AIAA 2011-6286 AIAA Guidance, Navigation, and Control Conference 08 - 11 August 2011, Portland, Oregon Improving Adaptation Performance For Systems With Slow Dynamics Jonathan A. Muse∗ Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 U.S. Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433 It is often diﬃcult to achieve good tracking performance in the presence of modeling error with the use of high adaptation gain when a system has slow modes. This leads to unnecessary high frequency control eﬀort that can excite unmodeled dynamics. This paper introduces an adaptive control architecture that allows fast adaptation for systems with both fast and slow modes. Fast adaptation is achieved using a high bandwidth state emulator to train the adaptive element. The state emulator allows the drift part of the adaptation dynamics to be set arbitrarily. This allows a control designer to shift the adaptive process dynamics to a more favorable set and represents a new strategy for improving the adaptation process in that no modiﬁcation terms need to be added to the adaptive law to improve adaptation. Though not required for system stability, the system tracking error is kept small via a low bandwidth feedback on the emulator tracking error. The usefulness of the architecture is illustrated on a nonlinear model for wing rock and a linear model of a Boeing 747. I. Introduction As adaptive control theory has advanced over the past few years, it has reemerged as a dominant force in the control research community and has sparked much interest in industry.1–7 Model reference adaptive control(MRAC) has numerous advantages over modern linear model-based control design methods. Classical methods are limited by uncertainties and nonlinearity. Robust control design reduces the eﬀect of uncertainty and nonlinearity at the expense of reduced performance. Adaptive control oﬀers the possibility of achieving a much higher degree of robust performance, particularly in applications that are dominated by the presence of uncertain ﬂexible dynamics.8, 9 However, a major disadvantage of adaptive control is that it lacks an accepted means of quantifying the behavior of the control signal apriori. Hence, most adaptive control laws will require a more extensive veriﬁcation and validation process due to the time varying and nonlinear manner in which its gains are adapted. This process can lead to unacceptable transients during adaptation, which can be made worse by actuator limitations10 and can yield a transient response that exceeds the practical limits of the plant. One case where adaptive control can lead to unacceptable control input is when fast adaptation is desired for a system with both fast and slow modes. In this case, it is often diﬃcult to achieve good tracking performance in the presence of modeling error with the use of high adaptation gain. This leads to unnecessary high frequency control eﬀort that can excite unmodeled dynamics. This paper introduces a generalization of the adaptive control architecture for slow reference models previously developed.11, 12 This architecture also allows fast adaptation for systems with slow reference models. Fast adaptation is achieved using a high bandwidth state emulator to train the adaptive element. The state emulator allows the drift part of the adaptation dynamics to be set arbitrarily. This allows a control designer to shift the adaptive process dynamics to a more favorable set and represents a new strategy for improving the adaptation process in that no modiﬁcation terms need to be added to the adaptive law to improve adaptation. Though not required for system stability, the system tracking error is kept small via a low bandwidth feedback on the emulator tracking error. The previous concept11, 12 only allowed the adaptation drift dynamics to be altered through the range of the control eﬀort. However, signiﬁcantly shifting a slow mode in a dynamic system through the range of the control may require large gain which can degrade performance in a real system (even though the eﬀect of the gain only enters the ∗ Research Aerospace Engineer, Control Design and Analysis Branch, Member AIAA. 1 of 20 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. American Institute of Aeronautics and Astronautics control channel at low frequencies). It turns out that constructing stable adaptation dynamics using standard adaptive laws does not require the diﬀerence between the drift dynamics of the system and the drift dynamics of adaptation error dynamics to be in the range of the control. The usefulness of the new architecture for systems with fast and slow modes is illustrated on the same nonlinear model for wing rock and a linear model of a Boeing 747 used in the previous work. This allows a comparison that shows that similar performance is possible from the new architecture without the potential limitations of the old architecture. II. Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 II.A. Mathematical Preliminaries Projection Operator The adaptive architecture derived, with minor modiﬁcations, can use an adaptive control law of choice. However, the projection operator13 allows one to obtain stronger results since asymptotic convergence of the system to the reference model can be retained for perfect parameterization while the adaptive weights are guaranteed bounded with a known bound. The following deﬁnition of the projection operator is taken from.13, 14 Definition II.1. Consider a convex compact set with a smooth boundary given by Ωc ≡ {θ ∈ Rn : f (θ) ≤ c}, 0≤c≤1 (1) where f : Rn 7→ R is the following smooth convex function 2 f (θ) = 2 ∥θ∥ − θmax 2 ϵθ θmax (2) where θmax is the norm bound imposed on the parameter vector θ, and ϵθ denotes the convergence tolerance of our choice. Let the true value of the parameter θ, denoted by θ∗ , belong to Ω0 , i.e. θ∗ ∈ Ω0 . The projection operator is defined as if f (θ) < 0 y P roj(θ, y) = y (3) if f (θ) ≥ 0 and ∇f T y ≤ 0 ∇f ∇f T y − ∥∇f ∥ ⟨ ∥∇f ∥ , y⟩f (θ) if f (θ) ≥ 0 and ∇f y > 0 The next lemma13 allows one to guarantee that adaptive parameters updated with the adaptive law θ̇(t) = P roj(θ(t), y(t)) are contained in a compact invariant set for all t ≥ 0. Lemma II.1. The projection operator P roj(θ, y) as defined in (3) does not alter y if θ belongs to the set Ω0 ≡ {θ ∈ Rn : f (θ) ≤ 0}. In the set {0 ≤ f (θ) ≤ 1}, if ∇f T y > 0, the projection operator subtracts a vector normal to the boundary of Ω1 = {θ ∈ Rn : f (θ) = c} so that there is a smooth transformation from the original vector field y to an inward or tangent vector field for c = 1. Thus, if θ is the adaptive parameter, and θ̇(t) = P roj(θ(t), y(t)), then θ(t) can never leave Ωc . The next lemma is useful for proving that a chosen Lyapunov function candidate’s time derivatives are non-positive for the adaptive laws applied in this paper. Lemma II.2. Given the vectors y = [y1 , · · · , yn ] ∈ Rn , θ = [θ1 , · · · , θn ] ∈ Rn , and θ∗ = [θ1∗ , · · · , θn∗ ] ∈ Rn where θ∗ is the constant true value of the parameter θ. Then T T T (θ − θ∗ ) (P roj(θ, y) − y) ≤ 0 T (4) Proof. Note that 0 ∗ T (θ − θ ) (P roj(θ, y) − y) = 0 ∗ ∇f ∇f (θ − θ) ∥∇f ∥ ⟨ ∥∇f ∥ , y⟩f (θ) f (θ) < 0 f (θ) ≥ 0, ∇f T y ≤ 0 (5) f (θ) ≥ 0, ∇f y > 0 T Since the angle between θ∗ − θ and ∇f is greater than 90 degrees by deﬁnition of the projection operator, we have the result. 2 of 20 American Institute of Aeronautics and Astronautics This property can be applied when θ and y are matrices.14 θ and y will become matrices in this paper when the uncertainty has dimension greater than one. In this case, the projection operator is deﬁned column-wise as P roj(Θ, Y ) = (P roj1 (θ1 , y1 ) ... P rojN (θN , yN )) (6) where ...yN ) ∈ RnxN Y = (y1 and Θ = (θ1 ...θN ) ∈ RnxN Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 and yi and θi are vectors in Rn . Note that each vector θi may have a diﬀerent projection bound θmax . To see how the previous Lemma is applied with this new deﬁnition of the projection operator in a stability proof, consider the following equation ( ) f T (t)β(x(t)) + 2tr W f T (t)Γ−1 W ḟ (t) (7) −2b eT (t)P DW f is the weight estimation error matrix, W c is the uncertainty weight estimate where eb(t) is the error state, W matrix, β(x) is a set of basis functions, P and D are matrices, and Γ is a constant positive deﬁnite matrix. This arrangement of variables will appear in this paper. Equation (7) can be rewritten in the form [ ( )] ḟ − β(x(t))b f T (t) Γ−1 W eT (t)P D 2tr W If the adaptive law is deﬁned as ( ) ċ (t) = ΓP roj W c , β(x(t))b W eT (t)P D then using the properties of the trace operator and the projection operator one has that [ ( )] f T (t) Γ−1 W ḟ − β(x(t))b 2tr W eT (t)P D = N ∑ ( ( ) ( ) ) fjT (t) P rojj (W cj , β(x(t))b W eT (t)P D j ) − β(x(t))b eT (t)P D j ≤ 0 j=1 This is a fundamental property of the projection operator and will be used in the paper. II.B. Useful L Stability Properties Proofs in this paper use input-output stability to show stability and bounded transients. Excellent references for L Stability theory include but are not limited to.15, 16 Lp stability studies input-output maps of the form y(t) = Hu(t) where H is an operator that describes y : [0, ∞) 7→ Rn in terms of u : [0, ∞) 7→ Rm . A piecewise continuous function signal, u(t) exists in the space Lp if (∫ ∥u∥Lp = ∞ 0 p ∥u(t)∥q ) p1 dt < ∞, 1≤p<∞ or for p = ∞, if ∥u∥L∞ = sup ∥u(t)∥q < ∞ t≥0 where q speciﬁes the spatial norm deﬁned by 1 ∥x∥q = (∥x1 ∥q + · · · + ∥xn ∥q ) q , x ∈ Rn , 1≤q<∞ 3 of 20 American Institute of Aeronautics and Astronautics or when q = ∞ ∥x∥∞ = max ∥xi ∥, x ∈ Rn i=1...n The input-output mapping H : Lm → 7 Lm cannot be deﬁned properly for all signals(i.e. unstable systems). Usually, H : Lm 7→ Lm is deﬁned as a mapping from the extended Lem space to the extended Lem space where Lem = {u : uτ ∈ Lm , Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 and ∀τ ∈ [0, ∞)} u(t) 0 ≤ t ≤ τ uτ (t) = 0 t>τ The next Lemma16 shows that the L1 signal norm of a system’s convolution kernel allows for an upper bound on all signals in Lp . Lemma II.3. Consider the system defined by the casual convolution operator ∫ t y(t) = h(t − σ)u(σ)dσ 0 where y(t) is the system output, h(t) is the convolution kernel, and u(t) is the system input. If u ∈ Lp and h ∈ L1 then ∥y∥Lp ≤ ∥h∥L1 ∥u∥Lp where p ∈ [1, ∞]. III. Slow Systems Architecture Consider the uncertain nonlinear dynamical system deﬁned by ẋ(t) = Ax(t) + BΛu(t) + Bf (x(t)) where A ∈ Rnxn , B ∈ Rnxm is a matrix with full column rank, x(t) ∈ Rn is the system state, u(t) ∈ Rm is the system control input, Λ ∈ Rmxm is a constant unknown positive deﬁnite matrix, and f (x) : Rn 7→ Rm is an unknown function of the system state. In this case, it is assumed