Nonlinear order reduction of structural dynamic aircraft models

Aerosp. Sci. Technol. 5 (2001) 55–68  2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S1270-9638(00)01086-5/FLA

Nonlinear order reduction of structural dynamic aircraft models Michael Kordt a,1 , Helmut Lusebrink b a German Aerospace Research Center (DLR), Institute of Robotics and Mechatronics, Oberpfaffenhofen, 82234 Wessling, Germany b EADS Airbus GmbH, Kreetslag 10, 21129 Hamburg, Germany

Received 2 September 1999; revised and accepted 9 November 2000

Abstract

New methods for nonlinear order reduction of structural dynamic aircraft models described by nonlinear differential equations have been developed. The methods are based on singular perturbations, weak coupling and cost functionals for selecting the physical states of the reduced order model, computing and analyzing the reduced order model. By introducing a formally affine nonlinear model structure, the cost functional vector optimization yields closed expressions for the reduced order system. Via Lagrange multipliers, even constraints with regard to the reduced order system can be considered, only extending these closed expressions. This leads to the notion of smaller and control-oriented realizations. Moreover, a unified approach to structural dynamic aircraft modeling for loads analysis and for structural dynamic control is developed, which allows us to apply the new reduction methods directly to both problems. Here, the methods are applied to a structural dynamic aircraft model in order to achieve a very low order model, suited for structural dynamic controller design.  2001 Éditions scientifiques et médicales Elsevier SAS nonlinear model reduction / structural dynamic aircraft model / singular perturbation / multidisciplinary aircraft control / loads analysis

Zusammenfassung

Nichtlineare Ordnungsreduktion strukturdynamischer Flugzeugmodelle. Neue Methoden zur nichtlinearen Ordnungsreduktion strukturdynamischer Flugzeugmodelle, die durch nichtlineare Differentialgleichungen beschrieben werden, sind entwickelt worden. Die Verfahren basieren auf der singulären Perturbation, auf der Identifikation schwach gekoppelter Teilsysteme und auf Gütemaße zur Auswahl der physikalischen Zustände des reduzierten Systems, zur Berechnung und Analyse der ordnungsreduzierten Modelle. Durch die Einführung einer formal affinen nichtlinearen Modellstruktur führt die Gütevektoroptimierung auf geschlossene Formeln für das reduzierte System. Mittels Lagrange’scher Multiplikatoren können Nebenbedingungen an das reduzierte System gestellt werden, die lediglich zu einer Erweiterung der geschlossenen Formeln führen. Dies führt zum Konzept der kleineren und der entwurfsorientierten Realisierung. Außerdem wird ein vereinheitlichter Ansatz zur Lastenanalyse und zum strukturdynamischen Reglerentwurf entwickelt. Er ermöglicht es, die neuen Reduktionsmethoden direkt auf beide Problemstellungen anzuwenden. Die Methoden werden hier auf ein strukturdynamisches Flugzeugmodell angewandt, um ein Modell sehr niedriger Ordnung zu erhalten, das für den strukturdynamischen Reglerentwurf geeignet ist.  2001 Éditions scientifiques et médicales Elsevier SAS nichtlineare Modellreduktion / strukturdynamisches Flugzeugmodell / singuläre Perturbation / multidisziplinäre Flugzeugregelung / Lastenanalyse

1 Correspondence and reprints. Present address: EADS Airbus GmbH, Loads Department, Kreetslag 10, 21129 Hamburg, Germany. Email address: [email protected] (M. Kordt)

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M. Kordt, H. Lusebrink / Aerosp. Sci. Technol. 5 (2001) 55–68

1. Introduction Modern structural dynamics of aircraft covering design loads and aeroelastics is not only concerned with nonlinear, fully flexible dynamic response and flutter analysis, but also with the design of the corresponding active control functions as a slight and short-term extension of the standard electronic flight control system, primarily designed for handling quality purposes. The active control functions aim at alleviating the maneuver, gust and engine out loads and at structural mode control for flutter margin augmentation and comfort [2,6,8,13,15,16,18,20, 24–26]. To apply control design methods to these problems, very low order models restricted to the most essential nonlinearities and corresponding nonlinear reduction methods are required; e.g., in case of flutter margin augmentation from a pure viewpoint of modeling, several nonlinearities have to be considered [3,19,30]: stiffness nonlinearities in case of large bending deflections, actuator nonlinearities after their operational limits have been exceeded, free-play nonlinearities (slackness) in case of control linkages, hysteresis when friction loads affect linkage dynamics, etc. Using a bottom-up modeling approach, Stragnac et al. [2,13], try to consider only as small a number of nonlinearities as possible for the design of a flutter margin augmentation system. Their bottom-up modeling approach had to be experimentally validated [2]. For the design of load alleviation systems, the models are often more complex than in case of flutter margin augmentation. They cover the nonlinear rigid body motion in six degrees of freedom and a large number of flexible modes hardening a bottom-up approach for derivation of synthesis models for the design of load alleviation systems: McLean [25] had to consider a model of the 29th order for the longitudinal and the lateral aircraft motion to design a gust load alleviation system. Structural dynamic engine out controller design aims at reducing loads at vertical tailplane, in consequence of engine failure and thus induced pilot panic or overreaction [16, 24,31]. It requires a model which covers the steady-state effect of all modes which are relevant for structural dynamics. Usually about 1000 modes have to be considered. Maneuver load alleviation system design for the wing via aeroservoelastic control requires a model which covers not only the rigid body and the elastic aircraft motion, but also the nonlinear actuator dynamics [20]. In dynamic response analysis for design gust loads, so far fully flexible linear models have been considered, whereas for maneuver loads (one engine out, yawing or roll maneuver) quasi-flexible nonlinear models are used [24,31]. Starting from an original structural mechanical model of an order n > 1000, a high precision assessment covering the whole flight envelope and considering all parameter modifications during aircraft design and in service requires an order reduction to bring about a reasonable computational effort without affecting the accuracy of the loads computation. However, up until now the gust and ma-

