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Lateral Acceleration Control Design of a Non-Linear Homing Missile
Copyright Cl IFAC Nonlinear Control Systems Design, Enschede, The Netherlands, 1998
LATERAL ACCELERATION CONTROL DESIGN OF A NON-LINEAR HOMING MISSILE B.A. White A. Tsourdos and A.Blumel
Department of Aerospace, Power and Sensors, Royal Military College of Science, Cranfield University, Shrivenham,SN6 BLA, England, UK brian, tsourdos and blumel @rmsc.cranfield.ac. uk
Abstract. Input-output approximate linearisation of a non-linear fourth order system has been studied. A method for controlling the non-linear system that is ijo linearisable is examined that retains the order of the system in the linearisation process, hence producing a linearised system with no internal or zero dynamics. Desired tracking performance for lateral acceleration of the missile is achieved. Simulation results are shown that exercise the final design and show that the linearisation and controller design are satisfactory. Copyright © 1998 IFAC Keywords. non-linear control, feedback linearisation, trajectory control, missile system
1. INTRODUCTION
earised systems that have no internal dynamics, techniques which preserve the dynamic order of the system are needed. Several approaches are possible to the avoidance of internal or zero dynamics. One approach, named the g-modification technique neglects terms in input derivatives until the required system order is reached (John Hauser, 1992). Another is to pre-compensate the system to increase the system relative degree artificially, and thus having limited some authority over the stability of the internal dynamics (Isidori, 1985). Designing systems with unstable zero dynamics can also be achieved (X. Y. Lu, 1997) provided the input to the system remains bounded under feedback . A fourth way is to choose an output which has the required relative degree, and which is related to the required control output in some manner. A paper that looks at the use of sideslip velocity as an approximation to lateral acceleration (A. Tsourdos, 1998) deals with the use of an output of the required relative degree after some small terms are neglected. This paper looks at a better approximation that does not require any terms to be neglected.
This paper looks at the application of feedback linearisation to a missile model that is described by lookup tables that define non-linear characteristics of the aerodynamics. One of the main problems with applying feedback linearisation techniques is that the process produces a system with the same relative degree as the original system, but usually with an order that is less (H. Hahn, 1994), (Johan Suykens, 1995), (Michael A. Henson, 1990), (Scott Bezick, 1995). The linearised system order is the same as the relative degree unless pre-compensators are used to artificially change the order and relative degree. This process results in zero or internal dynamics, which are modes that are effectively rendered unobservable by the linearisation process. H the system is non-minimum phase, then the zero dynamics are unstable. The analogy with linear systems is that a zero-pole system is linearised into an all-pole system by selecting the pole-zero excess as the order of the approximating system. In order to produce lin-
681
The model used is developed by Matra-British Aerospace as a research model for non-linear system design (M.P.Horton, 1992). Both linear spot point designs have been examined by Horton (M.P.Horton, 1992), and a quasi-linear approach by White (B.A.White, 1997). The aim of this paper is to track the missile lateral acceleration demand in both the pitch and yaw plane. The tracking and non-linear controllers are designed by defining lateral acceleration as an output augmented in such a manner that the body acceleration rather than the c.g. acceleration is used as the controlled output. Applying standard Input/Output linearisation the relative degree Of the lateral acceleration of the missile is Tt + TZ = O. This results in an equivalent linear system with second order unstable internal dynamics. Rather than apply the g-modification technique (John Hauser, 1992) at this stage, another output that defines the body acceleration component of the lateral acceleration is chosen as the controlled output. The missing component is the acceleration generated by the control fins. This will always be small in well designed missiles, as the main effect of the fins is to produce a turning moment to generate incidence for body lift; the lateral force component is usually small compared to body lateral force.
Fig. 1. Airframe axes can be representE!q by polynomials which can be fitted to the set of curvei taken from a look-up tables for different flight conditions. Moments and forces are related by:
= SmCyv Cn( = S,Cy(
C nv
C mw = -SmCzw Cm 1) = -S,CZ 1) (5)
where:
2. MISSILE MODEL The missile model used in this study derives from a non-linear model produced by Horton of Matra-British Aerospace (M.P.Horton, 1992) . It describes a 5 DOF model in parametric format with severe cross-coupling and non-linear behaviour. This study will look at the reduced problem of a 4 DOF controller for the pitch and yaw planes without roll coupling. The angular and translational equations of motion of the missile airframe are given by:
(6)
A detailed description of the model can be found in (A.Tsourdos, 1997) . The aerodynamic forces and moments acting on the airframe and defined in equations (5) and (6), are nonlinear functions of Mach number, longitudinal and lateral velocities, control surface deflection, aerodynamic roll angle and body rates.
