Asymptotic rejection of nonvanishing disturbances despite plant-model mismatch

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2012; 26:1090–1110Published online 2 May 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/acs.2292

Asymptotic rejection of nonvanishing disturbances despiteplant–model mismatch

Ph. Mullhaupt 1, S. Valentinotti 1,‡, B. Srinivasan 2 and D. Bonvin 1,*,†

1Laboratoire d’Automatique, EPFL, CH-1015 Lausanne, Switzerland2Department of Chemical Engineering, Ecole Polytechnique Montreal, Montreal, Canada

SUMMARY

A direct adaptive control methodology for the rejection of unmeasured nonvanishing disturbances isproposed. The approach uses the framework of polynomial RST controllers and relies on the internal modelprinciple with additional degrees of freedom provided by the Q parametrization. The parameters of theQ polynomial are adapted using minimization of the closed-loop output error. Asymptotic rejection ofunmeasured nonvanishing disturbances is guaranteed under the assumption that the closed-loop plantremains stable. A simulation example illustrates the theoretical developments. Copyright © 2012 John Wiley& Sons, Ltd.

Received 24 December 2010; Revised 6 December 2011; Accepted 15 March 2012

KEY WORDS: disturbance rejection; RST control; internal model principle; Q parametrization;Youla–Kucera parametrization; direct adaptive control; unmodeled dynamics

1. INTRODUCTION

Disturbance rejection is an important issue in the control of real plants. Depending on their nature,disturbances can be classified as stochastic or deterministic. Stochastic disturbances are randomin nature and are often attributed to process and measurement noises. Deterministic disturbancesare representative of process drifts and unmodeled dynamics. A common approach to disturbancerejection is to model the disturbances as the output of a disturbance model, called exosystem, excitedby stationary white noise (for stochastic disturbances) or a Dirac impulse (for deterministic distur-bances). If the plant and disturbance models are known linear time-invariant systems, linear controldesign methods such as linear quadratic regulator can be applied. When the disturbance modelis unknown or time varying, adaptive control can be used. This paper considers the rejection ofunmeasured nonvanishing deterministic disturbances.

According to the internal model principle (IMP), nonvanishing disturbances can only be rejectedif the plant is operated in closed loop and the marginally stable or unstable part of the exosystemis included in the controller [1, 2]. This idea can be applied in a straightforward manner when thedisturbances are known. However, because this is rarely the case, it is necessary to identify the dis-turbance model online. If only the disturbance model is identified, the approach is referred to asdirect [3, 4]. By contrast, if the plant model is also identified, the approach is referred to as indi-rect [5–7]. The available techniques for disturbance rejection using direct adaptive control require afairly accurate plant model because the model is not adapted. On the other hand, the main drawbackof the indirect approach is that sufficient excitation must be provided to identify all the parameters.

*Correspondence to: D. Bonvin, Laboratoire d’Automatique, EPFL, CH-1015 Lausanne, Switzerland.†E-mail: [email protected]‡S. Valentinotti is currently with Direccion de Desarrollo de Productos, Sanofi-aventis, Mexico D.F.

Copyright © 2012 John Wiley & Sons, Ltd.

ASYMPTOTIC REJECTION OF NONVANISHING DISTURBANCES 1091

Tay and Moore [8] suggested to use the Q parametrization, a special case of the Youla–Kuceraparametrization, to incorporate the disturbances in the controller. The idea is to add degrees offreedom to the controller via the Q parametrization and adapt them online using least squares min-imization. This way, these authors were able to minimize the effects of stochastic disturbances andimprove the performance of the existing controller. Amara et al. [9] proposed adaptive feedbackcontrol based on IMP and Youla–Kucera parametrization to asymptotically cancel out sinusoidaldisturbances with unknown frequency. Recently, Wang et al. [10] combined an adaptive internalmodel with online frequency identification and an adaptive finite impulse response filter to rejectstochastic and deterministic sinusoidal disturbances. Valentinotti [11] proposed to use IMP and Qparametrization in the context of RST controller to reject nonvanishing deterministic disturbances.Landau et al. [12] used the same framework to reject narrow-band sinusoidal disturbances in anactive suspension system. The same authors also proved the stability of the direct adaptive schemefor the case of a perfect plant model. The present paper is based on Valentinotti’s thesis [11] andproposes a direct adaptive control scheme based on IMP and Q parametrization to asymptoticallyreject unmeasured nonvanishing deterministic disturbances. The results hold in the presence ofplant–model mismatch.

The framework adopted in this paper is the two-degree-of-freedom RST polynomial controller,which helps address independently the problems of setpoint tracking and closed-loop stability [13].As mentioned previously, the rejection of unmeasured nonvanishing disturbances requires adaptingthe internal model of the controller in real time. This is carried out by (i) adding degrees of free-dom to the controller via the Q parametrization [14, 15] and (ii) adjusting the coefficients of the Qpolynomials via least squares minimization. The main contribution of this paper will be to showthat, as long as the closed-loop plant remains stable, the proposed scheme guarantees asymptoticdisturbance rejection despite plant–model mismatch.

The proposed approach has been motivated by the control and optimization of a yeast fed-batchfermentation process [16]. Therein, the problem of controlling the ethanol concentration at a con-stant setpoint is addressed, with the exponential biomass growth being considered as an unstabledeterministic disturbance to be rejected. The details of the application are provided in [16], whereasthe theoretical results are established here. Another application of the same approach has been pre-sented recently and concerns the rejection of narrow-band unmeasured disturbances in an activesuspension system [17, 18].

The paper is organized as follows. Background material and notations are introduced in Section 2.The main result on direct adaptive disturbance rejection is presented in Section 3, where sufficientconditions for convergence are provided. Section 4 presents a simulation that illustrates the rejectionof unmeasured nonvanishing disturbances. Conclusions are provided in Section 5, whereas technicalmaterial used in the proofs of Section 3 is given in the Appendix.

2. PRELIMINARIES

2.1. RST controller

Let the single-input single-output discrete-time plant involving the manipulated input uk and themeasured output yk be

yk D1

A�.q�1/

�B�.q�1/uk C vk

�, (1)

where B�=A� is the plant transfer function and vk the disturbances in equation error form.These disturbances can be expressed in terms of the input disturbances vi,k and the outputdisturbances vo,k :

vk D A�.q�1/vo,k CB�.q�1/vi,k . (2)

Let the controller be

R.q�1/uk D�S.q�1/yk C T.q�1/rk , (3)

Copyright © 2012 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2012; 26:1090–1110DOI: 10.1002/acs

1092 PH. MULLHAUPT ET AL.

where R, S, and T are polynomials and rk is the reference signal.§ The closed-loop output yk canbe expressed as

yk DB�TA�c

rk CRA�cvk , (4)

with the characteristic polynomial A�c D A�RCB�S.Because A� and B� are unknown, the model polynomials A and B are used for control design by

solving the Diophantine equation Ac D ARC BS. Note that it is possible to arbitrarily choose thecharacteristic polynomial Ac, provided the polynomials A and B are coprime [19]. Let the desiredoutput predicted by the model be

yd,k D

�BTAc

�rk . (5)

The effect of the disturbances vk and model mismatch .A� ¤ A/ can be expressed as the closed-loopoutput error (CLOE):

ek WD yk � yd,k DRAc

�Ac

A�cvk C

.B�A�A�B/TA�c

rk

�, (6)

with the first term representative of the effect of the disturbances and the second term correspondingto the effect of model mismatch.

