Document
Bounded Linear Stability Margin Analysis of Nonlinear Hybrid
Adaptive Control
Nhan T. Nguyen, Jovan D. Boskovic Abstract — This paper presents a bounded linear stability the tracking error. The study shows that the hybrid adaptive analysis for a hybrid adaptive control that blends both direct control potentially can offer better tracking performance and and indirect adaptive control. Stability and convergence of can prevent problems with high-gain control using direct nonlinear adaptive control are analyzed using an approximate MRAC alone.
linear equivalent system. A stability margin analysis shows that a large adaptive gain can lead to a reduced phase margin. This method can enable metrics-driven adaptive control whereby the adaptive gain is adjusted to meet stability margin requirements.
I. I NTRODUCTION Adaptive control is nonlinear and stability of adaptive control cannot be analyzed by the traditional phase and gain Fig. 1 - Hybrid Adaptive Control Architecture margins. These margins are used for linear control laws to provide robustness in the presence of system uncertainties.
II. H YBRID A DAPTIVE C ONTROL The lack of stability metrics for adaptive control is a major challenge to certifying adaptive control for safety-critical Given a plant model as systems. Metrics-driven adaptive control introduces a notion ˙ x = A x + B u (1) p p that adaptation should be driven by some stability metrics to achieve robustness [1]. A bounded linear stability analysis n n where x ∈ R is a state vector, u ∈ R is a control vector, method is introduced for analyzing adaptive control in terms n × n and A , B ∈ R are unknown.
p p of the linear stability concept by establishing an approximate The objective is to produce a controller that enables the linear equivalent system as a function of persistent excitation.
plant to follow a reference model described by This linear equivalent system is only used for analysis and not for actual adaptation, and can provide estimates of ˙ x = A x + B r (2) m m m m relative stability of nonlinear adaptive control for a given n × n n × n where A ∈ R is Hurwitz and given, B ∈ R is also m m adaptive gain. By adjusting the adaptive gain during the n given, and r ∈ R ∈ L is a bounded command vector with ∞ adaptation to meet certain stability margin requirements, n ˙ r ∈ R ∈ L also bounded.
∞ the adaptive law is thus made to be metrics-driven. The Defining an estimator model bounded linear stability analysis is studied in a framework of a hybrid adaptive control which blends both direct and > ˙ ˆ x = Ax + Bu + Θ Φ + u (3) ad indirect adaptive control to improve tracking performance [ ] n × n > n × 2 n ˆ ˆ [2], as shown in Fig. 1. where A , B ∈ R are known, Θ = ∆ A ∆ B ∈ R , [ ] > In recent years, direct model-reference adaptive con- > > 2 n n Φ = x u ∈ R , and u ∈ R is a direct adaptive ad trol (MRAC) using neural networks has been a topic of signal.
great research interests [3], [4], [5]. Indirect adaptive con- Defining the tracking error as ˜ x = x − x , the goal is m trol achieves adaptation by means of system identification to determine a controller that results in lim ‖ ˜ x ‖ = 0.
t → ∞ of plant parameters or uncertainties based on certainty- A dynamic inversion controller is designed from Eq. (3) equivalence control schemes [6], [7]. In this study, a re- to give the tracking error a second-order response with a cursive least-squares (RLS) indirect adaptive law is used as proportional-integral feedback control as a parameter estimation technique to reduce the modeling ( ) ∫ t error, while a direct MRAC law achieves a reduction in − 1 ˆ ˆ u = B ˙ x − A x + K ˜ x + K ˜ xd τ − u (4) m p p i ad p Nhan T. Nguyen is with the Intelligent Systems Division at NASA Ames Research Center, Moffett Field, CA 94035 USA (e-mail: ˆ ˆ ˆ ˆ where A = A + ∆ A and B = B + ∆ B are estimates p p Nhan.T.Nguyen@nasa.gov).
of A and B , K = diag ( k , . . . , k ) > 0, and K = p p p p , 1 p , n i Jovan D. Boskovic is with Scientific Systems Company, Inc., Woburn, MA 01801 USA (e-mail: Jovan.Boskovic@ssci.com). diag ( k , . . . , k ) > 0.
i , 1 i , n The tracking error dynamics are expressed as then it can be shown that ∫ 1 1 t − 1 > > ˙ R Θ + ΦΦ Θ = Φ ε (18) > ˙ 2 2 ˜ x = − K ˜ x − K ˜ xd τ + Θ Φ + u − ( A − A ) x − ( B − B ) u p i p p ad m m which results in Eq. (13). Differentiation of the identity (5) [ ] ∫ > t − 1 2 n R R = I also yields Eq. (14).
