IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 9

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 9, SEPTEMBER 2007 1543

Adaptation and Parameter Estimation in SystemsWith Unstable Target Dynamics and

Nonlinear ParametrizationIvan Yu. Tyukin, Member, IEEE, Danil V. Prokhorov, Senior Member, IEEE, and Cees van Leeuwen

Abstract—In this paper, we propose a solution to the problemof adaptive control and parameter estimation in systems withunstable target dynamics. Models of uncertainties are allowedto be nonlinearly parameterized, and required to be smooth andmonotonic functions of linear functionals of the parameters. Themere assumption of existence of nonlinear operator gains for thetarget dynamics is sufficient to guarantee that system solutionsare bounded, reach a neighborhood of the target set, and themismatches between the modeled uncertainties and their compen-sator converge to zero. With respect to parameter convergence,a standard persistent excitation condition suffices to ensure thatit is exponential. When a weaker, nonlinear version of persistentexcitation is satisfied, asymptotic convergence is guaranteed.The spectrum of possible applications ranges from tyre-road slipcontrol to asynchronous message transmission in spiking neuraloscillators.

Index Terms—Adaptive control, exponential convergence,monotone functions, nonequilibrium dynamics, nonlinearparametrization, (nonlinear) persistent excitation, parameterestimation, unstable.

I. INTRODUCTION

ADAPTIVE control and system identification theoriesare broadly applied in control engineering, and are of

growing interest to physics and biology [1]. To further promotetheir application in these fields, we will have to confront cer-tain problems engineering has been able to avoid. Nonlinearparametrization of uncertainties is intrinsic to physical andbiological models [2]–[6]. Many, in addition, have nongloballystable target dynamics, in which multiple attractors coexist[7]–[10]. Some use instability, e.g., for signal amplification[11], unstable synchrony, e.g., for solving image segmentation[12] or binding problems [13] in visual systems, or chaoticdynamics, e.g., in solving path finding problems in mazes[14]. The ”standard” requirements of adaptive control theory,

Manuscript received June 30, 2005; revised September 6, 2006. Recom-mended by Associate Editor A. Garulli.

I. Yu. Tyukin is with the Laboratory for Perceptual Dynamics, Brain ScienceInstitute, RIKEN (Institute for Physical and Chemical Research), Wako-shi,Saitama 351-0198, Japan (e-mail: [email protected]). He is also withthe Department of Mathematics, the University of Leicester, Leicester, LE17RH, U.K., and the Department of Automatic Control, Saint-Petersburg StateUniversity for Electrical Engineering, Saint-Petersburg, 197 376, Russia.

D. V. Prokhorov is with the Toyota Technical Center, Ann Arbor, MI 48105USA (e-mail: [email protected]).

C. van Leeuwen is with the Laboratory for Perceptual Dynamics, Brain Sci-ence Institute, RIKEN (Institute for Physical and Chemical Research), Wako-shi, Saitama, 351-0198, Japan (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAC.2007.904448

viz. that 1) system uncertainties be linearly parametrized and2) target dynamics be globally asymptotically stable, with aLyapunov function available for the design [15]–[17], thereforepose conspicuous limitations for expansion in these fields.

In direct adaptive control, solutions to the first problemhave been subject to convexity constraints [18]. Therefore,dominance of the nonlinearly parameterized terms is generallyused [19], [20]. This method inevitably overcompensates theuncertainties and eliminates the nonlinearities inherent in thesystem’s target behavior. More gentle control is needed: onethat operates through modification of the system’s intrinsic mo-tions rather than through compensation. In identification-basedapproaches, solutions are restricted to Hammerstein (Wiener)models [21], [22], in which the dynamics is linear and the non-linearities are static input (output) maps. Extensions involvinglocal modelling techniques result in models that are not alwaysadequate [23]. Altogether, a satisfactory solution to the firstproblem is yet to be obtained.

