AN ADAPTIVE TAKAGI-SUGENO FUZZY GUIDANCE LAW WITH HIGH GAIN OBSERVER CONSIDERING MISSILE UNCERTAINTIES AND TARGET MANOEUVRE

AN ADAPTIVE TAKAGI-SUGENO FUZZY GUIDANCE LAW WITH HIGH GAIN OBSERVER CONSIDERING MISSILE UNCERTAINTIES AND TARGET MANOEUVRE Mostafa Abedi, Hossein Bolandi, Farhad Fani Saberi

Iran University of Science and Technology

Abstract: This paper proposes an observer based adaptive Takagi-Sugeno fuzzy guidance law for a surface to air missile. An integrated guidance-control model is derived to consider target manoeuvre and missile uncertainties in the design process of the proposed guidance law, also use of adaptive intelligent system for approximation of unknown dynamic parts of the mentioned model and a high gain observer for estimation of required states caused that this approach, in contrast to guidance approaches introduced by now, use the minimum number of inputs for a robust missile guidance. Presented proofs and simulated results show the practical application and effective performance of designed guidance law. Keywords: Guidance, missile, adaptive, Takagi-Sugeno, fuzzy, target, uncertainty, robust.

1. INTRODUCTION Much research has been done in the category of missile guidance, including proportional navigation (PN) (Mahapatra and shukla, 1989), true proportional navigation (Ghose, 1994) and optimal guidance law (Ke, 2003) . All guidance laws that mentioned above, however, are designed with the assumption of ideal performance of missile control system or availability of target acceleration, but in real situations, we can not expect the ideal performance of the control system or accurate measuring of target acceleration, so the performance of these laws are decreased. Therefore in this article an integrated guidance and control model, including dynamic and parametric uncertainties of missile, has been used for extracting of guidance law, also the target acceleration is considered as an unknown disturbance input. So far different guidance laws have been introduced for compensating the effect of uncertainties and target manoeuvre, such as robust adaptive back stepping control (Chaw and Choi, 2003) and dynamic surface control (Swaroop, et al., 2000). Above methods, however, in spite of robustness against uncertainties

usually have complicated structure that needs a large number of inputs and so implementation of them is difficult. So in this article, we utilize the universal approximation capability of adaptive-Takagi-Sugeno fuzzy systems for approximating of unknown dynamic parts. In addition a high gain observer is used together with the guidance law which causes that this approach needs only the missile-target distance and the line of sight (LOS) angle as its input variables. Also for ensuring the closed-loop stability and reducing the tracking error, due to approximation error of adaptive-fuzzy system, a compensating part is augmented to the guidance law that measures the error bound in accordance with an estimation law. So this approach, in addition to robustness against uncertainties and target manoeuvre, use the minimum number of inputs for missile guidance. We prove the stability of guidance–control loop by lyapunov method and by performing different simulations, the effective performance of designed guidance law is shown.

2. DERIVATION OF GUIDANCE-CONTROL MODEL In this section an integrated guidance-control model is obtained. Fig. 1 shows the relation between guidance and control loops. The control loop consists of missile system and nonlinear controller, and guidance loop produces the reference acceleration for control loop. Since the considered missile in this article is with type of S.T.T.1 that has Y and Z symmetry, so guidance and control problems can be considered separately for missile movement in each of yaw and pitch planes (Blackelock, 1991). In the sequel, only movement in the yaw plane is considered. With assumptions that usually considered in the design of control system of these missiles, the yaw dynamic is obtained as:

y component

of

 y = a1 y + a2 y + a3 y + a4 - k 3wn 2u

(1) where

missile

Dy , Dn show uncertainties in modelling of these coefficients. In order to design the controller, we use a procedure similar to (Chaw and Choi, 2002). According to this approach, control inputs are obtained such that above mentioned dynamic equations are approximately linearized. By applying these control inputs to (1), we can show that the yaw dynamic follows a reference model as: Ay + 2xw n A y + w n 2 A y = w n 2 A y c + D y c

(2)

where A yc is the reference acceleration that is produced by the guidance loop, Dyc is an uncertainty from

missile

(3)

R

where R y is the y component of R . By repeated differentiating of equation (3) and substituting A

uncertainties Dy , Dn and

wn , 2xw are design parameters.