On the other hand, upon considering the equation error,

wk WD Ayk �Buk , (7)

together with the control law (3) and the definition of Ac, one can write

yk DBTAcrk C

RAcwk . (8)

It is interesting to compare (4) with (8). The former expresses the measured output in terms of theplant, the controller, and the two exogeneous signals rk and vk , whereas the latter expresses yk interms of the plant model, the controller, the reference signal rk , and the equation error wk .

With the use of the closed-loop output expression (4), the control laws (3), and the definition ofthe characteristic polynomials A�c and Ac, the equation error (7) can be rewritten as

wk DAc

A�cvk C

.B�A�A�B/TA�c

rk . (9)

With the use of this expression in (6), the CLOE can be conveniently expressed as

ek DRAcwk , (10)

that is, in terms of the measured signals uk and yk and the known polynomials R, Ac, A, and B.

2.2. Q parametrization

It has been proposed in [20] and [15] to rewrite the control law (3) using the additional term Qwk toexplicitly take into account the closed-loop error expressed here in terms of the equation error wk :

Rouk D�Soyk C Trk �Qwk , (11)

where So and Ro represent a particular solution of the Diophantine equation Ac D ARCBS and Qis a polynomial of degree ıQ. The controller polynomials R and S can be rewritten as [19]

RD Ro �QB, SD SoCQA. (12)

§All operators are polynomials in the backward-shift operator q�1, for example, yk�1 D q�1yk . In the remainder, thedependency on q�1 will be omitted as much as possible.



B A

BRo

1T

Q

So

rk u

ky

k

wk

vk

- -

-

A

1*

*

PlantRST Controller

Figure 1. Schematic of the controlled plant showing the plant .A�, B�/, the model .A, B/, the RST controllerwith the Q parametrization (11), the equation error wk , and the exogenous signals rk and vk .

Because the polynomial Q is used to reformulate the controller polynomials, this parametrizationis often termed the Q parametrization. It can be shown to be a particular case of the Youla–Kucera parametrization [21]. The plant (A�, B�) and the controller (11) are shown in Figure 1.The characteristic polynomial of the controlled plant is given by

A�c D A�RoCB�SoCQ.B�A�A�B/. (13)

RemarkBy construction, the Q parametrization does not change the poles of Ac. Furthermore, in the case ofa perfect model, A D A�, B D B�, and thus A�c D Ac, the Q parametrization does not modify theclosed-loop poles of the plant. Otherwise, A�c changes with the amount of plant–model mismatch.

2.3. Internal model principle

The equation error wk , which is one of the signals driving the control law (11), is computed fromuk and yk using the model (A, B). It can also be obtained by filtering vk and rk according to (9).Alternatively, one can express the equation error as a filtered impulse response as follows:

wk DN

A�c D ık DN

A�c Ds Duık , (14)

where N and D are coprime polynomials of degrees ıN and ıD, respectively, Ds and Du are thestable and unstable parts of D, and ık is the unit pulse at the kth time instant. Note that, because wkin (9) contains A�c in the denominator, this term is written explicitly in (14). Combining (10) and(14) gives the CLOE expression

ek DRAc

NA�cDsDu

ık . (15)

If Du could be included as a factor of the controller denominator, that is, R D CDu where C isany arbitrary polynomial, then the effect of the unstable part of the equation error wk on the CLOEek would be rejected. It turns out that this can be achieved by setting RD CDu in the first equationof the Q parametrization (12) and solving the Diophantine equation¶

QBCCDu D Ro (16)

¶Without loss of generality, it will be assumed that the polynomials C, Du, and Ro are monic. Solution of the Diophantineequation requires the polynomials B and Du to be coprime [19].



for Q and C, with NQ and NC denoting the corresponding solutions. This property, which is commonlyknown as the IMP [22], will be proven next for the case of plant–model mismatch.

Theorem 1 (IMP for the case of plant–model mismatch)Consider the plant (1) with the controller (11) designed on the basis of the model .A, B/. Despiteplant–model mismatch, the CLOE (10) goes to zero asymptotically if and only if Q is solution of(16) and A�c is stable.

ProofConsider the CLOE Nek that results from (15) by setting RD CDu and solving (16) for NQ and NC:

Nek DNC

Ac

NA�cDs

ık . (17)

If part: If Q is solution of (16), QD NQ and CD NC. Hence, the CLOE will correspond to Nek and goasymptotically to zero because Ac and Ds are stable by construction and A�c is stable by assumption.Only if part: Proof by contradiction. The assumption Q ¤ NQ implies QBC CDu ¤ Ro. It followsthat Du will not be canceled out in the CLOE ek given by (15). Unstable roots will also be presentin the denominator of ek if A�c is unstable (unless they happen to be canceled out by R or N , casesthat are ruled out from a genericity point of view). Hence, if either Q ¤ NQ or A�c is unstable, theCLOE ek will grow unbounded. Therefore, Q D NQ and A�c stable are necessary conditions for theCLOE to go to zero asymptotically. �

Remarks1. This result is well known in the absence of plant–model mismatch. Theorem 1 indicates that a

similar result also holds in the presence of plant–model mismatch.2. With Du being a factor of R, the unstable modes seen in wk (see (14)) are not felt by the con-

trolled variable yk (see (8)). However, these unstable modes are felt by the input uk , which ismanipulated according to the equation error wk (see (11)).

Proposition 1 (Uniqueness)Let NQ be solution of (16). The solution corresponding to the minimal degree of NQ is unique and hasdegree ıQ D ıDu � 1, where ıDu is the degree of Du. For any ıQ > ıDu � 1, there are infinitelymany solutions.

The proof is given in [11].

3. ADAPTIVE REJECTION OF UNMEASURED NONVANISHING DISTURBANCES

The objective is to estimate a polynomial NQ that satisfies (16) despite the fact that Du is unknown. Itis first shown that identifying the unknown polynomial Du and solving (16) for NQ and NC is equivalentto minimizing the CLOE in the least squares sense on the basis of the measured inputs and outputs.Conditions that ensure persistency of excitation are also provided. Then, a recursive version of theleast squares minimization algorithm will be proposed and its convergence will be established.

3.1. Minimization of closed-loop output error

With the use of the Q parametrization, the CLOE (10) reads

ek DRo �QB

Acwk D �1,k � '

Tk � , (18)

with

'k WD��2,k �2,k�1 : : : �2,.k�ıQ/

�T,

� WD��0 �1 : : : �ıQ

�T,



and the signals �1,k WD .Ro=Ac/wk and �2,k WD .B=Ac/wk that can be computed from knowledgeof the plant model (A and B) and the controller (Ro and So) and from the measured uk and yk . Theparameters �0, �1, : : : , �ıQ are the coefficients of the polynomial Q. Let n� D ıQ C 1 denote thenumber of unknown � parameters.