As for the second problem, the restriction to globally stabletarget dynamics has partially been lifted, for linear systems withlinear parameterization and neutrally stable target dynamics[24]. For nonlinear systems with nonlinear parameterizationand, possibly, unstable target dynamics these problems requirefurther development.

The present paper provides a unified approach for adaptivecontrol and parameter estimation in systems with nonlinearlyparameterized uncertainties and potentially unstable target dy-namics. When the target dynamics is stable knowledge of theLyapunov function of the desired motions1 is not required.

The proposed method applies to a class of nonlinear param-eterizations satisfying a specific monotonicity constraint. Thisclass covers a broad variety of models in physics, mechanics,biology and neural computation. It includes models of stiction,slip, and surface dependent friction, nonlinearities in dampers,smooth saturation, dead-zones in mechanical systems, and non-linearities in models of bio-reactors [2]–[6].

The method employs an operator formalism in functionalspaces rather than conventional techniques.2 We consider thedesired dynamics in terms of input-output mappings in func-tional spaces that are required to be locally bounded. Their in-puts are initial conditions, parameters, and mismatches betweenthe uncertainty and a compensator. The outputs are the state

1This solves the issue mentioned in [17] as an open theoretical challenge.2In particular the common practice to fit the derivative of the goal functionals

(Lyapunov candidates) to specific algebraic inequalities leading to the propertyof Lyapunov stability of the closed loop system.

0018-9286/$25.00 © 2007 IEEE

1544 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 9, SEPTEMBER 2007

and an error function , not necessarily definite in state.Adaptation thus becomes a problem of regulation of mismatchesto specific functional spaces. This is followed, if possible, byminimization of their functional norm. Because continuity ofthe mappings of the target dynamics is not required, its stability,which in many cases is synonymous to continuity of the flowwith respect to initial conditions and/or inputs [25], is not re-quired either. That may not be definite allows us to liftconventional restrictions on the goal functionals.3

Under standard conditions of persistent excitation of the func-tional of state, the proposed algorithms solve the problem of pa-rameter estimation for nonlinearly parameterized uncertainties.In this case convergence is exponential and its rate is estimatedusing the results of [26]. In case standard persistent excitationdoes not hold, a nonlinear persistent excitation condition [27]guarantees asymptotic convergence of the estimates to the ac-tual values of unknown parameters.

The paper is organized as follows. Section II describes nota-tions and conventions used. Section III formulates the problem.Section IV contains the main results. Section V provides practi-cally relevant applications, and Section VI concludes the paper.

II. NOTATIONAL CONVENTIONS

• Symbol defines the field of real numbers, and symbolstands for the following set ;

• Symbol defines the set of natural numbers.• Symbol stands for an -dimensional linear space over

the field of reals.• denotes the space of functions that are at least times

differentiable.• Symbol denotes the class of all strictly increasing func-

tions such that ; symboldenotes the class of all functions such that

.• denotes the Euclidian norm of .• Notation stands for the absolute value of a scalar.• Symbol denotes the signum-function.• By symbol , where , , , we

denote the space of all functions such that

.

• Symbol denotes the -norm of .• By , , we denote the space of

all functions such that; stands for

the norm of .• Let be given. Function

is said to be locally bounded if for any ,there exists constant such that .

• Let be an square matrix, then denotes apositive definite (symmetric) matrix and be the inverseof . By we denote a positive semi-definite matrix.

• We reserve symbol to denote the quadratic form:, where and is the transpose of .

• Symbols , stand for the minimal andmaximal eigenvalues of , respectively.

3Which are usually defined as positive-definite and radially unbounded func-tions of state [17].

• By symbol we denote the identity matrix.• The solution of a system of differential equations

, , passingthrough point at will be denoted for as

, or simply as if it is clear from thecontext what the values of are and how the function

is defined.• Let be a function of state ,

parameters , and time . Let in addition both and befunctions of . Then, in case the arguments of are clearlydefined by the context, we will simply write insteadof .

