Worst Case Analysis of a Saturated Gust Loads Alleviation System

Transcription

1 Worst Case Analysis of a Saturated Gust Loads Alleiation System Andreas Knoblach Institute of System Dynamics and Control, German Aerospace Center (DLR), Weßling, Germany Harald Pfifer and Peter Seiler Department of Aerospace Engineering & Mechanics, Uniersity of Minnesota, Minneapolis In order to determine a guaranteed upper bound for orst case 1-cosine gust loads of flexible aircrafts, the usage of the orst case energy-to-peak gain of linear parameterarying (LPV) systems has been recently proposed. A limitation of this approach is that it cannot deal ith nonlinearities. This paper uses integral quadratic constraints (IQCs) to circument this restriction and to consider the saturation of a gust loads alleiation system. Based on the dissipation inequality frameork, a linear matrix inequality constraint hich bounds the orst case energy-to-peak gain of saturated LPV systems is gien. In order to reduce the conseratism, an iteratie procedure to refine local IQCs is proposed. The conseratism of this approached is analyzed at the example of a to-dimensional thin airfoil in combination ith a saturated gust loads alleiation system. I. Introduction In order to certify a ne aircraft, it must be proed that it can ithstand loads caused by turbulence and gusts. 1 According to the CS-25 of the European Aiation Safety Agency (EASA), 2 to types of excitations hae to be considered: discrete 1-cosine gusts and continuous turbulence. In the first case, an aircraft model is excited ith a single 1-cosine gust profile and the model outputs, e. g. the ing root bending moment (WRB) or the ertical accelerations, are computed and the simulation is repeated for seeral gust lengths. Moreoer, the complete process has to be performed for many flight points such that the entire flight enelope (defined by elocity, altitude, loading etc.) is coered. Finally, the maximum and minimum peak of eery output for all these simulations define the limit loads due to discrete gusts. The other type of excitation namely continuous turbulence considers the stochastic nature of turbulence. Here, the on Karman ind turbulence model is used to excite the aircraft and the results are ealuated by stochastic means. The limit loads are deried from the root mean square (RMS) alue of the model response. A detailed description of discrete gusts and continuous turbulence scenarios can be found in Ref. 1. Since these millions of simulations are ery time consuming, there is a need for fast and reliable algorithms to identify orst case flight points and to compute maximum loads. To that end, it has been recently proposed in Ref. 3 to use the orst case energy-to-peak gain or mathematically speaking the induced L 2 -L norm. This approach allos an extremely fast computation of an upper bound for the peak loads. Hoeer, this method is restricted to parametrically uncertain or linear parameter-arying (LPV) systems. Since a nonlinear gust load alleiation system has to be considered in many cases, the applicability of using the orst case energy-to-peak gain is limited. The major nonlinearity of gust load alleiation systems is saturation. Since the interconnections of linear time inariant (LTI) systems and saturation occur ery often in practice, they are intensiely studies in the literature. Most approaches can be put into the context of integral quadratic constraints (IQCs), see e. g. Refs. 4, 5, and 6. Recent results in Ref. 7 extend the IQC frameork to LPV systems, hich allos the analysis of saturated LPV systems. Research Associate, andreas.knoblach@dlr.de. Postdoctoral Associate, hpfifer@umn.edu. Assistant Professor, seiler@aem.umn.edu. 1 of 12