that Λ can be decomposed as Λ = I + δΛ (8) where δΛ ≤ I and that {A, B} is a stabilizable pair. The previous assumptions are suﬃcient conditions for system controllability. It is also assumed that f (x) can be linearly parameterized to within a bounded approximation error as f (x) = W T β(x) + ϵ(x), ∀x ∈ Rn (9) where β : Rn 7→ Rj is a set of known locally Lipschitz functions, W ∈ Rj×m is a set of constant but unknown ideal weights, and ϵ : Rn 7→ Rm is unknown, locally Lipschitz continuous, and bounded by ϵ∗ ∈ R+ as ∥ϵ(x)∥ ≤ ϵ∗ < ∞, ∀x ∈ Rn (10) Suppose that the total control eﬀort is deﬁned as u(t) = un (t) − uad (t) (11) un (t) = −Kx x(t) + Kr r(t) (12) where uad (t) will be deﬁned shortly and 4 of 20 American Institute of Aeronautics and Astronautics is an existing nominal control law that achieves the desired response of the system assuming that Λ = I and the system uncertainty is zero. This nominal control law deﬁnes the ideal system behavior as ẋm (t) = Am x(t) + Bm r(t), xm (0) = x(0) (13) where Am = A − BKx is assumed to be Hurwitz and Bm = BKr (the reference model often used in MRAC architectures). A state emulator (structurally similar to a series parallel model18 ) will be used to separate the adaptation process from the control realization. Let the state emulator in this paper be deﬁned as Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 b c T (t)β(x(t)) x(t) ḃ = Ā (b x(t) − x(t)) + Ax(t) + B Λu(t) + BW (14) c is the uncertainty weight where x b ∈ Rn is the state emulator state, Ā ∈ Rn×n is any Hurwitz matrix, W mxm b estimate, and Λ ∈ R is an estimate for Λ (the term estimate is used loosely). If the state emulator tracking error is deﬁned as, eb(t) = x(t) − x b(t), the weight update laws are deﬁned as ( ) ċ (t) = −ΓW P roj W c (t), β(x(t))b W eT (t)P B (15) ( ) ḃ = −ΓΛ P roj δ Λ(t), b δ Λ(t) B T P eb(t)uT (t) where P roj(·, ·) is the projection operator13 deﬁned using a known bound on the unknown weights, P ∈ Rn×n such that P = P T > 0 is from the solution of the Lyapunov equation ĀT P + P Ā + Q = 0 (16) b = I + δ Λ. b Using these deﬁnitions, the adaptive where Q ∈ Rn×n is any matrix such that Q = QT > 0, and Λ control signal is given by ( )−1 [ ] b b n (t) + W c T (t)β(x(t)) + uad (t) uad (t) = I + δ Λ δ Λu (17) s where b Uads (s) = Fc (s)E(s) (18) ( )−1 T ( ) Fc (s) = Gc (s) B T B B sI − Ā (19) Fc (s) is deﬁned as and Gc (s) is a low pass ﬁlter with the following realization ẋc (t) = Ac xc (t) + Bc (t)uc (t) yc (t) = Cc xc (t) (20) Since Gc (s) is strictly proper, Fc (s) is proper and realizable. This deﬁnes the complete adaptive control architecture. Next, the state emulator error dynamics are examined. Let the weight estimation errors be deﬁned as f=W c − W and Λ e=Λ b − Λ, then the state emulator error dynamics can be expressed as W ( ) f T (t)β(x(t)) + δ Λu(t) e (21) e(t) ḃ = Āb e(t) − B W − ϵ(x(t)) The above dynamics are used to train the adaptive weights. Note how the above dynamics diﬀer from the those typically used in adaptive control. In this case, the arbitrarily selected matrix Ā replaces the matrix Am in a standard set of adaptation error dynamics. This gives a degree of freedom in the adaptive design that is fundamentally diﬀerent that most modiﬁcations to adaptive laws. Most modiﬁcations developed to improve the performance of an adaptation process achieve the modiﬁcation by changing how the weights update via modiﬁcations to the weight update law (i.e. by modifying equation (15)). This is in contrast to this method which changes the adaptive law training signal to improve adaptation performance. The following theorem shows that eb(t) remains bounded. 5 of 20 American Institute of Aeronautics and Astronautics Theorem III.1. Consider the uncertain nonlinear dynamical system in equation (8), the state emulator defined in equation (14), the adaptive weight update laws defined in equation (15), and the control signal defined in equations (11), (12), and (17). Let Q ∈ Rn×n be such that Q = QT > 0 and let P ∈ Rn×n be the solution of the Lyapunov equation in equation (16). Then the emulator error, eb(t), is bounded. c b and ∥δΛ∥F ≤ δ Λ Moreover, if eb(0) = 0 and the unknown ideal weights satisfy ∥W ∥F ≤ W F,max F,max 2 2 c b where ∥·∥F represents the Frobenius norm and W and δ Λ are the maximum allowed Frobenius F,max F,max Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 c (t) and δ Λ(t) b norms of W set by each respective projection operator, then eb(t) is bounded ∀t ≥ 0 by √ 2 2 2 ∥W ∥F,max ∥δΛ∥F,max λmax (P ) ∥P B∥F (ϵ∗ )2 + ∥b e(t)∥2 ≤ 2 + λmin (P ) λ2min (Q) λmin (ΓW ) λmin (ΓΛ ) Proof. First, boundedness of eb(t) is shown. Consider the Lyapunov candidate function ( ) ( ) fT e f (t)) = ebT (t)P eb(t) + tr δ Λ(t)Γ e eT e V (b e(t), δ Λ(t), W Λ δ Λ (t) + tr W (t)ΓW Λ(t) (22) (23) where P is the solution of the Lyapunov equation in equation (16). Computing the Lyapunov derivative of e f (t)) along the system trajectories, one has that V (b e(t), δ Λ(t), W ( ) ( ) f , δ Γ) e = 2b f T (t)Γ−1 W ḟ (t) + 2tr δ Λ e T (t)Γ−1 δ˙Λ(t) e V̇ (b e(t), W eT (t)P e(t) ḃ + 2tr W (24) W W Substituting the emulator error dynamics in equation (21) and applying properties of the trace operator, equation (24) becomes ( ) f (t), δ Λ(t)) e V̇ (b e(t), W = ebT (t) ATm P + P Am eb(t) + 2b eT (t)P Bϵ(x(t)) [ ( )] f T (t) Γ−1 W ċ (t) + β(x(t))b +2tr W eT (t)P B W (25) [ ( )] T e ḃ +2tr