Polynomial Eigenstructure Assignment, Application to Missile Autopilot

Lilian Bruyeere, Antonios Tsourdos and Brian A. White

Abstract— This paper suggests another approach to eigen-structure assignment in a polynomial matrices framework. Inparticular, the design of eigenstructure assignment is performedwithout selecting initial eigenvalues. The eigenspaces of thesystem are handled in polynomial matrices format, where theclosed-loop transfer function of the system is expressed as acoprime factorisation. The controller structure is left to thedesigner for more flexibility. Scalar and dynamic controllers aretackled alike. Like with classical eigenstructure assignment, thedesign objectives may include decoupling of modes, robustnessand performance properties. This paper presents the techniquein the framework of linear parameter-varying and multi-inputmulti-output systems. An example of LPV autopilot is designedfor a model of a tactical missile. This tail-controlled missile inthe cruciform fin configuration is modelled as a second-orderquasi-linear parameter-varying system. Simulations show goodtracking with fast responses over the full envelope.

I. INTRODUCTION

There are different approaches to obtain a linearparameter-varying (LPV) representation of a nonlinear sys-tem. For instance, when the first order tangent system pro-vides ”sufficient” information characterising various equilib-rium points, the scheduling of the set of LTI systems is oftenused. One would thus use LTI design and analysis tools andextend the results to its scheduled system or controller. LTIsystems have been used in designing control systems fornonlinear systems. In fact, the first order tangent system of anonlinear system provides a useful ground to tackle its designand analysis. As a result, LTI techniques are still considered,investigated often in the objective to extend these approachesto LPV systems. Like many others H2 and H∞ have beenapplied successfully to LPV systems. Here an eigenstructureassignment approach is adapted to LPV systems and thusenhance further flexibility as algebraic computation andparameter-dependent controller are obtained without need ofinterpolations. The controller structure is left to the designer.However, alike eigenstructure assignment, the technique suf-fers the same limitations in the number of assignmentspossible. In the context of linear parameter-varying systems,the approach leads to a parameter-dependent controller offixed structure and order. It is exhibited as the modes ofunderlying linear systems and enabling self-scheduling ofthe controller according to the LPV nature of the systemand avoiding classic gain scheduled approaches. The ad-vantage is that like in aerospace applications and otherareas, criteria are often interpreted as linear time-invariantcharacteristics like modes. Modes commonly accepted to

The authors are working for the Department of Aerospace, Power& Sensors, Cranfi eld University, Shrivenham, Swindon SN6 8LA, UnitedKingdom. a.tsourdos@cranfield.ac.uk

provide insight to the physical behaviour of the system as itgives access to robustness, sensitivity, decoupling, actuatormotions properties as well as describes transient responses.In fact, eigenstructure assignment has been of interest formany years focusing on linear time-invariant systems and hasbeen a useful tool both for analysis and design. A survey ofeigenstructure assignment [1] presents early research worksand classical algorithms. However, it is seldom used in theLPV framework due to the rather tedious scheduling processleading not necessarily to satisfactory controller.

In this paper another approach to eigenstructure assign-ment, a technique based on LTI systems and using a polyno-mial matrix framework, developed in recent research work[2], [3] is used to design a parameter-dependent controllerfor a broader class of systems, LPV systems. In particular,the design of eigenstructure assignment is performed withoutselecting initial eigenvalues. The eigenspaces are handledas polynomial matrices and the system open-loop transferfunction is expressed as a coprime factorisation. The con-troller structure is left to the designer. Scalar and dynamiccontrollers are tackled alike. Like classical eigenstructure as-signment, the control system design may include decouplingof modes, robustness and performance properties.