y = s , u = A yc

and

by

(4) assuming

R 2R 1 k1 = , k2 = , k 3 = , we obtain coefficients of R R R

(4) as: a1 = - k 2

(5)

a2 = -( k 1 + 2k2 ) a3 = -( k2 + 2k1 )

(6) (7)

a4 = A y (-k3 + wn 2 k 3 ) + A y (-2k3 + 2xwn k 3 ) + k3 AT + 2k3 AT + k 3 AT - k 3 Dyc

acceleration, c y , c n are aerodynamic coefficients and

arising

Ry

from (2) we have:

where Q is dynamic pressure, M m is mach number, S is aerodynamic reference area, D is the length of missile, m is missile mass, U is missile velocity, b is side slip angle, d r is deflection of yaw control fin, r is the y component of angular velocity vector of the

s =

y

ì QS ï b = -r + Um (c y ( M m , b , d r ) + Dy ) ï ï  QSD (c n ( M m , b , d r ) + Dn ) ír = Im ï ï QS (c y ( M m , b , d r ) + Dy ) ïA y = î m

missile, A y is

Fig. 2 shows the relative position of missile and target in yaw plane. In this figure R is missile-target distance and s is line of sight (LOS) angle. The base that is used in the proposed adaptive fuzzy guidance law is that of well-known PN law. According to this guidance law, the acceleration command is generated in the direction of making the rate of rotation of the LOS angle ( s ) be equal to zero. In reality the purpose of the suggested guidance law is that the rate of rotation of the LOS angle tends to zero in spite of all existing uncertainties in the control loop and target manoeuvre. Since s is a small value, it can be defined as:

(8)

where AT is the target acceleration. By selecting X = [ y y y]T as state vector, (4) is transformed into an integrated guidance and control model as (9) and (10) which have the standard structure of a regulation problem: X = AX + B (f ( x , t ) + g (t )u + d ) (9) y =CT X

(10)

Where é0 1 0ù é0 ù é1 ù A = êê0 0 1úú , B = êê 0úú , C = êê 0úú êë0 0 0úû êë1 úû êë0úû

f (X ,t ) = a1 y + a2 y + a3 y

(11) (12) (13)

g (t ) = -k 3wn 2 , d = a4

2

Guidance Law

Acceleration Command

Guidance Loop

Nonlinear Controller

Fin Demand

Control Loop

Fig.1. Guidance and control loops 1

-Skid To Turn

Y

T

Missile

M

R

s X

Fig. 2. Relative position of missile and target in yaw plane

3.

Adaptive Takagi-Sugeno Fuzzy Guidance Law

In this section, the design procedure of the adaptive Takagi-Sugeno fuzzy guidance law is explained and shown that the output (the line of sight angle rate), can be made to be sufficiently small even in the presence of uncertainties and the estimation error. Initially we obtain error equations as: e = Ae + B ( g (t )u + f (e , t ) + d ) (14) e1 = C T e

where the error vector q considered as:  ˆ qf = qf * - qf

Theorem 1: If the guidance law is designed as (18), in which the adaptation law of adaptive fuzzy system is obtained by (26) and the adaptive compensator is designed as (27) with the estimation law of the error range in (28), then asymptotic stability of the guidance system can be guaranteed :



(15)

qˆf = g 1e T PB f (e )

T

where e = [e1 e 2 e 3 ] and error variables have been considered as: ìe1 = y = x 1 ï (16) íe 2 = y = x 2 ïe = y = x 3 î 3 We choose a matrix k such that A m = A - Bk becomes Hurwitz, and rewrite (14) as: e = A e + B (ke + g (t )u + f (e , t ) + d ) (17) m

