American Institute of Aeronautics and Astronautics1

Adaptive Disturbance Rejection Control using SystemInput-Output Data

Nilesh V. Kulkarni*

QSS Group, Inc., NASA Ames Research Center, Moffett Field, CA - 94035

A stable discrete-time adaptive disturbance rejection control law is designed for linearsystems. The disturbance signal is assumed to be unknown, but can be described by a sumof periodic signals. The baseline control law is based on dynamic inversion withproportional-integral augmentation. Dynamic inversion is carried out using a system input-output model that implicitly contains the effects of the external disturbance. The input-output model is realized in an off-line manner using the available input-output data. On-lineadaptation of this control law is carried out by providing a parameterized augmentationsignal to the dynamic inversion. The parameters of this augmentation signal are updated toachieve the nominal desired error dynamics. Contrary to the Lyapunov-based approachesthat guarantee only stability, the current approach guarantees stability as well asperformance. Simulation results are provided to illustrate the overall control approach.

I. Introductionisturbance rejection forms an important control problem in various aerospace systems ranging from airplaneand space structures to aircraft combustors1-4. The external disturbances affecting the system, while unknown

in nature, can typically be represented as a sum of periodic signals over a relevant frequency domain. Variousapproaches have been proposed towards disturbance rejection in literature. Phan, Juang, and Eure5 and Darling andPhan6 have illustrated the applicability of the dynamic input-output data-based predictive control approach forsimultaneous feedback stabilization and disturbance rejection. The appealing nature of this approach lies in that itdoes not need knowledge or explicit identification of the external disturbances. At the same time, the approachdirectly uses system input-output data in realizing the controller, without explicit knowledge of the system input-state transfer function matrices. The approach uses system input-output history to compute multi-step-ahead input-output predictive models, and realizes control gains that minimize a receding horizon cost function.

In this research the problem of adaptive disturbance rejection control is addressed using system input-outputdata. The input-output data-based disturbance rejection approach outlined in references [5] and [6] is off-line innature. For on-line adaptation, the issue of stability is a key concern. Designing an adaptation law for the predictive-control-based disturbance rejection approach is challenging, given that it minimizes a receding horizon costfunction. Section II outlines a novel input-output data-based disturbance rejection approach that uses dynamicinversion with proportional augmentation. This approach uses system input-output data to identify a single stepahead dynamic input-output model. An error-based proportional controller computes the control input by invertingthe identified dynamics. This corresponds to the baseline implementation that is off-line in nature. Section IIIformulates the adaptive control approach. The off-line controller is adapted on-line by giving an additional signal tothe dynamic inversion. This signal is parameterized as the function of the system inputs, outputs, and tracking errors.This adaptive formulation is similar to the Intelligent Flight Control (IFC) Architecture being developed at theNASA Ames Research Center, which is based on the work of Rysdyk and Calise7. The IFC adaptation lawformulates a Lyapunov function of the system error and the augmented signal parameter error. The augmentedsignal parameters are then updated so that the Lyapunov function is non-increasing with time However, the currentapproach looks at the performance error, which is computed if the system error does not follow its desired dynamics.Unlike the Lyapunov-based-approach, the present approach adapts the augmented signal parameters only if thisperformance error is non-zero. The analysis provides guarantees on the boundedness of the performance andparameter errors. Section IV presents simulation results that illustrate the adaptive disturbance rejection approach ona three degree-of-freedom spring-mass-damper system. Section V provides the conclusions of this study, andoutlines directions for future research.

* Scientist, M/S 269-2, AIAA Member, nilesh@email.arc.nasa.gov

D

AIAA Guidance, Navigation, and Control Conference and Exhibit21 - 24 August 2006, Keystone, Colorado

AIAA 2006-6414

Copyright © 2006 by Nilesh Kulkarni. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

American Institute of Aeronautics and Astronautics2

II. Off-line Dynamic Inversion based Control DesignA traditional control approach involves two steps: estimating the state of the system from the given system input-

output data, and using this information of the estimated state to design a state feedback control. Both these stepsimplicitly assume the knowledge of the model of the system. For real physical systems, a model derived fromphysical principles, even if available, can show significant differences from the working physical system. This canlead to errors in the control design. If the system suffers any damages or failures during operation, state estimation,which uses the nominal system model, can produce erroneous results. This adds on to the problem of having an ill-tuned controller. A data-based approach, on the other hand, strives to design an input-output dynamic feedbackcontroller directly from the system input-output data. Phan and Juang8 have illustrated the use of data-basedpredictive controllers for feedback stabilization. Phan, Lim, and Longman9 as well as Kulkarni and Phan10 haveillustrated these data-based approaches, and their relation to the state-feedback control design.