q = ~I;/PVoSd(~dCmqq + Cmww + VoCm1)1J) (1) . 1 W = 2mPVoS(CzwW + VoC Z 1)1J)
+ Uq
Control of the missile will be accomplished in this paper by controlling an augmented version of lateral acceleration. The dynamic equation for lateral acceleration can be derived (A.Tsourdos, 1997) and is given by:
(2)
r = ~ 1;'/ pVoSd( ~dCnrr + Cnvv + VoCn«()
(3)
v = 2~PVoS(Cl/vv + VoCI/«() - Ur
(4) o=v+Ur
where the variables are defined in Figure 1. Equations 2,4 describe the dynamics of the body rates and velocities under the influence of external forces (e.g. Czw ) and moments (e.g. Cmq ), acting on the frame. These forces and moments are derived from wind tunnel measurements and by using polynomial approximation algorithms Cl/V, CI/(' Cnr , Cnv and C n( (A.Tsourdos, 1997)
0=
VO(Cl/Vv + VoCl/«()
I Dv +Vo(Cl/(o + Cl/(MM + Cl/(" I IKl =VO[(Cl/VOv + Cl/V" I v I v
= VO[(C IIVo + CllvMM + Cl/V"
(J'
(J'
+VoCl/(o( + VoCI/(" I v I Cl
682
=cp(v) + 1/J(v, ()
t.iJ
(7)
= fw(q, w) + gw(q , w)l1
(10)
which can be written in matrix format. As the pitch and yaw equations are not coupled in this example, and as the missile is symmetric in both planes, only one plane (the yaw plane) need be treated as shown in equation (ll).
where the Mach number M, and the total velocity Vo are slowly varying and where:
where:
x= U
[Xl
X2] T
=
[v
r ]T
= [ud T = [(r
The functions fv-r(x) and gv-r(x) are given by equations (12) :
fv(v,r) From equation (7), the output contains the input control fin deflection (by virtue ofthe term 'I/J(v, (). This makes the lateral acceleration have a relative degree of zero. This term , however, can be shown to be the lateral force developed by the fin . The fin 's main contribution to the dynamics of the missile is to develop a turning moment, by virtue of the term ~I;}pV;SDCn«( in equation (3) and the equivalent term in equation (1) . If this term is included in the output equation, then:
= 0: -1/J(v , () = cp(v) = VO[(CyVOv + Cyv" I v I v]
Ur
gv(v , r) = VOVo(Cy(o + Cy(" I v I 1 fr(v,r) = -RO«dxcPOCyVO + (xcpoCyv"
(8)
Q
= VO(Cyvo + Cyv" I v I)v -
+xcp"Cyvo) I v I +xcp"CyV" I v 1
-
I)v
-
+ Cnr" I v Dr gr(v , r) = RO S, Vo(Cy(o + Cy(" I v I) +2dRO(Cnro
(12)
4. APPROXIMATE INPUT-OUTPUT LINEARISATION
(9)
The state-space form of the non-linear system of the home missile can now be written in a compact parametric format, as:
The augmented or body acceleration Q is used for lateral control instead of the lateral acceleration 0:. The difference between the two outputs 0: and Q is now just the lateral acceleration developed by the control fin, and as such will not introduce much error in the control of the lateral acceleration.
+ a2Xr + a3X2 + (a4 x I + a5)uI = blX~ + b2Xr + b3XI + b4XIX2 + b5 X 2
Xl = alXI X2
+(b6 xI
+ ~ )UI (13)
3. NON-LINEAR STATE-SPACE MODEL FOR LATERAL DYNAMICS
The augmented acceleration
Q
is given by:
The equations (2),(4) describing the angular and translational dynamics of the non-linear system, have also been recast in polynomial format, to give:
(14) or in matrix form:
+ gv(v, r)( r = fr(v, r) + gr(v , r)( q = fq(q, w) + gq(q, w)l1
v=
fv(v, r)
X = f(x)
+ g(x)u y = h = [hI] = [yI]
683
(15)
This equation is now in standard form and input-output linearisation techniques can be applied to it . In order to apply the input-output linearisation and to retain the system order in the linearisation process, an approximate input-output linearisation technique must be applied to the missile model. Modification of the 9 function detailed in (John Hauser, 1992) is used to produce a relative degree of 2 for the augmented acceleration output.
(17) Hence the output Yl possesses a relative degree rl of 2 and have no internal dynamics. Similarly, for the pitch plane, let
Using this approximation technique, terms are discarded in order to retain an approximate system with an equivalent order and relative degree. In other words the 9 vector field is modified. This is achieved by neglecting the terms 'rh (x , Ul) shown in (16) Let .
6
6
6
=~4
~4
= 02 + {32u2 = V2(X, u)
The output Y2 also possesses a relative degree with no internal dynamics.
= hI (x) . Then:
of 2
I,
..
+ "2a~xr + 2a2a3xlx2,
.
