Discrete-time Lyapunov stable-staggered estimator MIMO adaptive interference cancelation with...

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2010; 24:275–286Published online 3 June 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/acs.1131

Discrete-time Lyapunov stable-staggered estimator MIMO adaptiveinterference cancelation with experimental verification

Jawad Arif∗,†,‡ and Suhail Akhtar

Department of Avionics, Institute of Avionics and Aeronautics, Air University, E-9 Islamabad, Pakistan

SUMMARY

Adaptive interference cancelation is of vital importance in a broad array of scientific and engineering disciplines. In this paperwe develop a closed-loop discrete-time interference cancelation algorithm. The novel features of this algorithm are its abilityto deal with multiple channels being affected by interferences with different frequency spectrums. Also we provide a proofof Lyapunov stability of closed-loop system and asymptotically perfect interference cancelation for a class of interferencesignals. Furthermore, we introduce a new approach for updating the estimator through the use of staggered estimate. Thegoal of staggered estimation is to minimize the total number of estimates/calculations done within a time period whileensuring that there is no estimator aliasing. Finally, the proposed algorithm is implemented on an TMS320C6713 DSP Kitand an experimental verification is obtained. Copyright q 2009 John Wiley & Sons, Ltd.

Received 20 July 2007; Revised 16 April 2009; Accepted 16 April 2009

KEY WORDS: adaptive interference cancelation; discrete time; multi-variable feedback; Lyapunov stable

1. INTRODUCTION

Traditionally, notch filters have been used to elimi-nate sinusoidal interferences in an information bearingsignal. A fixed notch filter can eliminate the noise whenits distribution is centered exactly at the frequency forwhich the filter is designed [1]. If the frequency ofinterference signal is unknown or if it drifts slowlythen an adaptive notch filter is required [2]. Adaptive

∗Correspondence to: Jawad Arif, Department of Avionics, Instituteof Avionics and Aeronautics, Air University, E-9 Islamabad,Pakistan.

†E-mail: [email protected], [email protected]‡Currently, he is a PhD student in the Department of Electricaland Electronics Engineering, Imperial College London, U.K.

Contract/grant sponsor: Institute of Avionics and Aeronautics,Air University, Islamabad, Pakistan.

notch filters can be implemented using a feed forwardor a feedback structure [3]. While the feed-forwardstructure alleviates stability concerns, it requires themeasurement of the interference or a signal correlatedwith the interference [2, p. 24]. In the feedback structuredirect measurement of the disturbance is not requiredbut stability is not guaranteed and needs to be demon-strated.

In this paper we consider the problem of simulta-neous multiple channel interference cancelation viaa multi-variable feedback control law. The Lyapunovstability of the closed-loop system is demonstratedby using the Lyapunov function candidate whosedifference is shown to be nonpositive. Asymptotically,perfect interference cancelation for a class of distur-bances is demonstrated through the use of a lemma onmaximally monotonically increasing subsequences.

From the algorithm implementation point of view,computation of the parameter estimate at each step

Copyright q 2009 John Wiley & Sons, Ltd.

276 J. ARIF AND S. AKHTAR

entails the use of a sophisticated and therefore expen-sive digital signal processor. In order to make use ofless capable and therefore less expensive componentsfor implementation, it is some times desirable not toupdate the parameter estimates at each time step but toallow a number of time step before an update is made.This scheme can however lead to estimator aliasingin which although the parameter error is growingperiodically it is zero at the update instants. To circum-vent this problem we propose staggered estimatorupdating in this paper.

The contents of the paper are as follows. InSection 2 we describe controller model. In Section 3we formulate the closed-loop system. The adaptivealgorithm, Lyapunov stability of the closed-loop systemand asymptotically perfect disturbance rejection forbounded time-invariant interference are demonstratedin Section 4. In Section 5 we discuss an algorithmwith staggered estimator updating and also present itsstability analysis. In Section 6, we use the proposedalgorithm to reject tonal and amplitude modulatedinterferences in a computer simulation. Finally, inSection 7 we implement the proposed algorithm ona DSP kit and obtain experimental verification of itsworkability.

2. ADAPTIVE CONTROLLER ARCHITECTURE

The structure of the proposed control scheme isdepicted in Figure 1, where w,u, z represent the inter-ference, the control and the performance, respectively.