Let e = ˜ xd τ ˜ x ∈ R , then Proposition 1: The hybrid adaptive law can be shown to ( ) > ˙ e = A e + b Θ Φ + u − ε (6) be stable and result in bounded signals.
c ad ∗ ∗ ˜ Proof: Let Θ , W be constant ideal weights, and Θ = ∗ ∗ ˙ where ε = ˙ x − Ax − Bu is an estimation error which is ˜ ˜ ˙ Θ − Θ , W = W − W be weight variations, then Θ = Θ and ˙ assumed to be measurable and ˜ ˙ W = W . Consider the following Lyapunov candidate function [ ] [ ] ( ) 0 I 0 > > − 1 > − 1 ˜ ˜ ˜ ˜ A = , b = (7) V = e Pe + trace W Γ W + Θ R Θ (19) c − K − K I i p ˙ V is evaluated as The direct adaptive signal is parameterized by a linear-in- ( ) ( ) parameter matched uncertainty as > > > > ˙ V = e A P + PA e + 2 e Pb Θ Φ + W β − ε c c [ > u = W β ( x ) (8) ( ) ad > > > > > ˜ + trace − 2 ˜ W β e Pb − Θ Φ Φ Θ − ε m × n m 2 m where W ∈ R is a weight matrix and β ∈ R is a basis ] vector with Lipschitz properties ( ) d > − 1 ˜ ˜ + Θ R Θ (20) ‖ β ( x ) − β ( x ) ‖ ≤ C ‖ x − x ‖ (9) dt 0 0 − 1 for some constant C > 0, which implies a bounded derivative Since R R = I , then ∥ ∥ ( ) ( ) d d 1 ∥ ∥ − 1 − 1 − 1 > ∂ β ( x ) ˙ R R + R R = R R − ΦΦ R = 0 (21) ∥ ∥ ≤ L (10) ∥ ∥ dt dt m ∂ x So ( ) d 1 for some constant L > 0.
− 1 > R = ΦΦ (22) The adaptive law is given by dt m Using the trace property trace ( AB ) = BA , one then obtains > ˙ W = − Γ β e Pb (11) ( ) > > > ˙ ˜ V ≤ − e Qe + 2 e Pb Θ Φ + ∆ 2 n × 2 n 2 where Γ > 0 ∈ R is an adaptive gain and P > 0 ∈ R ( ) 2 1 solves the Lyapunov equation > > > > > ˜ ˜ ˜ ˜ − Φ Θ − ∆ Θ Φ + Φ Θ Θ Φ (23) 2 2 m m > ∣ ∣ ∣ ∣ PA + A P = − Q (12) c c ∗> ∗> ∣ ∣ ∣ ∣ where ∆ = sup ε − Θ Φ and ∆ = sup W β − ∆ 1 2 1 x , u x , u where Q > 0 is a symmetric positive-definite matrix. are approximation errors.
ˆ ˆ ˙ ∆ A and ∆ B are estimated by an indirect adaptive law based V is bounded by (∥ ∥ ) on the recursive least-squares (RLS) method ∥ ∥ 2 > ˜ ˙ V ≤ − λ ( Q ) ‖ e ‖ + 2 λ ( P ) ‖ e ‖ ∥ Θ Φ ∥ + ‖ ∆ ‖ min max 2 ( ) > > ˙ ∥ ∥ ∥ ∥ Θ = − R Φ Φ Θ − ε (13) 1 2 2 ∥ ∥ ∥ ∥ > > m ˜ ˜ − Θ Φ + ‖ ∆ ‖ Θ Φ ∥ ∥ ∥ ∥ 2 2 m m > ˙ = − ‖ e ‖ [ λ ( Q ) ‖ e ‖ − 2 λ ( P ) ‖ ∆ ‖ ] R = − R ΦΦ R (14) min max 2 [ ] m ∥ ∥ ∥ ∥ 1 2 ∥ ∥ ∥ ∥ > > ˜ ˜ 2 > − ∥ Θ Φ ∥ ∥ Θ Φ ∥ − 2 λ ( P ) ‖ e ‖ − ‖ ∆ ‖ (24) max 1 where m = 1 + Φ R Φ ∈ R is a normalization factor, R > 2 2 m m 2 n × 2 n 0 ∈ R is a covariance matrix.