2 The aim of this paper is to inestigate ho IQCs can be used for the analysis of a saturated gust loads alleiation system. The paper is structured as follos: The interconnection of LPV an system and saturation is described in Section II. A brief summary on IQCs is proided in Section III. The focus lies on the description of saturation by IQCs. The analysis interconnection and an upper bound for the orst case energy-to-peak gain of saturated LPV systems is gien in Section IV. Additionally, an iteratie procedure to refine local IQCs is proposed. The effectieness of the proposed method is inestigated in Section V. This inestigation is pursued at the example of a to-dimensional thin airfoil in combination ith a gust load alleiation system. II. Saturated LPV Systems e sat G ρ d σ σ σ σ (a) Feedback interconnection of an LPV system and saturation (b) Saturation nonlinearity Figure 1. Saturated LPV system The paper considers a system under saturation gien by the feedback interconnection, denoted F u (G ρ, sat), of an LPV plant G ρ and a saturation nonlinearity sat as shon in Figure 1(a). The saturation = sat() is defined by if < σ = sat() = (1) sgn() σ otherise, here σ denotes the saturation limit. The saturation nonlinearity is depicted in Figure 1(b). LPV systems are a class of linear systems hose state space matrices dependent on a scheduling parameter ector ρ: R R nρ. The parameter is assumed to be a continuously differentiable function of time and the admissible trajectories are restricted, based on physical considerations, to a compact subset P R nρ. In addition, the parameter rates of ariation ρ are assumed to lie ithing the hyperrectangle Ṗ defined by { } Ṗ := q q i ν i i {1,..., n ρ }, (2) here ν i are nonnegatie numbers. The set of all admissible trajectories is defined as { } A := ρ ρ(t) P, ρ(t) Ṗ t. (3) The state-space matrices of an LPV system are continuous functions of the parameter, e. g. A: P R nx nx. An n th x order LPV system G ρ is defined by (t) ẋ(t) = A(ρ(t))x(t) + B(ρ(t)) (4) d(t) (t) = C(ρ(t))x(t) + D(ρ(t)). (5) (t) e(t) Throughout the remainder of the paper, the explicit dependence on t is occasionally suppressed to shorten the notation. d(t) III. Integral Quadratic Constraints for Saturation This section describes constraints on the saturation function that can be incorporated in the input/output analysis. The constraints are taken from Ref. 4 and fit in the more general frameork of IQCs, hich bound the input/output signals of a bounded causal operator = (). These constraints can be expressed in the frequency domain as ell as in the time domain. First, a definition for an IQC in the frequency domain is gien. 2 of 12

3 Definition 1: Let Π : jr C (n+n) (n+n) be a measurable Hermitian-alued function. A bounded, causal operator : L n 2e Ln 2e satisfies an IQC defined by Π if the folloing inequality holds for all Ln 2, = (), and T ˆ(jω) ˆ(jω) Π(jω) dω. (6) ŵ(jω) ŵ(jω) In (6), ˆ and ŵ are Fourier transforms of and, respectiely. The multiplier Π can be factorized as Π(jω) = Ψ(jω) MΨ(jω) (7) hich allos to connect the frequency domain formulation to a time-domain formulation. 4, 8, 9 This leads to Definition 2 for an IQC in the time domain. It should be noted, that there are some technical restrictions in the factorization of IQCs, see Refs. 4, 8, and 1. Hoeer, these restrictions do not apply to the IQCs considered in this paper. Definition 2: Let M be a symmetric matrix, i. e. M = M T R nz nz, and Ψ a stable linear system, i.e. Ψ RH nz (n+n). A bounded, causal operator : L n 2e Ln 2e satisfies an IQC defined by (Ψ, M) if the folloing inequality holds for all L n 2, = (), and T here z is the output of the linear system Ψ T z(t) T Mz(t)dt, (8) ẋ Ψ = A Ψ x Ψ + B Ψ1 + B Ψ2, x Ψ () = (9a) z = C Ψ x Ψ + D Ψ1 + D Ψ2. The notation IQC(Ψ, M) is used if satisfies the IQC defined by (Ψ, M). (9b) Ψ z Figure 2. Graphical interpretation of the IQC Figure 2 proides a graphical interpretation of a time domain IQC. The input and output signals of are filtered through Ψ. If IQC(Ψ, M), then the output signal z satisfies the (time-domain) constraint in (8) for any finite-horizon T. Three classical IQCs from Ref. 4 for saturation are gien in the folloing subsections. III.A. Memoryless Sector Bounded Nonlinearity σ σ σ σ Figure 3. Sector constraint ( ) for the saturation nonlinearity ( ) It is easy to see in Figure 3, that saturation lies in the sector defined by to lines ith slope α = and β = 1, denoted as sector, 1. More general, a memoryless nonlinearity Φ: R R lies in sector α, β if ((t) α(t)) (β(t) (t)) (1) 3 of 12

