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Lateral Acceleration Flight Control for a Quasi-Linear Parameter Varying Missile Model Via Pseudolinearisation
Copyright@ IFAC Robust Control Design, Prague, Czech Republic, 2000
LATERAL ACCELERATION FLIGHT CONTROL FOR A QUASI-LINEAR PARAMETER VARYING MISSILE MODEL VIA PSEUDOLINEARISATION A. Tsourdos R. Zbikowski and B. A. White
Department of Aerospace, Power and Sensors Cmnfield University RMCS Shrivenham Swindon SN6 8LA, England, UK A, [email protected]
Abstract. A lateral augmented acceleration autopilot is designed for a model of a tactical missile and robust stability of the closed-loop system investigated. The tail-controlled missile in the cruciform fin configuration is modelled as a second-order quasi-linear parameter-varying system. This nonlinear model is obtained from the Taylor linearised model of the horizontal motion by including explicit dependence of the aerodynamic derivatives on a state (sideslip velocity) and external parameters (longitudinal velocity and roll angle) . The autopilot design is based on input-output pseudolinearisation, which is the restriction of input-output feedback linearisation to the set of equilibria of the nonlinear model. The design makes Taylor linearisation of the closed-loop system independent of the choice of equilibria. Thus, if the operating points are in the vicinity of the equilibria, then only one linear model will describe closed-loop dynamics, regardless of the rate of change of the operating points. Simulations for constant lateral acceleration demands show good tracking with fast response time. Parametric stability margins for uncertainty in the controller parameters and aerodynamic derivatives are analysed using Quadratic Lyapunov approach. Copyright @2000 IFAC Keywords. Pseudolinearisation, autopilot, missile, quadratic Lyapunov functions
1. MISSILE MODEL
derived from wind-tunnel measurements (Horton 1992). The nonlinear model (Tsourdos et al. 1998), (Horton 1992) ofthe horiwntal (xy plane in Figure 1) motion is:
Missile autopilots are usually designed using linear models of nonlinear equations of motion and aerodynamic forces and moments (Horton 1995), (Wise 1992). This paper proposes a robust design of a sideslip (yaw) velocity autopilot for a nonlinear missile model. This model describes a quite realistic airframe of a tail-controlled tactical missile in the cruciform fin configuration (Figure 1). The aerodynamic parameters in this model are
(1)
347
dence is a nonlinear function of the sideslip velocity and longitudinal velocity, u = u(v, U).
p
The Mach number is obviously defined as M = Vola, where a is the speed of sound. Since Vo = v'v 2 + U2, the Mach number is also a nonlinear function of the sideslip velocity and longitudinal velocity, M = M(v, U). 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. For an equilibrium (vo, ro, (0) it is possible to derive from (1) a linear model iJ,1 incremental variables, v == v - Vo, f == r - ro and ( == ( - :-(0. In particular, for the straight level flight (with gravitY' influence neglected), we have (vo,ro , (o) = (0,0,0), so that the incremental and absolute variables are numerically identical, although conceptually different.
Fig. 1. Airframe axes. 11
I Interpolated formula C Yv C II , Cn• C nv
Cn ,
0.5[(-25 + M - 601(11)(1 + cos 4'>')+ (-26 + 1.5M - 301(11)(1 - cos 4.>.)] 10 + 0.5[( -1.6M + 21(11)(1 + cos 4'>')+ (-lAM + 1.51(1 1)(1 - cos 4.>.)] -500 - 30M + 2001£11 smClIv' where: Sm = d- 1 [1.3 + O.lM + 0.2(1 + cos 4).)1£11+ 0.3(1 - cos 4'>')1£11- (1.3 + m/500)] slCy{, where: sf = d- 1 [2.6 - (1.3 + m/500)]
2. INPUT-OUTPUT PSEUDOLINEARlSATION Feedback linearisation has several limitations (Zbikowski et al. 1994), 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.
Table 1. Coefficients in nonlinear model (1). where the variables are defined in Figure 1. Here v is the sideslip velocity, r is the body rate, ( the rudder fin deflections, Yv, y( semi-non-dimensional force derivatives due to lateral and fin angle, nv , n(, nr semi-nondimensional 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 - 0.094h air density (Po = 1.23 kgm-3 is the sea level air density and h the missile altitude in km), Vo the total velocity in ms- 1 , S = 7rcP /4 = 0.0314 m 2 the reference area (d = 0.2 m is the reference diameter) and Iz = 67.5 kgm2 is the lateral inertia. For the coefficients eyv ' ell" en., env ' en, only discrete data points are available, obtained from wind tunnel experiments. Hence, an interpolation formula, involving the Mach number M E [0.6,6.0], roll angle .A E [4.5° , 45°] and total incidence u E [3°, 30°], has been calculated with the results summarised in Table 1.
Consider the nth order nonlinear system with m inputs and q outputs: x::!: f(x,u) y=h(x),
(2)
where f: X x U --t Rn and h: X --t Rq are smooth and X eRn, U C Rm are open sets. The set of equilibria of (2) is assumed to depend on parameters pEP C RP, P open, and is denoted as :
E(P)
= {(xo(P),uo(P))1
f(xo(P),ua(P))
= O},
(3)
where Xo: P --t X and uo : P --t U are at least differentiable. It is important to note that parameters p need not be external, i.e., p may depend on x and/or u. In particular, p may depend on both state x and external parameters (J, a fact that is used in Section 3.
v.,
The total velocity vector is the sum of the longitudinal velocity vector {j and the sideslip velocity vector V, i.e. = {j + V, with all three vectors lying on the xy plane (see Figure 1). We assume that U » v, so that the total incidence u, or the angle between (j and v." can be taken as u = v/Vo, as sinu ~ u for small u. Thus, we have u = v /Vo = v / v'v 2 + U2, so that the total inci-
v.,
Let x(P) == x-xo(P), il(p) == u-uo(P) and y(P) == h(x)h(xo(P)) be the incremental variables arising from Taylor linearisation of the open-loop system (2) at an equilibrium from E(P) . Setting A(P) == fJf /fJx!<.:o(p),uo(p)),
348
B(P) == 81/8ul(%o(p),uo(p)) and C(P) == 8h/8xl(%o(p) ,uo(p)) , the corresponding linearised system is:
x(p) y(P)
=A(p)x(p) + B(P)u(P) =C(p)x(p)
(4)
with the additional assumption that (4) is completely controllable and observable and has relative degree r for all points from E(P) and all pEP.