The adaptive fuzzy guidance law is considered as: (18) u = (-1/ g (t ))(ke + fˆ (e | qˆ ) + u ) f

add

 where fˆ (e | qˆf ) is output of adaptive fuzzy system which estimates f (e ,t ) as: ˆ ˆ fˆ (e | qf ) = qf T f (e ) (19) where q T = [q1T q 2T ,... ql T ...,q M T ] is a vector of

coefficients of consequence part of fuzzy rules that adjust with an adaptation law and estimates the optimal parameter vector defined as:  q * = arg min[ sup fˆ (e | q ) - f (e , t ) ] (20) 

f

in

Me,M f

f

e
qf < M f

the

above equation are

ranges

of e and vectors respectively. q f T (e ) = [f 1 (e ), f 2 (e ),...f l (e ),..., f M (e ) ] in (19) is a fuzzy basis function vector which defined as:

u add

f l (e ) =

j =1 M

j

S ( P mF l (e j ))

l =1 j =1

where mF l (e j ) is the membership value of the fuzzy j

variable e j . u add shows the output of an adaptive controller that has been augmented to guidance law for compensating of estimation error of adaptive fuzzy system and existing uncertainties in the model. Adding and subtracting fˆ (e | q f * ) = q f *f (e ) in (17) yields:  e = A m e + B (ke + g (t )u + f (e | q f * ) + w ) (22) where w is the fuzzy system approximation error and defined as: w = f (e , t ) - fˆ (e | qf * ) + d (23) Substituting u in (22) yields:  e = A e + B (q f (e ) - u m

f

add

+w )

(24)

(28)

¥

obtained as: ì1 x > 0 sgn(x ) = í î -1 x < 0 Proof: We define a lyapunov function as: 1 1 T  1 V = e T Pe + qf qf + w m 2 2 2g 1 2g 2

(29)

(30)

where w m = w m -wˆ m and P Î R 3´3 is a positive definite symmetric matrix that is obtained from the lyapunov equation as: PA m + A mT P = -Q (31) where Q Î R 3´3 is also a positive definite symmetric matrix. Taking the derivative of the lyapunov function, it is obtained that: 1 1 1   1 V = eT Pe + e T Pe + qf T qf + w m (32) 2 2 g1 g2 Substituting (24) and (31) into (32) and using   q = -qˆ , w = -wˆ yields: f

f

m

m

1 V = - e T Qe + e T PB (-u add + w ) + 2

1  ˆ 1  + e T PBq f T f (e ) - q f T q f - w mwˆ m (33)

g1

By

j

(27)

where g 1 and g 2 are positive constants and wˆ m show the estimated value of w m which is the maximum value of w ( w £ w m ), also the sgn(.) function is

(21)

n

(26)

= wˆ m sgn(e T PB )

wˆ m = g 2 e T PB

n

P mF l (e j ).[1, e T ]

(25)

g2

substituting

(26) into (33) and using w m = w m -wˆ m we have: 1 V = - e T Qe - e T PBu add + e T PBw 2 -

1

g2

(w m - wˆ m )wˆ m

(34)

So the following inequality is obtained for V : 1 V £ - e T Qe - e T PBu add + 2 + e T PB w

¥

-

1

g2

(w m - wˆ m )wˆ m

(35)

Substituting (27) and (28) into (35) yields: 1 V £ - e T Qe - e T PB (w m - w ¥ ) 2 1 1 £ - e T Qe £ - lmin (Q ) e 2 2

2

£0

(36)

where lmin (Q ) show the minimum eigenvalue of Q . Since V (t ) £ 0 so V (t ) is negative semi definite (V (t ) £ V (0) ) and e will be bounded ( e Î L¥ ). By integrating two sides of ¥ ò

0

e

(36) we have:

e

2

V (0) -V (¥) dt £ 1 lmin (Q ) 2

(37)

Since the right hand side of (37) is bounded so e Î L 2 and by Barbalet lemma it can be shown that: lim e = 0

(38)

t ®¥

Therefore lim e1 = 0 and so lim ( y = s ) = 0 . t ®¥

t ®¥

In general, parameter adaptive laws like (26) can not  ensure the bounded ness of qf in the process of adaptation. In order to guarantee it bounded, we introduce projection operator (Wang, et al., 2002) for (26). For this purpose we initially define following closed set: W = {q f | qf £ M f } (39) where M f is specified by the designer. The improved adaptation law is:   ìg 1e T PB f q £ M f or ( q = M f and ï   ï e T PB q f T f £ 0) (40) qf = í   ï T T q = M f and e PB q f f > 0 ïîPr oj [.] where the projection operator defined as:   f

f

f

Proj [.] = g 1e T PB f - g 1e T PB

qf qf T f 

qf

2

(41)



It can be verified that this law satisfies q f Î W for all 

t ³ 0 if q f (0) Î W , and also the asymptotic stability condition (38). We assume that all the initial conditions are 

bounded. In particular, q f (0) Î W and e (0) Î E 0 , where E 0 is a compact set and can be chosen large enough to cover any given bounded initial condition. 