Defining a compact set V as The proof of the RLS indirect adaptive law is as follows: { Proof: Consider the following cost functional to be mini- 2 λ ( P ) ‖ ∆ ‖ max 2 n > n ˜ V = e ∈ R , Θ Φ ∈ R : ‖ e ‖ ≥ r = , mized ∫ ∥ ∥ t λ ( Q ) 2 min 1 1 ∥ ∥ > } J ( Θ ) = ∥ Θ Φ − ε ∥ d τ (15) 2 ∥ ∥ 2 m ∥ ∥ > 2 ˜ ∥ Θ Φ ∥ ≥ r = 2 r m λ ( P ) + 2 ‖ ∆ ‖ (25) 2 1 max 1 The necessary condition is obtained as ∫ ∫ t t 1 1 and a complementary compact set S which contains e = 0 > > > ∇ J = 0 ⇒ ΦΦ Θ d τ = Φ ε d τ (16) Θ 2 2 ˜ m m and Θ = 0, then V increases in S but all trajectories of 0 0 > ˜ e and Θ Φ will stay inside of S . It follows by LaSalle’s By letting ∫ t extensions of the Lyapunov method that e and ˜ Θ are bounded, − 1 > R = ΦΦ d τ (17) 2 and so are x and u .
m ∣ ∣ ∣ ∣ > ∣ ∣ ∣ ∣ > Φ R Φ > ˙ ˜ ˙ ˜ III. B OUNDED L INEAR S TABILITY A NALYSIS Let ε = sup ∣ W β ∣ and ε = sup ∣ ∆ + Θ Φ ∣ 1 2 1 x , u , t x , u , t 2 m for t ∈ ( t − T , t ] . These error terms come from the actual Stability of nonlinear adaptive control is usually analyzed 0 0 adaptive laws (11) and (13) and thus act as bounded distur- by the Lyapunov method. The traditional linear stability mar- bances. Upon integration, one gets gin concept may be extended to nonlinear adaptive control ∫ t if it could be represented by some linear approximations.
> > z ( t ) − z ( t − T ) ≤ − b Pe β Γ β dt + ε T (32) 1 0 1 0 1 To obtain an equivalent LTI system, the adaptive law can t − T be linearized at a certain point in time when the weights ∫ t > are at a steady state, usually long after initial transients have z ( t ) − z ( t − T ) ≤ − z Φ R Φ dt + ε T (33) 2 0 2 0 2 2 m t − T settled down. However, transient responses during adaptation can be important and the adaptive law should be designed in The mean value theorem for integration states that a way that would prevent large initial transients which can ∫ ∫ b b compromise system robustness. The bounded linear stability F ( t ) G ( t ) dt = F ( c ) G ( t ) dt (34) a a analysis seeks a piecewise linear equivalent approximation where c ∈ [ a , b ] and g ( t ) ≥ 0. If G = 1, then the special case of nonlinear adaptive control in terms of a persistent excita- of the mean value theorem for integration is obtained as tion (PE) over a short, moving time window during which ∫ the LTI concept of stability margins could be analyzed to b F ( t ) dt = F ( c ) ( b − a ) (35) provide a method for adjusting the adaptive gain for the a next time window. The linear equivalent approximation is Applying the mean value theorem for integration then not a replacement of an adaptive law but rather is used in yields conjunction with the adaptive law for the stability analysis ∫ t purpose.
> > z ( t ) − z ( t − T ) ≤ − Γ b Pe ( t ) β β dt + ε T (36) 1 0 1 0 1 1 Theorem 1: The hybrid adaptive law and the tracking t − T error dynamics can be approximated by a piecewise linear ∫ t > representation as z ( t ) − z ( t − T ) ≤ − z ( t ) Φ R Φ dt + ε T (37) 2 0 2 0 2 1 2 m t − T e A b b e c where t ∈ ( t − T , t ] .