Comparison of (5) with normal forms obtained by feedback linearisation shows that input-output pseudolinearisation may be interpreted as the restriction of feedback linearisation to the parameterised family of sets of equilibria {E(P)}PEP. The focus is on a small portion of the p-parameterised (x, u)-space, i.e., on linearisation along the parameterised family of curves {E(P) }PEP. Thus, it suffices to investigate the tangents 8c) /8x == T of c) and 8k/8x == F, 8k/8v == G of k along {E(P)}PEP, rather then global properties of c) and k. 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. The two conditions on k were implicitly used in footnote 2, but explicit knowledge of k-or c)-is not necessary even there, as vo(P)-and zo(P)-is never explicitly needed. This is because the starting point of the design is (4), which is then transformed into (5).
The problem of input-output pseudolinearisation is to find for system (2) the restriction of a transformation z 4>(x) to E(P) and the restriction of a feedback law u k(x, v) 1 to E(p), so that Taylor linearisation of the resulting closed-loop system is independent of the choice of equilibrium from E(P) and parameter p from P. It should be emphasised that, unlike for feedback linearisation, we are not looking for a global transformation z = 4>(x) and a global feedback law u = k(x, v). We seek only their restrictions to the parameterised family of curves {E(P)}PEP, so that we need not find the whole of 4> and the whole of k . This simplifies the design considerably, but the resulting control law will be applicable only in the immediate neighbourhood of {E(P)}PEP, not the whole X x U. However, Taylor linearisation of the resulting closed-loop system will be independent of p and thus of all equilibria of E(P) . In this way, if the operating points are in the vicinity of the equilibria, then one and only one linear model will describe closed-loop dynamics, regardless of the rate of change of the operating points. Note that the design cannot guarantee anything beyond the immediate neighbourhood of {E(P)}PEP.
= =
The formula for the tangent of transformation 4> along {E(P)}PEP is (Lawrence and Rugh 1994, Lawrence 1998): Z
= Z2
Z2
= Z3
Zr-l
= zr
T(P)
where rows T i , i
= a;+l (xo(p), uo(P))z
=
C(p)Ar-1(p) Tr+l(P)
= r + 1, ... , n
(8)
(5) which is a system of n - r linear equations in (n - r)n unknowns.
where only the n-dimensional vectors ar+l, ... ,an still depend on equilibria from E(P) and parameters from P .
For the tangent of feedback law k along {E(P)}PEP the formula is (Lawrence and Rugh 1994, Lawrence 1998):
With v to be detennined as a function of z and reference signal. 2 Here z ~ z - iI>(xo(p» and ti ~ v - vo(p) with uo(p) k(xo(p), vo(p» . 1
(7)
can be obtained from
Zn = a~(xo(P), uo(P))z y = Zl,
(6)
C(p) C(p)A(P)
zr =V Zr+l
= T(P)x
with T given by
Thus, for a SISO system (4) the restriction of transformation z = c)(x) and feedback law u = k(x, v) to {E(P) }PEP should give the following Taylor linearisation in the (z, v) space: 2 Z1
Since the dynamics defined by a r +1, . .. , an are unobservable (and therefore must be at least stable), the behaviour of (5) from the input-output, v-y, viewpoint is linear of order r and remains the same, no matter what the current values of xo, tLo and p are. This should be contrasted with Taylor linearisation (4) of the open-loop system (2).
=
u = F(P)x + G(p)v,
349
(9)
where
On the other hand, (14) is also of the form (4), allowing the use of pseudo-linearisation.
F(P)
= - (C(P)Ar - 1(p)B(p))
G(p)
= (C(p)Ar-1 (P)B(P)
-1
C(p)Ar(p) , (10)
r1.
We use incremental lateral acceleration av as output. Because av = v + Ur , we can use (12) to obtain a v = YvV + yC;( . Since the term YC;( will be small, we can use the approximation av ~ YvV, which is the augmented lateral acceleration. The use of the augmented lateral acceleration can be justified on the grounds that the acceleration provided directly by the rear control surface Y(( is usually small in comparison to the main lift acceleration. It also causes the minimum phase zero to occur as the lift force developed by the fin by the fin appears directly on the centre of gravity acceleration variable. By removing this term'·;; the effect of the fin is cancelled from the accelerometer signal.
(11)
Formulae (6)-(11) transform a SISO (4) into (5) . Design of a stabilising control law (9) is complete when ii = K1i1 + ... + Krir , 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 - Zl and its derivatives e(i) = - ( i) - Z- i+1, t. = 1, .. . , r - 1. H ence t h e track'mg controI Yd law will be ii = K1 e + ... + K r e(r-1) . Putting z = T(P)x and ii in (9) gives the feedback law in terms of X, so that transformation T can be viewed as an auxiliary tool for designing feedback control law (9) . Thus derived law (9) should be substituted in (4).
