A Robust Output-Feedback Guidance Law for Homing Missiles Using �1-Norm Optimization Theory

Patrick Cadotte1, Dany Dionne2 andHanna Michalska3

Centre for Intelligent MachinesMcGill University, 3480 University Street

Montreal, H3A 2A7, Canada1pcadot@cim.mcgill.ca2dany.dionne@gmail.com3michalsk@cim.mcgill.ca

Abstract— The paper presents the first application of �1

optimization theory to terminal guidance. The novel designapproach accounts for a bounded disturbance representing theunknown evasive tactic of the target, a noise signal affectingthe measurements, and a saturation limit on the actuators ofthe pursuer. The resulting �1 guidance law is a linear output-feedback controller. The performance of the guidance law isassessed using a pre-selected interception scenario of a ballisticmissile. Robust homing performance is demonstrated.

I. INTRODUCTION

This paper adresses the terminal interception problem ofa randomly manoeuvring target by a guided missile in theenvironment of noise corrupted measurements. The missileterminal guidance phase is defined by a control problem withimperfect information and a very short horizon. The objectiveis to minimize the miss distance between the target and themissile.

The classical approach to the solution of this problem em-ploys a linearized model of the dynamical system consistingof the missile and the target. The classical approach thenemploy a state estimator and a control law, both designedindependently of each other and based on the linearizedmodel. The separate design of estimation and control assumesthe validity of the separation theorem, cf. [1], and commonlyemploys the certainty equivalence principle. However, thedesign of a controller with respect to a nominal modelor to a fixed disturbance often exhibits poor performanceor even instability in systems with unknown parameters ordisturbances, cf. [2].

There are many techniques that address system uncer-tainties and exogenous disturbances. The H∞ control, thesliding mode control, and the recently developed �1-normcontrol are examples of such techniques. Guidance lawsbased on H∞ control were already presented in Refs. [2],[3], [4], [5]. In Ref. [4], a nonlinear kinematic model isconsidered but the solution requires solving a set of cross-coupled Hamilton-Jacobi partial differential inequalities. InRefs. [2], [3], approximate solutions that do not require thesolution of the Hamilton-Jacobi inequalities are proposed. InRef. [5], a linearized kinematic model with a constraint onthe interception angle is considered and a two-loop LQR-H∞ solution is presented. Guidance laws based on sliding

mode control were presented in Refs. [6], [7], [8], [9].These laws meet some robust stability criteria, but usuallyneglect the uncertainty in the measurements and do notguarantee the satisfaction of pre-specified bounds on thesystem performance.

The problem of designing a robust �1-optimal controllerwas thoroughly investigated by Dahleh; see Ref. [10] andthe references therein for an extensive summary of resultspertaining to �1-oriented control problems. The key charac-teristic which distinguishes the �1 approach from most othercontrol strategies is the ability to handle several performanceobjectives and operating conditions involving time-domainsignals bounded in magnitude. Such requirements are oftenencountered in guidance problems.

This paper presents a novel linear output feedback terminalguidance law based on the �1-norm control. For robustness,the guidance law is derived assuming: (i) a hard boundon the control, (ii) a bounded disturbance representing the(unknown) evasive manoeuvers of the target, and (iii) anoise signal affecting the measurements. The objective of theguidance law is to maintain the lateral separation betweenthe interceptor and the target within a given bound. Theinterception is hence ensured when the bound on the lateralseparation is smaller than the lethal radius of the interceptor.To the best knowledge of the authors, this paper is the firstapplication of the �1-norm optimization theory to guidance.

II. NOTATION

Consider a system S partitioned as follows

S �[

S11 S12

S21 S22

]. Given a system Q of dimension com-

patible with S22, let

Fl(S,Q) � S11 + S12Q(I − S22Q)−1S21 (1)

denote the lower linear fractional transformation between Sand Q.