{2=t/>1 ( x )
+ (a2a4Xr + (ala4 + a2a5)x~ + a l a5xr)u1 "
,
'"
The effect of neglecting the term '1/11(x, ur) in equation (16)is to eliminate a non-linear zero in the system within the model description, It had be shown in (B ,A,White, 1997) and (A.Tsourdos, 1997), this will not affect the performance of the control design in a significant manner as the zero can be approximated by:
tPt{Xl ,Ul) 4
= ...(6a 2 + 2a2a3bl)xl
~
"'1
+ ...(12ala~ + a1a3bl + 2a2a3b2)Xr, "'1
+ ...(a~ + 6aia2 + ala3b2 + 2a2a3b3)xi,
'1/11 (x) z ~ - {31(X)
...
J
"'1
+ ...(2a2a~)x~ + (aia3 + ala3b5)x2' + (6aia4Xr + 2a2(3a1a4 + 3a2a5 + a3b6)xi), Ul
...
~
PI
+ .((6 a 1a2a5 + ai a 4 + ala3b6 + 2a2a3b7)xr), Ul ...
P1
Equations (17), (18) represent a direct relationship between the outputs hi and the inputs Ui (Wang, 1994) . The required static state feedback for decoupled closed loop input/output behaviour (Costas Kravaris, 1990) is given by as:
+ (2a2a3(a4 x l + a5)x2»U1 ...
(19)
Examination of equation (19) shows that z is always positive if the fin moment arm is greater than the static margin. This is true in all well designed missiles as the fin moment arm gives the small fin force sufficient turning moment to overcome the lift induced moment from the static margin. Hence the non-linear zero will always be in the stable left half s plane. It will tend to enhance the stability of the closed loop system, rather than detract from it; hence if the linearisation takes place without taking the zero into account, the resulting system should be more stable.
+ (8a1a2a3 + ala3b4 + 2a2a3b5)XIX2
...
r2
The total relative degree (Jean-Jacques E. Slotine, 1991) of the system which is equal with the summation of the rl and r2 , is now 4, and has the same order as the original system, Therefore there are no internal dynamics , Since the total relative degree is equal with the order of the system, fully linearisation of the non-linear system can now be achieved.
2 2 = ...(alxl + 3ala2xl + ala3 X2 "
3
= h3(x). Then:
(18)
{2=t{X )
6.
6
I
P1
+ ...(a1(ala5 + a3b7» , U1
.
(31
(16) U
= E- 1 { V - [~~]}
(20)
or where E is the characteristic ((Costas Kravaris, 1990» or decoupling ((Jean-Jacques E . Slotine, 1991» matrix
6=6
684
each channel, kl = 2(wn and k2 = w;, where Wn = 60(rad/ sec) and ( = 0.65. The speed of response is significantly faster than the open loop missile response and so should exercise the dynamics of the non-linear missile sufficiently for meaningful conclusions to be drawn. .
of the system, and is given by:
E-
1 = [1/{31
o
0]
1/{32
(21)
which is nonsingular.
The results of a 10 m/sec 2 and 50 m/sec 2 demand in acceleration is shown in Figures 3, 4 for the 10 m/sec 2 case, and Figures 5, 6 for the 50 m/sec 2 case. The figures show almost identical step responses for both 10,50 m/sec 2 with some variation in peaks and steady state values for the body rate, the actuator movement and the lateral velocity. The difference between the lateral acceleration and the body acceleration shows that there is a good match between the two and that steady state values are very close. This illustrates the small effect that the fin force has on the missile acceleration and justifes the use of the augmented body acceleration. The results also show that the actuator does not significantly affect the design. The non-linear approach is also shown to be reasonably accurate, as the predicted and actual performance are very close.
The linearised closed loop system is now given by:
ih
= Vi
(22)
Where V is the new linearised system input (Wang, 1994). Now choose the new control input to be: (23)
where e == Y - Yd . The close-loop system is thus characterised by: (24)
where kl and k2 are chosen such that all roots of S2 + kl S + k2 = 0 are in the open left-half plane, which ensures limt-HX) e(t) = 0 (Wang, 1994).
IIt::: :::1 r~1J:: :: ; :1
It can be said that now the tracking control problem for the non-linear system described by equations (13), (14) has been solved using the control law in equations (20), (22) and (23) . Indeed, since equation (24) has the same order as the non-linear system, there is no part of the system dynamics which is rendered "unobservable" in the approximate input-output linearisation. Since there are no zero dynamics in the linearised system, the stability of the linearised system can be guaranteed and the tracking problem has been solved (Isidori, 1985), (JeanJacques E . Slotine, 1991), (John Hauser, 1992).
o
0 .05
0 .1
0 .1 5
0 .2 Time(lMC)
0 .25
0 .3
0 .35
0 .•
o
O.OS
0 .1
0 .' 5
0 .2 Tlme(eec)
0 .25
0 .3
0 .35
0 ."