Our design of the augmented lateral acceleration autopilot takes the second order (n = 2) model (14)-(15) as the starting point. Noting that the relative degree r is 2, we then use formulae (6)- (11) . Since in our case r = n , we have no need for (8) , so that
3. DESIGN OF LATERAL ACCELERATION AUTOPILOT
T
As explained in Section I , the missile model given by (1) and Table 1 can be represented as:
(P) _ [ C(P) ] _ [Yv(P) - C(P)A(P) - y~(P)
o (P)]
-P2Yv
. (16)
The control law will be
v = Yv(V ,U, A)V - Ur + ydV ,U,A)( r = nv(v, U, A)V + nr(v, U, A)r + ndv , U, A)(.
'ii.
(12)
= F(P)x + G(p)w,
where w is the fictitious input still to be determined in terms of x and the reference signal. Matrix F is
Setting
x"[~l · 'ii.
== (
and
p"m ·
Y == yv(V,U,A)V ,
F(P)
oX
= ~ [-b 1(P) -~(P)] G(p)
= (C(P)A(p)B(P)) - 1 = a~)'
(19)
(14)
where
where
b1(P) = y~(P) - nv(p)yv(p)P2, ~(P) = -y~(P)P2 - nr (P)Yv (P)P2' a(p) = y~(P)y((P) - nC;(p)Yv(P)p2' Thus, representation (12) can be seen as the quasi-linear paraIlleter-varying QLPV model,3 since p in (13) comprises both a state (incremental side-slip velocity v) and external paraIlleters (longitudinal velocity U and roll angle A) , so that matrices (15) depend both on x and 8. QLPV dynamics is: :i: = A(x , 8)x
(18)
and scalar G
= A(P)x + B(P)'ii.
Y = C(P)x,
= - (C(P)A(P)B(P)) -1 C(P)A2(P)
(13)
equations (12) lead to the following description:
3
(17)
(20)
Substituting (18)-(19) to (17), the resulting pseudolinearising control is:
+ b(x , 8)u.
which still requires defining
350
w to ensure tracking.
Let e == a Vd - av be the side-slip velocity error, where aVd is the augmented lateral acceleration demand and = -av with aVd = 0 so that the demand does not need differentiation. Then the final form of the control law is:
e
1 (= a(p) _
x (-
b,(p)y((p) y.(p)
x
0.05
0.1
O. I 5
0.2
0.25
0.3
0.35
0.4
TIme(sec)
~:~~ av - ~(p)r + Kte + K 2e) .
(22)
w;
The constants Kt, K2 in (22) are chosen as Kt = and K2 = 2~wn , where Wn = 60 rad/sec and ~ = 0.65, so that the error equation is + 2~wne + w~e = o. This should give a 3- 4 times faster response than in the open loop.
e
i
Simulation results for 100 m/sec 2 constant demands in lateral acceleration are shown in Figure 2. The observed steady-state error in lateral acceleration in Figure 2 is about 3%.
-:rc. .
.~
u
• :•
•
••• • • • • • • • • • • •
• ••• • • •
• ••• ••• • • • ••• • •
• • • • • • •••. • • • • • • • • • • • • ••• • • • • • • • • • •
~
..... ...... __ ... .....,.: ......... .. :.. ..........:... ........ : ... ....... . : ......... .. : .. ... ..... _
~-10
~ : : . ." -15 L--~--c".,---:c':-:---:'::----:-7::---:':---:-=---7 iii 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Recall that the augmented lateral acceleration demand was derived approximately from the normal lateral acceleration demand. The approximation neglected the fin force term y« . The effect of this term is evident in the non-minimum phase characteristic in normal lateral acceleration a v and it is clearly visible in Figure 2; note also that the initial fin angle ( < O. This is quickly overcome by the side-slip force as incidence builds up. The steady-state error represents the fin force contribution that was neglected in the analysis.
!":VC : : 'm~~I : • ' 1 i ltJ .:: <. .; :. ' ·· 1 e
o
0.05
~. 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.25
0.3
0.35
0.4
TIm~sac)
0
0.1
0.15
0.2 TIme(sec)
4. ROBUST STABILITY ANALYSIS
Fig. 2. Simulation for constant lateral acceleration demand of 100 ms- 2 •
In this section we analyse stability of the autopilot designed in Section 3 with uncertainties of mixed type (Djaferis 1995) with the objective of quantifying, as nonconservatively as possible, the sizes of perturbations that can be tolerated by the closed-loop system.
Assuming that the derivative cl Vd of the side-slip velocity demand is zero, we have = -a v . Then the closed loop equations (23) can be rewritten as:
e
E(P)x fj
4.1 Nominal model
with
x,
= Ac(P)x + Bc(P)Vd
= C(P)x
(24)
fj and p as in (13), matrix Ac is:
The side-slip velocity autopilot closed-loop characteristic equation can be obtained by substituting control law (22) into equation (12). This yields :
(25)
where:
v = Yv(P)v -
P2r
+ a~) (-bt(P)v - b2(p)r
+ Kte + K 2e)
= yv(p)a(p) -
yv(P)y«P)(bt (P) - Kt)
a ct2 a c2t
= -P2a(p) - y«(p)~(p) = nv(p)a(p) - Yv(p)n«p)(bt(p) -
ac22
= nr(P)a(p) -
n«(p)~(p)
acll
r = nv(p)v + nr(P)r + :~)(-bt(P)v-b2(p)r+Kte+K2e). (23)
351
Kt)
(26)
and matrices E, Bc, C given as:
C(P)
= [Yv(P)
0] .
systelllS similar to the one encountered in this paper. The test defines sufficient conditions for stability, suitable for systems such as quasi-linear parameter varying, where state matrices e.g. A(P(t)) are functions of the parameters p(t). The test requires that A(P(t)) be modelled as a convex polytope of matrices, from the possible set of values of the A(p(t)). The quadratic stability test seeks a matrix P > 0 such that V(x) = x T Px is a Lyapunov function for the differential inclusion (24) . That is, such that
(27)
For the purposes of robust stability analysis we assume that (24)-(27) represent an uncertain LT! model, whose uncertainty is caused by variations in p (recall that PI is a state), mass and air density.
