Document
Application of Bounded Linear Stabilit y Analysis
Metho d for Metrics-Driv en A daptiv e Con trol
∗
Mary am Bakh tiari-Nejad
†
Nhan T. Nguy en
‡
Kalmanje Krishnakumar
NASA A mes R ese ar ch Center, Moett Field, CA 94035
This pap er presen ts the application of Bounded Linear Stabilit y Analysis (BLSA) metho d
for metrics-driv en adaptiv e con trol. The b ounded linear stabilit y analysis metho d is used for analyzing stabilit y of adaptiv e con trol mo dels, without linearizing the adaptiv e la ws.
Metrics-driv en adaptiv e con trol in tro duces a notion that adaptation should b e driv en b y some stabilit y metrics to ac hiev e robustness. By the application of b ounded linear stabilit y analysis metho d the adaptiv e gain is adjusted during the adaptation in order to meet cer- tain phase margin requiremen ts. Analysis of metrics-driv en adaptiv e con trol is ev aluated nd for a 2 order system that represen ts a pitc h attitude con trol of a generic transp ort air-
craft. The analysis sho ws that the system with the metrics-conforming v ariable adaptiv e
gain b ecomes more robust to unmo deled dynamics or time dela y . The eect of analysis
time-windo w for BLSA is also ev aluated in order to meet the stabilit y margin criteria.
I. In tro duction
A daptiv e con trol la ws are generally nonlinear and therefore stabilit y robustness of adaptiv e con trol cannot b e analyzed b y linear stabilit y metrics in terms of phase and gain margins. These margins are designed in to linear con trol la ws to pro vide robustness to accoun t for system uncertain ties suc h as mo deling errors and exogenous disturbances. The lac k of stabilit y metrics for adaptiv e con trol is a ma jor c hallenge to enable adaptiv e con trol la ws to b e adopted in pro duction con trol systems. Metrics-driv en adaptiv e con trol in tro duces a notion that adaptation should b e driv en b y some stabilit y metrics to ac hiev e robustness. The b ounded linear stabilit y analysis metho d is applied in order to analyze adaptiv e con trol in terms of the linear stabilit y concept b y establishing an appro ximate linear equiv alen t system as a function of a mean square v alue of teh input fucn tion to the adaptiv e la w. The metho d uses an error b ound analysis to extract the dominan t linear comp onen ts of the nonlinear adaptiv e la ws without linearization. The idea is to seek a linear represen tation that b ounds a nonlinear adaptiv e la w. This metho d can pro vide an understanding of the stabilit y margin of nonlinear adaptiv e con trol that can b e used to establish limits on adaptiv e gains during adaptation to ensure system robustness, and th us the adaptation is made to b e metrics-driv en.
I I. A daptiv e Con trol of a Second-Order System
Figure 1 sho ws the mo del-reference adaptiv e con trol (MRA C) arc hitecture for a pitc h attitude con trol of nd a generic transp ort mo del (GTM) of a 2 order system of aircraft. The plan t mo del is giv en as
˙ x = Ax + Bu (1)
[ ] [ ] θ 0 1 ˙ where x = is the state v ector, θ and θ are pitc h angle and pitc h rate resp ectiv ely , A = ˙ θ − M M q ˙ α α ∗
A erospace Engineer, In telligen t Systems Division, Mail Stop 269-1, AIAA Mem b er
† Researc h Scien tist, In telligen t Systems Division, Mail Stop 269-1, AIAA Senior Mem b er ‡ Researc h Scien tist, In telligen t Systems Division, Mail Stop 269-1, AIAA Asso ciate F ello w
1 of 11
American Institute of A eronautics and Astronautics
[ ] is assumed to b e unkno wn, and B = is kno wn.
M δ e ( )
3 ¯ c QS ¯ c QS ¯ c
A ccording to R.C. Nelson, M = M + M = C + C , and M = C , where C , q ˙ α q ˙ α m q m ˙ α δ e δ e m q 2 u I I 0 y y C , and C are the pitc hing momen t deriv ativ es, ¯ c is the mean aero dynamic c hord, S is the wing area, u m δ 0 ˙ α e is the aircraft v elo cit y along the x b o dy axis, , and I is the mass momen t, and Q = ρu , where ρ is the air y densit y .
