HARD AND SOFT COMPUTING IN THE ROBUST FLIGHT CONTROL ...

Appl. Comput. Math. 5 (2006), no.2, pp.166-180

HARD AND SOFT COMPUTING IN THE ROBUST FLIGHT CONTROL SYSTEMS A.A. TUNIK†, M.A. TOUAT‡, § Abstract. In the paper adaptive robust autopilot with combined structure is considered. This structure uses the ,,crisp” feedback with hard computing and fuzzy controller with soft computing. The structure of the crisp prototype of autopilot is designed via robust H2 H∞ - optimization. The sensitivity theory is applied to justify, which part of control system to be fuzzified. The fuzzy controller designed has the ability of learning from the reference model in order to adjust its parameters. As case study, the longitudinal channel of small UAV autopilot is considered to show the efficiency of the proposed method. Key words: robust control, adaptive control, fuzzy control, sensitivity theory, fuzzy learning, H2 /H∞ -optimization.

1. INTRODUCTION The problem of the robust flight control is very actual in the last time [1, 6, 7], especially in the area of unmanned aerial vehicles (UAV) [9, 10, 11]. It is explained by the fact, that parameters of small UAV dynamic models are very vulnerable towards the changeable atmosphere conditions; therefore there is significant uncertainty of controlled plant’s models as well as of the spectral properties of the exogenous disturbances. One of the effective ways to overcome these difficulties is the application of the fuzzy control principles [6, 7]. Fuzzy control became a good challenge to the classical control approaches [1], especially in the cases, when controlled plant and external disturbances possess uncertainty, time-varying dynamic characteristics, incomplete measurement of the state space vector components etc. However, the realization of fuzzy controller imposes the choice of many parameters by the designer, such as the shape of membership functions, inference strategy etc. But one of the most difficult problems is the high order of models, describing the dynamics of UAV or any other aircraft. It requires including in the flight control law a large amount of state space variables in order to stabilize this plant and to achieve desirable performance and complicates significantly the knowledge base of the fuzzy autopilot. In [6, 7] it was proposed to use combined principles of “crisp” and fuzzy control (or hard and soft computing), when the simplest dynamic elements of feedback (in [6, 7] there were integrals and derivatives) were used as “crisp” elements, meanwhile the static gains were fuzzyfied. It is supposed that the entire “crisp” structure was determined by application of proper method of modern control theory and then one part of feedback would remain unchanged; meanwhile another part would be fuzzyfied. This paper is devoted to the substantiation of this principle in general case via application of the sensitivity theory, which has to facilitate the choice of fuzzyfied part of the entire robust “crisp” structure. This “crisp” robust controller would be used as prototype for creation of the combined controller. In order to increase the zone of uncertain parameters of aircraft model with preserving the performance and robustness of control system, it is necessary to augment this system with the †National Aviation University,1, Prospect Komarova, 03058, Kiev, Ukraine, Professor of the Flight Control Systems Dept., Doct of Sci.(Eng), tel. : +380-44-497-8186, home : +380-44-486-4762, E-mail : [email protected], [email protected] ‡National Aviation University,1, Prospect Komarova, 03058, Kiev, Ukraine, Flight Control Systems Dept., Post-graduate student, tel. : +380-44-497-8186. E-mail : mohand [email protected] §Manuscript received November 03, 2006. 166

A.A. TUNIK, M.A. TOUAT: HARD AND SOFT COMPUTING ...

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means of adaptation. That is why in this paper it is proposed the method of fuzzy controller adaptation via application of the fuzzy inverse model. The “crisp” analog of this method is well-known method of reference model adaptive control, which realizes the searchless real-time adaptation of controller’s parameters to the changing of plant’s model during process of control. So the first of all it is necessary to consider briefly the procedure of the “crisp” robust controller synthesis, which is used for determination of the dynamic feedback of the “combined” controller and simultaneously for creation of the reference model for fuzzy controller tuning. 2. BRIEF DESCRIPTION OF THE “CRISP” CONTROLLER DESIGN PROCEDURE. This procedure is based on the robust parametricH2 /H∞ - optimization of the control system with given structure. Description of this procedure can be facilitated using case study of the small UAV longitudinal motion control in an altitude-hold mode [7]. The standard structure of this control system [8] is shown in Fig.1. The navigation system provides the longitudinal channel of autopilot with the

