be chosen as simple as possible, for example as piecewise continuous functions. In the case of the airplane landing system the X i in (1) are defined as follows:

Error in h(t) = max (An is An2)] O R

[•aij(Xi)],

Maximum rate o f ascent = max dh/dt te[0, T I

Pitch angle at touchdown = O( T) Maximum angle o f attack = max tr(t) te[0, T I

551

Technical Communique 1.5

2

1

0

-1

g

-2

g

-0.5

-3

-1,5

-4

-5

5

0

-5

10

pitch angle at touchdown

-2

20

15

2

8

4

~

12

14

16

18

(h)

(a)

~

tO

8

anllle of attack [delreem]

[degrees]

1.5

1.5

1

1

0.5

0.5

o

0

-0.5

-0.5

-1

-1

-1.5

\

-1.5

\ -2

-2

0

10

5

15

20

maximum

25

absolute

30

35

altitude

error

40

45

50

i

,

,

,

,

,

,

,

2

3

4

5

6

7

8

9

maximum

[ft]

a b s o l u t e r a t e of a s c e n t e r r o r [ f t / s ]

(d)

(0 1.5

1

0.5

o

~

-0.5

-1

-3

-1.5

-4

-2 14

16

18

20

24

22

time at touchdown

26

28

-5

30

-4

-3

-2

[s]

-1

0

1

r a t e of a s c e n t a t t o u c h d o w n

(el

2 Ift/s]

(0 2

0

-2

-3

-'-,

-3

-2

-1 maximum

O

~

~

~

r a t e of a s c e n t [ f t / s ]

(g) FIG. 1. Membership functions for the aircraft landing problem: (a) Angles positive less than 15°. (b) Angles of attack below stall value. (c) L o w altitude error. (d) L o w rate of ascent error. (e) Touchdown at about 20sec. (f) Slightly negative rates of ascent at touchdown. (g) Negative or about zero maximum rates of ascent.

3

20

552

T A B L E 2. SUMMARY OF RESULTS Order No.

Initial

Determined

guess

parameters

function

iterations

1

0 - 1

0.2837

34

- 1

-0.4311 - 1.0304 -2.4636 - 1.2422

2

0 -0.5 -1.5 - 1

-0.1171 -0.6675 -1.7064 -0.7630

0.1519

25

3

-0.7158 -0.7663 -3.5855 -0.8224

- 1.0087 - 1.3194 -7.6266 - 1.8916

0.3333

28

-2

TABLE 1. INITIAL CONDITIONS

dO/dt

AO

dh/dt

h

0 0 0

-0.0625 -0.0781 -0.0938

- 16 -20 -24

120 100 80

Number

of

r a t e of a s c e n t [ f t / s ]

a l t i t u d e [ft] 120

Preference

.

.

.

.

lO0 y' jjj

z ' " ~

80 60

40

I::/

20 -20 0

--25

-20 0

5

10

15

20

r "" ,"

0

25

10

I

20

t i m e [s]

t i m e [s]

(a) 1

I

15

(b) p i t c h a n g l e [degrees]

e l e v a t o r d e f l e c t i o n [degrees] 4

0

r

,.:_L--Z2. . . . . . ::_:::_=::_:: .......... -

*'/'"2:,

',

3 /]

- 1

-2

,;'i 0

-3/! 9

-i 0

/:•

8

5

10

15

20

70

25

5

10

15 t i m e [s]

t i m e Is]

(d)

(c) 18 a n g l e of a t t a c k [degrees]

[7

~' \,

16

','

~4 ii

12

0

i

",:,,

5

10

15

20

t i m e [s]

(e) F I o . 2. L a n d i n g p e r f o r m a n c e f o r p a r a m e t e r set 2.

25

20

25

Technical Communique

553

r a t e of a s c e n t [ f t / s l

a l t i t u d e [ft] 0

120

. . . . . . . . . . . . .

I00

-5

\ flO

I0

60

40

15

// //

20

J/,

20 0

,/ 25

-20

5

10

15

20

25

lO

time [~]

(a) 2 elevator deflection

15

2o

25

IS

20

25

t i m e [s]

(b) 5

[degrees]

0

-2

p i t c h a n g l e [degrees]

4

,/- "k ,[ 'k

3

r

',

2

i / " ""'-..

l

!/

""--'_--L --:'-y'::'':::

...........

