SIMPLE ADAPTIVE CONTROL - A STABLEDIRECT MODEL REFERENCE ADAPTIVE

CONTROL METHODOLOGY - BRIEF SURVEY

Itzhak Barkana 1

Abstract: In spite of successful proofs of stability and even successful demon-strations of performance, the eventual use of Model Reference Adaptive Control(MRAC) methodologies in practical real world systems has met a rather strongresistance from practitioners and has remained very limited. Apparently, thepractitioners have a hard time understanding the conditions that can guaranteestable operations of adaptive control systems under realistic operational envi-ronments. Besides, it is difficult to measure the robustness of adaptive controlsystem stability and allow it to be compared with the common and widely usedmeasure of phase margin and gain margin that is utilized by present, mainlyLTI, controllers. Furthermore, recent counterexamples seem to show that adaptivesystems may diverge even when all required conditions are fulfilled. This paperattempts to revisit the fundamental qualities of the common direct model referenceadaptive control methodology based on gradient and to show that some of itsbasic drawbacks have been addressed and eliminated within the so-called SimpleAdaptive Control methodology. The sufficient conditions that guarantee stabilityare clearly stated and lead to similarly clear proofs of stability, and the previouscounterexamples to MRAC become just simple, successful, and stable applicationsof SAC. Copyright c©2007 IFAC.

Keywords: Control systems, stability, passivity, uncertain systems, almost strictpassivity (ASP), adaptive control

1. INTRODUCTION

We will consider the square, multiple-input-multiple-output, system with the realization

x(t) = Ax(t) + Bu(t) (1)

y(t) = Cx(t) (2)

Here, x is the n-dimensional state vector, u isthe m-dimensional input vector and y is the m-dimensional output vector, and A, B, and C arematrices of corresponding dimensions. Becauseonly a nominal model of the real-world plantis usually available for the control design and,

1 Kulicke & Soffa Industries, Inc. 1005 Virginia Drive, FortWashington, PA 19034, USA. Phone: 215-784-6279, FAX:215-784-6001, Email: ibarkana@kns.com

furthermore, plant parameters may vary undervarious operational and environmental conditions,adaptive control methodologies seemed to be thenatural solution for the problem. However, thecustomary stationary controllers have developedverification techniques that allow the designer toevaluate the stability robustness of the controlledsystem. Even if the actual gain deviates fromthe nominal control gains Knom, the designerhas a very clear idea about the admissible do-main Kmin < K < KMAX of possible devia-tion of the actual constant gain K that wouldnot result in destruction of system stability. Theadaptive control methodologies, as any other con-trol techniques that use nonstationary controllers,cannot guarantee stability if the stationary gainK is replaced by an arbitrary, nonstationary,gain K(t) even if it follows the same condition

Kmin < K(t) < KMAX . Therefore, new rulesand specialized methods for the proofs of stabilityand measure of its robustness had to be devel-oped. Section 2 gives a short review of classicalModel Reference Adaptive Control (MRAC) andits many contributions to stability analysis, alongwith some implied drawbacks related to the so-called “unmodeled dynamics” and to the need for“sufficient excitation” that have prevented MRACfrom being used in many real-world applicationsin spite of its qualities. Section 3 presents thevarious passivity and “almost passivity” condi-tions that are used to proof stability with adaptivecontrollers. Section 4 mentions some counterex-amples that diverge under MRAC, and thereforeseem to demonstrate that previous assumptionsmay not be sufficient to guarantee stability withMRAC. Section 5 presents the Simple AdaptiveControl (SAC) methodology which initially wasdeveloped as a simplified and modest alternativeto MRAC, yet ultimately seems to eliminate thedrawbacks related to MRAC. Section 6 showsthat the almost passivity conditions are indeedsufficient to guarantee robust stability with SAC.Because the control community at large tries toavoid using adaptive controllers, the example ofSection 7 illustrates the danger involved with thebelief that LTI system theory and its safety gainand phase margin can be used in real-world chang-ing environments. Section 8 then shows how ba-sic stabilizability properties of plants and parallelfeedforward can be used so SAC can be appliedwith systems that do not inherently satisfy the ba-sic “almost passivity” conditions. Finally, Section9 then shows how a simple σ-term can be usedto add the necessary robustness so SAC can beapplied in those situations where perfect trackingis not possible.

2. MODEL REFERENCE ADAPTIVECONTROL

First attempts at using adaptive control tech-niques were developed during the sixties andwere based on intuitive and even ingenious ideas(Whitaker, 1959), (Osborn, Whitaker and Kezer,1961), yet they ended in failure, mainly becauseat the time there was not very much knowledgeof stability analysis with nonstationary parame-ters. Modern methods of stability analysis thathad been developed by Lyapunov at the startof the 19th century were not broadly known,much less used, in the West (Hahn, 1967). Af-ter the initial problems with adaptive controltechniques of the sixties, stability analysis hasbecome a center point in new developments re-lated to adaptive control. Participation of someof the leading researchers in the control com-munity at the time, such as Narendra, Landau,

Astrom, Kokotovic, Goodwin, Morse, Grimbleand many others, added a remarkable contri-bution to the better modeling and to the un-derstanding of adaptive control methodologies(Monopoli, 1974), (vanAmerongen and TenCate,1975), (Feuer and Morse, 1978), (Morse, 1980),(Landau, 1974), (Landau, 1979), (Narendra andValavani, 1978), (Narendra and Valavani, 1979),(Narendra, Lin and Valavani, 1980), (Narendraand Annaswami, 1989), (Goodwin, Ramadge andCaines, 1980), (Goodwin and Sin, 1984), (Astrom,1983), (Astrom and Wittenmark, 1989), (Ioannouand Kokotovic, 1983), (Moir and Grimble, 1984),(Mareels, 1984), (Kreiselmayer and Anderson,1986), (Ortega and Yu, 1987), (Sastri and Bodson,1989), (Ioannou and Sun, 1996), (Bitmead, Geversand Wertz, 1990), (Wellstead and Zarrop, 1991),(Krstic, Kanellakopoulos and Kokotovic, 1995).New tools and techniques have been developedand used and they finally led to successful proofsof stability, mainly based on the Lyapunov sta-bility approach. The standard methodology wasthe Model Reference Adaptive Control approachwhich, as its name states, basically requires thepossibly “bad” plant to follow the behavior of a“good” Model Reference.

˙xm(t) = Amxm(t) + Bmum(t) (3)

ym(t) = Cmxm(t) (4)

The control signal that feeds the plant is a linearcombination of the Model state variables

u(t) =∑

kixmi(t) = Kxm(t) (5)

If the plant parameters were fully known, onecould compute the corresponding controller gainsthat would force the plant to asymptotically fol-low the Model, or

x(t) → xm(t) (6)

and correspondingly

y(t) → ym(t) (7)

Because the entire plant state ultimately behavesexactly as the model state, MRAC is sometimesinterpreted as Pole-Zero placing. However, in thisreport we only relate to MRAC in relation to itsmain aim, namely, the plant output should followthe desired behavior represented by the modeloutput.

When the plant parameters are not (entirely)known, one is naturally lead to use adaptivecontrol gains. The basic idea is that the plant isfed a control signal that is a linear combinationof the model state through some gains. If allgains are correct, the entire plant state vector

U

ey=ym-y

X mX

Yp

1

1/s

1/s

Kx

Ku

Cm

C

Bm

B

Am

A

Um

1

Fig. 1. Schematic representation of MRAC

would follow the model exactly. If, however, notall gains are correct, the measured plant outputdiffers from the output of the Model Reference.The resulting “tracking error”

ey(t) = ym(t)− y(t) (8)

can be monitored and used to generate adaptivegains. The basic idea of the adaptation is likethat: assume that one component of the controlsignal that is fed to the plant is coming from thevariable xmi through the gain kxi. If the gain isnot perfectly correct, this component contributesto the tracking error and therefore the trackingerror and the component xmi are correlated. Thiscorrelation is used to generate the adaptive gain

kxi(t) = γiey(t)xmi(t) (9)

where γi is a parameter that affects the rateof adaptation. The adaptation should continuesuntil the correlation diminishes and ultimatelyvanishes and therefore the gain derivative tendsto zero and the gain itself is (hopefully) supposedto ultimately reach a constant value. In vectorialform,

Kx(t) = Σγiey(t)xmi(t) = ey(t)xTm(t)Γx (10)

u(t) = Σkxixmi(t) = Kx(t)xm(t) (11)

As Figure 1 below shows, there are various othercomponents that can be added to improve theperformance of the MRAC system such as

Ku(t) = ey(t)uTm(t)Γu (12)

so the total control signal is

u(t) = Kx(t)xm(t) + Ku(t)um(t) (13)

Many other elements, such as adaptive observers,etc., can be added to this basic MRAC scheme andcan be found in the reference cited above, yet here

we want to pursue just the basic Model Referenceidea.

This approach was able to generate some rigorousproofs of stability that showed that not only thetracking error but even the entire state error

ex(t) = xm(t)− x(t) (14)

asymptotically vanishes. This result implied thatthe plant behavior would asymptotically repro-duce the stable model behavior and would ul-timately achieve the desired performance repre-sented by the ideal Model Reference. In partic-ular, the Lyapunov stability technique revealedthe prior conditions that had to be satisfied inorder to guarantee stability and allowed gettingrigorous proofs of stability of the adaptive controlsystem. Because along with the dynamics of thestate or the state error, adaptive control systemshave also introduced the adaptive gains dynamics,the positive definite quadratic Lyapunov functionhad to contain both the errors and the adaptivegains and usually had the form

V (t) = eTx (t)Pex(t)

+ tr

[(K(t)− K

)Γ−1

(K(t)− K

)T]

(15)

Here, K is a set of the ideal gains that couldperform perfect model following if the parameterswere known, and that the adaptive control gainswere supposed to asymptotically reach. Yet, inspite of successful proofs of stability, very littleuse has been made of adaptive control techniquesin practice.