d 1 0 0 2 > z ≤ − Γ β b P 0 0 z > > 1 1 But R Φ Φ ≤ Φ R Φ , hence dt z 0 0 − a z 2 2 R b ∆ z ( t ) − z ( t − T ) ≤ − z ( t ) × 2 0 2 0 2 1 > 1 + R Φ ( t ) Φ ( t ) 0 1 1 + ε (26) ∫ t ε > × Φ Φ dt + ε T (38) t − T over a semi-open time interval t ∈ ( t − T , t ] , where z , z ∈ 0 0 1 2 R Φ n 0 2 2 Applying the mean value theorem for integration once R , a = > 0, R = λ ( R ) , and β , Φ ∈ R are 0 min 0 0 1 + R Φ 0 more gives persistent excitation values defined as ∫ t ∫ > t > Φ Φ dt = Φ ( t ) Φ ( t ) T (39) 2 2 2 > β = β β dt (27) 0 t − T T t − T ∫ If T is sufficiently small, then t ≈ t ≈ t ∈ ( t − T , t ] so t 1 2 0 0 2 > Φ = Φ Φ dt (28) that T t − T ∫ t > > > 2 > n > n Φ ( t ) Φ ( t ) ≈ Φ ( t ) Φ ( t ) = Φ Φ dt = Φ (40) ˜ ˜ 1 1 2 2 Let z = W β ∈ R and z = Θ Φ ∈ R . Then 1 2 T t − T > > > ˜ ˙ ˙ z = − b Pe β Γ β + W β (29) 1 and ( ) 1 1 z ( t ) − z ( t − T ) > ∗> > > 1 0 1 0 2 > ˜ ˙ ˙ z = − z Φ R Φ + ε − Θ Φ Φ R Φ + Θ Φ (30) ˙ z ≈ ≤ − Γ β b Pe + ε (41) 2 2 1 0 1 2 2 m m T Since β satisfies the Lipschitz condition and ˙ x is bounded z ( t ) − z ( t − T ) 2 0 2 0 ∂ β ˙ z ≈ ≤ − az + ε (42) ˙ 2 2 2 because x and u are bounded, then β = ˙ x is therefore T ∂ x [ ] > > > ˙ bounded. Also, Φ is bounded since ˙ Φ = ˙ x ˙ u and The tracking error dynamics can also be written as ˙ u can be shown to be bounded by differentiating Eq. (4) as ˙ e ≤ A e + b ( z + z + ∆ ) (43) c 1 2 2 [ − 1 > Remark 1: The piecewise linear approximation of the ˆ ˆ ˙ u = B A ˙ x + B ˙ r − A ˙ x − b A ˙ e m m m p c p nonlinear adaptive laws and the tracking error dynamics ] over a moving time window enables the adaptive control ( ) > > > > > ˙ + Θ Φ − ε Φ R Φ + b Pe β Γ β − W β (31) to be analyzed in the context of an equivalent LTI system m from which system robustness can be assessed via the linear stability margin concept during that time window. The the tracking error dynamics. On the other hand, the direct window width T can be adjusted to sufficiently capture initial MRAC interacts intimately with the tracking error which can transients for analyzing system robustness. affect robustness of the direct adaptive law. For each loop, 2 2 Remark 2: The persistent excitation values β and Φ the characteristic equation is 0 0 ( ) may be a more suitable choice than the standard persis- 3 2 2 2 ∫ t ( s + a ) s + k s + k s + Γ β p s + Γ β p = 0 (49) 0 > p i 22 12 0 0 tent excitation definition which would be β β dt t − T T ∫ t 1 0 > and ΦΦ dt , respectively. The persistent excitation For brevity, the subscript i is dropped. By factorization T t − T matrices are singular and so are not invertible. On the other with residue, the characteristic equation can be written as 2 2 hand, β and Φ are zero only if β = 0 and Φ = 0. It { 0 0 [ ( ) ( ) can be shown that the tracking error depends on β and 2 2 2 2 ( s + a ) s + Γ β α s + k − Γ β α s + k + Γ β p p i 0 0 0 22 the approximation error of the direct adaptive law, while } Φ affects the parameter convergence of the RLS indirect ] ( ) 2 2 adaptive law.
