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Robust Lateral Augmented Acceleration Flight Control Design
ELSEVIER
Copyright © IFAC Robust Control Design Milan, Italy, 2003
IFAC PUBLICATIONS www.elsevier.comllocatc/ifac
ROBUST LATERAL AUGMENTED ACCELERATION FLIGHT CONTROL DESIGN Lilian Bruyere· Antonios Tsourdos· Brian A.White •
• Department of Aerospace, Power and Sensors Royal Military College of Science, Cranfield University Shrivenham, SN6 BLA, UK
Abstract: A lateral augmented acceleration autopilot is designed for a model of a tactical missile. The tail-controlled missile in the cruciform fin configuration is modelled as a second-order quasi-linear parameter-varying system. The autopilot design is based on input-output pseudolinearisation, which is a restriction of inputoutput feedback linearisation to the set of equilibria of the nonlinear model. Robust autopilot design taking account parametric stability margins for uncertainty aerodynamic derivatives is implemented using convex optimisation and linear matrix inequalities - LMI. Simulations for constant lateral acceleration demands including uncertainties show satisfactory robust performance. Copyright © 2003 IFAC Keywords: Missiles, linearisation, robustness, stability, performance, pole assignment
1. INTRODUCTION: MISSILE MODEL
where the variables are defined in Figure 1. Here
Missile autopilots are usually designed using linear models of nonlinear equations of motion and aerodynamic forces and moments (Wise, 1992). The objective of this paper is robust design of a lateral augmented acceleration autopilot for a nonlinear missile model. This model describes a reasonably realistic airframe of a tail-controlled tactical missile in the cruciform fin configuration, Figure 1. The aerodynamic parameters in this model are derived from wind-tunnel measurements (Horton, 1992).
Fig. 1. Airframe axes.
The starting point for mathematical description of the missile is the following nonlinear model (Horton, 1992) of the horizontal motion: 1 v= "2m-lpVoS(Cyuv+ VoCYc () -
v is the sideslip velocity, r is the body rate, ( the rudder fin deflections, Yv,Ye semi-non-dimensional force derivatives due to lateral and fin angle, n v , ne, n r semi-non-dimensional moment derivatives due to sideslip velocity, fin angle and body rate. Finally, U is the longitudinal velocity. Furthermore, m = 125 kg is the missile mass, P = Po O.094h air density (Po = 1.23 kg.m -3 is the sea
Ur
r = ~I;l PVoSdGdCnrr + C nu v + VoC nc <}l) 34\
Interpolated formula
11
C yv C y< C nr C nv
C n<
{(xo(p), uo(P»1 f(xo(p), uo(P» = O}. It is important to note that parameters p need not be external. In particular, p may depend on both state x and external parameters ().
11
0.5[(-25 + M - 601(71)(1 + cos 4>")+ (-26 + 1 5M - 301(71)(1 - cos 4>..)J 10 + 0 5[( -1.6M + 21(71)(1 + cos 4>")+ (-l.4M + 1 51(71)(1 - cos4>..)J -500 - 30M + 2001<71 SmCyv' where: Sm = d- 1 [1.3 + O.lM + 0.2(1 + cos 4>")1<71+ 0.3(1 - cos4>")j<7l- (1.3 + m/500)J S/Cy<, where: sf = d- 1 [2.6 - (1.3 + m/5OO)J
y(p)
~
ables arising from Taylor linearisation of the open-loop system (2) at an equilibrium from E(P). Setting A(P) ~ of/ax!
Table 1. AerodynamIc coefficIents of the nonlinear missile model (1).
x(P) = A(p)x(P) + B(P)u(P) y(P) = G(p)x(P)
level air density and h the missile altitude in km),
Vo the total velocity in m.s- I , S = rrd 2 /4 =
0.0314 m 2 the reference area (d = 0.2 m is the reference diameter) and I z = 67.5 kg.m 2 is the lateral inertia. For the aerodynamic coefficients Gyv ' Gy <, Gnr , Gnv ' Gn < only discrete data points are available, obtained from wind tunnel experiments. Hence, an interpolation formula, involving the Mach number M E [2,3.5], roll angle A E [4.5°,45°] and total incidence (J" E [3°,30°], has been calculated with the results summarised in Table 1.
(3)
with the additional assumption that (3) is completely controllable and observable and has relative degree r for all points from E(P) and all pEP. The problem of input-output pseudolinearisation is to find for system (2) the restriction of a transformation z =
The total velocity vector Vo is composed of the longitudinal velocity vector U and the sideslip velocity vector v. We assume that U » v, so that the total incidence (J", can be taken as (J" = viVo, as sin (J" :::::: (J" for small (J". The Mach number is obviously defined as M = Vola, where a is the speed of sound. It follows from the above discussion that all coefficients in Table 1 can be interpreted as nonlinear functions of three variables: sideslip velocity v, longitudinal velocity U and roll angle A.
Thus, for a SISO system (3) the restriction of transformation z =
2. PSEUDOLINEARISATION Feedback linearisation has several limitations some of which can be overcome if the requirement is relaxed to linearise the system only along its set of equilibria, not the whole state-space. Such an approach (Reboulet and Champetier, 1984; Lawrence and Rugh, 1994; Lawrence, 1998) is called pseudolinearisation and may be viewed as applying the principles of feedback linearisation to gain scheduling.