Minimization of the CLOE (18) consists in adjusting � to minimize the following leastsquares criterion:

Jk.�/D1

2N

N�1XiD0

e2k�i .�/. (19)

The criterion (19) uses a moving horizon of lengthN , with the window extending from .k�N C1/to k, the current time instant. There should be at least as many data points, N , as there are param-eters to be identified, that is, N > n� . Minimization of this criterion at time k gives the parameterestimates O�k . As k ! 1, O�k converges to N� . The next theorem shows that N� corresponds to NQsolution of the Diophantine equation (16).

Theorem 2 (IMP as a least squares problem)The vector N� minimizes the least squares criterion J. N�/ if and only if the corresponding polynomialNQ is solution of the Diophantine equation (16) and A�c is stable.

ProofIf part: If NQ is solution of the Diophantine equation and A�c is stable, then, from Theorem 1, Nek! 0

as k ! 1, thus leading to J. N�/ D 0, which is the smallest value that J can have because of itspositivity.Only if part: By contradiction, if NQ is not solution of (16) or if A�c is unstable, then, from Theorem 1,

Nek 6! 0 as k!1; thus, J. N�/¤ 0. This is not optimal because, as seen in the ‘if part’ of the proof,there exists one solution that makes J. N�/D 0. �

Remarks1. The aforementioned results are independent of model accuracy and can also be used

with an imperfect model. However, in the presence of plant–model mismatch, 'k willdepend on � (via wk and Q); thus, the minimization of Jk.�/ is not a standard leastsquares problem. In this case, there does not exist a closed-form solution, and the solu-tion needs to be obtained numerically using optimization techniques. A recursive algo-rithm for solving this minimization problem will be presented in the next subsection.The main difficulty in proving convergence lies in the fact that it is not a standard leastsquares problem.

2. As with the Diophantine equation, the smallest degree of Q is ıQ D ıDu � 1, for whichthe optimal solution is unique. On the other hand, if ıQ > ıDu � 1, the minimization prob-lem will have infinitely many solutions. Note that if ıDu is unknown, it is not possible tochoose the minimal degree; thus, a conservative choice needs to be made to be able to solvethe problem. However, with a conservative choice, the optimal solution is not necessarilyunique. To avoid this issue, it will be assumed in the rest of the paper that the degree ıDu

is known.3. Identification of the vector � requires persistent excitation of w through the disturbances v

or the reference r . We assume here that the disturbances contains ıDu unstable and persis-tently excited modes. In the case of a perfect model, w D v, and persistent excitation ofv implies persistent excitation of w. In the presence of plant–model mismatch, w is a fil-tered version of v and r . Persistency of excitation can be ensured as per Lemma 1 given inthe Appendix.



3.2. Recursive algorithm for minimization of closed-loop output error

A recursive version of the least squares minimization algorithm is used next [23]. The degreeıQD ıDu � 1 is assumed to be known and constant. Consider the update law

O�k D O�k�1C � Pk'k

��1,k � '

TkO�k�1

�, (20)

P�1k D .1� �/P�1k�1C � 'k'

Tk , (21)

where Pk is a (n� � n� )-dimensional matrix and � 2 R is a positive constant gain. Notethat persistency of excitation implies that Pk is full rank [23]. The convergence of O�k to N� isestablished next.

3.2.1. Idea of the convergence proof. The convergence proof uses the auxiliary signal ´k , whichrepresents the CLOE that would result if the plant were only driven by the stable part of the dis-turbances or, in other words, if the controller were able to perfectly reject the unstable part of thedisturbances. Parameter convergence is shown by constructing a Lyapunov-like function Vk . Thedifference Vk � Vk�1 has two terms, a negative term and a positive term proportional to ´2

k. It is

shown that, if ´k converges exponentially to zero, Vk also converges to zero despite the presence ofthe positive term. The decrease of Vk is not strictly monotonic as this would be the case for a trueLyapunov function, and peaking can occur during convergence.

Because the convergence of Vk rests on the exponential convergence of ´k , conditions guarantee-ing this must be enforced. Exponential convergence of ´k requires that (i) a polynomial containingthe converged values of the parameters be Hurwitz and (ii) the parameters remain sufficiently closeto these converged values.

The convergence proof deals separately with (i) the convergence of Vk under the condition that´k goes to zero and (ii) the convergence of ´k to zero when the parameters are close to the con-verged values. Inductive reasoning is used to show that the parameter update mechanism does notinduce a peaking of the parameters beyond the value for which exponential convergence of ´k canbe guaranteed. Hence, convergence O�k! N� , ´k! 0, and ek! 0 can be proven.

3.2.2. Auxiliary signal and hypothesis. The proof is constructed around the auxiliary signal ´k thatrepresents the CLOE that would result with a controller capable of perfectly rejecting the unstablepart of the disturbances. To construct ´k , RD NCDu is included in the numerator of the CLOE ek in(15) to absorb the effect of the unstable part of the equation error wk :

´k WDNCN

AcA�cDsık . (22)

Defining the stable, exponentially decreasing signal

vs,k WDNCN

AcDsık (23)

and expressing the closed-loop characteristic polynomial in terms of Q according to (13) gives

´k D1

A�c .Q/vs,k D

1

A�RoCB�SoCQ.B�A�A�B/vs,k . (24)

Note that, upon convergence, ´k D Nek .The transfer function (24) depends on Q, which indicates that variations of the parameters O�k

(the coefficients of Q) might lead to instability even though, for frozen parameter values, the polesof this transfer function are all within the unit circle (cf. the stability of time-varying systems [24]).Hence, we need to show that Q in the denominator of (24) will eventually converge to NQ and thus´k to Nek . The proof rests on the following hypothesis.



Hypothesis 1The closed-loop characteristic polynomial A�c . NQ/ is stable.

Although Hypothesis 1 guarantees that ´k converges exponentially to zero for Q D NQ, it is notobvious that ´k converges to zero when Q is updated. Variations of Q may lead to instability of thetime-varying transfer function (24).

The proof will use two preliminary results formulated as propositions. The first result ensures that,if the parameters O�k remain sufficiently close to the values N� that correspond to NQ, the signal ´k willconverge to zero exponentially. This is established through the existence of a common Lyapunovfunction associated with the realizations of ´k . The second result ensures parameter convergenceonce a condition on exponential convergence of ´k holds. It requires a Lyapunov function associ-ated with the parameters. These preliminary results are given in the next two subsections beforebeing combined to prove the convergence O�k! N� , ´k! 0, and ek! 0 in the third subsection.

3.2.3. Convergence ´k ! 0 when the parameters O�k remain close to N� . A common Lyapunovfunction will be used to assess the stability of ´k in relation to that of Nek .

Definition 1 (Common Lyapunov matrix)The positive definite matrix P , solution to NATP NA�P C I D 0, is a common Lyapunov matrix forthe set of matrices AD f NACE W kEk< �g if

ATPA�P C I < 0 8A 2A.

