L 1 Adaptive Output Feedback Controller for Non Strictly Positive Real Reference Systems with Applications to Aerospace Examples

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1 AIAA Guidance, Navigation and Control Conference and Exhibit August 28, Honolulu, Hawaii AIAA L 1 Adaptive Output Feedback Controller for Non Strictly Positive Real Reference Systems with Applications to Aerospace Examples Chengyu Cao University of Connecticut, Storrs, CT 6269 Naira Hovakimyan University of Illinois at Urbana-Champaign, Urbana, Illinois 6181 This paper presents an extension of the L 1 adaptive output feedback controller to systems of unknown relative degree in the presence of time-varying uncertainties without restricting the rate of their variation. As compared to earlier results in this direction, a new piece-wise continuous adaptive law is introduced along with the low-pass filtered control signal that allows for achieving arbitrarily close tracking of the input and the output signals of the reference system, the transfer function of which is not required to be strictly positive real SPR. Stability of this reference system is proved using small-gain type argument. The performance bounds between the closed-loop reference system and the closed-loop L 1 adaptive system can be rendered arbitrarily small by reducing the step size of integration. Simulations verify the theoretical findings. I. Introduction This paper extends the results of Ref. 1 by considering reference systems that do not verify the SPR condition for their input-output transfer function. Similar to Ref., 1 the L -norms of both input/output error signals between the closed-loop adaptive system and the reference system can be rendered arbitrarily small by reducing the step-size of integration. The key difference from the results in Ref. 1 is the new piece-wise continuous adaptive law. The adaptive control is defined as output of a low-pass filter, resulting in a continuous signal despite the discontinuity of the adaptive law. We notice that adaptive algorithms achieving arbitrarily improved transient performance for system s output were reported in Refs References 1, 17 presented the opportunity to regulate also the performance bound for system s input signal, by rendering it arbitrarily close to the corresponding signal of a bounded reference LTI system. This paper presents the adaptive output feedback counterpart of the results in Refs., 1, 17 without enforcing the SPR condition on the input/output transfer function of the desired reference system, which typically appears in conventional adaptive 18, 19 output feedback schemes. Application of the methodology presented in this paper can be found in Refs. The paper is organized as follows. Section II gives the problem formulation. In Section III, the closed-loop reference system is introduced. In Section IV, some preliminary results are developed towards the definition of the L 1 adaptive controller. In Section V, the novel L 1 adaptive control architecture is presented. Stability and uniform performance bounds are presented in Section VI. In Section VII, simulation results are presented and two aerospace applications are discussed, while Section VIII concludes the paper. The small-gain theorem and some basic definitions from linear systems theory used throughout the paper are given in Appendix. Unless otherwise mentioned, will be used for the 2-norm of the vector. Research is supported by AFOSR under Contract No. FA and NASA under contracts NNX8AB97A and NNX8AC81A. Assistant Professor, Department of Mechanical Engineering, AIAA Member, ccao@engr.uconn.edu Professor, Department of Mechanical Science and Engineering, and AIAA Associate Fellow, nhovakim@illinois.edu Copyright 28 by C. Cao and Naira Hovakimyan. Published by American the American Institute Institute of Aeronautics of Aeronautics and and Astronautics Astronautics, Inc., with permission. 1 of 16

2 II. Problem Formulation Consider the following single input single output SISO system: ys = Asus + ds, y =, 1 where ut R is the input, yt R is the system output, As is a strictly proper unknown transfer function of unknown relative degree n r, for which only a known lower bound 1 < d r n r is available, ds is the Laplace transform of the time-varying uncertainties and disturbances dt = ft, yt, while f is an unknown map, subject to the following assumptions. Assumption 1 There exist constants L > and L > such that the following inequalities hold uniformly in t : ft, y 1 ft, y 2 L y 1 y 2, ft, y L y + L. Assumption 2 There exist constants L 1 >, L 2 > and L 3 > such that for all t : dt L 1 ẏt + L 2 yt + L 3, 2 where the numbers L, L, L 1, L 2, L 3 can be arbitrarily large. Let rt be a given bounded continuous reference input signal. The control objective is to design an adaptive output feedback controller ut such that the system output yt tracks the reference input rt following a desired reference model, i.e. ys Msrs, where Ms is a minimum-phase stable transfer function of relative degree d r > 1. We rewrite the system in 1 as: ys = Msus + σs, 3 σs = As Msus + Asds /Ms. 4 Let A m, b m, c m be the minimal realization of Ms, i.e. it is controllable and observable and A m is Hurwitz. The system in 3 can be rewritten as: ẋt = A m xt + b m ut + σt, 5 yt = c m xt, x = x =. III. Closed-loop Reference System Consider the following closed-loop reference system: y ref s = Msu ref s + σ ref s 6 σ ref s = As Msu refs + Asd ref s 7 Ms u ref s = Csrs σ ref s, 8 where d ref t = ft, y ref t, and Cs is a strictly proper system of relative order d r, with its DC gain C = 1. We notice that there is no algebraic loop involved in the definition of σs, us and σ ref s, u ref s. For the proof of stability of the reference system in 6-8, the selection of Cs and Ms must ensure that Hs = AsMs/ CsAs + 1 CsMs 9 is stable and Gs L1 L < 1, 1 where Gs = Hs1 Cs. This in its turn restricts the class of systems As in 1, for which the proposed approach in this paper can achieve stabilization. However, as discussed later in Remark 3, the class of such systems is not empty. Letting As = A ns A d s, Cs = C ns C d s, Ms = M ns M d s, 11 2 of 16