[E(p)-I A(p)]T P
P> I
In this section the stability margins associated with two kinds of parametric uncertainty: (1) varying controller parameters: a, bI , b2 ; (2) varying aerodynamic parameters: Yv, P2, n r , nv, n( are analysed. The problem can be simplified by making the same approximations that were used in the design process by neglecting the fin angle force Y(.
In order to obtain the worst case stability margin for each controller parameter, we have to maximise their quadratic stability region. For each controller parameter a(t) E [Q, a] denote the centre and radius of each interval by J.L = ~(Q+a) and 6 = HQ-a), respectively. The maximal quadratic stability region is defined as the largest portion of the parameter box where the quadratic stability can be established. In other words, the largest dilatation factor 0 such that the system is quadratically stable whenever a(t) E [J.L - 06, J.L + 06]. Computing this largest factor is a gen~ralised eigenvalue minimisation problem. This optimisation is performed with existing LMI software (Gahinet et al. 1995). It was found that the stability margins for the controller parameters are 60% for a, ~ and' b1 .
4.2 .1. Controller stability analysis To assess the robustness to controller parameters errors in a, bI , ~ we first define their nominal values: ,
2
(28)
The error model will be of the form
+ Ab I =b2 + Ab2 a = ii + Aa.
(30)
along all parameter tI;ajectories. In other words, it suffices that P be positive definite and satisfy the Lya, punov inequality at each corner of the parameter box. This reformulation has the merit of reducing a problem with infinitely many constraints to a finite set of matrix inequalities. The resulting LMI system (30) is then readily solved numerically with existing LMI software (Gahinet et al. 1995). The conservative assumption of the quadratic Lyapunov stability test is that the system can change infinitely fast.
4.2 Worst case parametric stability margin
b1 = Yv - n v P2 , 2 b2 = -Y vP2 - Yv n vP2 ii = -Yv(p)n(P2.
+ P[E(p)-I A(p)] < 0
bI = bI
~
(29)
4.2 .2. Aerodynamic derivatives stability analysis
For real time-varying parameters, there are essentially two tractable tests of robust stability: quadratic stability and real J.L upper bound with constant D, G scaling matrices, the second being more conservative in the case of affine real parametric uncertainty. Note that both criteria guarantee robustness against arbitrarily fast time variations of the parameters. The quadratic stability test is also applicable to constant uncertain parameters, yet at the expense of overly conservative answers in general.
The uncertainty in aerodynamic derivatives can also be studied using Quadratic Lyapunov FUnctions techniques. The approach used here assumes that the true values and the design values are in error. The error model is thus:
=
Yv Yv + Ayv nv =nv + Anv nr =nr + Anr n( =n( + An,
Since our approach builds upon quadratic stability, we briefly review the method. The quadratic Lyapunov test is a general, although conservative test applicable to the
P2
352
= ft2 + AP2.
(31)
It should be noted that most of the aerodynamic coef-
Horton, M. P. (1995). Autopilots for tactical missiles: An overview. IMechE: Journal of Systems and Control Engineering 209, 127-139. 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). Input-output pseudolinearization for nonlinear systems. IEEE Transactions on Automatic Control 39(11), 22072218. Reboulet, C. and C. Champetier (1984). A new method for linearizing nonlinear systems: The pseudolinearization. International Journal of Control 40(4),631-638. Tsourdos, A., A. Blumel and B. A. White (1998). Trajectory control of a nonlinear homing missile. In: 14th IFAC Symposium on Automatic Control in Aerospace. Korea. 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. Zbikowski, R., K. J. Hunt, A. Dzielinski, R. MurraySmith and P. J . Gawthrop (1994). A review of advances in neural adaptive control systems. Technical Report of the ESPRIT NACT Project TPl. Glasgow University and Daimler-Benz Research. (Available from FTP server ftp.mech.gla.ac.uk as PostScript file /nact/nact_tpl.ps) .
ficients are multi-affine functions of the uncertain variables. This clearly will give conservative results in the stability analysis. Following the same approach with the one for the controller parametric stability analysis we found that the aerodynamic derivative nv has little influence on the performance and the robustness. Both can be varied by up to 70% without approaching instability. The 1>2 terms and the ne; terms represent the most sensitive parameters and have a similar effect on stability since for these parameters a reduced maximum variation of 30% is found for them. For the remaining aerodynamic derivatives, the quadratic stability can be established for parameter variation of 60%.
5. CONCLUSIONS It has been shown that a reasonably realistic missile model can be described as a QLPV system. This suggests that the quasi-linear, time-varying dynamics description (of which QLPV is a special case) is not only relevant, but also offers the proper level of generality for nonlinear representation of missiles. QLPV models are amenable to input-output pseudolinearising design, which results in a controller which depends both on the state and external parameters. The pseudolinearising autopilot is free of the difficulties associated with gain scheduling, as it consists of one controller only and "scheduling" is done automatically by feedback, giving total independence of the operating point . However, the pseudolinearising design is only valid (as gain scheduling is) in the vicinity of the equilibria and the state transformation matrix may be ill-conditioned. An effective robustness analysis was possible by treating the closed-loop system as an uncertain LTI model. Finally, the choice of output allowed having the relative degree equal to the order of the system, so that no unobservable dynamics was involved, which would have required their stability analysis and could have diminished closed-loop performance.
6. REFERENCES Djaferis, T. E . (1995). Robust Control Design - A Polynomial Approach. Kluwer Academic Publishers. Gahinet, P., A. Nemirovski, A.J. Laub and M. Chilali (1995). LMI Control Toolbox. The MathWorks Partner Series. The Math Works Inc. Horton, M. P. (1992). A study of autopilots for the adaptive control of tactical guided missiles. Master's thesis. University of Bath.