δ Λ(t) Γ−1 eT (t)xT (t)P B Λ δ Λ (t) + u(t)b Next, applying the adaptive laws from equation (15) and making use of the properties of the projection operator from Section II.A, the derivative of equation (23) along the system trajectories is bounded by f (t), δ Λ(t)) e V̇ (b e(t), W ≤ −b eT (t)Qb e(t) + 2b eT (t)P Bϵ(x(t)) (26) Since ∥ϵ(x)∥2 ≤ ϵ∗ ∀x ∈ Rn , ∥b e(t)∥ > 2 ∥P B∥F ϵ∗ λmin (Q) (27) f (t), δ Λ(t)) e implies that V̇ (b e(t), W ≤ 0 and whence eb(t) is ultimately bounded. Next, the bound for eb(t) is derived.11 The adaptive are bounded due to the projection operator. weights f e The projection operator ensures that there exists a W and δ Λ such that F,max f f W (t) ≤ W F and F,max F,max e e δ Λ(t) ≤ δ Λ 2 (28) F,max where ||X||F,max is the maximum Frobenius norm of X(t), ∀t ≥ 0 . This implies that the Lyapunov-like f (t), δ Λ(t)) e function derivative satisﬁes V̇ (e(t), W ≤ 0 outside the compact set } { 2 ∥P B∥F ϵ∗ f , δ Λ) e = (b f , δ Λ) e : ∥b α(b e, W e, W e∥2 ≤ λmin (Q) } ∩{ f f , δ Λ) e : f (b e, W W ≤ W F F,max { } ∩ e f , δ Λ) e : e (b e, W Λ ≤ Λ δ δ 2 2,max 6 of 20 American Institute of Aeronautics and Astronautics f (t), δ Λ(t)) e f (t), δ Λ(t)) e V (b e(t), W cannot grow outside this set. Hence, the evolution of V (b e(t), W is upper bounded by f (t), δ Λ(t)) e V (b e(t), W ≤ max f ,δ Λ)∈α e (b e,W f , δ Λ), e V (b e, W t≥0 (29) From the deﬁnition of the Lyapunov-like candidate in equation (23), ) ( ) ( −1 e T f (t), δ Λ(t)) e f T (t)Γ−1 W f (t) + tr δ Λ(t)Γ e V (b e(t), x(t), W =b eT (t)P eb(t) + tr W W Λ δ Λ (t) (30) Therefore, from equation (29) ∀t ≥ 0, f (t), δ Λ(t)) e V (b e(t), W ≤ Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 ≤ max f ,δ Λ)∈α e (b e,W max f ,δ Λ)∈α e (b e,W f , δ Λ) e V (b e, W [ ] 2 f T Γ−1 W f ) + tr(δ Λ e T Γ−1 δ Λ) e λmax (P ) ∥b e∥2 + tr(W W Λ (31) Using properties of the trace operator, 1 fT W f) tr(W λmin (ΓW ) 1 e T Γ−1 δ Λ) e ≤ e T δ Λ) e tr(δ Λ tr(δ Λ Λ λmin (ΓΛ ) f T Γ−1 W f) ≤ tr(W W (32) Applying these inequalities, from the deﬁnition of the set α(·, ·), f (t), δ Λ(t)) e V (b e(t), x(t), W ≤ 2 ∥P B∥F (ϵ∗ )2 4λmax (P ) λ2min (Q) 2 f W + F,max λmin (ΓW ) 2 e δ Λ + F,max (33) λmin (ΓΛ ) Noting that f c (t) W (t) ≤ ∥W ∥F + W F F , c ≤ 2 W ∀t ≥ 0 (34) ∀t ≥ 0 (35) F,max and e b δ Λ(t) ≤ ∥δΛ∥F + δ Λ(t) 2 F , b ≤ 2 δ Λ F,max c where W F,max b and δ Λ are known bounds set by the projection operator, it is found that F,max ∥P B∥F (ϵ∗ )2 4 4 2 2 f (t), δ Λ(t)) e + ∥W ∥F,max + ∥δΛ∥F,max V (b e(t), W ≤ 4λmax (P ) 2 λmin (Q) λmin (ΓW ) λmin (ΓΛ ) 2 (36) Since equation (30) implies that f (t), δ Λ(t)) e λmin (P ) ∥b e(t)∥ ≤ V (e(t), x(t), W 2 (37) the result in equation (22) is obtained. Using the previous theorem showing that the emulator error is bounded, it can be shown that the system tracking error, e(t) = xm (t) − x(t), is bounded and converges to zero asymptotically when ϵ(x) = 0. Towards this goal, the system dynamics in equation (8) can be rewritten as ( ) ẋ(t) = Ax(t) + BΛu(t) + B W T β(x(t)) + ϵ(x(t)) ( ) = Am x(t) + Bm r(t) − Buad (t) + BδΛu(t) + B W T β(x(t)) + ϵ(x(t)) 7 of 20 American Institute of Aeronautics and Astronautics Deﬁning the tracking error dynamics between the reference model and the system state as e(t) = xm (t)−x(t), the tracking error dynamics are computed as [ ] ė(t) = Am e(t) + Buad (t) − B δΛu(t) + W T β(x(t)) + ϵ(x(t)) The choice of uad (t) in equation (17) implies the following relationship b c T (t)β(x(t)) + uad (t) uad (t) = δ Λ(t)u(t) +W s (38) Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 which implies that the tracking error dynamics, e(t), can be expressed as [ ] e f T (t)β(x(t)) − ϵ(x(t)) ė(t) = Am e(t) + Buads (t) + B δ Λ(t)u(t) +W From the state emulator error dynamics in equation (21) and the fact that B has full column rank, the following relationship is obtained ) ( )−1 T ( f T (t)β(x(t)) + δ Λu(t) e (39) W − ϵ(x(t)) = − B T B B e(t) ḃ − Āb e(t) This implies that ) ( )−1 T ( ė(t) = Am e(t) + Buads (t) − B B T B B e(t) ḃ − Āb e(t) This system is linear and its Laplace transform exists. Assuming that eb(0) = 0, in the Laplace domain ( )−1 T ( ) b sE(s) = Am E(s) + BUads (s) − B T B B sI − Ā E(s) (40) b where E(s), E(s), and Uads (s) is the Laplace transform of e(t), eb(t), and uads (t) respectively. Using some algebra and the deﬁnition of Uads (s), this simpliﬁes to ( )−1 T ( ) b E(s) = Am E(s) + B (Gc (s) − I) B T B B sI − Ā E(s) or equivalently −1 E(s) = (sI − Am ) ( )−1 T ( ) b B (Gc (s) − I) B T B B sI − Ā E(s) (41) −1 The above transfer function is proper since (sI − Am ) B is strictly proper and the matrix polynomial ( T )−1 T ( ) B B B sI − Ā does not contain any polynomial of s with order greater than 1. Using equation (41), the desired properties of e(t) are derived in the following theorem. Theorem III.2. Consider the uncertain nonlinear dynamical system in equation (8), the state emulator defined in equation (14), the adaptive weight update laws defined in equations (15), and the control signal defined in equations (11), (12), and (17). Assume that the statements and conditions of Theorem III.1 hold and let F be a realization for equation (41) such that [ ] AF BF F = (42) CF DF Then assuming that the initial state of F is zero, the system tracking error, e(t), is bounded as [ ] e∥L∞ ∥e∥L∞ ≤ max ∥fi ∥L1 + ∥DFi ∥1 ∥b i=1...n (43) where DFi is the ith row of DF , the convolution kernel is defined as f (t) = CF eAF t BF and fi is the ith row of f . Moreover, if ϵ(x) = 0 and r(t) is bounded, eb(t) → 0 as t → ∞ and e(t) → 0 as t → ∞. 