Fig. 3. Acceleration and lateral velocity for 5. TRAJECTRORY CONTROLLER DESIGN
ad
= 10
fJ\: m: . " ' ] '"
0
0.05
0.1
0.15
.. 02
0.2
n:S8C)
0.25
0.3
0.35
0.4
t.V\m :,mJmm, l ' .. ]
Fig. 2. Trajectory control design Figure 2 shows the non-linear controller structure. A fast linear actuator with natural frequency of 250 rad/sec has been included in the non-linear system. The trajectory controller performs by defined a desired acceleration as a demand. The error dynamics are constructed using the ad signal and the feedback of the actual states - velocity, rate, acceleration and jerk.
o
0.05
0.1
0.15
0.01
02
:S8C)
0.25
0.3
0.35
0.4
J 'VF~~ : ,---:-,:--:,'~,I -O.Ol~
o
0.05
0.1
0.15
0.2
0.25
0.3
Tino(eec)
The error coefficients in (24) are chosen to satisfy a Hurwitz polynomial. For the second order error equation in
Fig. 4. Rate and fin angle for
685
ad
= 10
0.35
0.4
7. REFERENCES
FE :: ::
A. Tsourdos, A. Blumel, B.A.White (1998). Trajectory control of a non-linear homing missile. In: Proceedings of the 1998 IFAC Symposium on Automatic Control in Aerospace. A.Tsourdos, A.Blumel, B.A.White (1997). Non-linear horton missile model. Technical report. The Royal Military College of Science,Cranfield University. DAGS,SEAS,RMCS,CU ,Shrivenham,Wilts,SN6 8LA. B.A.White (1997). Non-linear control of a missile. Technical report. The Royal Military College of Science,Cranfield University. DAGS,SEA-S,RM CS, CU ,Shrivenham,Wilts,SN 6 BLA. ' t. Costas Kravaris , Masoud Soroush (1990). Syntesis of multivariable nonlinear controllers by input-output linearization. AIChE Journal 36(2) , 249-264. H. Hahn, A.Piepenbrink, K.D. Leimbach (1994) . Inputoutput linearization control of an electro servohydraulic actuator. In: Proceedings of the 1994 IEEE Conference on Control Applications. Vo!. 2. pp. 995-1000. Isidori, Alberto (1985) . Nonlinear Control Systems. Springer-Verlag. Jean-Jacques E . Slotine, Weiping Li (1991) . Applied Nonlinear Control. Prentice Hall. Johan Suykens, Joos Vandewalle (1995). Feedback linearization of non linear distortion in electrodynamic loudspeakers. Journal of the Audio Engineering Society 43(9), 690-694. John Hauser, Shankar Satry, Petar Kokotovic (1992) . Nonlinear control via appriximate input-output linearization: The ball and beam example. IEEE Transactions on Automatic Control 37(3), 392-398. Michael A. Henson, Dale E . Seborg (1990). Inputoutput linearization of general nonlinear processes. AIChE Journal 36(U), 1754-1757. M.P.Horton (1992) . A study of autopilots for the adaptive control of tactical guided missiles. Master's thesis. University of Bath. Scott Bezick, llan Rusnak, W. Steven Gray (1995). Guidance of a homing missile via nonlinear geometric control methods. Journal of Guidance, Control and Dynamics 18(3), 441-448. Wang, Li-Xin (1994) . Design of adaptive fuzzy controllers for nonlinear systems by input-output linearization. In: Proceedings of the 1994 1st International Joint Conference of NAFIPS IFIS NASA . pp. 89-93. X. Y. Lu, S. K. Spurgeon, I Postlethwaite (1997) . Robust variable structure control of a pvtol aircraft. International Journal of Systems Science 28(6), 547558.
1
~-~O~~07. . 0'~~0~'--~0~.~~5~0.2~~0.2~'--~0~ .3---0~ . ~--~0.' TIft'Ht(MC)
l~lE o
0 .05
0 .1
::
0 .15
0.2
:
0 .25
•
0 .3
Fig. 5. Acceleration and lateral velocity for
LlS::,
0
0.05
0.1
0.15
0.05
0.1
0 .'
0.15
0.05
ad
= 100
l ,,] 0.2
0.25
I':~ , ,'~~) : o
1
!
0 .35
Tm.{MC)
0.2 rlt11~sec)
0.25
0.2
0.25
':
0.3
0.3
0.35
0.35
0.4
1 0.4
t:------:-::----f'~ mm :'- - - -:'- - .•- ~:j o
0.05
0.1
0.15
0.3
0.35
0.4
rwne(sec)
Fig. 6. Rate and fin angle for
ad
= 100
6. CONCLUSIONS This paper presents the lateral acceleration control design of non-linear missile model that examines outputs that retain the order of the linearised system, resulting in a linear equivalent system with no internal or zero dynamics, and with a design of a trajectory control which gives small tracking errors for the lateral acceleration. The design used augmented lateral acceleration which assumed that the direct acceleration produced by the fin is small compared to the body acceleration. It also showed that a neglected zero during the linearisation process was minimum phase. If lateral acceleration had been chosen as the linearisation output, a non-minimum phase system would result with the associated unstable internal dynamics. Other linearisation techniques are being examined which retain system order as well as relative degree. This approach will again not produce internal dyanmics and can thus be used on non-minimum phase systems.