Fig.1. Block diagram of the UAV control system in the altitude-hold mode signals of three sensors: altitude h, pitch angle ϑ and rate q for attitude stabilization. The digital control law has the following form: δe(z) = Wa (z) · [Wh (z), Kϑ , Kq ] · [h, ϑ, q]T , Tdh Tda (z − 1) , Wh (z) = Kh + (z − 1) , (1) T ·z T ·z where: Kϑ , Kq , Kh are pitch angle, rate and altitude gains respectively, Wa (z)and Wh (z) are the dynamic compensators (PD-controllers) for the attitude (ϑ) and altitude (h) loops respectively, Tda , Tdh are time constants of the 1st difference elements in these loops, T is the sampling period (0.02 sec),δeis the deflection of elevator. The thrust control in this UAV is absent [11]. In accordance with control law (1) vector of the autopilot’s adjustable parameters P~n , which has to be determined from optimization procedure, has the following components: Wa (z) = 1 +

P~n = [Kh , Kϑ , Kq , Tdh , Tdϑq ] .

(2) 1 Tac s+1 .

Actuator “Act” can be described with the transfer functionWact (s) = In this case Tac = 0.5 sec . The purpose of the optimization procedure is sustaining the performance and robustness levels of the closed loop system with the same control law for “nominal” plant (true airspeed V=70 m/sec) and parametrically disturbed plant (V=55.5 m/sec). The state space description of these plants is represented with the standard equations: X˙ = AX + Bu + w, Y = CX, with following state and control matrices A and B respectively:

(3)

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APPL. COMPUT. MATH., VOL.5, NO.2,DECEMBER 2006

   A=      Ap =   

−0.0345 −0.0041 0 0.0033 0

6 −1.76 0 −25.7 −69.4

−9.78 0 0 0 69.4

0 0.99 1 −2.19 0

0 0 0 0 0

−0.0273 −0.0064 0 0.0036 0

6 −1.76 0 −16.1 −55.6

−9.78 0 0 0 55.6

0 1 1 −1.73 0

0 0 0 0 0

   ,  

B=

£

0.36 −0.16 0 −31.1 0

¤T

,

(4)

   ,  

Bp = [ 0.36 −0.13 0 −19.9 0 ]T . (5)

Variable w in (3) denotes the perturbation – turbulent wind, which can be described with standard Dryden model [3], state vector is equal to X = [V, α, ϑ, q, h],where V is the true airspeed, αis the angle of attack. In accordance with block diagram depicted in the Figure 1, observation matrix C used for actual feedback is equal to  0 0 1 0 0 C =  0 0 0 1 0 . 0 0 0 0 1 

(6)

The performance-robustness index (PRI), which has to be minimized with optimization procedure, would have the following form: J = λod Jod + λos Jos + λpd Jpd + λps Jps + λ∞ T∞ + λp∞ Tp∞ ,

(7)

where: J0d , J0s stand respectively for deterministic and stochastic partial performance indices (PPI) of nominal system, λ0d , λ0s stand for corresponding weight factors, while the same symbols with subscripts ‘p’ stand for the same values of perturbed system. Symbols T∞ (Tp∞ ) and λ∞ (λp∞ ) denote H∞ −norms for complementary sensitivity functions of the closed loop systems (measures of robustness [8]) and weight factors for the nominal and perturbed systems respectively. Deterministic PPI’s were calculated as the H2 -norms of the nominal and perturbed closed loop systems, meanwhile the stochastic PPI’s were calculated as the H2 -norms of the series connection of the Dryden filters [3] with the nominal and perturbed closed loop systems. This procedure is described in details in [10] along with the considerations concerning the choice of weight factors λ0d , λ0s ,λpd , λps λ∞ , λp∞ . Running this procedure with various values of these weight factors [10] gives eventually the following numerical values for vector of adjustable parameters of autopilot: P~n =[-9.2, -1, -0.05, 0.14, 0.008]. The closed loop system having the controller with parameters defined by this vector P~n can be considered as the reference model. In order to comply with the standard block diagram for application of learning approach proposed in [2,4], we could consider the attitude stabilization loop including elements Kϑ , Kq , Wa (z) (see Fig.1) as the inner loop for the stability and performance augmentation. So we can apply the principles of fuzzy learning control to the generalized plant including actuator, UAV and the inner attitude stabilization loop. It is useful to notice that this method of parametric synthesis of the robust control system can be replaced with many other existing methods of advanced control theory. For example, it is possible to use structural – parametric optimization [9], where structure was determined on the basis of the separation theorem with further “robustization” based on the aforementioned procedure. So far as the choice of method of the “crisp” system’s synthesis is not the ultimate goal of this paper, for the sake of clarity we selected one of the simplest but nevertheless effective methods.