We have already seen that q f (t ) Î W for all t ³ 0 . Let 1   c1 = max e T Pe , c2 = max q f T q f (42) e ÎE 0 2g 1 qf ,qf *ÎW def

e . With the goal of recovering the performance achieved under state feedback, we use the high gain observer as:  q (43) e i = i i-1 , 1 £ i £ 3

{

}

and c 3 > c1 + c 2 .Then e (t ) Î E = e (t ) | e T Pe £ c 3 for  all t ³ 0 . Since the control input u (e , qf ) defined by (18) and the vector f (e ) defined by (21) are  continuous functions of e and qf , so they are bounded on compact sets of these variables. 4. High Gain Observer (H.G.O) To implement the adaptive fuzzy guidance law using the output feedback ( y = e1 ), we need to estimate

where

e qi = q i +1 + a i (e1 - q1 ), 1 £ i £ 2 (44) e q3 = a 3 (e1 - q1 ) (45)

where e is a small positive parameter to be specified. The positive constants a i are chosen such that the roots of : s 3 + a1s 2 + a 2 s 1 + a 3 = 0 (46) have negative real parts. The system (44), (45) is a singularity perturbed model and will not exhibit peaking if the input e1 and the initial conditions are bounded functions of e . By taking  e -e xi = i i -1 i , 1 £ i £ 3 (47)

e

T

and x = [x1 , x 2 , x3 ] , we represent the closed loop system (22) in the standard singularity perturbed form:  e = A e + B (ke + g (t )u (e - D (e )x ,q ) + m

f

+ f (e - D (e )x )qf * + w )



(48)

ex = ( A - HC )x + e B ( g (t )u (e - D (e )x , q f ) + + f (e - D (e )x )qf * + w )

(49)

where H = [a1 , a 2 , a 3 ]T , and the characteristic equation of A - HC is (46), hence it is Hurwitz, D (e ) is a diagonal matrix with e 3-i ( 1 £ i £ 2 ) as the

 ith diagonal element, also we substitute e with e - D (e )x in the above equations.

The e -dependent scaling (47) causes an impulsive– like behaviour in x as e ® 0 , but since x enters the slow equation (48) through bounded functions  u (e - D (e )x , q f ) and f (e - D (e )x )q f * , slow variables  (e , q f ) do not exhibit a similar impulsive like

behaviour. Theorem 2: Consider the closed loop system (48) and 

(49), suppose that q f (0) Î W , e (0) Î E 0 and q (0) be bounded. Then there exists e 0 > 0 such that for all 0 < e < e 0 , all state variables of the closed –loop system are bounded. Since the proof of Theorem 2 is completely similar to (Khalil, 1993) so for abbreviation of paper we don’t state it again. We can establish that for sufficiently small e ® 0 , the output feedback adaptive fuzzy guidance law can recover the performance achieved by its state feedback.

vector component in this table represents the value along the y and z axis respectively. In all scenarios the target initially travels at constant speed and then make step changes in acceleration. Final scenario is performed in the presence of uncertainty in aerodynamic coefficients. Tables 2 and 3 compare the miss distance and flight time of AFGL with that of P.N. under each scenario. We can see that the proposed AFGL has better performance than P.N. in all scenarios especially for targets with high velocity or in the presence of uncertainty in missile dynamics.

5. Simulation Results In this section, simulation results of the designed adaptive fuzzy guidance law (AFGL), which uses states estimated by the high gain observer (HGO), for a six degree of freedom missile are expressed. Since when missile reaches the final phase of interception, minimization of miss distance and flight time have a great importance, so we select above parameters as performance measures. The performance of the proposed method is compared with P.N. which considered as: (50) u PN = NV c s

Figs. 3-6 compare the three-dimensional missiletarget trajectory, reference and real missile accelerations and the line of sight rate ( y = e1 = s ) of the AFGL with that of PN in the second and sixth scenarios. Figs. 3.b and 5.b show that the estimated values of the line of sight rate produced by the high gain observer track its real value after a short transient time. Above Figs. show that the AFGL with states that is estimated by the high gain observer apply more reasonable acceleration commands which, in contrast to PN case, keeps the line of sight rate in a limited bound around the zero and so leads to a more suitable interception trajectory. We performed simulations for other different scenarios as well and found that the proposed one is effective in most cases.