neuver loads models were condensed by different methods, simplifications and approximations. Now, considering the design of future large aircraft, elastic and nonlinear effects might become more critical, because the frequency gap between rigid body and elastic modes vanishes and new airframe properties occur, due to new wing control surfaces or tail design. This requires revising and validating the present modeling approach and the inherent approximations. However, instead of partly extending gust and maneuver load analysis models with regard to so far neglected nonlinear and elastic effects, using a unified nonlinear fully flexible model for both disciplines is more straightforward. One requirement for such an approach is the availability of a nonlinear model reduction process, which extends the commonly used methods by Karpel [12] and Anderson [1]. Based on a careful and systematic choice of the dominant elastic dynamics, the method should yield very low order models that are restricted to the most essential nonlinearities. For this systematic choice of the dominant dynamics, physical reasoning should be fully included. This paper is organized as follows: section 2 presents the nonlinear structural dynamic model which is the starting point for both loads analysis and structural control function design. Section 3 suggests a linear structural dynamic model reduction suited for a unified gust and maneuver loads analysis model, which can be applied for the physical formulation of the equations. However, the method involves some heuristics concerning nonlinearities. Therefore, two general nonlinear order reduction methods are presented in section 4: a ‘cost functional based approach’ and an extension of the approach by Karpel [12] and of nonlinear singular perturbation to weakly coupled systems. In section 5, the challenging example of a very low order reduction for the design of structural control functions is presented.

2. Structural dynamic modeling Structural dynamic modeling considers both the rigid body and the elastic motion of an aircraft. The rigid body motion is described by the three velocity components uB , vB , wB , the three angular velocity components pB , qB , rB with regard to a body-fixed frame B and the three Euler angles φ, θ , ψ [6]. The elastic motion is described relative to the rigid body motion. According to a standard FEM model reduction, like ‘Guyan reduction or condensation’ [10], and structural dynamic engineering judgement, only a finite number of degrees of freedom, usually less than 1000, have to be considered as a starting point for a consistent structural dynamic order reduction. The corresponding physical coordinates x 1,E are divided into two substate vectors: x 1,Er describes the elastic degrees of freedom, which are retained in the reduced order model and x 1,Eo describes those which can be omitted in consequence of the order reduction: x 1,E =

M. Kordt, H. Lusebrink / Aerosp. Sci. Technol. 5 (2001) 55–68

[x 1,Er x 1,Eo ]T . For brevity, lag states for the approximate unsteady aerodynamic modeling in the time domain and the modeling of the electronic flight control system are not considered here. The order reduction method does not require specific coordinates like physical or generalized coordinates. Defining x 1 = [x 1,R x 1,Er x 1,Eo ]T yields the equations of motion [6,8,21]: M x¨ 1 + D  x˙ 1 + (C  + K)x 1 +F g(x 1 , x˙ 1 , u1 ) = Ru1 + E  e + G g  + · · · , (1) D :

with M: mass matrix, aerodynamic and structural damping matrix, C  : aerodynamic stiffness matrix, K: structural stiffness matrix, R: control surface deflection effectiveness matrix, 2 F : effectiveness matrix of the nonlinearities, E  : effectiveness matrix of the engines, G : simplified gust effectiveness matrix, 3 u1 : control surface deflection angles (aileron, elevator, rudder,. . .), e: engine states, e.g. thrust, g  : simplified gust states. The dots indicate further input or disturbance terms due to more detailed linear or nonlinear modeling of engines, control surfaces and corresponding actuators, electronic flight control system, aircraft design gust or failure cases. Written in detail:   M RR M REr M REo     M =  M Er R M Er Er M Er Eo  ,   M Eo R M Eo Er M Eo Eo     RR FR         R =  R Er  , F =  F Er  ,     F Eo R Eo and C  , D  , K, E  and G are defined analogously. All nonlinear terms are included in the vector: 4   sin θ       cos θ cos φ − 1      (qB sin φ + rB cos φ) tan θ  g(x 1 , x˙ 1 , u1 ) =   . (2)     1  (qB sin φ + rB cos φ)    cos θ   .. . 2 It could also be termed gain or influence matrix. 3 The gust penetration effect for a fully flexible aircraft is approximated in the time domain by considering in G as many columns and

thereby gust input signal functions as grid points with different distances from the gust field. 4 For brevity, only some of the rigid body nonlinearities are given here: two gravity, two kinematic and two Euler terms.

57

The matrix F describes the influence of the nonlinearities on the system dynamics. Nonlinearities, which are identical up to a constant factor, only have to be considered once in g(x˙ 1 , x 1 , u1 ). For some applications, only nonlinearities in x 1,R have to be considered [6,8], i.e.   FR     F g(x 1 , x˙ 1 , u1 ) =  0  g(x 1,R , x˙ 1,R , u1 ). (3)   0 The rare relevance of nonlinearities including elastic states, in particular nonlinear inertial coupling terms, are discussed in [5]. The fully flexible local loads (dynamic forces) at the considered grid points of the aircraft are defined by    x 1,R K RR K REr K REo       f Loads =  K Er R K Er Er K Er Eo   x 1,Er  . (4)    K Eo R K Eo Er K Eo Eo x 1,Eo They are computed according to the force summation method [24], i.e. equation (1) has to be solved for f Loads : f Loads = −M x¨ 1 − D  x˙ 1 − C  x 1 − F g(x 1 , x˙ 1 , u1 ) + R  u1 + E  e + G g  + · · · .

(5)

By substituting x¨ 1 = [x¨ 1,R x¨ 1,E ] by the expression of equation (1), the equation of the local loads can be written as an extended nonlinear output equation in state-space form [11]:   x1 f Loads = C Loads   + D Loads u x˙ 1 + H Loads h(x 1 , x˙ 1 , u), g  ]T .