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3. CHOICE OF THE FUZZIFIED PARAMETERS It was stated in [1, 2, 4, 6, 7] and many other references, that usage of fuzzy control allows to achieve very weak dependence of the control system performance on the parameters of controlled plant’s model, i.e. this principle of control accepts great uncertainty of model. So it is obvious, that first of all it is necessary to make fuzzy those parameters of control law, which possess the highest sensitivity function. So it is necessary to determine the sensitivity functions for all tuned parameters of flight control law, which are the entries of vector P~n defined by expression (2). Denote the components of this vector as pi ,i = 1, 2, ..., 5. The state space model of plant is defined by the quadruple of matrices (4, 5, 6): [A, B, C, 03×1 ] and the state space model of the controller is defined by another quadruple: [Au (P~n ), Bu (P~n ), Cu (P~n ), Du (P~n )], which can be easily derived from control law (1). The state and control matrices AC , BC of the closed loop system would be defined by the following expressions: ³ ´  ³ ´   ³ ´  ³ ´ ³ ´ BDu P~ A − B · Du P~ · C −Bu P~ · C ³ ´ . ³ ´ ³ ´  , Bc P~ =  (8) Ac P~ =  ~ ~ Au P Bu P~ Bu P · C The sensitivity function vector with respect to the parameter pi : ∂X (t, pi ) X˙ pi = ∂pi would be described by state space equation [11]:

(9)

³ ´ ~ ´ ³ ´ ∂B c P ∂Ac (pi ) X˙ pi = Ac (pi ) Xpi + X P~ , t + Y P~ , t . (10) ∂pi ∂pi The input signals to this system X(t, P~n )and Y (t, P~n ) is described by the state space equation of the closed loop system: ³ ´ ³ ´ ³ ´ X˙ P~ , t = Ac P~ · X P~ , t + Bc · R (t) , ³ ´ ³ ´ (11) Y P~ , t = Cc · X P~ , t , ³

where: Cc = [C, 0m ], (12) and 0m is zero matrix with appropriate size. So the sensitivity functions with respect to the controller’s parameters are defined by systems (11), (12) with corresponding initial conditions. Analytical expressions for these sensitivity functions would be very complicated, that is why it is expedient to use numerical procedures. The sensitivity functions (9) are describing additional motions of system, which were caused by variations of the tuned parameters pi . So it is easy to use simulation of the closed loop system with nominal and disturbed parameters of controller and estimate these additional motions after parameter’s variations. Results of this simulation for nominal plant (4, 5, 6) and nominal parameters of controller P~n =[-9.2, -1, -0.05, 0.14, 0.008] are given in the Table 1. Here it is denoted: Kdh = TTdh ; Kda = TTda . In the table only maximum values of sensitivities of the output variable of system (altitude) with respect to the parameters of controller are represented. It is obvious, that sensitivities with respect to the parameters of the altitude stabilization loop (Kh , Kdh ) are the highest, so it is necessary to use fuzzy PD-controller in the altitude stabilization loop. The attitude stabilization would be used as the inner loop encompassing the controlled plant, the block diagram is depicted in Fig.2.

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Table 1. Sensitivity of altitude output to the feedback gains It can be easily explained from the physical viewpoint. The attitude longitudinal motion is so called short-period (fast) motion, meanwhile the change of altitude is a long- period (slow) motion [3] that is why they can be considered separately.