Remark 2.1Notice that there are no plant dynamics and as suchthe architecture does not fall into the classical controltheoretic disturbance rejection paradigm. However,such a situation arises in communication systems wherethe removal of a narrow band interference is desiredfrom the intermediate frequency or base band signalin the receiver, e.g. multiple carrier-based communi-cation systems such as power-line communicationssuffer from mutual carrier interference which can bemodeled as narrow band interference and is difficultto cancel with radio frequency filters because of the

Figure 1. Adaptive controller architecture.

high currents and voltages involved [4]. Similarly,radio frequency interference in broad band orthogonalfrequency-division multiplexing cable communicationsystems using unshielded wire can be modeled asnarrow band interference in the receiver [5].

Let the strictly proper and instantaneously linearcontroller Gc be represented by the MIMO ARMAmodel

u(k)=−nc∑j=1

� j (k)u(k− j)+nc∑j=1

� j (k)z(k− j) (1)

for all k�0, where u∈Rm, z∈Rl and nc is the orderof the controller Gc. Also the controller parametermatrices �1, �2 . . . , �nc ∈Rm×m and �1, �2 . . . , �nc ∈Rm×l are determined by a yet unspecified controllerparameter estimator.

Next, define the control horizon vector

U (k)�=

⎡⎢⎢⎢⎣

u(k−1)

...

u(k−nc)

⎤⎥⎥⎥⎦∈Rq1 (2)

the performance horizon vector

Z(k)�=

⎡⎢⎢⎢⎣

z(k−1)

...

z(k−nc)

⎤⎥⎥⎥⎦∈Rq2 (3)

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2010; 24:275–286DOI: 10.1002/acs

DISCRETE TIME LYAPUNOV STABLE-STAGGERED ESTIMATOR 277

the regressor

�(k)�=

[U (k)

Z(k)

]∈Rq3 (4)

and the controller parameter matrix

�(k)�=[−�1(k) . . . −�nc(k)�1(k) . . . �nc(k)] (5)

where �(k)∈Rm×q3,q1=ncm,q2=ncl and q3=q1+q2. With this notation, (1) can be written as

u(k)=�(k)�(k) (6)

To vectorize the matrix �we use the Kronecker productidentity [6, Proposition 7.1.9, p. 249]

vec(I �(k)�(k))=(�T(k)⊗ I )vec�(k) (7)

to obtain

u(k)=�T(k)�(k) (8)

where

�T(k)�=�T(k)⊗ I, ∈Rm×q4 (9)

is the regressor matrix and

�(k)�=vec�(k), ∈Rq4 (10)

is the estimated controller parameter vector. Also,

q4�=mq3.

3. CLOSED-LOOP SYSTEM

From Figure 1 the closed-loop performance is given by

z(k)=w(k)−u(k) (11)

From (8) and (11) we have

z(k)=w(k)−�T(k)�(k) (12)

Assumption 3.1The disturbance w(k) is generated by the free responseof a Lyapunov stable LTI system of order nw�nc.

Remark 3.1Assumption 3.1 is satisfied by the linear combinationsof sinusoidal and step signals.

Remark 3.2Assumption 3.1 implies that ∃�∗ ∈Rm×q3 such thatw(k)=�T(k)vec�∗. The existence of such a �∗ isguaranteed via Corollary 4.1 of [7].

Now define

�∗ �=vec�∗ ∈Rq4

Then from (12)

z(k) = �T(k)vec�∗−�T(k)�(k)

= �T(k)�∗−�T(k)�(k)

= �T(k)[�∗− �(k)] (13)

Define the parameter error vector

�(k)�=�∗− �(k) (14)

then

z(k)=�T(k)�(k) (15)

4. ADAPTIVE ALGORITHM AND ITSSTABILITY ANALYSIS WITH

REGULAR ESTIMATOR UPDATING

4.1. Adaptive algorithm

The adaptive feedback mechanism in Figure 1 consistsof an instantaneously linear controller Gc given by (1)and a parameter update law that modifies the controllerparameters at each time step k.