Therefore, we will first discuss some of the prob-lems that are inherent to the classical MRAC ap-proach and that are emphasized when one intendsto use adaptive methods with such applicationsas large flexible space structures and similar largescale systems. First, the fact that the entire plantstate vector is supposed to follow the behavior ofthe model state vector immediately implies thatthe model is basically supposed to be of the sameorder as the plant. If this is not the case, variousproblems have been shown to appear, includingtotal instability. As real world plants are usually ofvery high order when compared with the nominalplant model, a so-called “unmodeled dynamics”must inherently be considered in the context ofthis approach. The developers of adaptive controltechniques were able to show that the adaptivesystem still demonstrates stability robustness inspite of the “unmodeled dynamics,” yet to thisend they required that the “unmodeled dynam-ics” be “sufficiently small.” Furthermore, if anystate variable of the Model reference is zero, thecorresponding adaptive gain is also zero. Also, ifthe model reaches a steady state, some of the

various adaptive gains loose their independence,and this point raises the need for some “persistentexcitation” or “sufficient excitation.” It should beemphasized that the need for sufficiently largeModels, sufficiently small “unmodeled dynamics”and “sufficient excitation” appear even if one onlyintends to guarantee the mere stability of theplant, before even mentioning performance. Fi-nally, when all these basic conditions are satisfied,the stability of the adaptive control could initiallybe proved only if the original plant was StrictlyPassive (SP), which in LTI systems implies thatits input-output transfer function is Strictly Posi-tive Real (SPR). Passivity-like conditions appearin various forms in different presentations, so theydeserve a special section.

3. ON VARIOUS PASSIVITY CONDITIONSREQUIRED IN ADAPTIVE CONTROL

Positive Realness or, more precise, Passivity ofsystems is a useful systems property that hasbeen first introduced in networks (Cauer, 1958),and probably introduced to dynamic systems byKalman (Kalman, 1964) in the context of opti-mality. It has also shown its usefulness in thecontext of “absolute stability” (Popov, 1962). Aswe already mentioned, Positive Realness has alsobeen shown to be useful for the proof of stabilitywith adaptive controllers. Here, we present thestate-space representation of the SPR conditionswhich seems to be the most useful for successfulproofs of stability using Lyapunov stability theory.

Definition 1. A linear time-invariant system witha state-space realization {A, B, C}, where A ∈Rn∗n, B ∈ Rn∗m, C ∈ Rm∗n, with the m*mtransfer function T (s) = C(sI − A)−1B, is called“strictly passive (SP)” and its transfer function“strictly positive real (SPR)” if there exist twopositive definite symmetric (PDS) matrices, Pand Q, such that the following two relations aresimultaneously satisfied:

PA + AT P = −Q (16)

PB = CT (17)

The relation between the strict passivity condi-tions (16)-(17) and the strict positive realnessof the corresponding transfer function has beentreated elsewhere (Ioannou and Tao, 1987), (Wen,1988). Relation (16) is the common algebraic Lya-punov equation and shows that an SPR systemis asymptotically stable. One can also show thatconditions (16)-(17) also imply that the systemis strictly minimum-phase, yet simultaneous satis-faction of both conditions (16)-(17) is far from be-ing guaranteed even in stable and minimum-phase

systems, and therefore the SPR condition seemedmuch too demanding. (Indeed, some colleaguesin the general control community use to ask: ifthe system is already asymptotically stable andminimum-phase, why would one need adaptivecontrollers?)

For a long time, the passivity condition had beenconsidered very restrictive (and rather obscure)and at some point the adaptive control communityhas been trying to drop it and to do withoutit. The passivity condition has been somewhatmitigated when it was shown that stability withadaptive controllers could be guaranteed evenfor the non-SPR system (1)-(2) if there exists aconstant output feedback gain (unknown and notneeded for implementation), such that a fictitiousclosed-loop system with the system matrix

AK = A−BKeC (18)

is SPR, namely, it satisfy the passivity conditions(16)-(17). Because in this case the original system(1)-(2) was only separated by a simple constantoutput feedback from strict passivity, it was called“Almost Strictly Positive Real (ASPR)” or “Al-most Strictly Passive (ASP)” (Barkana and Kauf-man, 1985), (Barkana, 1987). Note that such ASPsystems are sometimes called (Fradkov, 2003),(FradkovHill, 1998) “feedback passive” or “passi-fiable.” However, as we will show that any stabiliz-able system is also passifiable via parallel feedfor-ward, those systems that are only at the distanceof a constant feedback gain from Strict Passivitydeserve a special name.

At the time, this “mitigation” of the passivity con-ditions did not make a great impression, becauseit was still not clear what systems would satisfythe new conditions. (Some even claimed that ifSPR seemed to be another name for the void classof systems, the “new” class of ASPR was onlyadding the boundary.) Nonetheless, some ideaswere available. Because a constant output gainfeedback was supposed to stabilize the system, itseemed apparent that the original plant was notrequired to be stable. Also, because it was knownthat SPR systems were minimum-phase and thatthe product CB is Positive Definite Symmetric(PDS), it was intuitive to assume that minimum-phase systems with Positive Definite Symmet-ric CB were natural ASPR candidates (Barkanaand Kaufman, 1985). Indeed, simple Root-locustechniques were sufficient to proof this result inSISO systems, and many examples of minimum-phase MIMO systems with CB product PDS wereshown to be ASPR (Barkana and Kaufman, 1985),(Barkana, 1987). However, it was not clear howmany of such MIMO system actually were ASPR.Because the ASPR property can be stated as asimple condition and because it is the main con-

dition needed to guarantee stability with adaptivecontrollers, it is useful to present here the ASPRtheorem for the general multi-input-multi-outputsystems:

Theorem 1. Any linear time-invariant system withthe state-space realization {A, B, C}, where A ∈Rn∗n B ∈ Rn∗m ,C ∈ Rm∗n, with the m*mtransfer function T (s) = C(sI − A)−1B, that isminimum-phase and where the matrical productCB is PDS, is “almost strictly passive (ASP)” andits transfer function “almost strictly positive real(ASPR).”

Although the original plant is not SPR, a (ficti-tious) closed-loop system satisfies the SPR condi-tions, or in other words, there exist two positivedefinite symmetric (PDS) matrices, P and Q, anda positive definite gain such that the following tworelations are simultaneously satisfied:

P (A−BKeC) + (A−BKeC)T P = −Q (19)

PB = CT (20)

As a matter of fact, a proof of Theorem 1 had beenavailable in the Russian literature (Fradkov, 1976)since 1976 yet it was not known in the West. Here,many other works have later independently redis-covered, reformulated, and further developed theidea (see (Barkana, 2004) and references thereinfor a brief history and for a simple and direct, al-gebraic, proof of this important statement). Evenas late as 1999, this simple ASPR condition wasstill presented as some algebraic condition (Huanget al., 1999) that might look obscure to the con-trol practitioner. On the other hand, (Huang etal., 1999) managed to add an important con-tribution and emphasize the special property ofASPR systems by proving that if a system cannotbe made SPR via constant output feedback, nodynamic feedback can render it SPR.

Theorem 1 has thus managed to explain the ratherobscure passivity conditions with the help of newconditions that could be understood by controlpractitioners. It is useful to notice an importantproperty that may makes an ASPR system to bea good candidate for stable adaptive control: if aplant is minimum-phase and its input-output ma-trical product CB is Positive Definite Symmetric(PDS) it is stabilizable via some static PositiveDefinite (PD) output feedback. Furthermore, ifthe output feedback gain is increased beyond someminimal value, the system remains stable even ifthe gain increase is nonstationary. The requiredpositivity of the product CB could be expected,as it seemed to be a generalization of the sign ofthe transfer function that allows using negativefeedback in SISO systems. However, although at

the time it seemed to be absolutely necessary forthe ASPR conditions, the required CB symmetryproved to be rather difficult to fulfill in practice, inparticular in adaptive control systems where theplant parameters are not known.

After many attempts that have ended in failure,a recent publication has managed to eliminatethe need for a symmetric CB. First, it was easyto observe that the Lyapunov function remainspositive definite if the gain term is rewritten asfollows:

V (t) = eTx (t)Pex(t) (21)

+ tr

[S

(K(t)− K

)Γ−1

(K(t)− K

)T

ST

]

Here, S is any nonsingular matrix. This new for-mulation allowed the extension of useful passivity-like properties to a new class of systems that wascalled W-ASPR, where W = ST S, through thefollowing definition:

Definition 2. Any linear time-invariant systemwith state-space realization {A, B, C}, whereA ∈ Rn∗n, B ∈ Rn∗m, C ∈ Rm∗n, with them*m transfer function T (s) = C(sI − A)−1B, iscalled “W-almost strictly passive (WASP)” andits transfer function “W-almost strictly positivereal (WASPR),” if there exist three positive defi-nite symmetric (PDS) matrices, P, Q, and W, anda positive definite gain Ke such that the followingtwo conditions are simultaneously satisfied:

P (A−BKeC) + (A−BKeC)T P = −Q (22)

PB = CT WT (23)

This new definition can be used with the followingtheorem (Barkana, Teixeira and Hsu, 2006):

Theorem 2. Any minimum-phase LTI system witha state-space realization {A, B, C}, where A ∈Rn∗n, B ∈ Rn∗m, C ∈ Rm∗n, with the m*mtransfer function T (s) = C(sI−A)−1B, where thepositive definite and not necessarily symmetricmatrical product CB is diagonalizable is WASPin accord with Definition 2.

Thus, (Barkana, Teixeira and Hsu, 2006) hadmanaged to mitigate a result that has been aroundfor more than 40 years. Nevertheless, it was verytempting to try to eliminate any restriction be-sides the positivity of CB. This new result was ”al-most” made possible by observing that, althoughthe product of two PD matrices is not necessarilyPD, the trace of the product is PD if at leastone of the two matrices is PDS. Therefore, the

Lyapunov function remains positive definite if thesecond term in it is again rewritten as follows:

V (t) = eTx (t)Pex(t)

+ tr

[W

(K(t)− K

)Γ−1

(K(t)− K

)T]

(24)

even if W is only positive definite yet not necessar-ily symmetric. However, we will show that in orderto allow the Lyapunov derivative to be negativedefinite or semidefinite, the proof of stability doesrequire the symmetry of W.

As we showed above, new developments havesimplified the (sufficient?) conditions that, alongwith limits on the “unmodeled dynamics” andwith “sufficient excitation,” would be sufficient toallow rigorous and successful proofs of stabilitywith adaptive controllers. Still, it appears thatthe ASPR condition was not sufficient to alwaysguarantee the stability of MRAC. Besides, al-though successful proofs of stability usually endedshowing that the following errors vanish asymp-totically, it is rather commonly accepted that theadaptive gains may not converge to any specificlimit at all, even if they are guaranteed to bebounded. Furthermore, some recent counterexam-ples seem to show that MRAC systems may di-verge even when all previously assumed sufficientconditions are satisfied (Hsu and Costa, 1999).