= Z2 Z2 = Z3 ZI
Zr-I = zr
Consider the nth order nonlinear system with m inputs and q outputs: = f(x, u) y = h(x),
~ x - xo(p), u(P) ~ u - uo(p) and h(x) - h(xo(P» be the incremental vari-
Let x(p)
zr
Zr+l
= ii = a:+l (xo(p), uo(p»z
:i;
Zn =
(2)
a; (xo(p), uo(p»z
y= ZI,
where f and h are smooth on their definition sets X eRn, U c Rffi. The set of equilibria of (2) is assumed to depend on parameters pEP C RP, P open, and is denoted as E(p) =
(4)
where only the n-dimensional vectors ar+I,' .. ,an still depend on equilibria from E(p) and parameters from P. Since the dynamics defined by
342
3. AUGMENTED ACCELERATION AUTOPILOT
ar+I, ... ,an are unobservable (and therefore must be at least stable), the behaviour of (4) from the input-output, v-Y, viewpoint is linear of order r and remains the same, no matter what the current values of xo, Uo and p are. This should be contrasted with Taylor linearisation (3) of the openloop system (2).
As explained in Section 1, the missile model given by (1) and Table 1 can be represented as follows and the block diagram representation of the missile autopilot is shown in Figure 2
It suffices to investigate the tangents oip / Ox ~ T of ip and ok/ox ~ F, ok/ov ~ G of k along {E(P)}PEP' In particular (Lawrence and Rugh, 1994; Lawrence, 1998), it is required that T(xo(P)) is invertible for all PEP, feedback law u = k(x, v) is smooth, satisfies uo(P) = k(xo(p), vo(P)) and G(xo(p), vo(p)) is invertible for all pEP.
v = Yv(v, U, >.)v -
+ Ydv, U, >.)( (10) r = nv(v, U, >.)v + nr(v, U, >.)r + n«v, U, >.)(
(M, A) ~-------
,
The formula for the tangent of transformation ip along {E(p) }PEP is (Lawrence and Rugh, 1994; Lawrence, 1998): 2
= T(p)x
Ur
Augme~ e
latax
..
Lmeansed system
------------------.
lII(v, T, K(.»
_
(5)
Nonlinear controller
External parameters
U
= (
: : ,
Outer-loop
Fig. 2. Block diagram of the missile autopilot with T given by Setting
C(P) C(p)A(p) T(p) =
C(p)Ar-I(p) T r +! (p)
x ~ [V ii ~
(6)
For the tangent offeedback law k along {E(P)}PEP the formula is (Lawrence and Rugh, 1994; Lawrence, 1998):
+ G(p)v,
(7)
r
I .
Thus, representation (10) can be seen as the quasilinear parameter-varying (QLPV) model since pin (11) comprises both a state (incremental sideslip velocity v) and external parameters (longitudinal velocity U and roll angle >.). Pseudolinearisation approach has been successfully applied to the Horton missile (Bruyere and al., 2002). Our design for lateral augmented acceleration, Yv(p)v, autopilot takes the second order (n = 2) model (3) and (12) as the starting point. Noting that the relative degree r is 2 and thus r = n, we then use formulae (5)-(9) so that:
F(P) = - ( C(p)Ar-I(p)B(p)) - I C (p)A r (p), (8) G(p) = (C(p)A r - I (p)B(p)
(11)
equations (10) lead to the following description of (3) where
where rows T i , i = r + 1, ... ,n can be obtained from [Tr+!(p) ... Tn(p) 1B(p) = 0 which is a system of n - r linear equations in (n - r)n unknowns.
ii = F(p)x
r ]T, P ~ [V U >.]T, and Y ~ Yv(v, U, >.)v
C,
(9)
Formulae (5)-(9) transform a SISO (3) into (4). Design of a stabilising control law (7) is complete when v = KI21 +.. .+Kr 2r , where K i , i = 1, ... , r are constants. If the desired output Yd is non-zero, then the tracking error is e ~ Yd - Y = Yd - 21 and 't d enva . t'Ives e-(i) -- Yd _(i) . - 1 1. 1 S - Zi+l, Z , .•• , r Hence the tracking control law will be v = K 1 e + ... + Kre(r-l). Putting 2 = T(p)x and iJ in (7) gives the feedback law in terms of x, so that transformation T can be viewed as an auxiliary tool for designing feedback control law (7).
The control law will be ii
= F(p)x + G(p)w where
b1(p) = y~(p) - n v (p)Yv(P)P2, b2(p) = -y~(p)P2 - n r (p)Yv(P)P2, a(p) = y~(p)Ydp) - ndp)Yv(P)P2'
343
(14)
invariant system. The equivalent error dynamics equation easily state the performance where Zl is the lateral augmented acceleration where K 1 and K 2 still need to be determined.