-4

-6

oii

-8

°

I

-10 -12 0

5

IO

20

15 ume

25

0

5

10

[,]

(c)

(d) tfl a n g l e of a t t a c k

[degrees]

16

14

13

12 5

10

15

20

25

t i m e Is]

(e) FIG. 3. Landing performance for parameter set 3.

The membership functions for the A i (low, slightly negative, etc.) are given in Fig. 1. Membership value 1 corresponds to fully satisfactory performance with respect to X i. Negative membership characterizes completely unacceptable situations. Membership in the range [0, 1] is therefore acceptable and membership 1 is ideal. Even the casual reader is generally used to membership values in the interval [0, 1] only. However, it is easily seen that for our purpose the exact membership function image is not relevant, as long as it clearly expresses realistic membership ranking. Those who dislike negative membership values may map the proposed

function onto the interval [0, 1] and continue thereafter to achieve the same results, with an obvious increase in computational expense. For the landing problem, controlled system specifications are given independently of initial conditions, which may vary within the given ranges. But system behavior strongly depends on the initial conditions. Thus for performance evaluation purposes the nominal and the two worst case initial conditions are considered. The preference function is evaluated for those three cases, and the worst value (the lowest) then taken. This procedure is adequate since the two

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Technical Communique

worst case initial conditions are easily identified as those corresponding to the situation in which the aircraft is 20% above or below the desired flare-path at initial time and diverges at the largest permissible rate. The three initial conditions are given in Table 1.

5. Comments on the optimization process The controller selection procedure consists of finding a set of controller parameters which gives a good, possibly the best, preference function value. Starting at an initial guess, a methodic optimization procedure is used to reach a local optimum. Such optimization procedure should not rely on gradient values. The chosen algorithm should be started at several different initial guesses to ensure that a good value of the performance function is reached. The Nelder-Mead procedure was adopted to determine the controller parameters for the aircraft landing system. At each step system performance was obtained through simulation for the three initial conditions of Table 1. Based on these results, the preference function was then evaluated as described in Section 4.

deflection. Nevertheless it is seen that restriction (f) (Section 2) is not critical. As indicated by the performance function value, parameter set 3 really is the best since it keeps the aircraft the furthest from stall. It should be noted that the solution for the landing problem in this contribution is considerably simpler than that determined by Ellert and Merriam (1963) and Tou (1964) and yields better (qualitative) performance.

7. Conclusions It was shown how fuzzy sets and related concepts may help to solve controller design problems via optimization. The illustrated approach is particularly useful when design specifications do not relate directly to quadratic performance indices, system eigenvalues, singular values or other system parameters manipulated by analytic design methods. If in a particular application certainty factors are involved and/or the X i in (1), (Section 4) take values which are not singletons, the approach of this contribution needs refinements and extension using the concept of plausible approximate reasoning (see Kienitz, 1990).

6. Results The Neider-Mead algorithm was started at three initial guesses. These initial guesses were determined from root-locus and pole-placement considerations in order to ensure at least system stability. For all three guesses the algorithm located satisfactory controller parameters. The initial guesses, determined parameters, locally optimal values of the preference function and the number of iterations needed by the algorithm to reach the solution are found in Table 2. Figures 2 and 3 show landing performance for the initial conditions of Table 1 and the calculated parameter sets 2 and 3 of Table 2. Solid curves depict landing performance for the first initial condition of Table 1, dashed curves depict it for the second and dot-dashed curves for the third. Dotted curves are desired performance. Although the system performances in Figs 2 and 3 are similar, the controller gains differ considerably, which clearly influences elevator

References Ellert, F. J. and C. W. Merriam (1963). Synthesis of feedback controls using optimization theory--an example. IEEE Trans. Aut. Control, AC-8, 89-103. Etkin, B. (1982). Dynamics of Flight Stability and Control, 2nd ed. John Wiley, New York. Himmelblau, D. M. (1972). Applied Nonlinear Programming. McGraw-Hill, New York. Kandel, A. (1986). Fuzzy Mathematical Techniques with Applications. Addison-Wesley, Reading, MA. Kienitz, K. H. (1990). Plausible approximate reasoning. Cybernetics and Systems, 21, 647-654. Tou, J. (1964). Modern Control Theory. McGraw-Hill, New York. Zimmermann, H.-J., L. A. Zadeh and B. R. Gaines (Eds) (1984). Fuzzy Sets and Decision Analysis. North Holland, Amsterdam.