adaptive output rejection of unmatched input disturbances

Systems & Control Letters 47 (2002) 25–35www.elsevier.com/locate/sysconle

Adaptive output rejection of unmatched input disturbances�

Gang Taoa ;∗, Xidong Tanga, Suresh M. Joshib

aDepartment of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22903, USAbMail Stop 161, NASA Langley Research Center, Hampton, VA 23681, USA

Received 16 October 2000; received in revised form 25 March 2002

Abstract

To adaptively reject the e2ect of certain unmatched input disturbances on the output of a linear time-invariant system, atransfer function matching condition is needed. A lemma which presents a novel basic property of linear systems is derivedto characterize system conditions for such transfer function matching. An adaptive disturbance rejection control scheme isdeveloped for such systems with uncertain dynamics parameters and disturbance parameters. This adaptive control techniqueis applicable to control of systems with actuator failures whose failure values, failure time instants, and failure patterns areunknown. A solution is presented to this adaptive actuator failure compensation problem, which ensures closed-loop stabilityand asymptotic output tracking, in the presence of any up to m−1 uncertain failures of the total m actuators. Desired adaptivesystem performance is veri8ed by simulation results. c© 2002 Elsevier Science B.V. All rights reserved.

Keywords: Actuator failure; Adaptive control; Disturbance rejection; Output tracking; Stability; Unmatched disturbances

1. Problem statement

Consider the linear time-invariant plant

x(t) = Ax(t) + biui(t) + bjuj(t); y(t) = cx(t);(1.1)

where A∈Rn×n; bi; bj ∈Rn; c∈R1×n are constant pa-rameter matrices, the state vector x(t)∈Rn is availablefor measurement, ui(t); uj(t)∈R are two actuating in-puts, and y(t)∈R is the plant output. The basic prob-lem is to design one actuating (control) input ui(t) tocancel the e2ect of the other actuating (disturbance)input uj(t) on the plant output y(t).Let the desired behavior of y(t) be given by the

output ym(t) of the reference model

ym(t) =Wm(s)[r](t); Wm(s) =1

Pm(s); (1.2)

� This research was supported by the NASA Langley ResearchCenter under grant NCC-1342.

∗ Corresponding author.

where Wm(s) is the reference model transfer functionwith Pm(s) being a stable monic polynomial of degreen∗, and r(t) is a bounded and piecewise continuousreference input.For the case of known plant parameters, the ideal

controller structure is

ui(t) = u∗i (t) = k∗T1i x(t) + k∗2ir(t) + f∗i (t); (1.3)

where k∗1i ∈Rn; k∗2i ∈R, and f∗i (t)∈R, which leads to

the closed-loop system

x(t) = (A+ bik∗T1i )x(t)

+bik∗2ir(t) + bif∗i (t) + bjuj(t);

y(t) = cx(t): (1.4)

If (A; bi) is controllable and all zeros of (c; A; bi) arestable, there exist k∗1i and k∗2i such that

c(sI − A− bik∗T1i )−1bik∗2i =Wm(s) (1.5)

0167-6911/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S 0167 -6911(02)00166 -4

26 G. Tao et al. / Systems & Control Letters 47 (2002) 25–35

with stable zero-pole cancellations, that is, A+ bik∗T1ihas all stable eigenvalues. With this choice ofk∗1i and k∗2i, the closed-loop system (1.4) has thefrequency-domain expression

y(s) = ym(s) + c(sI − A− bik∗T1i )−1x(0)

+1k∗2i

Wm(s)f∗i (s)

+c(sI − A− bik∗T1i )−1bjuj(s): (1.6)

Based on (1.6), rejection of three classes of distur-bances may be studied: (i) uj(t)= Kuj ∈R is a constant(that is, uj(s) = Kuj=s), (ii) uj(t) =

∑ql=1(�l sin!lt +

�l cos!lt) for some constants �l; �l and !l (that is,uj(s) =

∑ql=1(�l!l=(s2 + !2

l ) + �ls=(s2 + !2l ))), and

(iii) uj(t) =∑q

l=1 �lgl(t), where gl(t); l=1; 2; : : : ; q,are bounded and continuous functions, and �l; l =1; 2; : : : ; q, are constant parameters.For class (i) disturbances, with the choice off∗

i (t)=�∗ (that is, f∗

i (s) = �∗=s), for any bj ∈Rn and Kuj ∈R(that is, uj(s) = Kuj=s), there exists a constant �∗ suchthat

lims→0

s(

1k∗2i

Wm(s)f∗i (s)

+c(sI − A− bik∗T1i )−1bjuj(s)

)= 0: (1.7)

For class (ii) disturbances, with the choice off∗

i (t)=∑q

l=1(cl sin!lt+dl cos!lt) (that is, f∗i (s)=∑q

l=1(cl!l=(s2 + !2l ) + dls=(s2 + !2

l ))), for any bj

and uj(t) in the speci8ed form, there exist constantscl and dl; l = 1; 2; : : : ; q, such that (1.7) also holds.In both cases the function s( 1

k∗2iWm(s)f∗

i (s) + c(sI −A − bik∗T1i )