353
LATERAL ACCELERATION FLIGHT CONTROL FOR A QUASI-LINEAR PARAMETER VARYING MISSILE MODEL VIA PSEUDOLINEARISATION A. Tsourdos R. Zbikowski and B. A. White
Department of Aerospace, Power and Sensors Cmnfield University RMCS Shrivenham Swindon SN6 8LA, England, UK A, [email protected]
Abstract. A lateral augmented acceleration autopilot is designed for a model of a tactical missile and robust stability of the closed-loop system investigated. The tail-controlled missile in the cruciform fin configuration is modelled as a second-order quasi-linear parameter-varying system. This nonlinear model is obtained from the Taylor linearised model of the horizontal motion by including explicit dependence of the aerodynamic derivatives on a state (sideslip velocity) and external parameters (longitudinal velocity and roll angle) . The autopilot design is based on input-output pseudolinearisation, which is the restriction of input-output feedback linearisation to the set of equilibria of the nonlinear model. The design makes Taylor linearisation of the closed-loop system independent of the choice of equilibria. Thus, if the operating points are in the vicinity of the equilibria, then only one linear model will describe closed-loop dynamics, regardless of the rate of change of the operating points. Simulations for constant lateral acceleration demands show good tracking with fast response time. Parametric stability margins for uncertainty in the controller parameters and aerodynamic derivatives are analysed using Quadratic Lyapunov approach. Copyright @2000 IFAC Keywords. Pseudolinearisation, autopilot, missile, quadratic Lyapunov functions
1. MISSILE MODEL
derived from wind-tunnel measurements (Horton 1992). The nonlinear model (Tsourdos et al. 1998), (Horton 1992) ofthe horiwntal (xy plane in Figure 1) motion is:
Missile autopilots are usually designed using linear models of nonlinear equations of motion and aerodynamic forces and moments (Horton 1995), (Wise 1992). This paper proposes a robust design of a sideslip (yaw) velocity autopilot for a nonlinear missile model. This model describes a quite realistic airframe of a tail-controlled tactical missile in the cruciform fin configuration (Figure 1). The aerodynamic parameters in this model are
(1)
347
dence is a nonlinear function of the sideslip velocity and longitudinal velocity, u = u(v, U).
p
The Mach number is obviously defined as M = Vola, where a is the speed of sound. Since Vo = v'v 2 + U2, the Mach number is also a nonlinear function of the sideslip velocity and longitudinal velocity, M = M(v, U). 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. For an equilibrium (vo, ro, (0) it is possible to derive from (1) a linear model iJ,1 incremental variables, v == v - Vo, f == r - ro and ( == ( - :-(0. In particular, for the straight level flight (with gravitY' influence neglected), we have (vo,ro , (o) = (0,0,0), so that the incremental and absolute variables are numerically identical, although conceptually different.
Fig. 1. Airframe axes. 11
I Interpolated formula C Yv C II , Cn• C nv
Cn ,
0.5[(-25 + M - 601(11)(1 + cos 4'>')+ (-26 + 1.5M - 301(11)(1 - cos 4.>.)] 10 + 0.5[( -1.6M + 21(11)(1 + cos 4'>')+ (-lAM + 1.51(1 1)(1 - cos 4.>.)] -500 - 30M + 2001£11 smClIv' where: Sm = d- 1 [1.3 + O.lM + 0.2(1 + cos 4).)1£11+ 0.3(1 - cos 4'>')1£11- (1.3 + m/500)] slCy{, where: sf = d- 1 [2.6 - (1.3 + m/500)]
2. INPUT-OUTPUT PSEUDOLINEARlSATION Feedback linearisation has several limitations (Zbikowski et al. 1994), 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.
Table 1. Coefficients in nonlinear model (1). where the variables are defined in Figure 1. Here v is the sideslip velocity, r is the body rate, ( the rudder fin deflections, Yv, y( semi-non-dimensional force derivatives due to lateral and fin angle, nv , n(, nr semi-nondimensional 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 - 0.094h air density (Po = 1.23 kgm-3 is the sea level air density and h the missile altitude in km), Vo the total velocity in ms- 1 , S = 7rcP /4 = 0.0314 m 2 the reference area (d = 0.2 m is the reference diameter) and Iz = 67.5 kgm2 is the lateral inertia. For the coefficients eyv ' ell" en., env ' en, only discrete data points are available, obtained from wind tunnel experiments. Hence, an interpolation formula, involving the Mach number M E [0.6,6.0], roll angle .A E [4.5° , 45°] and total incidence u E [3°, 30°], has been calculated with the results summarised in Table 1.
Consider the nth order nonlinear system with m inputs and q outputs: x::!: f(x,u) y=h(x),
(2)
where f: X x U --t Rn and h: X --t Rq are smooth and X eRn, U C Rm are open sets. The set of equilibria of (2) is assumed to depend on parameters pEP C RP, P open, and is denoted as :
E(P)
= {(xo(P),uo(P))1
f(xo(P),ua(P))
= O},
(3)
where Xo: P --t X and uo : P --t U are at least differentiable. It is important to note that parameters p need not be external, i.e., p may depend on x and/or u. In particular, p may depend on both state x and external parameters (J, a fact that is used in Section 3.
v.,
The total velocity vector is the sum of the longitudinal velocity vector {j and the sideslip velocity vector V, i.e. = {j + V, with all three vectors lying on the xy plane (see Figure 1). We assume that U » v, so that the total incidence u, or the angle between (j and v." can be taken as u = v/Vo, as sinu ~ u for small u. Thus, we have u = v /Vo = v / v'v 2 + U2, so that the total inci-
v.,
Let x(P) == x-xo(P), il(p) == u-uo(P) and y(P) == h(x)h(xo(P)) be the incremental variables arising from Taylor linearisation of the open-loop system (2) at an equilibrium from E(P) . Setting A(P) == fJf /fJx!<.:o(p),uo(p)),
348
B(P) == 81/8ul(%o(p),uo(p)) and C(P) == 8h/8xl(%o(p) ,uo(p)) , the corresponding linearised system is:
x(p) y(P)
=A(p)x(p) + B(P)u(P) =C(p)x(p)
(4)
with the additional assumption that (4) is completely controllable and observable and has relative degree r for all points from E(P) and all pEP.