686
LATERAL ACCELERATION CONTROL DESIGN OF A NON-LINEAR HOMING MISSILE B.A. White A. Tsourdos and A.Blumel
Department of Aerospace, Power and Sensors, Royal Military College of Science, Cranfield University, Shrivenham,SN6 BLA, England, UK brian, tsourdos and blumel @rmsc.cranfield.ac. uk
Abstract. Input-output approximate linearisation of a non-linear fourth order system has been studied. A method for controlling the non-linear system that is ijo linearisable is examined that retains the order of the system in the linearisation process, hence producing a linearised system with no internal or zero dynamics. Desired tracking performance for lateral acceleration of the missile is achieved. Simulation results are shown that exercise the final design and show that the linearisation and controller design are satisfactory. Copyright © 1998 IFAC Keywords. non-linear control, feedback linearisation, trajectory control, missile system
1. INTRODUCTION
earised systems that have no internal dynamics, techniques which preserve the dynamic order of the system are needed. Several approaches are possible to the avoidance of internal or zero dynamics. One approach, named the g-modification technique neglects terms in input derivatives until the required system order is reached (John Hauser, 1992). Another is to pre-compensate the system to increase the system relative degree artificially, and thus having limited some authority over the stability of the internal dynamics (Isidori, 1985). Designing systems with unstable zero dynamics can also be achieved (X. Y. Lu, 1997) provided the input to the system remains bounded under feedback . A fourth way is to choose an output which has the required relative degree, and which is related to the required control output in some manner. A paper that looks at the use of sideslip velocity as an approximation to lateral acceleration (A. Tsourdos, 1998) deals with the use of an output of the required relative degree after some small terms are neglected. This paper looks at a better approximation that does not require any terms to be neglected.
This paper looks at the application of feedback linearisation to a missile model that is described by lookup tables that define non-linear characteristics of the aerodynamics. One of the main problems with applying feedback linearisation techniques is that the process produces a system with the same relative degree as the original system, but usually with an order that is less (H. Hahn, 1994), (Johan Suykens, 1995), (Michael A. Henson, 1990), (Scott Bezick, 1995). The linearised system order is the same as the relative degree unless pre-compensators are used to artificially change the order and relative degree. This process results in zero or internal dynamics, which are modes that are effectively rendered unobservable by the linearisation process. H the system is non-minimum phase, then the zero dynamics are unstable. The analogy with linear systems is that a zero-pole system is linearised into an all-pole system by selecting the pole-zero excess as the order of the approximating system. In order to produce lin-
681
The model used is developed by Matra-British Aerospace as a research model for non-linear system design (M.P.Horton, 1992). Both linear spot point designs have been examined by Horton (M.P.Horton, 1992), and a quasi-linear approach by White (B.A.White, 1997). The aim of this paper is to track the missile lateral acceleration demand in both the pitch and yaw plane. The tracking and non-linear controllers are designed by defining lateral acceleration as an output augmented in such a manner that the body acceleration rather than the c.g. acceleration is used as the controlled output. Applying standard Input/Output linearisation the relative degree Of the lateral acceleration of the missile is Tt + TZ = O. This results in an equivalent linear system with second order unstable internal dynamics. Rather than apply the g-modification technique (John Hauser, 1992) at this stage, another output that defines the body acceleration component of the lateral acceleration is chosen as the controlled output. The missing component is the acceleration generated by the control fins. This will always be small in well designed missiles, as the main effect of the fins is to produce a turning moment to generate incidence for body lift; the lateral force component is usually small compared to body lateral force.
Fig. 1. Airframe axes can be representE!q by polynomials which can be fitted to the set of curvei taken from a look-up tables for different flight conditions. Moments and forces are related by:
= SmCyv Cn( = S,Cy(
C nv
C mw = -SmCzw Cm 1) = -S,CZ 1) (5)
where:
2. MISSILE MODEL The missile model used in this study derives from a non-linear model produced by Horton of Matra-British Aerospace (M.P.Horton, 1992) . It describes a 5 DOF model in parametric format with severe cross-coupling and non-linear behaviour. This study will look at the reduced problem of a 4 DOF controller for the pitch and yaw planes without roll coupling. The angular and translational equations of motion of the missile airframe are given by:
(6)
A detailed description of the model can be found in (A.Tsourdos, 1997) . The aerodynamic forces and moments acting on the airframe and defined in equations (5) and (6), are nonlinear functions of Mach number, longitudinal and lateral velocities, control surface deflection, aerodynamic roll angle and body rates.