3 it follows from 9 that where Hs = C dsm n sa n s H d s, 12 H d s = C n sa n sm d s + M n sa d sc d s C n s. 13 A strictly proper Cs implies that the order of C d s C n s and C d s is the same. Since the order of A d s is higher than that of A n s, the transfer function Hs is strictly proper. The next Lemma establishes the stability of the closed-loop system in 6-8. Lemma 1 If Cs and Ms verify the condition in 1, the closed-loop reference system in 6-8 is stable. Proof. It follows from 7-8 that u ref s = Csrs CsAs Msu ref s + Asd ref s/ms, and hence u ref s = CsMsrs CsAsd refs CsAs + 1 CsMs. 14 From 6-7 one can derive y ref s = Asu ref s + d ref s, which upon substitution of 14, leads to y ref s = Hs Csrs + 1 Csd ref s. 15 Since Hs is strictly proper and stable, Gs is also strictly proper and stable and therefore y ref L HsCs L1 r L + Gs L1 L y ref L + L. Using the condition in 1, one can write y ref L ρ, 16 where ρ = HsCs L 1 r L + Gs L1 L 1 Gs L1 L Hence, y ref L is bounded, and the proof is complete. <. 17 IV. Preliminaries for the Main Result Let H s = As/ CsAs + 1 CsMs H 1 s = As MsCs CsAs + 1 CsMs 19 H 2 s MsCs = CsAs + 1 CsMs 2 H 3 s = HsCs/Ms Using the expressions from 11 and 13, H s and H 1 s can be rewritten as: H s = C d sa n sm d s/h d s, H 1 s = C n sa n sm d s C n sa d sm n s /H d s. 22 Since the degree of C d s C n s is larger of C n s by d r, the degree of M n sa d sc d s C n s is larger of C n sa d sm n s by d r. Since the degree of A d s is larger of A n s by d r, while the degree of M n s is larger of M d s by d r, the degree of M n sa d sc d s C n s is larger than that of C n sa n sm d s. Therefore, H 1 s is strictly proper with relative degree d r. We notice from 12 and 22 that H 1 s has the same denominator as Hs, and therefore it follows from 1 that H 1 s is stable. Using similar arguments, it can be verified that H s is proper and stable. Similarly, H 3 s is strictly proper and stable. 3 of 16