(6)

where u = [u1 e Computing the required time histories in x 1 and x˙ 1 = [x˙ 1,R x˙ 1,Er x˙ 1,Eo ] from equation (1), the contributions of x˙ 1,Eo to f Loads are expected to be small. The order reduction is required in this case to yield the dimension of x 1,Eo or equivalently the order n, ˜ to which the system, equation (1), can be reduced. Moreover, the nonlinear order reduction should compute the parameters of the reduced order system such that the resulting trajectories [x˜ 1,R x˜ 1,Er ] approximate those of the original system as well as possible in spite of neglecting the dynamics of x 1,E0 . Finally, it should compute reconstruction matrices W i , i = 0, 1, yielding good approximations x 1,approx = W 0 [x˜ 1,R x˜ 1,Er ]T and x˙ 1,approx = W 1 [x˙˜ 1,R x˙˜ 1,Er ]T of the full original vector x 1 and its time derivative x˙ 1 . The approximations have to be good in the sense that peak and steady state loads are approximated either conservatively or to the required accuracy by

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f˜ Loads

    x˜ 1 = C Loads W 0 W 1   + D Loads u x˙˜ 1 + H Loads h(W 0 x˜ 1 , W 1 x˙˜ 1 , u).

(7)

negligible energy. Next, the terms in q˙ Eo and q¨ Eo are neglected in equation (10), which is based on the approach by Karpel [12]. Then, the remaining algebraic equation (Cˆ  Eo Eo + Kˆ Eo Eo )q Eo

For structural dynamic control, very low order models are required, that represent the dynamics from certain inputs u to certain measured quantities y, used for feedback control. Such measured quantities are, e.g., local accelerations at certain points of the aircraft, containing sufficient information about the flexible effect under consideration. They are described by the same type of output equation as the loads equation:   x1 y = C   + Du + H h(x 1 , x˙ 1 , u). (8) x˙ 1

has to be solved for q Eo . Next, all nonlinear effects with respect to the omitted elastic states are neglected:

As in the case of the loads computation, the order reduction is required here to compute a very low order dynamic model and the reconstruction matrices W 0 and W 1 such that    x˜ 1  y˜ = C W 0 W 1   + Du x˙˜ 1

(12) This approximation ensures uniqueness and simplifies the solution of equation (12):  ˆ ˆ ˆ q Eo = S −1 (13) Eo Eo R Eo u1 + E Eo e + GEo g

− Dˆ  Eo R q˙ R − Dˆ  Eo Er q˙ Er − Cˆ  Eo Er q Er ,

+ H h(W 0 x˜ 1 , W 1 x˙˜ 1 , u)

(9)

yields a good approximation of y. Comparing equations (7) and (9) shows that a unified mathematical modeling approach for structural controller design and loads analysis has been achieved. It allows us to develop reduction methods, which directly cover both problems. 3. Structural dynamic model reduction techniques A classical approach to structural dynamic model reduction is based on a modal transformation x = T q of the undamped structural mechanical system, described by the homogeneous vector differential equation M x¨ 1 + Kx 1 = 0. Applying the modal transformation to equation (1) yields ˆ q¨ + Dˆ  q˙ + (Cˆ  + K)q ˆ + Fˆ g( M ˆ q˙ , q , u1 )   ˆ ˆ + ··· ˆ 1 + E e + Gg = Ru

(10)

Substituting x = T q and x˙ = T q˙ into equation (6) and equation (9) yields the loads equation and the output equation for the measured quantities in terms of the generalized coordinates. The differential equations describing high frequency elastic dynamics beyond a certain frequency bound are to be omitted in the reduced model. In this case, q Eo corresponds to the highest frequencies in the model. The frequency bound requires a physical justification; e.g., in case of gust loads analysis, it is defined by the fact that beyond this bound, the Dryden or the von Kármán spectrum contains only

ˆ Eo g  − Dˆ E R q˙ R = Rˆ Eo u1 + Eˆ Eo e + G o  − Dˆ Eo Er q˙ Er − Cˆ  Eo Er q Er − Fˆ Eo g(q, ˆ q, ˙ u1 ) (11)



Fˆ R

  Fˆ g( ˆ q, ˙ q, u1 ) =  Fˆ Er  0

   ˆ q˙ R , q˙ Er , q R , q Er , u1 ).  g( 

S Eo Eo = Cˆ  Eo Eo + Kˆ Eo Eo .

(14)

Subsequently, substituting this expression in the differential equations for q R and q Er realizes the static correction of the dynamics of q R and q Er with regard to the omitted modes and yields the reduced order model:       q ˙ q ¨ q ˆˆ  R  + Dˆˆ   R  + (Cˆˆ  + K) ˆˆ  R  M q¨ Er q˙ Er q Er ˆˆ  ˆˆ ˆ ˆˆ  + · · · + Fˆ g( ˆ q, ˙ q, u1 ) = Ru 1 + E e + Gg where, e.g.   ˆ ˆ ˆˆ   D RR D REr  D = Dˆ  Er R Dˆ  Er Er  ˆ Cˆ  REo S −1 Eo Eo D Eo R − ˆ Cˆ  Er Eo S −1 Eo Eo D Eo R

ˆ Cˆ  REo S −1 Eo Eo D Eo Er ˆ Cˆ  Er Eo S −1 Eo Eo D Eo Er

 .

According to the approximation, equation (12), the nonlinearities   Fˆ R ˆˆ  g( ˆ q, ˙ q, u1 ) F g( ˆ q, ˙ q, u1 ) =  Fˆ Er remain unchanged. The reconstruction to the full state vector for evaluating the loads equation and the output equation for structural dynamic control is achieved via

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x = T [q R q Er 0]T . The method improves the standard modal reduction and the Guyan reduction [10] by considering the complete structural dynamic aircraft model instead of the elastomechanical model. However, the method has the following drawbacks: (1) The approximation with respect to the nonlinearities, equation (12), is rough in the following sense: (i) nonlinearities in the omitted states cannot be considered at all, which might become a problem in the transonic domain for future large aircraft; (ii) the procedure makes no suggestion how to handle mixed nonlinearities containing both retained and omitted states; (2) The selection of the dynamics to be omitted is not covered by the reduction method; (3) Within the computation process, one has no influence on the truncation error. By extending the reduction method to a more general nonlinear reduction method, these problems can be solved consistently, which will be done in the following section. 4. Cost functional based nonlinear order reduction Consider the original system of nonlinear differential equations 5 of the order n in a formally affine representation (i.e. the system is not only affine in u, but also in g(x, u)): x˙ = A x + B u + F g(x, u), x˙ = E m,

(15)

where E = [A B F ], m = [x u Hence, the nonlinear reduced order system computation becomes a cost functional vector optimization of the parameters of the reduced order system, that is an optimization, aspiring a Pareto optimum for several cost functionals. Equation (1) can be transformed to this form by defining the state vector x as x = [x 1 x˙ 1 ]T . Next, the dominant state vector x d and thereby the dimension n˜ < n of the reduced order system has to be chosen (in the following referred to as ‘User Option: selection of dominant states’). The dominant state vector contains: (i) those states having high impact on the structural dynamic performance under consideration; (ii) the measured or commanded states, e.g. local accelerations at certain points of the aircraft or flight mechanical quantities like yaw rate or sideslip angle; (iii) states on which the significant nonlinearities depend. States having a significant impact on structural dynamic performance are not always obvious. Therefore, additional cost functionals will be introduced to systematize the identification of the dominance structure of the system. g(x, u)]T .