In the presence of disturbances, the attraction of such data-based approaches increases furthermore. Thealgorithms outlined in references [5] and [6] have illustrated the applicability of the dynamic input-output data-basedapproaches for simultaneous feedback stabilization and disturbance rejection. The only additional feature for thedisturbance rejection problem corresponds to increasing the time-length of the past input-output data needed for thefeedback control design. The following analysis formulates the baseline disturbance rejection problem using systeminput-output data.

Consider a linear time-invariant system with external disturbances as:

( 1) ( ) ( ) ( )

( ) ( )

k k k k

k k

+ = + +=

dx Ax Bu B d

y Cx(1)

To help with the notations, a supervector, ( )skz is defined for a vector z as the concatenation of its value for s time

indices starting from time index k.

( )

( 1)( )

( 1)

s

k

kk

k s

+ = + −

z

zz

z

M(2)

Goodzeit and Phan11 prove that if the disturbance d can be represented by a discrete number of cyclic frequencies,the system dynamics can again be represented as a dynamic input-output model.

1( 1) ( 1) ( 1) ( )q qk k q k q k−+ = − + + − + +y yu yy A y A u B u (3)

The condition on the data length q is now given equivalent to Eq. (18) as

2 1ql n f> + + , (4)

where f corresponds to the number of distinct frequencies in the external disturbance. The disturbance informationis, thereby, subsumed in the dynamic input-output model, and the disturbance signal does not appear explicitly in thesystem equations.

Given sufficient system input-output data history, the input-output model given by Eq. (3) can be computedin an off-line manner using least squares methods. Equation (3) can be written for multiple data samples in anexpanded form as:

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1 2

1 21 2

1 2

1 2

1 2

1 2

( 1) ( 1) ( 1)

( 2) ( 2) ( 2)( 1) ( 1) ( 1)

( 1) ( 1) ( 1)

( 1) ( 1) ( 1)

( 2) ( 2) ( 2)

( 1) ( 1) (

p

pp

p

p

p

k q k q k q

k q k q k qk k k

k k k

k q k q k q

k q k q k q

k k

− + − + − +

− + − + − + + + + = − − −

− + − + − +

− + − + − ++

− −

yu

y

u u u

u u uy y y A

u u u

y y y

y y yA

y y y

K KM M M

KM M M

1 2( ) ( ) ( )

1)

p

p

k k k

k

+ −

yB u u uK

(5)

Defining the data matrices as

1 2

1 21 2

k+1 past

1 2

1 2

11 2

k past

1

( 1) ( 1) ( 1)

( 2) ( 2) ( 2)( 1) ( 1) ( 1) ,

( 1) ( 1) ( 1)

( 1) ( 1)

( 2) (( ) ( ) ( ) ,

( 1)

p

pp

p

p

k q k q k q

k q k q k qk k k

k k k

k q k q

k q kk k k

k

− + − + − +

− + − + − + = + + + = − − −

− + − +

− + − = =

−

y y y

y y yY y y y Y

y y y

u u

u uU u u u U

u

K KM M M

KM

2

2

( 1)

2) ( 2)

( 1) ( 1)

p

p

p

k q

q k q

k k

− +

+ − + − −

u

u

u u

KM M

,

Equation (5) can be written as:

past

k+1 past

k

=

yu y y

U

Y A A B Y

U

(6)

The system matrices, yuA , yA , and yB , can now be computed by inverting Eq. (6) as:

1

past past past

future past past past

future future future

T T − =

yu y y

U U U

A A B Y Y Y Y

U U U

(7)

Equations (5-7) represent the identification of a one-step-ahead model. References [5-6], equivalently, identifyone through r-step-ahead models, and then find the control gains that minimize an r-step-ahead receding horizonquadratic cost. In this paper, with on-line adaptation in mind, a different approach is outlined. The one-step aheadinput-output model, with implicit disturbance effects, can also be inverted with proportional or proportional-integralaugmentation to track a reference signal.