−1bjuj(s)) does not have any pole inRe[s]¿ 0. Therefore, from the 8nal value theoremof Laplace transform, the time-domain function of(1=k∗2i)Wm(s)f∗

i (s)+ c(sI −A− bik∗T1i )−1bjuj(s) con-

verges to zero exponentially fast as time t goes toin8nity, which means that limt→∞(y(t)− ym(t)) = 0exponentially fast, that is, output rejection of theunmatched input disturbance uj(t) of class (i) andclass (ii) can be achieved. When the plant parame-ters A; bi; c and the disturbance parameters Kuj or �l

and �l; l = 1; 2; : : : ; q, are unknown, adaptive esti-mates of the controller parameters in (1.3) can begenerated from some stable adaptive laws such that

closed-loop stability is ensured and asymptotic track-ing limt→∞(y(t)− ym(t)) = 0 is achieved.

For class (iii) disturbances, since gl(t); l =1; 2; : : : ; q, are arbitrary, the existence of an f∗

i (t) ofthe same components gl(t) to meet the above 8nalvalue theorem condition may be impossible unlessthe following transfer function matching is ensured:

1k∗2i

Wm(s)k∗3i + c(sI − A− bik∗T1i )−1bj = 0 (1.8)

for some constant k∗3i ∈R. Under this condition,the choice of f∗

i (t) = k∗3iuj(t) in (1.3) (that is,f∗

i (s) = k∗3iuj(s)) ensures that the plant-model outputmatching: y(s) = ym(s), is achieved for x(0) = 0, andlimt→∞(y(t)−ym(t))=0 exponentially, for x(0) �=0.Clearly, if bj = �jibi for some constant �ji ∈R (thatis, the case of matched input disturbances), then k∗3iexists to satisfy (1.8) which is crucial for adaptiverejection of the class (iii) disturbances uj(t) when theparameters �l; l = 1; 2; : : : ; q, are unknown. What ifbi and bj are not parallel? This is the 8rst question tobe answered in this paper: what class of (c; A; bi; bj)can satisfy the matching equations (1.5) and (1.8)simultaneously? In Section 2, we derive a necessaryand suNcient condition for (1.5) and (1.8), based onwhich we develop an adaptive scheme in Section 3for rejecting the class (iii) disturbances uj(t) withunknown parameters �l, l = 1; 2; : : : ; q. In Section 4,we make use of this adaptive disturbance rejectionresult to derive a solution to the actuator failure com-pensation problem: both ui(t) and uj(t) are subject toa failure model described by

uk(t) = uk +q∑

l=1

Kdklfkl(t); t¿ tk ; k ∈{i; j}(1.9)

such that either one of ui(t) and uj(t) fails as de-scribed by (1.9) or both ui(t) and uj(t) do not fail,where Kuk ∈R is an unknown constant, Kdkl ∈R are someunknown constants, and fkl(t)∈R are some knownbounded signals, k = i; j; l = 1; 2; : : : ; q. The controlobjective of failure compensation is to design an adap-tive control scheme for the plant (1.1), which en-sure closed-loop stability and output tracking, for allthree possible unknown situations: (a) ui(t) = vi(t)and uj(t) = Kuj +

∑ql=1

Kdjlfjl(t), (b) uj(t) = vj(t) andui(t) = Ku i +

∑ql=1

Kdilfil(t), and (c) ui(t) = vi(t) anduj(t) = vj(t), where vi(t) and vj(t) are the applied

G. Tao et al. / Systems & Control Letters 47 (2002) 25–35 27

control signals generated from an adaptive feedbackdesign. In Section 5, we discuss extensions of theseresults and some related work in the literature.

2. A lemma for output matching

The following lemma gives a characterization ofthe set (c; A; bi; bj) for the existence of k∗1i ∈Rn andk∗2i ; k

∗3i ∈R to satisfy the matching equations (1.5) and

(1.8).

Lemma 2.1. Given (A; bi) is controllable. There existconstant k∗1i ∈Rn; k∗2i ; k

∗3i ∈R such that

c(sI − A− bik∗T1i )−1bik∗2i =Wm(s) =

1Pm(s)

; (2.1)

1k∗2i

Wm(s)k∗3i + c(sI − A− bik∗T1i )−1bj = 0; (2.2)

where Pm(s) is a monic polynomial of degree n∗; ifand only if the two systems (c; A; bi) and (c; A; bj)have the same relative degree n∗.