Comparison of (5) with normal forms obtained by feedback linearisation shows that input-output pseudolinearisation may be interpreted as the restriction of feedback linearisation to the parameterised family of sets of equilibria {E(P)}PEP. The focus is on a small portion of the p-parameterised (x, u)-space, i.e., on linearisation along the parameterised family of curves {E(P) }PEP. Thus, it suffices to investigate the tangents 8c) /8x == T of c) and 8k/8x == F, 8k/8v == G of k along {E(P)}PEP, rather then global properties of c) and k. 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. The two conditions on k were implicitly used in footnote 2, but explicit knowledge of k-or c)-is not necessary even there, as vo(P)-and zo(P)-is never explicitly needed. This is because the starting point of the design is (4), which is then transformed into (5).
The problem of input-output pseudolinearisation is to find for system (2) the restriction of a transformation z 4>(x) to E(P) and the restriction of a feedback law u k(x, v) 1 to E(p), so that Taylor linearisation of the resulting closed-loop system is independent of the choice of equilibrium from E(P) and parameter p from P. It should be emphasised that, unlike for feedback linearisation, we are not looking for a global transformation z = 4>(x) and a global feedback law u = k(x, v). We seek only their restrictions to the parameterised family of curves {E(P)}PEP, so that we need not find the whole of 4> and the whole of k . This simplifies the design considerably, but the resulting control law will be applicable only in the immediate neighbourhood of {E(P)}PEP, not the whole X x U. However, Taylor linearisation of the resulting closed-loop system will be independent of p and thus of all equilibria of E(P) . In this way, if the operating points are in the vicinity of the equilibria, then one and only one linear model will describe closed-loop dynamics, regardless of the rate of change of the operating points. Note that the design cannot guarantee anything beyond the immediate neighbourhood of {E(P)}PEP.
= =
The formula for the tangent of transformation 4> along {E(P)}PEP is (Lawrence and Rugh 1994, Lawrence 1998): Z
= Z2
Z2
= Z3
Zr-l
= zr
T(P)
where rows T i , i
= a;+l (xo(p), uo(P))z
=
C(p)Ar-1(p) Tr+l(P)
= r + 1, ... , n
(8)
(5) which is a system of n - r linear equations in (n - r)n unknowns.
where only the n-dimensional vectors ar+l, ... ,an still depend on equilibria from E(P) and parameters from P .
For the tangent of feedback law k along {E(P)}PEP the formula is (Lawrence and Rugh 1994, Lawrence 1998):
With v to be detennined as a function of z and reference signal. 2 Here z ~ z - iI>(xo(p» and ti ~ v - vo(p) with uo(p) k(xo(p), vo(p» . 1
(7)
can be obtained from
Zn = a~(xo(P), uo(P))z y = Zl,
(6)
C(p) C(p)A(P)
zr =V Zr+l
= T(P)x
with T given by
Thus, for a SISO system (4) the restriction of transformation z = c)(x) and feedback law u = k(x, v) to {E(P) }PEP should give the following Taylor linearisation in the (z, v) space: 2 Z1
Since the dynamics defined by a r +1, . .. , an are unobservable (and therefore must be at least stable), the behaviour of (5) from the input-output, v-y, viewpoint is linear of order r and remains the same, no matter what the current values of xo, tLo and p are. This should be contrasted with Taylor linearisation (4) of the open-loop system (2).
=
u = F(P)x + G(p)v,
349
(9)
where
On the other hand, (14) is also of the form (4), allowing the use of pseudo-linearisation.
F(P)
= - (C(P)Ar - 1(p)B(p))
G(p)
= (C(p)Ar-1 (P)B(P)
-1
C(p)Ar(p) , (10)
r1.
We use incremental lateral acceleration av as output. Because av = v + Ur , we can use (12) to obtain a v = YvV + yC;( . Since the term YC;( will be small, we can use the approximation av ~ YvV, which is the augmented lateral acceleration. The use of the augmented lateral acceleration can be justified on the grounds that the acceleration provided directly by the rear control surface Y(( is usually small in comparison to the main lift acceleration. It also causes the minimum phase zero to occur as the lift force developed by the fin by the fin appears directly on the centre of gravity acceleration variable. By removing this term'·;; the effect of the fin is cancelled from the accelerometer signal.
(11)
Formulae (6)-(11) transform a SISO (4) into (5) . Design of a stabilising control law (9) is complete when ii = K1i1 + ... + Krir , 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 - Zl and its derivatives e(i) = - ( i) - Z- i+1, t. = 1, .. . , r - 1. H ence t h e track'mg controI Yd law will be ii = K1 e + ... + K r e(r-1) . Putting z = T(P)x and ii in (9) gives the feedback law in terms of X, so that transformation T can be viewed as an auxiliary tool for designing feedback control law (9) . Thus derived law (9) should be substituted in (4).
Our design of the augmented lateral acceleration autopilot takes the second order (n = 2) model (14)-(15) as the starting point. Noting that the relative degree r is 2, we then use formulae (6)- (11) . Since in our case r = n , we have no need for (8) , so that
3. DESIGN OF LATERAL ACCELERATION AUTOPILOT
T
As explained in Section I , the missile model given by (1) and Table 1 can be represented as:
(P) _ [ C(P) ] _ [Yv(P) - C(P)A(P) - y~(P)
o (P)]
-P2Yv
. (16)
The control law will be
v = Yv(V ,U, A)V - Ur + ydV ,U,A)( r = nv(v, U, A)V + nr(v, U, A)r + ndv , U, A)(.