The output of the dynamic in v ersion u is de ned as
( ) ( ) − 1 T T T
u = b B b ˙ x − b A x − u + u (2)
n m n ad pd [ ] [ ] [ ] 0 0 0 0
where b = , B = and A = con tain the nominal parameters for
n n 1 M − M − M δ q ˙ α α en n n n the dynamic in v ersion, ˙ x is a mo del-reference acceleration, u is the adaptiv e con trol signal, and u is a m ad pd prop ortional-deriv ativ e (PD) con troller.
Fig.1 MRA C Arc hitecture for Pitc h Mo del of Aircraft.
The PD con troller u is de ned as
pd
u = ke (3)
pd [ ]
where k = is a gain v ector with k and k b eing the prop ortional and deriv ativ e gains
k k p d p d resp ectiv ely , and e = x − x is the trac king error.
m
The adaptiv e signal U is parametrized b y a linear-in-parameters matc hed uncertain t y
ad >
u = W x (4)
ad where W is a w eigh t matrix and x is the basis function, whic h in the case of this example is the same as the state v ector.
The adaptiv e la w giv en b elo w can b e sho wn to b e stable based on the Ly apuno v stabilit y pro of.
> ˙
W = − Γ xe P b (5)
where Γ > 0 is an adaptiv e gain, and P solv es the Ly apuno v equation as follo ws
T P A + A P = − Q c c [ ] 0 1 where A = is Hurwitz, and Q = I > 0 is a symmetric p ositiv e-de nite matrix.
c − k − k p d Figure 2 sho ws the trac king p erformance of the adaptiv e con trol system. In order to sim ulate an un- certain t y , a 30% reduction in the pitc h damping is imp osed. A constan t adaptiv e gain of Γ = 100 is used.
Figure 2 sho ws that the trac king p erformance of the adaptiv e system with the parameteric failure is p erfectly ac heiv ed.
2 of 11
American Institute of A eronautics and Astronautics
1.2 no adaptaion output 0.8 with adaptation command 0.6 0.4 0.2 0 2 4 6 8 10 Fig.2 T rac king P erformance with Reduced Pitc h Damping.
I I I. Bounded Linear Stabilit y Analysis (BLSA)
Stabilit y of nonlinear adaptiv e con trol is usually analyzed b y the Ly apuno v metho d. The traditional linear stabilit y margin, phase margin and gain margin concept, ma y b e extended to nonlinear adaptiv e con trol if it could b e represen ted b y some linear appro ximations.The b ounded linear stabilit y analysis seeks a piecewise linear equiv alen t appro ximation of nonlinear adaptiv e con trol in terms of a mean square v alue of the input function to the adaptiv e la w o v er a short time windo w during whic h the linear time in v arian t (L TI) concept of stabilit y margins could b e analyzed to pro vide a metho d for adjusting the adaptiv e gain for the next time windo w.
The stabilit y of the adaptiv e la w can b e analyzed b y a conserv ativ ely b ounded linear stabilit y using a linear equiv alen t adaptiv e la w as follo ws.
∗ ∗ ˜ Let W b e the constan t ideal w eigh t and W = W − W b e the w eigh t v ariation, therefore ( ) d > > ˜
W x ≤ − Γ α b P e + ∆ (6)
0 1 dt where ∆ > 0 is a constan t that represen ts a b ound, and α > 0 is a mean square v alue of the input 1 0
function to the adaptiv e la w suc h that
ˆ n − 1 t + T ∑ 1 1 > T
α ≤ x x dτ ≈ x x ( t + kT ) (7)
T n t k =0 where T = n ∆ t is the analysis time windo w and n is the n um b er of time steps ∆ t .