Polynomial eigenstructure assignment is presented to-gether with a particular controller structure chosen for thepurposes of this paper. The design of a velocity/rate autopilotis presented for a quasi-linear parameter-varying (QLPV)tactical missile model. The scheduling is directly performedby a QLPV controller and does not need form of inter-polation [2]. This paper thus demonstrate the capabilitiesof polynomial eigenstructure assignment to LPV systemsboth SISO and MIMO in a computationally attractive way.Simulations show the overall nonlinear missile is performingsimilarly to previous studies based on dynamic inversiontechniques.

II. POLYNOMIAL EIGENSTRUCTURE ASSIGNMENT

Assuming the LTI system,

x = Ax + Bu, (1a)

y = Cx + Du, (1b)

where A is the state matrix, B is the input matrix, C theoutput matrix and D the feedforward matrix, x the statevector, y the output vector and u the input vector.

Many approaches solving eigenstructure assignment (EA)[1], [4], [5] problems require solve the following null spaceas [

A− λiI B] [

Zi

Pi

]= 0, (2)

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

WA6.4

0-7803-9354-6/05/$20.00 ©2005 IEEE 1355

for some chosen eigenvalue λi giving an eigenvector spacenoted Zi and its associated eigenvector space noted Pi. Fromthese spaces, eigenvectors zi and associated eigenvectors pi

can be selected such as to satisfy criteria for decoupling,robustness, sensitivity properties. However, in general alleigenvalues and eigenvectors cannot be simultaneously as-signed and hence the process becomes iterative.

A. Overview of Polynomial Eigenstructure Assignment

Polynomial eigenstructure assignment (PEA) was devel-oped in recent research work by White [2], [3]. The initialmotivation was to preserve the eigenstructure assignmentapproach in many ways while giving better access to theeigenspaces without initially selecting closed-loop eigen-values. Instead, it requires polynomial matrix computationleading to a one-pass algorithm.

In fact, EA usually requires a given partial eigenstructurein order to proceed to controller design, however in practicethis is a rather tedious process since slight changes ineigenvalues/eigenvectors may drastically affect the reach-able closed-loop eigenstructure. Although there are eigen-value/eigenvector sensitivity criteria to quantify and reducethe effect, it stays difficult to chosen an initial eigenstructure.Designers customarily resort to several design iterations.Expressing eigenspaces and associated spaces as eigenvaluefunctions, the selection is postponed while the system prop-erties are reflected in the polynomial eigenstructure coun-terparts. For instance, the open-loop transfer function iswritten as a coprime factorisation of polynomial matrices [6].The approach is moreover directly applicable to multi-inputsmulti-outputs (MIMO) systems, to output feedback controland static and dynamic controller designs are tackled alike.Ultimately, PEA still suffers similar limitations to EA.

Successful applications of PEA to VSTOL [3], [2], [7]and unmanned underwater vehicle [8] display a design en-compassing multi-objectives including stability, settling time,damping, the decoupling of channels, actuation dynamicslimitation, the robustness to disturbances using appropriatemetrics.

In this presentation, the system is assumed controllableand observable and like often in aerospace application withthe same number of controlled outputs and inputs. Results arevalid for SISO and MIMO systems and static and dynamiccontrollers alike.

B. The Mechanic of PEA

Like classic eigenstructure assignment approach, the samesteps have been carried over in the framework of polynomialmatrices. Assuming LTI system in (1a) and (1b) the nullspace equation (2) is written using variable s instead of λi,[

(A − sI) B] [

Z(s)P(s)

]= 0, (3)

where s plays the role of the eigenvalue together with theLaplace variable. Z(s) and P(s) represent the eigenvectorspace and its associated control vector space, respectively.In the sequel the eigenvectors are referred as their corre-sponding polynomial matrices without mentioning.

The transfer function Gy(s),

Gy(s) = C(sI− A)−1B + D, (4)

becomes a function of the eigenvector and its associatedspace as described in (3) and takes the form,

Gy(s) = (CZ(s) + DP(s))P(s)−1, (5a)

= Z0(s)P(s)−1. (5b)

Thus specifying a particular controller structure wouldenable the designer to change the dynamics of the open-loop system without having to cancel out the zero-dynamics.The next section introduces a particular controller structurepresenting sufficient flexibility for the purpose of aerospaceapplication under consideration.

