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Lateral flight control design for a highly

flexible aircraft using nonsmooth optimization

Alberto M. Sim˜ oes∗

ONERA, Toulouse, France

Pierre Apkarian†

ONERA & UPS, Toulouse, France

Daniel Alazard‡

Universit´ e de Toulouse-ISAE, France

Dominikus Noll§

Universit´ e Paul Sabatier, Toulouse, France

This paper describes a nonsmooth optimization technique allowing to

design a lateral flight control law for a highly flexible aircraft.Flexible

modes and high-dimensional models pose a major challenge to modern con-

trol design tools. It is shown that the nonsmooth approach offers potent

and flexible alternatives in this difficult context. More specifically, the pro-

posed technique is used to achieve a mix of frequency domain as well as

time domain requirements for a set of different load conditions.

I.Introduction

The synthesis of flight control laws for modern aeronautics and space applications re-

mains a challenging task whenever aeroservoelastic phenomena significantly affect the control

bandwidth. Such phenomena are especially critical when demanding specifications including

performance and robustness constraints of different natures must be achieved. Performance

specifications, for instance, are normally related to control objectives like tracking and de-

coupling and are naturally expressed in terms of time-domain constraints such as limited

overshoot, short settling- or rise-times, small steady-state error and amplitude limitation.

Flexible modes, on the other hand, are frequently dealt with via frequency-domain criteria

or modal specifications (prescribed damping ratios). A further complication is related to

∗Researcher, Control Systems Department.

†Researcher & Professor, Control Systems Department & Institut de Math´ ematiques.

‡Professor, Department of Mathematics, Computer and Control Sciences.

§Professor, Institut de Math´ ematiques.

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structural constraints imposed on the controller. Simpler controllers are generally sought to

facilitate on-board implementation and management.

The classical approach in which a control law is designed for the rigid dynamics and

a low-pass filter is inserted a posteriori to avoid or reduce spillover effects is no longer a

valid scheme for such applications. The reason is that in order to meet appropriate level of

performance, the controller bandwidth should overlap with the frequency range of flexible

modes which represents a core issue of such problems.

Traditional H2or H∞syntheses20do not provide suitable answers to these difficulties.

First of all, time-domain specifications should be addressed indirectly via nontrivial tuning

of weighting filters. Secondly, these methods produce full-order controllers and therefore rely

on model reduction techniques to derive simple controllers which is always prone to failure.

Design methods based on the Youla parametrization8offer some flexibility to handle

both time- and frequency-domain specifications. The resulting controllers however suffer

from substantial order inflation and are hardly amenable to numerical implementation.

Different approaches have been reported in the literature trying to exploit eigenstruc-

ture assignment methods to design problems involving lightly-damped flexible modes.13,14,17

Eigenstructure assignment methods are interesting because time-domain specifications can

be captured through modal shaping. Unfortunately, as noted in Ref. 13, determining ap-

propriate eigenspaces associated with flexible modes remains an inherent difficulty.

Nonsmooth optimization techniques have been used recently to solve a number of dif-

ficult structured controller design problems involving time- or frequency-domain specifica-

tions.2,3,7,9,12,18The nonsmooth design method considered here bear the following appealing

features. First, time-domain specifications are addressed directly, thus dispensing with the

use of auxiliary tuning parameters such as weighting filters. Moreover, frequency-domain

constraints such as those related to flexible modes are easily incorporated within the same

framework. Secondly, such techniques remain operational even for large size plants, and

thus allow to short-circuit risky model reduction phases. Finally, they encompass arbitrary

controller structures which make them methods of choice when implementation constraints

are important.

The central aim of the present work is to illustrate the efficiency and the flexibility of

nonsmooth design methods in solving difficult structured control design problems like large

size flexible transport aircraft.

The paper is organized as follows. Section II discusses the multi-objective control design

problem, while Section III outlines the key ingredients of the proposed nonsmooth optimiza-

tion technique. The difficult design problem of lateral flight control for a highly flexible

aircraft subject to turbulence and multiple load conditions is addressed in Section IV.

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II.Multi-objective controller design via nonsmooth optimization

P ∈ P

K(κ)

uy

wz

Figure 1. Closed-loop synthesis interconnection

To begin with, consider the synthesis interconnection given by the standard form in Fig. 1

with u ∈ Rm2and y ∈ Rp2and where the multivalued plant P(s) takes values in a finite family

of linear plants P := {P1,...,Pp} representing, for instance, multiple operating conditions

or faulty modes. Each plant P ∈ P is described by a minimal state-space realization of the

form

y(t)

C2

D21

˙ x(t)

z(t)

=

AB1

D11

B2

D12

D22

C1

x(t)

w(t)

u(t)

,

(1)

where plant indexing has been removed for simplicity. In order to address practical controller

structures we introduce a state-space parametrization of the form

κ ∈ Rq→ K(κ) :=

AK(κ) BK(κ)

CK(κ) DK(κ)

(2)

with corresponding frequency-domain representation

K(s,κ) = CK(κ)(sI − AK(κ))−1BK(κ) + DK(κ),

where AK ∈ Rk×k. In the above description, κ designates the decision vector of design

variables in the controller. Note the case of a static controller (k = 0) is a particular

instance. The mapping K : Rq→ R(m2+k)×(p2+k)is assumed to be continuously differentiable

but otherwise arbitrary.