4 holds for all (t) R and (t) = φ((t)). This can be equialently expressed as T (t) 2αβ (t) α + β α + β (t). (11) 2 (t) Since (11) has to hold pointise in time, it also holds hen integrated oer time T T (t) 2αβ (t) α + β α + β (t) dt (12) 2 (t) for any T. Note that the conerse is not true. Hence, the saturation function satisfies the IQC defined by 2αβ α + β Ψ = I 2 and M =, (13) α + β 2 ith α = and β = 1. Note, that the sector IQC holds also for time arying, memoryless nonlinearities (t) = φ((t), t). III.B. Popo IQC In order to capture the time inariance of saturation, the Popo IQC can be used. Consider a memoryless and time inariant nonlinearity (t) = φ((t)), here φ: R R is a continuous function ith () =, and and are square integrable. Then, and satisfy the IQC defined by Π(jω) = ± jω jω. (14) Hoeer, since Π is not bounded on the imaginary axis, it is not a proper IQC. As a remedy, the loop transformation from Figure 4 can be used. The operator satisfies the IQC defined by jω 1 jω Π(jω) = ± jω, (15) 1+jω In order to reert the lo-pass filter, the non-proper filter s + 1 must be added to the interconnection (see Figure 4). This requires that the plant is strictly proper. The combination of the Popo IQC and of the sector constraint IQC yields the classical Popo criterion. 4 s s+1 Ψ z Figure 4. Popo IQC: loop transformation to yield a proper IQC III.C. Zames-Falb IQC Finally, saturation is described by a monotonic, slope restricted, and odd function hich is the last property satisfying a classical IQC. Consider (t) = φ((t)), here φ: R R is a monotonic and odd function and its slope is bounded by φ, k for a constant k >. Then and satisfy the IQC 1 + H(jω) Π(jω) = (16) 1 + H(jω) (2 + H(jω) + H(jω) ) /k here H(s) is an arbitrary rational transfer function hose impulse response has an L 1 norm not larger than one. Details can be found in Ref of 12

5 IV. Robust Performance Analysis of Saturated LPV Systems IV.A. Uncertain LPV Systems In Ref. 7, it is proposed to describe an uncertain LPV system by the feedback interconnection of an LPV system G ρ ith an nonlinear and/or uncertain operator. This corresponds to an upper linear fractional transformation (LFT), hich is denoted F u (G ρ, ). The exact relation = () is replaced by an IQC such that IQC(Ψ, M). The dynamics of the resulting analysis interconnection (see Figure 5) is ruled by ẋ = A(ρ)x + B 1 (ρ) + B 2 (ρ)d z = C 1 (ρ)x + D 11 (ρ) + D 12 (ρ)d e = C 2 (ρ)x + D 21 (ρ) + D 22 (ρ)d. (17a) (17b) (17c) Note that x includes the states of G ρ and Ψ. In the folloing analysis, the uncertainty is remoed and is treated as an external signal constrained by (8). Ψ z e G ρ d Figure 5. Analysis interconnection IV.B. Energy-to-Peak Gain of Uncertain LPV Systems Although the induced L 2 -L 2 norm (or for LTI system also H norm) is commonly used in the robust performance analysis frameork, the induced L 2 -L is more suited for loads analysis since it allos to compute an upper bound for the output peaks. The induced L 2 -L norm of an uncertain LPV system is defined as In (18), x L2 = F u (G ρ, ) L2 L := sup ( x(t) T x(t) ) 1 IQC(Ψ,M) sup d L 2\{} ρ A 2 dt and x L = sup t e L d L2. (18) ( x(t) T x(t) ) 1 2 represent the L 2 (or energy) signal norm and the L (or peak) signal norm, respectiely. The notations x L 2 and x L are used to refer to signals ith finite L 2 and L norm. The induced L 2 -L norm represents hence the orst case energy-to-peak gain oer all uncertainties that satisfy the IQC (Ψ, M) and oer all admissible parameter trajectories. Using the interconnection in (17), a theorem based on the dissipation inequality frameork can be formulated hich bounds the induced L 2 -L norm. The analysis leads to conditions that include the parameter and the parameter rate at a single point in time, i. e. (ρ(t), ρ(t)). In order to emphasize that the conditions depend only on the sets P and Ṗ, the parametric notation (p, q) P Ṗ is introduced. The folloing theorem is a minor modification of the orst case condition for the induced L 2 -L 2 norm gien in Ref. 7. Theorem 1: The induced L 2 -L norm from d to e of an uncertain LPV system F u (G ρ, ) L2 L ith D 21 (ρ) = and D 22 (ρ) = is smaller than a performance index γ if the interconnection is ell posed for all IQC(Ψ, M) and if there exists a continuous differentiable Lyapuno matrix function X : P R nx nx ith X T = X and a scalar λ > s. t. for all (p, q) P Ṗ (19) X(p) >, (2a) 5 of 12