C. Controller Structure

The controller structure choice is driven by the considera-tions of improving stability, performance, tracking, sensitiv-ity and robustness of the system. In this attempt the chosenstructure is here shown in Figure 1 which unlike classicfeedback gives more design flexibility. The controller Ka(s)

+

− −−

+d

yi

ycuI

s

Ki(s)

Kc(s)

Ku(s)−1 Gx(s)

Fig. 1. Controller structure chosen for aerospace applications introducingdynamic gains Ku(s), Ka(s), Kc(s) and Ki(s) with yc the controlledoutputs, yi the additional measured outputs.

shapes the tracking response of the closed-loop system to thedesired demands, the controller Ku(s) shapes the inputs tothe plant, Ki(s) feedback the extras measurements availableyi and Kc(s) feedback the controlled outputs yc shapingthus transient response [3]. Altogether gains Ku(s), Ka(s),Kc(s), Ki(s) compose a dynamic controller, where yc repre-sents the controlled outputs and yi represents the additionalmeasured outputs. Note that the controller structure is notrestricted solely to the one presented in this work.

The gains Ka(s), Ku(s), Kc(s), Ki(s) are genericMIMO transfer functions. However in any case, additionalstructure is required without which it would be difficult tosolve the controller. In some previous work [2], [3], furthercontroller structure is introduced where each gain functions isitself a coprime factorisation [6], and further decompositionof each matrices of the coprime factorisation is done intosingular value decomposition (SVD) [7]. Here, these gainsare chosen to be polynomial matrices.

The resulting system interconnection leads to the closed-loop system summed up as follows,

Ty(s) = Gc

˘I + Ku

−1 [(Ka + Kc)Gc + KiGi]¯−1

Ku−1Ka,

(6)

where Ty(s) is the reference input response.

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Due to the particular case of controlled and measuredoutputs, there are two transfer functions respectively of type(4). Each of them can be written like (5) and thus lead to twoeigenvectors composing Z0(s), namely Zc

0(s) and Zi0(s),

Z0(s) =[Zc

0(s)Zi

0(s)

], (7)

related to their respective open-loop transfer functions,Gc(s) and Gi(s). Consequently, for controllable and ob-servable systems, their coprime factorisation is derived as,

Gc(s) = Zc0(s)P(s)−1, (8a)

Gi(s) = Zi0(s)P(s)−1. (8b)

Using this last equation (8) the transfer functions (6) isexpanded accordingly using solely eigenvector polynomialforms, Zc

0(s) and Zi0(s), and its common polynomial as-

sociate space P(s). The reference input response transferfunction, Ty(s), is thus developed as,

Ty(s) = Zc0

hKuP + (Ka + Kc)Z

c0 + KiZ

i0

i−1

Ka. (9)

D. Matching Conditions

When the closed-loop system eigenstructure is correctlychosen, eigenstructure assignment is obtained by matchingthe closed-loop system (9) to a desired one (10)

Tdy(s) = Nd(s)Dd

−1(s), (10)

where Dd(s) and Nd(s) are polynomial matrices. WhileDd(s) comprises the poles of the system, Nd(s) includesthe open-loop zeros, and thus the closed-loop objectives aretranslated in a desired transfer function.

However, restrictions apply and are summed up in (11).The open-loop zeros are transfered to the closed-loop (11a),the feedback loop does not attempt to cancel open-loop zerosunlike other dynamic inversion. From Figure 1, this systemhas a unit static gain and thus restrict the selection of Dd(0)as in Equation (11b). Finally for a system with same numberof inputs as controlled outputs, matching Ty(s) to Td

y(s)lead to (11c)

|Nd| = |Zc0|, (11a)

Zc0(0) = Dd(0), (11b)

Dd(s)Nd+(s) =

hsKu(s)P(s) + sKc(s)Z0(s)

+ sKi(s)Zi0(s) + Zc

0(s)iZc

0+(s), (11c)

where the superscript + designates the polynomial adjointmatrix.