5 Notice that structural dynamics distinguish velocity and accelera-

tion, whereas system dynamics only considers state derivatives.

59

Cost functionals are the core of the reduction methods, constituting a cost functional vector j = [j1 , . . . , jN ]T and estimating the deviations between the dynamics of the original and the reduced order model. This approach was first introduced by Lohmann [23]. The cost functionals are evaluated for a finite number of input signal/initial condition pairs, representing the operation domain of the system (in the following referred to as ‘User Option: selection of input signal/initial condition pairs’). Concerning the number and type of input signals in general, one can take advantage of the analogy between order reduction and identification of dynamic systems: signals used for parameter identification also apply to the order reduction problem. In order to cover the whole operating domain of the system as closely as possible, the signals can be either chosen due to engineering judgement or in a more mathematical approach as a complete function system like sinoids, pulses or special polynomials (e.g., Legendre polynomials). As the systems being examined are nonlinear, both amplitude parameters and initial conditions have to be varied. Here, design and operational maneuvers [6,24] have to be considered, but signals used for aerodynamic parameter identification are also suited to keep the number of signals small. Concerning simulation and sample time, a tradeoff has to be found between a reasonable amount of computation time and the necessity of covering all relevant information of the system by the simulation data. Eigenfrequencies of the system and higher order harmonics help to choose simulation and sample time. The cost functionals are minimized with regard to the parameters of the reduced order system under constraints, concerning the physical and structural properties of the reduced order system (‘User Option: Cost functional and constraint selection’). Such properties are: (i) linear and nonlinear coupling or decoupling properties between subsystems; (ii) steady state accuracy of the reduced order model; (iii) the number of nonlinearities; or (iv) the contribution of the linear terms and the nonlinear terms within the differential equations. These constraints are particularly suited to keep only the most relevant nonlinearities in the system. The method is iterative in the sense that the reduced order system will be computed based on representative signals: in a detailed assessment the reduced system has to be simulated for different input signals, covering the whole operating domain densely. The cost functionals can be formulated for each individual test signal, e.g. as quadratic cost functionals for a good time-averaged approximation of the dominant states: Ti j1,i =

!

˜ d 2 dt = min(E), ˜ q 2 (t) x˙ d − Em

(16)

0

˜ B ˜ = [A ˜ F˜ ] inwhere md = [x d u g(W x d , u)]T and E cludes the matrices of the reduced order system time-

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M. Kordt, H. Lusebrink / Aerosp. Sci. Technol. 5 (2001) 55–68

weighting, . is the standard Euclidean norm, q 2 (t) is a positive function for time-weighting. By iterating the choice of q 2 (t), the initial or transient behaviour in specific time intervals or the stationary performance of the trajectories of the reduced order system can be improved. The matrix W reconstructs the full state x out of x d , so that the nonlinearities can be transferred unchanged to the reduced order system. It can be computed via a second quadratic cost functional: Ti j2,i =

!

2 qW (t) x − W x d 2 dt = min(W ).

(17)

0 2 (t) qW

is a positive function for time-weighting, cf. comments on q 2 (t). The quadratic cost functional approach allows to cover several signals i = 1, . . . , N with corresponding finite simulation time Ti by one cost functional (here tr abbreviates trace): T1 +···+T  N

j1 = 0

˜ d )QQT (X ˙ d − EM ˜ d )T = tr (X˙ d − EM ! ˜ = min(E),

(18)

j2 = tr (X − W Xd )QW QTW (X − W Xd )T !

= min(W ),

(19)

where the cost functionals have been evaluated discretely:  M d = md (t11 ), . . . , md (t1,n1 = T1 ), . . . , md (tN,1 ), . . . ,  md (tN,nN = TN ) , i.e. the column vectors md , evaluated for each discrete time step and each simulation, are concatenated in a fat 6 matrix, equivalently:   ˙ d = x˙ d (t11 ), . . . , x˙ d (tN,nN = TN ) , X Xd = x d (t11 ), . . . , x d (tN,nN = TN )



and  Q = diag q(t11 ), . . . , q(t1,n1 = T1 ), . . . ,  q(tN,1 ), . . . , q(tN,nN = TN ) . Consequently, a whole simulation for one input signal can be weighted more strongly then another simulation via Q. According to this simulation based formulation of the cost functionals, not only closed nonlinear functions, but also nonlinearities given as data arrays and hysteresis 6 Fat means that there are more column than rows.

Using Lagrange multipliers, constraints of the type ˜ − L = 0 can be included in the cost functional GEH equation (18), still allowing us to compute the reduced order system via a closed expression ˜ =E ˜ R + [L − E ˜ R H ][ΩH ]−1 Ω, E

(21)



−1 and Ω = H T M d QQT M Td

˜R =X ˙ d QQT M Td M d QQT M Td −1 . E

(22)

where

˜ d 2 dt q 2 (t) x˙ d − Em



can be considered in g(x, u). The cost functional vector optimization problem equation (18) and (19) is solved in two steps: firstly, the reconstruction matrix W is analytically computed from the minimum condition corresponding to equation (19); secondly, by substituting this matrix W into M d , the optimization problem for the parameters ˜ can be solved analytically of the reduced order system E from the minimum condition corresponding to equation (18), yielding a closed expression for the system matrices of the reduced order system