4. COUNTEREXAMPLES TO MODELREFERENCE ADAPTIVE CONTROL

In the examples of (Hsu and Costa, 1999), a 2*2stable plant with CB positive definite is requiredto follow the behavior of a stable model of sameorder. In fact both the plant and the modelhave the same diagonal system matrices withnegative eigenvalues, and only the input-outputmatrix differentiates between the two. The plant,that appears in a 2D adaptive robotic visualservoing with uncalibrated camera, is defined bythe system matrices

A =[−a 0

0 −a

]; B =

[cosϕ sinϕ−hsinϕ hcosϕ

](25)

C =[

1 00 1

](26)

It is shown (Hsu and Costa, 1999) that standardMRAC systems become unstable even though theMRAC system was supposed to be stable becausethere was no “unmodeled dynamics,” there was“sufficient excitation,” and the assumably “suf-ficient” passivity conditions were also satisfied.We note that (Hsu and Costa, 1999) shows waysto avoid the problem and, using various kinds ofprior knowledge, other solutions have also beenproposed.

5. SIMPLE ADAPTIVE CONTROL (SAC), ORTHE SIMPLIFIED APPROACH TO MODEL

REFERENCE ADAPTIVE CONTROL

Various kinds of additional prior knowledge havebeen used and many solutions and additions havebeen proposed to overcome some of the variousdrawbacks of the basic MRAC algorithm. How-ever, this paper sticks to the very basic idea ofModel Following. Next sections will show thatthose basically ingenious adaptive control ideasand the systematic stability analysis they intro-duced had finally led to adaptive control sys-tems that can guarantee stability robustness alongwith superior performance when compared withalternative, non-adaptive, methodologies. In thissection we will first assume that at least one ofthe passivity conditions presented above holdsand will deal with a particular methodology thatmanaged to eliminate the need for the plant orderand therefore can mitigate the problems relatedto “unmodeled dynamics” and “persistent excita-tion.” Subsequent sections will then extend thefeasibility of the methodology to those real-worldsystems that do not inherently satisfy the passiv-ity conditions.

The beginning of the alternative adaptive con-trol approach can be found in the intense activ-ities at Rensselaer (RPI) during 1978-1983. Atthat time, such researchers as Kaufman, Sobel,Barkana, Balas, Wen, and others (Sobel, Kauf-man and Mabus, 1982), (Kaufman et al., 1981),(Barkana and Kaufman, 1982), (Barkana, Kauf-man and Balas, 1983), (Barkana, 1983), (Wen andBalas, 1989) were trying to use customary adap-tive control techniques with large order MIMOsystems, such as planes, large flexible structures,etc. It did not take long to realize that it wasimpossible to even think of controllers of the sameorder as the plant, or even of the order of a “nom-inal” plant. Besides, those were inherently MIMOsystems, while customary MRAC techniques atthe time were only dealing with SISO systems.Because now the very reduced-order model couldnot be considered to be even close to the plant, onecould not consider full model state following, sothis aim was naturally replaced by output modelfollowing. Furthermore, as the (possibly unstable)large-order plant state could not be comparedwith the reduced-order model state, the modelcould not be thought to guarantee asymptoticstability of the plant any longer.

In order to allow stability of the reduced orderadaptive control system, new adaptive controlcomponents that were not deemed to be needed bythe customary MRAC had to be considered. Wewill show that this “small” addition had an as-tonishing effect towards the successful applicationof the modified MRAC. In brief, as it was known

that stability of adaptive control systems requiredthat the plant be stabilizable via a constant gainfeedback, the natural question was why not usingthis direct output feedback.

Following this idea, an additional adaptive outputfeedback term was added to the adaptive algo-rithm that otherwise is very similar to the usualMRAC algorithms, namely,

u(t) = Keey(t) + Kxxm(t) + Kuum(t)

= K(t)r(t) (27)

where we denote the reference vector

K(t) =[Ke(t) Kx(t) Ku(t)

](28)

Subsequently in this paper, it will be shown thatthe new approach uses the model as a CommandGenerator and therefore it is sometime calledAdaptive Command Generator Tracker. Becauseit also uses low-order models and controllers, itwas ultimately called Simple Adaptive Control(SAC). Before we discuss the differences betweenthe new SAC approach and to adaptive controlclassical MRAC, it is useful to first dwell over thespecial role of the direct output feedback term.If the plant parameters were known, one couldchoose an appropriate gain Ke and stabilize theplant via constant output feedback control

u(t) = −Key(t) (29)

As we already mentioned above, it was knownthat an ASPR system (or, as we now know, aminimum-phase plant with appropriate CB prod-uct) could be stabilized by a positive definiteoutput feedback gain. Furthermore, it was knownthat ASPR systems are high-gain stable, so sta-bility of the plant is maintained if the gain valuehappens to go arbitrarily high beyond some mini-mal value. Whenever one may have sufficient priorknowledge to assume that the plant is ASPR, yetdoes not have sufficient knowledge to choose agood control gain, one can use the output itselfto generate the adaptive gain by the rule:

Ky(t) = y(t)yT (t)Γy (30)

and the control

u(t) = Ky(t)y(t) (31)

In the more general case when the plant is re-quired to follow the output of the model, onewould use the tracking error to generate the adap-tive gain

Ke(t) = ey(t)eTy (t)Γe (32)

and the control

u(t) = Ke(t)ey(t) (33)

We will show how this adaptive gain additionis able to avoid some of the most difficult in-herent problems related to the standard MRACand to add robustness to its stability. Althoughit was developed as a natural compensation forthe low-order models and was successfully appliedat Rensselaer as just one element of the Sim-ple (Model Reference) Adaptive Control method-ology, it is worth mentioning that, similarly tothe first proof of the ASPR property, the ori-gins of this specific adaptive gain can again befound in an early Fradkov’s work (Fradkov, 1976)in the Russian literature. Besides, later on thisgain received a second birth and became verypopular after 1983 in the context of adaptivecontrol “when the sign of high-frequency gainis unknown.” In this context (Nussbaum, 1983),(Morse, 1984), (Heyman, Lewis and Meyer, 1985)and after a very rigorous mathematical treatment(Byrnes and Willems, 1984), it also received anew name and it is sometimes called the Byrnes-Willems gain. Its useful properties have been thor-oughly researched and some may even call this oneadaptive gain Simple Adaptive Control as theywere apparently able to show that it can do “al-most” everything (Ilchman, Owens and Pratzel-Wolters, 1987), (Mareels and Polderman, 1996).Indeed, if an ASPR system is high-gain stable,it seems very attractive to let the adaptive gainincrease to even very high values in order toachieve good performance that is represented bysmall tracking errors. However, although at firstthought one may find that high gains are veryattractive, a second thought and some more en-gineering experience with the real world applica-tions make it clear that high gains may lead tosaturations and may excite vibrations and otherdisturbances. These disturbances may not haveappeared in the nominal plant model that wasused for design and may not be felt in the real-world plant unless one uses those very high gains.Furthermore, as the motor or the plant dynamicswould always require an input signal in order tokeep moving and tracking the desired trajectory,it is quite clear that the tracking error cannotbe zero or very small unless one uses very highgains indeed. Designers of tracking systems knowthat feedforward signals that come from the de-sired trajectory can help achieving low-error oreven perfect tracking without requiring the useof dangerously high gains (and, correspondingly,exceedingly high bandwidth) in the closed loop.In the non-adaptive world, feedforward could beproblematic because unlike the feedback loop, anyerrors in the feedforward parameters are directlyand entirely transmitted to the output trackingerror. Here, the adaptive control methodology candemonstrate an important advantage on the non-adaptive techniques, because the feedforward pa-rameters are finely tuned by the very tracking er-

U

ey=ym-y

X mX

Yp

1

1/s

1/sKe

Kx

Ku

Cm

C

Bm

B

Am

A

Um

1

Fig. 2. Schematic representation of SAC

ror they intend to minimize. The issues discussedhere and the need for feedforward again seem toshow the intrinsic importance of the basic ModelFollowing idea, and again point to the need for amodel. However, the difference between the modelused by SAC and the Model Reference used by thestandard MRAC is that this time the so-called“Model” does not necessarily have to reproducethe plant besides incorporating the desired input-output behavior of the plant. At the extreme, itcould be just a first-order pole that performs areasonable step-response, or otherwise a higherorder system, just sufficiently high to generatethe desired trajectory. As it generates the com-mand, this “model” can also be called “CommandGenerator” (Brousard and Berry, 1978) and thecorresponding technique “Command GeneratorTracker (CGT).” In summary, the adaptive con-trol system monitors all available data, namely,the tracking error, the model states and the modelinput command and uses them to generate theadaptive control signal (Figure 2)

Ke(t) = ey(t)eTy (t)Γe (34)

Kx(t) = ey(t)xTm(t)Γx (35)

Ku(t) = ey(t)uTm(t)Γu (36)

that using the concise notations (27)-(28) gives

K(t) = ey(t)rT (t)Γ (37)

and the control

u(t) = K(t)r(t) (38)

It is worth noting that, initially, SAC seemedto be a very modest alternative to MRAC withapparently very modest aims and that also seemedto be very restricted by new conditions. Althoughat the time it probably was the only adaptivetechnique that could have been used in MIMOsystems and with such large systems as large

flexible structures, and therefore was quite imme-diately adopted by many researchers and practi-tioners, the SAC approach got a cold receptionand for a long time has been largely ignored bythe mainstream adaptive control. In retrospective(besides some lack of good selling) at the time thiscold reception had some good reasons. Althoughit was called “simple” as it was quite simple toimplement, the theory around SAC was not simpleand many tools that were needed to support itsqualities and that, slowly and certainly, revealedthemselves over the year, were still missing. Itsubsequently not only required developing newanalysis tools but also, probably even more im-portant, better expertise at understanding theirimplications before they could be properly usedso that they ultimately managed to highlight thevery useful properties of SAC. Finally, based ondevelopments that had spanned over more than 25years, we will attempt to show that SAC is in factthe stable MRAC, because right from the begin-ning it avoids some difficulties that are inherent inthe standard MRAC. First, it is useful to noticethat because there is no attempt at comparisonbetween the order or the states of the plant andthe model, there is no “unmodeled dynamics.”Also, because basically the stability of the systemrests on the direct output feedback adaptive gain,the model is immaterial in this context and ofcourse there is no need to mention “sufficientexcitation.” Besides, as we will later show and as itwas observed by almost all practitioners that havetried to use it, SAC proved to be good control.While the standard MRAC may have to explainwhy it does not work when it is supposed to work,SAC may have to explain why it does work even incases when the (sufficient) conditions are not fullysatisfied. Although, similarly to any nonstationarycontrol, in Adaptive Control it is very difficult tofind the very minimal conditions that would keepthe system stable, it can be shown why SAC maydemonstrate some robustness even when the basicsufficient conditions are not satisfied. We notethat this last point is just an observation basedon experience, yet we must also note that in thosecases when the basic conditions are fulfilled, theyare always sufficient to guarantee the stability ofthe adaptive control system, with no exceptionsand no counterexamples. In this respect, one canshow that the MRAC “counterexamples” becomejust trivial, stable, and well behaving examples forSAC.