To obtain the parameter update law for �, we definea priori performance as

z(k)�=�T(k)�(k+1) (16)

and the cost function

J (k, �(k+1)) � 12 z

T(k)z(k)= 12 [�T(k)�(k+1)]T

×[�T(k)�(k+1)] (17)



which is quadratic in a priori performance �T(k)�(k+1). Then we use a recursive least-squares esti-mate of �(k+1) to minimize (17); for details see, forexample, [8]. The recursive least-square estimate of�(k+1) is given by

�(k+1)= �(k)−P(k+1)�(k)z(k) (18)

P(k+1)=P(k)−G(k)�T(k)P(k) (19)

where

G(k)=P(k)�(k)[I +�T(k)P(k)�(k)]−1

Using (14), (15), (18) and (19) the closed-loop errorsystem is given by

�(k+1)= �(k)−P(k+1)�(k)�T(k)�(k) (20)

P(k+1)=P(k)−G(k)�T(k)P(k) (21)

where P(0)>0.

4.2. Stability analysis

To demonstrate that every equilibrium of the system(16), (20) and (21) is Lyapunov stable we require thefollowing lemma:

Lemma 4.1

VP(P)�= trP2 (22)

�VP(k)�= tr [P2(k+1)−P2(k)] (23)

V�(�,P)�= �

TP−1� (24)

and�V�(k)

�= �T(k+1)P−1(k+1)�(k+1)

−�T(k)P−1(k)�(k) (25)

Then,�VP(k)�0 (26)

�V�(k) = −�T(k)�(k)[I +�T(k)P(k)�(k)]−1

×�T(k)�(k) (27)

�−‖z(k)‖22

1+�∑nc

i=0[‖u(k−i)‖22+‖z(k−i)‖22](28)

where

��=�max[P(0)] (29)

Furthermore,

limk→∞�V�(k)=0 (30)

limk→∞ �(k) and limk→∞P(k) exist and z(k)→0 ask→∞.

ProofThe results (26), (28), (30), and the convergence of{�(k)}∞k=0 and {P(k)}∞k=0 follows from the standardproperties of recursive least square, see [9, p. 60], [10,p. 22], [11, p. 58] and [12, p. 202]. From (26) it followsthat:

P(k)�P(k−1)

Now we demonstrate the asymptotic convergence ofthe performance to zero. From (9) it follows that

�T(k)�(k) = [�T(k)⊗ I ][�(k)⊗ IT]= �T(k)�(k)⊗ I IT

= �T(k)�(k)⊗ I (31)

where I ∈Rm×m . From (31) it follows that:

‖�(k)‖2=‖�(k)‖2 (32)

Using (32) in (4) we have

�T(k)�(k)=nc∑i=1

[‖u(k−i)‖22+‖z(k−i)‖22] (33)

It follows from (11), (27) and (33) that:

�V�(k) � −‖z(k)‖221+�

∑nci=1[‖u(k−i)‖22+‖z(k−i)‖22]

� −‖z(k)‖221+�

∑nci=1[‖u(k−i)‖22+‖z(k−i)‖22]+M(k)

(34)

where

�=�max[P(0)]>�max[P(k)], �>0 (35)



and

M(k)=�‖z(k)‖22+�‖u(k)‖22 (36)

From (11) it follows that:

‖u(k)‖�‖w(k)‖+‖z(k)‖ (37)

From (34), (36) and (37), we have

�V�(k)�−‖z(k)‖22

1+�∑nc

i=0[‖w(k−i)‖22+2‖z(k−i)‖22]FromAssumption (3.1), it follows thatw(k) is bounded,therefore ∃c∈R+ such that

∑nci=0 ‖w(k−i)‖22<c.

Consequently

�V�(k) �−‖z(k)‖22

1+�c+2�∑nc

i=0 ‖z(k−i)‖22

= −‖z(k)‖22c1+c2

∑nci=0 ‖z(k−i)‖22

(38)

where c1�=1+�c and c2

�=2�. Now suppose that{‖z(k)‖}∞k=0 is unbounded. Then it follows fromLemma (A.1) that there exist c3>0 and c4>0 such thatthe maximal monotonically increasing subsequence{‖zi (k)‖}∞i=0 satisfies

c1+c2nc∑i=0

‖z(k−i)‖22<c3+c4‖z(k)‖22

which implies that

�V�i(k)�

−‖zi (k)‖22c3+c4‖zi (k)‖22

(39)

for all i=1,2, . . . . Furthermore, since by (30)limi→∞ �V�i

(k)=0, it follows that:

limi→∞

−‖zi (k)‖22c3+c4‖zi (k)‖22

=0 (40)