˜ = X˙ d QQT M Td M d QQT M Td −1 . (20) E

For brevity, G has been chosen as the unity matrix here. In general, it only has to have maximum row rank, 7 H has to have maximum column rank. G and H select ˜ The matrix L contains the the desired elements from E. desired values of these elements. Hereby, the structural and physical properties (i)–(iv) can be imposed on the reduced order system. The approach is open to arbitrary cost functionals, which do not necessarily need to be analytically solvable. The only consequence is that an iterative optimization becomes necessary (‘User Option: selection of optimization procedure’). An example is a maximum norm cost functional ˜ d =! min(E) ˜ ji = max x˙ d − Em t ∈[0,Ti ]

(23)

to cover peak phenomena exactly, which is of particular interest for maximum loads in structural dynamics. Within such an iterative optimization, more general con˜
const, i = 1, . . . , Nc , can be used. straints |fc,i (E)| Depending on the optimization algorithm and setup, many norms and very nonlinear functions can be used. Thereby, a setup has been created which allows us to achieve a so-called smaller and a control-oriented realization, which are defined in the following and which are of high practical interest in nonlinear control system design and understanding of nonlinear systems (‘User Option: special realizations’). 7 Solving the optimization problem in this case requires the Kronecker-product and yields more lengthy closed expression.

M. Kordt, H. Lusebrink / Aerosp. Sci. Technol. 5 (2001) 55–68

 D EFINITION (Nonlinearity index). – A system α : x˙ α = Aα x α + B α uα + F α g α (W α x α , uα ) has the nonlinearity index Nα , if and only if, it has Nα non-vanishing entries in the matrix F α .  D EFINITION (Smaller realization). – A system β : is termed x˙ β = Aβ x β + B β uβ + F β g β (W β x β , uβ )  smaller realization with respect to the system α , if and only if: (a) it has a smaller order: nβ < nα ; (b) the nonlinearity index does not increase: Nβ
Nα . D EFINITION (Control-oriented realization). – The  system β is termed control-oriented realization of α , if and only if:  (a) the order of the system β does not increase: nβ
nα ; (b) the matrices Aβ , B β , F β , W β match predefined properties required for the controller design. Such smaller or control-oriented realizations correspond to zero elements in the matrix F˜ . They can be ˜ = 0 (constraints for analytachieved by requiring GEH ically closed optimization) or fc (F˜ ) = F˜i21 j1 + F˜i22 j2 + · · · < ε, where ε is a small number. F˜i1 j1 are certain elements chosen out of F˜ . Instead of the Euclidean norm, a wide range of inequality formulations according to optimization theory can be used for the iterative optimization. Control-oriented realizations may require not only ˜ constraints with respect to F˜ , but also with respect to A ˜ and B. The essential advantage for the controller design is that a synthesis model structure identification can be made, i.e. the question can be answered whether an approximate model of the given system matches a model structure of a classified system suited for a particular robust nonlinear controller design method like [9,11,17,29]. In contrast to a minimal realization [11], which is very hard to construct, for achieving smaller realizations, now, constructive methods have been developed here. If the choice of the dominant states is not obvious, the cost functional, equation (19), for the reconstruction matrix W can be extended to allow an iterative choice of the dominant states (‘User Option: systematic choice of dominant states’): as physical coordinates are often not chosen according to the question of dominant or nondominant states, a state transformation  

 zd (t)



=

znd (t)    z(t)

Vd

x approx(t) = W zd (t), = W V d x(t)

!

= min(W , Z d ).

(25)

Via the diagonal matrix S x , an additional weighting of the individual states is possible, allowing us to consider the already available knowledge of the dominance structure of the system. To solve the optimization problem without numerical iterations, a singular value decomposition of the matrix product S x XQ = U x ΣP T   Σd 0 PT = Ux  0 Σ nd

(26)

is used. U x and P T are orthogonal matrices. Σ is a diagonal matrix, in which the singular values σi occur in decreasing magnitudes, which implies the following approach for the unknown product:   Σd 0 PT S x W Zd Q = U x     0 0 Xapprox    Σd 0   P T . (27) = U xd U xnd  0 0 Now, substituting equation (26) and equation (27) into equation (25) and using matrix orthogonality properties yields: n 

jD = trace Σ nd Σ nd T = σi 2 . i=n+1 ˜

Obviously, σi2 is the contribution to the deviation between the original simulation data and the reduced simulation data, generated by omitting the simulation data for the state zi . Therefore, σi2 is a suitable dominance measure for the state zi . Now, by choosing n˜ such that n˜ 

 x(t)

i=1

is introduced such that the dominant states constitute the upper n˜ states. Consequently, the reconstruction matrix is now applied to zd :

(24)

and thereby the cost functional, equation (19), becomes

jD = trace S x (X − W Z d )QQT (X − W Z d )T S Tx



V nd    V

61

σi2 

n 

σi2 ,

i=n+1 ˜

the claim in equation (25) can be fulfilled. In order to derive closed expressions for W and V , equation (27) is rewritten, using equation (26):   S x W Z d Q = U xd 0 ΣP T    X approx

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  = U xd 0 U Tx S x XQ

the reduced order system can be computed according to equation (32). In order to combine both methods and simultaneously answer the question of their applicability, the assumption of small x˙ nd is withdrawn: then substituting equation (30) in equation (29) yields



−1 x˜˙ d = A11 − A12 A−1 22 A21 x d + B 1 A12 A22 B 2 u

+ F 1 − A12 A−1 (33) 22 F 2 g(x d , u),

= U xd U Txd S x XQ, and thereby W Z d = S x −1 U xd U Txd S x X.    X approx Finally, comparing with equation (24) yields W = S −1 x U xd

and V d = U Txd S x .

