Multi-Objective Frequency Loop-Shaping of a Lateral Missile Autopilot

ELSEVIER

Copyright © IFAC Automatic Control in Aerospace, Saint-Petersburg, Russia, 2004

IFAC PUBLICATIONS www.elsevier.comllocate/ifac

MULTI-OBJECTIVE FREQUENCY LOOP-SHAPING OF A LATERAL MISSILE AUTOPILOT T. Sreenuch, A. Tsourdos, E. J. Hugbes and B. A. White

Department ofAero:,pace, Power and Sensors Cranfield University (RMO; Shrivenham, S .....indon, !:,·''/V68LA. UK

Abstract: The paper presents the lateral acceleration control design of non-linear missile model using the multi-objective evolutionary optimisation. The interpolated controller design is carried out by minimising frequency domain based performance objectives subject to actuator limitations (position and rate). TIle actuator constraint functions are formulated in term of closed-loop actuator impulse response. The controller's trade-ofls are analysed using the obtained Pareto-optimal solutions (corresponding to a given set of target vectors). The non-linear simulation result., show that the selected interpolated controller is indeed a robust tracking controller for all possible perturbations. In addition, the resulting fin responses are accurately determined. Copyright~) 2004 TFAC

2. MISSILE MODEL AND AUTOPILOT REQUIREMENTS

1. INTRODUCTION

2. f Non-Linear Model

This paper continues the work of gain-scheduling missile autopilot design using multi-objective evolutionary optimisation described in (Sreenuch et al., 2004). The design will be carried out by simultaneously shaping the inner and outer loop results in the desired open and closed-loop frequency responses (QFT-like). However, the demanded fast closed-loop response is constrained by the actuator limitations (i.e. position and rate limits), Instead of basing on the gain frequency product, the actuator constraint functions \Nill be formed using the closed-loop actuator impulse response. This means to improve on the conservatism. To ensure the stability of the interpolated controller, the controllers' operating regions are intuitively suggested to be overlapped.

The missile model used in this study is taken from Horton's MSc thesis (Borton, 1992). It describes a 5 DOF model in parametric format with severe crosscoupling and non-linear behaviour. This study will look at the reduced problem of a 2 DOF controller for the lateral motion (on the xy plane in Fig. I). The [J

This paper is organised as follows: The missile's lateral dynamics and autopilot requirements arc described in section 2. The problem formulation, the stability constraints and frequency domain based objective functions, is stated in section 3. The implication of trade-ofts and simulation results are presented ill section 4.

Fig. 1. Airframe axes and nomenclature. airframc is roll stabilised (,\ = 45°), and no coupling is assumed between pitch and yaw channels. With these assumptions, the equations of motion are given by 577

v = Yv(M , O")v + yc,(M, oX -

the lateral accelerometer dynamics and la = 0.9 m (0.8 m) is the accelerometer moment arm when full (all burnt). The open-loop transmission

Ur ,

1

= 2mPVS(Cyvv + VCy, () - Ur, r = nv(M, O")v + nr(M, O")r + ne; (M, 0")( , 1

= 2Iz pVSd(Cnvv

1

+ '2dCnrr + VCn,()

n(s-(nC;Yv-nvYC;) d Gr(s)= )an , S2 - (Yv + nr)s + (Unv + Yvnr

(I)

Go. (s)

where the variables are defined in Fig. I. Here v is the side-slip velocity, r is the body rate, ( is the rudder fin deflections, Yv, Ye; are semi-non-dimensional force derivatives due to lateral velocity and fin angle, n v , n r , ne; are semi-non-dimensional force derivatives due to side-slip velocity, body rate and fin angle. U is the forward velocity. Furthermore, m = 150 kg (125 kg) is the missile mass when full (all burnt), P = Po 0.094h is the air density (Po = 1.23 kg/m 3 is the sea level air density and h is the missile altitude in km), V is the total velocity in m/s, S = 7rd 2 /4 = 0.0314 m2 is the reference area (d = 0.2 m is the reference diameter) and 1z = 75 kg · m2 (60 kg . m2 ) is the lateral inertia when full (all burnt). For the coefficients Cy., Cy" C n., C nr , C n, only discrete data points are available, obtained from wind tunnel experiments. The interpolation formulas, involving the Mach number M and incidence 0", have been evaluated with the results summarised in Table I. V = v'U2 + v 2 is to

I

Ae~ynamic denvatlve C llv

C lle C nr C nv

Cn,

I

Vr

= YC;s2 -

y(nrs - U(n<;yv - nvYc;) nC;s - (nC; yv - nvYC;)

(2)

represent the missile dynamics, obtained by linearising (I).

2.3 Closed-Loop Performance Specifications

The autopilot is required to track a lateral acceleration demand aYd over the whole flight envelope (see 500 m/s 2 is constrained by Fig. 3), whom lalldl limitations on the airframe's structural integrity. Note

:s

-

...