Proof. Su8ciency: Given that (A; bi) is controllable;and (c; A; bi) andWm(s)=1=Pm(s) have the same rela-tive degree n∗; from pole placement theory; there existconstant k∗1i ∈Rn and k∗2i = 1=cAn∗−1bi such that (2.1)is satis8ed. The main task now is to use the condition

c(sI − A− bik∗T1i )−1bi = �iiWm(s);

�ii = cAn∗−1bi; Wm(s) =1

Pm(s)(2.3)

to show that there exists k∗3i ∈R such that (2.2) is met;that is; for some �ij;

c(sI − A− bik∗T1i )−1bj = �ijWm(s) (2.4)

such that k∗3i can be chosen as k∗3i =−�ij=�ii.For KA = A + bik∗T1i ∈Rn×n; c∈R1×n and b∈Rn×1,

the resolvent formula

c(sI − KA)−1b=N (s)

det(sI − KA); (2.5)

det(sI − KA) = sn + ansn−1 + · · ·+ a2s+ a1; (2.6)

N (s) = sn−1cb+ sn−2(ancb+ c KAb) + · · ·+sn−n∗(an−n∗+2cb+ an−n∗+3c KAb+ · · ·

+anc KAn∗−2

b+ c KAn∗−1

b) + · · ·+s(a3cb+ a4c KAb+ · · ·

+anc KAn−3

b+ c KAn−2

b)

+a2cb+ a3c KAb+ · · ·+anc KA

n−2b+ c KA

n−1b (2.7)

will be used to simplify some polynomial equationsin deriving the desired result (2.4).Since c(sI − A)−1bi and c(sI − A)−1bj have the

relative degree n∗, we have that cAkbi = 0; cAkbj =0; k =0; : : : ; n∗ − 2, and cAn∗−1bi �=0; cAn∗−1bj �=0.Without loss of generality, we let cAn∗−1bi=1 (whichimplies that �ii = 1 in (2.3)), and set KA; c and bi asKA= A+ bik∗T1i

=

0 1 0 0 · · · 0

0 0 1 0 · · · 0

· · · · · ·0 0 · · · 0 0 1

−a1 −a2 · · · −an−2 −an−1 −an

∈Rn×n;

c = [c1; c2; : : : ; cn−n∗ ; 1; 0; : : : ; 0]∈R1×n;

bi = [0; : : : ; 0; 1]T ∈Rn×1; (2.8)

because (A; bi) is controllable. From (2.3) with �ii=1,we obtain

c(sI − KA)−1biPm(s) = 1; (2.9)

where Pm(s) has the form

Pm(s) = sn∗+ a∗n∗s

n∗−1 + a∗n∗−1sn∗−2

+ · · ·+ a∗2s+ a∗1 : (2.10)

Hence, we have

c = [c1; c2; : : : ; cn−n∗ ; 1; 0; : : : ; 0];

c KA = [0; c1; c2; : : : ; cn−n∗ ; 1; 0; : : : ; 0];

· · ·c KA

n∗−1= [0; 0; : : : ; 0; c1; c2; : : : ; cn−n∗ ; 1];

c KAn∗

= [− a1;−a2; : : : ;−an∗ ; c1 − an∗+1;

c2 − an∗+2; : : : ; cn−n∗ − an]: (2.11)


From the fact that c(sI − KA)−1bi and c(sI − A)−1bi

have the same zero polynomial sn−n∗ + cn−n∗sn−n∗−1

+ · · ·+ c2s+ c1, we have

c(sI − KA)−1bi

=sn−n∗ + cn−n∗sn−n∗−1 + · · ·+ c2s+ c1

sn + ansn−1 + · · ·+ a2s+ a1: (2.12)

Using this expression in (2.9), we get

[a1; a2; : : : ; an; 1; 0; : : : ; 0]

=[0; : : : ; 0; c] + [0; : : : ; 0; c; 0]a∗n∗

+[0; : : : ; 0; c; 0; 0]a∗n∗−1 + · · ·

+[c; 0; : : : ; 0]a∗1 ∈R1×(n+n∗): (2.13)

Combining (2.11) and (2.13), we conclude that

c( KAn∗

+ a∗n∗ KAn∗−1

+ a∗n∗−1KAn∗−2

+ · · ·+ a∗2 KA+ a∗1 I)

=0: (2.14)

It follows from (2.14) that

c KAkbi =−a∗n∗c KA

k−1bi − a∗n∗−1c KA

k−2bi − · · ·

−a∗2c KAk−n∗+1

bi − a∗1c KAk−n∗

bi; (2.15)

c KAkbj =−a∗n∗c KA

k−1bj − a∗n∗−1c KA

k−2bj − · · ·

−a∗2c KAk−n∗+1

bj − a∗1c KAk−n∗

bj (2.16)

for k = n∗; n∗ + 1; : : : ; n− 1.From the condition that cAkbi = 0; cAkbj = 0; k =

0; 1; : : : ; n∗ − 2, we get

c KAkbi =

1�ij

c KAkbj = 0; k = 0; 1; : : : ; n∗ − 2; (2.17)

independent of �ij. Choosing the parameter �ij as

�ij =c KA

n∗−1bj

c KAn∗−1

bi

= c KAn∗−1

bj; c KAn∗−1

bi = 1; (2.18)

we guarantee that

c KAn∗−1

bi =1�ij

c KAn∗−1

bj: (2.19)