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Figure 1 illustrates the proposed disturbance rejection control architecture. This architecture follows the oneoutlined in references [7, 12]. A reference model takes the tracking signal, and computes a smoothened signal alongwith its desired value for the next time-step based on an appropriately chosen first or second order dynamics. For afirst order reference model, the reference signal is given by

( ) ( ) ( )ref ref ref com1k k k+ = +y K y y (8)

The controller is designed in such a way that the closed-loop output error follows a desired response. For a firstorder response, this desired error dynamics is given by

( ) ( )1 pek k+ =e K e (9)

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

des ref ref

des ref ref

1 1

1 1

pe

pe

k k k k

k k k k

+ − + = − ⇒ + = + + −

y y K y y

y y K y y(10)

( )des 1k +y represents the desired value of the output at the next time step for the error to follow the desired

dynamics given by Eq. (9). The identified input-output model given by Eq. (9) can now be inverted with this desiredvalue, ( )des 1k +y , to compute the control input, ( )ku , at the current time step.

( )( ) ( ) ( ){ }

1des 1

1ref ref 1

( ) 1 ( 1) ( 1)

( ) 1 ( 1) ( 1)

q q

p e q q

k k k q k q

k k k k k q k q

−−

−−

= + − − + − − +

⇒ = + + − − − + − − +

y y yu

y y yu

u B y A y A u

u B y K y y A y A u(11)

The control design given by Eqs (8-11) uses a first order reference model and a first order error dynamics. Ifnecessary, these can be substituted by second order dynamics, and the corresponding control law can be formulated.

III. Adaptive Control FormulationThe control design in section II inverts the input-output model that is obtained from the observed system data.

For any changes in the system dynamics or disturbance inputs, the pre-identified input-output model will differ fromthe actual model and lead to errors in the inversion process. An additional parameterized augmentation signal isinput to the dynamic inversion to account for this inversion error. Figure 2 illustrates the control architecture for theadaptive control formulation.

ReferenceModel

Proportional-Augmentation Dynamic

Inverse-+

com ( 1)k +y

ref ( 1)k +y

ref ( )ky

des ( 1)k +y+

+( )ky

Physicalplant

( )ku ( 1)k +y

( )kd

Figure 1: Disturbance rejection control design based on dynamic inversion with proportional augmentation

1z−

Kp ( 1)k +y

ref ( 1)k +y

( )ke

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The desired output value going into the dynamic inverse is now augmented as:

( ) ( ) ( ) ( )pedes ref K AA1 1 1 1k k k k+ = + + + − +y y y y , (12)

where

( ) ( ) ( )peK ref1 pek k k+ = − y K y y ,

and ( )AA 1k +y is the adaptive augmentation signal.

In order to get the form of this adaptive augmentation signal, consider Eq (11).

( )1des 1

ˆ ˆˆ( ) 1 ( 1) ( 1)q qk k k q k q−−

= + − − + − − + y y yuu B y A y A u (13)

The pre-identified input-output model matrices ˆyA , ˆ

yuA and ˆyB are now given with a hat over the original

representation since the actual values yA , yuA and yB are different, and not known. Substituting Eq. (12) in Eq. (13)

gives:

( ) ( ) ( ){ }pe

1ref K AA 1

ˆ ˆˆ( ) 1 1 1 ( 1) ( 1)q qk k k k k q k q−−= + + + − + − − + − − +y y yuu B y y y A y A u (14-a)

The control input, to provide the same desired error dynamics given by Eq. (9), however, needs to be

( ) ( ){ }pe

1ref K 1( ) 1 1 ( 1) ( 1)q qk k k k q k q−

−= + + + − − + − − +y y yuu B y y A y A u (14-b)

Equating the control input in Eqs. (14-a) and (14-b) provides the form for the adaptive augmentation signal.