• The (forward complete) systemis said to have an , gain( , ) with respect to itsinput if and only iffor any and there exists a function

, such that the fol-lowing holds: .Function is assumed to be non-decreasing in , and locally bounded in itsarguments.

• When dealing with vector fields and partial derivatives wewill use the following extended notion of the Lie deriva-tive of a function. Let it be the case that and canbe partitioned as follows: , where ,

, , ,, and denotes concatenation of two vectors.

Define such that ,where , ,

, . Thensymbol , denotes the Lie deriva-tive of function with respect to vector field :

.

III. PRELIMINARY ASSUMPTIONS AND PROBLEM FORMULATION

Let the following system be given:

(1)

where ,, ; is

a vector of unknown parameters; is a closed bounded subsetof ; is the control input; , are the initialconditions, functions , ,

, are locally bounded. Vectoris a state vector. Vectors , are referred to as un-

certainty-independent and uncertainty-dependent partitions of, respectively.For the sake of compactness we will also use the following

description of (1):

(2)

where ,.

TYUKIN et al.: ADAPTATION AND PARAMETER ESTIMATION IN SYSTEMS 1545

Our objective is to derive the control input as a function ofstate variable , controller parameters ,and time , e.g., such that for a class ofnonlinearly parameterized , all and

: 1) all trajectories of the system withare bounded; 2) state converges to a given bounded targetset; 3) ensuring, if possible, that the estimate converges tounknown asymptotically.

As required for many physical and biological systems, weenvisage to allow desired (target) motions in the system to beglobally unstable. When the target dynamics is stable we donot require knowledge of the corresponding Lyapunov function.Prior to providing the formal problem statement let us brieflyreview conventional assumptions on target sets and dynamics,and substitute alternatives.

Usually the target set is specified in terms of the zeroes of agoal, or error, function

(3)

where the function should additionally satisfy some(algebraic) metric restrictions. In particular, it is required that

be (positive) definite with respect to the target set .For example, in problems of tracking a reference trajectory

, , these restrictions lead to thefollowing constraints:

(4)

Furthermore, function should be a Lyapunov candidatefor the closed loop system at . Knowledge ofis explicitly used in standard certainty-equivalence approaches[17]. Finding such a Lyapunov candidate is not a trivial task,even more so when the desired trajectories are underspec-ified or the target set is a nontrivial subset of . For sys-tems with globally Lyapunov-unstable target dynamics no ap-propriate goal function could be given.

We will deal with these issues by replacing standard alge-braic restriction (4) with operator relations. We replace standardnorms in in (4) with functional norms ,

in the functional spaces , ,. Thus, we keep function as a measure of closeness

of trajectories to the desired set without imposingstate-metric or definiteness restrictions (4) on . On theother hand, we will be able to derive bounds for from thevalues of -norms of the function . Let us for-malize this requirement as follows.

Assumption 1 (Target Operator): For a given functionand all the following property holds:

(5)

where is a nonnegative, locallybounded, and nondecreasing in function.

Assumption 1 can be interpreted as unbounded observability[28] of (1) with respect to the “output” . Clearly, itincludes conventional definiteness requirements on as

a special case. A large class of systems obeys Assumption 1without requiring the function to be definite. Consider,for instance, the equations of a spring-mass system:

(6)

where , is a nonlinear dampingterm, an unknown parameter. Let ,

, and suppose that , i.e.,, . Rewriting (6) in accordance

with this constraint results in the following description:

(7)

It follows immediately from (7) thatand

Therefore, the following estimate holds:

and, hence, Assumption 1 is satisfied. For a more general classof systems

(8)

and a class of nondefinite functions, , checking Assumption 1 amounts

to establishing a bounded input – bounded state property forthe following cascade:

Assumption 1 is validated for (6), (8) and functionsindependently of knowledge of the domain . Even thoughsuch knowledge may generally be necessary for a still widerclass of systems and nondefinite functions , Assumption1 allows us to exploit a much broader spectrum of criteria forensuring system state boundedness than standard conditions (4).