4 Let H 3 s L1 = H 1 s L1 r L + H s L1 L ρ + L + γ H 1 s/ms L1 + L H s L1, 1 Gs L1 L where γ > is an arbitrary constant. Since both H 1 s and Ms are stable and strictly proper with relative degree d r, and Ms is minimum-phase, H 1 s/ms is stable and proper. Hence, H 1 s/ms L1 exists. Therefore is bounded. Further, since A m is Hurwitz, there exists P = P > that satisfies the algebraic Lyapunov equation A m P + PA m = Q, Q >. From the properties of P it follows that there exits non-singular P such that P = P P. Given the vector c m P 1, let D be a n 1 n matrix that contains the null space of c m P 1, i.e. c m Dc m P 1 =, 23 and further let Λ = D P. Lemma 2 For any ξ = y z R n, where y R and z R n 1, there exist p 1 > and positive definite P 2 R n 1 n 1 such that ξ Λ 1 PΛ 1 ξ = p 1 y 2 + z P 2 z. Proof. Using P = P P, one can write ξ Λ 1 PΛ 1 ξ = ξ PΛ 1 PΛ 1 ξ. 24 We notice that Λ P 1 = c m P 1 D. Letting q 1 = c m P 1 c m P 1, Q 2 = DD, it follows from 23 that Λ P 1 Λ P 1 q = Q 2 Non-singularity of Λ and P implies that Λ P 1 Λ P 1 is non-singular, and therefore Q 2 is also nonsingular. Hence, PΛ 1 PΛ 1 = Λ P 1 Λ P 1 1 = Λ P 1 PΛ 1 q 1 = 1 Q Letting p 1 = q1 1 and P 2 = Q 1 2, it follows from 24 that ξ PΛ 1 PΛ 1 ξ = p 1 y 2 + z P 2 z, 27 which completes the proof. Let T be any positive constant and 1 1 R n be the basis vector with first element 1 and all other elements zero. Let φt R n 1 be a vector which consists of 2 to n elements of 1 1 eλamλ 1T and let Further, let κt = T 1 1 eλamλ 1 T τ Λb m dτ. 28 ςt = α φt + κt, λ max P 2 29 α = 2 Λ λ max Λ PΛ 1 2 Pb m λ min Λ QΛ 1. 4 of 16

5 Letting 1 1 e ΛAmΛ 1t = η 1 t η2 t, 3 where η 1 t R and η 2 t R n 1 contain the first and 2 to n elements of the row vector 1 1 e ΛAmΛ 1t, respectively, we introduce the following definitions β 1 T = β 2 T = max η 1t, 31 t, T max η 2t. 32 t, T Further, let ΦT be the n n matrix and let ΦT = T e ΛAmΛ 1 T τ Λdτ, 33 where Finally, let Lemma 3 η 3 t = η 4 t = t t β 3 T = β 4 T = max η 3t, 34 t, T max η 4t, 35 t, T 1 1 e ΛAmΛ 1 t τ ΛΦ 1 Te ΛAmΛ 1T 1 1 dτ, 1 1 e ΛAmΛ 1 t τ Λb m dτ. α γ T = β 1 TςT + β 2 T λ max P 2 + β 3TςT + β 4 T. 36 lim γ T =. 37 T Proof. Notice that since β 1 T, β 3 T, α and are bounded, it is sufficient to prove that lim ςt =, T 38 lim 2T =, T 39 lim 4T =. T 4 Since lim 1 1 e ΛAmΛ 1T = 1 1, then lim φt = n 1, which implies lim φt =. Further, it follows from T T T the definition of κt in 28 that lim κt =. Since α and are bounded, then lim ςt =, which proves 38. T T Since η 2 t is continuous, it follows from 32 that lim β 2 T = lim η 2 t. Hence, the upper bound in 39 follows T t from lim 1 1 e ΛAmΛ 1t = 1 1. Thus, lim η 2 t =, which proves 39. Similarly lim β 4 T = lim η 4 t =, t t T t α which proves 4. Boundedness of α, β 3 T and implies that lim β 1 TςT+β 2 T T λ max P 2 +β 3TςT+ β 4 T =, which completes the proof. V. L 1 Adaptive Output Feedback Controller We consider the following state predictor or passive identifier: ˆxt = A mˆxt + b m ut + ˆσt 41 ŷt = c mˆxt, ˆx = x =, 5 of 16