q = ~I;/PVoSd(~dCmqq + Cmww + VoCm1)1J) (1) . 1 W = 2mPVoS(CzwW + VoC Z 1)1J)
+ Uq
Control of the missile will be accomplished in this paper by controlling an augmented version of lateral acceleration. The dynamic equation for lateral acceleration can be derived (A.Tsourdos, 1997) and is given by:
(2)
r = ~ 1;'/ pVoSd( ~dCnrr + Cnvv + VoCn«()
(3)
v = 2~PVoS(Cl/vv + VoCI/«() - Ur
(4) o=v+Ur
where the variables are defined in Figure 1. Equations 2,4 describe the dynamics of the body rates and velocities under the influence of external forces (e.g. Czw ) and moments (e.g. Cmq ), acting on the frame. These forces and moments are derived from wind tunnel measurements and by using polynomial approximation algorithms Cl/V, CI/(' Cnr , Cnv and C n( (A.Tsourdos, 1997)
0=
VO(Cl/Vv + VoCl/«()
I Dv +Vo(Cl/(o + Cl/(MM + Cl/(" I IKl =VO[(Cl/VOv + Cl/V" I v I v
= VO[(C IIVo + CllvMM + Cl/V"
(J'
(J'
+VoCl/(o( + VoCI/(" I v I Cl
682
=cp(v) + 1/J(v, ()
t.iJ
(7)
= fw(q, w) + gw(q , w)l1
(10)
which can be written in matrix format. As the pitch and yaw equations are not coupled in this example, and as the missile is symmetric in both planes, only one plane (the yaw plane) need be treated as shown in equation (ll).
where the Mach number M, and the total velocity Vo are slowly varying and where:
where:
x= U
[Xl
X2] T
=
[v
r ]T
= [ud T = [(r
The functions fv-r(x) and gv-r(x) are given by equations (12) :
fv(v,r) From equation (7), the output contains the input control fin deflection (by virtue ofthe term 'I/J(v, (). This makes the lateral acceleration have a relative degree of zero. This term , however, can be shown to be the lateral force developed by the fin . The fin 's main contribution to the dynamics of the missile is to develop a turning moment, by virtue of the term ~I;}pV;SDCn«( in equation (3) and the equivalent term in equation (1) . If this term is included in the output equation, then:
= 0: -1/J(v , () = cp(v) = VO[(CyVOv + Cyv" I v I v]
Ur
gv(v , r) = VOVo(Cy(o + Cy(" I v I 1 fr(v,r) = -RO«dxcPOCyVO + (xcpoCyv"
(8)
Q
= VO(Cyvo + Cyv" I v I)v -
+xcp"Cyvo) I v I +xcp"CyV" I v 1
-
I)v
-
+ Cnr" I v Dr gr(v , r) = RO S, Vo(Cy(o + Cy(" I v I) +2dRO(Cnro
(12)
4. APPROXIMATE INPUT-OUTPUT LINEARISATION
(9)
The state-space form of the non-linear system of the home missile can now be written in a compact parametric format, as:
The augmented or body acceleration Q is used for lateral control instead of the lateral acceleration 0:. The difference between the two outputs 0: and Q is now just the lateral acceleration developed by the control fin, and as such will not introduce much error in the control of the lateral acceleration.
+ a2Xr + a3X2 + (a4 x I + a5)uI = blX~ + b2Xr + b3XI + b4XIX2 + b5 X 2
Xl = alXI X2
+(b6 xI
+ ~ )UI (13)
3. NON-LINEAR STATE-SPACE MODEL FOR LATERAL DYNAMICS
The augmented acceleration
Q
is given by:
The equations (2),(4) describing the angular and translational dynamics of the non-linear system, have also been recast in polynomial format, to give:
(14) or in matrix form:
+ gv(v, r)( r = fr(v, r) + gr(v , r)( q = fq(q, w) + gq(q, w)l1
v=
fv(v, r)
X = f(x)
+ g(x)u y = h = [hI] = [yI]
683
(15)
This equation is now in standard form and input-output linearisation techniques can be applied to it . In order to apply the input-output linearisation and to retain the system order in the linearisation process, an approximate input-output linearisation technique must be applied to the missile model. Modification of the 9 function detailed in (John Hauser, 1992) is used to produce a relative degree of 2 for the augmented acceleration output.
(17) Hence the output Yl possesses a relative degree rl of 2 and have no internal dynamics. Similarly, for the pitch plane, let
Using this approximation technique, terms are discarded in order to retain an approximate system with an equivalent order and relative degree. In other words the 9 vector field is modified. This is achieved by neglecting the terms 'rh (x , Ul) shown in (16) Let .
6
6
6
=~4
~4
= 02 + {32u2 = V2(X, u)
The output Y2 also possesses a relative degree with no internal dynamics.
= hI (x) . Then:
of 2
I,
..
+ "2a~xr + 2a2a3xlx2,
.