Fig.2. Block diagram of the UAV control system with fuzzy controller The fuzzy controller represented here has the ability of adaptation of its parameters using the aforementioned method. 4. FUZZY MODEL REFERENCE LEARNING CONTROL (FMRLC) The functional diagram of the FMRLC [2, 4], which is shown in Fig. 3, is similar to the structure of known “crisp” model reference adaptive control (MRAC) [2, 4]. It has four main parts: the generalized plant (GP), which is described in the previous chapter, the fuzzy controller (FC), the reference model (RM) and the learning mechanism, which uses the fuzzy inverse model (FIM). The Generalized plant (GP) consists of the UAV, which is encompassed with inner ,,crisp” control loop. The goal of this method is to synthesize fuzzy controller and to adjust its membership functions in order to withstand to the action of parametric disturbances in the controlled plant. The term learning is used instead of adaptive only to distinguish the adjusting of the fuzzy controller’s membership functions in the control process from a simple adaptive conventional control. In the first case the tuning processes include joint variations of input and output membership functions along with action of the inference mechanism. In the second case the adaptation process is reduced to the variations of the simple numerically adjusted parameters of “crisp” controller. The expected performance of the overall closed loop system is specified in the reference model. Next, we describe aforementioned components of the FMRLC more in detail. 5. FUZZY CONTROL OF THE ALTITUDE The input to the FC is the error e(kT) between the reference input r(kT) and system output h(kT) –altitude of UAV. The FC output u(kT) is the reference value for the inner Attitude Control Loop (ACL). In accordance with control law (1) and fuzzy control ideology described in [2, 4], FC uses two input variables - error and change of error (the 1st difference): e(kT ) = r(kT ) − h(kT ), e (kT ) − e (kT − T ) c (kT ) = . T

(13) (14)


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Fig.3. Block diagram of FMRLC In FC each universe of discourse for each plant input is normalized to the interval [-1 +1] by means of constant scaling factors. The gains ge , gc and gu were used to normalize the universe of discourse for the error e (kT ) and change in error c (kT ), and controller output u (kT ) respectively. We apply inference mechanism (IM) of Mamdani type, which is recommended for aircraft usage in [6, 7]. It can be expressed in the form of IF-THEN rules as follows: IF e˜ is Ei AN D c˜ is Cj T HEN u ˜ is Uk ,

(15)

where e˜, c˜ and u ˜ are the linguistic variables of the error, change in error and output, respectively. Ei , Cj and Uk , i = 1, m, j = 1, n and k = 1, l are the linguistic values of the error, change in error and the output, these are implemented respectively in Block of Input Membership Function BIMF1 , BIMF2 and Block of Output Membership Function BOMF1 . Each rule is associated with fuzzy relationRk [5], considered as a fuzzy intersection of the premise and consequent fuzzy sets: Rk = (Ei ∩ Cj ) ∩ Uk .

(16)

Using logical operators, the above relation can be transformed to: Rk (˜ e, c˜, u ˜) = (Ei (˜ e) ∧ Cj (˜ c)) ∧ Uk (˜ u) ,

(17)

the fuzzy relation Rk , (k = 1, l), associated with individual rules are aggregated using fuzzy union, resulting in the fuzzy relation R associated with the fuzzy controller: R=

[ k

(Ei ∩ Cj ∩ Uk ),

substituting the fuzzy operators the above relation becomes:

(18)

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l

l

k=1

k=1

R (˜ e, c˜, u ˜) = ∨ Rk (˜ e, c˜, u ˜) = ∨ (Ei (˜ e) ∧ Cj (˜ c) ∧ Uk (˜ u)) .

(19)

The aforementioned fuzzy relation can be summarized in the following algorithm [5]: 1. For each rule compute the Degree Of Firing (DOF) denoted τk τk = (∨e˜,˜c [Ei ∧ Cj ] ) ; 2. Find the fuzzy set Fk inferred by the

(20)

k th

Fk (u) = τk ∧ Uk (˜ u) ; 3. Aggregate the inferred fuzzy sets Fk by using the max operator l

F (u) = ∨ Fk (˜ u) .

(21)

(22)

k=1

4. End In formulas (17,19-22) the fuzzy disjunction ∨ and fuzzy conjunction ∧ are used in the Zadeh’s sense [4, 5]. The last step of the fuzzy controller is to convert the fuzzy output resulting from aggregation method to a crisp output, which represents the control action; this is implemented in DF1 block and uses the Center Of Gravity (COG) defuzzification method expressed as

u∗n (kT ) =

l P Aˆkn (kT ) cˆn (kT ) k

l P Aˆkn (kT )