Let us express the closed-loop characteristic polynomials A�c . NQ/ and A�c .Q/ as 1CPıiD1N�iq�i

and 1CPıiD1 �iq

�i , respectively, with �i D N�i C �i . Correspondingly, the signals Nek and ´k can beexpressed using the following canonical realizations:

qkC1 D NAqk CB vs,k , (25)

Nek D C qk ,

and

pkC1 D Apk CB vs,k , (26)

´k D C pk ,

with

NA WD

0BBBBB@

0 1 : : : 0 0

0 0 : : : 0 0...

. . ....

...0 0 : : : 0 1

�N�ı �N�ı�1 : : : �N�2 �N�1

1CCCCCA , (27)

A WD

0BBBB@

0 1 : : : 0 0

0 0 : : : 0 0...

. . ....

...0 0 : : : 0 1

��ı ��ı�1 : : : ��2 ��1

1CCCCA , (28)



B D�0 0 : : : 1

T, and C D

�1 0 : : : 0

. The perturbed canonical matrix A can be

written as AD NACE, where

E WD

0BBBB@

0 0 : : : 0 0

0 0 : : : 0 0...

......

...0 0 : : : 0 0

��ı ��ı�1 : : : ��2 ��1

1CCCCA . (29)

Furthermore, consider the common Lyapunov function Wk WD pTkPpk , with P satisfying the

Lyapunov equation NATP NA� P C I D 0. Under Hypothesis 1, the Lyapunov function constructedfor the canonical realization NA becomes a common Lyapunov function for all realizations corre-sponding to A. To see this, let us define N� WD max�.P / and � WD min�.P /. The signal ´2

kcan be

related to this common Lyapunov function in the following way:

´2k D p2k,1 6 p

Tk pk D

1

��pT

k pk

6 1�pTk Ppk D

1

�Wk , (30)

with pk,1 the first element of pk .

Proposition 2 (Close parameters imply exponential convergence of ´k)

(A1) Let the stable disturbances vs,k evolve according to v2s,k 6 .vs,0/

2�k .(A2) Let pk evolve according to (26), with possible variations of Q at any time instant k, but

such that maxi j�i j6 �, with � a positive real number.(A3) Let N� WD maxi j�i j, Na WD k NAk2 D

pmax�. NAT NA/, ı D dim pk , and � be chosen such that

0 < ..1��/=ı N�/ < 1.Then, ´2

k6 ˛ k , 8k 2N, with

˛ D1

�

W0C

v2s,0

�

N�C . NaC �/2

ı N�

��

!1

ln � ln�e

ln��ln�ln�Cln�

!,

D ı

��2C 2 N� �C 1�

1��

ı N�

�.

ProofThe proof is first sketched. The common Lyapunov function Wk is computed for all realizations of´k . The state of these realizations is pk . The decrease rate of Wk is related to the driving stabledisturbances vs,k . These disturbances create a peaking measured by ˛.

The proof is given next. Consider the common Lyapunov function Wk D pTkPpk with P sat-

isfying NATP NA � P C I D 0. Let M WD ATPA � P and express the decrease rate of Wkas follows:

WkC1 �Wk D pTkC1PpkC1 � p

Tk Ppk

D .Apk CBvs,k/TP.Apk CBvs,k/� p

Tk Ppk

D pTk A

TPApk � pTk Ppk C 2B

TPApkvs,k CBTPBv2s,k

D pTkMpk C 2B

TPApkvs,k CBTPBv2s,k .



Using (55) in the Appendix, one can write pTkMpk 6 ı .maxi jpk,i j/

2.ı N� �2 C 2ı N� N� � � 1/.Furthermore, the following relationships also hold,|| BTPApkvs,k 6 N�. NaC �/

p.1=�/Wkjvs,kj and

BTPB 6 N�, thus leading to

WkC1 �Wk 6 ı .maxijpk,i j/

2.ı N� �2C 2ı N� N� � � 1/C 2BTPApkvs,k CBTPBv2s,k

6 ı .maxijpk,i j/

2.ı N� �2C 2ı N� N� � � 1/C 2 N�. NaC �/

s1

�Wkjvs,kj C N�v

2s,k .

Furthermore, ı.maxi jpk,i j/2 > pT

kpk D .1= N�/ N�pT

kpk > .1= N�/pT

kPpk D .1= N�/Wk and, because

from Lemma 3, .ı N� �2C 2ı N� N� � � 1/ < 0, one can write

WkC1 �Wk 6 ı��2C 2 N� � �

1

ı N�

�Wk C 2 N�. NaC �/

s1

�

pWkjvs,kj C N�v

2s,k .

With � chosen such that 0 < ..1��/=ı N�/ < 1 and upon completing the squares, one has

WkC1 �Wk 6

0@ N�. NaC �/

sı N�

��jvs,kj �

r�

ı N�

pWk

1A2

C ı

��2C 2 N� � �

1

ı N�

�Wk

C 2 N�. NaC �/

s1

�

pWkjvs,kj C N�v

2s,k

D ı

��2C 2 N� � �

1��

ı N�

�Wk C N�

. NaC �/2 ı N�2

��C 1

!v2s,k .

Sending Wk to the right-hand side and writing the relationship for the previous iteration leads to

Wk 6 Wk�1C v2s,k�1, (31)

with WD ı��2C 2 N� �C 1� ..1��/=ı N�/

and WD N�

�.. NaC �/2 ı N�2=��/C 1

. Note that,

because vs,k represent stable disturbances, there exists a constant vs,0 and a constant 0 6 � < 1

such that v2s,k 6 .vs,0/

2�k .Inequality (31) resembles inequality (46) given in the Appendix for Vk D Wk , �1 D , �2 D ,

and vk�1 D vs,k�1. With ˛ D v2s,0, ˇ D �,** and ˇ� D ,†† inequality (47) gives

Wk 6 W0C

v2s,0

�

N�C . NaC �/2

ı N�

��

!1

ln � ln�e

ln��ln�ln�Cln�

!k . (32)

The constant � must be chosen such that (57) holds. �

3.2.4. Convergence O�k! N� when ´k converges exponentially to zero. In this subsection, we estab-lish the second preliminary result that ensures parameter convergence O�k ! N� when ´k convergesexponentially to zero. Exponential bounds on the parameters are obtained in the process. Thesebounds depend on the exponential bound of ´k and the value of the gain � chosen for updatingthe parameters.

||1. BTPApkvs,k 6 kBTkkP kkAkkpkkjvs,kj6 1 � N� � kAk � kpkkjvs,kj. Next, with the use of the triangle inequality,kAk 6 k NACEk 6 NaC � because kEk D �. Also, from

pWk D

ppTk Ppk >

ppTk pk� D kpkk

p�, it follows

that kpkk6q1N�Wk . Altogether, BTPApkvs,k 6 N�. NaC �/

q1

�Wkjvs,kj.

2. BTPB corresponds to the .ı , ı/-element of P and therefore BTPB 6max �.P /D N�.**ˇ D � because the largest peaking corresponds to the slowest stable disturbance.††It is possible to choose � so that ˇ� D �, which means that �D ln�

lnˇD ln�

ln�.



Let us define the parameter error Q�k WD O�k � N� and the Lyapunov function Vk WD Q�Tk P�1kQ�k .