-

_ • .,' EIIwIIIen

.

u

J'

i

Interpolated formula

-26 + 1.5M - 60lul -10 + lAM - 1.51ul -500 - 30M + ·200Iul SmCII. ' where Sm = d- 1 [(1.3 + m/500) -(1.3 + O.lM + 0.31001)]

.

DO

l1Int(l)

Fig. 3. Velocity operating envelope.

s,GII , , where sf = d- 1 [(1.3 + m/500) - 2.6]

that the lateral acceleration ay at the center of gravity is defined by ay = v + Ur. It must also be as robust to the variation in mass (the propellant is constantly burnt) and uncertainty in aerodynamic derivatives (boc 1111 ,boc...( , bocn ,bocn r ,bo c n( = ±5%). A list of performance specification (for a step input) is given in the time-domain using familiar figures as follows :

. . . Table I. AerodynamiC denvatlves of the non-linear model (A = 45°).

total velocity. It is assumed that U » v, so that the total incidence 0" can thus be taken as 0" = v/U, as sin 0" ;:::: 0" for small 0". Finally, the Mach number is obviously defined as M = V/a, where a = 340 m/s is the speed of sound.

\I

< 0.2 s) provided that the system at high frequency is not upset by actuator saturation (1(1 :s S = 0.1 rad) and sluggishness (1(1 :s R = 10 rad/s), Settling time variation lOt. 1 :s 0.05 s, Steady state error eBs :s 10%, Damping ratio (a v ;:::: 0.7, Gain margin GM ;::: 9 dB, Phase margin PM;::: 40°.

I. Settling time t. is small (t.

2.2 2 DOF Autopilot Corifiguration 2. 3. 4. 5.

The lateral autopilot configuration used in this paper is shown in Fig. 2, where A(s) = 98700/(S2 +

3. DESIGN OF LATERAL MISSILE AUTOPILOT Fig. 2. 2 DOF autopilot configuration.

3.1 Stability Requirements

445s + 98700) is the fin servo dynamics, Hr(s) = 253000/(s2 + 710s + 253000) is the rate gyro dynamics, Ho.v(s) = 394800/ (s2 + 890s + 394800) is

3.1.1. Internal Stability The sufficient and necessary condition for the robust stability of the closedloop systems (depicted in Figure 2) is that

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I. T(s) = ay(s)/ayd(s) is stable. 2. A non-minimum phase zero of Gall <(s) is not canceIed by an unstable pole of C(s).

3.2.1. Gain and Phase Margins Based Cost Function A look at the Nichols Chart qualitatively reveals that gain and phase margin can be defined in term of wax IT(jw)l, where T(s) is for now defined as w a unity-feedback closed-loop transmission ratio (Sidi, 2001). For instance, if IT(jw)1 Mp = 3 dB, then GM > 4 dB and PM > 45° are guaranteed (see Figure 6).

In this work, the stability of T(s) is determined by solving the roots of the characteristic polynomial. To ensure intemal stability, a minimum and stable C(s) is desired. This can guarantee no RHP pole (and zero) cancellations.

:s

3.1.2. Stability Preserving Linear Interpolated controller To obtain the non-linear controller, K r , C(s) and F(s) must be designed for each operating point. Suppose a fixed control structure is used, the controllers can then be interpolated by interpolation of poles, zeros and gains. However, to ensure the stability of the resulting interpolated controller, these operating regions are suggested to be overlapped.

! 1.

J

s 0

L.

o

....

."

By trial and error, 9 subdivisions of the operating range Q of the Mach number M and incidence Ij shown in Fig. 5 is sufficient for a reasonable required control structure complexity. Note that QiS' propor-

·~~~.'~"--.~m~~.m=--7..'m~-.='~--~--~~~~·

0.---.,... ...

Fig. 5. Nichols chart. Adopting these relationships, the gain and phase margin based cost function can be given by ifT1'j (s) is stab

_

_

_

_

_. ~

...,

~

~

~

~

where Tl(S) = ay (s)/a Ydl (s), j is the Q/s vertices index, Mpl is the peak magnitude constraint, Mo is -6 dB constant gain contour, and n is some large number.

u

By using a standard block diagram reduction rules, L(s) can be written as

Fig. 4. Operating range Q of the Mach number M and incidence Ij corresponding to ay = 2 max ±500 m/s . Recall that Ij = v/U, Ijmax can thus be found by equating (1) to 0 and using the relationship a yss = Ur ss .

om

:s

But then IlajwGr<(jw) + Ga.<(jw)1 IGa.<(jw)1 and L-(la(jw)Gr<(jw) + Ga./jw» ~ L-Ga.<(jw) (see Fig. 6), thus the achieved gain and phase margins

tions are sensible as ex: M - 2.4 (3.7 - M) for 2.4 M < 3.0 (3.0 M 3.7) and Oy". , .. . ,on<. ex: M[;. - Mr., where M L • and Mu, are Qi'S lower and upper velocity bounds (see section 2.1 and 2.3).