( ) T ∗ ˜
Since u = W + W x , then the linear b ound to the adaptiv e la w (Eq.5) is rewritten as
ad d > ( u ) = − Γ α b P e + ∆ ad 0 1 dt
or
P s + P 22 12 u ( s ) = − Γ α Θ ( s ) + ∆ ad 0 e 2 s where in the Laplas transform Θ ( s ) = Θ ( s ) − Θ( s ) is the error, Θ ( s ) is the command, and ∆ is a constan t e c c 2 b ound .
Using the b ounded linear stabilit y analysis approac h the op en-lo op transfer function b et w een the output Θ( s ) and the error Θ ( s ) could easily b e deriv ed as sho wn b elo w. In deriving the op en-lo op transfer function, e the parametric uncertain t y in the pitc h damping ξ is included.
3 of 11
American Institute of A eronautics and Astronautics
Substituting Eq. in to Eq.2 and then in to Eq.1 and taking the Laplace transform results in [ ] [ ] [ ] [ ] P s + P 0 0 Θ 22 12 e
sX = ( A − A ) X + Γ α × Θ ( s ) + × + sx + ∆ (8)
n 0 e k k m 2 p d ︸ ︷︷ ︸ s 1 1 s Θ e ∆ ︸ ︷︷ ︸ ︸ ︷︷ ︸ u u ad pd [ ] Θ( s )
Substituting for X ( s ) = , Eq.8 b ecomes
s Θ( s ) ( ) P s Θ( s ) = − ( M − M + W ( t )) s Θ( s ) − ( W ( t )) Θ( s ) + Γ α + P Θ ( s ) + ( k + k s ) Θ ( s ) + ∆ q ˙ α q ˙ α 2 1 0 22 e p d e 3 n n s
(9)
where W ( t ) , W ( t ) are the adaptiv e w eigh ts, and ξ con tains the failure that w ould ev en tually b e corrected 1 2 b y the adaptiv e w eigh t W ( t ) .
The system op en-lo op transfer function is th us obtained as
Θ k s + ( k + Γ α P ) s + Γ α P d p 0 22 0 12
G ( s ) = ( s ) = (10)
3 2
Θ s + ( M − M + W ( t )) s + ( W ( t )) s
e q ˙ α q ˙ α 2 1 n n If the adaptation is suc h that it results in a p erfect mo del matc hing condition so that W ( t ) = M − 2 q ˙ α M , then the op en-lo op transfer function in Eq. (10) b ecomes an ideal transfer function: q ˙ α n n k s + ( k + Γ α P ) s + Γ α P d p 0 22 0 12 ∗
G ( s ) = (11)
s W e note that b oth transfer functions are functions of the Γ and α . The third p ossible op en-lo op transfer function is obtained when the adaptiv e w eigh ts reac h their steady state v alues, assuming that the adaptation con v erges. Then the transfer function with constan t w eigh ts is obtained as: k s + k d p
G ( s ) = (12)
s + (2 ξω − 2 ξ ω + W ) s + W n n n 2 1 n The frequency resp onse of the three op en-lo op transfer functions (Eq. 10,11,12) is sho wn in gure 3.
The adaptiv e w eigh ts in the transfer functions of equations 10 and 12 are c hosen after the adaptation is con v erged. A daptiv e con trol system with constan t adaptiv e gain Γ = 100 is sho wn later to only tolerate less than 0.1sec of time dela y and ha v e a phase margin as lo w as 5deg. Figure 3 indicates that the op en-lo op transfer function with constan t w eigh ts G ( s ) pro vides a non-practical prediction for phase margin and time dela y margin. Therefore G ( s ) should not b e used for the purp ose of metrics-driv en adaptiv e con trol analysis.
∗ On the other hand, b oth the op en lo op transfer function G ( s ) and the ideal transfer function G ( s ) pro vide v ery close frequency resp onses, and b oth predict v ery small phase margin of ab out 4 deg corresp onding to 0 . 01 sec time dela y margin.