Hence, the resulting pseudolinearising control still requires to define w to ensure tracking. Let e ~ a Vd - a v be the acceleration error, where a Vd is the lateral augmented acceleration demand and e = -av with aVd = 0 so that the demand does not need differentiation. Then the final form of the control law is: (=
1
a( ) _ P
b ()
()
1 P y( P
(
b1 (p) --(p)a v
Yv
-
_~)
b2(P)r + K1e
+ Kze
It has to be noticed that the missile is limited by both its natural behaviour and its actuators performance. These actuators are modelled as second order systems with damping 0.7 and natural frequency 250 rad/s, with an angle range of ±0.3 rad. Consequently, it has been chosen to keep the response of the pole placement controller within these constraints so that the actuators do not operate above their cut off frequency. This is done by limiting the frequency response of the pole placement controller less than 100 rad/s and keeping the damping ratio above the critic. Additionally, the desired performance requires that the system performs within 0.1 s of rising time. The resulting D-stability region is shown on the next Figure 3, where the poles have to stay in the cone defined with half-angle 7r/ 4 and with pole real part in the range between -40 and -100.
Yv(p)
(15)
The constants K 1 , K 2 in (15) determine the tracking properties for instance K 1 = and K 2 = 2~wn, where W n = 60 rad/sec and ~ = 0.7, so that the error equation is + 2~wne + w;e = O. This should give a 3-4 times faster response than the open loop. Simulation for 100 m/sec 2 lateral acceleration de-
w;
e
mand is shown by the continuous curves in Figure 5. The observed steady-state error in lateral acceleration is about 5% due to the neglected fin force contribution, yC;(, in the design. This approximation or non-minimum phase, is clearly visible. Note also that the initial fin angle ( < 0 is quickly overcome by the side-slip force as incidence builds up. Note: An accelerometer is used to measure lateral acceleration. If the accelerometer is placed at the missile centre of gravity, the resulting system becomes non-minimum phase, as the accelerometer measures the effect of both the body aerodynamic force and the almost instantaneous fin force. This also has the effect of making the relative degree zero. To overcome both of these effects, an augmented acceleration signal is used. As a v = YvV + YC;c" augmentation can be obtained by mixing the accelerometer signal with the fin angle to eliminate the dependence of a v on C, to give av ;::::: YvV. This approximation will not result in significant error in the control design, because the fin force contribution is small in a well designed airframe. The same effect can be obtained by moving the accelerometer forward from the centre of gravity to the centre of rotation. This also removes the dependence on ( and makes the system minimum phase.
Fig. 3. The D-stability region for the pole placement controller to guarantee performance on the transient
5. ROBUST PERFORMANCE DESIGN
The pseudolinearisation enables to attain the desired level of performance through the choice of coefficients K 1 and K 2 • In this section, the idea is to estimate these coefficients of the pole placement controller according to the desired level of performance by using a pole placement criteria.
The parametric uncertain system which has been investigated represents a system with poor accuracy in its aerodynamic coefficients. Aside with these aerodynamic coefficients, the variation in forward velocity, which is explicit in differential equation (10) will also be considered as an uncertainty. Now, the vector of uncertain parameters can be written as p = [Yv, U, n v , nn nc;]T, where P2, i.e. U, represents the forward velocity. The following work presents a state feedback design through Lyapunov theory for a parametric uncertain system.
The performance of the system is better stated in the [Zl, z2jT-space in which the nominal system is arbitrarily chosen as second order linear time
The lateral augmented acceleration autopilot of control law (15) brings the closed-loop system (10) in the following form
4. NOMINAL DESIGN
344
iJ
= Yv(P)v - P2r
f =
+ a~~) (-b1(p)v nv(p)v + nr(p)r + a7~) (-b1(p)v -
b2(p)r
The nominal plant considered is an LTI, however, when it comes to consider parametric uncertainties the uncertain system is not anymore so, since it depends on the operating point (22). The uncertainties are moreover involved in a multi-affine form. The uncertain model has been brought to affine parametric form by introducing some overbounding and consequently the uncertain parameter is now extended to p = [Yv, U, n v , n r , n(, p~JT where each parameter is multiplicative uncertain.
+ K1e + K 2e) (17)
b2(p)r
+ K1e + K 2e).
Assuming that the derivative of the lateral augmented acceleration, aVd ' is zero, we have = -av . Then the closed loop equations (17) can be rewritten as:
e
E(P)x = Ae(P)x + Be(p)vd ij = C(p)x with
x,
ij and p as in (11), matrix
The uncertain input matrix 18, increases computation complexity and it is chosen not to take it into account in the following design. Then the following model (24) is used instead where the controller is set for the nominal model.
(18)
A e as
acl2 ]
a e 22
(19)
where
= yv(P)a(p) -
Yv (P)ydp)(b 1(p) - K 1) acl2 = -P2a(P) - y( (P)~ (P) a e 21 = nv(p)a(p) - yv(P)ndp)(b1(P) - Kt) ae22 = nr(p)a(p) - n«p)b 2(p) (20) aell
At each operating point a controller, coefficients K 1 and K 2 , is designed for the local parametric uncertain system and ensure that the nonlinearities of the system are captured. The robust performance of the closed-loop system is achieved at each operating point (a, M, A) by solving the LMI (23). The set of controllers over the whole flight envelope constitutes a scheduled controller. The carpet plots of these coefficients is shown in Figure 4. It is a complex problem to design these
and matrices E, Bc, C given as
E(P) = [a(p) + yv(p)y«P)K2 0] yv(P)n«(p)K2 a(p) Be(p)
=
[~~~~~],
C(p)
= [Yv(P)
(21)
0].