Figure 2: Adaptive disturbance rejection control architecture

ReferenceModel

Proportional-Augmentation Dynamic

Inverse

+

-

+

+Physical

plant

AdaptiveAugmentation

1z−

-

( )ke

AA ( 1)k +y

com ( 1)k +y

ref ( 1)k +y

peK ( 1)k +y

ref ( 1)k +y

des ( 1)k +y

( )ky

ref ( )ky

ref ( 1)k +y

( )ky

( )ku

( )kd

( 1)k +y

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( ) ( ) ( )( ) ( ) ( )

pe

1 1AA u u 1

1ref K

ˆ ˆˆ ˆ1 ( 1) ( 1)

ˆ 1 1

q qk k q k q

k k

∗ − −−

−

+ = − − + + − − +

+ − + + +

y y y y y y y y

y y

y B B A A y B B A A u

I B B y y(15)

This adaptive augmentation signal, given that the actual values of the input-output model matrices yA , yuA and yB

are unknown, is represented as a parameterized signal.

( )AA AA1 ( ) ( )Tk k k+ =y W β (16)

AA ( )kW represents the unknown parameters or the weight vector that needs to be computed for the vector of basis

functions, ( )kβ . Given the form identified in Eq. (15) for this augmentation signal, the basis functions are chosen to

be

( ) ( )pe

1

ref K

( 1)

( ) ( 1)

1 1

q

q

k q

k k q

k k

−

− + = − + + + +

y

β u

y y

(17)

The update law for the weight vector AA ( )kW is typically obtained by considering a Lyapunov function that weights

the system and parameter error vectors. The parameter vector is then updated so that the Lyapunov function is non-increasing. This approach, while guaranteeing stability, does not address performance. The following proposedupdate law addresses stability as well as performance. The weight update is driven towards achieving the desirednominal error dynamics.

Let the modeling error be defined as:

( ) 1ˆ ˆ ˆ( 1) 1 ( 1) ( 1) ( )q qk k k q k q kε − + = + − − + + − + + y yu yy A y A u B u (18)

Substituting the expression for the control from Eq. (14),

( )( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

pe

pe

1

1ref K AA 1

ref K AA

ref pe ref AA

ˆ ˆ( 1) ( 1)( 1) 1

ˆ ˆˆ ˆ 1 1 1 ( 1) ( 1)

1 1 1 1

1 1 K ( ) ( )

q q

q q

T

k q k qk k

k k k k q k q

k k k k

k k k k k k

−

−−

− + + − + + = + − + + + + − + − − + − − + = + − + − + + +

= + − + − − +

y yu

y y y yu

A y A uy

B B y y y A y A u

y y y y

y y y y W β

ε

(19)

Rearranging,

( ) ( ) ( ) ( )ref pe ref AA

pe AA

1 1 K ( 1) ( ) ( )

( 1) K ( ) ( 1) ( ) ( )

T

T

k k k k k k k

k k k k k

+ − + − − = + −

⇒ + − = + −

y y y y W β

e e W β

ε

ε(20)

Eq. (20) implies that with errors in the dynamic inverse, the output error can still follow the prescribed first orderdynamics if the adaptive augmentation signal can cancel the modeling error. A performance error is correspondinglydefined as:

pe( 1) ( 1) K ( )k k k+ = + −E e e (21)

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Equations (15), (16), (20), and (21) illustrate that

( )AA 1 ( 1)k k∗ + = +y ε , (22)

and the performance error is consistently zero when ( ) ( )AA AA1 1k k∗+ → +y y . Thus,

AA AA( 1) ( ) ( ) ( )T Tk k k k∗+ = −E W β W β . (23)

Let ( 1)iE k + correspond to the ith element of the l-element vector performance error ( 1)k +E . Equation (23) can

thus be written element-wise as:

AA AA( 1) ( ) ( ) ( )i i

T TiE k k k k∗+ = −W β W β , (24)

where AA i

∗W and AA iW are the ith column vectors of the weight matrices AA

∗W and AAW respectively. The update

law is given for each of these column vectors of the weight matrix as:

AA AA( )* ( )* ( )

( 1) ( )1 ( ) ( )i i

iT

a k E k kk k

k k+ = +

+

βW W

β β (25)

( )a k corresponds to the learning rate that needs to satisfy the condition:

0 ( ) 2a k< < (26)

Reference [13] proves that this weight update law [Eqs.(25-26)] guarantees ( ) 0iE k → and

AA AA( )i i

k ∗→W W 1i l∀ = K as k → ∞ . For completeness, the proof is included in the Appendix section.