Bode Diagram Magnitude (dB) −50 −100 −90 constant weight G(s) −135 −180 ideal G*(s) −225 Phase (deg) G(s) −270 −315 −3 −2 −1 0 1 2 3 10 10 10 10 10 10 10 Frequency (rad/sec)
4 of 11
American Institute of A eronautics and Astronautics
Fig3. F requency Resp onse Comparison at the end of A daptation.
Although the adaptiv e la w pro vides a viable trac king p erformance for a system with parametric un- certain t y as sho wn in gure 2, the system is not su cien tly robust to other t yp es of uncertain t y suc h as unmo deled dynamics or time dela y due to the v ery lo w predicted stabilit y margins. By adjusting the adap- tiv e gain in order to ac hiev e certain criteria for phase margin, adaptation is made to b e metrics-driv en. Since ∗ G ( s ) con tains the unkno wn parameter M − M + W ( t ) due to the parametric failure, G ( s ) is most q ˙ α q ˙ α 2 n n suited for adjusting the adaptiv e gain and the pro cess is explained in more details in the next section.
IV. Metrics-Driv en A daptiv e Con trol
The ideal transfer function (Eq.11) con tains all kno wn parameters and therefore it could b e used in order to adjust the adaptiv e gain Γ during the adaptation in order to meet certain phase margin requiremen ts.
The frequency resp onse of the ideal transfer function (Eq.11) is used to de ne the phase margin φ and the m corresp onding gain corsso v er frequency ω .
g By de nition, the phase margin is describ ed as: ∗ φ = arg [ G ( jω )] + Π m g
or
( ) ( k + Γ α P ) ω p 0 22 g
φ = arctan − π/ 2 (13)
m Γ α P − k ω 0 12 d g and the corresp onding gain-crosso v er frequency ω is de ned b y the follo wing expression: g ∗ | G ( jω ) | = 1 g
or
√ √ ( ) √ 2 2 Γ α P − k ω + ( k + Γ α P ) ω 0 12 d p 0 22 √ g g
= 1 (14)
4 6 ω + ω g g The α pareameter is computed according to Eq.7 within a giv en time windo w T . Using this v alue and a desired phase-margin φ , equations 13 and 14 are solv ed together using a nonlinear ro ot searc h to calculate m ω and the appropriate adaptiv e gain Γ . The calculated Γ is then used for adaptation for the next time g windo w. With the purp ose of reac hing a desired phase-margin φ , this pro cess is rep eated for eac h time m windo w un til Γ should reac h a steady state v alue. Therefore, the adjusted adaptiv e gain Γ( t ) instead of the constan t gain Γ is used in the adaptiv e con trol. The analysis result is illustrated in the sim ulation section.
Ho w ev er, analysis of the con v ergence of the system of nonlinear equations (13 and 14) sho ws that a solution for Γ and ω alw a ys exist for the selection of the giv en phase margins of φ ≤ 65 deg , and there g m w ould b e no solution for Γ or ω with the phase margins of φ > 65 deg . This fact is sho wn in gure 4, g m whic h demonstrates that only the graphs for the range of phase margins of φ = 10 − 65 deg in tersect with m the graph for the corresp onding gain crosso v er frequency ω and no solution is obtained out of this range.
g This analysis in gure 4 is sho wn for α = 1 , while same result is obtained for direten t v alues of α .
0 0
5 of 11
American Institute of A eronautics and Astronautics
φ =90 m φ =10 m solution for 0dB−magnitude or gain crossover frequency, ω g φ =20 Γ m φ =30 m adaptive gain, solutions for φ =40 phase margins φ m m φ =50 m φ =60 m φ =70 m φ =80 m 0 1 2 3 4 5 gain crossover frequency, ω g Fig. 4 Limit on the Desired φ Criteria for the Metrics-Driv en A daptiv e Con trol.