III. PROBLEM STATEMENT

A constant velocity planar interception scenario with lin-earized kinematics is considered. The manoeuvring dynamicsof the pursuer and evader is approximated by first-order trans-fer functions with time constants τP and τE , respectively. It

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

WA6.2

0-7803-9354-6/05/$20.00 ©2005 IEEE 1343

Fig. 1. The block diagram of the terminal interception problem.

is assumed that the following system captures the linearizeddynamics of the terminal interception problem⎡

⎣ luy

⎤⎦ � G

⎡⎣ a

nu

⎤⎦ (2)

where l and y are the true and the measured lateral sep-arations between the pursuer and the evader, respectively,n is a disturbance on the measurements, |a(t)| ≤ amax isthe bounded acceleration command of the evader, and u isthe acceleration command of the pursuer. In practice, theactuators may saturate as it is required that |u(t)| ≤ umax. Ablock diagram of G is provided in Fig. 1 and a continuous-time state-space realization of G is given by

x(t) = Ax(t) + B[a(t) n(t) u(t)]T (3a)

[l(t) u(t) y(t)]T = Cx(t) + D[a(t) n(t) u(t)]T (3b)

with

A =

⎡⎢⎢⎣

0 1 0 00 0 1 −10 0 −1

τE0

0 0 0 −1τP

⎤⎥⎥⎦ B =

⎡⎢⎢⎣

0 0 00 0 01

τE0 0

0 0 1τP

⎤⎥⎥⎦

C =

⎡⎣ 1 0 0 0

0 0 0 01 0 0 0

⎤⎦ D =

⎡⎣ 0 0 0

0 0 10 1 0

⎤⎦

(3c)

where τP = 0.2 [s] and τE = 0.2 [s]. The measurementsin Eq. (3b) are delivered to the controller K in discrete-time. The sampling time interval, ∆, is assumed to be ∆ =0.01 [s]. Moreover, it is assumed that umax = 30 [g] andamax = 15 [g].

With respect to the above requirements, the followingperformance objectives are sought:

i) the nominal lateral separation, l, remains bounded asfollows: |l| ≤ lmax = 3.75 [m],

ii) the nominal command signal always avoids saturation,iii) and the peak-to-peak gain of the controller is mini-

mized to increase robustness against the measurementnoise.

IV. CONTROLLER SYNTHESIS

A. Preliminary Notions

The �1 norm of a system is the maximal peak-to-peakgain that can be generated by the system itself. The peak-to-peak gain is an output/input ratio between both absolutepeak values achieved by an admissible pair of output andinput signals. Consequently, in the presence of any giveninput signal which remains absolutely bounded in magnitudeby one, the maximal absolute magnitude of the output signalis guaranteed to be bounded by the �1-norm value of thesystem considered.

More rigorously, let �n∞ denote the space of all infinite

sequences {s(k)}∞k=0 of vectors of length n, s(k) ∈ Rn,equipped with the norm ‖s‖∞ � sup

k≥0max

i∈{1,...,n}|si(k)|. Given

a bounded operator S : �n∞ �→ �m

∞ with s �→ S(s), let

‖S‖∞−ind � sups�=0

‖S(s)‖∞‖s‖∞ (4)

be the induced ∞-norm of S. In the case when S islinear, causal, and time-invariant, S(s) is determined by the

convolution (S ∗ s)(k) �k∑

l=0

S(l)s(k − l), where {S(l)}∞l=0

is the impulse response of S. Then, it is known that, seeRef. [10], ‖S‖∞−ind = ‖S‖1, where

‖S‖1 � maxi∈{1,...,m}

n∑j=1

∞∑k=0

|Sij(k)| (5)

where Sij is the ijth entry of system S. Moreover, let S(z) �∞∑

k=0

S(k)z−k denote the z-transform of the impulse response

of S. The explicit dependence on (z) is dropped when clearfrom the context. Additionally, define the A-norm of S as

‖S‖A � ‖S‖1 (6)

B. The �1 Optimization Problem

Assuming a sampling-time interval of 0.01 [s], the z-transform of G is

G =

⎡⎣ −Gyu 0 Gyu

0 0 1−Gyu 1 Gyu

⎤⎦ (7a)

where

Gyu =g(z − z1)3

(z − p1)2(z − p2)(7b)

with g ≈ −6.10e−7, z1 = −1, p1 = 1, and p2 ≈ 0.95, andwhere the indexed variable (·)yu denotes a transfer functionfrom u to y.