Proposition 3 (Exponential convergence ´k! 0 implies parameter convergence O�k! N� )Consider the update law (20)–(21) and assume the following:

(A1) ´2k6 ˛k , with ˛ > 0 and 06 < 1.

(A2) The update gain satisfies

1� < � <min

1,

1

'TkPk 'k

!. (33)

(A3) The disturbances are persistently exciting, which implies that P�1k

is full rank and thusadmits a lower bound � 2RC on its eigenvalues, that is,

0 < �6min��.P�1k /, 8 k 2N.

Then, the parameters are bounded exponentially as

k O�k � N�kL1 6s1

�

rV0C

� ˛

ln.1� �/� ln e

ln.1��/�ln�ln.1��/Cln�

�p1� �

�k. (34)

ProofThe parameter update can be written for the parameter error Q�k ,

Q�k D�I � �Pk'k'

Tk

Q�k�1C �Pk'k´k ,

and the Lyapunov function Vk can be developed as

Vk D Q�Tk�1P

�1kQ�k�1 � 2� Q�

Tk�1'k'

TkQ�k�1C �

2 Q�Tk�1'k'Tk Pk'k'

TkQ�k�1 (35)

C 2 .�´k/'Tk

�I � �Pk'k'

Tk

Q�k�1C .�´k/

2'Tk Pk'k .

With the use of the definition of Vk�1 and the update law (21), the increment (Vk � Vk�1) becomes

Vk � Vk�1 D � � Q�Tk�1P

�1k�1Q�k�1 � � Q�

Tk�1

�'k'

Tk � �'k.'

Tk Pk'k/'

Tk

Q�k�1 (36)

C 2 .�´k/�1� �'T

k Pk'k'TkQ�k�1C .�´k/

2'Tk Pk'k .

This equation can be rearranged by considering that 'TkPk'k is a scalar, which can be factorized out:

Vk � Vk�1 D��Vk�1 � ��1� �'T

k Pk'kQ�Tk�1'k'

TkQ�k�1

C 2 .�´k/�1� �'T

k Pk'k'TkQ�k�1C �´

2k.�'

Tk Pk'k/. (37)

Then, add �´2k

and ��´2k

without affecting the result and regroup the newly added term ��´2k

withthe last term of (37) to obtain

Vk � Vk�1 D��Vk�1 � ��1� �'T

k Pk'kQ�Tk�1'k'

TkQ�k�1

C 2 .�´k/�1� �'T

k Pk'k'TkQ�k�1 � �´

2k

�1� �'T

k Pk'k/C �´2k . (38)

Next, notice that the middle terms of (38), namely the terms of the right-hand side between ��Vk�1and C�´2

k, can be written as ��.1� �'T

kPk'k/.a

2 � 2ab C b2/ with the scalars a WD 'TkQ�k�1 D

Q�Tk�1

'k and b WD ´k . Then, using the relation a2 � 2abC b2 D .a� b/2 gives

Vk � Vk�1 D��Vk�1 � �.1� �'Tk Pk'k/.'

TkQ�k�1 � ´k/

2C �´2k . (39)

Choosing the gain such that .1��'TkPk'k/ > 0, considering .a�b/2 > 0,8a, b 2R and grouping

the Vk�1 terms give

Vk 6 .1� �/Vk�1C �´2k . (40)



Inequality (40) resembles inequality (46) given in the Appendix for �1 D 1 � � , �2 D � , andvk�1 D ´k . Then, inequality (47) gives

Vk 6 .1� �/kV0C˛ �

ˇ.� lnˇ/.1��/e��1�C1 .ˇ�/

kC18�, 06 � < 1. (41)

With ˇ�=ˇ 6 1, for 06 � < 1, one can finally write

Vk 6 .1� �/kV0C˛ �

.� lnˇ/.1��/e��1�C1 .ˇ�/

k8�, 06 � < 1. (42)

Choose 1 � � < , which implies ˇ D maxf1 � � , g D . Hence, an admissible � in the range0 6 � < 1 can be found such that � D 1� � or, equivalently, � D ln.1� �/=ln . The inequality(42) becomes the exponential bounding envelope:

Vk 6�V0C

˛ �

ln.1� �/� ln e

ln.1��/�ln�ln.1��/Cln�

�.1� �/k . (43)

Assumption A3 guarantees that . O�k � N�/T. O�k � N�/ D .1=�/�. O�k � N�/T. O�k � N�/ 6 .1=�/Vk .

Furthermore, exponential convergence of Vk to zero implies exponential convergence of O�k to N�according to (34), thus completing the proof. �

3.2.5. Convergence O�k ! N� , ´k ! 0 and ek ! 0. We finally prove that the parameters convergeto N� and both the signals ´k and ek converge to zero. This means that the unstable part of the dis-turbances eventually does not affect the output of the system. We also establish that the parametersconverge to values that ensure building an internal model of these disturbances. This requires thedisturbances to be persistently exciting.

The two preliminary results (Propositions 2 and 3) are combined to achieve parameter and CLOEconvergence without dependence upon each other. Induction is used to achieve this. The importantpoint to notice is that it is not possible to guarantee a priori that the assumptions associated witheach of the preliminary results hold. Take, for instance, the first preliminary result (Proposition 2).Because parameter update occurs at each iteration, it is not possible to guarantee a priori closenessof O�k and N� . The best that can be guaranteed is that the initial parameter values are close to N� . Thisaccounts for the initialization of the induction argument.

Then, a Gedankenexperiment is used to proceed, namely the parameters are considered as notbeing updated after k D 0 so that the result of Proposition 2 can be invoked, the conclusion ofwhich validates the assumptions of Proposition 3 regarding exponential convergence of the param-eters. This means that updating the parameters at time k D 1 will keep the parameters close toN� . This accounts for the second step of the induction. The key point is that, regardless of whetherthe parameter update law is used (next Gedankenexperiment) or not (current Gedankenexperiment),neither the states p1, nor ´1 and its exponential envelope, will be affected as long as the new A.Q/is compatible with the common Lyapunov function. The aforementioned reasoning is then repeatedfor the induction step by supposing that O�k converges exponentially up to k D N and is then heldconstant for all k > N (holding the parameters constant is Gedankenexperiment number N ). Thisguarantees the assumption of closeness to N� so that ´k can be ensured (in Gedankenexperiment N )to converge exponentially with the same bounds as for the initial step. Exponential convergence of´k then ensures that �NC1 will be closer to N� than �N , thus proving the induction step by validatingGedankenexperiment N C 1. Again, key to the transition between Gedankenexperiments N andN C 1 is the fact that the states pNC1 will not change regardless of whether the parameters areupdated or not at step N C 1. Finally, because the exponential envelope for ´k is never requiredthroughout the inductive reasoning, the envelope used initially remains valid, that is, convergence isconditioned only upon the value of the initial parameters �0.



Theorem 3 (Convergence O�k! N� and ´k! 0)

(A1) Let � satisfy 0 < � < ..1��/=ı N�/, where ı is the order of the realization in Proposi-tion 2 and N� is the largest eigenvalue of the common Lyapunov matrix associated with thisrealization.