Under assumptions of (11a) and (11b), the controller gainsare solved from (11c) has the following left null space,⎡

⎢⎢⎣Ku(s)Kc(s)Ki(s)

I

⎤⎥⎥⎦

T ⎡⎢⎢⎣

P(s)Zc0+(s)

Zc0(s)Zc

0+(s)

Zi0(s)Zc

0+(s)

1s

(Zc

0(s)Zc0+(s) − Dd(s)Nd

+(s))⎤⎥⎥⎦ = 0.

(12)

Note that if only partial feedback is available then thedesired closed-loop transfer function is further restricted. Thepolynomial null space is computed by simply reorganisingthe polynomial matrices and applying a classic null spacealgorithm in some extent. This formulation enables to specifythe desired order for each controller gain independently.

When the null space can be row reduced using eventuallyGaussian elimination and the last term ensured as identityI , then the controller space is found. A suitable controllercan then be extracted from a combination of controller rowswhile additional criteria can be accounted for.

III. THE QLPV MISSILE MODEL

The Horton missile model describes a reasonably realisticairframe of a tail-controlled tactical missile in the cruciformfin configuration.

The aerodynamic coefficients in this model are derivedfrom wind-tunnel experiments [9] and interpolation formu-lae, involving Mach number M ∈ [2, 3.5], roll angle λ ∈[4.5◦, 45◦] and incidence σ ∈ [3◦, 17◦], have been calculated[9], [10], [11]. These interpolations are polynomials of firstorder with respect to Mach number and incidence angle. Thetotal velocity vector �Vo is the sum of the longitudinal veloc-ity vector �U , the sideslip velocity �v and pitch velocity vector�w, i.e. �Vo = �U+�v+�w. The Mach number is defined as M =Vo/a. Since Vo =

√U2 + v2 + w2, the Mach number is also

a nonlinear function of the velocities, M = M(U, v, w).The total incidence σ is arcsin

(√v2+w2

Vo

). It follows from

the above discussion that aerodynamic coefficients can beinterpreted as nonlinear functions of sideslip velocity v, pitchvelocity w, Mach number M and roll angle λ.

The multi-input multi-output (MIMO) dynamics is derivedfrom [9] in plane horizontal (v − r) and vertical (w − q)plane together with roll, p. The 5 DOF system is the Taylorlinearisation of the missile for perturbed flight. The dynamicof the system can be restated in the form of a quasi-linearparameter-varying system (QLPV),

v = yv(p)v + yr(p)r + yζ(p)ζ (13a)

r = nv(p)v + nr(p)r + nζ(p)ζ + nξ(p)ξ (13b)

w = zw(p)w + zq(p)q + zη(p)η (13c)

q = mw(p)w + mq(p)q + mη(p)η + mξ(p)ξ (13d)

p = lp(p)p + lζ(p)ζ + lη(p)η + lξ(p)ξ (13e)

where p is now[v w M λ

]T. Note that additional

parameters like mass m, inertias Iz , Ix, altitude h could havebeen considered as well but for the sake of simplicity are notincluded here. The terms yv, yζ , nv, nr, nζ , nξ and othersare semi-non-dimensional derivatives, while yr is equal to−U and similarly zq in the (w−q) plane. This model is stillnonlinear and its form is non unique however it presentssimilarity with linear parameter-varying systems.

IV. SIDESLIP VELOCITY CONTROL APPLICATION

This section presents how polynomial eigenstructure as-signment can be applied to SISO LPV systems. A veloci-ties/roll rate controller is designed for a specific controller

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structure and order (static in this case). The controllerobtained has the same parametrisation as the LPV plant andrenders the closed-loop system almost linear time-invariantfor the wide flight envelope and hence uniform performanceare expected.