Finally according to (2.15) and (2.16) and the fact in(2.17) and (2.19), we can prove that

c KAkbi =

1�ij

c KAkbj; (2.20)

from k = n∗ to k = n− 1. In summary, we have

c KAk(bi − 1

�ijbj

)= 0; k = 0; 1; : : : ; n− 1: (2.21)

Using this result and applying (2.5) to c(sI− KA)−1(bi−1�ij

bj), we obtain

c(sI − KA)−1(bi − 1

�ijbj

)= 0: (2.22)

In view of (2.3) (with �ii = 1), we see that (2.3) isequivalent to (2.4): c(sI−A−bik∗T1i )

−1bj=�ijWm(s).This result is true for all j = 1; 2; : : : ; m; j �= i.Necessity: Since Wm(s) has relative degree n∗, it

follows from (2.1) and (2.2) that C(sI − KA)−1bi andC(sI − KA)−1bj have the same relative degree n∗,where KA = A + bik∗T1i . Using (2.5)–(2.7) for b = bi

and b=bj, we can derive that c KAkbi=0; c KA

kbj=0; k=

0; 1; : : : ; n∗−2, and c KAn∗−1

bi �=0; c KAn∗−1

bj �=0, whichimplies that cAkbi = 0; cAkbj = 0; k = 0; 1; : : : ;n∗−2, and cAn∗−1bi �=0; cAn∗−1bj �=0, which meansthat C(sI − A)−1bi and C(sI − A)−1bj have relativedegree n∗.

The essence of Lemma 2.1 is that given any bi

and bj such that cAkbi = CAkbj = 0; k = 0; : : : ;n∗ − 2; cAn∗−1bi = 1 and cAn∗−1bj �=0, if c(sI −A−bik∗T1i )

−1bi=1=Pm(s) for Pm(s) in (2.10) and somek∗1i ∈Rn, then c(sI − A − bik∗T1i )

−1bj = −k∗3ic(sI −A − bik∗T1i )

−1bi for some k∗3i ∈R. On the other hand,if this equality holds, then (c; A; bj) has relativedegree n∗. For closed-loop stability, all zeros of Pm(s)and (c; A; bi) should be stable, that is, all eigenvaluesof A+ bik∗T1i are stable.

3. Adaptive disturbance rejection

In this section, we design an adaptive controlscheme for the plant (1.1) of unknown parameters(c; A; bi; bj), with one control actuator ui(t)=vi(t) andone disturbance actuator uj(t) =

∑ql=1 �lgl(t), where

gl(t); l = 1; 2; : : : ; q, are known, bounded and con-tinuous signals, and �l; l = 1; 2; : : : ; q, are unknown


constant parameters. The goal is to adaptively rejectthe e2ect of uj(t) by an adaptive feedback design vi(t),to ensure closed-loop stability and asymptotic track-ing: limt→∞(y(t) − ym(t)) = 0, when (A; bi) is con-trollable and all zeros of (c; A; bi) are stable.Recall that, for plant (1.1), from Lemma 2.1,

if (c; A; bi) and (c; A; bj) have the same rela-tive degree as that of Wm(s) in (1.2), and theplant parameters (c; A; bi; bj) are known, and soare the disturbance parameters �l and signalsgl(t); l=1; 2; : : : ; q, then there exist k∗1i ∈Rn; k∗2i ∈R,and f∗

i (t) =∑q

l=1 �∗l gl(t)∈R; �∗l = k∗3i�l, calcu-

lated from (1.5) and (1.8), such that controller (1.3)leads to the desired plant-model output matching:limt→∞(y(t)− ym(t)) = 0 exponentially.When the plant parameters (c; A; bi; bj) and distur-

bance parameters �l; l= 1; 2; : : : ; q, are unknown, weuse the following adaptive version of controller (1.3):

ui(t) = vi(t) = kT1i(t)x(t) + k2i(t)r(t) + fi(t);

fi(t) =q∑

l=1

�l(t)gl(t); (3.1)

where k1i(t); k2i(t); �l(t) are the estimates of k∗1i ; k∗2i ; �

∗l ,

respectively, l= 1; 2; : : : ; q.Introduce the parameter vectors $ ∗ and $ as

$ ∗ = [k∗T1i ; k∗2i ; �

∗1 ; : : : ; �

∗q ]

T;

$= [kT1i ; k2i ; �1; : : : ; �q; ]T (3.2)

and the regressor signal vector !(t) as

!(t) = [xT(t); r(t); g1(t); : : : ; gq(t)]T: (3.3)

We express the control signal ui(t) as

ui(t) = u∗i (t) + $T(t)!(t); $(t) = $(t)− $ ∗ (3.4)

and the closed-loop system as

x(t) = (A+ bik∗T1i )x(t) + bik∗2ir(t) + bi$T(t)!(t)

+bif∗i (t) + bjuj(t); y(t) = cx(t): (3.5)