,

(23)

k=1

where Aˆln (kT ) and cˆn (kT ) are the areas and the centers of areas, associated with the resulted fuzzy set (22). 6. THE REFERENCE MODEL (RM) The reference model generates the desired performance of the overall process. In general, the reference model may be any type of dynamical system. The performance of the overall system is computed with respect to its output hm (kT ) by generating an error signal, he (kT ) = hm (kT ) − h (kT ) . (24) In our case the reference model is the closed loop control system with “crisp” controller, produced by the robust parametric H2 /H∞ -optimization procedure described above. We only used its simple approximation by the 2nd order model with the same rise time and small overshoot in order to facilitate its usage. 7. THE LEARNING MECHANISM As previously mentioned, the learning mechanism performs the function of modifying the knowledge base (membership functions) of the direct fuzzy controller so, that the closed loop system behaves like the reference model. These knowledge base modifications are made on the basis of observing data from the controlled process, the reference model and the fuzzy controller. In accordance with Fig. 3 the learning mechanism consists of two parts: a fuzzy inverse model (FIM) and a knowledge base modifier (KBM). FIM performs the function of mapping the error he (kT ) and the change of error hc (kT ), to the changes of the FC output membership function’s parameters P = [p1 , · · · , pr ]T in BOMF1 , which are necessary to force he (kT ) to zero. These parameters are the shifts of positions of the membership function’s centers. KBM performs the


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function of modifying the fuzzy controller’s knowledge base to perform the needed change in the process inputs. More details of this process are discussed next.

8. THE FUZZY INVERSE MODEL (FIM) A fuzzy system is used to represent the inverse plant dynamics. It is not necessary for fuzzy control to accurately characterize the inverse dynamics; only approximate representation is needed [2,3]. As it is shown in the Fig. 3, FIM simply maps he (kT ) and hc (kT ) to the necessary changes in the plant inputs, that is why it is called “fuzzy inverse model”. Hence, FIM is used to characterize how to change the plant inputs to force the plant output h (kT ) to hm (kT ) as close as possible. Likewise to the fuzzy controller, the FIM shown in Fig. 3 contains normalizing scaling factors, namely ghe , ghc and gp for each universe of discourse. Selection of the normalizing gains can impact the overall performance of the system. The knowledge base for the fuzzy inverse model is generated from fuzzy rules of the form: ˜ e is H j AND h ˜ c is H k THEN p˜ is P j,k , IF h e c

(25)

P j,k (kT ) is the necessary variation of position of the membership function’s center, decreasing he (kT ). The structure of fuzzy inverse model is the same as the fuzzy controller developed in the last chapter.

9. THE KNOWLEDGE BASE MODIFIER (KBM) KBM performs the function of modifying the FC rule base to achieve better performance. Given the information about the necessary changes in the plant input, which are represented by p (kT ), to force the error he (kT ) to zero, the knowledge base modifier change the FC rule-base so that the previously computed control action u (kT − T ) would be modified at the next step as follows: u (kT − T ) + p (kT ). By modifying the fuzzy controller’s knowledge base, we may force the fuzzy controller to produce a desired output, which we should put in at time kT − T to make he (kT ) smaller. Then the next time we get similar values for the error and change in error, the input to the plant will be one that will reduce the error between the reference model and the plant output. Assume that we use symmetric output membership function for the fuzzy controller, and let bl ˜l . Knowledge base modification denote the center of the membership function associated with U ˜l , is performed by shifting centers bl of the membership function of the output linguistic valueU which are associated with the fuzzy controller rules that contributed to previous control action u (kT − T ). This is two-step process: • Find all FC rules (16), which satisfy the following condition: τk (e (kT − T ) , c (kT − T )) Â 0 ,

(26)

k = 1,¯ l, defining the set of the Degrees of Firing of rules at time kT − T , also called “active set” • Let bl denote the center of lth output membership function at time kT . For all rules in the active set, use bl (kT ) = bl (kT − T ) + pl (kT ) to modify the output membership function centers. Rules that are not in the active set do not have their membership function modified. KBM includes also the storage (see Fig. 3), which preserves the results of tuning of membership functions, in order to be used when the output of the FC at time kT is the same at kT − 1.