6 A(p) T X(p) + X(p)A(p) + n ρ X(p) i=1 p i q i X(p)B 1 (p) X(p)B 2 (p) B 1 (p) T X(p) B 2 (p) T X(p) γi C 1 (p) T +λ D 11 (p) T M D 12 (p) T C 1 (p) D 11 (p) D 12 (p) <, (2b) X(p) C 2 (p) Proof: The proof is based on defining a storage function V : R nx P R + C 2 (p) T >. (2c) γi V (x, ρ) := x T X(ρ)x. (21) Left and right multiply (2b) by x T T d T and its transpose, respectiely, to sho that V satisfies the dissipation inequality V (t) + λz(t) T Mz(t) γd(t) T d(t) (22) The dissipation inequality can be integrated from t = to t = T ith the initial condition x() =. Along ith the IQC condition (8) and λ > this leads to T V (T ) γ d(t) T d(t)dt. (23) A Schur complement on (2c) and the left and right multiplication ith x T and x yield Combine (23) and (24) and take the supremum. r. t. T to conclude A detailed proof of the analogous induced L 2 norm can be found in Ref γ e(t )T e(t ) V (T ). (24) e L γ d L2. (25) An upper bound for the orst case energy-to-peak gain can be obtained from Theorem 1 by soling a semidefinite program in the ariables X, λ and γ. In order to arrie at numerically tractable conditions, a gridding based approximation of the parameters space is used and basis functions are assigned for X(p), see Refs. 12 and 13 for details. Note that for simplification the theorem is only gien in terms of a single IQC (Ψ, M). It can be easily extended to handle multiple IQCs. 7 IV.C. Analysis using Local IQCs In Section III, standard constraints from literature 4, 11 for the saturation function hae been gien. While all these IQCs guarantee global robust performance of a system under saturation, all of them are independent of the saturation leel. Hence, they are expected to be ery conseratie for many practical applications. It is ell understood in literature, that performance results that only hold locally can greatly improe the analysis of systems under saturation by incorporating the actual saturation leel, see Refs. 14, 15, and 16. If the global performance requirement is relaxed to only consider local performance, a modified, less conseratie ersion of the sector bound IQC can be used. The approach pursued in this paper is based on the notion of local analysis of IQCs. 17 There, a general frameork for local analysis of systems under perturbations described by IQCs is gien. In this paper, the specific nature of the perturbation, namely the saturation function, is exploited to obtain local conditions. The conditions are similar to the IQCs for saturations gien in Ref. 5. The global sector constraint for the saturation function is defined by the sector, 1. It is easily seen that the sector bound holds independent of the size of the input, see Figure 6. If an L norm bound on the input is knon, i.e. R, then a refined sector constraint can be gien. Instead of checking oer the hole sector, 1, it is sufficient to only consider the smaller sector σ/r, 1. Recall that σ is the saturation 6 of 12