6. PROOF OF STABILITY OF SIMPLEADAPTIVE CONTROL

One can easily prove that the WASP conditionsare sufficient to prove stability using just the sim-ple adaptive output feedback gain (32) (Barkana,

Teixeira and Hsu, 2006). However, in order toavoid any misunderstandings related to the role ofthe unknown matrix W , here we chose to presenta rigorous proof of stability for the general outputmodel tracking case. As usual in adaptive con-trol, one first assumes that the underlying fullydeterministic output model tracking problem issolvable. A recent publication (Barkana, 2005a)shows that if the Model Reference uses a stepinput in order to generate the desired trajectory,the underlying tracking problem is always solv-able. If, instead, the model input command isitself generated by an unknown system of ordernu, the model is required to be sufficiently largeto accommodate this command (Barkana, 1983),(Kaufman, Barkana and Sobel, 1998), or

nm + m ≥ nu (39)

We assume that the plant to be controlled isminimum-phase and that the CB product is Pos-itive Definite and diagonalizable though not nec-essarily symmetric. As we showed, the plant isWASP according to Definition 2, so it satisfiesconditions (22)-(23). Under these assumptions onecan use the Lyapunov function (24). Differenti-ating (24) and using the W-passivity relations,finally leads to the following derivative of theLyapunov function (Appendix A)

V (t) = −eTx (t)Qex(t) (40)

One can see that V (t) in (40) is negative def-inite with respect to ex(t), yet only negativesemidefinite with respect to the entire state-space {ex(t),K(t)}. A direct result of Lyapunovstability theory is that all dynamic values arebounded. According to LaSalle’s Invariance Prin-ciple (Kaufman, Barkana and Sobel, 1998), allstate-variables and adaptive gains are boundedand the system ultimately ends within the domaindefined by V (t) ≡ 0. Because V (t) is negative def-inite in ex(t), the system thus ends with ex(t) ≡ 0,that in turn implies ey(t) ≡ 0. In other words, theadaptive control system demonstrates asymptoticconvergence of the state and output error andboundedness of the adaptive gains.

6.1 On gain convergence, basic conditions forstability, optimality, robustness, etc.

Some particularly interesting questions may ariseduring the proof of stability. First, although theLyapunov function was carefully selected to con-tain both the state error and the adaptive gains,the derivative only contains the state error. Itappears as if the successful proof of stability has“managed” to eliminate any possibly negative ef-fect of the adaptive gains. One is then entitled to

ask what positive role the adaptive gains play (be-sides not having negative effects). This is just onemore illustration of the difficulties related to theanalysis of nonlinear systems. Indeed, althoughLyapunov stability theory manages to prove sta-bility, it cannot and does not provide all answers.Besides, as potential counterexamples seem toshow, although the tracking error and the deriva-tive of the adaptive gains tend to vanish, this mereresult does not necessarily imply, as one mighthave initially thought, that the adaptive gainswould reach a constant value or even a limit atall. If the adaptive gain happens to be a functionsuch as k(t) = sin(ln t) (suggested to us by MarkBalas), its derivative is k(t) = cos(ln t)/t. In thisexample one can see that although the derivativetends to vanish in time, the gain k(t) itself doesnot reach any limit at all. Therefore, the commonopinion that seems to be accepted among expertsis that the adaptive gains do not seem to convergeunless the presence of some “sufficient” excitationcan be guaranteed. This seem to imply that evenin the most ideal, perfect following, situations, theadaptive control gains may continue wandering forever.

However, recent results have shown that theseopen questions and problems are only apparent.First, even if it is not a direct result of Lyapunovanalysis, one can show that the adaptive controlgains always perform a steepest descent minimiza-tion of the tracking error (Barkana, 2005a). Al-though this “minimum” could still increase with-out bound in general, if the stability of the systemwere not guaranteed, yet this is not the case withSAC.

Second, with respect to the final gain values,when one tests an adaptive controller with agiven plant, one first assumes that an underlyingLTI solution for the ideal control gains exists,and then the adaptive controller is supposed tofind those gains at the end of the adaptation.If the plant is known, one can first solve thedeterministic tracking problem and find the idealcontrol gains. Then, the designer proceeds withthe implementation of the adaptive controller andexpects it to converge to the pre-computed idealsolution. In practice, however, one observes that,even though the tracking errors do vanish, theadaptive gains do not seem to converge. Evenin those cases when the gains do demonstrateconvergence, their final values are far from theideal solution that was computed for the LTI case,and this happens even when the LTI solution is“unique.”

This point may give the practitioner pause, be-cause if there is such an uncertainty about theadaptive gains even in the ideal situations, whatcan be said in more difficult situations, in the pres-

ence of noise, etc.? This question has proved to bea very serious challenge and has remained openfor more than 30 years, yet recently it was finallyanswered. It is worth mentioning that althoughextensions of LaSalle’s Invariance Principle fornon-autonomous systems have been around forquite some time (Artstein, 1977), (LaSalle, 1981),besides very few exceptions they don’t seem tobe widely used in Adaptive Control or in non-linear control systems in general. This fact couldbe partially explained by the fact that the re-sults are of a very general character. However,their proper interpretation and application to-wards the development of new basic analysis toolssuch as combining a Modified Invariance Principlewith Gromwall-Bellman Lemma (Barkana, 1983),(Barkana, 2005a), (Kaufman, Barkana and So-bel, 1998), finally managed to provide the solutionto this problem.

It was shown that if the adaptive control gainsdo not reach the “unique” solution that the pre-liminary LTI design seemed to suggest, it is notbecause something was wrong with the adaptivecontroller, but rather because the adaptive controlcan go beyond the LTI design. The existence of a“general” LTI solution is useful in facilitating andshorting the proof of stability, yet it is not neededfor the convergence of the adaptive controller.While the sought after stationary controller mustprovide a fixed set of constant gains that would fitany input commands, the adaptive controller onlyneeds that specific set of control gains that corre-spond to the particular input command. Even inthose cases when the general LTI solution doesnot exist, the particular solution that the adap-tive controller needs does exist (Barkana, 2005a).However, it complicates the stability analysis be-cause it was shown that those particular solutionsmay allow perfect following only after a transientthat adds supplementary terms to the differen-tial equations of motion. As a consequence, thestability analysis may end with the derivative ofLyapunov function being

V (t) = W1(t) + W2(t)

=−eTx (t)Qex(t) + F (x, t)eAmt (41)

Although the derivative (41) still contains thenegative definite term with respect to the errorstate, it also contains a transient term that isnot negative, so the derivative is not necessarilynegative definite or even semidefinite. Apparently,(41) cannot be used for any decision on stability.However, although the decision on stability isnot immediate, the Modified Invariance Principlereveals that all bounded solutions of the adaptivesystem reach asymptotically the domain definedby

W1(t) = −eTx (t)Qex(t) ≡ 0 (42)

Therefore, one must find out what those “boundedtrajectories” are and it is the role of Gromwall-Bellman Lemma to actually show that, under theWASP assumption, all trajectories are bounded.Therefore, the previous conclusions on asymptot-ically perfect tracking remain valid.

Moreover, because the gains also reach that do-main in space where perfect tracking is possi-ble, this approach has also finally provided theanswer to the (previously open) question on theadaptive gain convergence. Even if one assumesthat the final asymptotically perfect tracking mayoccur while the adaptive gains continue to wan-der, one can show that the assumably nonstation-ary gains satisfy a linear differential equitationwith constant coefficients and their solution is asummation of generalized exponential functions((Barkana, 2005a) and Appendix B). This partialconclusion immediately shows that such nonlin-ear “counterexample” gains as that we presentedabove are, maybe, nice and tough mathematicalchallenges, yet they cannot be solutions of, andthus are actually immaterial for, the SAC track-ing problem. Furthermore, because the gains arebounded, they can only be combinations of con-stants and converging exponentials, so they mustultimately reach constant values. Therefore, wewere finally able to show (at least within the scopeof SAC) that the adaptive control gains do ulti-mately reach a set of stabilizing constant values atthe end of a steepest descent minimization of thetracking error ((Barkana, 2005a) and AppendixB).

A recent paper (Barkana, 2007) tests SAC witha few counterexamples for the standard MRAC(Hsu and Costa, 1999). The paper shows thatSAC not only maintains stability in all cases thatled to instability with standard MRAC, but alsodemonstrates very good performance.

Many practitioners that have tried it have beenimpressed with the ease of implementation ofSAC and with its performance even in large andcomplex applications. Many examples seem toshow that SAC maintains its stable operation evenin cases when the established sufficient conditionsdo not hold. Indeed, conditions for stability ofSAC have been continuously mitigated over theyears, as the two successive definitions of almostpassivity conditions presented in this paper mayshow. In order to get another qualitative estimateon SAC robustness, assume that instead of (1)-(2)the actual plant is

x(t) = Ax(t) + Bu(t) + f(x) (43)

y(t) = Cx(t) (44)

Assume that the nominal {A, B,C} system isWASP, while f(x) is some (linear or nonlinear)

component that prevents the satisfaction of thepassivity conditions. If one uses the same Lya-punov functions (24), instead of (40) one gets forthe stabilization problem

V (t) = −xT (t)Qx(t) + xT (t)Pf(x) (45)

and for the tracking problem

V (t) = −eTx (t)Qex(t) + eT

x (t)P [f(x)− f(x∗)](46)

where x∗ is the ideal trajectory, as defined inAppendix A. Note that the derivative of theLyapunov function remains negative definite interms of x(t) or ex(t), correspondingly, if thesecond term in the sum is not too large, asdefined (for example) by the inequality (Torresand Mehiel, 2006)

||f(x)|| ≤ ||Q||||P || ||x|| (47)

or

||f(x)− f(x∗)|| ≤ ||Q||||P || ||x− x∗|| (48)

While until very recently the main effort has beendedicated to the clarification and relaxation ofthe passivity conditions, similar effort is dedicatednow to clarifying the limits of robustness of SACwhen the basic passivity conditions are not en-tirely satisfied.