m
V. Sim ulation
In order to illustrate the metrics-driv en adaptiv e con trol with the BLSA metho d a sim ulation is p erformed nd for a 2 order pitc h attitude con trol for a generic transp ort mo del of aircraft at mac h M = 0 . 6 and altitude h = 20 , 000 f t . As men tioned in the previous section, the phase margin conforming adjusted adaptiv e gain Γ is obtained from the analysis of the ideal op en-lo op transfer function (Eq.11) with the purp ose of ac hieving a desired phase margin φ = 45 deg . The adjusted adaptiv e gain Γ instead of the constan t gain is then used m in the adaptiv e con trol. Figure 5 sho ws α that is obtained o v er the time-windo w of T = 1 . 5 sec , and also the adjusted adaptiv e gain Γ with the starting v alue of Γ = 100 , and the steady state v alue of Γ = 1 . 33 .
1.2 0.8 α 0.6 α 0.4 0.2 0 2 4 6 8 10 12 14 Γ Γ Adaptive Gain 0 2 4 6 8 10 12 14 time(sec) Fig5. α P arameter and Phase Margin Conforming V ariable A daptiv e Gain Γ .
Figure 6 sho ws the eect of the adjusted adaptiv e gain Γ v ersus the constan t adaptiv e gain Γ = 100 on the stabilit y margin analysis. The phase margin φ and time-dela y margin T D prediction b y BLSA are m m
6 of 11
American Institute of A eronautics and Astronautics
calculated from the op en-lo op transfer function G ( s ) (Eq.10). The calculated phase margin φ with the m v ariable Γ is sho wn to reac h the desired criteria φ = 45 deg . Also time-dela y margin T D is sho wn to b e m m m uc h higher with the v arying adaptiv e gain Γ , whic h indicates that the adaptiv e system with the adjusted Γ should b e able to tolerate m uc h more uncertain t y in the system.
In order to analyze the eectiv eness of the metrics-driv en adaptiv e con trol, system with added uncertain t y is ev aluated as time dela y is in tro duced in the sim ulink mo del righ t after the con trol la w and b efore the plan t.
In gure 7 the left plot sho ws that the adjusted Γ as compared to the constact Γ do es increase the time-dela y margin of the system up to 15 times. The righ t plot sho ws the result of the calculation of time-dela y margin b y BLSA metho d with the adjusted and xed Γ , and the analysis sho ws that BLSA pro vides a conserv ativ e y et practical prediction of time-dela y margin with resp ect to the actual tolerated time-dela y of the system.
(deg) m φ phase margin, 0 2 4 6 8 10 12 14 adjusted Γ fixed Γ =100 0.5 (sec) 0.4 m 0.3 0.2 0.1 time−delay margin, TD 0 2 4 6 8 10 12 14 Fig 6. Stabilit y Margin Comparison with the A djusted and Constan t A daptiv e Gain Γ .
max. time−delay time−delay margin 1.5 1.5 tolerance by the system calculated by BLSA 0.5 0.5 0 0 Γ =100 adjusted Γ Γ =100 adjusted Γ Fig 7. Analysis of Time Dela y Margin Prediction with BLSA, and Max. A dded Uncertain t y to the System.
Sim ulation of the system with the adjusted adaptiv e gain Γ as sho wn in gure 7, indicates that system can tolerate as m uc h as 1.48 sec of time dela y that is added after the con trol la w. Ho w ev er, with the constan t adaptiv e gain Γ = 100 , system can only tolerate up to 0.07 sec of added time dela y . Figure 8 illustrates the trac king p erformance with the in tro duced maxim um time dela y .
7 of 11
American Institute of A eronautics and Astronautics
1.4 fixed Γ 1.2 0.08 sec 0.8 added time−delay 0.6 0.4 0.2 0 5 10 15 20 25 30 1.4 adjusted Γ 1.2 0.8 0.6 1.7 sec 0.4 added time−delay 0.2 0 5 10 15 20 25 30 Fig 8. T rac king P erformance with the In tro duced Maxim um Time Dela y .
P erformance of the adaptiv e w eigh ts are sho wn in gure 9 to demonstrate the eect of the adjusted adaptiv e gain Γ .