Let us specify a class of control inputs which can ensureboundedness of solutions for every and

. According to (5), boundedness ofis ensured if we find a control input such that

. To this aim, consider the dynamics of (2) with re-spect to

(9)

According to (9) the dynamics of is affected by unknownthrough the following term:

(10)

We require a feedback that is capable of annihilatingthe influence of uncertainty on the dynamics of


Fig. 1. Illustration of the choice of goal functionals (x; t). (a) Conventional requirements. Function (x; t) should satisfy (4) in which boundedness of (x; t)explicitly implies boundedness of state x. (b) Requirements following from Assumptions 1 , 2 . Boundedness of (x; t) may not explicitly imply that state xis bounded. However, given that �(t) 2 L [t ;1], k (x(t); t)k � ( ;!!!; k�(t)k ) = M from Assumption 2 implies that the statex(t) is bounded and belongs to the sphere = fx 2 j x : kx(t)k � ~ (x ; ��;M) = �g. On the other hand, the state x(t) belongs to the domain = fx 2 j x : j (x; t)j �Mg. This ensures that the segments of trajectory x(t;x ; t ; ��;u(t)), for t � t will remain in the bounded domain \(a shaded volume).

and, in addition, ensures boundedness of . As-suming that the inverse exists everywhere4

and using notation (10), we define control input as follows:

(11)

where is a vector of known parameters of func-tion , and function stands for distur-bances due to measurement noise, unmodeled dynamics, etc.Unless stated otherwise, we assume that .

Feedback (11) renders (9) into the error model form [16]:

(12)

or, when

(13)

Let us now specify the desired properties of functionin (11), (12) and (13). Instead of global Lyapunov stability of(12) for , we propose that finite energy of the signal

, defined for example by itsnorm with respect to the variable , results in bound-

edness of and, hence, by Assumption 1, of the state. Formally this requirement is introduced in Assumption 2:

Assumption 2 (Target Dynamics Operator): Consider the fol-lowing system:

(14)

where and is from (12). For every(14) has gain with respect to input. In other words, there exists a locally bounded function

such that

(15)

4This assumption limits our choice of functions (x; t) to ones that sat-isfy the following constraint: sign( g (x)@ (x; t)=@x ) = const. Eventhough invertibility ofL (x; t) is not at all necessary for our approach andthe specific choice of feedback u(x; ��; t) is not the central topic of our presentcontribution, we sacrifice generality for the sake of constructive design

and function is nondecreasing in.

Assumption 2 does not require global asymptotic stability ofthe origin of unperturbed system (14), i.e., for . System(14) is allowed to have Lyapunov-unstable equilibria, multipleattractors, or no equilibria at all. For the latter case, see ExampleB in Section V. When the target dynamics is stable, a benefit ofAssumption 2 is that there is no need to know the particularLyapunov function of the unperturbed system.

Fig. 1 illustrates the differences between Assumptions 1, 2and conventional restrictions on the goal functionals as wellas approaches based on geometric representations [29]. Resultsbased on coordinate transformation around the target manifold(3) are applicable only in a subset of where doesnot depend explicitly on , and rank of is constant. Inthis respect, these results are local. Assumptions 1, 2 do not re-quire constant rank conditions and allow time-varyingand . They may, therefore, replace the conventionalones for systems with nonstationary target dynamics or ones thatare far away from equilibrium or target manifolds.

Using Assumptions 1, 2 we are ready to provide the formalstatement of the problem considered in the paper.