Besides, although much effort has been dedi-cated to clarification of passivity concepts inthe context of Adaptive Control of stationarycontinuous-time systems, similar effort has beendedicated to extending these concepts to discrete-time (Barkana, 2005c) and nonstationary andnonlinear systems (Barkana, 2005d), (Bobtsovand Nagovitsina, 2007).

7. ON “SAFE” GAIN SCHEDULING VS.LACK OF STABILITY NORMS FOR

ADAPTIVE CONTROL

Although the use of nonstationary gains thatwould adapt to the specific and momentary needsof the control systems is attractive, the some-what uncertain conditions that would guaranteestability, and in particular the lack of verifiablenorms for the stability robustness of adaptivecontrol systems has kept their use in real-worldapplication very low. On the other hand, thestability robustness of linear time invariant (LTI)controllers in an LTI world could always be mea-sured in such customary practical terms as gain-margin and phase margin. Even in a changingworld, when it is known that the plant parameterscould change as a function of internal or external

−2 −1.5 −1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Gs

Real Axis

Imag

inar

y A

xis

Fig. 3. Plant Root Locus

environmental and operational conditions, at thepresent time most designers would prefer usingconstant control gains and, if forced to, gain-scheduling as a function of some measure of thechanging operational conditions. Although it is, orshould be, quite clear to the designer that alongwith switching from one gain to another one loosesthe absolute estimator of stability given by gainmargin and phase margin, experience and, prob-ably even more, tradition, keep people preferring,using, and trusting those basically LTI designmethodologies. In order to show that, along withimproved performance, adaptive control can addto, not destroy, the stability robustness of systems,we here consider a simple plant, non-minimumphase yet otherwise open-loop stable system, amodel that fits some missiles and other examples.The transfer function is

G (s) =s2 − 0.36

s3 + 1.1s2 + s + 0.95(49)

Although the open loop is theoretically stable(Figure 3), one wants to use feedback gains inorder to improve the stability and performance ofthe system. To this end, one first checks and findsout that in a given environment, the closed-loopsystem remains stable for any positive constantgain less than 2.64. To make sure the system is farfrom instability, one decides to use the constantgain k=1.3. (One knows that the environmentalconditions may change, and when they do changeone would switch to a different gain). Even if oneknows that in the present conditions the actualcontrol gain might vary around the nominal valueof 1.3, as long the monitored conditions show thatthe gain stays within the “admissible” range 0 -2.64, there seems to be no reason to worry.

Many tests using constant and even various vari-able gains indeed seem to show that the closedloop system maintains its stability. Even whenthe missile maneuvering leads to sinusoidal time-variation around the mid-range value of 1.3,namely k(t) = 1.3 + asint, where the parameter’a’ is constant, the system shows stability and

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

Gain vs time

Time [sec]

Fig. 4. Sinusoidal Gain with a=0.7

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02vout vs yout

Fig. 5. Position-Velocity with a=0.7

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

Gain vs time

Time [sec]

Fig. 6. Sinusoidal Gain with a=1.15

satisfactory behavior even for a = 0.7 (Figures4-5).

However, when the sinusoidal variation reacheda = 1.15 (Figures 6-7), the non-stationary gainleads to total divergence although the gain re-mains within the so-called “admissible” range!

In spite of the example above, even if one maynot like the lack of robustness with stationarycontrollers in changing environments, AdaptiveControl methods do not seem to be an alterna-tive because the plant used with our example isnon-minimum-phase, fact that would make Adap-tive Control methodologies unusable. As we showbelow, a solution for plants that do not inher-

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06−0.06

−0.04

−0.02

0

0.02

0.04

0.06vout vs yout

Fig. 7. Position-Velocity with a=1.15ently satisfy the ASPR conditions, including non-minimum phase, has been available for quite awhile, so we will now present the Parallel Feed-forward methodology that can be used to allowapplicability of Adaptive Control with almost allreal-world systems.

8. PARALLEL FEEDFORWARD ANDSTABILITY OF SIMPLE ADAPTIVE

CONTROL

Using for illustration the example of Section VIII,assume that KMAX = 2.5 is an estimate of thehighest admissible constant gain that maintainsstability of the system. One would never use thisvalue because it would not be a good control gainvalue. Indeed, we only use the mere knowledgethat a (fictitious) closed-loop system using thehigh gain value of 2.5 would still be stable. Insteadof implementing constant output feedback we usethis knowledge in order to augment the systemwith a simple Parallel Feedforward Configuration(PFC) across the plant. If the original plant hastransfer function

G(s) =B(s)A(s)

(50)

the closed-loop system would be

GCL(s) =B(s)

A(s) + KB(s)(51)

and would be asymptotically stable. The aug-mented system using the inverse of the stabilizinggain D = 1

K is

Ga(s) =B(s)A(s)

+1K

(52)

or

Ga(s) =A(s) + KB(s)

KA(s)(53)

and if the closed-look system would be stable, onecan see that the augmented system is minimum-phase (Figure 8). Note that although we would

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Ga

Real Axis

Imag

inar

y A

xis

Fig. 8. Augmented Plant Root Locus

never suggest using direct input-output gains inparallel with the plant, this is a simple and usefulillustration that may facilitate the understandingof the basic idea. Also, although in this paper weonly dealt (and will continue to deal) with strictlycausal systems, for this specific case it is useful torecall that a minimum-phase plant with relativedegree 0 (zero) is also ASPR. As (53) shows,one could use the inverse of any stabilizing gainin order to get ASPR configurations. However,any such addition is added ballast to the originalplant output, so using the inverse of the maximalallowed gain adds the minimal possible alterationto the plant output. The augmented system looksas follows: The augmented system has three polesand three zeros and all zeros are minimum-phase.Such a system cannot become unstable, no mat-ter how large the constant gain k becomes, yetbecause it is ASPR one can also show that itwould also stay stable no matter how large thenonstationary adaptive gain k(t) becomes. Onecan easily see that the parallel feedforward hasmade the effective control gain that affects theplant to be:

keff (t) =k(t)

1 + Dk(t)=

k(t)

1 + k(t)KMAX

=1

1k(t) + 1

KMAX

(54)

One can see that the effective gain is always belowthe maximal admissible constant gain (Figure 9).While this qualitative demonstration intends toprovide some intuition to the designer that isused to estimate stability in terms of gain andphase margins, rigorous proofs of stability usingthe Lyapunov-LaSalle techniques and almost pas-sivity conditions are also available and providethe necessary rigorous proof of stability. As wealready mentioned above, the constant parallelfeedforward has only been presented here for afirst intuitive illustration. In practice, however,one does not want to use direct input-outputacross the plant that would require solving im-plicit loops that include the adaptive gain com-

0 20 40 60 80 100 120 140 160 180 200

0

0.5

1

1.5

2

2.5

Effective Control Gain

Adaptive Control Gain

Effe

ctiv

e C

ontr

ol G

ain

Fig. 9. Effective Gain

−2 −1.5 −1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Gpds

Real Axis

Imag

inar

y A

xis

Fig. 10. Plant with PD Controller

putations. Therefore, we go to the next step thattakes us to the ubiquitous PD controllers. In prac-tice, many control systems use some form of PDcontroller, along with other additions that maybe needed to improve performance. While theadditions are needed to achieve the desired per-formance, in many cases the PD controller aloneis sufficient to stabilize the plant. In our case, a PDcontroller H(s) would make the Root-locus plot tolook like (Figure 10) The system is asymptoticallystable for any fixed gain within the “admissible”range 0 - 2.66, so we again choose KMAX = 2.5as an estimate of the highest admissible constantgain that maintains stability of the system. Thistime however we use D(s) = 1/H(s), the inverseof the PD controller, as the parallel feedforwardacross the plant. The Root-locus of the resultingaugmented plant is shown in Figure 11. This is astrictly causal system with 4 poles and 3 strictlyminimum-phase zeros and is therefore, ASPR. Al-though the original plant was non-minimum phaseand this fact would usually forbid using adaptivecontrollers, here one can apply SAC and be surethat stability and asymptotically perfect trackingof the augmented system is guaranteed. The onlyopen question is how well the actual plant outputperforms. In this respect, the maximal admissi-ble gain with fictitious PD (or with any otherfictitious controller) defines how small the addedballast is and how close the actual output is to the

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Ga

Real Axis

Imag

inar

y A

xis

Fig. 11. Augmented Plant with strictly causalPFC

augmented output. The example here is a verybad system and was only used to illustrate theproblems one may encounter using constant gainin changing environments and cannot be expectedto result in good behavior without performingmuch more study and basic control design. Theexamples above have been used to present asimple principle: if the system can be stabilizedby the controller H(s), then the augmented sys-tem Ga(s) = G(s) + H−1(s) is minimum-phase.Proper selection of the relative degree of H−1(s)will thus render the augmented system ASPR(Barkana, 1987). This last statement implies that“passivability” of systems is actually dual to sta-bilizability. If a stabilizing controller is known,its inverse in parallel with the plant can makethe augmented system ASPR. When sufficientprior knowledge is available to design a stabiliz-ing controller, some researchers prefer to use thisknowledge and directly design the correspondingparallel feedforward (Iwai and Mizumoto, 1992)or ”shunt” (Fradkov, 1994). When the “plant” isa differential equation, it is easy to assume thatthe order or the relative degree is available andthen a stabilizing controller or the parallel feedfor-ward can be implemented. However, in real world,where the “plant” could be a plane, a flexiblestructure or a ship, the available knowledge is theresult of some wind-tunnel or other experimentaltests that may result in some approximate fre-quency response or approximate modeling, suffi-cient to allow some control design, yet in generaldo not provide reliable knowledge on the order orrelative degree of the real plant. On the other hand(although it may very much want some adaptivecontrol to help improving performance if it onlycould be trusted), the control community actuallycontinues to control real-world systems with fixedcontrollers. Therefore, in our opinion the ques-tion “How can you find a stabilizing controller?”should not be given any excessive emphasis. Inany case, if there is sufficient prior knowledge todirectly design the feedforward there is definitelysufficient information to design a stabilizing con-

figuration, and vice versa. Note that the exampleof this section is a bad system that was on purposeselected to provide a counterexample for the sta-bility with assumably “constant” gains. Althoughthe stability of the augmented system with adap-tive control is guaranteed, the plant output maynot behave very well, even with the added paral-lel feedforward. In any case, even in those caseswhen the parallel feedforward is too large to al-low good performance as monitored at the actualplant output, the behavior of the, possibly bothunstable and non-minimum phase, plant withinthe augmented system is stable and it was shownto allow stable identification schemes (Johanssonand Robertsson, 2002) and thus, lead to better un-derstanding of the plant towards better, adaptiveor non-adaptive, control design. Still, as recentlyshown with a non-minimumphase UAV example(Barkana, 2005b) and with many other realisticexamples (Kaufman, Barkana and Sobel, 1998),prior knowledge usually available for design allowsusing basic preliminary design and then very smalladditions to the plant that not only result inrobust stability of the adaptive control systemeven with originally non-minimum phase plants,but that also lead to performance that is ulti-mately superior to other control methodologies.A recent publication uses the parallel feedforwardcompensator for safe tuning of MIMO AdaptivePID Controllers (Iwai, Mizumoto, Nakashima andKumon, 2007) and another (Ben Yamin, Yaeshand Shaked, 2007) shows how to implement Sim-ple Adaptive Controllers with guaranteed H∞performance.