(A2) Let � be any value that satisfies 0 6 � <

q. NaC �/2C .1= N�/ � N� and �2 C 2 N�� C 1 �

..1��/=ı N�/ < 1.(A3) Let the initial parameters �0 and � be such that

s1

�

rV0C

� ˛

ln.1� �/� ln e

ln.1��/�ln�ln.1��/Cln� < �,

with ˛ given in Proposition 2. If necessary adjust � to achieve this inequality.Then, the recursive least squares algorithm (20)–(21) with � that satisfies

1� < � <min

(1,

1

'TkPk'k

)

guarantee that O�k converges exponentially to N� and that ´k converges exponentially to zero with a

rate proportional top, that is, j´kj6

p˛pk .

ProofConsider the following propositions, for which all the assumptions stated in this theorem hold exceptfor the fact that O�k is updated in a different way:

P0 W Setting O�k D �0 for all k > 0 implies j´kj6p˛pk for all k > 0.

P1 W Setting O�k D �0 for k D 0 and using (20) to compute �1 and then setting O�k D �1 for all

k > 1 imply j´kj6p˛pk for all k > 0.

......

PN W Setting O�k D �0 for k D 0 and using (20) to compute O�k for 1 6 k 6 N and then

maintaining O�k D �N for all k > N imply j´kj6p˛pk for all k > 0.

PNC1 W Setting O�k D �0 for k D 0 and using (20) to compute O�k for 1 6 k 6 N C 1 and then

maintaining O�k D �NC1 for all k > N C 1 imply j´kj6p˛pk for all k > 0.

......

Only Proposition P0 can be shown to be true using Proposition 2 because, for the other propositions,the assumptions of Proposition 2 are not met by those of this theorem. Proposition P0 accounts forthe initialization step of the induction.

Then, assume that PN holds so that Proposition 3 can be applied to guarantee that VNC1 is lessthan VN upon applying the update law (20)–(21) at step N C1. The state of the realization pNC1 isnot affected by whether or not the update law is applied at step N C 1. Hence, the assumptions ofProposition 2 are met, which leads to PropositionPNC1 being true. Applying the induction principleconcludes the proof of the theorem. �

It remains to prove that ek! 0 when ´k! 0 converges.

Corollary 1 (Convergence ek! 0)Let NQ and NC solve the Diophantine equation (16) so that Ro � NQB D NCDu. Compute N , the largestmodulus of the roots of Du. The output sequence fek W k 2 Ng will converge to zero exponentiallyif the update gain is chosen such that � > . N � 1/= N .



ProofComparing (15) and (22) gives

´k DNCDu

Rek , (44)

which, with (12) and (16), becomes

´k.Ro �QB/D ek�Ro � NQB

�.

If the assumptions of Proposition 3 are satisfied, the coefficients of

Ro �QBDpXiD0

hi ,kq�i

converge exponentially to those of

Ro � NQBDpXiD0

Nhiq�i ,

that is,

jhi ,k � Nhi j6 jhi ,0 � Nhi j.1� �/k i D 1, : : : ,p.

The proof follows upon invoking Lemma 4 given in the Appendix with NHD Ro � NQB andˇ Dp. �

Remarks1. Proposition 2 shows that the stable part of the disturbances, which is not modeled in Q, tends

to reduce the domain of possible initial guesses �0.2. A�c . NQ/must be stable for the algorithm to work, but this is impossible to check a priori because

the plant–model mismatch is typically unknown.3. The estimate of � is conservative; however, � is only needed in the convergence proof. In

practice, the adaptive control algorithm might work for a larger set of initial guesses �0.4. If Q is overparameterized with respect to the unstable modes in the disturbances, there might

exist many different NQ capable of canceling out the effect of the unstable disturbances onthe output.

5. With regard to Figure 1, direct adaptive control consists in constructing the equation error wkfrom knowledge of the signals uk and yk and the model .A, B/ and adapting the Q polynomial(i.e., the parameter vector �) using the update law (20)–(21).

4. ILLUSTRATIVE EXAMPLE

An example will show the capability of the Q parametrization approach to reject unmeasurednonvanishing disturbances in the case of a poorly identified model.

Let the plant be described by the transfer function

G�.q�1/D0.052q�1

1� 0.86q�1

and the plant model by

G.q�1/D0.1168q�1

1� 0.75q�1,

with the sampling period h D 0.1 s. With the desired closed-loop characteristic polynomial Ac D1�1.70q�1C0.71q�2 and a unity closed-loop static gain, the Diophantine equation ARCBSD Ac



Table I. At each step k, k D 1, 2, � � � , the equations in the right column are solved for the variables appearingin the left column. The equations are presented in the order required by the simulation. It is assumed that

both the disturbance signal vk and the reference signal rk are known for the simulation.

Variables Equations

Qk�1 Qk�1 D O�0,k�1CO�1,k�1q

�1C O�2,k�1q�2C � � � C O�ıQ,k�1q

�ıQ

vk vk D A�vo,k CB�vi ,kyk .A�RoCB�SoCQk�1.B�A�A�B// yk D B�Trk C .Ro �Qk�1B/vkuk .Ro �Qk�1B/uk D .SoCQk�1A/yk�1,k �1,k.ARoCBSo/D ARoyk �BRouk�2,k �2,k.ARoCBSo/D AByk �B2uk

O�k�1O�k�1 D

hO�0,k�1

O�1,k�1O�2,k�1 � � � O�ıQ,k�1

iT'k 'k D

��2,k �2,k�1 �2,k�2 � � � �2,k�ıQ

�TPk P�1

kD .1� �/P�1

k�1C � 'k'

Tk

O�kO�k D

O�k�1C � Pk'k.�1,k � 'TkO�k�1/

is solved. A particular solution gives the controller polynomials Ro D 1, So D �8.17C 6.10q�1,and TD 0.10.

The plant is perturbed by the following input and output disturbances vi and vo:

vi,k D1

1� 1.02q�1ık�k1 ,

vo,k D1

1� 2 cos.0.5/q�1C q�2ık�k2 ,

applied at k1h D 10 s and k2h D 20 s, respectively. These disturbances represent a growing expo-nential and a marginally stable harmonic signal. Note that the value of ıDu changes with time. Upto k1h D 10 s, a step input is applied. In the absence of an integral term, this would result in someoffset. However, the error arising from the step input can be eliminated by treating the step input as aconstant disturbance (i.e., and unstable disturbance with q D 0 and ıDu D 1). Between k1hD 10 sand k2hD 20 s, an exponential disturbance is present, thus leading to one additional unstable pole;thus, ıDu D 2. After k2hD 20 s, the harmonic disturbance adds two more unstable poles, thus lead-ing to ıDu D 4. Because it is necessary to define the degree of Q on the basis of the largest degree ofthe equation error, ıQD ıDu�1D 3 is chosen. The polynomial QD �0C�1q�1C�2q�2C�3q�3 isinitialized with O�0,0 D O�1,0 D O�2,0 D O�3,0 D 1. Parameter adaptation is carried out using the updatelaw (20)–(21), the update gain � D 0.8 and P0 D 10000I4. Table I summarizes the equationsrequired for simulating the plant and estimating the Q parameters.