6 where ˆσt R n is the vector of adaptive parameters. Notice that while σt R in 5, i.e. the unknown disturbance is matched, the uncertainty estimation in 41 is ˆσt R n, i.e. the estimation of it is unmatched. This is the key step of the solution and the subsequent analysis. Taking the Laplace transform of 41, gives Letting ỹt = ŷt yt, the update law for ˆσt is given by where ΦT is defined in 33, and The control signal is the output of the low-pass filter: ŷs = Msus + c msi A m 1ˆσs. 42 ˆσt = ˆσiT, t it, i + 1T 43 ˆσiT = Φ 1 TµiT, i =, 1, 2,, 44 µit = e ΛAmΛ 1T 1 1 ỹit, i =, 1, 2, 3,. 45 us = Csrs Cs c m si A m 1 b m c m si A m 1ˆσs. 46 The complete L 1 adaptive controller consists of 41, 44 and 46, subject to the L 1 -gain upper bound in 1. Let xt = ˆxt xt. The error dynamics between 5 and 41 are Lemma 4 Let et = yt y ref t. Then xt = A m xt + ˆσt b m σt 47 ỹt = c m xt, x =. e t L H 3s L1 1 Gs L1 L ỹ t L. 48 Proof. Let It follows from 46 that σs = Cs c msi A m 1 b m c msi A m 1ˆσs Csσs. 49 us = Csrs Csσs σs, 5 and the system in 3 consequently takes the form: ys = Ms Csrs + 1 Csσs σs. 51 Substituting us from 5 into 4, we have As σs = Ms Csrs Csσs σs + Asds /Ms, 52 and hence σs = As Ms Csrs σs + Asds Ms + CsAs Ms. 53 Using the definitions of H s and H 1 s in 18 and 19, we can rewrite σs into the following form: σs = H 1 srs H 1s Cs σs + H sds. 54 Substitution into 51 leads to ys = Ms Cs+H 1 s1 Cs rs σs Cs +H sms 1 Cs ds. Recalling the definition of Hs from 9, one can verify that MsCs + H 1 s1 Cs = HsCs, 6 of 16

7 Hs = H sms, which consequently implies ys = Hs Csrs σs + Hs1 Csds. Using the expression for y ref s from 15, one can derive es = Hs1 Csd e s σs, where d e s is introduced to denote the Laplace transform of d e t = ft, yt ft, y ref t. Lemma 5 and Assumption 1 give the following upper bound: where r 1 t is the signal with its Laplace transform being e t L L Hs1 Cs L1 e t L + r 1t L, 55 r 1 s = Hs σs. 56 Using the expression for σs from 49 along with the expressions for ys from 3 and ŷs from 42, one can derive ỹs = c m si A m 1ˆσs Msσs = Ms Cs Cs Ms c msi A m 1ˆσs Ms Ms Csσs = σs. 57 Cs Cs This implies that r 1 s can be rewritten as r 1 s = CsHs Ms Ms Cs σs = H 3sỹs, and hence r 1t L H 3 s L1 ỹ t L. Substituting this back into 55 completes the proof. VI. Analysis of L 1 Adaptive Controller In this section, we analyze the stability and the performance of the L 1 adaptive controller. Using the definitions from 11, H 2 s in 2 can be rewritten as H 2 s = C nsa d sm n s H d s, 58 where H d s is defined in 13. Since degc d s C n s degc n s = d r, it can be checked straightforwardly that H 2 s is strictly proper. We notice from 12 and 58 that H 2 s has the same denominator as Hs, and therefore it follows from 1 that H 2 s is stable. Since H 2 s is strictly proper and stable with relative degree d r, H 2 s/ms is stable and proper and, therefore its L 1 gain is finite. Consider the state transformation: ξ = Λ x. 59 It follows from 47 that where ξ 1 t is the first element of ξt. ξt = ΛA m Λ 1 ξt + Λˆσt Λbm σt 6 ỹt = ξ 1 t, 61 Theorem 1 Given the system in 1 and the L 1 adaptive controller in 41, 44, 46 subject to 1, if we choose T to ensure γ T < γ, 62 where γ is an arbitrary positive constant introduced in 23, then ỹ L < γ 63 y y ref L γ 1 64 u u ref L γ 2, 65 with γ 1 = H 3 s L1 γ/1 Gs L1 L, γ 2 = L H 3 s L1 γ 1 + H 2 s/ms L1 γ. 7 of 16

8 Proof. First we prove the bound in 63 by a contradiction argument. Since ỹ = and ỹt is continuous, then assuming the opposite implies that there exists t such that which leads to Since yt = y ref t + et, the upper bound in 16 can be used to arrive at ỹt < γ, t < t, 66 ỹt = γ, 67 ỹ t L = γ. 68 y t L y reft L + e t L ρ + CsHs/Ms L1 γ/1 Gs L1 L. It follows from 54 and 57 that σs = H 1 srs H 1 sỹs/ms + H sds, and hence 68 implies that σ t L H 1 s L1 r L + H 1 s/ms L1 γ + H s L1 L y t L + L, which along with 69 leads to It follows from 6 that Since ξit = ξit + t = e ΛAmΛ 1 t ξit + it+t it e ΛAmΛ 1 it+t τ ΛˆσiTdτ σ t L. 69 it+t t = e ΛAmΛ 1 t ξit + e ΛAmΛ 1 t τ ΛˆσiTdτ ỹit + zit it, it follows from 7 that e ΛAmΛ 1 it+t τ Λb m στdτ 7 t e ΛAmΛ 1 t τ Λb m σit + τdτ. ξit + t = χit + t + ζit + t, 71 where χit + t = e ΛAmΛ 1 t ζit + t = e ΛAmΛ 1 t ỹit zit + t t e ΛAmΛ 1 t τ ΛˆσiTdτ, 72 e ΛAmΛ 1 t τ Λb m σit + τdτ. 73 In what follows, we prove that for all it t one has ỹit ςt, 74 z itp 2 zit α, 75 where ςt and α are defined in 29. Since ξ =, it is straightforward that ỹ ςt, z P 2 z α. For any j + 1T t, we will prove that if then hold for j + 1 too. Hence, hold for all it t. Assume hold for j, and in addition, It follows from 71 that ỹjt ςt, 76 z jtp 2 zjt α, 77 j + 1T t. 78 ξj + 1T = χj + 1T + ζj + 1T, 79 8 of 16