− Γ β α k − Γ β α + r = 0 (50) p 0 0 Proof: Eliminating z and z in the linearly approximate 1 2 tracking error dynamics results in where a and the residue r are defined as ( ) ( ) ( ) R Φ − 1 0 2 2 2 > 0 α = k + Γ β p p (51) s − A s + Γ β bb P e ≤ b − z + ε + ε + ∆ i 22 12 c 2 1 2 2 0 1 + R Φ [ ( )] 2 2 2 2 2 (44) r = Γ β p − Γ β α k + Γ β p − Γ β α k − Γ β α 12 i 22 p 0 0 0 0 0 For R Φ 1, a ≈ 1, the solution of z is (52) 0 2 For Γ β p k , which corresponds to fast adaptation and − T 22 i z ( t ) ≤ [ z ( t − T ) − ε ] e + ε 2 0 2 0 2 2 or large persistent excitation, then so in the limit z converges to ( ) − 1 α = Γ β p p (53) 22 12 lim sup | z | = ε (45) 2 2 [ ( )] t → ∞ − 1 − 1 − 1 r = − p p k − p p k − p p (54) 12 i 12 p 12 22 22 22 Therefore, the convergence of the tracking error can be For the ideal tracking error response with A = A and p found by B = B , the characteristic equation is second-order with p b ( ε + ∆ ) 1 2 lim sup | e | = (46) a = 0 and Γ = 0 in Eq. (50). For the system to have good 2 > t → ∞ 0 Γ β λ ( bb P ) min damping characteristics, the closed-loop poles should be a k One should note that while increasing Γ β can help p complex-conjugate pair. This implies k ≥ in order for i reduce the tracking error, the system robustness may be Re [ − λ ( A )] to be largest. Then max c compromised when it is examined in the context of the LTI ( ) − 1 stability margins.
k p − 1 − 1 p p = k ( 1 + k ) ≤ k 1 + (55) 12 p i p To analyze the linear stability of the approximate tracking 22 error and the hybrid adaptive law, the characteristic equation Thus r is relatively small if k is sufficiently large and of closed-loop system is evaluated by the Schur complement p therefore can be neglected. Then, the approximate roots of formula as ( ) 2 > the characteristic equation (50) are ( ) Γ β bb P ¯ det sI − A = det ( sI + aI ) s det sI − A + c s s = − a (56) (47) 2 − 1 s = − Γ β α = − p p (57) where ¯ A is the state transition matrix in Eq. (26). 0 ( ) Since K and K are diagonal and represent individual loop p i 2 ¯ ¯ k k p p ¯ gains for the tracking error, the determinant can be evaluated s = − ± j k − (58) i 2 4 as ( ) n where ¯ k and ¯ k are the linearly approximate adaptive pro- p i ¯ det sI − A = ( s + a ) × portional and integral gains defined as n ( ) 3 2 2 2 × s + k s + k s + Γ β p s + Γ β p (48) − 1
∏ p , i i , i 22 , i 12 , i
0 0 ¯ k = k − p p (59) p p i = 1 ( ) ( ) 2 − 1 − 1 ¯ k = k + Γ β p − p p k − p p (60) − 1 − 1 − 1 i i 22 12 p 12 22 22 where p = qk and p = qk 1 + k , i = 1 , . . . , n , 12 , i 22 , i i , i p , i i , i Equation (58) reveals that as Γ β increases for fast adapta- are diagonal elements of partitioned matrices P and P of 12 22 P , which solves Eq. (12) with Q = 2 qI , where q > 0 is a tion and or large persistent excitation, the imaginary part of constant. the complex-conjugate poles becomes large. Consequently, The linear equivalent effect of the RLS indirect adaptive fast adaptation will result in high frequency oscillations in law is to add a pole at s = − a , but it does not interact with adaptive signals, a well-known fact in adaptive control [8].
This high frequency oscillation can result in excitation of V. SIMULATION unmodeled dynamics that may be present in the system and To illustrate the bounded linear stability analysis method, a therefore can lead to a possibility of instability. The approx- simulation was performed for a damaged twin-engine generic imate bounded linear stability method is able to capture this aircraft with 25% of the left wing missing [2], as shown in behavior of nonlinear adaptive control in the linear analysis Fig. 2. The hybrid adaptive control is implemented in a flight context. This method should be able to provide a method for control to track a pitch doublet.
assessing linear stability margins that can be used to adjust the adaptive gain.