A. Controller structure

The controller structure adopted is represented in Fig-ure 1 for a LTI system and equivalently in Figure 2 forthe LPV case. The gain Ka(s) becomes a pure integratorensuring zero steady state error. Gains Ku(s, p), Kc(s, p)and Ki(s, p) are now parametric dependent on p where p

represents the same parametrisation as of the missile modeland is in this case

[v M λ

]T. The input noted here

−−

+

−

+

QLPV controller

yi

ycd

p

1

sKu(s, p)−1 Gx(s, p)

Ki(s, p)

Kc(s, p)

Fig. 2. The LPV controller structure selected. It includes Ka(s) a pureintegrator, 1

sand controller gains Ku(s, p), Kc(s, p), Ki(s, p) scheduled

by parameter p.

d denotes the demand, i.e. sideslip velocity demand, vd,while yc denotes the controlled output v and yi denotes inthis SISO case the yaw rate, r. The yaw rate taken fromrate gyros is fed back in order to increase stability androbustness. In the context of PEA, sideslip velocity and yawrate feedback makes full feedback available and thus enablesto place all system closed-loop poles without which onlypartial assignment would be possible.

A dynamic controller would further populate the null spacesolution and hence provides further flexibility in the choiceof a suitable controllers, however, this beyond the scope ofthis paper and static controller gains are used here.

The closed-loop transfer function Ty(s, p) from Equation(9) becomes,

Ty(s, p) = Zc0(s, p)

ˆsKu(s, p)P (s,p) + sKc(s, p)Zc

0(s, p)

+ sKi(s, p)Zi0(s,p) + Zc

0(s, p)˜−1

.(14)

B. Polynomial matching

Like previously mentioned the matching of closed-looptransfer functions, (10) and (14) is performed if conditions(11) holds. They are rewritten for the LPV case as conditions(15),

Nd(s, p) = Zc0(s, p), (15a)

Zc0(0, p) = Dd(0, p), (15b)

Dd(s, p) =ˆsKu(s, p)P (s, p) + sKc(s,p)Z0(s,p)

+ sKi(s, p)Zi0(s,p) + Zc

0(s, p)˜, (15c)

where polynomial Dd(s, p) and Nd(s, p) are the LPVdesired closed-loop polynomial matrices replacing the onesin (10). Note that perfect matching is not always possibleas satisfying condition (15c) may not be possible, in whichcase different controller structure and/or higher controllerorder would be required. However here for this SISO systemsperfect match is achievable for a simple static controller. Anydesign objectives respecting (15) would therefore be met.

Equation (15c) is rewritten as a null space problem⎡⎢⎢⎣Ku(s, p)Kc(s, p)Ki(s, p)

I

⎤⎥⎥⎦

T ⎡⎢⎢⎣

P (s, p)Zc

0(s, p)Zi

0(s, p)1s (Zc

0(s, p) − Dd(s, p))

⎤⎥⎥⎦ = 0, (16)

where the left null space solution is identified as suitablecontroller gains.

C. Performance objectives

With static controller gains (except Ka(s)), the system isnow 3rd order, i.e. 2nd order plant and Ka(s), 1st order pureintegrator. The desired closed-loop transfer function can thusbe described by its characteristic polynomial of degree 3 andwritten as,

Dd(s, p) = c0 + c1s + c2s2 + s3 (17)

independent of p and its zeros Nd(s, p).Assuming the zeros in this case do not influence too much

the response, the closed-loop performance would be relatedto its characteristic polynomial. The requirement for theclosed-loop system transient step response is to have a peakovershoot less than 5% and settling time less than 0.2 s. Thecharacteristic polynomial is thus selected as the combinationof a second order polynomial with natural frequency wn =30 rad/s and damping ratio ζn = 0.7 and an additional firstorder polynomial with pole at −100. The controller does notintroduce additional zeros and does not attempt to cancelout any open-loop zeros for this non-minimum phase zerossystem. In fact by condition (15a), Nd(s, p) is constrainedto Zc

0(s, p) which indeed preserves the open-loop zeros.