Alike for pseudolinearisation, the y( term in (17) is neglected in the following. The approximate closed-loop equations (17) can be written in the transformed space (5) as Fig. 4. Gain K 1 (left) and K 2 (right) of the pole placement controller for whole flight envelope (and A = 0°). gain through the whole flight envelope, however approximating our model of uncertainties with a multi-affine model which capture the nonlinearities enables to design a pole placement controller at each knot (see carpet plots in Figure 4). It can be shown that the linear interpolation between these knots lead to a self-scheduled controller (Kt(a,M) and K 2(a,M)) and it would bring more confidence in the robust performance over the whole flight envelope. This is checked in the next section through an analysis of the closed-loop system. It should be emphasised that the methodology developed so far has for interest to give a systematic tool to estimate gains in the pole placement controller which verify some performance robustness properties (of transient).
Some modified Lyapunov equations are used to state this D-stability region - performance introduced in the previous section, they turn the usual Lyapunov equation to Linear Matrix Inequalities for which the LMI Toolbox (Gahinet and al., 1995) for MATLAB has been used. Assuming a region V = {z E Cl L + M z + M T Z < O}, the closed-loop matrix A + BK would have all its eigenvalues in this region V if there exists a symmetric positive definite matrix X and matrix Y (where Y = KX) satisfying the LMI (>.ijX
+ J.LijAX + J.LjiXAT + J.LijBY + J.LjiyT BT)ij < 0 (23)
345
6. ROBUST PERFORMANCE ANALYSIS
7. DISCUSSIONS & CONCLUSIONS
The robust pole placement controller validity is now investigated. Because the self-gain scheduled controller is built from linear interpolation between chosen gains (K 1 and K 2 ) and since it has been checked that the representative class of uncertain systems capture the nonlinearities, the self-scheduled controller designed previously is expected to give the whole system some robust performance. But some of the simplifications in the design, compare equations (22) with (24), are reconsidered in this section. For instance, the uncertain system described by equation (22) becomes for the closed-loop system,
A reasonably realistic missile model has been described as a QLPV system. The pseudolinearising autopilot consists of one controller only and "scheduling" is done automatically by feedback. However, the pseudolinearising design is only valid (like gain scheduling) in the vicinity of the equilibria. It states performance of the closed-loop system, but previous studies shown not very good robust performance. An attempt to achieve the desired transient response for a class of parametric uncertain systems has been carried on by estimating the pole placement controller coefficients. This is a more systematic tool to tune some coefficients for which the closed-loop system is more likely to be robust. Finally, the robust performance analysis has been done to validate this approach. In the context of polytopic approach the system nonlinearities can be very complex and if in principle the approximation can be as accurate as required a trade off was necessary in this approach. Because the Horton model and its uncertain parametric model can be approximated by a multi-linear system with respect to incidence and Mach number, the robust design/analysis introduces some overboundings of the uncertain class. For more complex polytopes current algorithms may fail to prove even feasibility of the LMIs. The presentation of this study is kept simple by neglecting some parameters. Further consideration could lead to a design/analysis less conservative for instance if bounds on the rate of change were considered.
For simplicity in this analysis, the term ai (P) is not taken into account. This uncertain closedloop system is then included in a polytope and a similar LMI to (23) is solved. The feasibility of which is confirmed for some 20% uncertainty on the uncertain parameters of p along the whole flight envelope. Here, the parametric uncertainty boundaries are limited by the most sensible parameter without finding the full extend of all parameters, also further analysis would extend individual parameter boundaries (via Monte Carlo simulation for instance). The nominal (continuous curves) and the uncertain (dashed curves) system responses for 100m/s2 lateral acceleration demand are presented on Figure 5. The frequency bandwidth, the critic damping and the rising time of these simulations are satisfactory.
REFERENCES
Bruyere, Lilian and al. (2002). Robust performance study for lateral autopilot of a quasilinear parameter-varying missile. In: American Control Conference. pp. 226-231. Gahinet, Pascal and al. (1995). LMI Control Toolbox User's Guide. The MathWorks. Horton, M. P. (1992). A study of autopilots for the adaptive control of tactical guided missiles. Master's thesis. University of Bath. Lawrence, D. A. (1998). A genereal approach to input-output pseudolinearization for nonlinear systems. IEEE Transactions on Automatic Control 43(10), 1497-1501. Lawrence, D. A. and W. J. Rugh (1994). Inputoutput pseudolinearization for nonlinear systems. IEEE Transactions on Automatic Control 39(11), 2207-2218. Reboulet, C. and C. Champetier (1984). A new method for linearizing nonlinear systems: The pseudolinearization. International Journal of Control 40(4), 631-638. Wise, K. A. (1992). Comparison of 6 robustness tests evaluating missile autopilot robustness to uncertain aerodynamics. Journal of Guidance, Control and Dynamics 15(4), 861-870.
150 ..
0.5
...... Sdestip Vebcily
YawRate
>.5
0.1
0flm. C!l3
0.4
0.5
Fig. 5. Simulation of the system with up to 20% uncertainties on aerodynamic coefficients for 100 m/s 2 lateral acceleration demand.