(28)

5. Nonlinear order reduction by compensation The second order reduction method, termed ‘reduction by compensation’, combines singular perturbation [14] and a reduction method for systems with weakly coupled subsystems. In contrast to the ‘cost functional based reduction’, the reduced order system can be analytically computed straight from the differential equations. Thereby, iterations in consequence of inadequate test signal sets are avoided. In order to derive the method, it is assumed that: (R1) the system can be partitioned in a dominant and a non-dominant part d and nd, where only the dominant states are to be retained in the reduced order system: x˙ d = A11 x d + A12 x nd + B 1 u + F 1 g(x d , u), (29) x˙ nd = A21 x d + A22 x nd + B 2 u + F 2 g(x d , u); (30) (R2) the vector g(x d , u) only contains dominant states. In case of singular perturbations, x˙ nd is assumed to be small, i.e. the corresponding dynamic is weakly excited and high frequent, e.g., due to small time constants, small masses or large gains [14]. If A22 is regular, the remaining algebraic equation 0 = A21 x d + A22 x nd + B 2 u + F 2 g(x d , u)

(31)

serves to eliminate x nd in equation (29)



−1 x˙ d = A11 − A12 A−1 22 A21 x d + B 1 − A12 A22 B 2 u

+ F 1 − A12 A−1 (32) 22 F 2 g(x d , u). Next, a reduction technique for weakly coupled systems is presented. In weakly coupled systems, all elements in the matrix A12 are small, i.e.: 8 |A12 x nd (t)|  |x˙ d (t)| ∀t, x initial, urelevant (meaning all relevant input signals of the system). This condition allows to neglect the dynamic effect of equation (30) in equation (29), if the time scales of the non-dominant systems are of the same order or smaller and in particular if x˙ nd is small. Consequently, 8 Via this condition, the case of different amplitude scaling in x and d x nd is covered.

x˙ nd = A21 x d + A22 x nd + B 2 u + F 2 g(x d , u), where x˜˙ d = x˙ d − A12 A−1 ˙ nd . (34) 22 x Now, from equation (34), all conditions can be read when a combined reduction, based on singular perturbation and weak coupling applies: (i) all coupling constants from the non-dominant to the dominant system are small (special case of reduction for a weakly coupled system). We emphasize that the case of same orders of time scales (or frequencies in the special case of linear systems) in the retained and the omitted dynamics can be tackled; (ii) A−1 ˙ nd is small (special case of singular 22 or x perturbation); (iii) both special cases interfere adequately and cause the vector A12 A−1 ˙ nd to be small. This interference 22 x allows condition (i) and (ii) to be violated to some extent, but in a complementary fashion: this is the essential case to which the applicability of both methods is extended. In all three cases, the reduced order system can be computed as in case of singular perturbations, equation (32). The method is termed ‘reduction by compensation’ because in the reduced order system the above listed effects are all compensated by additive corrections to the parameters of the vector differential equation of the retained states. Now, the most essential consequence of equation (34) is that it suggests a cost functional which is suited to answer the question, if one of the conditions (i) to (iii) is fulfilled. Remark (Applicability criterion for the reduction by compensation). – The system (29), (30) can be reduced to equation (32), if −1 T ˙ ˙T trace A12 A−1 22 X nd X nd (A12 A22 ) J=  1, (35)

˙ Td ˙ dX trace X i.e. the relative time-averaged contribution of A12 A−1 ˙ nd 22 x to x˙ d , is small. Notice that the cost functional (35) only requires simulation data of the original system and a suggestion of the dominant states. Both techniques are developed and will be further developed within the cost functional based approach.

M. Kordt, H. Lusebrink / Aerosp. Sci. Technol. 5 (2001) 55–68

The proof is obvious: introducing the time integral and using (34) yields:

 ∞ trace 0 (x˙˜ d − x˙ d )(x˙˜ d − x˙ d )T dt  ∞

 1, J= trace 0 x˙ d x˙ Td dt

(36)

i.e. the relative time-averaged deviations between x˙ d and x˙˜ d are small. An essential advantage of the criterion is that it is a matrix criterion, i.e. not all entries of the matrices A12 and A22 −1 have to be analyzed individually whether they constitute small constants for the whole operating domain. This is done via the criterion in one shot, i.e. simultaneously for all matrix entries and thereby for all coupling and time constants under consideration. In this sense, the criterion also supports and confirms the choice or the definition of small constants within singular perturbation [14]. Via linearization x˙ i = Ai x i + B i u of the system at a sufficiently dense set of operating points i, the cost functional can even be evaluated without simulations. The index i is dropped in the following. The idea is to consider a sufficiently dense set of initial conditions x 0 and step inputs of height u0 for each operating point i. Via partitioning x: x d = [I d 0]x,

x nd = [0 I nd ]x,

(37)

where I d and I nd are unity matrices of dimension n˜ and n − n, ˜ the integrals in the numerator and the denominator can be evaluated via the integral ∞

∞ x˙ x˙ dt = (Ax + Bu)(Ax + Bu)T dt. T

0

(38)

0

Now, solving the linearized differential equation at an arbitrary operating point i for a step input of height u0 and the initial condition x 0 , allows us to evaluate equation (38) without simulation: ∞ x˙ x˙ T dt = AS x 0 ,x 0 AT + AS x 0 ,Bu0 0

+ S Bu0 ,x 0 AT + S Bu0 ,Bu0 ,

(39)

63

if, and only if, all eigenvalues of A are stable where W ∗,• is defined in analogy to S ∗,• , e.g., W Bu0 ,Bu0 = Bu0 uT0 B T . Thereby, the cost functional can be written as         0 −1 T   trace A12 A−1 A ) $ (A 12 22 22 0 I nd   I nd    J= ,   Id  trace I 0 $   d 0 

Σ = AS x 0 ,x 0 AT +AS x 0 ,Bu0 +S Bu0 ,x 0 AT +S Bu0 ,Bu0 . It only contains the chosen vectors x 0 , u0 and the system matrices, so that it can be evaluated very efficiently by comparison with a coupling analysis based on simulations of the two coupled subsystems. The consistency of the formula is obvious from the quadratic term in A12 A−1 22 in the numerator. Finally, one can overcome the restrictions (R1) and (R2) by combining the method with the cost functional based approach: Ad (R2): the matrix W , computed according to (25) and (28), is used to approximate g(x, u) by g(W x d , u). Care has to be taken in case of transition layers [7], i.e. the problem of non-unique solutions of equation (31). Nevertheless, one should analyze in that case, if the problem can be approximately tackled by the reconstruction matrix at the level of the time-averaged approximation, based on representative signals of the considered operating domain, according to equation (25). Ad (R1) and (R2): use dominance analysis to find zd and znd and a corresponding reconstruction matrix. A particular advantage of restriction (R2) and overcoming it only approximately by the matrix W is that g(x, u) is not required to consist of analytically closed functions, but allows nonlinearities to be given in terms of data arrays. Moreover, hysteresis can be considered. The structural dynamic order reduction of section 3 is not only fully covered by this approach, but the terms according to equation (12) do not have to be neglected. Their impact on the dominant trajectories can be systematically approximated in the reduced order.

where ∞ S Bu0 ,Bu0 =

6. Application of the new nonlinear order reduction methods to the derivation of a synthesis model for structural dynamic controller design

T eAt Bu0 (Bu0 )T eAt dt.