{2=t/>1 ( x )
+ (a2a4Xr + (ala4 + a2a5)x~ + a l a5xr)u1 "
,
'"
The effect of neglecting the term '1/11(x, ur) in equation (16)is to eliminate a non-linear zero in the system within the model description, It had be shown in (B ,A,White, 1997) and (A.Tsourdos, 1997), this will not affect the performance of the control design in a significant manner as the zero can be approximated by:
tPt{Xl ,Ul) 4
= ...(6a 2 + 2a2a3bl)xl
~
"'1
+ ...(12ala~ + a1a3bl + 2a2a3b2)Xr, "'1
+ ...(a~ + 6aia2 + ala3b2 + 2a2a3b3)xi,
'1/11 (x) z ~ - {31(X)
...
J
"'1
+ ...(2a2a~)x~ + (aia3 + ala3b5)x2' + (6aia4Xr + 2a2(3a1a4 + 3a2a5 + a3b6)xi), Ul
...
~
PI
+ .((6 a 1a2a5 + ai a 4 + ala3b6 + 2a2a3b7)xr), Ul ...
P1
Equations (17), (18) represent a direct relationship between the outputs hi and the inputs Ui (Wang, 1994) . The required static state feedback for decoupled closed loop input/output behaviour (Costas Kravaris, 1990) is given by as:
+ (2a2a3(a4 x l + a5)x2»U1 ...
(19)
Examination of equation (19) shows that z is always positive if the fin moment arm is greater than the static margin. This is true in all well designed missiles as the fin moment arm gives the small fin force sufficient turning moment to overcome the lift induced moment from the static margin. Hence the non-linear zero will always be in the stable left half s plane. It will tend to enhance the stability of the closed loop system, rather than detract from it; hence if the linearisation takes place without taking the zero into account, the resulting system should be more stable.
+ (8a1a2a3 + ala3b4 + 2a2a3b5)XIX2
...
r2
The total relative degree (Jean-Jacques E. Slotine, 1991) of the system which is equal with the summation of the rl and r2 , is now 4, and has the same order as the original system, Therefore there are no internal dynamics , Since the total relative degree is equal with the order of the system, fully linearisation of the non-linear system can now be achieved.
2 2 = ...(alxl + 3ala2xl + ala3 X2 "
3
= h3(x). Then:
(18)
{2=t{X )
6.
6
I
P1
+ ...(a1(ala5 + a3b7» , U1
.
(31
(16) U
= E- 1 { V - [~~]}
(20)
or where E is the characteristic ((Costas Kravaris, 1990» or decoupling ((Jean-Jacques E . Slotine, 1991» matrix
6=6
684
each channel, kl = 2(wn and k2 = w;, where Wn = 60(rad/ sec) and ( = 0.65. The speed of response is significantly faster than the open loop missile response and so should exercise the dynamics of the non-linear missile sufficiently for meaningful conclusions to be drawn. .
of the system, and is given by:
E-
1 = [1/{31
o
0]
1/{32
(21)
which is nonsingular.
The results of a 10 m/sec 2 and 50 m/sec 2 demand in acceleration is shown in Figures 3, 4 for the 10 m/sec 2 case, and Figures 5, 6 for the 50 m/sec 2 case. The figures show almost identical step responses for both 10,50 m/sec 2 with some variation in peaks and steady state values for the body rate, the actuator movement and the lateral velocity. The difference between the lateral acceleration and the body acceleration shows that there is a good match between the two and that steady state values are very close. This illustrates the small effect that the fin force has on the missile acceleration and justifes the use of the augmented body acceleration. The results also show that the actuator does not significantly affect the design. The non-linear approach is also shown to be reasonably accurate, as the predicted and actual performance are very close.
The linearised closed loop system is now given by:
ih
= Vi
(22)
Where V is the new linearised system input (Wang, 1994). Now choose the new control input to be: (23)
where e == Y - Yd . The close-loop system is thus characterised by: (24)
where kl and k2 are chosen such that all roots of S2 + kl S + k2 = 0 are in the open left-half plane, which ensures limt-HX) e(t) = 0 (Wang, 1994).
IIt::: :::1 r~1J:: :: ; :1
It can be said that now the tracking control problem for the non-linear system described by equations (13), (14) has been solved using the control law in equations (20), (22) and (23) . Indeed, since equation (24) has the same order as the non-linear system, there is no part of the system dynamics which is rendered "unobservable" in the approximate input-output linearisation. Since there are no zero dynamics in the linearised system, the stability of the linearised system can be guaranteed and the tracking problem has been solved (Isidori, 1985), (JeanJacques E . Slotine, 1991), (John Hauser, 1992).
o
0 .05
0 .1
0 .1 5
0 .2 Time(lMC)
0 .25
0 .3
0 .35
0 .•
o
O.OS
0 .1
0 .' 5
0 .2 Tlme(eec)
0 .25
0 .3
0 .35
0 ."