'ii.
(12)
= F(P)x + G(p)w,
where w is the fictitious input still to be determined in terms of x and the reference signal. Matrix F is
Setting
x"[~l · 'ii.
== (
and
p"m ·
Y == yv(V,U,A)V ,
F(P)
oX
= ~ [-b 1(P) -~(P)] G(p)
= (C(P)A(p)B(P)) - 1 = a~)'
(19)
(14)
where
where
b1(P) = y~(P) - nv(p)yv(p)P2, ~(P) = -y~(P)P2 - nr (P)Yv (P)P2' a(p) = y~(P)y((P) - nC;(p)Yv(P)p2' Thus, representation (12) can be seen as the quasi-linear paraIlleter-varying QLPV model,3 since p in (13) comprises both a state (incremental side-slip velocity v) and external paraIlleters (longitudinal velocity U and roll angle A) , so that matrices (15) depend both on x and 8. QLPV dynamics is: :i: = A(x , 8)x
(18)
and scalar G
= A(P)x + B(P)'ii.
Y = C(P)x,
= - (C(P)A(P)B(P)) -1 C(P)A2(P)
(13)
equations (12) lead to the following description:
3
(17)
(20)
Substituting (18)-(19) to (17), the resulting pseudolinearising control is:
+ b(x , 8)u.
which still requires defining
350
w to ensure tracking.
Let e == a Vd - av be the side-slip velocity error, where aVd is the augmented lateral acceleration demand and = -av with aVd = 0 so that the demand does not need differentiation. Then the final form of the control law is:
e
1 (= a(p) _
x (-
b,(p)y((p) y.(p)
x
0.05
0.1
O. I 5
0.2
0.25
0.3
0.35
0.4
TIme(sec)
~:~~ av - ~(p)r + Kte + K 2e) .
(22)
w;
The constants Kt, K2 in (22) are chosen as Kt = and K2 = 2~wn , where Wn = 60 rad/sec and ~ = 0.65, so that the error equation is + 2~wne + w~e = o. This should give a 3- 4 times faster response than in the open loop.
e
i
Simulation results for 100 m/sec 2 constant demands in lateral acceleration are shown in Figure 2. The observed steady-state error in lateral acceleration in Figure 2 is about 3%.
-:rc. .
.~
u
• :•
•
••• • • • • • • • • • • •
• ••• • • •
• ••• ••• • • • ••• • •
• • • • • • •••. • • • • • • • • • • • • ••• • • • • • • • • • •
~
..... ...... __ ... .....,.: ......... .. :.. ..........:... ........ : ... ....... . : ......... .. : .. ... ..... _
~-10
~ : : . ." -15 L--~--c".,---:c':-:---:'::----:-7::---:':---:-=---7 iii 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Recall that the augmented lateral acceleration demand was derived approximately from the normal lateral acceleration demand. The approximation neglected the fin force term y« . The effect of this term is evident in the non-minimum phase characteristic in normal lateral acceleration a v and it is clearly visible in Figure 2; note also that the initial fin angle ( < O. This is quickly overcome by the side-slip force as incidence builds up. The steady-state error represents the fin force contribution that was neglected in the analysis.
!":VC : : 'm~~I : • ' 1 i ltJ .:: <. .; :. ' ·· 1 e
o
0.05
~. 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.25
0.3
0.35
0.4
TIm~sac)
0
0.1
0.15
0.2 TIme(sec)
4. ROBUST STABILITY ANALYSIS
Fig. 2. Simulation for constant lateral acceleration demand of 100 ms- 2 •
In this section we analyse stability of the autopilot designed in Section 3 with uncertainties of mixed type (Djaferis 1995) with the objective of quantifying, as nonconservatively as possible, the sizes of perturbations that can be tolerated by the closed-loop system.
Assuming that the derivative cl Vd of the side-slip velocity demand is zero, we have = -a v . Then the closed loop equations (23) can be rewritten as:
e
E(P)x fj
4.1 Nominal model
with
x,
= Ac(P)x + Bc(P)Vd
= C(P)x
(24)
fj and p as in (13), matrix Ac is:
The side-slip velocity autopilot closed-loop characteristic equation can be obtained by substituting control law (22) into equation (12). This yields :
(25)
where:
v = Yv(P)v -
P2r
+ a~) (-bt(P)v - b2(p)r
+ Kte + K 2e)
= yv(p)a(p) -
yv(P)y«P)(bt (P) - Kt)
a ct2 a c2t
= -P2a(p) - y«(p)~(p) = nv(p)a(p) - Yv(p)n«p)(bt(p) -
ac22
= nr(P)a(p) -
n«(p)~(p)
acll
r = nv(p)v + nr(P)r + :~)(-bt(P)v-b2(p)r+Kte+K2e). (23)
351
Kt)
(26)
and matrices E, Bc, C given as:
C(P)
= [Yv(P)
0] .
systelllS similar to the one encountered in this paper. The test defines sufficient conditions for stability, suitable for systems such as quasi-linear parameter varying, where state matrices e.g. A(P(t)) are functions of the parameters p(t). The test requires that A(P(t)) be modelled as a convex polytope of matrices, from the possible set of values of the A(p(t)). The quadratic stability test seeks a matrix P > 0 such that V(x) = x T Px is a Lyapunov function for the differential inclusion (24) . That is, such that
(27)
For the purposes of robust stability analysis we assume that (24)-(27) represent an uncertain LT! model, whose uncertainty is caused by variations in p (recall that PI is a state), mass and air density.