The final part of this analysis corresponds to investigating the behavior of the system error ( )ke . Let ( )iE k < δafter time k, where δ is some small positive scalar. This implies

( 1) ( )ii pe ie k K e k δ+ − < (27)

From Cauchy-Schwarz inequality,

( 1) ( ) ( 1) ( )i ii pe i i pe ie k K e k e k K e k+ − ≥ + − (28)

Equations (27) and (28) imply:

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( )

( )

2

3 2

2 1

( 1) ( )

( 1) ( )

( 2) ( 1)

( 2) ( )

( )

( 3) ( )

( ) ( ) 1

i

i

i

i i

i i

i i i

i i i i

i pe i

i pe i

i pe i

i pe pe i

pe i pe

i pe i pe pe

n n

i pe i pe pe pe

e k K e k

e k K e k

e k K e k

e k K K e k

K e k K

e k K e k K K

e k n K e k K K K

δ

δ

δ

δ δ

δ δ

δ δ δ

δ−

+ − <

+ < +

+ < + +

+ < + +

< + +

+ < + + +

+ < + + + + +

M

K

(29)

Since 1ipeK < , as k → ∞ , ( )ie k is bounded above as:

( )( )1

i

i

pe

e kK

δ<−

(30)

Thus, if the performance error is bounded, Eq. (30) establishes bounds on the output error.

IV. Simulation ResultsIn this section, the adaptive disturbance rejection control design is illustrated on a 3 degree-of-freedom spring-

mass-damper system. The dynamics of the system are given as:

1 2 1 2 2 21 1

1 1 1 1

2 2

3 3

2 3 2 3 3 32 24 4

2 2 2 2 2 25 5

6 6

3 3 3 3

3 3 3 3

0 1 0 0 0 0

0 0

0 0 0 1 0 0

0 0 0 0 0 1

0 0

k k c c k cx x

m m m mx x

x xdk k c c k ck cx xdt

m m m m m mx x

x xk c k c

m m m m

+ + − − = + + − − − −

11

2

1

2

31

42

5

6

0 0 0

1 0 0

0 0 0

0 1 5

0 0 0

0 0 0

0 1 0 0 0 0

0 0 0 1 0 0

ud

u

x

x

xy

xy

x

x

+ +

=

(31)

Equation (31) is used to create input-output data, which is used to identify the input-output model given by Eq.(3). For this system, the number of state elements, n, is 6, the number of outputs, l, is 2, and the number ofdisturbances, f, is 1. The lower integer bound for the past data length, q, is thus computed from Eq. (4) to be 5. Thusany value of q that is greater than 4 can capture this input-output model. For the current simulation studies, q ischosen as 8. The values of all the system constants, ( 1 2 3 1 2 3 1 2 3, , , , , , , ,m m m k k k c c c ) are each assumed to be 1. The

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disturbance signal is given by a sinusoidal signal of amplitude 5 and frequency 15 rad/s. Figure 3 presents theillustrates the open-loop dynamics of the system.

A. Regulation Results

In this set of results, the commanded output is set to be zero. The reference output follows the chosen first orderdynamics. Figure 4 presents the simulation results using the baseline dynamic inversion with proportionalaugmentation controller. The system outputs follow the reference signal, and the controller rejects the disturbance

Figure 4: System output and control input based on the dynamic inversion with proportional augmentationcontroller (red – reference output, blue – system output)

0 5 10 15 20-2

-1

0

1

2

3

4

5

time (s)

y 1(m

/s)

0 5 10 15 200

0.5

1

1.5

2

time (s)

y 2(m

/s)

0 5 10 15 20-150

-100

-50

0

50

100

150

time (s)

u1

(N)

0 5 10 15 20-20

-15

-10

-5

0

5

10

15

time (s)

u2

(N)

0 10 20 30 40 50-4

-3

-2

-1

0

1

2

3

time (s)

y 1(m

/s)

0 10 20 30 40 50-2.5

-2

-1.5

-1

-0.5

0

0.5

1

time (s)

y 2(m

/s)

0 10 20 30 40 50-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (s)

u1,u

2(N

)

0 10 20 30 40 50-5

-4

-3

-2

-1

0

1

2

3

4

5

time (s)d

(N)

Figure 3: System open-loop dynamics

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input. The control input effectively gets modulated in a sinusoidal fashion to reject the disturbance. Figure 5illustrates the behavior of the system with the same controller when the input disturbance frequency changes fromthe assumed value of 15 rad/s to 3 rad/s. The disturbance frequency change effectively alters the input-output modelused for inversion. The controller is, therefore, unable to reject the disturbance input to the system. The adaptation isnow turned on, with the parameterized augmentation signal given as input to the dynamic inversion. The parametersare updated as per the update law given in Eq. (25). Figure 6 illustrates the behavior of the system outputs andcontrol inputs with adaptation. The controller is again able to reject the disturbance signal and the system outputsconverge back to zero. Figure 7 shows the resulting elements of the weight vectors with adaptation. The weightsconverge within a few seconds.