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10. DESIGN AND IMPLEMENTATION OF FMLRC The total design procedure for the FMRLC, which is used in combination with “crisp” feedback, involves the following steps: • The specification of a direct fuzzy controller with consequent membership functions that can be tuned. This fuzzy controller can be chosen via conventional (heuristic) fuzzy control design techniques for the nominal plant. • Specifying the reference model of control system, which characterizes the desired system performance. The example of such specification is shown in a Chapter 2 of this paper. • Specifying the fuzzy inverse model, which characterizes how the inputs to the plant should be changed so, that the desired performance is achieved. • Selection of the normalizing gains for the fuzzy controller and the fuzzy inverse model. So far as the selection of the normalizing gains for both the fuzzy controller and the fuzzy inverse model can impact the overall performance, it is necessary to provide a procedure for choosing these parameters. Due to physical constraints for a given system, the range of values for the process inputs and outputs is generally known from a qualitative analysis of the process especially, when the crisp prototype of system is determined via some known procedure of control synthesis. As a result, we can determine the range of values or the universe of discourse for e (kT ), u (kT ), he (kT ) and p (kT ). Consequently, ge , gc , gu , ghe , and gp are chosen so that the appropriate universes of discourse are mapped to [-1, 1]. They could be determined on the basis of the “crisp” prototype by iteratively applying inputs to r (kT ), observing c (kT ), and finding scaling factors to map the universes of discourse to [-1, 1]. The coefficient gc and ghc , which are used for scaling the rate of errors in FIM and main control loop respectively can be additionally tuned using well-known properties of PD-controllers: • If undesirable oscillations in these loops occur, then it is necessary to increase corresponding gains, and • If the process in the one of these loops is over damped, then it is necessary to decrease corresponding gain. 11. CASE STUDY: FMRLC-BASED UAV AUTOPILOT DESIGN The inputs to the fuzzy controller are the altitude error and change in altitude error and the output is the pitch angle reference, which is compared with the actual pitch output of the UAV. In this fuzzy controller design 5 fuzzy sets are defined for both controller inputs (using the structure of Fig. 4.) such that the membership functions are triangular shaped and uniformly distributed on appropriate universe of discourse as it is shown in Fig. 4. The normalizing controller gains for the error, change in error and the controller output are chosen to be ge = 0.02, gc = 1.6 and gu = 0.3, respectively. The fuzzy set for the controller output is also assumed to be triangular shaped with width of 1 on the normalized universe of discourse. The knowledge base array was initially chosen with all zero entries.

Fig.4. Fuzzy sets on the universe of discourse for inputs signals The reference model for this plant was chosen to represent somewhat realistic performance specifications complied with the robust “crisp” prototype and is expressed by the following state space representation:


·

¸· ¸ · ¸ −2 −1.33 xr1 0.01 r, + 0.4 0 xr2 0 · ¸ £ ¤ xr1 hm = 0 133 + [0] r, (27) xr2 where xr1 , xr2 are the state variables of the reference model. Model (28) provides the altitude step response shown in the Fig. 5 with solid lines. The input to the fuzzy inverse model includes the error and change in error between the reference model and the UAV’s altitude output expressed as he (kT ) = hm (kT ) − h (kT ) and hc = he (kT )−hTe (kT −T ) respectively. For these inputs, 5 fuzzy sets are defined with triangular membership functions, which are evenly distributed on the appropriate universe of discourse (represented in Fig. 4). The normalizing gains associated with he (kT ) , hc (kT ) and p (kT ) are chosen to be ghe = 0.01, ghc = 1.2 and gp = 0.11 respectively. The rule base of the fuzzy controller is proposed in [2, 4] and shown in the Table 2 : x˙ r1 x˙ r2

¸

175

·

=

Table 2. Centers of the fuzzy controller output membership function The entries of the table represent the center values of triangular membership functions with base widths 1 as shown in the Fig. 4 for fuzzy sets Ul on the normalized universe of discourse. 12. SIMULATION RESULTS The simulation results for the FMRLC of the UAV with nominal and disturbed plant’s models (4-6) are provided in Figures 5-12 the comparison of dynamics properties of fuzzy and classical ,,crisp” autopilot is represented in the Tables 4-5 for step responses. These results could be summarized as follows: (1) Deflections of all UAV state variables are within tolerances, which are acceptable from the viewpoint of flight safety. (2) The step response performances of fuzzy system for nominal and parametrically perturbed plants are better than the same performances of the “crisp” robust prototype.