8 of 20 American Institute of Aeronautics and Astronautics Proof. Proof of the ﬁrst claim in equation (43) is immediate from equation (41) by applying standard L stability arguments (for details, see Chapter 4 in An H∞ Norm Minimization Approach For Adaptive Control 11 ). Now, suppose that ϵ(x) = 0. With ϵ(x) = 0, the emulator error dynamics become ( ) f T (t)β(x(t)) + δ Λu(t) e e(t) ḃ = Āb e(t) − B W (44) and the derivative along the system trajectories of the Lyapunov function in equation (23) reduces to f (t), δ Λ(t)) e V̇ (b e(t), W ≤ −λmin (Q) ∥b e(t)∥ From the deﬁnition of the Lyapunov function, this implies that Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 f (t), δ Λ(t)) e eb(t)T P eb(t) ≤V (b e(t), W (45) f (0), δ Λ(0)) e ≤V (b e(0), W f (t), δ Λ(t)) e Whence, V (e(t), W and e(t) are uniformly bounded. Integrating the bound from equation (45), a new bound is obtained as ∫ t f (0), δ Λ(0)) e (46) 0≤ ebT (τ )Qb e(τ )dτ ≤ V (b e(0), x(0), W 0 f (t), δ Λ(t)) e due to the fact that ebT Qb e is non-negative. Since x(t) is also bounded, e(x(t), ḃ eb(t), W is bounded uniformly in t for all t ≥ 0. Hence, eb(t) is uniformly continuous in t on [0, ∞). eb(t) uniformly continuous in t on [0, ∞) implies that ebT (t)Qb e(t) is uniformly continuous in t on [0, ∞). Hence, Barbalat’s Lemma16 implies T that eb (t)Qb e(t) → 0 as t → ∞. Which implies that eb ∈ L2 . From the previously derived norm bound, it is known that [ ] ∥e∥L2 ≤ max ∥fi ∥L1 + ∥DFi ∥1 ∥b e∥L2 (47) i=1...n Hence, e ∈ L2 which implies that e(t) → 0 as t → ∞. The previous theorem yields some important insight. First, the control signal uads (t) is not strictly necessary (i.e. one could choose uads (t) = 0). The use of the emulator error to train the adaptive law allows one to independently set the adaptation dynamics from the reference model dynamics without additional control eﬀort. However, in practice, uads (t) may be necessary to achieve tight tracking bounds. Another important thing to notice from the theorem is that Gc (s) − I in equation (41) acts as a high pass ﬁlter that can be used to attenuate the system tracking error relative to the emulator error. Note that the properties of the adaptive law in this paper are analogous to those possessed by many adaptive laws in the literature. IV. Wing Rock Example Consider the following idealized model of aircraft wing rock dynamics.19 ẋ = Ax + B(u + f (x)) y = Cx where [ A= ] 0 1 0 0 [ , B= ] 0 1 (48) [ , T C = ] 1 0 (49) u is the control moment, f (x) is a set of unknown system nonlinearities, x = [ϕ ϕ̇]T , and ϕ is the aircraft roll angle. The system nonlinearities are deﬁned as f (x) = 0.1 + ϕ + 2ϕ̇ − 0.6|ϕ|ϕ̇ + 0.1|ϕ̇|ϕ̇ + 0.2ϕ3 9 of 20 American Institute of Aeronautics and Astronautics (50) Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 This model choice assumes that the unknown high frequency gain is unity and is the same used in previous work.12 It is used again in this paper to allow a comparison between the old and new approach. Suppose that it is desired that the system behave like a second order system with natural frequency ωn = 1 rad/s and damping ratio ζ = 0.707. With this choice, the nominal control law deﬁned in (12) is given by Kx = [ωn2 2ωn ζ] and Kr = ωn2 . This choice of nominal control law is closed loop unstable when only the nominal control law is used. For comparison purposes, the performance of a standard adaptive architecture13 was also investigated. The equations for the standard adaptive control law used can be recovered by setting Ā = Am and uads (t) = 0 in equations (14) and (17) since the emulator dynamics collapse to the reference model dynamics in equation (13). The system nonlinearity representation used in both the standard adaptive law and the new architecture was a radial basis function neural network with 121 basis functions uniformly distributed over a unit cube and an added bias term. If the desired closed loop performance was fast enough, the standard adaptive law performed well. For illustration purposes, the closed loop system response and control eﬀort for the standard adaptive law with a reference model natural frequency of ωn = 5 rad/s is shown in Figures 1 and 2 for a 10 degree step command. However, when the reference model natural frequency was lowered to the desired natural frequency of ωn = 1 rad/s, the performance of the standard law degraded signiﬁcantly. One can vary the choice of adaptive gain over several orders of magnitude and achieve the same trend in the results. Good tracking and a low frequency control signal cannot be obtained simultaneously. An example of good tracking with a poor quality control signal was obtained when the adaptive gains are set to ΓW = 10000 and ΓΛ = 10. This result is shown in Figures 3 and 4. In this case, the control signal was highly oscillatory and was relatively large in amplitude. This causes the system to oscillate some around the reference model trajectory and could excite unmodeled dynamics in a real system. The new adaptive architecture was designed so that Ā has the form [ ] 0 1 Ā = (51) −wn2 −2ζωn where ωn = 10 rad/s and ζ = 0.707. Gc (S) from equation (20) was implemented as a ﬁrst order ﬁlter with the ﬁlter pole selected as 10 