Using (1.2), (1.5) and (1.8), for the tracking errore(t) = y(t)− ym(t), we have

e(t) = ce(A+bik∗T1i )tx(0) + &∗Wm(s)[$T!](t); (3.6)

where &∗ = 1=k∗2i. Introducing the auxiliary signals

'(t) =Wm(s)[!](t);

((t) = $ T(t)'(t)−Wm[$ T!](t); (3.7)

we de8ne the estimation error

)(t) = e(t) + &(t)((t); (3.8)

where &(t) is the estimate of &∗. From (3.6)–(3.8), itfollows that

)(t) = &∗$T(t)'(t) + &(t)((t) + )p(t); (3.9)

where &(t) = &(t)− &∗ and )p(t) = ce(A+bik∗T1i )tx(0) isan exponentially decaying term.Based on (3.9), we choose the adaptive laws for $

and & as

$(t) =− sign[&∗]+$'(t))(t)1 + 'T(t)'(t) + (2(t)

; +$ = +T$ ¿ 0;

(3.10)

&(t) =− �&((t))(t)1 + 'T(t)'(t) + (2(t)

; �& ¿ 0: (3.11)

This gradient adaptive scheme can be analyzed byusing the positive de8nite function

V ($; &) = 12 |&∗|$ T

+−1$ $+ 1

2�−1& &2 (3.12)

whose time-derivative of along the trajectories of(3.10) and (3.11) is

V =− )2(t)N 2(t)

+)(t))p(t)N 2(t)

6− )2(t)2N 2(t)

+)2p(t)

2N 2(t);

(3.13)

where N (t)=√1 + 'T(t)'(t) + (2(t). Since )p(t) de-

cays exponentially, it can be concluded from (3.13)that V ($; &)∈L∞ and )(t)=N (t)∈L2. It in turn impliesthat $(t); &(t)∈L∞. With (3.9) and (3.10), it is alsoobtained that )(t)=N (t)∈L∞, $∈L2∩L∞ and &∈L2∩L∞. From these desired properties of the adaptivescheme (3.10) and (3.11), the closed-loop stabilityand asymptotic tracking: limt→∞(y(t) − ym(t)) = 0,can be proved by using a standard analysis.

Remark 3.1. The conditions of Lemma 2.1 are crucialfor an adaptive disturbance rejection control schemefor plant (1.1) with unknown parameters (c; A; bi). Ifthe plant parameters (c; A; bi; bj) and disturbance pa-rameters �l and signals gl; l= 1; 2; : : : ; q; are known;from (1.6); the choice of f∗

i (s) = −k∗2iPm(s)c(sI −A− bik∗T1i )

−1bjuj(s) would lead to the desired outputmatching: y(s)=ym(s)+c(sI−A−bik∗T1i )

−1x(0);with-out needing the matching condition (1.8). However;the parametrization of this choice of f∗

i (s) needs theknowledge of (c; A; bi) when n∗ ¡n (that is; when


(c; A; bi) has 8nite zeros; in this case; they are in thedenominator of −k∗2iPm(s)c(sI − A− bik∗T1i )

−1bj; andare not convenient to be estimated); as well as thatof the derivatives of gl(t) when the relative degree ofc(sI − A− bik∗T1i )

−1bj is less than that of c(sI − A−bik∗T1i )

−1bi.

4. Adaptive actuator failure compensation

We now consider the adaptive actuator failurecompensation problem for plant (1.1) with up to oneactuator failure as modeled in (1.9): develop an adap-tive feedback control scheme to generate the appliedinput signals vi(t) and vj(t) such that closed-loop sta-bility and asymptotic output tracking are ensured forall three possible situations which are unknown to theadaptive controller: (a) ui(t) = vi(t) and uj(t) = vj(t),(b) ui(t)=vi(t) and uj(t) fails (that is, k= j in (1.9)),and (c) uj(t) = vj(t) and ui(t) fails (that is, k = i in(1.9)).It is important to note that the adaptive compensator

to be developed does not require the knowledge ofthe actuator failures: the failure values, failure timeinstants, and failure patterns (that is, which and howmany actuators have failed), also see the general casein Section 5.To develop a solution to this problem, we choose

the equal-control strategy:

vi(t) = vj(t) = v0(t) (4.1)

and assume that (c; A; bi); (c; A; bj) and (c; A; bi + bj)are all controllable and have the same relative degreen∗ as that of Wm(s) in (1.2), and their zeros are allstable.The controller structure for v0(t) is

v0(t) = kT10(t)x(t) + k20(t)r(t) + f0(t);

f0(t) = u 0(t) +j∑

k=i

q∑l=1

dkl(t)fkl(t); (4.2)

where u 0(t) is the estimate of

u∗0 =

0 if ui(t) = vi(t) and uj(t) = vj(t);

k∗3iu j if ui(t) = vi(t) and uj(t) fails;

k∗3ju i if uj(t) = vj(t) and ui(t) fails

(4.3)

with k∗3i and k∗3j de8ned from (1.8) for ui and uj, re-

spectively, dil(t) is the estimate of

d∗il =

{k∗3j Kdil if uj(t) = vj(t) and ui(t) fails

0 otherwise;(4.4)

and djl(t) is the estimate of

d∗jl =

{k∗3i Kdjl if ui(t) = vi(t) and uj(t) fails

0 otherwise;(4.5)

where Kuk ; Kdkl; k = i; j; l = 1; 2; : : : ; q, are de8ned inthe actuator failure model (1.9). The parameters k10(t)and k20(t) are the estimates of k∗10 and k∗20 which satisfythe matching equations