Performance specifications are given in most cases in terms of time-domain constraints

like limited overshoot, short settling- or rise-times, but also amplitude limitation in order

to guarantee decoupling properties or to avoid reaching operational limits of the system.

Such time-domain constraints are achieved by direct shaping closed-loop system responses

to fixed test input signals. More specifically, it is assumed that each plant in the family

P in feedback loop with the controller K(s,κ) is subject to one or several input signals w

selected in a finite signal generator set W := {w1,...,wd}. This gives rise to a finite family

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of closed-loop responses z ∈ Z, where Z := {z1,...,zr}. Each instance in Z is called a

scenario. Practically speaking, the signal generator set is made of typical deterministic test

inputs such as steps, ramps, sinusoids, etc.

The above description is flexible enough to reflect situations in which a single plant is

submitted to various test signals as in the case when decoupling properties must be exam-

ined, or when the response to a given test signal is to be considered for multiple operating

conditions or faulty modes. The proposed set-up also accepts more complicate formulations

where each plant in the family P is tested against several inputs.

uz

αz

t

0

lz

Figure 2. Envelope constraints on the step response

The goal is to compute κ ∈ Rqsuch that the closed-loop time responses z ∈ Z obtained

with controller K(κ) meet envelope constraints of the form

lz(t) ≤ z(t) ≤ uz(t), ∀t ≥ 0, ∀z ∈ Z,

(3)

where lzand uzare lower and upper bounds for z and are assumed piecewise constant in the

sequel. These bounds are illustrated as dashed lines in Fig. 2 for a step following specification

(αzstands for a coordinate of z).

On the other hand, design specifications including attenuation of exogenous bounded-

energy disturbances or robustness against unstructured uncertainties are known to be better

addressed by frequency-domain criteria involving bounds on the maximum singular value

norm of suitable closed-loop transfers. Therefore, in addition to the constraints in (3), the

designed controller K(κ) is required to achieve prescribed bounds for a finite set of closed-loop

transfers

?Fl(P(s),K(s,κ))?IP≤ γP, γP> 0, ∀P ∈ P∞⊂ P,

(4)

where Fl(·,·) denotes the traditional lower Linear Fractional Transformation, and ?.?IPde-

notes the peak value of the transfer function maximum singular value norm on a prescribed

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frequency interval IP:

?Fl(P(s),K(s,κ))?IP:= sup

ω∈IP

σ (Fl(P(jω),K(jω,κ))).

The frequency band IP is typically a closed interval IP = [ωP

finite union of intervals IP= [ωP

infinite values. Alternatively, a dynamic weight WP(s) can be included in (4) if necessary

1, ωP

2], or more generally, a

1, ωP

2] ∪ ... ∪ [ωP

q, ωP

q+1], where right interval tips may take

?WP(s)Fl(P(s),K(s,κ))?IP≤ 1, ∀P ∈ P∞⊂ P (5)

to stress the relative importance of each channel.

Finally, the most fundamental specification for a closed-loop system is internal stability.

Thus, the sought controller K(κ) must also guarantee negative upper bounds on the closed-

loop spectral abscissas (maximum real part of closed-loop eigenvalues)

α(AP(κ)) ≤ αP, αP< 0, ∀P ∈ P,

(6)

where AP(κ) is the state matrix of the closed-loop system Fl(P(s),K(s,κ)).

In summary, the considered multi-objective controller design problem may be stated as:

find controller variables κ ∈ Rqsuch that constraints (3)-(6) are satisfied. In what follows

this problem is addressed through a nonsmooth optimization technique. Notice, initially,

that the time-domain constraints in (3) are automatically met if the function

ft(κ) := max

z∈Zmax

t≥0{[z(κ,t) − uz(t)]+, [lz(t) − z(κ,t)]+}(7)

is non-positive, where the notation [.]+ applied to a vector v ∈ Rnis defined as [v]+ =

max{0, max

i=1,...,nvi}. Similarly, the frequency-domain constraints in (4) and the spectral con-

straints in (6) are satisfied if the functions

f∞(κ) := max

P∈P∞

?Fl(P(s),K(s,κ))?IP

γP

− 1(8)

and

gα(κ) := max

P∈P(α(AP(κ)) − αP) ,

(9)

are non-positive, respectively.

Our nonsmooth design method is thus based on solving the max-type optimization prob-

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