9. ROBUSTNESS OF SIMPLE ADAPTIVECONTROL WITH DISTURBANCES

The presentation so far showed that a simpleadaptive controller can guarantee stability of anysystem that is minimum-phase if the CB prod-uct is Positive Definite and diagonalizable if notsymmetric. In case these conditions do not inher-ently hold, basic knowledge on the stabilizabilityproperties of the plant, usually known, can beused to fulfill them via Parallel Feedforward Con-figurations. Therefore, the proposed methodologyseems to fit almost any case where asymptoticallyperfect output tracking is possible. However, afterwe presented the eulogy of the adaptive outputfeedback gain (32), it is about time to also presentwhat could become its demise, if not properlytreated. When persistent disturbances such asrandom noise or very high frequency vibrationsare present, perfect tracking is not possible. Evenwhen the disturbance is known and various vari-ations of the Internal Model Principle can bedevised (Fradkov and Andrievsky, 2007) to filterthem out, some residual tracking error may always

be present. While tracking with small final errorscould be acceptable, it is clear that the adaptivegain term (32) would, slowly but certainly, in-crease without limit. Indeed, theoretically, ASPRsystems maintain stability with arbitrarily highgains and in some cases (in case of missiles, forexample) the adaptive system mission could endeven before problems are even observed. How-ever, allowing the build-up of high gains that donot come in response to any actual requirementis not acceptable, because in practice they maylead to numerical problems and saturation effects.However, very early we observed how the robust-ness of SAC with disturbances can be guaran-teed by adding Ioannou’s σ-term (Ioannou andKokotovic, 1983) with the error adaptive gain thatwould now be

Ke(t) = ey(t)eTy (t)Γe − σKe(t) (55)

Finally, this new addition is literally making SACan adaptive controller (see (Barkana, 2005a) and(Kaufman, Barkana and Sobel, 1998) and ref-erences therein): while the control gains alwaysperform a steepest descent minimization of thetracking error, the error gain defined in (55) goesup-and-down fitting the right gain to the rightsituation in accord with the changing operationalneeds.

10. CONCLUSIONS AND FUTURE WORKS

This report presented the various components andmain properties of a special version of Model Ref-erence Adaptive Control, called Simple AdaptiveControl. Initially, because it was attempting touse low-order models with large order plants andalso because of some initial lack of mathematicaltools, SAC scope and performance seemed to bevery modest when compared with the customaryMRAC. However, with the development of specialmathematical analysis tools along with the grad-ually deeper understanding of its special qualities,SAC appears to have ultimately become the stableMRAC.

Appendix A. PROOF OF STABILITY

The underlying deterministic tracking problemassumes that there exists an “ideal control”

u∗(t) = Kxxm(t) + Kuum(t) (A.1)

that could keep the plant along an “ideal tra-jectory” that performs perfect tracking. In otherwords, the ideal plant

x∗(t) = Ax∗(t) + Bu∗(t) (A.2)

y∗(t) = Cx∗(t) (A.3)

moves along “ideal trajectories” such that

y∗(t) = ym(t) (A.4)

We assume that the underlying LTI problem issolvable and thus, that some ideal gains Kx andKu exist (Barkana, 2005a). Because the plantand the model can have different dimensions, the“following error” is defined to be the differencebetween the ideal and the actual plant state

ex(t) = x∗(t)− x(t) (A.5)

and correspondingly

ey(t) = ym(t)− y(t) = y∗(t)− y(t)

= Cx∗(t)− Cx(t) = Cex(t) (A.6)

Differentiating ex(t) gives:

ex(t) = x∗(t)− x(t) (A.7)

= Ax∗(t) + Bu∗(t)−Ax(t)−Bu(t)

ex(t) = Aex(t)−Bu(t) + Bu∗(t) (A.8)

ex(t) = Aex(t)−B(Keey(t) + Kxxm(t) + Kuum(t))

+ B(Kxxm(t) + Kuum(t)) (A.9)

Adding and subtracting BKeey(t) = BKeCex(t)above gives

ex(t) =(A−BKeC

)ex(t)−B

(K(t)− K

)r(t)(A.10)

where for convenience we denoted

K =[Ke Kx K

](A.11)

The derivative of the Lyapunov function (24) is

V (t) = eTx (t)P ex(t) + eT

x (t)Pex(t)

+tr[WK(t)Γ−1

(K(t)− K

)](A.12)

+tr[W

(K(t)− K

)Γ−1K(t)T

]

Using relations (22)-(23) gives

V (t) = xT (t)[PAK + AT

KP]x(t)

−xT CT W(K − K

)Cx− xT CT

(K − K

)WCx

+xT CT W(K − K

)Cx + xT CT

(K − K

)WCx

(A.13)

The last two terms in (A.13), originating in thederivative of the adaptive gain terms in V(t), can-cel the previous, possibly troubling, non-positive,terms and thus lead to the Lyapunov derivative

•V (t) = −eT

x (t)Qex(t) (A.14)

Appendix B. GAIN CONVERGENCE

Let the linear time-invariant plant (1)-(2) trackthe output of the model (3)-(4). In general, inthe past we have assumed that the model usesstep inputs to generate the desired command(Barkana, 2005a). For the more general commandfollowing case, we assume that the command itselfis generated by an unknown input generator

xu(t) = Auxu(t) (B.1)

u(t) = Cuxu(t) (B.2)

We want to check what the ultimate adaptivecontrol gains that perform perfect tracking couldbe. When the error is zero, the input controlto the plant is a linear combination of availablemeasures.

u(t) = Kxxm(t) + Kuum(t) (B.3)

Assume that the plant moves along such “idealtrajectories” and the nonstationary gains are suchthat the plant output y∗(t) = Cx∗(t) perfectlytracks the model output, namely, ey(t) = 0, or

Cx∗(t) = Cmxm(t) (B.4)

Differentiating (B.4) gives

Cx∗(t) = Cmxm(t) (B.5)

or

CAx∗(t) + CBu(t) = CmAmxm(t) + CmBmum(t)(B.6)

CAx∗(t) + CBKxxm(t) + CBKuum(t)

= CmAmxm(t) + CmBmum(t) (B.7)

CAx∗(t) = [CmAm − CBKx]xm(t)

+ [CmBm − CBKu] um(t) (B.8)

We assume that CA is maximal rank and get

x∗(t) =

(CA)T(CA (CA)T

)−1

[CmAm − CBKx(t)] xm(t)

+ (CA)T(CA (CA)T

)−1

[CmBm − CBKu(t)] um(t)+ x∗0(t)

(B.9)

or

x∗(t) = Sx(t)xm(t) + Su(t)um(t) + x∗0(t) (B.10)

Here,

Sx(t) = (CA)T(CA (CA)T

)−1

[CmAm − CBKx(t)](B.11)

Su(t) = (CA)T(CA (CA)T

)−1

[CmBm − CBKu(t)](B.12)

and x∗0(t) represents those functions that satisfy

CAx∗0(t) ≡ 0 (B.13)

and are solutions of the plant differential equation

x∗0(t) = Ax∗0(t) (B.14)

that result in

y∗0(t) = Cx∗0(t) = CAx∗0(t) ≡ 0. (B.15)

Note that the differential equation (B.14) of thesupplementary term x∗0(t) does not contain con-trol terms because those would be included in theother terms in (B.10). Because CB is nonsingularone gets from (B.4)

u(t) = (CB)−1 [CmAmxm(t) + CmBmum(t)− CAx∗(t)](B.16)

or

u(t) = Kx(t)xm(t) + Ku(t)um(t) (B.17)

Here

Kx(t) = (CB)−1 (CmAm − CASx(t)) (B.18)

Ku(t) = (CB)−1 (CmBm − CASu(t)) (B.19)

x∗(t) = Sx(t)xm(t) + Sx(t)xm(t)+Su(t)um(t) + Su(t)um(t) + x∗0(t)

(B.20)

x∗(t) = Sx(t)Amxm(t) + Sx(t)Bmum(t)+Sx(t)xm(t) + Su(t)CuAuxu(t)+Su(t)Cuxu(t) + x∗0(t)

(B.21)

x∗(t) = Ax∗(t) + Bu(t) = ASx(t)xm(t)+ASu(t)um(t) + Ax∗0(t)+BKx(t)xm(t) + BKu(t)um(t)

(B.22)

ASx(t)xm(t) + ASu(t)um(t) + Ax∗0(t)+BKx(t)xm(t) + BKu(t)um(t)= Sx(t)Amxm(t) + Sx(t)BmCuxu(t)+Sx(t)xm(t) + Su(t)CuAuxu(t)+Su(t)Cuxu(t) + x∗0(t)

(B.23)

The terms in x∗0(t) and x∗0(t) cancel each otherand we get

ASx(t)xm(t) + ASu(t)CuAuxu(t)+B (CB)−1 (CmAm − CASx(t))xm(t)+B (CB)−1 (CmBm − CASu(t)) CuAuxu(t)= Sx(t)Amxm(t) + Sx(t)BmCuxu(t)+Sx(t)xm(t) + Su(t)CuAuxu(t)+Su(t)Cuxu(t)