Therefore, the maximally monotonically increasingsubsequence zi (k)→0 as k→∞, which is a contra-diction. Hence, {‖z(k)‖}∞k=0 is bounded and it follows

from Lemma A.1 that there exist c5>0 and c6>0 suchthat, for all k�0,

�V�(k)�−‖z(k)‖22

c5+c6‖z(k)‖22(41)

Now using (30), we have

limk→∞

−‖z(k)‖22c5+c6‖z(k)‖22

=0 (42)

Therefore, z(k)→0 as k→∞. �

Remark 4.1In the algorithm described in this section, the controllerparameter vector is estimated using an estimator witha time-varying gain P(k) that is nonincreasing. Thischaracteristic does not allow the estimator to react tochanges in the disturbance spectrum once the P(k) issignificantly reduced. In order to overcome this problemwe have previously proposed an alternate method thatuses the projection algorithm to update the parameterestimates [13]. Since in this method the estimator gainis only a function of the instantaneous value of theregressor, the estimator is able to track a slowly time-varying interference.

5. ADAPTIVE ALGORITHM WITHSTAGGERED ESTIMATOR UPDATING

AND STABILITY ANALYSIS

In this section we consider the case where the estimatoris updated less frequently, i.e it is not updated at eachtime step. In Section 5.1, first we consider the sameadaptive algorithm described in Section 4 except thatthe parameter estimates remain unchanged for n steps.We then explain the problem encountered by adoptingthis methodology. Finally, we suggest the staggeredestimator updating as a means of allowing more timefor estimator computations while avoiding the problemof estimator aliasing.

5.1. Adaptive algorithm

The adaptive feedback mechanism in Figure 1 consistsof an instantaneously linear controller Gc given by (1)and a parameter update law that modifies the controller



parameters at time step k= jn, where j =1,2,3, . . .and n is a positive integer. For n=1, the regular esti-mator updates the controller parameters at each timestep, which is already described in Section 4. Here,we explain the case where estimator updates are infre-quent, i.e. n>1. Consider the control law (8) under theconstraint

�( jn)= �( jn−1)=, · · · ,= �( jn−n+1) (43)

That is, the gain estimate is not updated for n steps.To obtain the parameter update law for �, we definea priori performance as

z( jn)�=�T( jn)�( jn+1) (44)

and the cost function

J = 12 [�T( jn)�( jn+1)]T[�T( jn)�( jn+1)] (45)

which is quadratic in a priori performance�T( jn)�( jn).Then we use a recursive least-square estimate of�( jn+1) to minimize (45); for details see, [8, 9, 14].The recursive least-square estimate of �( jn+1) isgiven by

�( jn+1) = �( jn)

−P( jn+1)�( jn)z( jn) (46)

P( jn+1) =P( jn)

−G( jn)�T( jn)P( jn) (47)

where

G( jn)=P( jn)�( jn)[I +�T( jn)P( jn)�( jn)]−1

Using (14), (15), (46) and (47) the closed-loop errorsystem is given by

�( jn+1) = �( jn)−P( jn+1)

×�( jn)�T( jn)�( jn) (48)

P( jn+1)=P( jn)−G( jn)�T( jn)P( jn) (49)

where P(0)>0,

�( jn)= �( jn−1)=, · · · ,= �( jn+n−1)

and

P( jn)=P( jn−1)=, · · · ,=P( jn+n−1)

5.2. Stability analysis

To demonstrate that every equilibrium of the system(44), (48) and (49) is Lyapunov stable we require thefollowing lemma:

Lemma 5.1Define

VP(P)�= trP2 (50)

�VP( jn)�= tr [P2( jn+1)−P2( jn)] (51)

V�(�,P)�= �

TP−1� (52)

and

�V�( jn)�= �

T( jn+1)P−1( jn+1)�( jn+1)

−�T( jn)P−1( jn)�( jn) (53)

Then,

�VP( jn)�0 (54)

�V�( jn) = −�T( jn)�( jn)

×[I +�T( jn)P( jn)�( jn)]−1�T� (55)

�−‖z( jn)‖22

1+�∑nc

i=0[‖u( jn−i)‖22+‖z( jn−i)‖22](56)

where

��=�max[P(0)] (57)

Furthermore,

limj→∞�V�( jn)=0 (58)

lim j→∞ �( jn) and lim j→∞P( jn) exist and z( jn)→0as j →∞.