9 where χj + 1T = e ΛAmΛ 1 T ζj + 1T = e ΛAmΛ 1 T ỹjt zjt T + e ΛAmΛ 1 T τ ΛˆσjTdτ, 8 T e ΛAmΛ 1 T τ Λb m σjt + τdτ. 81 Substituting the adaptive law from 44 in 8, we have It follows from 73 that ζt is the solution of the following dynamics: Consider the following function: χj + 1T =. 82 ζt = ΛA m Λ 1 ζt Λb m σt, 83 ζjt =, t jt, j + 1T. 84 zjt V t = ζ tλ PΛ 1 ζt, 85 over t it, i + 1T. Since Λ is non-singular and P is positive definite, Λ PΛ 1 is positive definite and, hence, V t is a positive definite function. It follows from 83 that over t jt, j + 1T V t = ζ tλ PΛ 1 ΛA m Λ 1 ζt + ζ tλ A mλ Λ P Λ 1 ζt 2ζ tλ PΛ 1 Λb m σt = ζ tλ PA m + A m P Λ 1 ζt 2ζ tλ Pb m σt 86 = ζ tλ QΛ 1 ζt 2ζ tλ Pb m σt. Using the upper bound from 69 we can compute the following upper bound over t it, i + 1T V t λ min Λ QΛ 1 ζt ζt Λ Pb m. 87 Notice that for all t jt, j + 1T, if V t α, 88 α we have ζt λ max Λ PΛ 1 = 2 Λ Pb m λ min Λ QΛ 1, and hence the upper bound in 87 yields V t. 89 It follows from Lemma 2 and the relationship in 84 that V ζjt = z jtp 2 zjt, which further along with the upper bound in 77 leads to the following It follows from and 9 that V ζjt α. 9 V t α, t jt, j + 1T, 91 and therefore Since V j + 1T = ζ j + 1TΛ PΛ 1 ζj + 1T α. 92 ξj + 1T = χj + 1T + ζj + 1T, 93 then the equality in 82 and the upper bound in 92 lead to the following inequality ξ j +1TΛ PΛ 1 ξj + 1T α. Using the result of Lemma 2 one can derive that z i+1tp 2 zi+1t ξ i+1tλ PΛ 1 ξi+ 1T α, which implies that the upper bound in 77 holds for j of 16