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As depicted in Fig. 1, define the discrete-time closed-loopsystem by

T � Fl(G, K) (8)

where K is the z-transform of the controller. The perfor-mance objectives listed in §III are then captured by the �1

control problem stated below

minK stabilizing

‖Tun‖A (9a)

amax‖Tla‖A ≤ lmax (9b)

amax‖Tua‖A ≤ umax (9c)

Note that Tua = −Tln, so that ‖Tln‖A remains bounded aswell.

C. Parameterization of All Stabilizing Controllers

Let M , N , X , and Y be discrete stable transfer functionssatisfying

Gyu = M−1N (10a)

XM − Y N = 1 (10b)

Note that the solution for (10a) and (10b) is non-unique;see Refs. [10] and [11] for details on how to solve suchequations. A valid choice of M , N , X , and Y is givenin Appendix. Then, by Youla parameterization, cf. [11], theparameterization of all internally stabilizing controllers forthe system T with respect to the stable transfer function Qis given by

K = (Y − MQ)(X − NQ)−1 (11)

where Q has the following form

Q �

i=nQ∑i=0

aizi

i=nQ∑i=0

bizi

(12)

with ai, bi ∈ R for every i ∈ {1, ..., nQ}.

D. �1-Suboptimal Controller Synthesis

Introducing (10a) and (10b) in (8) gives

T =[ −NX Y N

−Y N M Y

]+

[ −N2Q −MNQ

MNQ −M2Q

](13)

which is affine with respect to the transfer function Q.Solving the optimization problem (9) with respect to a

stable Q (instead of K), by way of Eq. (13), guaranteesthe internal stability of T . The optimization is performedwith respect to the parameters of Q which comprise of thevariables ai and bi of Eq. (12).

It was shown in Refs. [12], [13], and [14] that, by virtue ofEq. (13) being an affine function in Q, the control problem(9) is convex in the parameters of Q whenever Q exhibits

a finite impulse response, i.e., when every bi�=nQ= 0 and

bnQ= 1. Note that under such assumption, every pole of Q

are at the origin and the value of every optimization variableai represents the magnitude of an impulse different of theimpulse response of Q. Hence, the bigger is the value ofnQ, the larger is the number of optimization variables ai,and the longer is the impulse response of Q.

Unfortunately, for the choice of M , N , X , and Y providedin the Appendix, the optimization problem (9) is minimizedby a Q which exhibits a very lengthy impulse response.The lengthy tail of the impulse response of the optimal Qis indicated by the slow convergence of problem (9) as nQ

increases. As a result, applying the convex approach of Refs.[12], [13], and [14] would require a prohibitive number ofoptimization variables ai.

Consequently, a suboptimal, but feasible approach to find-ing an adequate Q is adopted by solving the optimizationproblem (9) with respect to both the ai and bi variables inEq. (12). In this approach bi, i = {0, · · · , nQ} are consideredfree variables so that the transfer function Q may containnon-zero poles. It is hence possible to consider a muchsmaller value of nQ and still generate a good approximationof the optimal Q (as it is well known that transfer func-tions with non-zero poles naturally exhibit lengthy impulseresponses).

The proposed suboptimal approach renders the optimiza-tion process nonconvex for any value of nQ. Nevertheless,the computational effort required to obtain an acceptablesolution is reduced significantly, as compared to the optimalconvex approach, as the number of optimization variables ismuch smaller. Moreover, the smaller value of nQ associatedwith the suboptimal approach allows to recover a controllerof much smaller order than the one derived by the convexapproach.

The optimization problem (9) is non-differentiable due tothe nature of the �1-norm, see Eq. (5). Hence, the local solu-tions are computed using the non-differentiable optimizationMatlab toolbox Solvopt, cf. [15].

In the present example, setting nQ = 11 allows to computean adequate suboptimal Q for the optimization problem (9);refer to the Appendix for the complete numerical expressionof Q.