(B.24)

and finally

Mxxm(t) + Muxu(t) = 0 (B.25)

Here,

Mx = Sx(t)−ASx(t) + Sx(t)Am

−B (CB)−1 (CmAm − CASx(t))(B.26)

Mu = Su(t)Cu + Sx(t)BmCu + Su(t)CuAu

−ASu(t)CuAu

−B (CB)−1 (CmBm − CASu(t)) CuAu

(B.27)

We first consider the case when the signals xm(t)and xm(t) are “sufficiently rich” so the equationscan be separated and the differential equations ofSx(t) and Su(t) are

Sx(t)−[I −B (CB)−1

C]ASx(t) + Sx(t)Am

= B (CB)−1CmAm

(B.28)

Su(t)Cu +[I −

(I −B (CB)−1

C)

A]Su(t)CuAu

= −Sx(t)BmCu + B (CB)−1CmBmCuAu

(B.29)

One can see that (B.28) is a linear differentialequation with constant coefficients, excited by aconstant forcing function. Therefore, the solutionof (B.28) is a combination of generalized exponen-tials of the form

Sxij(t) =∑

ctmebt sin (βt + ϕ) (B.30)

Equation (B.29) is then a linear differential equa-tion with constant coefficients excited by exponen-tial and constant forcing functions, and its solu-tion is therefore also a combination of generalizedexponentials. Now, from (B.18) and (B.19) onecan see that the components of the control gainsare also combinations of constants and generalizedexponentials of the form

kij(t) = aij +∑

ctmebt sin (βt + ϕ) (B.31)

On the other hand, because one knows that theadaptive gains that perform asymptotically per-fect tracking are bounded, any divergent exponen-tials are excluded. What is left is a combination ofconverging exponentials, constants and eventuallystable sinusoidal functions. However, because weknow that the derivative of the adaptive gain van-ishes in time, steady sinusoidal functions are alsoexcluded. Therefore, those adaptive gains that canperform perfect tracking can only be combinationsof constants and converging exponentials that ul-timately tend to reach a constant limit as timegoes to infinity.

Now we must consider the more general casewhen the signals cannot be considered “suffi-ciently rich.” The solution for the model (3)-(4)supplied with the command (B.1)-(B.2) has theform

xm(t) = Exu(t) + eAmtd0 (B.32)

At t = 0 one gets

xm(0) = Exu(0) + d0 (B.33)

d0 = xm(0)− Exu(0) (B.34)

Substituting (B.32) in (3) gives

Exu(t) = EAuxu(t) = AmExu(t) + AmeAmtd0

+BmCuxu(t)(B.35)

(AmE − EAu + BmCu)xu(t) + AmeAmtd0 = 0(B.36)

Therefore, at steady state the model is

xm(t) = Exu(t) (B.37)

where E satisfies the equation

AmE − EAu + BmCu = 0 (B.38)

Substituting (B.38) in (B.28) gives

Mx1Exu(t) + Mx2xu(t) = 0 (B.39)

Here

Mx1 = Sx(t)−ASx(t) + Sx(t)Am

−B (CB)−1 (CmAm − CASx(t))(B.40)

Mx2 = Su(t)Cu + Sx(t)BmCu + Su(t)CuAu

−ASu(t)CuAu

−B (CB)−1 (CmBm − CASu(t)) Cu

(B.41)

Equation (B.41) becomes

M(t)xu(t) = 0 (B.42)

Here

M(t) = Sx(t)E + Sx(t)AmE + Sx(t)BmCu

−ASx(t)E + B (CB)−1CASx(t)E

+Su(t)Cu + Su(t)CuAu + ASu(t)CuAu

+B (CB)−1CASu(t)Cu

−B (CB)−1Cm (AmE + BmCu)

(B.43)

and after some algebra

M(t) = Sx(t)E + Su(t)Cu

+(Sx(t)E + Su(t)Cu)Au

−(I + B (CB)−1

C)

A (Sx(t)E − Su(t)Cu)

−B (CB)−1CmEAu

(B.44)

We first use relation (B.44) to again show that theperfect following problem has many solutions. Ifwe first choose those solutions that satisfy

Sx(t)E − Su(t)Cu = 0 (B.45)

Su(t)Cu = Sx(t)E = S1 (t) (B.46)

we get

M(t) = S1(t) + S1(t)−B (CB)−1CmEAu (B.47)

The equation

S1(t) + S1(t)−B (CB)−1CmEAu = 0 (B.48)

is a stable linear differential equation with con-stant coefficients. Therefore, is given by a com-bination of exponential function and ultimatelyreaches a constant limit, Sf . However, it nowimplies that only a linear combination of the ulti-mate gains satisfies a relation of the form

Sx(t)E = Su(t)Cu = Sf (B.49)

Similarly only a linear combination of the ultimateadaptive control gains satisfies a relation of theform

Kx(t)F + Ku(t)G = Kf (B.50)

While any set of constant gains that satisfy (B.50)would perform perfect tracking, nonstationarygains could also do. Moreover, in order to simplifythe equation and show that it has solutions, weonly considered those particular solutions thatsatisfy (B.49), yet the selection is almost arbitrary,and the equation

M(t)xu(t) = 0 (B.51)

has many more solutions than (B.48), in general.Therefore, any effort of proving ultimate conver-gence of the adaptive gains actually seems to endin failure. There is no doubt that, in principle, per-fect tracking can occur while the bounded time-varying gains keep wandering across some hyper-surface described, for example, by (B.50) or byany corresponding equation. However, althoughsuch solutions for the perfect tracking exist, onemay still ask whether those nonstationary gainscan be the ultimate values of the adaptationprocess. Can the steepest descent minimizationend with some ever wandering gains? As we con-clude below, most certainly, not. First, althoughit is hard to translate engineering intuition intorigorous mathematics, it is ”felt” that the lackof ”richness” that the perfect following equationshows does not express the ”richness” of signalsthat exists during the entire process of adaptationup to and until ”just before” perfect tracking. Yet,somewhat more rigorously, the same argumentthat seems to fail the Lyapunov-LaSalle approachcan now be used to redeem it. Along with equation(B.50) of the ultimate hyper-surface that containsthem, the ultimate adaptive gains are also locatedon the hyper-ellipsoid that corresponds to the finalvalue of the Lyapunov function. If the initial valueof the Lyapunov function is V(t=0) = V0, its finalvalue is given by

limt→∞

V (t) = V0 −∞∫

0

eTx (t)Qex (t) dt = Vf (B.52)

or, in case there are transient terms in the Lya-punov derivative

limt→∞

V (t) = V0

−∞∫

0

eTx (t)Qex (t) + transient

dt = Vf

(B.53)

As the errors ultimately vanish and the monotoni-cally increasing output gain Ke(t) reaches an idealstabilizing gain value, the adaptive control gainsare located on the hyper-ellipsoid defined by

trace[(Kx(t)− Kx)T Γ−1x (Kx(t)− Kx)

+ (Ku(t)− Ku)T Γ−1u (Ku(t)− Ku)] = Vf

(B.54)

with the set{

Kx, Ku

}at its center. Because any

set of constant gains that satisfy the perfect track-ing equation can play the role of ideal gains setthat is used in the Lyapunov function, choosingthe set

{Kx1 , Ku1

}finds the final gain on hyper-

ellipsoid with the center in{

Kx1 , Ku1

}, namely,

trace[(Kx(t)− Kx1)T Γ−1x (Kx(t)− Kx1)

+ (Ku(t)− Ku1)T Γ−1u (Ku(t)− Ku1)]

= Vf1(Ku(t)− Ku)] = Vf

(B.55)

However, assuming the fictitious set{

Kx2 , Ku2

}

finds the final gain on a different hyper-ellipsoidwith the center in

{Kx2 , Ku2

}. Therefore, for the

same adaptation process, that starts and endswith the same values, this thinking experimentfinds the final gains located at the intersection ofinfinitely many distinct hyper-ellipsoids, so theircommon intersection is a point or a ”line” ofLebesque measure zero. Although this argumentmay requires more polishing, it points to the factthat, ultimately, the adaptive gains do convergeto a limit. In some cases, the rate of convergencemay be slow and simulations may show the gainvarying for a long-long time. Hence, it is impor-tant to know the gains do not vary at random andthat, even if sometimes slowly, they certainly tendto reach their final bounded constant limit.

REFERENCES

Artstein, Z. (1977). Limiting equations and sta-bility of nonautonomous systems. Journal ofDifferential Equations 25, 184–202.

Astrom, K. J. (1983). Theory and Applicationsof Adaptive Control - A Survey. Automatica19(5), 471–486.

Astrom, K. J., and B. Wittenmark (1989).Adaptive Control, Addison Wesley, Reading,Massachusetts.

Barkana, Itzhak (1987). Parallel feedforwardand simplified adaptive control. Interna-tional Journal of Adaptive Control and SignalProcessing 1(2), 95–109.

Bar-Kana, I. (1983). Direct Multivariable Adap-tive Control with Application to Large Struc-tural Systems, Ph.D. Dissertation, ECSEDept., Rensselaer Polytechnic Institute, Troy,New York.

Barkana, I. (2004). Comments on ‘Design ofStrictly Positive Real Systems Using Con-stant Output Feedback’. IEEE Transactionson Automatic Control 49, 2091–2093.

Barkana, I. (2005a). Gain conditions and con-vergence of simple adaptive control. Interna-tional Journal of Adaptive Control and SignalProcessing 19( 1), 13–40.

Barkana, I. (2005b) Classical and Simple AdaptiveControl for Non-Minimum Phase AutopilotDesign. Journal of Guidance, Control, andDynamics 24(4), 631–638 (also in shorterform in Proceedings of the AIAA GN&C 2004,paper AIAA2004-4868).

Barkana, I. (2005c) On output feedback stabilityand passivity in discrete linear systems. Pro-ceedings of The 16th Triennial IFAC WorldCongress, Prague, July 2005.

Barkana, I. (2005d) Output Feedback Stabilityand Passivity in Nonstationary and NonlinearSystems. Proceedings of The 2005 Interna-tional Conference on Control and Automation(ICCA2005) June 27-29, 2005, Budapest,Hungary.