1/s. The performance of the new adaptive control architecture for the same 10 degree step command is shown in Figures 3 and 4 using the same choice for ΓW and ΓΛ . In this case, the tracking performance is similar but the control signal is smooth. The Fc transfer function used to compute uads remains low bandwidth. The frequency response of Fc (s) is shown in Figure 5. Note that a potentially better choice of Ā is possible but Abar was chosen to be the same as in previous work.12 This allows direct comparison between the results. V. Boeing 747 Example The next example, based on a model for a Boeing 747, introduces a set of dynamics that is diﬃcult for the standard projection based adaptive law to perform well. Consider the following model of the longitudinal dynamics of a Boeing 747.20 ẋ = Ax + B(δe + W T x) yt = Ct x where A= B= −0.006868 0.01395 −0.09055 −0.3151 0.0001187 −0.001026 0 0 −0.000187 −17.85 , CtT = −1.158 0 0 773.98 −0.4285 1 0 0 0 −32.2 0 0 0 , 1 10 of 20 American Institute of Aeronautics and Astronautics W is a set of unknown weights, x = [∆u w q ∆θ]T , ∆u is the change in the x-body velocity, w is the z-body velocity, q is the pitch rate, ∆θ is the change in pitch angle, and δe is the elevator command. yt and Ct describe the desired tracking variable, ∆θ. For demonstration purposes, suppose the unknown weights are given by [ ]T W = −0.01 −0.01 −0.01 −0.01 Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 where the known magnitude bound on each weight is 0.03. The baseline control design was computed from LQR theory. To create the LQR design, the dynamics in (52) were augmented with an integrator in the control design to ensure that the nominal system has zero steady state error in the tracking variables to a change in pitch command. The augmented matrices used in the design are given by [ ] 0 Ct Aaug = , (52) 0 A and [ ]T Baug = 0 B T From the deﬁnitions of these matrices, the augmented state vector for the system is given by x̄ = [xint xT ]T where xint is the integrator state. The LQR weighting matrices for the baseline LQR design are ( ) QAm = diag [1x106 0.1 6 1 1x103 ] , RAm = 1x105 (53) where QAm is the state weighting matrix and RAm is the control weighting matrix. In order to realize a tracking control law from the computed feedback gain, the integrator error dynamics are given by ẋint (t) = ∆θ(t) − r(t) (54) where r(t) is the desired change in pitch angle. This design is considered ﬁxed and unchangeable. Strictly speaking this does not ﬁt the form of the system model used in this paper. However, the modiﬁcations to the equations are simple and are often made in adaptive control. These modiﬁcations are sometimes referred to as the extended system dynamics. In this case, the nominal control law takes the form un (t) = −Kx x̄(t) (55) where Kx is the computed LQR gain and the state emulator takes the form b c T (t)β(x(t)) + Bm r(t) x(t) ḃ = Ā (b x(t) − x(t)) + Ax(t) + B Λu(t) + BW (56) [ ]T Bm = −1 Kx5 B T (57) with and Kx5 is the 5th element of the LQR feedback gain (i.e. the feedback on ∆θ) computed for this example (Bm is used to create the reference model in equation (13)). The previously derived theory still applies as the diﬀerential equations for the error signals e(t) and eb(t) remain the same when e(t) = xm (t) − x̄(t) and eb(t) = x̄(t) − x(t). The Ā matrix used for the new architecture is selected as the same Ā used in previous work12 to allow direct comparison (the new theory allows a more general selection of Ā if desired). To compute Ā an LQR design is computed using the same integrator augmented system. The computed LQR gain is referred to as K̄ and is computed using the following weighting matrices (Baug is used again in the LQR design) ( ) QĀ = diag [1x106 0 60 1 1x105 ] , RĀ = 10 (58) Using this gain, Ā = Aaug − Baug K̄. The main eﬀect of the K̄ design is to penalize the “control eﬀort” less. This in eﬀect speeds up the emulator error dynamics allowing one to increase the adaptation gain. The ﬁlter Gc (s) used in equation (20) was randomly chosen to be a ﬁrst order ﬁlter each with a pole at 30 1/s. This yields the frequency 11 of 20 American Institute of Aeronautics and Astronautics Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 response for Fc (s) shown in Figure 6. Since the uncertainty is linear, the standard adaptive law and the new adaptive architecture are implemented with β(x̄) = x̄ for simplicity (the standard adaptive law equations were implemented by choosing Ā = Am and uads = 0). This is the form used in classical adaptive control and ensures asymptotic convergence of the system tracking error. Evaluating the Lyapunov equation for each adaptive law, assuming that Q = I, the resulting P matrices are similar in that the ratio of the norms, ∥PAm ∥ / ∥PĀ ∥, is 1.3. This implies that the eﬀect of PAm and PĀ on β(x)eT P B was, loosely speaking, the same in terms of eﬀective gain. For each example, a square wave reference command with a 10◦ amplitude and a frequency of 0.1 Hz was given to ∆θ command channel. The control signal of the standard adaptive law oscillated with high frequency for all adaptive gains above Γstd = 1x10−7 . The tracking performance and control signal for this adaptive gain is shown in Figures 7 and 8 respectively. For this gain, the tracking performance was extremely poor. At an adaptive gain of Γstd = 1x10−2 , the