Time (s)

Con

trol

led

outp

ut y

k

step vi vo

Figure 2. Closed-loop step response. The exponential input disturbance vi and the harmonic outputdisturbance vo are applied after 10 and 20 s, respectively.



0 10 20 30 40 50 60-50

-40

-30

-20

-10

0

10

20

30

40

Q p

aram

eter

s

Time (s)

θ1

θ0

θ2

θ3

θ1

θ0

θ2

θ3

step vi vo

Figure 3. Parameter adaptation in response to plant–model mismatch and disturbances. Notice the changesin parameter values when the disturbances vi and vo enter the plant.

0 10 20 30 40 50 600

5

10

15

20

25

0 10 20 30 40 50 60-200

-150

-100

-50

0

50

Time (s)

Dis

turb

ance

vk

Inpu

t uk

uk

vk

step vi vo

Figure 4. The disturbance v D A�vo C B�vi in equation error form (left axis) and the input u needed fordisturbance rejection (right axis).

The plant step response in the presence of the two disturbances is shown in Figure 2. Initially, asseen in Figure 3, the controller needs to adapt in order to eliminate the effect of plant–model mis-match. As soon as the exponential disturbance enters the plant, the Q parameters undergo a secondadaptation in order to keep the output at the setpoint. Finally, a third adaptation occurs when theharmonic disturbance appears. The disturbance signal v D A�vo C B�vi and the resulting controlsignal u are shown in Figure 4. Oscillations in the control signal occur soon after the harmonicdisturbance is introduced. In order to eliminate the three disturbances, the controller denominator Rneeds to include as factors the poles of the harmonic disturbance vo and of the unstable disturbancevi. In addition, it needs to provide integral action in order to eliminate the offset due to plant–model mismatch. As a result, the control signal contains an exponential that increases in a directionopposite to vi.

5. CONCLUSIONS

In order to reject disturbances, the feedback loop needs to incorporate the disturbance model accord-ing to the IMP. If the plant and disturbance models are known LTI systems, control design tech-niques such as the linear quadratic regulator are excellent choices. However, if the disturbances areunknown, direct adaptive control that uses a least squares minimization to ‘identify’ the disturbancesare more appropriate.



A direct adaptive control methodology for the rejection of unmeasured nonvanishing disturbanceshas been presented in this work. This was performed using RST control and the additional degreesof freedom provided by the Q parameterization. It was first shown that the IMP can reject non-vanishing disturbances asymptotically also in the presence of plant–model mismatch. Then, it wasdemonstrated that the IMP can be implemented via the solution of a least squares problem. Finally,the convergence of a recursive algorithm for asymptotic disturbance rejection was proven. The maintechnical result is that, if parameter adaptation satisfies both .1� �/ < 1= N , where N is the largestmodulus of the unstable modes to be rejected, and .1 � �/ < , where is the converging rate ofthe stable auxiliary signal ´k , then the adaptive scheme will ensure that ek converges to zero. Theonly condition is that the characteristic polynomial A�c . NQ/ be stable. Hence, it is recommended touse robust control techniques to design the nominal controller.

APPENDIX—SUPPORTIVE LEMMAS

Lemma 1 (Persistency of excitation)If B does not share common roots with Du, then �2,k is persistently exciting of order at least ıDu.

ProofWith (14), �2,k can be written as

�2,k DBAc

NA�c Ds Du

ık . (45)

If B does not share common roots with Du, it cannot eliminate the unstable modes of Du that arepresent in wk . Hence, all ıDu unstable modes will be present in �2,k . If A�c is stable, there willbe exactly ıDu unstable modes in �2,k ; otherwise, there will be in addition the unstable modesof A�c . �

Lemma 2 (Bounding a sequence of positive real numbers)(A1) Let v2

kbe an exponentially decreasing sequence, that is, for vk 2 R, k 2 N, 9˛ 2 R�C and

9 2 Œ0, 1/ such that

v2k 6 ˛k , 8k 2N.

(A2) Let Vk 2RC, k 2N, be a sequence of positive real numbers obeying the implicit inequality

Vk 6 �1Vk�1C �2v2k�1, (46)

with the two constants �1 2 Œ0, 1/ and �2 2R�C.Then, the following explicit bound holds for all � 2 Œ0, 1/:

Vk 6 �k1V0C˛ �2

ˇ .� lnˇ/.1��/e��1�C1 .ˇ�/

k8�, 06 � < 1, (47)

with

ˇ Dmax f�1, g .

ProofBacksubstituting Vk in itself in (46) gives

Vk 6 �1Vk�1C �2v2k�1 6 �1.�1Vk�2C �2v2k�2/C �2v

2k�1. (48)

The resulting inequality tower is not broken because all factors and terms remain positive with�1 > 0. Iterating this process generates the following pattern:

Vk 6 �k1V0C �2kXiD1

� i�11 v2k�i . (49)



Using the fact v2k6 ˛k leads to

Vk 6 �k1V0C ˛�2kXiD1

� i�11 k�i . (50)

Defining ˇ WDmaxf�1, g allows bounding the terms in the summation in (50) separately by k termsall equal to ˇk�1; thus,

Vk 6 �k1V0C˛ �2

ˇkˇk . (51)

Furthermore, fkˇk W k 2 Ng is the positive integer sequence obtained upon evaluating the con-tinuous positive function .1=.� lnˇ//te�t at the discrete time instants t D .� lnˇ/k (note that.� lnˇ/ > 0 follows from 0 < ˇ < 1).

It remains to bound te�t exponentially. For this, choose any real � 2 R, 0 < � < 1, and find apositive constant C� 2 RC such that C�e��t � te�t > 0, 8t > 0. The first function C�e��t is amonotonically decreasing convex function, whereas the second function te�t has a single extremumfor t D 1, at which time the function changes from concave to convex. Hence, the difference of thesetwo functions has, at most, one extremum, and a sharp bound is obtained when both functions sharethe same value and the same derivative, thereby resulting in

te�t 6 1

1��e��1�C1 e��t , 8�, 06 � < 1. (52)

To go back to discrete samples, simply evaluate the bounds (52) at the discrete values t D .� lnˇ/kto obtain

kˇk 6 1

.� lnˇ/.1��/e��1�C1 .ˇ�/

k ,

which with (51) gives (47). �

Lemma 3 (Condition for common Lyapunov matrix)(A1) Consider the matrix NA in canonical form,

NAD

0BBBBB@

0 1 : : : 0 0

0 0 : : : 0 0...

. . ....

...0 0 : : : 0 1

�N�ı �N�ı�1 : : : �N�2 �N�1

1CCCCCA , (53)

and the associated perturbation matrix

E D

0BBBB@

0 0 : : : 0 0

0 0 : : : 0 0...

......

...0 0 : : : 0 0

��ı ��ı�1 : : : ��2 ��1

1CCCCA . (54)

(A2) Let the perturbation matrix satisfy j�i j< �, i D 1, : : : , ı, with

06 � <

sN�2C

1

ı N�� N� ,

where N�Dmax�.P /, N� Dmaxi j N�i j, and ı is the dimension of the square matrix NA.Then, all matrices A D NA C E share the common Lyapunov matrix P that is solution toNATP NA�P C I .