IV. M ETRICS -D RIVEN A DAPTIVE CONTROL Metric-driven adaptive control is an approach that ad- dresses stability and robustness of adaptive control in terms of quantifiable metrics. The goal of metrics-driven adaptive control is to achieve adaptation that satisfies a given set of metrics. Since adaptive control is nonlinear, the notion of metrics is not well established. Lacking of appropriate metrics for nonlinear adaptive control, the bounded linear sta- bility analysis method can provide a framework for metrics- Fig. 2 - Damaged Generic Aircraft driven adaptive control whereby the nonlinear adaptive law is approximated by a linear equivalent system. Stability of the 0.02 approximate LTI system can then be quantified in terms of Γ =0, R= 0 gain and phase margins. These margins define how close to −q (rad/sec) −0.02 m the verge of instability a control system is when subjected to 0 5 10 15 20 25 30 35 40 q −3 x 10 disturbances. The adaptive gain can then be estimated from 5 Γ =10 , R=0 the bounded linear stability analysis method to meet specified −q (rad/sec) −5 m margins and then used to drive the adaptation. Based on this 0 5 10 15 20 25 30 35 40 q −4 x 10 approach, the system transfer function is obtained by taking 4 4 Γ =10 , R=10 I the Laplace transform of the plant model as −5 −q (rad/sec) ( ) m 0 5 10 15 20 25 30 35 40 2 > q ( ) Γ β b P t (sec) ∗− 1 ∗ ∗− 1 > 0 ˆ ˆ ˆ sx = A − B B A x + B B − b A + e p p c p p p s ( ) Fig. 3 - Pitch Rate Tracking Error − az + ε + ε 2 1 2 ∗− 1 ∗> ˆ + B B sx − W β − (61) m p Figure 3 is a plot of the pitch rate tracking error. Without s adaptation ( Γ = 0 , R = 0), the tracking performance of the The open-loop transfer function between x and ˜ x is flight control is quite poor as the tracking error is large. With ( ) − 1 4 ∗− 1 ∗ ˆ ˆ the direct MRAC alone ( Γ = 10 , R = 0), the tracking error G ( s ) = sI − A + B B A × p p p p ( ) becomes smaller but high frequency contents also appear.
2 2 K + Γ β P Γ β P i 22 12 ∗− 1 0 0 ˆ × B B K + + (62) This is consistent with the closed-loop pole analysis. With p p p s s 4 4 the hybrid adaptive control ( Γ = 10 , R = 10 I ), the tracking The RLS indirect adaptive law results in convergence of error is significantly reduced along with the high frequency ∗ ∗ ˆ A → A and ˆ B → B if a → 1. Then, the transfer function p p p p contents. Thus, the hybrid adaptive control appears to be becomes more effective than the direct MRAC alone ( ) 2 2 2 K s + K + Γ β P s + Γ β P p i 22 12 0 0 ∗ G ( s ) ≈ (63) s ∗ G ( s ) can be broken into individual SISO transfer func- tions from which stability margins can be computed. Stability margins of G ( s ) can be evaluated by structured singular values, or by one loop at a time and then the worst- Imaginary Axis case stability margins could be estimated using multi-loop Increasing Γ −5 stability margin definitions [9]. The stability margins are 2 2 generally functions of Γ β . The persistent excitation β can −10 0 0 −2 −1.5 −1 −0.5 0 0.5 Real Axis be computed from Eq. (27) within a given time window.
Using this value, the adaptive gain Γ can be calculated and Fig. 4 - Root Locus for Pitch Loop used for adaptation for the next time window. This process is repeated until Γ should reach a steady state value when Figure 4 is the root locus plot of the characteristic equation the weights no longer vary. for a = 1 when R Φ 1. The root locus plot agrees well with the closed-loop pole analysis. The imaginary part of Figure 7 is a plot of the pitch rate doublet tracking and ◦ the complex-conjugate poles increases with increasing the roll and yaw rate responses to meet a phase margin of 45 adaptive gain Γ . This gives rise to high frequency oscillations with an adaptive gain Γ = Γ . The hybrid adaptive control max in the adaptive signals when Γ is large. ( Γ = Γ , R = 10 I ) clearly performs better than the direct max MRAC alone ( Γ = Γ , R = 0) , which suffers large initial max transients, although high frequency contents no longer appear Γ =10 , PM=64.1 Increasing Γ in the signals.