7 σ R σ R Figure 6. Local sector constraint IQC for saturation: Because of R, only the solid dran part of the saturation ( ) can be reached, but not the dashed one ( ). This allos to decrease the sector from ( ) to ( ). leel. The smaller, local sector is depicted in Figure 6. In order to compute a alid bound on, a bounded disturbance energy d L2 1 is assumed. Then, the induced L 2 -L norm from the performance input d to the saturation input proides a alid bound on. In order to reduce the conseratism of the results, a possibly small bound on should be determined. This can be achieed by the iteratie procedure in Algorithm 1. Algorithm 1 Iteratie refining of a local IQC 1: start ith global IQCs, specifically use the sector, 1 2: repeat 3: compute a bound on the saturation input L R i+1 4: refine the local sector IQC to the sector max(σ/r i+1, 1), 1 5: until R i is conerged 6: compute the induced L 2 -L norm of the performance channel In Line 1, the saturation is described by the globally alid IQCs from Section III. Specifically, the sector, 1 is used. Line 3 consists in computing the induced L 2 -L norm from d to using Theorem 1. In combination ith d L2 1, this norm proides an L norm bound R i on. This bound is used in Line 4 to define the locally alid sector constraint max(σ/r i+1, 1), 1. Lines 3 and 4 are repeated until R i is conerged. In the last step (Line 6), an upper bound for the orst case energy-to-peak gain of the performance channel is computed. Note that in the last step the more common orst case energy gain can be alternatiely considered using Theorem 2 in Ref. 7. V. Analysis of a Saturated Gust Loads Alleiation System The proposed approach is used to compute the orst case energy-to-peak gain of a simple aeroseroelastic system. A to degree of freedom airfoil in combination ith a gust load alleiation system is considered. The effects of different IQCs, seeral saturation limits and parameter rate bounds are inestigated. V.A. Aeroseroelastic System The considered aeroelastic system, taken from Ref. 18, is illustrated in Figure 7. The equations of motion are m m x cg ḧ k + h h L = (26) m x cg I α k α α τ. here h denotes the ertical deflection and α the pitch angle. a The lift L and the pitching moment τ are gien by L = 1 2 τ ρ2 Q eig (s) Q gust (s) h αβ, (27) The Theodorsen function Q eig (s) models the relation beteen the aerodynamic forces and the heae, the pitching and the flap moement, denoted h, α, and β, respectiely. The Sears function Q gust (s) maps the a All quantities are normalized and thus dimensionless. gust 7 of 12

8 α b z τ L k α b h x U x ar x cg x hp β k h Figure 7. Aeroseroelastic system ertical gust elocity gust on L and τ. Both transfer functions are non-rational in the Laplace domain. In order to yield a state space realization, R. T. Jones rational approximation is used. The functions depend on the free stream elocity U, the half chord length b, the location of the aerodynamic reference axis x ar, and the hinge position x hp. Q eig and Q gust are explicitly gien in Ref. 18. In this paper, the elocity U is considered as scheduling parameter. Ten equidistantly spaced grid points in the interal U.3,.9 are used. The rate bounds are U.1,.1. The other parameters are compiled in Table 1. Table 1. Model parameters Sym. Explanation Value k h translational stiffness.4 k α rotational stiffness.25 m mass 1. I inertia.25 x cg location of the center of graity.2 ρ air density.8 b half chord length 1. x ar location of the aerodynamic reference axis.2 x hp location of the hinge axis.8 The close loop configuration in Figure 8 is considered. The eighting filter { ẋ W Gust : W = 1 1 x W d (28) gust =x W is used to shape suitable gusts ith unit energy. 3 The performance outputs are the spring forces e e = h k = h h. (29) k α α e α The first oder lo pass filter G act : { ẋ act =x act + u =x act (3) seres as an actuator model. Since the flap deflection β is assumed to be limited, saturation β = = sat() (31) 8 of 12

9 is added. A simple gust load alleiation system hich is based on a gain scheduled (by U ) proportional controller { ( ) gust K : u = U e h (32) e α is used to close the loop. The controller is tuned using systune. 19 The tuning goal is to minimize the ariance of the performance output. Since the parameter dependence of the resulting state space model is highly inoled, it is not explicitly gien. Hoeer, it has ten states. A Bode diagram of the open and the nominal closed loop (i. e. ithout saturation) is depicted in Figure 9 for different parameter alues. d W gust gust e h u G Act e α K Figure 8. Closed loop of the aeroseroelastic system db(gρ) 4 8 Disturbance heae load f/hz 4 8 Disturbance pitch load f/hz Free stream elocity Figure 9. Bode diagram of the aeroseroelastic system: ( ) represents the open loop and ( ) the nominal closed loop. The color indicates the free stream elocity. V.B. Comparison of IQCs The effect of different IQCs is inestigated using the LTI model corresponding to the highest free stream elocity. The globally alid sector, 1 is considered separately, in combination ith the Popo IQC and in combination ith three different parameterizations of the Zames-Falb IQC. The Zames Falb IQC is parameterized by a first order lo pass filter 1 H(s) = 1 T ZF s + 1, (33) here the three time alues for the constants T ZF {.1,.1, 1} are used. Finally, the combination of the sector IQC, the Popo IQC, and all three Zames-Falb IQCs is considered. The results are compiled in Table 2. The best result is achieed using the sector IQC and the Zames-Falb IQC ith T ZF =.1. The results cannot be improed by adding the Popo IQC or combining seeral Zames-Falb IQCs. For that reason, the sector and the Zames-Falb IQCs ith T ZF =.1 are used belo. 9 of 12