346
Copyright © IFAC Robust Control Design Milan, Italy, 2003
IFAC PUBLICATIONS www.elsevier.comllocatc/ifac
ROBUST LATERAL AUGMENTED ACCELERATION FLIGHT CONTROL DESIGN Lilian Bruyere· Antonios Tsourdos· Brian A.White •
• Department of Aerospace, Power and Sensors Royal Military College of Science, Cranfield University Shrivenham, SN6 BLA, UK
Abstract: A lateral augmented acceleration autopilot is designed for a model of a tactical missile. The tail-controlled missile in the cruciform fin configuration is modelled as a second-order quasi-linear parameter-varying system. The autopilot design is based on input-output pseudolinearisation, which is a restriction of inputoutput feedback linearisation to the set of equilibria of the nonlinear model. Robust autopilot design taking account parametric stability margins for uncertainty aerodynamic derivatives is implemented using convex optimisation and linear matrix inequalities - LMI. Simulations for constant lateral acceleration demands including uncertainties show satisfactory robust performance. Copyright © 2003 IFAC Keywords: Missiles, linearisation, robustness, stability, performance, pole assignment
1. INTRODUCTION: MISSILE MODEL
where the variables are defined in Figure 1. Here
Missile autopilots are usually designed using linear models of nonlinear equations of motion and aerodynamic forces and moments (Wise, 1992). The objective of this paper is robust design of a lateral augmented acceleration autopilot for a nonlinear missile model. This model describes a reasonably realistic airframe of a tail-controlled tactical missile in the cruciform fin configuration, Figure 1. The aerodynamic parameters in this model are derived from wind-tunnel measurements (Horton, 1992).
Fig. 1. Airframe axes.
The starting point for mathematical description of the missile is the following nonlinear model (Horton, 1992) of the horizontal motion: 1 v= "2m-lpVoS(Cyuv+ VoCYc () -
v is the sideslip velocity, r is the body rate, ( the rudder fin deflections, Yv,Ye semi-non-dimensional force derivatives due to lateral and fin angle, n v , ne, n r semi-non-dimensional moment derivatives due to sideslip velocity, fin angle and body rate. Finally, U is the longitudinal velocity. Furthermore, m = 125 kg is the missile mass, P = Po O.094h air density (Po = 1.23 kg.m -3 is the sea
Ur
r = ~I;l PVoSdGdCnrr + C nu v + VoC nc <}l) 34\
Interpolated formula
11
C yv C y< C nr C nv
C n<
{(xo(p), uo(P»1 f(xo(p), uo(P» = O}. It is important to note that parameters p need not be external. In particular, p may depend on both state x and external parameters ().
11
0.5[(-25 + M - 601(71)(1 + cos 4>")+ (-26 + 1 5M - 301(71)(1 - cos 4>..)J 10 + 0 5[( -1.6M + 21(71)(1 + cos 4>")+ (-l.4M + 1 51(71)(1 - cos4>..)J -500 - 30M + 2001<71 SmCyv' where: Sm = d- 1 [1.3 + O.lM + 0.2(1 + cos 4>")1<71+ 0.3(1 - cos4>")j<7l- (1.3 + m/500)J S/Cy<, where: sf = d- 1 [2.6 - (1.3 + m/5OO)J
y(p)
~
ables arising from Taylor linearisation of the open-loop system (2) at an equilibrium from E(P). Setting A(P) ~ of/ax!
Table 1. AerodynamIc coefficIents of the nonlinear missile model (1).
x(P) = A(p)x(P) + B(P)u(P) y(P) = G(p)x(P)
level air density and h the missile altitude in km),
Vo the total velocity in m.s- I , S = rrd 2 /4 =
0.0314 m 2 the reference area (d = 0.2 m is the reference diameter) and I z = 67.5 kg.m 2 is the lateral inertia. For the aerodynamic coefficients Gyv ' Gy <, Gnr , Gnv ' Gn < only discrete data points are available, obtained from wind tunnel experiments. Hence, an interpolation formula, involving the Mach number M E [2,3.5], roll angle A E [4.5°,45°] and total incidence (J" E [3°,30°], has been calculated with the results summarised in Table 1.
(3)
with the additional assumption that (3) is completely controllable and observable and has relative degree r for all points from E(P) and all pEP. The problem of input-output pseudolinearisation is to find for system (2) the restriction of a transformation z =
The total velocity vector Vo is composed of the longitudinal velocity vector U and the sideslip velocity vector v. We assume that U » v, so that the total incidence (J", can be taken as (J" = viVo, as sin (J" :::::: (J" for small (J". The Mach number is obviously defined as M = Vola, where a is the speed of sound. It follows from the above discussion that all coefficients in Table 1 can be interpreted as nonlinear functions of three variables: sideslip velocity v, longitudinal velocity U and roll angle A.
Thus, for a SISO system (3) the restriction of transformation z =
2. PSEUDOLINEARISATION Feedback linearisation has several limitations some of which can be overcome if the requirement is relaxed to linearise the system only along its set of equilibria, not the whole state-space. Such an approach (Reboulet and Champetier, 1984; Lawrence and Rugh, 1994; Lawrence, 1998) is called pseudolinearisation and may be viewed as applying the principles of feedback linearisation to gain scheduling.