ProofDefine

M WD ATPA�P D�NACE

TP�NACE

�P

D NATP NA�P CETP NAC NATPE CETPE D�I CETP NAC NATPE CETPE.

Next, find a suitable � such that xTMx < 0 for all x:

xTMx D�xTxC xT�ETP NAC NATPE CETPE

x

6 �xTxC N�xT�ET NAC NATE CETE

x

D�

ıXiD1

x2i �N�

ıXiD1

ıXjD1

��i N�j C �j N�i � �i�j

xıC1�ixıC1�j

6 �ıXiD1

jxi j2C N� N�

ıXiD1

ıXjD1

�j�i j C j�j j

jxıC1�i jjxıC1�j j

C N�

ıXi

ıXj

j�i jj�j jjxıC1�i jjxıC1�j j

6 �ıXiD1

jxi j2C N�

�2� N� C �2

ıXiD1

ıXjD1

jxi jjxj j

6 �ı.maxijxi j/

2C ı2 N��2� N� C �2

.maxijxi j/

2

D ı.maxijxi j/

2�ı N��2C 2ı N� N�� 1/

. (55)

Because .maxi jxi j/2 > 0, and upon dividing by ı > 0, the inequality xTMx < 0 holds whenever

ı N� �2C 2ı N� N� � � 1 < 0, (56)

that is, when � satisfies the bound

06 � <

sN�2C

1

ı N�� N�. (57)

�

Lemma 4 (Combination of sequences)

(A1) Let NH DPniD0Nhiq�i be an unstable characteristic polynomial and assume that the n

sequences fhi ,k W k 2 Ng, i D 1, : : : n, converge asymptotically to the values Nhi at therate � , with 06 � < 1, that is, jhi ,k � Nhi j6 jhi ,0 � Nhi j�k .

(A2) Let ´k be a sequence that converges exponentially to zero, j´kj6 j´0jˇk , with 06 ˇ < 1.(A3) Define the exponentially diverging sequence fsk W k 2Ng as sk WD .1= NH/´k .Then, the sequence

ek D

nXiD0

hi ,kq�i

!sk

converges exponentially to zero for � < 1= N , where N is the largest modulus of the roots of NH.



ProofFrom the definition of the sequences sk and ek , one can write ´k D

PniD0Nhiq�isk and ek DPn

iD0 hi ,kq�isk , from which it becomes possible to construct the difference

ek � ´k D

nXiD0

.hi ,k � Nhi /q�isk ,

jek � ´kj6nXiD0

jhi ,k � Nhi jjsk�i j.

Using the assumption that jskj6 js0j Nk , for all k 2N, gives

jek � ´kj6 js0jnXiD0

jhi ,k � Nhi j Nk�i

6 js0jnXiD0

jhi ,0 � Nhi j�k Nk�i D js0j�

k NknXiD0

jhi ,0 � Nhi j N�i .

The left-hand side and thus also the right-hand side converge for � < 1= N . It follows thatjek � ´kj 6 c.� N /k , with c a constant term. Because f´k W k 2 Ng converges exponentially tozero, so does fek W k 2Ng. �

REFERENCES

1. Francis BA, Wonham WM. The internal model principe of control theory. Automatica 1976; 12:457–465.2. Bengtsson G. Output regulation and internal models – a frequency domain approach. Automatica 1977; 13:333–345.3. Tsypkin YZ, Nadezhdin PV. Robust continuous control systems with internal models. Control Theory and Advanced

Technology 1993; 9(1):159–172.4. Savaresi SM. Funnel filters: a new class of filters for frequency estimation of harmonic signals. Automatica 1997;

33(9):1711–1718.5. Johnson CD. Adaptive controller design using disturbance-accommodation techniques. International Journal of

Control 1985; 42(1):193–210.6. Feng G, Palaniswami M. A stable adaptive implementation of the internal model principle. IEEE Transactions on

Automatic Control 1992; 37(8):1220–1225.7. Palaniswami M. Adaptive internal model for disturbance rejection and control. IEE Proceedings-D 1993; 140(1):

51–59.8. Tay TT, Moore JB. Enhancement of fixed controllers via adaptive-Q disturbance estimate feedback. Automatica

1991; 27(1):39–53.9. Amara FB, Kabamba PT, Ulsoy AG. Adaptive sinusoidal disturbance rejection in linear discrete-time systems – Parts

1 and 2. Journal of Dynamic Systems, Measurement, and Control 1999; 121:648–659.10. Wang Y, Chu KC, Tsao TC. Adaptive rejection of stochastic and deterministic sinusoidal disturbances with unknown

frequency. ACC, St. Louis, Missouri, 2009; 5641–5646.11. Valentinotti S. Adaptive rejection of unstable disturbances: application to a fed-batch fermentation process. Ph.D.

Thesis, Nr. 2405, École Polytechnique Fédérale de Lausanne, Lausanne, 2001.12. Landau ID, Constantinescu A, Rey D. Adaptive narrow band disturbance rejection applied to an active suspension –

an internal model principle approach. Automatica 2005; 4:563–574.13. Åström KJ, Wittenmark B. Adaptive Control, (2nd edn). Addison-Wesley Longman Publishing Co., Inc.: Boston,

MA, 1994.14. Tsypkin YZ. Adaptive-invariant discrete control systems. In Foundations of adaptive control, Vol. 160, Thoma M,

Wyner A (eds), Lecture Notes in Control and Information Science. Springer-Verlag: Berlin, 1991; 239–268.15. Tsypkin YZ. Stochastic discrete systems with internal models. Journal of Automation and Information Sciences

1997; 4&5:156–161.16. Valentinotti S, Srinivasan B, Holmberg U, Bonvin D, Cannizzaro C, Rhiel M, von Stockar U. Optimal opera-

tion of fed-batch fermentations via adaptive control of overflow metabolite. Control Engineering Practice 2003;11(6):665–674.

17. Constantinescu A, Rey D, Landau ID. Rejection of narrow band unknown disturbances in an active suspensionsystem. ECC, Kos, Greece, 2007; 142–149.

18. Landau ID, Constantinescu A, Alma M. Adaptive regulation – rejection of unknown multiple narrow banddisturbances. 17th Mediterranean Conference on Control and Automation, Thessaloniki, Greece, 2009; 1056–1065.



19. Kucera V. Discrete Linear Control. The Polynomial Equation Approach. Wiley-Interscience: Chichester, 1979.20. Holmberg U, Myszkorowski P, Bonvin D. A pole-placement approach that matches identification and control

objectives. ECC 97, Brussels, 1997.21. Åström KJ, Wittenmark B. Computer Controlled Systems Theory and Design, Information and System Sciences.

Prentice Hall: Upper Saddle River, N.J., 1997.22. Ioannou PA, Sun J. Robust Adaptive Control. Prentice Hall: Upper Saddle River, N.J., 1996.23. Ljung L. System Identification. Theory for the User, Information and System Sciences. Prentice Hall: Englewood

Cliffs, New Jersey, 1987.24. Rugh WJ. Linear System Theory. Prentice-Hall: Upper Saddle River, N.J., 1993.


Asymptotic rejection of nonvanishing disturbances despite plant-model mismatch

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