10 It follows from 61, 82 and 93 that ỹj + 1T = 1 1 ζj + 1T, and the definition of ζj + 1T in 81 leads to the following expression: ỹj + 1T = 1 1 e ΛAmΛ 1 T zjt 1 1 T The upper bounds in 69 and 77 allow for the following upper bound: e ΛAmΛ 1 T τ Λb m σjt + τdτ. 94 T ỹj + 1T φt zit e ΛAmΛ 1 T τ Λb m σjt + τ dτ α φt + κt = ςt, 95 λ max P 2 where φt and κt are defined in 28, and ςt is defined in 29. This confirms the upper bound in 76 for j +1. Hence, hold for all it t. For all it + t t, where t T, using the expression from 7 we can write that t ỹit + t = 1 1 e ΛAmΛ 1 t ξit e ΛAmΛ 1 t τ ΛˆσiTdτ 1 1 t e ΛAmΛ 1 t τ Λb m σit + τdτ. The upper bound in 69 and definitions of η 1 t, η 2 t, η 3 t and η 4 t allow for the following representation ỹit + t η 1 t ỹit + η 2 t zit + η 3 t ỹit + η 4 t. 96 Taking into consideration and recalling the definitions of β 1 T, β 2 T, β 3 T, β 4 T in 31-35, for all t T and for any non-negative integer i subject to it + t t, we have α ỹit + t β 1 TςT + β 2 T λ max P 2 + β 3TςT + β 4 T 97 Since the right hand side coincides with the definition of γ T in 36, then for all t, t which along with our assumption on T introduced in 62 yields ỹt γ T, 98 ỹ t L < γ. 99 This clearly contradicts the statement in 68. Therefore, ỹ L < γ, which proves 63. Further, it follows from 48 that e t L H3s L 1 1 Gs L1 L γ, which holds uniformly for all t, and therefore leads to the upper bound in 64. It follows from 5 and 53 that us = MsCsrs Ms σs CsAsds. CsAs + 1 CsMs To prove the bound in 65, we use the expression of u ref s from 14 to derive us u ref s = H 3 sr 2 s + H 2s Cs σs = H 3sr 2 s + H 2s Ms σs, 1 Ms Cs where r 2 t = ft, yt ft, y ref t. It follows from 57 and 1 that u u ref L L H 3 s L1 y y ref L + H 2 s/ms L1 ỹ L, which along with leads to 65. The proof is complete. The main result can be summarized as follows. Theorem 2 Given the system in 1 and the L 1 adaptive controller in 41, 44, 46 subject to 1, we have: lim reft = T, t, 11 lim reft = T, t of 16

11 The proof follows from Theorem 1 directly. Thus, the tracking error between yt and y ref t, as well between ut and u ref t, is uniformly bounded by a constant proportional to T. This implies that during the transient one can achieve arbitrary improvement of tracking performance for both signals simultaneously by uniformly reducing T. Remark 1 Notice that the parameter T is the fixed time-step in the definition of the adaptive law. The adaptive parameters in ˆσt R n take constant values during it, i + 1T for every i =, 1,. Reducing T imposes hardware requirements and implies that the performance limitations are consistent with the hardware limitations. This in turn is consistent with the results in Refs., 1, 17 where improvement of the transient performance was achieved by increasing the adaptation rate in the continuous adaptive laws. Remark 2 We notice that the following ideal control signal u ideal t = rt σ ref t is the one that leads to desired system response y ideal s = Msrs by cancelling the uncertainties exactly. Thus, the reference system in 6-8 has a different response as compared to this one. It only cancels the uncertainties within the bandwidth of Cs, which can be selected comparable to the control channel bandwidth. This is exactly what one can hope to achieve with any feedback in the presence of uncertainties. Remark 3 We notice that stability of Hs is equivalent to stabilization of As by Cs Ms1 Cs. 13 Indeed, consider the closed-loop system, comprised of the system As and negative feedback of 13. The closedloop transfer function is: As. 14 Cs 1 + As Ms1 Cs Incorporating 11, one can verify that the denominator of the system in 14 is exactly H d s. Hence, stability of Hs is equivalent to the stability of the closed-loop system in 14. This implies that the class of systems As that can be stabilized by the L 1 adaptive output feedback controller 41, 44 and 46 is not empty. Remark 4 Finally, we notice that while the feedback in 13 may stabilize the system in 1 for some classes of unknown nonlinearities, it will not ensure uniform transient performance in the presence of unknown As. On the contrary, the L 1 adaptive controller ensures uniform transient performance for system s both signals, independent of the unknown nonlinearity and independent of As. VII. Simulations Numerical Example. As an illustrative example, consider the system in 1 with As = s + 1/s 3 s 2 2s + 8. We notice that As has poles in the right half complex plane. We consider L 1 adaptive controller defined via 41, 44 and 46, where 1 Ms = s s + 1, Cs = 1 s s + 1, T = 1 4. First, we consider the step response when dt =. The simulation results of L 1 adaptive controller are given in Figs 1a-1b. Next, we consider dt = ft, yt = sin.1ty 2 t + sin.4t, and apply the same controller without re-tuning. The control signal and the system response are plotted in Figs. 2a-2b. Further, we consider a time-varying reference input rt =.5 sin.3t and notice that without any re-tuning of the controller the system response and the control signal behave as expected, Figs. 3a-3b. Missile Longitudinal Autopilot Design. We consider the missile dynamics from Ref. 2, which is given by the following state-space representation: Z a 1 Z d M ẋt = a M d xt + Axt + ut + k m 1 xt ωa 2 2ζ a ω a ωa 2 yt = 1 xt, 11 of 16