0.25 0.2 0.15 0.1 Adjusted Γ 0.05 Fixed Γ =100 −0.05 −0.1 0 5 10 15 Fig 9. P erformance of the A daptiv e W eigh ts with A djusted and Fixed Γ .
No w, the c hoice of time-windo w T is ev aluated, o v er whic h the α parameter is obtained and the analysis is p erformed. The size of the analysis time-windo w T has a direct aect on ho w frequen t adaptiv e gain Γ is adjusted. As illustrated in gure 10, the c hoice of a v ery small time-windo w lik e T = 0 . 1 sec causes the adaptiv e gain Γ to b e c hanged v ery frequen tly . The v ery fast c hanges in the adaptiv e gain Γ cause a p o or con v ergence of the parameters since the adaptiv e la w will not ha v e enough time for learning. Ho w ev er, the analysis with a large time-windo w T is impractical and do es not capture the transien t. Therefore, a trade-o should b e made in c ho osing the prop er analysis time-windo w T .
8 of 11
American Institute of A eronautics and Astronautics
T=0.1sec T=1sec Γ 100 adjusted adaptive gain, 0 0.5 1 1.5 2 2.5 3 3.5 4 time (sec) Fig10. Eect of the Analysis Time-Windo w T on the A djusted A daptiv e Gain Γ .
As explained b efore, the ideal op en lo op transfer function (Eq. 11) that is obtained as a result of p erfect parameter estimation is used to calculate the v arying adaptiv e gain Γ in order to ac heiv e a certain phase margin φ criteria. The adjusted adaptiv e gain Γ is used in the adaptiv e con trol, and the true o-nominal m op en lo op transfer function (Eq. 10) is used to examine the stabilit y margins. Based on the c hoice of the time-windo w T , if the adjusted adaptiv e gain Γ do es not cause p erfect parameters con v ergence then the b eha vior of the system is aected and the desired phase margin is not reac hed. Figure 11 illustrates the eect of time-windo w T on the parametric failure related index in the denominator of the true o-nominal op en lo op transfer function , M − M + W ( t ) , whic h is supp osed to reac h zero after the adaptiv e q ˙ α q ˙ α 2 n n parameters are con v ergenced. The analysis with dieren t time-windo ws T sho ws that the b est parameter con v ergence is obtained with T = 1 . 5 sec .
−0.02 closest to zero −0.04 +W n α qn −M −0.06 α q M −0.08 −0.1 0 0.3 0.6 0.9 1.2 1.5 1.8 2 Time−Window, T (sec) Fig11. Eect of the Analysis Time-Windo w T on P arameter Con v ergence.
Figure 12 illustrates the eect of time-windo w T on the analysis of stabilit y margins, where the desired phase margin φ is completely ac hiev ed with T = 1 . 5 sec , and since the m uc h smaller time-windo w of m T = 0 . 1 sec do es not pro vide p erfect parameter con v ergence, therefore the desired stabilit y margin is not fully ac hiev ed.
9 of 11
American Institute of A eronautics and Astronautics
(deg) m φ phase margin, 0 5 10 15 T =1 sec T =0.1 sec 0.5 (sec) 0.4 m 0.3 0.2 0.1 time−delay margin, TD 0 5 10 15 Fig12. Eect of the Analysis Time-Windo w T on A c hieving the Desired Phase Margin.
Figure 12 demonstrates a similar analysis as sho wn in gure 7, but with dieren t time-windo ws T = 0 . 1 , 1 sec . The left plot illustrates that the max time-dela y system can tak e is ab out %25 larger with T = 1 sec as compared to T = 0 . 1 sec . Also, the righ t plot sho ws that the time-dela y margin calculated b y BLSA is almost %7 higher with T = 1 sec .
max. time−delay time−delay margin tolerance by the system calculated by BLSA 1.5 1.5 1 1 0.5 0.5 0 0 T=0.1 sec T=0.1 sec T=1 sec T=1 sec Fig.13 - Time-Dela y Margin Analsys with T w o Dieren t Time-Windo ws T = 0 . 1 , 1 sec .