Figure 5: System output and control input based on the dynamic inversion with proportional augmentationcontroller with different disturbance signal input (red – reference output, blue – system output)

Figure 6: System outputs and control inputs illustrating the results of adaptation (red – referenceoutput, blue – system output)

0 10 20 30 40 50-10

-5

0

5

10

time (s)

y 1(m

/s)

0 10 20 30 40 50-1

-0.5

0

0.5

1

1.5

2

2.5

time (s)

y 2(m

/s)

0 10 20 30 40 50-400

-300

-200

-100

0

100

200

300

time (s)

u1

(N)

0 10 20 30 40 50

-50

-25

0

25

time (s)

u2

(N)

0 10 20 30 40 50-10

-5

0

5

10

time (s)

y 1(m

/s)

0 10 20 30 40 50-0.5

0

0.5

1

1.5

2

2.5

time (s)y 2

(m/s

)

0 10 20 30 40 50-400

-200

0

200

400

time (s)

u1

(N)

0 10 20 30 40 50-50

-25

0

25

50

time (s)

u2

(N)

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B. Tracking Results

This set of results chooses the commanded outputs as sinusoid signals, and study the results with and withoutadaptation. Figure 8 presents the simulation results using the baseline dynamic inversion with proportionalaugmentation controller. The system outputs follow the sinusoid reference signal, and the controller rejects thedisturbance input. The control input effectively gets modulated to reject the disturbance. Figure 9 illustrates thebehavior of the system with the same controller when the input disturbance frequency changes from the assumedvalue of 15 rad/s to 3 rad/s. Again, the disturbance frequency change alters the input-output model used forinversion. The controller is, therefore, unable to reject the disturbance input to the system. The adaptation is nowturned where an additional parameterized augmentation signal is input to the dynamic inversion. The parameters areupdated as per the update law given by Eq. (25). Figure 10 illustrates the behavior of the system outputs and controlinputs with adaptation. The controller is again able to reject the disturbance signal, and the system outputs convergeback to the desired sinusoid signal.

0 20 40 60 80 100 120-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (s)

y 1(m

/s)

0 20 40 60 80 100 120-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (s)

y 2(m

/s)

0 20 40 60 80 100 120-100

-80

-60

-40

-20

0

20

40

60

80

100

time (s)

u 1(N

)

0 20 40 60 80 100 120-10

-8

-6

-4

-2

0

2

4

6

8

10

time (s)

u 2(N

)

Figure 8: System outputs and control inputs based on the dynamic inversion with proportional augmentationcontroller for sinusoidal commanded outputs (red – reference output, blue – system output)

0 5 10 15 20 25-0.3

-0.2

-0.1

0

0.1

0.2

time (s)

WA

A1

0 5 10 15 20 25-0.04

-0.02

0

0.02

0.04

time (s)

WA

A2

Figure 7: Evolution of the elements of the parameter vector of the augmenting signal for the two outputelements as a result of the update law

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0 20 40 60 80 100 120-5

-2.5

0

2.5

5

time (s)

y 1(m

/s)

0 20 40 60 80 100 120-1

-0.5

0

0.5

1

time (s)

y 2(m

/s)

0 20 40 60 80 100 120-100

-50

0

50

100

time (s)

u 1(N

)

0 20 40 60 80 100 120-20

-15

-10

-5

0

5

10

15

20

time (s)

u 2(N

)

Figure 9: System output and control input based on the dynamic inversion with proportional augmentationcontroller with different disturbance signal input for a sinusoidal tracking signal (red – reference output, blue –system output)

0 20 40 60 80 100 120-4

-3

-2

-1

0

1

2

3

4

time (s)

y 1(m

/s)