Problem 1: Consider (1), (12), and the goal functionsatisfying Assumptions 1, 2. Find a class of adaptation algo-rithms such that for all and a priori un-known

1) state and are bounded: ,;

2) signal has bounded-norm

(16)

3) the influence of uncertainty on the target dynamics van-ishes with time

(17)

4) and, possibly, ensuring that

(18)

Instead of considering general nonlinear parametrization ofwe find that a broad range of physical, mechanical


TABLE IEXAMPLES OF NONLINEARITIES SATISFYING ASSUMPTION 3 . PARAMETER � IS A POSITIVE CONSTANT

and biological phenomena can be described by a specific classof nonlinear functions, in particular:

(19)

where , ,are continuous functions and

is monotone in . Functions (19) naturally extend fromlinear to nonlinear parameterizations, covering a wide range ofpractically relevant models, as illustrated in Table I.

These observations motivated us to consider functionssatisfying the following assumption.

Assumption 3 (Monotonicity, Growth Rate in Parameters):For the given function in (12) there exists a function

and positive constantsuch that

(20)

(21)

Inequality (20) in Assumption 3 holds, for instance, forall functions (19). In this case, function can be givenas follows: , whereif is nondecreasing in and

if is nonincreasing in .Inequality (21) is satisfied if does not growfaster than a linear function in variable for every

. This requirement holds, for example, for functionsthat are globally Lipschitz in

Fig. 2. Illustration of the conditions of Assumption 3 for functions f (x; ��; t)which belong to class (19). Thick lines stand for the functionD��(x; t) ��,D =max jD (x; t)�(x; t)j in each block, respectively.

In this case, (20) and (21) hold for the functionsdefined by (19) with .Fig. 2 illustrates possible choices of function . Examplesof the relevant parameterizations in Table I also satisfy Assump-tion 3 . Their corresponding functions are listed in theright column.

In general, when , validation of Assumption3, amounts according to Hadamard’s lemma to findingand such that the following holds for all :

(22)

Assumption 3 bounds the growth rate of the differenceby the functional .

This will help us to find a parameter estimation algorithm suchthat the estimates converge to sufficiently fast for the solutionsof (1), (12) to remain bounded with nondominating feedback(11). On the other hand, parametric error can be inferred


from the changes in the variable , according to (12), onlyby means of the difference . Therefore,as long as convergence of the estimates to is expected, it isuseful to have the estimate of frombelow, as specified in Assumption 4.

Assumption 4: For a given function in (12) and afunction satisfying Assumption 3, there exists a positiveconstant such that

(23)

In problems of parameter estimation, effectiveness of thealgorithms often depends on how “good” the nonlinearity

is, and how predictable locally is the system’sbehavior. As a measure of goodness and predictability usuallysmoothness and boundedness are considered. Likewise, in ourstudy, we distinguish several specific properties of functions

and .H1: Function is locally bounded with respect to

, uniformly in .H2: Function , and is locally

bounded with respect to , uniformly in .H3: Let and be bounded. There exists

constant such that for every andAssumption 4 is satisfied with .

H4: Function is locally bounded in , uniformlyin .

The next section presents a solution to Problem 1 for func-tions satisfying Assumption 3. Depending on thespecific properties of the system (e.g., H1–H4, Assumption 4)we provide alternative characterizations of its asymptotic be-havior, including conditions which ensure convergence (18).

IV. MAIN RESULTS

Standard approaches in parameter estimation and adaptationproblems usually assume feedback and a parameter adjustmentalgorithm in the following form:

(24)

where is a function of the errorfunction , state and time . The favorite strategy offinding these is a two-stage design prescription, known asthe certainty-equivalence principle. First, construct uncer-tainty-dependent feedback , which ensuresboundedness of the trajectories . Second, replace with

in and, given the constraints (e.g., , cannot bemeasured explicitly while state is available), design function

which guarantees (16)–(18), and/or .With this strategy, design of feedback is generally

independent5 of the specific design of the parameter estimationalgorithm . This allows the full benefit of nonlinear

5In particular, it is a standard requirement that function u(x; ��; t) shouldguarantee Lyaponov stability of the system for �̂�� = ��, while parameter ad-justment algorithms use this property in order to ensure stability of the wholesystem. No other properties are required from the function u(x; ��; t).

control theory in designing feedback . On the otherhand, this strategy equally benefits from conventional param-eter estimation and adaptation theories, which provide a list ofready-to-be-implemented algorithms under the assumption thatfeedback ensures stability of the system.