VI. Discussion
By using the Bounded Linear Stabilit y Analysis (BLSA) concept, a metrics-driv en learning paradigm for adaptiv e con trol system is prop osed. With BLSA metho d a piecewise linear upp er b ound for the adaptiv e la w is formed, with whic h the true and ideal op en-lo op transfer functions are formed. With the goal of ac hieving certain phase margin φ criteria, the ideal transfer function is used in order to adjust the adaptiv e m gain Γ during adaptation. As demonstrated in gure ........, the analysis indicates that the adaptiv e gain Γ can alw a ys b e adjusted for a sp eci c desired phase margin on the range of φ ≤ 65 deg .
m By using the adjusted adaptiv e gain Γ during adaptation, the stabilit y margin analysis is p erformed for the true op en lo op transfer function (Eq.......... ). The analysis sho ws that the desired phase margin φ = 45 deg and a m uc h higher time-dela y margin T D = 0 . 48 sec is ac heiv ed, gure......... . While b y m m using a xed adaptiv e gain Γ = 100 , a phase margin and time-dela y margin as lo w as φ = 4 . 9 deg and m
10 of 11
American Institute of A eronautics and Astronautics
T D = 0 . 01 sec is obtained resp ectiv ely . F urthermore, b y adding a time-dela y in the sim ulink mo del b et w een m the con trol la w and plan t, system with the phase margin conforming adjusted adaptiv e gain Γ is sho wn to tak e as m uc h as 1 . 48 sec time-dala y and still remain stable. While system with the xed adaptiv e gain Γ = 100 only tak es as m uc h as 0 . 07 sec time-dela y . Therefore, system with the metrics-conforming v ariable adaptiv e gain Γ b ecomes up to 20 times more robust to unmo deled dynamics or time dela y .
The eect of the size of analysis time-windo w indicates that a trade-o should me made b et w een a large time-windo w that do es not prop erly capture transien ts and a small time-windo w that causes p o or con v ergence of the adaptiv e parameters, gure..... .
The analysis presen ted in this pap er
VI I. Conclusion
The application of b ounded linear stabilit y analysis on adaptiv e systems is ev aluated in order to construct a metho dology for establishing a metrics-driv en learning paradigm that preserv es margins during adaptation b y adjusting the adaptiv e gain. With the purp ose of ac hieving certain phase margin requiremen t, the adaptiv e nd gain is re-calculated o v er a time-windo w during adaptation. Analysis on the 2 order system with pitc h attitude con trol of the generic transp ort aircraft pro v es that with the adjusted adaptiv e gain Γ the system robustness to uncertain t y is increased signi can tly , and the desired phase margin is ac hiev ed. Analysis and ev aluation of the imp ortance of selecting the prop er time-windo w T is presen ted. F urther w ork will b e done on elab orating on the metrics-driv en learning paradigm, and also applying the metho d to realistic systems with dieren t failure scenarios.
References
J. T otah, K. Krishnakumar, and S. Vikien, In tegrated Resilien t Aircraft Con trol - Stabilit y , Maneuv erabilit y , and Safe Landing in the Presence of A dv erse Conditions, NASA A eronautics Researc h Mission Directorate A viation Safet y Program, April 13, 2007.
N.T. Nguy en, J. D. Bosk o vic, Bounded Linear Stabilit y Margin Analysis of Nonlinear Hybrid A daptiv e Con trol, American Con trol Conference, Seattle, W A, June 2008.
3 nd R.C. Nelson, Fligh t Stabilit y and Automatic Con trol, 2 edition, McGra w-Hill Science.
R.T. Rysdyk, and A.J. Calise, F ault T oleran t Fligh t Con trol via A daptiv e Neural Net w ork Augmen tation, AIAA Guid- ance, Na vigation, and Con trol Conference, AIAA-1998-4483, 1998.
W.K. Ho, O.P . Gan, E.B. T a y , and E.L. Ang, P erformance and Gain and Phase Margins of W ell-Kno wn PID T unning F orm ulas.
11 of 11