12 Time t a yt solid and rt dashed b Time-history of ut Figure 1. Performance for rt = 1 and dt = Time t a yt solid and rt dashed b Time-history of ut Figure 2. Performance for rt = 1 and dt = sin.1ty 2 t + sin.4t Time t a yt solid and rt dashed b Time-history of ut Figure 3. Performance for rt =.5sin.3t and dt = sin.1ty 2 t + sin.4t. where xt = A z q δ a δa is the system state, in which A z is the vertical acceleration, q is the pitch rate, δ a is the fin deflection, and δ a is the fin deflection rate, k m is the vector of matched parametric uncertainties, and A contains both unmatched and additional matched uncertainties. In simulations, the following numerical values have been used for the missile dynamics 2 : Z a = 1.346, Z d =.2142, M a = , M d = , K K K K A =, k m =.3 1, a 1 a 2 a 3 a 4 2 ζa ω a where K is the uncertainty scaling factor for unmatched uncertainties, and a 1, a 2, a 3, a 4 can assume arbitrary values, modeling additional matched uncertainties. For simulation of the L 1 adaptive output feedback controller, the following 12 of 16

13 a System response b Control history Figure 4. Simulation results for missile longitudinal autopilot design Ms and Cs have been selected, along with a sampling rate T =.1 s: Ms = 1 1/ωs 2 + 2ζωs + 1, Cs = 1 1/ω c s 2 + 2ζ c ω c s + 1. where ω = 1 rad s, ζ =.8, ω c = 15 rad s, and ζ c =.8. Figure 9 shows scaled response for scaled uncertainties, with K denoting the scaling factor for unmatched uncertainties. We notice that the system response does not change from variation of a 1, a 2, a 3, a 4. Flexible Wing Application We now apply the L 1 adaptive output feedback control architecture to a highly flexible semi-span wind tunnel model, considered earlier in the work of authors in Ref. 21 The wind-tunnel model is instrumented with accelerometers along the spar, strain gauges at the root and mid-spar, a rate gyro at the wing tip, a gust sensor vane in front of the wing, and a rate gyro and accelerometers at the tunnel attachment point. The accelerometers, strain gauges and rate gyro allow the control system to sense the bending modes and the structural stresses. The test objectives are, first, to control the first and second bending modes of the wing, while executing altitude hold and controlling pitch moment at the pivot point. The second control objective is GLA/flutter suppression, particularly around the frequency of the first bending mode. In the following figures, we compare the performance of L 1 adaptive output feedback controller to a baseline linear design that was provided with the model. Control signals are generated by the leading and the trailing edge control surfaces. In Figs. 5a-5b, we notice that the L 1 adaptive output feedback controller leads to the same performance as the baseline controller in the absence of turbulence. In the presence of turbulence, Figs. 6a-7b demonstrate that the L 1 adaptive output feedback controller improves its performance in the presence of turbulence. We further modify the original A matrix of the system by adding.1 to its every element. Figs. 8a-9b demonstrate that the L 1 adaptive controller outperforms the baseline controller for this class of uncertainty as well. 13 of 16

14 Vertical displacement, feet Output yt Attidue output Command Surface deflection, deg Control Signal ut LE TE1 TE2 TE3 TE a Tracking performance b Control history Figure 5. Simulation results of baseline and L 1 adaptive controller in the absence of turbulence Vertical displacement, feet Output yt Attidue output Command Surface deflection, deg Control Signal ut LE TE1 TE2 TE3 TE a Tracking performance b Control history Figure 6. Simulation results of baseline controller in the presence of turbulence Vertical displacement, feet Output yt Attidue output Command Surface deflection, deg Control Signal ut LE TE1 TE2 TE3 TE a Tracking performance b Control history Figure 7. Simulation results of L 1 adaptive controller in the presence of turbulence 14 of 16

15 Vertical displacement, feet Output yt Attidue output Command Surface deflection, deg Control Signal ut LE TE1 TE2 TE3 TE a Tracking performance b Control history Figure 8. Simulation results of baseline controller in the presence of additive uncertainty in the A matrix Vertical displacement, feet Output yt Attidue output Command Surface deflection, deg Control Signal ut LE TE1 TE2 TE3 TE a Tracking performance b Control history Figure 9. Simulation results of L 1 adaptive controller in the presence of additive uncertainty in the A matrix 15 of 16