tracking performance became acceptable but the control signal contained large high frequency content. The tracking performance and control signal for this adaptive gain is also shown in Figures 7 and 8 respectively. For the new adaptive control architecture, the adaptive gain was set at Γnew = 1x10−2 . Figures 9 and 10 show the tracking performance and the adaptive control eﬀort from the new control architecture. The control signal in this case appears perfectly smooth and tracking performance was better. A comparison of the frequency content between the standard adaptive law with Γstd = 1x10−2 and the new architecture is shown in Figure 11. Note that it was previously shown that σ-modiﬁcation21 and e-modiﬁcation22 do not improve performance11 of the standard adaptive law. These results are similar to previous results12 but do not rely on a feedback signal comprised of K̄ to achieve tight tracking. This is beneﬁcial since large K̄ in the previous work could cause the control law to operate poorly in a realistic system. One of the beneﬁts of being able to adapt fast, from a practical perspective, is the ability to compensate for faster varying uncertainty. Assuming that the ideal weights now vary as θnew (t) = θ + 2 θ squarewave(0.01πt), (59) the system controlled by the new architecture was simulated using the same adaptive parameters as used previously (where θ is the same weights deﬁned in equation (52)). The standard adaptive system, when it has a smooth control signal, cannot even compensate for constant unknown weights. However, the method in this paper allowed the control system to compensate for this harsh time varying uncertainty with a relatively smooth control signal. These results are shown in Figures 12 and 13. The frequency content of the control signal is shown in Figure 14. Even though there are large step changes in the unknown weights, the frequency content is similar to the case with constant weights. It is interesting to note that this same example was used in previous studies.12 However, in this case, the control signal was visibly smoother. VI. Conclusion When fast adaptation is desired for systems possessing slow dynamics, undesirable high magnitude oscillatory control signals can result. An adaptive control architecture is presented that allows the possibility of fast adaptation without oscillatory control signals and with smooth weight convergence. It oﬀers a new perspective for improving adaptive control system performance without using modiﬁcation terms. Simulation results using a model for wing rock and a Boeing 747 model illustrate the methods potential usefulness. References 1 Sharma, M. A., Calise, A. J., and Corban, E., “Application of an Adaptive Autopilot Design to a Family of Guided Munitions,” AIAA Guidance, Navigation, and Control Conference, Aug. 2000. 2 Calise, A., Lee, S., and Sharma, M., “Development of a reconfigurable flight control law for the X-36 tailless fighter aircraft,” AIAA Guidance, Navigation, and Control Conference, 2000. 3 Calise, A., Lee, S., and Sharma, M., “Development of a reconfigurable flight control law for tailless aircraft,” Journal of Guidance, Control, and Dynamics, Vol. 24, No. 5, 2001, pp. 896–902. 4 Brinker, J. and Wise, K., “Flight testing of reconfigurable control law on the x-36 tailless aircraft,” Journal of Guidance, Control, and Dynamics, Vol. 24, No. 5, 2001, pp. 903–909. 5 Wise, K., Lavretsky, E., Zimmerman, J., Francis Jr, J., Dixon, D., and Whitehead, B., “Adaptive flight control of a sensor guided munition,” AIAA Guidance, Navigation, and Control Conference, No. AIAA-2005-6385, 2005. 6 Johnson, E., Calise, A., and Corban, J., “Reusable launch vehicle adaptive guidance and control using neural networks,” AIAA Guidance, Navigation and Control Conference, Vol. 4381, Montreal, Canada: AIAA, 2001. 12 of 20 American Institute of Aeronautics and Astronautics Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 7 Johnson, E., Calise, A., El-Shirbiny, H., and Rysdyk, R., “Feedback linearization with neural network augmentation applied to X-33 attitude control,” Proceedings of the AIAA Guidance, Navigation, and Control Conference, 2000. 8 Yang, B. J., Adaptive Output Feedback Control of Flexible Systems, Ph.D. thesis, Georgia Institute of Technology, School of Aerospace Engineering, April 2004. 9 Muse, J. and Calise, A., “Adaptive Attitude and Vibration Control of the NASA Ares Crew Launch Vehicle,” AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, Hawaii, 2008. 10 Karason, S. and Annaswamy, A., “Adaptive control in the presence of input constraints,” IEEE Transactions on Automatic Control, Vol. 39, No. 11, 1994, pp. 2325–2330. 11 Muse, J., An H-Infinity Norm Minimization Approach for Adaptive Control, Ph.D. thesis, Georgia Institute of Technology, School of Aerospace Engineering, July 2010. 12 Muse, J. and Calise, A., “Adaptive Control for Systems with Slow Reference Models,” AIAA Infotech, Atlanta, Georgia, 2010. 13 Pomet, J. B. and Praly, L., “Adaptive nonlinear regulation: estimation from the Lyapunov equation,” Vol. 37, No. 6, 1992, pp. 729–740. 14 Lavretsky, E., “Adaptive Control,” Lecture notes for CDS 270 , 2010. 15 Desoer, C. and Vidyasagar, M., Feedback Systems: Input-Output Properties, Academic Press, 1975. 16 Khalil, H., “Nonlinear Systems, Prentice Hall,” Upper Saddle River, NJ , 2002. 