200 2 Γ =10 , PM=53.3 Γ =10 , PM=24.6 Γ =10 , PM=8.12 VI. C ONCLUSIONS Amplitude (dB) This paper has presented a bounded linear stability analy- −100 −90 sis method for analyzing approximate linear stability margins −135 of a nonlinear hybrid adaptive control that blends both direct −180 and recursive least-squares indirect adaptive laws. A piece- Increasing Γ −225 Phase (deg) wise, approximate linear equivalent system is formulated −270 −2 0 2 4 10 10 10 10 over a short, moving time windows within which the stability Frequency (rad/sec) margins are analyzed. The analysis relates the convergence of ∗ Fig. 5 - Bode Plot of G ( s ) of Pitch Loop the tracking error with the persistent excitation for the direct adaptive law, and the parameter convergence of the plant ∗ Figure 5 is the Bode plot of the transfer functions G ( s ) model with the persistent excitation for the indirect adaptive evaluated for the first 5 seconds. The Bode plot shows that as law. The closed-loop poles of the approximate linear equiva- Γ increases, the phase margin deteriorates. This is a typical lent system shows that increasing the adaptive gain results in behavior of a high-gain controller. Thus, while increasing Γ high-frequency oscillations in the adaptive signals. A margin leads to a better tracking performance, the relative stability analysis shows that increasing the adaptive gain causes the of the system is compromised, as high frequency signals can phase margin to decrease. Thus, there exists an upper bound excite unmodeled dynamics and lead to instability [7].
for an adaptive gain that satisfies a specified phase margin.
70 This adaptive gain is used to limit the direct adaptation in the hybrid adaptive control to provide robustness. The simulation shows that the metrics-driven hybrid adaptive control has a better tracking performance than the direct adaptive control alone.
Phase Margin (deg) R EFERENCES Γ max [1] Totah, J., Krishnakumar, K., and Vikien, S., “Integrated Resilient 0 500 1000 1500 2000 2500 3000 Aircraft Control - Stability, Maneuverability, and Safe Landing in the Γ Presence of Adverse Conditions”, NASA Aeronautics Research Mission Directorate Aviation Safety Program, April 13, 2007.
Fig. 6 - Phase margin [2] Nguyen, N., Krishnakumar, K., Kaneshige, J., and Nespeca, P., “Dy- namics and Adaptive Control for Stability Recovery of Damaged Asym- Figure 6 is a plot of the phase margin versus Γ . Increasing metric Aircraft”, AIAA Guidance, Navigation, and Control Conference , AIAA-2006-6049, 2006.
Γ causes the phase margin to decrease. MIL-F-9490 speci- [3] Rysdyk, R.T. and Calise, A.J., “Fault Tolerant Flight Control via fication for flight control systems typically requires a phase Adaptive Neural Network Augmentation”, AIAA Guidance, Navigation, ◦ margin of 45 and a gain margin of 6 dB. The adaptive gain Γ and Control Conference, AIAA-1998-4483, 1998.
[4] Johnson, E.N., Calise, A.J., El-Shirbiny, H.A., and Rysdyk, R.T., corresponding to this phase margin specification establishes “Feedback Linearization with Neural Network Augmentation Applied an upper bound Γ for metrics-driven adaptive control.
max to X-33 Attitude Control”, AIAA Guidance, Navigation, and Control Conference, AIAA-2000-4157, 2000.
[5] Hovakimyan, N., Kim, N., Calise, A.J., Prasad, J.V.R., and Corban, 0.05 E.J., “Adaptive Output Feedback for High-Bandwidth Control of an Unmanned Helicopter”, AIAA Guidance, Navigation and Control Con- q (rad) −0.05 ference, AIAA-2001-4181, 2001.
0 5 10 15 20 25 30 35 40 [6] Eberhart, R.L. and Ward, D.G., “Indirect Adaptive Flight Control 0.05 System Interactions”, International Journal of Robust and Nonlinear Control, Vol. 9, pp. 1013-1031, 1999.
p (rad) −0.05 [7] Ioannu, P.A. and Sun, J. Robust Adaptive Control , Prentice-Hall, 1996.
0 5 10 15 20 25 30 35 40 −3 [8] Cao, C., Patel, V.V., Reddy, C.K., Hovakimyan, N., Lavretsky, E., x 10 and Wise, K., “Are Phase and Time-Delay Margin Always Adversely Γ =212, R=0 Affected by High Gains?”, AIAA Guidance, Navigation, and Control Γ =212, R=10 I r (rad) −2 Conference, AIAA-2006-6347, 2006.
0 5 10 15 20 25 30 35 40 [9] Anderson, M.R., Rabin, U.H., and Vincent, J.H., “Evaluation Methods t (sec) for Complex Flight Control Systems”, AIAA Guidance, Navigation, and Control Conference, AIAA-1989-3502, 1989.
Fig. 7 - Metrics-Driven Hybrid Adaptive Control