0

The other integrals are abbreviated similarly: S x 0 ,x 0 , S x 0 ,Bu0 , S Bu0 ,x 0 . These four integrals S ∗,• can be evaluated using the so-called Lyapunov equation [28] AS ∗,• + S ∗,• AT = −W ∗,• ,

(40)

Consider structural control of fuselage bending modes. In case of a vanishing frequency gap between the elastic and the rigid body motion, an integral controller has to be designed, which incorporates the flight mechanical and the mode control [15]. For the controller design, a structural dynamic aircraft model of order 32 (aircraft type B52E [4,26]), is reduced to an order of n˜ = 16. The

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corresponding equations of motion including all numerical values are given in [4,26]. The goal is a model which adequately represents the dynamics of the lateral and the longitudinal load factors and the standard flight mechanical quantities. Via standard FEM, unsteady aerodynamics and linear order reduction methods, the model has already been condensed to an order of n = 32, see [4, 26]. An extensive structural dynamic controller design programme for the B52E including load alleviation and mode suppression was started as early as in 1965 by the US Air Force 9 [4]. The first task within the ‘cost functional based reduction’ is choosing the set of input signals, cf. ‘User Option: selection of input signal/initial condition pairs’ in section 4. Here, several combinations of ramps and doublets, which are well-suited for identification of aerodynamic parameters [27] are used for all control surface deflections. Moreover, a combined 2.5g- 10 and turn-maneuver is considered. This maneuver excites most of the nonlinear terms, in particular the nonlinear coupling terms between the longitudinal and the lateral motion. Next, the dominant states have to be found, using the cost functional of and equation (25), cf. ‘User Option: selection of dominant states’ in section 4. Here, the matrix S x allows the engineer to weight the significance of individual states. An obvious choice are the inverse amplitudes of the individual states to compensate different scales. Further choice is based on physical and engineering knowledge, e.g., that for the given aircraft, longitudinal modes are more significant than lateral ones. Here, the dominance analysis suggests a reduction to an order of n˜ = 16. Then, the reduced order system of order n˜ = 16 should be computed as a control-oriented realization, cf. ‘User Option: special realizations’ in section 4. The cost functionals 11 equations (18) and (19) are used for a good time-averaged approximation of the trajectories of the original system, cf. ‘User Option: Cost functional selection’ in section 4. A time-weighting using Q or QW (cf. equations (18) and (19)) turned out to be unnecessary here. The control-oriented realization here has to have the following properties: to have no additional nonlinear and no linear coupling terms from the longitudinal to the lateral motion in the reduced order system, the correspond˜ = 0 are introduced, cf. ‘User Oping constraints GEH tion: selection of constraints’ in section 4. This is essential for the structural dynamic controller design, because 9 The reason was the following: while on a low-level mission over the western USA, a B52H encountered severe turbulence of an estimated peak velocity of 35 m/s. The intensity of the gust saturated the yaw channel of the standard stability-augmentation system of the aircraft so that the acceleration of the vertical fin was such that almost 80% of the vertical fin broke off in flight. 10 g is the constant of gravity. 11 The advantage of these cost functionals is that they can be analytically optimized, cf. ‘User Option: selection of optimization procedure’ in section 4.

the standard approach is, first to linearize the model at a set of trim points, yielding decoupled, i.e. separate models for longitudinal and lateral motion. This separation into two models considerably simplifies the standard robust flight mechanical controller design. In order to design a structural dynamic controller which maintains the flight mechanical properties of an aircraft, it is reasonable to follow the same line. First step: design as many features as possible via linear methods, using linearized models of the reduced nonlinear system. Notice that the nonlinear order reduction here ensures correct stationary performance of the linearized model by covering the stationary aeroelastic effects correctly. Second step: analyze and appropriately extend the controller, using the nonlinear (reduced order) synthesis model and finally do the assessment on the original model. In order to accelerate the reduction process, the final assessment of a reduced order system for a large set of input signals and comparisons between the original model and different reduced order models can be done at different levels of precision via cost functionals. Cost functionals can be considered for (i) a whole set of N inputs, as in the case of equations (18) and (19), e.g.  T1 +···+TN J=

0

2 dt x d (t) − x(t) ˜

 T1 +···+TN 0

x d (t) 2 dt

,

(41)

(ii) one simulation  Tα Jα =

0

2 dt x d (t) − x(t) ˜ ,  Tα 2 0 x d (t) dt

(42)

and (iii) the individual states of one simulation, e.g.:  Tα (xd,i (t) − x˜i (t))2 dt Ji = 0  T . (43) α 2 0 (xi (t)) dt The cost functionals are illustrated in parallel coordinates: the values of the cost functionals can be read from the y-axis. On the x-axis in the case of (i) the index runs which numbers the reduction method or the iteration number within one reduction process with a fixed method. In the case of (ii), the index of the simulation and in the case of (iii), the index of the state are given on the x-axis. In case of (ii) for comparison of methods or iteration steps and in case of (iii) for comparison of many test signals, several values occur at each index on the x-axis (figures 1 and 2). Many test signals have to be analyzed in particular at the very beginning of an order reduction process and in the final assessment. To illustrate that conveniently, both parallel bars ( figure 1) and polygons ( figure 2) for graphical comparison are useful. Figure 1 shows a comparative assessment of the reduced order system of order 16 for ten important states in terms of parallel bars for four test signals. Polygons ( figure 2) are curves connecting the maximums of the bars

M. Kordt, H. Lusebrink / Aerosp. Sci. Technol. 5 (2001) 55–68

Figure 1. Analysis of the reduced systems for four input signals. The cost functionals Ji measure the time-averaged deviations according to equation (43) for the states i of the rigid body motion and the first bending mode: 1 = (u, 2 = w, 3 = q, 4 = θ , 5 = v, 6 = p, 7 = r, 8 = φ, 9 = ε1 , 10 = ν1 . Cost functional based reduction: left bar, linear reduction: center bar, reduction by compensation: right bar. In the upper left figure, a multi doublet is considered as the elevator input signal. This signal was also used for the computation of the reduced order system. In the case of the other three figures, the input signals have not been used for the cost functional based computation of the reduced order system.