10 Table 2. Results for seeral IQC combinations Sec Sec/Pop Sec/ZF Sec/ZF Sec/ZF Sec/Pop/ZF T ZF all Norm V.C. Conergence of Local IQCs The conergence behaior of Algorithm 1 is inestigated in this subsection. The LPV model from Section V.A is used and the six saturation limits σ {, 3,... 15} are considered. The Lyapuno function is parameterized by a quadratic function. The results are depicted in Figure 1, here the induced L 2 -L norm is plotted as a function of the iteration. In the left subplot, the norm from the disturbance to saturation input can be seen and, in the right one, the norm of the performance channel. First of all, a fast conergence after the third iteration can be recognized. The upper bound for the orst case energy-to-peak gain from the gust to the saturation input can be reduced by only 2 %. Hoeer, the proposed algorithm allos to reduce the upper bound for the performance channel by up to 69 % (depending on the saturation limit). In order to estimate the conseratism induced by the description of saturation ith IQCs, the robust analysis results are compared to standard LPV analysis results for the open and the nominal closed loop. A saturation limit of zero corresponds ith the open loop. Using Algorithm 1 for a zero saturation limit leads to conseratie results, as can be seen in Figure 1 (compare ( ) and ( )). The reason for this is that a zero saturation limit leads alays to the sector, 1. On the contrary, a ery high saturation limit corresponds to the nominal closed loop. Specifically, an L norm of the saturation input less than the saturation limit results in a neer saturated system. Consequently, the sector conerges to 1, 1 hich represents the smallest possible sector. In the considered example, the upper bound for the robust norm for a saturation limit of σ = 15 ( ) becomes een smaller than the result for the nominal closed loop ( ). The reason for this unexpected result is that the nominal LPV analysis is already conseratie. In case of the robust analysis, the additional state of the Zames-Falb IQC introduces additional decision ariables hich reduce the conseratism. Hoeer, using a third order test function for the Lyapuno matrix function ( ) yields consistent results. L2 L norm Disturbance saturation input Iteration Disturbance performance output Iteration Figure 1. Conergence of local IQCs: The induced L 2-L gain is depicted as function of the iteration number ( ) for seeral saturation limits indicated by the color. The results are compared to the open ( ) and the nominal closed ( ) loop results. In case of the closed loop, a cubic test function for the Lyapuno matrix ( ) is additionally considered Saturation limit V.D. Influence of the Parameter Rate Bounds In a last example, the influence of the parameter rate bounds is analyzed. To that end, Algorithm 1 is used to compute an upper bound of the induced L 2 -L norm for the saturation limits σ {, 3,..., 15} and for the rate bounds U.1ν,.1ν ith ν {.,.2,..., 1.}. The results are illustrated in Figure of 12