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Fig.5. UAV altitude output and model reference

Fig.6. UAV altitude output and model output for nominal plant reference output for disturbed plant

Fig.7. Angle of attack for the nominal plant


Fig.8. Angle of attack for the disturbed plant

Fig.9. Pitch angle for the nominal plant

Fig.10. Pitch angle for the disturbed plant

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178


Fig.11. Elevator deflection for the nominal plant

Fig.12. Elevator deflection for the disturbed plant

Table 3. Comparison of FMRLC and crisp

Table 4. Comparison of FMRLC and crisp controller step response for nominal plant controller step response for disturbed plant


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13. CONCLUSION (1) In order to provide robustness of flight control system with simulation preserving of their performance it is relevant to use fuzzy control systems, in which soft computing is implemented. However, high order of the mathematical model of aircraft requires application of fuzzy control with large amount of inputs. In this situation proper distribution of control functions between fuzzy control with soft computing and ,,crisp” control with hard computing can be considered as very promising approach. (2) This distribution can be done using analysis of sensitivity functions of flight control system with respect to the tuned parameters of ,,crisp” flight control law, which was chosen as a prototype for control system. This approach allows usage of the simple known fuzzy controller with 2 inputs, which are the fuzzy version of the ,,crisp” PDcontrollers. (3) Further increasing of robustness can be achieved using adaptation and learning principles for adjustment of fuzzy control parameters in a real-time in order to compensate unpredictable variations of plant’s parameters during the control process. In this case the application of the fuzzy inverse model is an effective toll for searchless adaptation. (4) Case study of longitudinal stabilization of small UAV demonstrates efficiency of proposed approach. References [1] Lin, C.F. , Advanced Control Systems Design. PTR Prentice Hall Inc., Englewood Cliffs, New Jersey, 1994.664 pages. [2] Laine, J.R., Passino K.M. Fuzzy model Reference Model Control. Journal of Intelligent and Fuzzy Systems.1996, Vol. 4, No. 1, pages 33-47. [3] McLean, D. Automatic Flight Control Systems. Prentice Hall Inc., Englewood Cliffs, 1990, 593 pages. [4] Passino, K.M., Yurkovich, S. Fuzzy Control. Addison-Wesley.- Menlo Park, Reading, Harlow, Berkley, Sidney, Bonn, Amsterdam, 1998. -502 pages. [5] Ronald R.Yager, Dimitar P.Filev. Essentials of Fuzzy Modeling and Control. John Wiley & sons, Inc, 1994. [6] Schram, G. , Kaymak, U. , Verbruggen, H.B. Fuzzy Logic Control. Chapter 13. “Robust Flight Control. Design Challenge”. Springer-Verlag.- Berlin, Heidelberg, New York, 1997.- pages 135-148. [7] Schram, G. , Verbruggen,H.B. A Fuzzy Control Approach. hapter 26 in the same issue. –pages 397-419. [8] Skogestad,S., Postlethwaite, I. Multivariable Feedback Control. Analysis and Design. John Wiley & Sons, Chichester, New York, Brisbane, Toronto, Singapore. 1997, 559 pages. [9] Tunik A., Galaguz T. Robust Stabilization and Nominal Performance of the Flight Control System for Small UAV. Appl.& Comput. Math. Vol.3, No.1, 2004, pp.34-45. [10] Tunik, A.A., Abramovich, E.A. Parametric Robust Optimization of the Digital Flight Control Systems. Proceedings of the National Aviation University.-2003.-2(17).- Kyiv. - pages 31-38. [11] Tunik,A.A., Ryu,H., Lee, H.C. Parametric Optimization Procedure for Robust Flight Control System Design. KSAS International Journal. Vol. 2, No. 2, Nov. 2001, pages 95-107. [12] Yusupov,R.M. Methods of Sensitivity Theory. In: Automatic Control Theory Handbook. A.A. Krasovsky, ed., ”Nauka”, Moscow, 1987, pp. 604-636, (in Russian).

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A.A. TUNIK, for a photograph and biography, see Apl. Compt. Math. (2004),vol.3., no.1., p.45.

M.A. Touat - was born in Algeria, Tizi-Ouzou city, in 1977. Graduated from Mouloud Mammeri University (Algeria) in 2002, receiving diploma of Engineer in automatic system, now he is Post-graduate Student of the National Aviation University, Kiev, Ukraine.