Figure 2. Comparison of reduction results in parallel coordinates for an early iteration step of a cost functional based reduction for 20 test signals. Via polygons, a large number of test signals can be considered that were not used to compute the reduced order system. Bad performing signals can easily be picked for the next iteration step of the order reduction process. The cost functionals measure the time-averaged deviations according to equation (43) for the following states i of the rigid body motion and the first bending mode: 1 = (u, 2 = w, 3 = q, 4 = θ , 5 = v, 6 = p, 7 = r, 8 = φ, 9 = ε1 , 10 = ν1 . Here, the three marked signals exceed a deviation of 20% for one or more states and are therefore used in the next iteration step.

65

from left to right via straight lines for the simulation of one test signal in the case of (iii) and for one reduction method or one iteration step within the ‘cost functional based reduction process’ in the case of (ii). Now, each reduction method or each reduction step in case of (ii) and each test signal simulation in case of (iii) corresponds to one curve or polygon. Hereby, many simulations can be depicted in one diagram. In case of (ii), the worst and the best iteration step or reduction method for each test signal (figure 2) and in case of (iii), the worst test signal for each state of the reduced order system, can be easily grasped by considering the envelope of all polygons. Figure 2 shows an assessment at the very beginning of the order reduction process. Concerning the deviations between the original and the reduced order system, three input signals constitute the upper envelope of all polygons. They are chosen for the computation of the reduced order system. They correspond to the bold polygons, marked by symbols in figure 2. Now, the results of the order reduction of the structural dynamic aircraft model, having used both reduction methods combined with the nonlinear dominance analysis, are briefly presented in terms of parallel bars and time histories. In figure 1, usually measured flight mechanical quantities and the longitudinal bending mode with the lowest frequency are depicted. These are the controlled quantities for integral flight mechanical and mode control. In the upper left of figure 1, a multi doublet [27] was chosen as elevator input. According to its frequency characteristic, this signal has turned out to be well-suited for parameter identification of aircraft. As system identification and order reduction are very closely related, this signal was used for the computation of the reduced order system. The three other test signals, which are depicted in the other three subplots of figure 1, were not considered for the computation of the reduced order system. The ‘cost functional based reduction’ and the ‘reduction by compensation’, which incorporates the structural dynamic model reduction of section 3, are compared with a reduction method for linear models [22]: here, a linear reduction method was applied to the linear part of the model and the nonlinearities of the original system were transferred unchanged to the reduced order system. It turned out that the ‘cost functional based order reduction’ and the ‘reduction by compensation’ yield much better results than the linear reduction (figure 1) shows the necessity and the benefits of nonlinear order reduction methods. In figure 3 (left), the corresponding time domain simulation of the difference velocity (u = u − u0 for the cost functional based reduction is shown, where u0 is the trim velocity at the starting time. Figure 3 (right) shows the necessity of constraints by giving the results of the lateral velocity v for an order reduction without constraints. Figure 3 (left) also shows that the constraints have no impact on the longitudinal performance.

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Figure 3. Comparison of the time histories of the velocity components in body-fixed x- and y-directions for a multi doublet as the elevator input, i.e. a pure longitudinal maneuver. Left: Cost functional based reduction (–·) with the above mentioned constraints for coupling suppression compared with the original system (—). Right: Cost functional based reduction (–·) without the above mentioned constraints compared with the original system (—).

Figure 4. Comparison of longitudinal and lateral load factor nz , ny for a combined turn- and 2g-maneuver: original system (–·), cost functional based reduction (–), reduction by compensation (- -), linear reduction (··). Top line: complete time histories, bottom line: initial dynamics.

In figure 4, the time histories of the longitudinal and the lateral load factors ny , nz for a combined turnand 2g-maneuver are considered. This is an assessment maneuver to confirm that it was sufficient to consider only the combined turn- and 2.5g-maneuver for the computation of the reduced order system. Both nonlinear methods yield a better approximation of the transient amplitudes of nz than the linear method.

7. Conclusion Two general nonlinear order reduction methods have been developed: The ‘cost functional based model reduction’ and the ‘reduction by compensation’, which extends the singular perturbation method to weakly coupled systems and incorporates a structural dynamic model reduction for a unified gust and maneuver loads analysis

M. Kordt, H. Lusebrink / Aerosp. Sci. Technol. 5 (2001) 55–68

model. Both methods were applied to a structural dynamic model reduction problem. It could be shown that the methods yield very low order nonlinear models suited for integral flight mechanical and structural dynamic controller design. The achieved accuracy is higher than that achieved by standard linear reduction methods. From the viewpoint of applying the new model reduction methods there is no difference between a model reduction for structural dynamic control and a model reduction for loads analysis. As the loads analysis allows to consider more states in the reduced order model than structural control, an even better reduction result can be achieved for a loads analysis. Availability of several nonlinear order reduction methods is necessary because either the methods are restricted to certain classes of systems (as in the case of the reduction by compensation) or the reduced order systems are only optimal in the sense of the considered cost functionals, but not in a universal sense (as in the case of the cost functional based reduction), which are standard problems in nonlinear system theory. The methods can be extended to nonlinear models with varying parameters by tackling them as multi-models, i.e. by introducing a grid in the varying parameters. Here, constraints can be used to transfer certain properties from the original system with varying parameters to the reduced multi-model system. For further evaluation of the methods, they should be applied to models in which the flexibility of the structure plays a more significant role, e.g., in the case of nonlinear vibrations.

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