Wm(s)

=

c(sI − A− (bi + bj)k∗T10 )−1(bi + bj)k∗20

if ui(t) = vi(t) and uj(t) = vj(t);

c(sI − A− bik∗T10 )−1bik∗20

if ui(t) = vi(t) and uj(t) fails;

c(sI − A− bjk∗T10 )−1bjk∗20

if uj(t) = vj(t) and ui(t) fails:

(4.6)

Under the above assumption for (c; A; bi; bj), suchmatching parameters k∗10 and k∗20 exist, and the result-ing zero-pole cancellation in (4.6) is stable.It is clear that the matching parameters k∗10; k∗20; u∗0

and d∗kl; k = i; j; l= 1; : : : ; q, are piecewise constant,

with one possible jump in their values as there is onepossible actuator failure.Introduce the parameter vectors $ ∗ and $ as

$ ∗ = [k∗T10 ; k∗20; u

∗0 ; d

∗i1; : : : ; d

∗iq; d

∗j1; : : : ; d

∗jq]

T;

$= [kT10; k20; u 0; di1; : : : ; diq; dj1; : : : ; djq]T (4.7)

and the regressor signal vector !(t) as

!(t) =

[xT(t); r(t); 1; fi1(t); : : : ; fiq(t); fj1(t); : : : ; fjq(t)]T:

(4.8)

Di2erent from that in (3.6), in this case, the trackingerror equation is

e(t) = ce KAtx(0) +Wm(s)[&∗$T!](t) (4.9)


where KA= A+ Kbk∗T10 with Kb= bi + bj; bi or bj for thethree cases in (4.6), and &∗ = 1=k∗20.Introducing the auxiliary signals

'(t) =Wm(s)[!](t); ((t) = $ T(t)'(t)−Wm[$ T!](t);

(4.10)

we de8ne the estimation error

)(t) = e(t) + &(t)((t); (4.11)

where &(t) is the estimate of &∗. From (4.9)–(4.11),it follows that

)(t) = &∗$T(t)'(t) + &(t)((t) + )a(t); (4.12)

where &(t) = &(t)− &∗; )a(t) = e KAtx(0) + )b(t) with

)b(t) = &∗($ ∗T(t)'(t)−Wm(s)[$ ∗T!](t))

+Wm(s)[&∗$T!](t)− &∗Wm(s)[$

T!](t):

(4.13)

Based on (4.13), we choose the adaptive laws for$ and & as

$(t) =− sign[&∗]+$'(t))(t)1 + 'T(t)'(t) + (2(t)

; +$ = +T$ ¿ 0;

(4.14)

&(t) =− �&((t))(t)1 + 'T(t)'(t) + (2(t)

; �& ¿ 0: (4.15)

To implement this design, we need to assume that k∗20in (4.6) has a known and constant sign.The sign of &∗ is equivalent to the sign of the sys-

tem’s high-frequency gain. This sign information canbe obtained from the physics of a system. Knowingthe sign of the system’s high frequency gain is a ba-sic assumption for model adaptive reference control,which can be relaxed by using a more complicatedcontrol law using a well-known Nussbaum gain [10].This gradient adaptive scheme can be analyzed by

using the positive de8nite function

V ($; &) = 12 |&∗|$ T

+−1$ $+ 1

2�−1& &2 (4.16)

whose time-derivative of along the trajectories of(4.14) and (4.15) is

V =− )2(t)N 2(t)

+)(t))a(t)N 2(t)

6− )2(t)2N 2(t)

+)2a(t)2N 2(t)

; t �= tf; (4.17)

where N (t) =√

1 + 'T(t)'(t) + (2(t), and tf is thetime instant when one of the two actuators possiblyfails. At t= tf, the matching parameters in $ ∗ de8nedin (4.7) change their values, causing a 8nite jumpingin $ ∗ and V ($; &) as well. It can be shown that )b(t) de-cays exponentially, and so does )a(t) as e

KAtx(0) does.Then, it follows from (4.17) that V ($; &)∈L∞ (thatis, $(t); &(t)∈L∞) and )(t)=N (t)∈L2. From (4.13)and (4.14), we also have )(t)=N (t)∈L∞, $∈L2∩L∞

and &∈L2∩L∞. The closed-loop stability and asymp-totic tracking: limt→∞(y(t)− ym(t))= 0, can be alsoestablished.