:s

:s

:s

3.2 Frequency Domain Performance Requirements

In this paper, the tracking performance specifications are modeled in the frequency domain requirements which have a convenient graphical interpretation in terms of tracking ratios. With the design objectives given in section 2.3, the controller's performances can then be measured by evaluating the following robustness assessment functions:

Fig. 6. Frequency responses of the Qi 'S nominal -Ga.« s) and -(lasGr«s) + Ga.«s».

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Using these results, the actuator rate limit based cost function can be given by

will always be greater than or equal to the designed values.

(7)

3.2.2. Tracking Boundaries Based Cost Function The system's tracking performance specifications are based upon satisfying all of the frequency forcing functions IBu(jw)1 and IBdjw)1 (QFT-like) shown in Fig. 8. They represent the upper and lower bounds

where TA(S) = ((s)/ayd(s) . The factor 2 is needed as aYd is allowed to sweep from to atdmu and vice versa.

a;d

%

mQ

3.2.4. Actuator Position Limit Based Cost Function Following the analysis in section 3.2.3, t

ly(t)1

J :s IJ =I

h(t - r)u(r)drl

o

00

h(t)dtl (u(t) = 1 for 0

,,'

:s

If lu(t)1 1 is a general bounded input, then ly(t)1 is then less than or equal Iooo Ih(t)ldt (Sanchez-Pena and Sznaier, 1998).

Fig. 7. Frequency domain response specifications. of tracking perfonnance specifications whom an acceptable response IT(jw) Imust lie within (Houpis and Rasmussen, 1999).

Note that ay .. is directly related to (ss, thus the actuator position limit based constraint function can be given by

Following this design concept, the tracking boundaries based cost function can be defined by,

hJ = "'L~~"'U

l1ij(jw)I-ITo(jw)1 { IBu(jw)I-ITo(jw)l' ITo(jw)I-ITij(jw)1 ITo(jw)I-IBdjw)1

JSij =

}'S)

where h(t)

= h(t),

oo tAij (t)dtl

I Io

1

S/ ayd mu otherwise.

1

> , (9)

Basic scheme of the multi-objective evolution strategy «(J.L + '\)-ES) used in this paper is as that described in (Hughes et al., 2003; Hughes, 2003). Instead of, using non-dominated ranking, finding all Pareto solutions, it locates some specific solutions on the Pareto front corresponding to a given set of target vectors (e.g. weighted Min-Max) V = {VI ,' .. , VT}. Each generation, T weighted Min-Max distances are evaluated for all J.L + Asolutions, whose results are held in a matrix S = (Sij). Note that

(10) where W(k) = 1/V(k) and o'k) is ith individual's kth J J , objective value. Each column of the matrix S is then ranked, with the best score population member on the corresponding target vector being given a rank of I, and the worst a rank of J.L + A. The rank values are stored in a matrix R. Now R can be used to rank the population based on the number of target vectors that are satisfied the best.

y(t) ~

Fig. 8. Block diagram ofa linear plant H(s).

y(t)

{o

S/ ayd mn

'f

4.1 Multi-Objective Evolutionary Optimisation Algorithms

3.2.3. Actuator Rate Limit Based Cost Function Racall that the fin servo is subjected to rate limit (for position case will be treated in section 3.2.4). Imposing by the restriction on power source capacity, the resulting design's fin rate is therefore demanded to be small (and, more importantly, cannot exceed the limit R). Now let's consider a linear plant H(s) in Fig. 8. Suppose u(t) = 1(t), then sY(s) = H(s). Thus,

H(s)

IIt tAij (t)dtl

4. OPTIMISATION

where To(jw) is the nominal tracking ratio. Note that Wu is defined to be the frequency at which ITo (jw) I = -12 dB. This frequency range is considered to be sufficient for the resulting time response approximation.

U(t)~1

:s t < (0). (8)

o

(6)

The primary advantages of this method is such that the target vectors can be arbitrary generated focusing on

= .c-I{H(s)}.

580

the interested regions. Also the limits of the objective space and discontinuities within the Pareto set can be identified by observing the distribution of the angular errors (Bi = COS-IV) . Oi ) across the total weight set.