Where N is navigation constant chosen as 4 and V c is closing velocity. We consider real missile dynamic equations and environmental conditions for simulations. The saturation bound of guidance commands is selected as u £ 40g . Design parameters for controller are x = .7,wn = 15 and missile control start time is 1s . Considered values for AFGL are g 1 = 80, g 2 = 15 , Q = diag ([5,5,5]), k = [15,17, 7] and e = .05, H = [16, 70,100]T have been chosen for the high gain observer. For performance evaluation of AFGL, we consider scenarios expressed in Table 1. Each

Table 1. Considered target maneuvers

Time (s)

Time (s)

0 0 1 0 1 0 0 1

At=[0,-5] At=[8,-8] At=[-8,8] At=[-10,5] At=[0,-7] At=[8,-8] At=[15,-10] At=[-15,15]

2.5 2 3 2.5 2.5 2 2 3

At=[6,-6] At=[10,-12] At=[-4, 4] At=[,-15,10] At=[15,-15]

0

At=[8,-8]

2

-

Table 3. Flight Time

40

PN

1 2 3 4 5 6 7 8 9

5.933 6.95 5.688 5.642 5.71 6.05 6.51 6.233 6.956

6.532 7.65 5.77 5.83 6.87 6.2 6.73 6.42 7.2

Ay Ayc

20

Ay[g]

AFGL

0 -20 -40

0

1

2

3

4

5

6

20 0 -20 1

2

3

PN

1 2 3 4 5 6 7 8 9

.6428 1.13 1.872 2.67 2.967 4.833 5.436 7.39 2.1

.779 2.8 4.12 10.742 10.42 15.37 25.54 33.4 6.92

real estimated

0 -0.1 -0.2

0

1

2

3

4

5

6

7

time[sec] Az Azc

0

AFGL

0.1

time[sec]

-40

Scenario

0.2

7

40

Az[g]

Scenario

LOS RATE(y)[rad/sec]

250

evasive

4

5

6

0.2

7

real estimated

0.1 0 -0.1 -0.2

time[sec]

0

1

2

3

4

5

time[sec]

(a)

(b) 2000

M T

1500

T

M Z[m]

1 2 3 4 5 6 7 8 9 (with uncertainty)

evasive

(m/ s2) 250 250 250 250 350 350 350 350

Table 2. Miss Distance

Second Target Acceleration (m/ s2)

Second

LOS RATE(z)[rad/sec]

Vt0

Scenario

First Target Acceleration (m/ s2)

First

1000

T T

T

T

M M

500

M 0 3000

M 2000

3000

M M

1000

2000

0

Y[m]

1000 -1000

0

X[m]

(c)

Fig. 3. Performance of AFGL in engagement scenario 2 (a) Acceleration of missile (b) LOS rate (c) Threedimensional trajectory of missile and target

6

7

40

Ay[g]

0 -20 -40

0

1

2

3

4

5

6

7

0.2

LOS RATE(y)[rad/sec]

Ay Ayc

20

0.1 0 -0.1 -0.2

8

0

1

2

3

Az[g]

0 -20 -40

0

1

2

3

4

5

6

7

5

6

7

8

5

6

7

8

0.2

LOS RATE(z)[rad/sec]

Az Azc

20

4

time[sec]

time[sec] 40

0.1 0 -0.1 -0.2

0

1

2

3

8

4

time[sec]

(b)

time[sec]

(a)

2000

T M T

Z[m]

1500 1000

T

T

T

M

500

M

0 3000

M 3000

M

2000

2000

M

1000 0

Y[m]

1000

0

(c)

X[m]

-20 -40

0

1

2

3

4

5

6

7

real estimated

0.1

-40

0

1

2

time[sec]

-20 1

2

3

4

5

6

7

0

1

2

4

5

6

7

4

5

6

0

7

20 0

-40

0

1

2

time[sec]