5. Discussion

The above results can be extended to the generalcase with a linear time-invariant plant

x(t) = Ax(t) + Bu(t); y(t) = cx(t); (5.1)

where A∈Rn×n; B = [b1; : : : ; bm]∈Rn×m (that is,bi ∈Rn; i = 1; 2; : : : ; m), c∈R1×n are unknown con-stant parameter matrices, and with a disturbance(actuator failure) model

uk(t) = uk +q∑

l=1

Kdklfkl(t);

t¿ tk ; k ∈{1; 2; : : : ; m}; (5.2)

where Kuk ∈R is an unknown constant, Kdkl ∈R are someunknown constants, and fkl(t)∈R are some knownbounded signals, k = 1; : : : ; m; l= 1; : : : ; q; q¿ 1.Disturbance rejection: The disturbance rejection

problem is to design a feedback control law forone particular actuator ui(t) among the m actuatorsfor plant (5.1) with unknown parameters (A; B) toachieve closed-loop stability and asymptotic outputtracking: limt→∞(y(t) − ym(t)) = 0, when all otheractuators uj(t); j=1; 2; : : : ; i−1; i+1; : : : ; m, are dis-turbance signals in form (5.1) with unknown param-eters Kuj; Kdjl, and known signals fjl(t); l = 1; : : : ; q.This problem has a solution under the condition that(c; A; bi) is controllable and has all its zeros stable,and (c; A; bj); j = 1; 2; : : : ; m, have the same relativedegree n∗ as that of Wm(s) in (1.2).Actuator failure compensation: The actuator fail-

ure compensation problem is to design a set of mfeedback control signals vi(t); i = 1; 2; : : : ; m, such


that for all cases when there are up to m− 1 actuatorfailures (that is, when uj(t) = Kuj +

∑ql=1

Kdjlfjl(t),j∈{j1; : : : ; jp}⊂{1; 2; : : : ; m}; ∀p∈{1; 2; : : : ; m− 1},and uj(t) = vj(t), j �∈ {j1; : : : ; jp}), as well aswhen there is no failure (that is, when uj(t) =vj(t); j = 1; 2; : : : ; m), the closed-loop stability andasymptotic output tracking are ensured. This prob-lem can be solved by an equal-control designvi(t) = v0(t); i = 1; 2; : : : ; m, under the condition that(c; A;

∑j �=j1 ;:::;jp bj); p∈{0; 1; : : : ; m−1}, are control-

lable, and have relative degree n∗ and all their zerosstable.Related work. Actuator disturbances or failures

cause control system performance deteriorations. Toimprove system reliability in the presence of uncertainactuator disturbances or failures, robust or adaptivecontrol designs are desirable. For disturbance rejec-tion, e2ective adaptive designs have been developedin [2,5] for matched input disturbances, and in [3]for output disturbances as well, rejecting disturbanceswith even unknown frequencies. In this paper, themain issue with disturbance rejection is about un-matched input disturbances in a possibly unstableplant.There have been several approaches for control

of systems with actuator failures including mul-tiple model, switching and tuning designs [8,6],adaptive (LQ, indirect, or failure compensation)designs [1,4,7], fault diagnosis [14], and adaptiveobserver-based design [15]. In [13], adaptive statefeedback control schemes were designed to en-sure that, despite the uncertainties of Kuj; Kdjl; tj,j∈{1; 2; : : : ; m} in actuator failures and that ofthe plant parameters (A; B), all closed-loop sig-nals are bounded and the plant state vector x(t)asymptotically tracks the state vector xm(t) of areference system xm(t) = AMxm(t) + bMr(t), underthe condition that there exist constant vectorsk∗s1i ∈Rn and scalars k∗s2i ∈R such that A + bik∗Ts1i =AM ; bik∗s2i = bM : This state tracking condition ismore restrictive than that for output tracking (thatis, (c; A;

∑j �=j1 ;:::;jp bj); p∈{0; 1; : : : ; m − 1}, are

controllable, and have relative degree & and alltheir zeros stable). Adaptive output tracking ac-tuator failure compensation control designs, basedon the less restrictive output tracking condition,are proposed in [12] using state feedback and in[11] using output feedback, for plant (5.1) with

unknown Kuj; tj; j∈{1; 2; : : : ; m} in actuator failuresand unknown plant parameters (A; B; c), in the casewhen Kdkl=0, k=1; 2; : : : ; m; l=1; 2; : : : ; q, in the ac-tuator failure model (5.2). In this paper, we have de-rived a solution to the adaptive output tracking actu-ator failure compensation problem with Kdkl �=0; k =1; 2; : : : ; m; l=1; 2; : : : ; q, in the actuator failure model(5.2), and developed a solution to the adaptive rejec-tion of the unmatched input disturbances. The outputmatching lemma given in Section 2 is the key to thederivation of these new results.