0 20 40 60 80 100 120-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

time (s)

y 2(m

/s)

0 20 40 60 80 100 120-100

-50

0

50

100

time (s)

u 1(N

)

0 20 40 60 80 100 120-50

-40

-30

-20

-10

0

10

20

30

40

time (s)

u 2(N

)

Figure 10: System outputs and control inputs illustrating the results of adaptation for the tracking case (red –reference output, blue – system output)

American Institute of Aeronautics and Astronautics13

V. ConclusionsA novel discrete-time adaptive disturbance rejection algorithm is outlined, which uses dynamic inversion with

proportional augmentation. A stable adaptation law is given for updating the parameters of the basis functions. Thebasis functions are chosen in an appropriate way, such that changes in system constants or disturbance profile can beeffectively accommodated. The control architecture is motivated by the NASA Ames Research Center IntelligentFlight Control (IFC) architecture, which is based on the work of Rysdyk and Calise7. However, the adaptation lawfollows a different approach. The IFC adaptation law is derived to have a non-increasing Lyapunov function thatguarantees boundedness of the system errors and the weights of the basis functions. The current approach proposesan adaptation law that updates the weights of the basis function only if the error does not follow its desireddynamics. This guarantees performance along with stability. Simulation results illustrate the approach on regulationand tracking problems. Future work will investigate the performance-error-based adaptation law for the control ofnon-linear systems.

VI. AcknowledgmentsThe author would like to acknowledge inspiring conversations on this topic with Kalmanje Krishnakumar, John

Kaneshige, and Nhan Nguyen of NASA Ames Research Center.

References1Addington, S., and Johnson, C. D., “Dual Mode Disturbance Accommodating Pointing Controller for Hubble SpaceTelescope,” Journal of Guidance, Control, and Dynamics, Vol. 18, No. 2, 1995, pp. 200-207.2Goodzeit, N. E., and Phan, M. Q., “System and Periodic Disturbance Identification for Feedforward-Feedback Control ofFlexible Spacecraft, AIAA-1997-682, 35th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 1997.3Gawronski, W., “Predictive Controller and Estimator for NASA Deep Space Network Antennas,” Journal of DynamicSystems, Measurement, and Control, Vol. 116, 1994, pp. 241-248.4Kulkarni, N. V., Krishnakumar, K., McIntosh, D., Kopasakis, G., “Data-Based Predictive Combustion Control,” AIAA-2005-3594, 41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Tucson, Arizona, July, 2005.5Phan, M. Q., Juang, J.-N., and Eure, K., “Design of Predictive Controllers for Simultaneous Feedback Stabilization andDisturbance Rejection from Disturbance Corrupted Data,” Proceedings of the 137th ASA Meeting and the 2nd EAA

Convention, Berlin, Germany, March 1999.6Darling, R. S. and Phan, M. Q., “Data-Based Design of Predictive Controllers for Periodic Disturbance Rejection,” AIAA-2003-5636, AIAA Guidance, Navigation and Control Conference and Exhibit, Austin, Texas, August 2003.7Rysdyk, R. T., and Calise, A. J., “Fault Tolerant Flight Control via Adaptive Neural Network Augmentation,” AIAA 98-4483, AIAA Guidance, Navigation, and Control Conference and Exhibit, Boston, MA, August 1998.8Phan, M. Q., and Juang, J.-N., “Predictive Controllers for Feedback Stabilization,” Journal of Guidance, Control, andDynamics, Vol. 21, No. 5, September-October, 1998, pp. 747-753.9Phan, M. Q., Lim, R. K., and Longman, R. W., “Unifying Input-Output and State-Space Perspectives of Predictive Control,”submitted to the Journal of Vibration and Control (preprint available at http://www.dartmouth.edu/~mqphan).10Kulkarni, N. V., and Phan, M. Q., “Data-Based Cost-to-go Design for Optimal Control,” AIAA-2002-4668, AIAAGuidance, Navigation, and Control Conference and Exhibit, Monterey, California, August 2002.11Goodzeit, N. E., and Phan, M. Q., “System Identification in the Presence of Completely Unknown Periodic Disturbances,”Journal of Guidance, Control, and Dynamics, Vol. 23, No. 2, March-April 2000, pp. 251-259.12Kaneshige, John and Gundy-Burlet, Karen, “Integrated Neural Flight and Propulsion Control System,” AIAA-2001-4386,AIAA Guidance, Navigation, and Control Conference and Exhibit, August 2001.13Goodman, G. C., Ramadge, P. J., and Caines, P. E., “Discrete-time Multivariable Adaptive Control,” IEEE Transactions onAutomatic Control, Vol. 25, No. 3, June 1980, pp. 449-456.