D. Sideslip velocity controllerIn this paper, the sideslip velocity controller for the

horizontal motion of the Horton missile is obtained afterremoving Cnξ

ξ, wp, pq terms from Equations (13a) and(13b), while p is as described earlier. The controlled outputis thus yc = v and the measured output is yi = r. Theseoutputs lead to polynomial matrices Zc

0(s, p) and Zi0(s, p)

while Z(s, p) and P (s, p) are determined from the open-loop equations. Since the outputs are the state Z0(s, p) isequal to Z(s, p) as,

Z0(s, p) =

24 c0(nr(p)yζ(p)−yr(p)nζ(p)−yζ(p)s)

nr(p)yζ(p)−yr(p)nζ(p)c0(−nv(p)yζ(p)−yv(p)nζ(p)−nζ(p)s)

nr(p)yζ(p)−yr(p)nζ(p)

35 , (18a)

P (s, p) =h

c0(yr(p)nv(p)−yv(p)nr(p)+yv(p)s−nr(p)s−s2)nr(p)yζ(p)−yr(p)nζ(p)

i,

(18b)

Solving the null space of (16) for this case results insuitable controller gains which ensure the closed-loop desired

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performance as described in the previous subsection. Thecomputation is done with MATHEMATICA and provide asymbolic polynomial annihilator which make it possible tokeep the parametrisation in p. The advantage here is that nointerpolation process is required as the controller is designedas an LPV controller.

After row reduction the null space takes the vector form[Ku(p) Kc(p) Ki(p) I

]and each controller gain can

thus be identified as follows,

Ku = (nζyr − nryζ) [nζ(nζyr − nryζ) + yζ(nζyv − nvyζ)] ,(19a)

Kc = −c0nζyζ + nζ(nζyr − nryζ) [c1 + yv(yv + c2)]

+ nv(nζyr − nryζ) [nζyr − yζ(c2 + nr + yv)] , (19b)

Ki = (nζyr − nryζ)ˆnζyr(yv + nr) − yζ(c1 + nvyr + nr

2)˜

+ c0yζ2 + c2(nζyr − nryζ)

2, (19c)

where parametrisation in p is dropped for ease and acommon dividing factor omitted.

Substitution of the coefficients c0, c1, c2 of the desiredclosed-loop denominator of the system as well as of the LPVterms yv(p), yr(p), yζ(p), nv(p), nr(p), nζ(p) lead to thefinal form of desired controller,

ζ = Ku(p)−1(

(v−vd)s − Kc(p)v − Ki(p)r

). (20)

E. Simulations

Simulations of the closed-loop system where parameterp includes both Mach number ranging in [2, 3.5], sideslipvelocity, implicitly incidence angle, ranging in [−0.3, 0.3],for a roll angle, λ = 0, are presented in Figure 3. Thesideslip velocity demand is pre-filtered with a second orderfilter (ζ = 0.7, wn = 50 rad/s) and shown as dashed curves.While a constant lateral acceleration demand of 100 m.s−2 isrequested, the sideslip velocity demand changes as the Machnumber changes, due to the change of missile dynamicsacross the flight envelope. Responses to nominal system

Fig. 3. Simulation for a sideslip velocity controller responding to a lateralacceleration demand of 100 m.s−2 with Mach number varying between 2to 3.5.

show no steady-state error in sideslip velocity althoughsome lag in the tracking of sideslip velocity ramp incurs

a steady-state error in lateral acceleration. In practice, thelateral acceleration is controlled and to this aim, the lateralacceleration demand is controlled indirectly through sideslipvelocity. While this is common practice, performance arenot necessarily transferable to lateral acceleration, as thenonlinear relationship, av = yv(p)v+yζ(p)ζ, stands betweensideslip velocity and lateral acceleration. Finally, one cannotice the non-minimum phase effect in lateral acceleration.In order to tackle these issues direct lateral acceleration oraugmented lateral acceleration control should be preferredbut is beyond the scope of this paper.