ProofThe results (54), (55), (58), and the convergence of{�( jn)}∞j=0 and {P( jn)}∞j=0 follows from standardproperties of recursive least square, see [9, p. 60],[10, p. 22], [11, p. 58] and [12, p. 202]. From (54)follows that:

P( jn)�P( jn−1)

Now we demonstrate the asymptotic convergence ofthe performance to zero. From (9) it follows that:

�T( jn)�( jn) = [�T( jn)⊗ I ][�( jn)⊗ IT]= �T( jn)�( jn)⊗ I IT

= �T( jn)�( jn)⊗ I (59)

where I ∈Rm×m . Equation (59) follows that:

‖�( jn)‖2=‖�( jn)‖2 (60)

Using (60) in (4) we have

�T( jn)�( jn)=nc∑i=1

[‖u( jn−i)‖22+‖z( jn−i)‖22] (61)

It follows from (11), (56) and (33) that:

�V�( jn) � −‖z( jn)‖221+�

∑nci=1

[‖u( jn−i)‖22+‖z( jn−i)‖22]

� −‖z( jn)‖221+�

∑nci=1

[‖u( jn−i)‖22+‖z( jn−i)‖22]+M( jn)

(62)

where

�=�max[P(0)]>�max[P( jn)], �>0 (63)

and

M( jn)=�‖z( jn)‖22+�‖u( jn)‖22 (64)

From (11) it follows that:

‖u( jn)‖�‖w( jn)‖+‖z( jn)‖ (65)

From (62), (64) and (65), we have

�V�( jn)�−‖z( jn)‖22

1+�∑nc

i=0[‖w( jn−i)‖22+2‖z( jn−i)‖22]

From Assumption 3.1, it follows thatw( jn) is bounded,therefore ∃c∈R+ such that

∑nci=0 ‖w( jn−i)‖22<c.

Consequently

�V�( jn) �−‖z( jn)‖22

1+�c+2�∑nc

i=0 ‖z( jn−i)‖22

= −‖z( jn)‖22c1+c2

∑nci=0 ‖z( jn−i)‖22

where c1�=1+�c and c2

�=2�.Now suppose that {‖z( jn)‖}∞j=0 is unbounded. Then

it follows from Lemma A.1 that there exist c3>0 andc4>0 such that the maximal monotonically increasingsubsequence {‖zi ( jn)‖}∞i=0 satisfies

c1+c2nc∑i=0

‖z( jn−i)‖22<c3+c4‖z( jn)‖22which implies that

�V�i( jn)�

−‖zi ( jn)‖22c3+c4‖zi ( jn)‖22

for all i=1,2, . . . . Furthermore, since by (30)limi→∞ �V�i

( jn)=0, it follows that:

limi→∞

−‖zi ( jn)‖22c3+c4‖zi ( jn)‖22

=0

Therefore, the maximally monotonically increasingsubsequence zi ( jn)→0 as j →∞, which is a contra-diction. Hence, {‖z( jn)‖}∞j=0 is bounded and it followsfrom Lemma (A.1) that there exist c5>0 and c6>0such that, for all j�0

�V�(n)�−‖z( jn)‖22

c5+c6‖z( jn)‖22Now using (58) we have

limj→∞

−‖z( jn)‖22c5+c6‖z( jn)‖22

=0 (66)

Therefore, z( jn)→0 as j →∞. �

Remark 5.1Although Lemma 5.1 ensures convergence of theperformance to zero at the time instants when the



0 5 10 15 20 25

0

1

–1

2

–2

3

–3

4

–4

5

6

k

z (k

)

z1(k)update instant

Figure 2. Estimator aliasing.

parameter estimate is updated, estimator aliasing canstill lead to a periodic and growing state vector. Weexplain estimator aliasing in the following example.

Example 5.1Consider the closed-loop performance (11) with thecontrol law (1) and the gain update (48)–(49). Supposethat n=4 so that the estimator update occurs atk= jn=5,9,13,17, . . . . Furthermore, suppose that�(0)=0 and at the update instants z(4 j+1)=0for j =1,2,3, . . . as shown in Figure 2. Therefore,�(k=4 j)=�(0)=0 for all k�0, i.e. the gain matrix�(k=4 j) is not updated. Consequently, P growsunbounded even in closed loop.