Barkana, I. (2007), Adaptive model tracking withmitigated passivity conditions, IFAC Work-shop on Adaptation and Learning in Controland Signal Processing (ALCOSP 2007). SaintPetersburg, RUSSIA August, 2931, 2007.

Bar-Kana, I., and H. Kaufman (1982). Model Ref-erence Adaptive Control for Time-VariableInput Commands. Proceedings 1982 Confer-ence on Informational Sciences and Systems,Princeton, New Jersey, 208–211.

Barkana, I., H., Kaufman and M. Balas (1983).Model reference adaptive control of largestructural systems. AIAA Journal of Guid-ance 6(2), 112–118.

Barkana, I., and H. Kaufman (1985). Global Sta-bility and Performance of an Adaptive Con-trol Algorithm. Int. J. Control 46(6), 1491–1505.

Barkana, I., M. C.-M. Teixeira and Liu Hsu(2006). Mitigation of symmetry conditionfrom positive realness for adaptive control.AUTOMATICA 42(9), 1611–1616.

Ben Yamin, R., I., Yaesh and U. Shaked (2007),Simplified adaptive control with guaranteedH∞ performance, IFAC Workshop on Adap-tation and Learning in Control and Signal

Processing (ALCOSP 2007). Saint Peters-burg, RUSSIA August, 2931, 2007.

Bitmead, R., M. Gevers, and V. Wertz (1990).Adaptive Optimal Control, The ThinkingMan’s GPC, Prentice Hall, Englewood Cliffs,New Jersey.

Bobtsov A., and A. Nagovitsina (2007), Adaptiveoutput controlof linear time-varying systems,IFAC Workshop on Adaptation and Learningin Control and Signal Processing (ALCOSP2007). Saint Petersburg, RUSSIA August,2931, 2007.

Broussard J., and P. Berry (1978). CommandGenerator Tracking - The Continuous TimeCase, TASC, Tech. Rep, TIM-612-1.

Byrnes, C. I., and J. C. Willems (1984). AdaptiveStabilization of Multivariable Linear Systems.Proc. 23rd IEEE Conf. on Decision and Con-trol, 1547–1577.

Cauer, W. (1958). Synthesis of Linear Communi-cation Networks, McGraw-Hill, New York.

Feuer, A., and A.S. Morse (1978). AdaptiveControl of Single-Input, Single-Output Lin-ear Systems. IEEE Trans. Automatic ControlAC-23, 557–569.

Fradkov, A. L. (1976). Quadratic Lyapunov Func-tion in the Adaptive Stabilization Problem ofa Linear Dynamic Plant. Siberian Mathemat-ical Journal 2, 341–348.

Fradkov, A. L. (1994). Adaptive stabilizationof minimal-phase vector-input objects with-out output derivative measurements. Physics-Doklady 39(8), 550–552

Fradkov, A. L. (2003). Passification of non-squarelinear systems and feedback Yakubovich -Kalman - Popov lemma. European Journal ofControl 6, 573-582.

Fradkov, A. L., and B. Andrievsky (2007),Passification-Based Adaptive Control withimplicit reference model IFAC Workshop onAdaptation and Learning in Control and Sig-nal Processing (ALCOSP 2007). Saint Peters-burg, RUSSIA August, 2931, 2007.

Fradkov, A. L., and D. Hill (1998). Exponentialfeedback passivity and stabilizability of non-linear systems. Automatica 34(66), 697–703.

Goodwin, G. C., and K. Sin (1984). AdaptiveFiltering, Prediction and Control, PrenticeHall, Englewood Cliffs, New Jersey.

Goodwin, G. C., P.J. Ramadge and P.E. Caines(1980). Discrete Time Multivariable Adap-tive Control. IEEE Trans. Automatic ControlAC-25, 449–456.

Hahn, W. (1967). Stability of Motion, Springer-Verlag, New York.

Heyman, M., J. H. Lewis and G. Meyer (1985).Remarks on the Adaptive Control of LinearPlants with Unknown High Frequency Gain.Systems and Control Letters 5, 357–362.

Hsu, L., and R. Costa (1999). MIMO direct adap-tive control with reduced prior knowledge ofthe high-frequency gain. Proceedings of 38thIEEE Conference on Decision and Control,Phoenix, AZ, 3303–3308.

Huang, C.-H., P.A. Ioannou, J. Maroulas, andM.G. Safonov (1999). Design of S. IEEETransactions on Automatic Control44, 569–573, 1999.

Ilchman, A., D. Owens and D. Pratzel-Wolters(1987). Remarks on the Adaptive Control ofLinear Plants with Unknown High FrequencyGain. Systems and Control Letters 8, 397–404.

Ioannou, P. and P. Kokotovic (1983) Adaptive Sys-tems with Reduced Models, Springer-Verlag,Berlin.

Ioannou, P., and J. Sun (1996). Robust AdaptiveControl, Prentice-Hall, Upper Saddle River,New Jersey.

Ioannou, P. A., and G. Tao (1987). Frequencydomain conditions for strictly positive realfunctions. IEEE Transactions on AutomaticControl 32, 53–54.

Iwai Z., and I. Mizumoto (1992). Robust and sim-ple adaptive control systems (1992). Interna-tional Journal of Control 55, 1453–1470.

Iwai Z., I. Mizumoto, Y. Nakashima, and M. Ku-mon (2007). Adaptive MIMO PID Controllerwith Model-based Parallel Feedforward Com-pensator (2007). IFAC Workshop on Adap-tation and Learning in Control and SignalProcessing (ALCOSP 2007). Saint Peters-burg, RUSSIA, August 2931, 2007.

Johansson, R., and A. Robertsson (2002).Observer-Based Strict Positive-Real (SPR)Feedback Control System Design. Automat-ica 38(9), 1557–1564.

Kalman, R. (1964). When is a Linear SystemOptimal? Transactions of ASME, Journal ofBasic Engineering, Serries D 86, 81–90.

Kaufman, H., M. Balas, I. Bar-Kana, and L.Rao (1981). Model Reference Adaptive Con-trol of Large Scale Systems. Proceedings 20thIEEE Conference on Decision and Control,San Diego, California, 984–989.

Kaufman, H., I. Barkana and K. Sobel 1998.Direct Adaptive Control Algorithm, 2nd ed.,Springer-Verlag, New York.

Kreiselmayer, G., and B. Anderson (1986) RobustModel Reference Adaptive Control. IEEETransactions on Automatic Control AC-31(2), 127–133.

Krstic, M., I. Kanellakopoulos and P. Kokotovic(1995). Nonlinear and Adaptive Control De-sign, John Wiley, New York.

Landau, I. D. (1974). A Survey of Model Refer-ence Adaptive Techniques: Theory and Appli-cations. Automatica 10, 353–379.

Landau, I. D. (1979). Adaptive Control - TheModel Reference Approach, Marcel Dekker,New York.

LaSalle, J. (1981). Stability of non-autonomoussystems. Nonlinear Analysis Theory Methodsand Applications 1(1), 83–90.

Mareels, I. A Simple Selftuning Controller for Sta-ble Invertible Systems. Systems and ControlLetters 4, 5–16.

Mareels, I., and J. W. Polderman (1996). AdaptiveSystems: An Introduction, Birkhauser.

Moir, T., and M. Grimble (1984) Optimal Self-Tuning Filtering, Prediction, and Smoothingfor Discrete Multivariable Processes. IEEETransactions on Automatic Control 29(2),128–137.

Monopoli, R. V. (1974). Model Reference Adap-tive Control with an Augmented Error Sig-nal. IEEE Transactions on Automatic Con-trol AC-19(5), 474–484.

Morse, A.S. Global Stability of Parameter Adap-tive Control Systems (1980). IEEE Transac-tions on Automatic Control AC-25, 433–439.

Morse, A. S. (1984). New directions in ParameterAdaptive Control Systems. Proc. 23rd IEEEConf. on Decision and Control, 1566–1568.

Narendra, K. S., and A. Annaswami (1989). StableAdaptive Systems, Prentice Hall, EnglewoodCliffs, New Jersey.

Narendra, K. S., and L.S. Valavani (1978). Adap-tive Controller Design - Direct Control. IEEETransactions on Automatic Control AC-23,570–583.

Narendra, K. S. and L.S. Valavani (1979) Directand Indirect Model Reference Adaptive Con-trol. Automatica 15, 653–664.

Narendra, K. S., Y.H. Lin, and L.S. Valavani(1980). Stable Adaptive Controller Design -Part II: Proof of Stability. IEEE Transactionson Automatic Control AC-25, 440–448.

Nussbaum, R. O. (1983) Some Remarks on aConjecture in Parameter Adaptive Control.Systems and Control Letters 3, 243–246, 1983.

Ortega, R., and T. Yu (1987). Theoretical Resultson Robustness of Direct Adaptive Controllers.Proc. IFAC Triennial World Conference 10,1–15.

Osborn, P. V., H. P. Whitaker and A. Kezer(1961). New developments in the design ofModel reference Adaptive Control Systems.Inst. Aeronautical Sciences, Paper 61-39.

Popov, V. M. (1962). Absolute Stability of Non-linear Control Systems of Automatic Control.Automation and Remote Control 22.

Sastri, S., and M. Bodson (1989). Adaptive Con-trol: Stability, Convergence and Robustness,Prentice-Hall, Upper Saddle River, New Jer-sey.

Sobel, K., H. Kaufman and L. Mabus (1982).Adaptive Control for a Class of MIMO Sys-

tems. IEEE Transactions on Aerospace 18,576–590.

Torres, S., and E. Mehiel (2006). Nonlinear DirectAdaptive Control and Disturbance Rejectionfor Spacecraft. Proceedings of the AIAA GNC2006, paper AIAA2006-6038.

van Amerongen, J., and A. U. Ten Cate (1975).Model Reference Adaptive Controller forShips. Automatica 11, 441–449.

Wellstead, P., and M. Zarrop (1991). Self-TuningControl Systems, Wiley, Chichester, UK.

Wen, J. T. (1988). Time-domain and frequencydomain conditions for strictly positive real-ness. IEEE Transactions on Automatic Con-trol 33, 988–992.

Wen, J., and M. Balas (1989). Finite-dimensionaldirect adaptive control for discrete-timeinfinite-dimensional Hilbert space. Journal ofMathematical Analysis & Applications 143,1–26.

Whitaker, H. P. (1959). An Adaptive Performanceof Aircraft and Spacecraft. Inst. AeronauticalSciences, Paper 59-100.