11 Regarding the saturation limits, the results from the preceding subsection hold for all parameter rate bounds. For high saturation limits, the norm bounds conerge to the nominal LPV analysis results ( ). As expected, reducing the parameter rate bounds leads to a smaller upper bound. If the rate bound approaches zero, the robust analysis conerges to the maximum LTI norm. r. t. all ten nominal closed loop models ( ). The latter norm is knon to be not conseratie and proides consequently a loer bound for the orst case performance. Disturbance performance output L2 L norm Saturation limit Rel. rate bound Figure 11. Influence of the parameter rate bounds: The induced L 2-L norm is computed for seeral rate bounds and saturation limits and compared to norms of the nominal closed loop ( ). Finally, ( ) indicates the maximum LTI norm of the nominal closed loop. Summarizing it can be said that the results are conseratie if a lo saturation limit is considered and the system is most of the time saturated. On the contrary, for relatiely high saturation limits, Algorithm 1 conerges to the nominal case and is hence only slightly conseratie. VI. Conclusions The analysis of saturated LPV systems using IQCs as discussed in this paper. In order to reduce the conseratism of the analysis results, an iteratie procedure to refine local IQCs as proposed. This method as intensiely studied at the example of a to-dimensional thin airfoil in combination ith a gust load alleiation system. While this approach orks ell in case of relatiely high saturation limits, the results are supposed to be conseratie if the system is most of the time in saturation. Hoeer, this approach allos to compute guaranteed upper bounds for gust loads of flexible aircraft ith a nonlinear gust load alleiation system. The meaningfulness of the nominal orst case energy-to-peak gain has been already demonstrated in Ref. 3. Whether the conseratism in terms of gust loads analysis is still acceptable or is to large for meaningful conclusions, has to be assessed using a generic aircraft model. This ill be done in future ork. Another point is to include further IQCs from Refs. 5 and 6. References 1 Hoblit, F. M., Gust loads on aircraft: concepts and applications, AIAA, EASA, Certification Specifications and Acceptable Means of Compliance for Large Aeroplanes CS-25, Knoblach, A., Robust Performance Analysis Applied to Gust Loads Computation, Journal of Aeroelasticity and Structural Dynamics, Vol. 3, No. 1, 213, pp Megretski, A. and Rantzer, A., System Analysis ia Integral Quadratic Constraints, IEEE Transactions on Automatic Control, Vol. 42, No. 6, 1997, pp Fang, H., Lin, Z., and Rotea, M., On IQC approach to the analysis and design of linear systems subject to actuator saturation, System & Control Letters, Vol. 57, 28, pp Materassi, D. and Salapaka, M. V., Less conseratie absolute stability criteria using Integral Quadratic Constraints, American Control Conference, IEEE, 29, pp Pfifer, H. and Seiler, P., Robustness Analysis of Linear Parameter Varying Systems Using Integral Quadratic Constraints, American Control Conference, 214, accepted and to be presented. 8 Megretski, A., KYP Lemma for Non-Strict Inequalities and the associated Minimax Theorem, arxic, of 12

12 9 Pfifer, H. and Seiler, P., Robustness Analysis of Linear Parameter Varying Systems Using Integral Quadratic Constraints, submitted to the International Journal of Robust and Nonlinear Control. 1 Seiler, P., Nonlinear Stability Analysis ith Dissipation Inequalities and Integral Quadratic Constraints, submitted to the IEEE Transactions on Automatic Control. 11 Zames, G. and Falb, P., Stability Conditions for Systems ith Monotone and Slope-Restricted Nonlinearities, SIAM Journal on Control, Vol. 6, No. 1, 1968, pp Wu, F., Yang, X. H., Packard, A., and Becker, G., Induced L 2 -Norm Control for LPV Systems ith Bounded Parameter Variation Rates, International Journal of Robust and Nonlinear Control, Vol. 6, 1996, pp Knoblach, A., Assanimoghaddam, M., Pfifer, H., and Saupe, F., Robust Performance Analysis: a Reie of Techniques for Dealing ith Infinite Dimensional LMIs, International Conference on Control Applications Part of the Multi-Conference on Systems and Control, IEEE, 213, pp Hindi, H. and Boyd, S., Analysis of Linear Systems ith Saturation using Conex Optimization, Conference on Decision and Control, IEEE, 1998, pp Hu, T., Lin, Z., and Chen, B. M., An analysis and design method for linear systems subject to actuator saturation and disturbance, Automatica, Vol. 38, 22, pp Hu, T. and Lin, Z., Exact Characterization of Inariant Ellipsoids for Single Input Linear Systems Subject to Actuator Saturation, IEEE Transactions on Automatic Control, Vol. 47, No. 1, 22, pp Summers, E. and Packard, A., L 2 Gain Verification for Interconnections of Locally Stable Systems Using Integral Quadratic Constraints, Conference on Decision and Control, IEEE, 21, pp Bisplinghoff, R., Ashley, H., and Halfman, R., Aeroelasticity, Doer Pubns, Apkarian, P. and Noll, D., Nonsmooth H Synthesis, IEEE Transactions on Automatic Control, Vol. 51, No. 1, 26, pp of 12