= Z2 Z2 = Z3 ZI
Zr-I = zr
Consider the nth order nonlinear system with m inputs and q outputs: = f(x, u) y = h(x),
~ x - xo(p), u(P) ~ u - uo(p) and h(x) - h(xo(P» be the incremental vari-
Let x(p)
zr
Zr+l
= ii = a:+l (xo(p), uo(p»z
:i;
Zn =
(2)
a; (xo(p), uo(p»z
y= ZI,
where f and h are smooth on their definition sets X eRn, U c Rffi. The set of equilibria of (2) is assumed to depend on parameters pEP C RP, P open, and is denoted as E(p) =
(4)
where only the n-dimensional vectors ar+I,' .. ,an still depend on equilibria from E(p) and parameters from P. Since the dynamics defined by
342
3. AUGMENTED ACCELERATION AUTOPILOT
ar+I, ... ,an are unobservable (and therefore must be at least stable), the behaviour of (4) from the input-output, v-Y, viewpoint is linear of order r and remains the same, no matter what the current values of xo, Uo and p are. This should be contrasted with Taylor linearisation (3) of the openloop system (2).
As explained in Section 1, the missile model given by (1) and Table 1 can be represented as follows and the block diagram representation of the missile autopilot is shown in Figure 2
It suffices to investigate the tangents oip / Ox ~ T of ip and ok/ox ~ F, ok/ov ~ G of k along {E(P)}PEP' In particular (Lawrence and Rugh, 1994; Lawrence, 1998), it is required that T(xo(P)) is invertible for all PEP, feedback law u = k(x, v) is smooth, satisfies uo(P) = k(xo(p), vo(P)) and G(xo(p), vo(p)) is invertible for all pEP.
v = Yv(v, U, >.)v -
+ Ydv, U, >.)( (10) r = nv(v, U, >.)v + nr(v, U, >.)r + n«v, U, >.)(
(M, A) ~-------
,
The formula for the tangent of transformation ip along {E(p) }PEP is (Lawrence and Rugh, 1994; Lawrence, 1998): 2
= T(p)x
Ur
Augme~ e
latax
..
Lmeansed system
------------------.
lII(v, T, K(.»
_
(5)
Nonlinear controller
External parameters
U
= (
: : ,
Outer-loop
Fig. 2. Block diagram of the missile autopilot with T given by Setting
C(P) C(p)A(p) T(p) =
C(p)Ar-I(p) T r +! (p)
x ~ [V ii ~
(6)
For the tangent offeedback law k along {E(P)}PEP the formula is (Lawrence and Rugh, 1994; Lawrence, 1998):
+ G(p)v,
(7)
r
I .
Thus, representation (10) can be seen as the quasilinear parameter-varying (QLPV) model since pin (11) comprises both a state (incremental sideslip velocity v) and external parameters (longitudinal velocity U and roll angle >.). Pseudolinearisation approach has been successfully applied to the Horton missile (Bruyere and al., 2002). Our design for lateral augmented acceleration, Yv(p)v, autopilot takes the second order (n = 2) model (3) and (12) as the starting point. Noting that the relative degree r is 2 and thus r = n, we then use formulae (5)-(9) so that:
F(P) = - ( C(p)Ar-I(p)B(p)) - I C (p)A r (p), (8) G(p) = (C(p)A r - I (p)B(p)
(11)
equations (10) lead to the following description of (3) where
where rows T i , i = r + 1, ... ,n can be obtained from [Tr+!(p) ... Tn(p) 1B(p) = 0 which is a system of n - r linear equations in (n - r)n unknowns.
ii = F(p)x
r ]T, P ~ [V U >.]T, and Y ~ Yv(v, U, >.)v
C,
(9)
Formulae (5)-(9) transform a SISO (3) into (4). Design of a stabilising control law (7) is complete when v = KI21 +.. .+Kr 2r , where K i , i = 1, ... , r are constants. If the desired output Yd is non-zero, then the tracking error is e ~ Yd - Y = Yd - 21 and 't d enva . t'Ives e-(i) -- Yd _(i) . - 1 1. 1 S - Zi+l, Z , .•• , r Hence the tracking control law will be v = K 1 e + ... + Kre(r-l). Putting 2 = T(p)x and iJ in (7) gives the feedback law in terms of x, so that transformation T can be viewed as an auxiliary tool for designing feedback control law (7).
The control law will be ii
= F(p)x + G(p)w where
b1(p) = y~(p) - n v (p)Yv(P)P2, b2(p) = -y~(p)P2 - n r (p)Yv(P)P2, a(p) = y~(p)Ydp) - ndp)Yv(P)P2'
343
(14)
invariant system. The equivalent error dynamics equation easily state the performance where Zl is the lateral augmented acceleration where K 1 and K 2 still need to be determined.