Fig. 3. Acceleration and lateral velocity for 5. TRAJECTRORY CONTROLLER DESIGN
ad
= 10
fJ\: m: . " ' ] '"
0
0.05
0.1
0.15
.. 02
0.2
n:S8C)
0.25
0.3
0.35
0.4
t.V\m :,mJmm, l ' .. ]
Fig. 2. Trajectory control design Figure 2 shows the non-linear controller structure. A fast linear actuator with natural frequency of 250 rad/sec has been included in the non-linear system. The trajectory controller performs by defined a desired acceleration as a demand. The error dynamics are constructed using the ad signal and the feedback of the actual states - velocity, rate, acceleration and jerk.
o
0.05
0.1
0.15
0.01
02
:S8C)
0.25
0.3
0.35
0.4
J 'VF~~ : ,---:-,:--:,'~,I -O.Ol~
o
0.05
0.1
0.15
0.2
0.25
0.3
Tino(eec)
The error coefficients in (24) are chosen to satisfy a Hurwitz polynomial. For the second order error equation in
Fig. 4. Rate and fin angle for
685
ad
= 10
0.35
0.4
7. REFERENCES
FE :: ::
A. Tsourdos, A. Blumel, B.A.White (1998). Trajectory control of a non-linear homing missile. In: Proceedings of the 1998 IFAC Symposium on Automatic Control in Aerospace. A.Tsourdos, A.Blumel, B.A.White (1997). Non-linear horton missile model. Technical report. The Royal Military College of Science,Cranfield University. DAGS,SEAS,RMCS,CU ,Shrivenham,Wilts,SN6 8LA. B.A.White (1997). Non-linear control of a missile. Technical report. The Royal Military College of Science,Cranfield University. DAGS,SEA-S,RM CS, CU ,Shrivenham,Wilts,SN 6 BLA. ' t. Costas Kravaris , Masoud Soroush (1990). Syntesis of multivariable nonlinear controllers by input-output linearization. AIChE Journal 36(2) , 249-264. H. Hahn, A.Piepenbrink, K.D. Leimbach (1994) . Inputoutput linearization control of an electro servohydraulic actuator. In: Proceedings of the 1994 IEEE Conference on Control Applications. Vo!. 2. pp. 995-1000. Isidori, Alberto (1985) . Nonlinear Control Systems. Springer-Verlag. Jean-Jacques E . Slotine, Weiping Li (1991) . Applied Nonlinear Control. Prentice Hall. Johan Suykens, Joos Vandewalle (1995). Feedback linearization of non linear distortion in electrodynamic loudspeakers. Journal of the Audio Engineering Society 43(9), 690-694. John Hauser, Shankar Satry, Petar Kokotovic (1992) . Nonlinear control via appriximate input-output linearization: The ball and beam example. IEEE Transactions on Automatic Control 37(3), 392-398. Michael A. Henson, Dale E . Seborg (1990). Inputoutput linearization of general nonlinear processes. AIChE Journal 36(U), 1754-1757. M.P.Horton (1992) . A study of autopilots for the adaptive control of tactical guided missiles. Master's thesis. University of Bath. Scott Bezick, llan Rusnak, W. Steven Gray (1995). Guidance of a homing missile via nonlinear geometric control methods. Journal of Guidance, Control and Dynamics 18(3), 441-448. Wang, Li-Xin (1994) . Design of adaptive fuzzy controllers for nonlinear systems by input-output linearization. In: Proceedings of the 1994 1st International Joint Conference of NAFIPS IFIS NASA . pp. 89-93. X. Y. Lu, S. K. Spurgeon, I Postlethwaite (1997) . Robust variable structure control of a pvtol aircraft. International Journal of Systems Science 28(6), 547558.
1
~-~O~~07. . 0'~~0~'--~0~.~~5~0.2~~0.2~'--~0~ .3---0~ . ~--~0.' TIft'Ht(MC)
l~lE o
0 .05
0 .1
::
0 .15
0.2
:
0 .25
•
0 .3
Fig. 5. Acceleration and lateral velocity for
LlS::,
0
0.05
0.1
0.15
0.05
0.1
0 .'
0.15
0.05
ad
= 100
l ,,] 0.2
0.25
I':~ , ,'~~) : o
1
!
0 .35
Tm.{MC)
0.2 rlt11~sec)
0.25
0.2
0.25
':
0.3
0.3
0.35
0.35
0.4
1 0.4
t:------:-::----f'~ mm :'- - - -:'- - .•- ~:j o
0.05
0.1
0.15
0.3
0.35
0.4
rwne(sec)
Fig. 6. Rate and fin angle for
ad
= 100
6. CONCLUSIONS This paper presents the lateral acceleration control design of non-linear missile model that examines outputs that retain the order of the linearised system, resulting in a linear equivalent system with no internal or zero dynamics, and with a design of a trajectory control which gives small tracking errors for the lateral acceleration. The design used augmented lateral acceleration which assumed that the direct acceleration produced by the fin is small compared to the body acceleration. It also showed that a neglected zero during the linearisation process was minimum phase. If lateral acceleration had been chosen as the linearisation output, a non-minimum phase system would result with the associated unstable internal dynamics. Other linearisation techniques are being examined which retain system order as well as relative degree. This approach will again not produce internal dyanmics and can thus be used on non-minimum phase systems.
686