American Institute of Aeronautics and Astronautics14

VII. Appendix

Proof of stability for the update law given in Eq. (25) 13:

The weight error is defined as:

iAA AA AAˆ

i i

∗= −W W W (32)

Substituting in Eq. (25) gives:

i iAA AA( )* ( 1)* ( )ˆ ˆ( 1) ( )

1 ( ) ( )iT

a k E k kk k

k k

++ = +

+

βW W

β β (33)

The performance error is given by Eq. (24) as:

AA AA( 1) ( ) ( ) ( )i i

T TiE k k k k∗+ = −W β W β (34)

The weight error update can thus be given as:

i i

i

i

i

AA AAAA AA

AA AAAA

AAAA

( )* ( ) ( ) ( ) * ( )ˆ ˆ( 1) ( )

1 ( ) ( )

( )* ( ) ( ) * ( )ˆ ( )

1 ( ) ( )

ˆ( )* ( ) ( )* ( )ˆ ( )

1 ( ) ( )

i i

i i

T T

T

T

T

T

T

a k k k k kk k

k k

a k k k kk

k k

a k k k kk

k k

∗

∗

− + = + +

− = − +

= − +

W β W β βW W

β β

β W W βW

β β

β W βW

β β

(35)

The norm of the weight error is given as:

i i i

2

AA AA AAˆ ˆ ˆ( 1) ( 1) ( 1)Tk k k+ = + +W W W (36)

Substituting Eq. (35) gives:

i i

i i i

i i

i i

2 AA AAAA AA AA

AA AA

2AA AA

ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ˆ ˆ ˆ( 1) ( ) ( )

1 ( ) ( ) 1 ( ) ( )

ˆ ˆ( ) ( )

ˆ ˆ2 ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 ( ) ( )

TT T

T T

T

T T T

T

a k k k k a k k k kk k k

k k k k

k k

a k k k k k a k k

k k

+ = − − + +

=

− + +

β W β β W βW W W

β β β β

W W

W β β W β ββ β

i i

i i

i

AA AA

2 AA AAAA

ˆ ˆ( ) ( ) ( ) ( )( )

1 ( ) ( ) 1 ( ) ( )

ˆ ˆ( ) ( ) ( ) ( )( ) ( ) ( )ˆ ( ) ( ) 21 ( ) ( ) 1 ( ) ( )

T T

T T

T TT

T T

k k k kk

k k k k

k k k ka k k kk a k

k k k k

+ + = + − +

+ +

W β β W

β β β β

W β β Wβ βW

β β β β

(37)

American Institute of Aeronautics and Astronautics15

i

i i

22 2 AA

AA AA

ˆ ( ) ( )( ) ( ) ( )ˆ ˆ( 1) ( ) ( ) 21 ( ) ( ) 1 ( ) ( )

TT

T T

k ka k k kk k a k

k k k k

+ − = − + + +

W ββ βW W

β β β β (38)

For the condition given by Eq. (26),

i i

2 2

AA AAˆ ˆ( 1) ( ) 0k k+ − <W W (39)

The weight vector error norm, being bounded and decreasing, converges to zero as k → ∞ . This implies that

AA AA( )i i

k ∗→W W . Equation (38) also gives:

i

i i

i

22 2 AA

AA AA

2

AA

2

ˆ ( ) ( )( ) ( ) ( )ˆ ˆlim ( 1) ( ) lim ( ) 2 01 ( ) ( ) 1 ( ) ( )

ˆ ( ) ( )lim 0

1 ( ) ( )

lim ( ) 0

TT

T Tk k

T

Tk

ik

k ka k k kk k a k

k k k k

k k

k k

E k

→∞ →∞

→∞

→∞

+ − = − + = + +

⇒ = +

⇒ =

W ββ βW W

β β β β

W β

β β (40)

Thus ( ) 0iE k → as k → ∞ .