6. Simulation study

To evaluate the adaptive actuator failure compen-sation control system performance, we consider a lin-earized Boeing 747 lateral motion dynamicmodel withactuator failures in the airplane’s segmented rudderservomechanism. With two augmented actuation vec-tors b2u2 and b3u3 to simulate a three-piece rudder, thelinearized Boeing 747 lateral dynamic equation [9] isexpanded as

x(t) =

−0:0558 −0:9968 0:0802 0:0415

0:598 −0:115 −0:0318 0

−3:05 0:388 −0:465 0

0 0:0805 1 0

x(t)

+

0:00729 0:01 0:005

−0:475 −0:5 −0:3

0:153 0:2 0:1

0 0 0

u(t);

y(t) = [ 0 1 0 0 ]x(t): (6.1)

Given m= 3, there are 7 possible 8nal failure pat-terns of up to 2 actuator failures: one for no failure,three for one failure, and three for two failures. Foreach two-failure case, there are 3 sub-patterns: oneactuator fails after another or they fail at the sametime. A desirable adaptive controller should ensureclosed-loop stability and output tracking for any ofthese 13 failure patterns. Such a controller can be de-rived from our design based on the stated conditionsin Section 5.


0 50 100 150 200 250 300−0.005

0

0.005

0.01

0.015

0.02

0.025

(a) Plant output y(t) (solid) and reference output ym

(t) (dashed) (rad/s), vs. time t (sec.)

0 50 100 150 200 250 300 −0.03

−0.02

−0.01

0

0.01

0.02

(b) Inputs u1(t) (solid), u

2(t) (dashed), u

3(t) (dotted) (rad), vs. time t (sec.)

Fig. 1. System response for failure pattern (i).

0 50 100 150 200 250 300−0.005

0

0.005

0.01

0.015

0.02

0.025



0 50 100 150 200 250 300 0.03

0.02

0.01

0


2(t) (dashed), u


−

−

−

Fig. 2. System response for failure pattern (ii).


0 50 100 150 200 250 300−0.005

0

0.005

0.01

0.015

0.02

0.025



0 50 100 150 200 250 300−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015


2(t) (dashed), u


Fig. 3. System response with a linear feedback design.

0 50 100 150 200 250 300−0.005

0

0.005

0.01

0.015

0.02

0.025



0 50 100 150 200 250 300 −0.03

−0.02

−0.01

0

0.01

0.02


2(t) (dashed), u


Fig. 4. System response with an adaptive design without failure compensation.


Two failure patterns are compensated in this study:(i) u1(t) does not fail, while u2(t) = Ku 2 + Kd21f21(t)for t¿ 50 s and u3(t)= Ku 3 + Kd31f31(t) for t¿ 100 s,and (ii) u1(t) does not fail, while u3(t) fails at t=50 sand u2(t) fails at t = 100 s (with the above failuremodel). The failure values are shown in Fig. 1(b) andFig. 2(b), as parts of the input signals.For simulation, Wm(s) = 1=(s + 0:5); r(t) = 0:01,

+$ = diag{10; 10; 10; 10; 10; 1; 105; 105}, �& = 10;x(0) = [0;−5 × 10−3; 0; 0]T, $(0) = [0:8; 1:2; 0:1;−0:02;−2; 0; 0; 0]T, &(0) =−0:5.Fig. 1(a) shows the output tracking error e(t) =

y(t)−ym(t) for failure pattern (i), and Fig. 2(a) showsthe error e(t) for failure pattern (ii). Figs. 1(b) and2(b) show the corresponding input signals (includingu1(t) = v0(t) and two failed input signals u2(t) andu3(t)). In both cases, the tracking error converges tosmall values after a transient response caused by ac-tuator failures, verifying the desired control systemperformance. As a comparison, for failure pattern (i),Fig. 3 shows the performance of a linear feedback de-sign based on the knowledge of (c; A; b1; b2; b3) butwithout consideration of actuator failures. Fig. 4 showsthe performance of an adaptive control design for un-known (c; A; b1; b2; b3) without failure compensation.

7. Concluding remarks

A main issue in adaptive control of plants with un-matched disturbances or actuator failures is the de-velopment of a controller parametrization capable ofachieving desired plant and reference model match-ing, when implemented with the knowledge of theplant and disturbance=failure parameters. In this pa-per, we established a necessary and suNcient condi-tion for plant-model output matching in the presenceof arbitrary unmatched input disturbances or actuatorfailures. We developed adaptive control schemes forplants with unknown parameters and such unknowndisturbances or actuator failures, and veri8ed the adap-tive control system performance by simulation results.

References

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[2] D.S. Bayard, A modi8ed augmented error algorithm foradaptive noise cancellation in the presence of plantresonances, Proceedings of the 1998 American ControlConference, Philadelphia, PA, pp. 137–141.

[3] M. Bodson, Performance of an adaptive algorithm forsinusoidal disturbance rejection in high noise, Automatica 37(7) (2001) 1133–1140.

[4] M. Bodson, J.E. Groszkiewicz, Multivariable adaptivealgorithms for recon8gurable Vight control, IEEE Trans.Control Systems Technol. 5 (2) (1997) 217–229.

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adaptive output rejection of unmatched input disturbances

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