Hence, the resulting pseudolinearising control still requires to define w to ensure tracking. Let e ~ a Vd - a v be the acceleration error, where a Vd is the lateral augmented acceleration demand and e = -av with aVd = 0 so that the demand does not need differentiation. Then the final form of the control law is: (=
1
a( ) _ P
b ()
()
1 P y( P
(
b1 (p) --(p)a v
Yv
-
_~)
b2(P)r + K1e
+ Kze
It has to be noticed that the missile is limited by both its natural behaviour and its actuators performance. These actuators are modelled as second order systems with damping 0.7 and natural frequency 250 rad/s, with an angle range of ±0.3 rad. Consequently, it has been chosen to keep the response of the pole placement controller within these constraints so that the actuators do not operate above their cut off frequency. This is done by limiting the frequency response of the pole placement controller less than 100 rad/s and keeping the damping ratio above the critic. Additionally, the desired performance requires that the system performs within 0.1 s of rising time. The resulting D-stability region is shown on the next Figure 3, where the poles have to stay in the cone defined with half-angle 7r/ 4 and with pole real part in the range between -40 and -100.
Yv(p)
(15)
The constants K 1 , K 2 in (15) determine the tracking properties for instance K 1 = and K 2 = 2~wn, where W n = 60 rad/sec and ~ = 0.7, so that the error equation is + 2~wne + w;e = O. This should give a 3-4 times faster response than the open loop. Simulation for 100 m/sec 2 lateral acceleration de-
w;
e
mand is shown by the continuous curves in Figure 5. The observed steady-state error in lateral acceleration is about 5% due to the neglected fin force contribution, yC;(, in the design. This approximation or non-minimum phase, is clearly visible. Note also that the initial fin angle ( < 0 is quickly overcome by the side-slip force as incidence builds up. Note: An accelerometer is used to measure lateral acceleration. If the accelerometer is placed at the missile centre of gravity, the resulting system becomes non-minimum phase, as the accelerometer measures the effect of both the body aerodynamic force and the almost instantaneous fin force. This also has the effect of making the relative degree zero. To overcome both of these effects, an augmented acceleration signal is used. As a v = YvV + YC;c" augmentation can be obtained by mixing the accelerometer signal with the fin angle to eliminate the dependence of a v on C, to give av ;::::: YvV. This approximation will not result in significant error in the control design, because the fin force contribution is small in a well designed airframe. The same effect can be obtained by moving the accelerometer forward from the centre of gravity to the centre of rotation. This also removes the dependence on ( and makes the system minimum phase.
Fig. 3. The D-stability region for the pole placement controller to guarantee performance on the transient
5. ROBUST PERFORMANCE DESIGN
The pseudolinearisation enables to attain the desired level of performance through the choice of coefficients K 1 and K 2 • In this section, the idea is to estimate these coefficients of the pole placement controller according to the desired level of performance by using a pole placement criteria.
The parametric uncertain system which has been investigated represents a system with poor accuracy in its aerodynamic coefficients. Aside with these aerodynamic coefficients, the variation in forward velocity, which is explicit in differential equation (10) will also be considered as an uncertainty. Now, the vector of uncertain parameters can be written as p = [Yv, U, n v , nn nc;]T, where P2, i.e. U, represents the forward velocity. The following work presents a state feedback design through Lyapunov theory for a parametric uncertain system.
The performance of the system is better stated in the [Zl, z2jT-space in which the nominal system is arbitrarily chosen as second order linear time
The lateral augmented acceleration autopilot of control law (15) brings the closed-loop system (10) in the following form
4. NOMINAL DESIGN
344
iJ
= Yv(P)v - P2r
f =
+ a~~) (-b1(p)v nv(p)v + nr(p)r + a7~) (-b1(p)v -
b2(p)r
The nominal plant considered is an LTI, however, when it comes to consider parametric uncertainties the uncertain system is not anymore so, since it depends on the operating point (22). The uncertainties are moreover involved in a multi-affine form. The uncertain model has been brought to affine parametric form by introducing some overbounding and consequently the uncertain parameter is now extended to p = [Yv, U, n v , n r , n(, p~JT where each parameter is multiplicative uncertain.
+ K1e + K 2e) (17)
b2(p)r
+ K1e + K 2e).
Assuming that the derivative of the lateral augmented acceleration, aVd ' is zero, we have = -av . Then the closed loop equations (17) can be rewritten as:
e
E(P)x = Ae(P)x + Be(p)vd ij = C(p)x with
x,
ij and p as in (11), matrix
The uncertain input matrix 18, increases computation complexity and it is chosen not to take it into account in the following design. Then the following model (24) is used instead where the controller is set for the nominal model.
(18)
A e as
acl2 ]
a e 22
(19)
where
= yv(P)a(p) -
Yv (P)ydp)(b 1(p) - K 1) acl2 = -P2a(P) - y( (P)~ (P) a e 21 = nv(p)a(p) - yv(P)ndp)(b1(P) - Kt) ae22 = nr(p)a(p) - n«p)b 2(p) (20) aell
At each operating point a controller, coefficients K 1 and K 2 , is designed for the local parametric uncertain system and ensure that the nonlinearities of the system are captured. The robust performance of the closed-loop system is achieved at each operating point (a, M, A) by solving the LMI (23). The set of controllers over the whole flight envelope constitutes a scheduled controller. The carpet plots of these coefficients is shown in Figure 4. It is a complex problem to design these
and matrices E, Bc, C given as
E(P) = [a(p) + yv(p)y«P)K2 0] yv(P)n«(p)K2 a(p) Be(p)
=
[~~~~~],
C(p)
= [Yv(P)
(21)
0].