Poles, Zeros and Gains Kr Kp

4.2 Robust Gain-Scheduled Controller

zp

PP

Using the fixed-structure controller (i.e. Kro C(s) = Kp(s + zp) / (s + pp) and F(s) = Kj(s + Zj)/(s + p j) in this case), instead of using piecewise linear interpolation, the controller's poles, zeros and gains are relatively simpler formulated as linear function of Mach number M and incidence u (K r =

Kt z/

PI

Interpolated fonnula

0.0557 + 0.00248M +(0.0169 - 0.OOO145M)lul 0.000202 + 3.541 x 10- M +( -1.195 x 10- 5 + 4.665 x 1O- 5 )lul 53.792 + O.l11M + (2.204 - 0.178M)lul 8 .24 + 9.022M + (-1.22 - 0.261M)lul 0.0159 + 3.427 x 10 - 0 M +(0.00343 + 0.000453M)ul 1879.732 + 21.948M +(266.025 + 18.354M)lul 24.155 + 0.283M + (3.448 + 0.122M)lul

Table 2. Poles, Zeros and Gams Interpolated Formulas.

k~O) + k~
.......

optimisation variables. Now recall that the small settling time ts is required besides acceptable stability margins, tracking responses and fin rate (see section 2.3). Thus, addition optimisation variable tSd and objective function J I = tSd/t'maz are needed, where tSd is the demanded settling time and t smaz = 0.2 s. Hence, (1) 1 x -- [t 'd ' k(O) r , .. . , p j M

(11)

is therefore formed a variables vector.

Fig. 10. Qi' nominal open-loop frequency responses.

Pursuing the method described in section 4.1, the Pareto-optimal solutions (where 0.8 ::; w(k) ::; 1.0) for the linear interpolated controller is shown in Fig. 9. The obtained trade-off solutions indicated that the

..

-~7 . ------7.------..7 . ------,~~----~,~ f...-q-.' ..... )

Fig. 11 . Qi' nominal closed-loop frequency responses.

Fig. 9. Cumulative trade-off graph of the linear interpolated controller at 500th -generation. closed-loop system is not upset by the fin saturation. Note that the prefer solution will depend on the designer's preferences. Now let's choose the solution (dash line) that has slower settling time, however better stability margins, transient responses and control activities. Its results corresponding to the minimising objectives are in Table 2 and Fig. 10-12. Notice that the influence of the non-minimum phase zero on the fin angle frequency response (occurrence of the sidelope) is clearly visible (see Fig. 12). Employing the non-linear 2 DOF model described in section 2.1, the

Fig. 12. Qi' nominal closed-loop actuator demand frequency responses.

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.~':.:! -

-!----c,,:---:---;':,,--c,:---:,C:.• --".------:,C:.'i --:' 1:~.1.:'

I,,)

Fig. 13. Lateral acceleration control responses of the perturbed systems. time responses of the interpolated controller are shown in Fig 13. The simulation results show that the resulting controller is indeed a robust tracking controller for all possible perturbations. Note also that the resulting fin rates are as predicted (sce Fig. 9).

REFERENCES Horton, M . P. (1992). A study of autopilots for the adaptive control of tactical guided missiles . Master's thesis. University of Bath. Bath, UK. Houpis, C. H. and S. J. Rasmussen (1999). Quantilath'e Feedback Theory: Fundamentals and Applicatiolls. Control Engineering. Marcel Dekker. New York. Hughes, E. J. (2003). Multiple single objective pareto sampling. To appear in the C'ongress on Evolutionary Computations, Canberra. Hugilcs, E. J., A. Tsourdos and B. A. White (2003). Multi-objective fuzzy design of a lateral autopilot for a quasi-linear parameter varying missile . In: Preprint of the iFAC C01~re,.ence on intelligent Control Systems and Signal Processing. Vol. I . AIgrave . pp. 239··244. Sanchez-Pcna, R. S. and M. Sznaier (1998). Rohust Systems: Theory and Applications. Adaptive and Learning Systems for Signal Processing Communications, and Control. John Wiley and Sons. New York. Sidi, M . (2001). Design of Robust Control Systems: From Classical to Modern Practical Approaches. Krieger. Malabar, Florida. Sreenuch, T. , A. Tsourdos, E. J. Hughes and B. A. White (2004). Lateral acceleration control design of a non-linear homing missile using multiobjective evolution strategies. Submitted to the American Control Conference, Boston.

5. CONCLUSIONS The paper presents the lateral acceleration control design of a non-linear missile model using the multiobjective evolutionary optimisation method. The tracking perfomlance specifications are modeled in the frequency domain rcquiremcnt..<; . The fin rate and position constraint functions are formulated in term of the fin angle impulse responses. Using the prcfound feasible controller stnlcture, the Pareto-optimal solutions of the linear interpolated controller are determined corresponding to the given target vectors (possible to analyse). TIley indicate that the closed-loop system is neither upset by the fin rate nor position saturation. The non-linear simulation results show that the selected interpolated controller is indeed a robust tracking controller for all possible perturbations. Moreover, the resulting fin angular and angular rate arc accurately determined.

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