Example 5.1 suggests the use of a staggeredupdating; i.e. the time between two consecutivegain updates be varied instead of updating after afixed interval. We explain staggered updating in thefollowing example.

Example 5.2Consider again the system of Example 5.1 and supposethat instead of updating the gain at time jn+1 weupdate the gain at time �( j), where �( j) is determinedrecursively as follows:

�( j+1)=�( j)+5 if j is odd

�( j+1)=�( j)+4 if j is even

z1(k)update instant

0 5 10 15 20 25

0

1

–1

2

–2

3

–3

4

–4

5

6

k

z (k

)

Figure 3. Staggered estimator updating.

where �(1)=4. With this interval update schemethe controller gains are updated at k=�( j+1)=4,8,13,17,22, . . . as shown in Figure 3. Since the Pis not identically zero at the update instants, the gainmatrix is updated in a direction that minimizes thenorm of the states at the update instants.

Having demonstrated via Example 5.2, how stag-gered updating helps prevent estimator aliasing, we nowpresent the result formally.

Lemma 5.2Let n be a positive integer and let

�( j+1)=�( j)+n+1 if j is odd

�( j+1)=�( j)+n if j is even

where j =1,2, . . . and �(1)=n. Furthermore, letk=0,1,2, . . . then there does not exist an integer T>1such that for all k=�( j)

sin

(2�k

T+�

)=0 (67)

where 0��<2�.

ProofTo satisfy (67) for all k=�( j),T/2 must divide

n,2n+1,3n+1,4n+2,5n+2,6n+3, . . .

Therefore, T cannot be an integer. �



Using Lemmas 5.2 and 5.1 we state the followinglemma.

Lemma 5.3Define the gain update

�(�( j+1)) = �(�( j))+P(�( j)+1)

×�(�( j))�T(�( j))�(�( j))

P(�( j)+1) =P(�( j))−P(�( j))

×�(�( j))L(�( j))�T(�( j))P(�( j))

P(0) > 0 (68)

�(�( j)−1)= �(�( j)−2)=·· ·= �(�( j−1)+1) (69)

P(�( j)−1)=P(�( j)−2)=·· ·=P(�( j−1)+1) (70)

and

�( j+1)=�( j)+n+1 if j is odd (71)

�( j+1)=�( j)+n if j is even (72)

where

L(�( j))=[In+�T(�( j))P(�( j))�(�( j))]−1 (73)

also j =1,2, . . . and �(1)=n. Then all parameters ofthe closed-loop system (11), (68)–(72) remain bounded,and z(k)→0 as k→∞.

ProofFrom Lemma 5.1 with jn+1 replaced by �( j) we havex(�( j))→0 as j →∞. Now we use Lemma 5.2 toconclude that z( jn)→0 as j →∞. �

6. COMPUTER SIMULATION

All the following examples are initialized with zerocontroller, that is, �(0)=0 and �(0)=0. The estimatorgain is initialized as P(0)=2I

Example 6.1In this example, the interference is a single tone at0.8rad/s, and the controller is of the order nc=4. A plotof the performance ‘z’ versus time is shown in Figure 4.

0 50 100 150 200 250 300

0

0.2

–0.2

0.4

–0.4

0.6

–0.6

0.8

Out

put

Time

Figure 4. Performance for the interference at 0.2 rad/s(Example: 6.1).

0

10

–10

20

–20

30

40

Mag

nitu

de (

dB)

10010–110–2 1010

90

180

270

360

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Figure 5. Bode plot of controller obtained after convergesof parameter with frequency at 0.8 rad/s (Example: 6.1).

A bode plot of the controller after the controller param-eters has converged is shown in Figure 5. Note that thecontroller has high gain at the interference frequencyof 0.8rad/s. In this regard our proposed method canbe viewed as an adaptive internal model control (IMC)scheme.



0 50 100 150 200 250 300 350 400 450 500

0

0.2

–0.2

0.4

–0.4

0.6

–0.6

0.8

–0.8

1

Time

Per

form

ance


0 50 100 150 200 250 300 350 400 450 500

0

0.5

–1.5

–0.5

1

–1

Per

form

ance

Time


Example 6.2In this example, two channels are subjected to interfer-ence signals. The first channel is affected by a singletone interference at 0.2rad/s, while on the secondchannel the interference frequency is 0.9rad/s. Plotsof the closed-loop performance ‘z’ versus time for twochannels are shown in Figures 6 and 7, respectively. Inthis example the order of controller is nc=4.