V. MIMO APPLICATION

In this section polynomial eigenstructure assignment isdemonstrated for the Horton MIMO missile model (13). Themotion for this missile is nonlinear, however its quasi-linearform enhances applying PEA. A velocity/roll rate controlleris designed where rudder, ζ and elevator, η are togetherinducing roll rate. The attempt is to decouple these fromthe roll rate.

A. The controller design

The controller structure of Figure 2 is kept with controlledoutputs yc being vector

[v w p

]Tmeasured outputs yi

being vector[r q

]Tand input u being vector

[ζ η ξ

]T.

The approach to MIMO case is the same as described in thePEA section and transforming Equations (9) and conditions(11) to the LPV framework.

The objective is to decouple the rudder and elevator fromthe roll rate while ensuring zero-steady state error, and fasttransient in controlled outputs. The desired transfer functionis thus selected in the form of,

Nd(s, p) =

⎡⎣n1(s, p) 0 0

0 n2(s, p) 00 0 n3(s, p)

⎤⎦ , (21)

Dd(s, p) =

⎡⎣d1(s) 0 0

0 d2(s) 00 0 d3(s)

⎤⎦ , (22)

where p is the parameter vector[v w M λ

]Tand

n1(s, p) = pvωvn

2Zv0 (s, p), (23a)

n2(s, p) = pwωwn

2Zw0 (s, p), (23b)

n3(s, p) = ωpn2Zp

0 (s, p), (23c)

d1(s) = (s − pv)(s2 + 2ζvωvns + ωv

n2), (23d)

d2(s) = (s − pw)(s2 + 2ζwωwn s + ωw

n2), (23e)

d3(s) = s2 + 2ζpωpns + ωp

n2, (23f)

and where Zv0 (s, p), Zw

0 (s, p), Zp0 (s, p) represent the un-

changed open-loop zeros, or equivalently polynomial eigen-vectors, of the system for the horizontal, vertical and rollchannels respectively. The controller structure includes anintegrator on each channel, Ka(s), which ensures zero-steady state error.

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B. Simulations

Simulations are presented for a missile flying at 6 kmaltitude, all burn fuel, and with total speed varying drasticallyduring manoeuvre. Fast response is obtained by selecting ap-propriately the poles for each channel, an arbitrary selectionis done with pv = −60, ωv

n = 30, ζv = 0.7, pw = −60,ωw

n = 30, ζw = 0.7, ωpn = 60, ζp = 0.7 and lead to

simulations of Figure 4.

0 0.5 1 1.5 2 2.5 3Time, [s]

0

50

100

150

200

250

Lat

eral

aug

men

ted

acce

lera

tion,

[m

^2/s

]

v

w

0 0.5 1 1.5 2 2.5 3Time, [s]

-3

-2

-1

0

1

2

3

Rat

e, [

rad/

s]

p

r

q

0 0.5 1 1.5 2 2.5 3Time, [s]

-0.05

0

0.05

0.1

0.15

Fin

angl

e, [

rad]

η

ζ

ξ

Fig. 4. Simulation for velocity/roll rate control of the MIMO QLPVHorton missile with Mach number varying linearly from 2 to 3.5 in 3 s andconstant altitude 6 km. Demand in horizontal velocity steps from 100 m/sto 250 m/s, demand in vertical velocity steps from 200 m/s to 50 m/sand roll rate is maintained to 0 rad/s.