16 VIII. Conclusions We presented the L 1 adaptive output feedback controller for reference systems that do not verify the SPR condition for their input-output transfer function. The new piece-wise constant adaptive law along with low-pass filtered control signal ensures uniform performance bounds for system s both input/output signals simultaneously. The performance bounds can be systematically improved by reducing the integration time-step. References 1 C. Cao and N. Hovakimyan. L 1 adaptive output feedback controller for systems of unknown dimension. IEEE Transactions on Automatic Control, 53: , April K.S. Tsakalis and P.A. Ioannou. Adaptive control of linear time-varying plants. Automatica, 23: , R. H. Middleton and G. C. Goodwin. Adaptive control of time-varying linear systems. IEEE Trans. Autom. Contr., 331:15 155, G. Bartolini, A. Ferrara, and A. A. Stotsky. Robustness and performance of an indirect adaptive control scheme in the presence of bounded disturbances. IEEE Trans. Autom. Contr., 444: , April J. Sun. A modified model reference adaptive control scheme for improved transient performance. IEEE Trans. Autom. Contr., 387: , July D.E. Miller and E.J. Davison. Adaptive control which provides an arbitrarily good transient and steady-state response. IEEE Trans. Autom. Contr., 361:68 81, January R. Costa. Improving transient behavior of model-reference adaptive control. Proc. of American Control Conference, pages , B.E. Ydstie. Transient performance and robustness of direct adaptive control. IEEE Trans. Autom. Contr., 378: , August M. Krstic, P. V. Kokotovic, and I. Kanellakopoulos. Transient performance improvement with a new class of adaptive controllers. Systems & Control Letters, 21: , R. Ortega. Morse s new adaptive controller: Parameter convergence and transient performance. IEEE Trans. Autom. Contr., 388: , August Z. Zang and R. Bitmead. Transient bounds for adaptive control systems. Proc. of 3 th IEEE Conference on Decision and Control, pages , December Y. Zang and P. A. Ioannou. Adaptive control of linear time-varying systems. In Proc. of IEEE 35th Conf. Decision Contr., pages , A. Datta and P. Ioannou. Performance analysis and improvement in model reference adaptive control. IEEE Trans. Autom. Contr., 3912: , December K. S. Narendra and J. Balakrishnan. Improving transient response of adaptive control systems using multiple models and switching. IEEE Trans. Autom. Contr., 399: , September R. Marino and P. Tomei. An adaptive output feedback control for a class of nonlinear systems with time-varying parameters. IEEE Trans. Autom. Contr., 4411: , A. M. Arteaga and Y. Tang. Adaptive control of robots with an improved transient performance. IEEE Trans. Autom. Contr., 477: , July C. Cao and N. Hovakimyan. Design and analysis of a novel L 1 adaptive control architecture with guaranteed transient performance. IEEE Transactions on Automatic Control, 53: , March J. Wang, C. Cao, N. Hovakimyan, R. hindman, and B. Ridgely. L 1 Adaptive Controller for a Missile Longitudinal Autopilot Design. AIAA Guidance, Navigation, and Control Conference and Exhibit, AIAA , E. Kharisov, I. Gregory, C. Cao, and N. Hovakimyan. L 1 Adaptive Control Law for Flexible Space Launch Vehicle and Proposed Plan for Flight Test Validation. AIAA Guidance, Navigation, and Control Conference and Exhibit, K. Wise. Robust Stability Analysis of Adaptive Missile Autopilots. AIAA Guidance, Navigation, and Control Conference and Exhibit, I. Gregory, V. Patel, C. Cao, and N. Hovakimyan. L 1 adaptive control laws for flexible wind tunnel model of high-aspect ratio flying wing. In Proc. of AIAA Guidance, Navigation & Control Conf., Hilton Head Island, SC, 27. AIAA H. K. Khalil. Nonlinear Systems. Prentice Hall, Englewood Cliffs, NJ, 22. IX. Appendix Definition 1 22 For a signal ξt, t, ξ R n, its truncated L and L norms are ξ t L ξ L = max sup i=1,..,n τ ξ iτ, where ξ i is the i th component of ξ. = max sup ξ i=1,..,n iτ, τ t Definition 2 22 The L 1 gain of a stable proper SISO system is defined Hs L1 = ht dt, where ht is the impulse response of Hs. Lemma 5 22 For a stable proper multi-input multi-output MIMO system Hs with input rt R m and output xt R n, we have x t L Hs L1 r t L, t. 16 of 16