E. Reduced-Order �1 Suboptimal Controller

The order of the controller developed in the previoussubsection is in general greater than nQ by a few orders dueto the nature of Eq. (11). To facilitate the implementation ofthe �1 guidance law, the following heuristic order-reductionprocedure is performed. First, the order of the controller Kobtained in the previous subsection is truncated by keepingonly the four most dominant pairs of poles and zeros so thatthe truncated K is of the same order as G. Then, using (8),the parameters of the truncated K are fine-tuned throught theoptimization problem (9) to recover the performance level

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achieved by the original K. As a trade-off to the order-reduction, the cost function (9a) may increase which resultsin a degradation in the robustness to measurement noise. Inthe present case, the degradation is found to be small and theinternal stability is preserved. The final resulting �1 guidancelaw has the following state-space realization

ξ(k + 1) = Fuξ(k) + Guy(k) (14a)

u(k) = Huξ(k) + Duy(k) (14b)

with

Fu =

⎡⎢⎢⎣

f11 f12 f13 f14

4 0 0 00 4 0 00 0 1 0

⎤⎥⎥⎦ Gu =

⎡⎢⎢⎣

8000

⎤⎥⎥⎦

Hu =[

h1 h2 h3 h4

]Du = d1

(14c)

where ξT (0) = [0 0 0 0]T , f11 = 1.6625331163,f12 = 0.0052275973, f13 = −0.0817745995, f14 =0.0370760550, h1 = 9.7831583408, h2 = −1.7066380641,h3 = −1.7350455211, h4 = 1.4245396526, and d1 =548.315884882.

V. NUMERICAL RESULTS

Two sets of simulations are carried out. The first setof simulations assesses the nominal performance of the �1

controller for the linearized closed-loop system (8) in anoise free environment, (i.e., n = 0). The second set ofsimulations assesses the statistical homing performance ofthe �1 guidance law in a more realistic interception scenariowith measurement noise and nonlinear kinematics.

A. Nominal Performance

Two types of evasive manoeuvres are considered (seeFig. 2a): a bang-bang manoeuvre from -15 [g] to +15 [g]occurring at 2 [s] and a sinusoidal manoeuvre with amplitude,frequency, and phase of 15 [g], 0.7 [rad/s], and −π/6 [rad],respectively. The results are illustrated in Figs. 2b and 2c anddemonstrate that the closed loop system with the �1 controllermeets the performance objectives (i) and (ii) in § III. In bothevasion attempts, the command signal u does not cross the±30 [g] bound and thus avoids saturation, while the lateralseparation l remains below ±3.75 [m]. The above simulationsare performed in ideal conditions in order to emphasize thefact that the �1 approach enforces hard bounds on the systemperformance.

B. Statistical Homing Performance

The homing accuracy of the novel �1 guidance law iscompared to that of the PN law, cf. [16], and of theDGL/0 law, cf. [17], using a specific example of a terminalplanar engagement between an interceptor (the pursuer) anda manoeuvrable ballistic missile (the evader). The exampleemploys a planar nonlinear kinematic model described inRef. [18] and requires the introduction of several additional

Fig. 2. Nominal simulation results using the �1 controller. Panel (a):Evader’s command acceleration, “a”. Panel (b): Pursuer’s command acceler-ation, “u”. Panel (c): Lateral separation, “y”. Solid line - Bang-bang evasivemanoeuvre; Dashed line - Sinusoidal evasive manoeuvre.

parameters: the pursuer velocity, VP , the evader velocity, VE ,and the distance between the pursuer and evader (the range),r. The value of the parameters of the simulations are providedin Table I where σ denotes the standard deviation of theangular measurement noise. The covariance of the linearizedmeasurement noise, Qη, is calculated from σ assuming therange, r, is known at any current time instant k:

Qη(k) = (r(k)σ)2 (15)

The evasive strategy is a bang-bang manoeuvre with a singleswitch, at a random time instant, tgosw, over the time intervalof the engagement.

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Fig. 3. Required lethal radius to guarantee SSKP=0.95. Solid line - �1 law.Dotted line - PN law. Dashed line - DGL/0 law.

The PN and the DGL/0 laws are state-feedback guidancelaws in that they require as inputs the derivative of the mea-surements in addition to the measurements themselves. Thederivative of the measurements is calculated using a Kalmanfilter with a shaping filter. The shaping filter is a Wienerprocess acceleration model, cf. [19], whose covariance, Qa,is selected following the formula recommended by Ref. [20],i.e.,

Qa = 4(amax

)2/tf (16)

where tf is the final time instant of the engagement.The PN law is a linear guidance law extensively employed

in practical guidance problems. It can be shown that the PNlaw is optimal with respect to some guidance criteria whenthe evader is not manoeuvring, cf. [16]. The DGL/0 law is ofthe bang-bang type and is a game-theoretic optimal guidancelaw, i.e., a saddle-point solution of a pursuit-evasion gameopposing a pursuer to a manoeuvring evader.