[E(p)-I A(p)]T P
P> I
In this section the stability margins associated with two kinds of parametric uncertainty: (1) varying controller parameters: a, bI , b2 ; (2) varying aerodynamic parameters: Yv, P2, n r , nv, n( are analysed. The problem can be simplified by making the same approximations that were used in the design process by neglecting the fin angle force Y(.
In order to obtain the worst case stability margin for each controller parameter, we have to maximise their quadratic stability region. For each controller parameter a(t) E [Q, a] denote the centre and radius of each interval by J.L = ~(Q+a) and 6 = HQ-a), respectively. The maximal quadratic stability region is defined as the largest portion of the parameter box where the quadratic stability can be established. In other words, the largest dilatation factor 0 such that the system is quadratically stable whenever a(t) E [J.L - 06, J.L + 06]. Computing this largest factor is a gen~ralised eigenvalue minimisation problem. This optimisation is performed with existing LMI software (Gahinet et al. 1995). It was found that the stability margins for the controller parameters are 60% for a, ~ and' b1 .
4.2 .1. Controller stability analysis To assess the robustness to controller parameters errors in a, bI , ~ we first define their nominal values: ,
2
(28)
The error model will be of the form
+ Ab I =b2 + Ab2 a = ii + Aa.
(30)
along all parameter tI;ajectories. In other words, it suffices that P be positive definite and satisfy the Lya, punov inequality at each corner of the parameter box. This reformulation has the merit of reducing a problem with infinitely many constraints to a finite set of matrix inequalities. The resulting LMI system (30) is then readily solved numerically with existing LMI software (Gahinet et al. 1995). The conservative assumption of the quadratic Lyapunov stability test is that the system can change infinitely fast.
4.2 Worst case parametric stability margin
b1 = Yv - n v P2 , 2 b2 = -Y vP2 - Yv n vP2 ii = -Yv(p)n(P2.
+ P[E(p)-I A(p)] < 0
bI = bI
~
(29)
4.2 .2. Aerodynamic derivatives stability analysis
For real time-varying parameters, there are essentially two tractable tests of robust stability: quadratic stability and real J.L upper bound with constant D, G scaling matrices, the second being more conservative in the case of affine real parametric uncertainty. Note that both criteria guarantee robustness against arbitrarily fast time variations of the parameters. The quadratic stability test is also applicable to constant uncertain parameters, yet at the expense of overly conservative answers in general.
The uncertainty in aerodynamic derivatives can also be studied using Quadratic Lyapunov FUnctions techniques. The approach used here assumes that the true values and the design values are in error. The error model is thus:
=
Yv Yv + Ayv nv =nv + Anv nr =nr + Anr n( =n( + An,
Since our approach builds upon quadratic stability, we briefly review the method. The quadratic Lyapunov test is a general, although conservative test applicable to the
P2
352
= ft2 + AP2.
(31)
It should be noted that most of the aerodynamic coef-
Horton, M. P. (1995). Autopilots for tactical missiles: An overview. IMechE: Journal of Systems and Control Engineering 209, 127-139. 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). Input-output pseudolinearization for nonlinear systems. IEEE Transactions on Automatic Control 39(11), 22072218. Reboulet, C. and C. Champetier (1984). A new method for linearizing nonlinear systems: The pseudolinearization. International Journal of Control 40(4),631-638. Tsourdos, A., A. Blumel and B. A. White (1998). Trajectory control of a nonlinear homing missile. In: 14th IFAC Symposium on Automatic Control in Aerospace. Korea. 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. Zbikowski, R., K. J. Hunt, A. Dzielinski, R. MurraySmith and P. J . Gawthrop (1994). A review of advances in neural adaptive control systems. Technical Report of the ESPRIT NACT Project TPl. Glasgow University and Daimler-Benz Research. (Available from FTP server ftp.mech.gla.ac.uk as PostScript file /nact/nact_tpl.ps) .
ficients are multi-affine functions of the uncertain variables. This clearly will give conservative results in the stability analysis. Following the same approach with the one for the controller parametric stability analysis we found that the aerodynamic derivative nv has little influence on the performance and the robustness. Both can be varied by up to 70% without approaching instability. The 1>2 terms and the ne; terms represent the most sensitive parameters and have a similar effect on stability since for these parameters a reduced maximum variation of 30% is found for them. For the remaining aerodynamic derivatives, the quadratic stability can be established for parameter variation of 60%.
5. CONCLUSIONS It has been shown that a reasonably realistic missile model can be described as a QLPV system. This suggests that the quasi-linear, time-varying dynamics description (of which QLPV is a special case) is not only relevant, but also offers the proper level of generality for nonlinear representation of missiles. QLPV models are amenable to input-output pseudolinearising design, which results in a controller which depends both on the state and external parameters. The pseudolinearising autopilot is free of the difficulties associated with gain scheduling, as it consists of one controller only and "scheduling" is done automatically by feedback, giving total independence of the operating point . However, the pseudolinearising design is only valid (as gain scheduling is) in the vicinity of the equilibria and the state transformation matrix may be ill-conditioned. An effective robustness analysis was possible by treating the closed-loop system as an uncertain LTI model. Finally, the choice of output allowed having the relative degree equal to the order of the system, so that no unobservable dynamics was involved, which would have required their stability analysis and could have diminished closed-loop performance.
6. REFERENCES Djaferis, T. E . (1995). Robust Control Design - A Polynomial Approach. Kluwer Academic Publishers. Gahinet, P., A. Nemirovski, A.J. Laub and M. Chilali (1995). LMI Control Toolbox. The MathWorks Partner Series. The Math Works Inc. Horton, M. P. (1992). A study of autopilots for the adaptive control of tactical guided missiles. Master's thesis. University of Bath.
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