Alike for pseudolinearisation, the y( term in (17) is neglected in the following. The approximate closed-loop equations (17) can be written in the transformed space (5) as Fig. 4. Gain K 1 (left) and K 2 (right) of the pole placement controller for whole flight envelope (and A = 0°). gain through the whole flight envelope, however approximating our model of uncertainties with a multi-affine model which capture the nonlinearities enables to design a pole placement controller at each knot (see carpet plots in Figure 4). It can be shown that the linear interpolation between these knots lead to a self-scheduled controller (Kt(a,M) and K 2(a,M)) and it would bring more confidence in the robust performance over the whole flight envelope. This is checked in the next section through an analysis of the closed-loop system. It should be emphasised that the methodology developed so far has for interest to give a systematic tool to estimate gains in the pole placement controller which verify some performance robustness properties (of transient).
Some modified Lyapunov equations are used to state this D-stability region - performance introduced in the previous section, they turn the usual Lyapunov equation to Linear Matrix Inequalities for which the LMI Toolbox (Gahinet and al., 1995) for MATLAB has been used. Assuming a region V = {z E Cl L + M z + M T Z < O}, the closed-loop matrix A + BK would have all its eigenvalues in this region V if there exists a symmetric positive definite matrix X and matrix Y (where Y = KX) satisfying the LMI (>.ijX
+ J.LijAX + J.LjiXAT + J.LijBY + J.LjiyT BT)ij < 0 (23)
345
6. ROBUST PERFORMANCE ANALYSIS
7. DISCUSSIONS & CONCLUSIONS
The robust pole placement controller validity is now investigated. Because the self-gain scheduled controller is built from linear interpolation between chosen gains (K 1 and K 2 ) and since it has been checked that the representative class of uncertain systems capture the nonlinearities, the self-scheduled controller designed previously is expected to give the whole system some robust performance. But some of the simplifications in the design, compare equations (22) with (24), are reconsidered in this section. For instance, the uncertain system described by equation (22) becomes for the closed-loop system,
A reasonably realistic missile model has been described as a QLPV system. The pseudolinearising autopilot consists of one controller only and "scheduling" is done automatically by feedback. However, the pseudolinearising design is only valid (like gain scheduling) in the vicinity of the equilibria. It states performance of the closed-loop system, but previous studies shown not very good robust performance. An attempt to achieve the desired transient response for a class of parametric uncertain systems has been carried on by estimating the pole placement controller coefficients. This is a more systematic tool to tune some coefficients for which the closed-loop system is more likely to be robust. Finally, the robust performance analysis has been done to validate this approach. In the context of polytopic approach the system nonlinearities can be very complex and if in principle the approximation can be as accurate as required a trade off was necessary in this approach. Because the Horton model and its uncertain parametric model can be approximated by a multi-linear system with respect to incidence and Mach number, the robust design/analysis introduces some overboundings of the uncertain class. For more complex polytopes current algorithms may fail to prove even feasibility of the LMIs. The presentation of this study is kept simple by neglecting some parameters. Further consideration could lead to a design/analysis less conservative for instance if bounds on the rate of change were considered.
For simplicity in this analysis, the term ai (P) is not taken into account. This uncertain closedloop system is then included in a polytope and a similar LMI to (23) is solved. The feasibility of which is confirmed for some 20% uncertainty on the uncertain parameters of p along the whole flight envelope. Here, the parametric uncertainty boundaries are limited by the most sensible parameter without finding the full extend of all parameters, also further analysis would extend individual parameter boundaries (via Monte Carlo simulation for instance). The nominal (continuous curves) and the uncertain (dashed curves) system responses for 100m/s2 lateral acceleration demand are presented on Figure 5. The frequency bandwidth, the critic damping and the rising time of these simulations are satisfactory.
REFERENCES
Bruyere, Lilian and al. (2002). Robust performance study for lateral autopilot of a quasilinear parameter-varying missile. In: American Control Conference. pp. 226-231. Gahinet, Pascal and al. (1995). LMI Control Toolbox User's Guide. The MathWorks. Horton, M. P. (1992). A study of autopilots for the adaptive control of tactical guided missiles. Master's thesis. University of Bath. Lawrence, D. A. (1998). A genereal approach to input-output pseudolinearization for nonlinear systems. IEEE Transactions on Automatic Control 43(10), 1497-1501. Lawrence, D. A. and W. J. Rugh (1994). Inputoutput pseudolinearization for nonlinear systems. IEEE Transactions on Automatic Control 39(11), 2207-2218. Reboulet, C. and C. Champetier (1984). A new method for linearizing nonlinear systems: The pseudolinearization. International Journal of Control 40(4), 631-638. Wise, K. A. (1992). Comparison of 6 robustness tests evaluating missile autopilot robustness to uncertain aerodynamics. Journal of Guidance, Control and Dynamics 15(4), 861-870.
150 ..
0.5
...... Sdestip Vebcily
YawRate
>.5
0.1
0flm. C!l3
0.4
0.5
Fig. 5. Simulation of the system with up to 20% uncertainties on aerodynamic coefficients for 100 m/s 2 lateral acceleration demand.
346