3

4

5

6

time[sec]

(b)

(a)

7

0.1 0 -0.1 -0.2

0

1

2

3

0

1

2

3

4

5

6

7

5

6

7

time[sec] Az Azc

-20

-0.1

0.2

time[sec]

real estimated

0.1

-0.2

3

40

0.2

Az[g]

0

4

5

6

7

0.2 0.1 0 -0.1 -0.2

0

1

2

3

4

time[sec]

time[sec]

(b)

(a) 1500

1500

T

T

T

T

1000

M

0 3000

M 3000

M

2000 1000

Y[m]

M 0

T

T

M

M 0 3000

T

T

M

500

M

500

T M

Z[m]

T

1000

Z[m]

Az[g]

LOS RATE(z)[rad/sec]

Az Azc

20

0

3

time[sec]

40

-40

0 -20

-0.1 -0.2

Ay Ayc

20

0

LOS RATE(y)[rad/sec]

Ay[g]

0

40

0.2

LOS RATE(z)[rad/sec]

Ay Ayc

20

Ay[g]

40

LOS RATE(y)[rad/sec]

Fig. 4. Performance of PN in engagement scenario 2 (a) Acceleration of missile (b) LOS rate (c) Three-dimensional trajectory of missile and target

0

(c)

2000

M 3000

2000

M 1000

1000

X[m]

Fig. 5. Performance of AFGL in engagement scenario 6 (a) Acceleration of missile (b) LOS rate (c) Three-dimensional trajectory of missile and target

6. CONCLUSION We propose an observer based adaptive TakagiSugeno fuzzy guidance law to improve the overall performance of guidance and control missile system. The proposed approach, when compared with existing results, is novel in that the guidance law uses of an adaptive intelligent system to approximate unknown nonlinear parts together with a high gain observer to estimate higher order derivatives of line of sight rate. So our approach suggests a robust guidance method which uses the minimum number of inputs. Utilizing of RBF neural networks to direct approximation of a feedback linearization guidance aw in PN missiles need to be studied in future. 7.REFERENCES Blackelock J. H. (1991). Automatic Control of Aircraft and Missiles. pp. 229-259. John Wiley, New York. Chaw D. and J. Y. Choi (2002). New parametric Affine Modelling and Control for Skid-to-Turn Missiles. IEEE Transaction on Control Systems Technology, vol. 9, no. 2, pp. 335-347. Chaw D. and J. Y. Choi (2003). Adaptive Nonlinear Guidance Law Considering Control Loop Dynamics. IEEE Transaction on Aerospace and Electronic Systems, vol. 39, no. 4, pp. 11341143.

Y[m]

M 0

0

2000 1000

X[m]

(c)

Fig. 6. Performance of PN in engagement scenario 6 (a) Acceleration of missile (b) LOS rate (c) Threedimensional trajectory of missile and target

Ghose D. (1994). True Proportional Navigation with Manoeuvring Target. IEEE Transaction on Aerospace and Electronic Systems, vol. 30, no 1, pp. 229-237. Ke F.. (2003). Design of Optimal Midcourse Guidance Sliding-Mode Control for Missiles with TVC. IEEE Transaction on Aerospace and Electronic Systems, vol. 39, no. 3, pp. 824-837. Khalil H. K. (1993). Asymptotic Regulation of Minimum Phase Nonlinear Systems Using Output Feedback. in Processing of American Control Conference, San Francisco, CA, pp. 1490-1494. Mahapatra P. R. and U. S. Shukla (1989). Accurate Solution of Proportional Navigation for Manoeuvring Targets. IEEE Transaction on Aerospace and Electronic Systems, vol. AES25, no. 1, pp. 81-89. Swaroop D., J. K. Hedrick, P. P. Yip, and J. C. Gerdes (2000). Dynamic Surface Control for a Class of Nonlinear Systems. IEEE Transaction on Automatic Control, vol. 45, no. 10, pp. 18931899. Wang, C. H., H. L. Liu and T. C. Lin (2002). Direct Adaptive Fuzzy-Neural Control with State Observer and Supervisory Controller for Unknown Nonlinear Dynamical Systems. IEEE Transaction on Fuzzy Systems, vol. 10, no. 1, pp. 39-49.