8 10 12 14 16 18 20

0

200

–600

–200

400

–400

Per

form

ance

Time

Figure 8. Performance for the AM interference at frequency10Hz (Example: 6.3).

Example 6.3In this example we again consider the two channel inter-ference problem with the first channel being affectedby a tone at the frequency 5Hz and the second channelbeing affected by an amplitude modulated signal whosecarrier frequency is a 100Hz and the modulating signalat 10Hz . The controller order used for this example hasnc=8. A plot of closed-loop performance ‘z’ versustime is shown in Figure 8.

7. HARDWARE IMPLEMENTATION

7.1. Experimental setup

The controller was implemented on a real-timeTMS320C6713 DSP Kit at a sampling rate of 8000Hz.The interference signals were generated via a externalsignal generator and the output of the DSP Kit wassent to an PC via the audio card where the performancevariable ‘z’ was recorded using the Windows audiorecorder.

7.2. Experiment 1

In the first experiment the interference was a tone at5 kHz. The plot of closed loop ‘z’ versus time is shownin Figure 9. Initially, the interference is allowed to effectthe performance and then the adaptive algorithm was



0 2 4 6 8 10 12 14 16 18

x 104

0

0.2

–0.2

0.4

–0.4

0.6

–0.6

0.8

–0.8

1

–1

Samples

Per

form

ance

Figure 9. Performance for the tone interference at frequency5 kHz (Experiment: 1).

0.5 1–1

1.5 2 2.5 3 3.5 4 4.5 5

x 105

0

0.2

–0.2

0.4

–0.4

0.6

–0.6

0.8

–0.8

Samples

Per

form

ance

Figure 10. Performance for the tone interference atfrequency 5 kHz (Experiment: 2).

switched on manually, via a DIP switch on the DSPkit, at approximately 5 s. The adaptive algorithm thenadapted the controller parameters to reject the tonaldisturbance.

7.3. Experiment 2

In this experiment two interference channels are consid-ered. Channel one being affected by a tonal disturbance

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

0

0.2

–0.2

0.4

–0.4

0.6

–0.6

0.8

–0.8

–1

Samples 105

Per

form

ance

Figure 11. Performance for the AM interference atfrequency 1KHz (Experiment: 2).

of 5 kHz and channel 2 by an amplitude modulatedwave at 1 kHz. The plots of closed loop ‘z’ vs time areshown in Figures 10 and 11, respectively.

8. CONCLUSIONS

In this paper we proposed an algorithm for multiplechannel interference cancelation via a multi-variablefeedback control law. The theoretical foundation forthe proposed algorithm was established via a compre-hensive proof of Lyapunov stability of the closed-loopsystem. Effectiveness of the algorithm as an inter-ference canceler was demonstrated via simulationsand experiments where different types of interfer-ences such as fixed frequency tones and amplitudemodulated waveforms were used. Furthermore, wesuggest a method for implementing the algorithm usinginfrequent estimator updating.

APPENDIX A

Lemma A.1Let {(k)}∞k=0 be a sequence of positive scalars. Let Nbe a positive integer, let g1>0,g2>0, and define

L(k)�=g1+g2

N∑j=0

(k− j) (A1)



Also, define the maximal monotonically increasingsubsequence {i (k)}∞i=0 such that (k)<i (k). Then thefollowing statement holds:

1. If {(k)}∞k=0 is bounded, then there existg3>0,g4>0 such that for all k�0

L(k)�g3+g4(k) (A2)

2. If {(k)}∞k=0 is unbounded then there existg3>0,g4>0 such that for all i=1,2, . . . themaximal monotonically increasing subsequence{i (k)}∞i=0, satisfies

Li (k)�g3+g4i (k) (A3)

ProofIf {(k)}∞k=0 is bounded, then (A2) is satisfied with g3=g1+(N+1)g2 supk�0 (k) and g4>0. Now, supposethat {(k)}∞k=0 is unbounded, then (A3) is satisfied withg3=g1 and g4=N+1. �

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