17 Haddad, W. and Chellaboina, V., Nonlinear dynamical systems and control: A Lyapunov-based approach, Princeton University Press, 2008. 18 Narendra, K. and Parthasarathy, K., “Identification and Control of Dynamical Systems Using Neural Networks,” IEEE Transactions on Neural Networks, Vol. 1, No. 1, 1990, pp. 4–27. 19 Volyanskyy, K., Calise, A., and Yang, B., “A novel Q-modification term for adaptive control,” Proceedings of the American Control Conference, 2006, pp. 5. 20 Etkin, B. and Reid, L., “Dynamics of Flight - Stability and Control,” New York: John Wiley & Sons, Inc, 1996., 1996. 21 Ioannou, P. and Kokotovic, P., “Instability analysis and improvement of robustness of adaptive control,” Automatica, Vol. 20, No. 5, 1984, pp. 583–594. 22 Narendra, K. and Annaswamy, A., “A new adaptive law for robust adaptation without persistent excitation,” IEEE Transactions on Automatic Control, Vol. 32, No. 2, 1987, pp. 134–145. 23 Muse, J. A., “A Method For Enforcing State Constraints in Adaptive Control,” AIAA Guidance, Navigation, and Control Conference, Portland, Oregon, August 2011. 24 Muse, J. A., “An Adaptive Law With Tracking Error Dependent Adaptive Gain Adjustment Mechanism,” AIAA Guidance, Navigation, and Control Conference, Portland, Oregon, August 2011. 13 of 20 American Institute of Aeronautics and Astronautics 15 10 0 −5 −10 Commanded Angle Reference Model Angle Standard Adaptive Angle −15 0 1 2 3 4 5 6 Time − sec 7 8 9 10 Figure 1. Roll track response of the standard projection based adaptive law with a reference model natural frequency of 5 rad/s. 5 4 3 Control Moment (N−m) Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 Roll Angle (deg) 5 2 1 0 −1 −2 −3 −4 −5 0 1 2 3 4 5 6 Time − sec 7 8 9 10 Figure 2. Adaptive control eﬀort of the standard projection based adaptive law with a reference model natural frequency of 5 rad/s. 14 of 20 American Institute of Aeronautics and Astronautics 15 10 0 −5 Commanded Angle Reference Model Angle Emulator Angle Old Adaptive Law Angle New Adaptive Law Angle −10 −15 0 1 2 3 4 5 6 Time − sec 7 8 9 10 Figure 3. Roll track response of the old adaptive architecture verses the new adaptive architecture with a reference model natural frequency of 1 rad/s. 1 Old Adaptive Law New Adaptive Law 0.8 0.6 Control Moment (N−m) Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 Roll Angle (deg) 5 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 1 2 3 4 5 6 Time − sec 7 8 9 10 Figure 4. Adaptive control eﬀort of the old adaptive architecture verses the new adaptive architecture with a reference model natural frequency of 1 rad/s. 15 of 20 American Institute of Aeronautics and Astronautics From: e1 From: e2 40 35 To: Uad 20 15 10 5 0 0 To: Uad s Magnitude (dB) ; Phase (deg) 25 −45 −90 −1 10 0 1 10 10 2 10 3 −1 0 10 10 10 Frequency (rad/sec) 1 2 10 3 10 10 Figure 5. Frequency response of Fc (s) for the wing rock example. From: e From: e 1 From: e 2 From: e 3 From: e 4 5 100 To: Uad s 50 0 −50 −100 225 s 180 To: Uad Magnitude (dB) ; Phase (deg) Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 s 30 135 90 0 10 5 10 0 10 5 10 0 5 0 10 10 10 Frequency (rad/sec) 5 10 0 10 Figure 6. Frequency response of Fc (s) for the Boeing 747 example. 16 of 20 American Institute of Aeronautics and Astronautics 5 10 30 20 Pitch Angle (deg) 0 −10 −20 Commanded Angle Adaptive Angle, Γ = 1e−2 std Adaptive Angle, Γ std −30 0 5 = 1e−7 10 15 Time − sec Figure 7. Pitch track performance for the standard adaptive law with Γstd = 1x10−2 and Γstd = 1x10−7 . 10 Adaptive Control, Γstd = 1e−2 Adaptive Control, Γ 8 std = 1e−7 6 4 ad 2 u Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 10 0 −2 −4 −6 −8 −10 0 5 10 15 Time − sec Figure 8. Adaptive control eﬀort for the standard adaptive law with Γstd = 1x10−2 and Γstd = 1x10−7 . 17 of 20 American Institute of Aeronautics and Astronautics 30 20 Pitch Angle (deg) 0 −10 −20 −30 0 Commanded Angle Reference Model Angle Emulator Angle New Adaptive Law Angle 5 10 15 Time − sec Figure 9. Pitch track performance for the new architecture with Γnew = 1x10−2 . 10 8 6 4 ad 2 u Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 10 0 −2 −4 −6 −8 −10 0 5 10 15 Time − sec Figure 10. Adaptive control eﬀort for the new architecture with Γnew = 1x10−2 . 18 of 20 American Institute of Aeronautics and Astronautics 0.25 Standard Law New Architecture 0.2 |δe(f)| 0.15 0.05 0 0 5 10 15 20 25 30 Frequency (Hz) 35 40 45 50 Figure 11. Single-sided amplitude spectrum of δe for Γnew = 1x10−2 and Γstd = 1x10−2 . 30 20 10 Pitch Angle (deg) Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 0.1 0 −10 −20 −30 0 Commanded Angle Reference Model Angle Emulator Angle New Adaptive Law Angle 5 10 15 Time − sec Figure 12. Pitch track performance for the new architecture with Γnew = 1x10−2 and time varying ideal weights. 19 of 20 American Institute of Aeronautics and Astronautics 10 8 6 4 u ad 2 0 −4 −6 −8 −10 0 5 10 15 Time − sec Figure 13. Adaptive control eﬀort for the new architecture with Γnew = 1x10−2 and time varying ideal weights. 0.25 0.2 0.15 |δe(f)| Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6286 −2 0.1 0.05 0 0 5 10 15 20 25 30 Frequency (Hz) 35 40 45 50 Figure 14. Single-sided amplitude spectrum of δe for Γnew = 1x10−2 with time varying unknown system weights. 20 of 20 American Institute of Aeronautics and Astronautics

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