The statistical performance criterion is the single shotkill probability (SSKP) and is evaluated through MonteCarlo simulations. The SSKP is defined as the probabilityof a successful interception. An interception is assumed

TABLE I

SIMULATION PARAMETERS

Pursuer velocity VP = 2 300 m/sEvader velocity VE = 2 700 m/sPursuer max. acc. umax = 30 gEvader max. acc. amax = 15 gPursuer time constant τP = 0.2 sEvader time constant τE = 0.2 sInitial range r(0) = 20 000 mMeasurement freq. f = 100 HzStd. dev. ang. noise σ = 0.1 mrad

successful when the miss distance, l(tf ), is within the lethalradius of the interceptor. The Monte Carlo simulations repeatthe pursuit-evasion scenario 20 000 times. Each repetition ischaracterized by a specific noise realization and a specificonset time instant tgosw for the bang-bang evasive manoeu-vre.

The minimum lethal radius for the pursuer to achieveSSKP=0.95 is shown in Fig. 3 as a function of the onsettime of the evasive manoeuvre. The �1 law consistentlyachieves a smaller miss distance than the PN law. The betterperformance of the �1 law is attributed to the ability of the�1 approach to account for both the disturbance afflictingthe measurements and the bounded manoeuvrabilities of boththe pursuer and the evader. As compared to the DGL/0 law,the �1 law requires a similar lethal radius, except whenthe onset of an evasive manoeuvre occurs in the intervaltgosw ∈ [0.3, 1.5] [s]. However, the bang-bang nature of theDGL/0 law renders its application difficult due to chattering,while such drawback is avoided with the �1 design as itdelivers a linear control law.

VI. CONCLUDING REMARKS

The paper presents a first application of the �1 optimizationtheory to terminal guidance. The new �1 law is shown to berobust with respect to bounds on the manoeuvring capabil-ities of both the pursuer and the evader. The presence of adisturbance in the measurements is also taken into account.The novel �1 law is demonstrated to achieve a significantlysmaller miss distance than the commonly employed PN lawin the example problem considered.

The most important ability of the �1 law is to enforce thelateral separation to the evader to remain within the lethalradius of the pursuer rather than to achieve a zero missdistance. The �1 design favors the use of command signals oflimited magnitude which reduces the likelihood of actuatorsaturation.

The �1 procedure can be employed to extend the robustnessof the guidance law to other types of uncertainties, such asthe uncertainty in the dynamics of the autopilot. Moreover,the �1 design approach is compatible with other controltechnique, e.g., multiloop control and mixed-norm control.Combination of techniques may lead to further performanceimprovement.

APPENDIX

The transfer functions

M ≈ (z3 − 2.95z2 + 2.90z − 0.95)/z3 (17a)

N ≈ − (6.10e−7z3 + 1.83e−6z2 + 1.83e−6z + 6.10e−7)z3

(17b)

X ≈ (0.65z3 + 0.98z2 + 0.60z + 0.14)/z3 (17c)

Y ≈ (5.81e5z3 − 2.29e5z2 − 3.62e5z + 2.15)/z3 (17d)

yield a valid solution to Eqs. (10a) and (10b).

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The smallest local minimum computed for the �1 optimiza-tion problem (9) yields a cost of 3183 at

Q = 1e6A/B (18)

where A ≈ 0.58z11 − 2.16z10 +1.98z9 +2.36z8 − 5.53z7 +2.00z6 +3.15z5 −3.19z4 +0.34z3 +0.86z2 −0.46z +0.076and B ≈ 1.00z11 − 6.28z10 +15.44z9 − 15.95z8 − 2.40z7 +23.46z6−22.55z5+4.16z4+8.15z3−7.20z2+2.50z−0.33.The corresponding controller K follows from Eqs. (11) and(17).

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