All simulations show zero-steady state error and fastresponse, however coupling is taking place in the transients.This is not a surprise since semi-non-dimensional derivativesare linked to total incidence angle or equivalently v and wand are not decoupled in this LPV approach. Peak overshootand settling time are satisfied in velocities.

VI. CONCLUSIONS & DISCUSSIONS

PEA for LPV systems is presented as an extension toclassical eigenstructure assignment. Such result is madepossible solving symbolic null space equations. PEA isdemonstrated suitable to MIMO systems and is applied to

the MIMO QLPV Horton missile for lateral velocities androll rate controls. When the designer translates his objectivesto an eigenstructure, the desired closed-loop eigenstructureis selected and accordingly its closed-loop transfer function.For a unique closed-loop eigenstructure, the closed-loopsystem is thus rendered independent of operating points.This suggests that QLPV models are amenable to dynamicinversion design using this approach.

The proposed QLPV controller is free of the difficultiesassociated with classical gain scheduling, as it consistsof one controller only and “scheduling” is automatic byfeedback, giving total independence to the operating point.The scheduling is here performed by a QLPV controllerwhich does not require interpolation process. Nevertheless,polynomial eigenstructure assignment design is only valid (asgain scheduling is) in the vicinity equilibria. Additionally, theapproach does not require to cancel out zero-dynamics, andthus applies to a wider class of systems including quasi-linearsystems and non-minimal phase systems in some extent.

Regarding the MIMO case, decoupling is achieved in someextent and reformulation of the quasi-linear model could helpreducing the incidence angle coupling effects in equations.Dynamic controllers would further improve the closed-loopsystem properties and different controller structures couldbe considered to relax some of the null space conditions andhence ease the matching. Finally, multi-objectives designsincluding decoupling, sensitivity, robustness, disturbance re-jection, partial feedback and actuator dynamics need to beextended to the LPV framework.

REFERENCES

[1] B. A. White, “Eigenstructure Assignment: A Survey,” IMechE, Systemsand Control Engineering, vol. 209, no. I1, pp. 1–11, 1995.

[2] ——, “Flight Control of a VSTOL Aircraft Using Polynomial,” in IEEUKACC, International Conference on Control 96, vol. 2, September1996, pp. 758–763.

[3] ——, “Robust Polynomial Eigenstructure Assignment Using DynamicFeedback Controllers,” IMechE, Systems and Control Engineering, vol.211, no. I1, pp. 35–51, 1997.

[4] G. P. Liu and R. J. Patton, Eigenstructure Assignment for ControlSystem Design. Chichester: John Wiley & Sons, 1998.

[5] M. T. Soylemez, Pole Assignment for Uncertain Systems. Baldock:Research Studies Press, 1999, UMIST Control Systems Center Series.

[6] T. Kailath, Linear Systems. London: Prentice-Hall, 1980.[7] B. A. White, “Robust Flight Control of a VSTOL Aircraft Using

Polynomial Matching,” in Proceedings of the American Control Con-ference, vol. 2, June 1998, pp. 1133–1137.

[8] ——, “Robust Control of an Unmanned Underwater Vehicle,” inProceedings of the 37th Conference on Decision & Control, vol. 3,December 1998, pp. 2533–2534.

[9] M. P. Horton, “A Study of Autopilots for the Adaptive Control ofTactical Guided Missiles,” Master’s thesis, University of Bath, 1992.

[10] A. Tsourdos, R. Zbikowski, and B. A. White, “Robust Design ofSideslip Velocity Autopilot for a Quasi-Linear Parameter-VaryingMissile Model,” Journal of Guidance, Control, and Dynamics, vol. 24,no. 2, pp. 287–295, 2001.

[11] L. Bruyere, A. Tsourdos, R. Zbikowski, and B. A. White, “RobustPerformance Study for Lateral Autopilot of a Quasi-Linear Parameter-Varying Missile,” in Proceedings of the American Control Conference,vol. 1, 2002, pp. 226–231.

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