Ironically, the power of the certainty-equivalence prin-ciple—simplicity and independence of the design stages—isalso its Achilles heel. It ignores the possibility of advantageousinteractions between control and parameter estimation proce-dures. There are numerous reports [29]–[32] that an additionalinteraction term added to the param-eters in function : introducesnew properties to the system. Unfortunately, straightforwardintroduction of this term as a new variable of the design nega-tively affects its simplicity and so much favored independenceof the design stages.

An alternative strategy is proposed in [33] and [34]. It in-troduces a new design paradigm, in which the adaptation algo-rithms in (24) are initially allowed to depend on unmeasurablevariables

(25)

For this reason, such algorithms are called virtual algorithms. Ifthe desired properties (16)-(18) are ensured with (25) then theunrealizable algorithm (25) is converted into integrodifferential,or finite, form [35]

(26)

Under the following condition, finite form representation (26)is equivalent to (25):

(27)

According to this strategy, design of the adaptation algo-rithms requires, first, finding appropriate virtual algorithms

and, second, solving (27) for ,. This approach preserves the convenience of the

certainty-equivalence principle, as the feedbackcould, in principle, be built independently of the subsequentparameter adjustment procedure. At the same time, it providesin the systematic way the necessary interaction term ,ensuring the required properties (16)–(18) of the closed-loopsystem even if function in (12) is nonlinear in .

We shall seek for solution of Problem 1 in the following classof virtual adaptation algorithms6:

(28)

6This choice is motivated by our previous study of derivative-dependent al-gorithms for systems with nonlinearly parameterized uncertainties [32], [36].


where , . As acandidate for finite form realization (26) of algorithms (28) weselect the following set of equations:

(29)

where function ,satisfies Assumption 5

Assumption 5: There exists a function such that

(30)

where is either zero or, ifis differentiable in , satisfies condition:

Functionin (29) is given as follows:

(31)

Functions and are introduced into

(29) in order to shape the derivative to fit (28). Therole of function in (29) is to compensate for the un-certainty-dependent term , and (30) is thecondition for such a compensation to be possible.7 With thefunction we eliminate the influence of theuncertainty-independent vector fields , and

on the desired form of the time-derivative . In this sense,Assumption 5 specifies the condition for solvability of (27) forthe class of virtual algorithms (28).

The properties of (1), (12) with adaptation algorithm (29) and(31) are summarized in Theorem 1 and Theorem 2.8

7Technical relevance and issues related to validation of Assumption 5 are dis-cussed after the formulation of Theorem 1.

8In the statements of our results, we refer to the closed loop system as to (1),(12), (29), (31) instead of (1), (11), (29), (31), replacing explicit mentioningof the feedback u(x; �̂��; t) defined by (11) with error models (12) or (13). Thisis done in order to be able to extend the results of Theorems 1, 2 to a widerrange of problems, including parameter identification and observation. In theseproblems, inclusion of the observers can render the resulting system into theform defined by (1), (12), (29), (31), without the requirement that input u(t)satisfies (11).

Theorem 1 (Boundedness): Let (1), (12), (29), and (31) begiven and Assumptions 3, 4, and 5 be satisfied. Then the fol-lowing properties hold.

P1: Let for the given initial conditions , and pa-rameter vector , interval , be the (maximal)time-interval of existence of solutions in the closed loop system(1), (12), (29), and (31). Then

(32)

In particular

(33)

where

(34)

In addition, if Assumptions 1 and 2 are satisfied then the fol-lowing holds.

P2: , and

(35)

where .P3: If properties H1, H4 hold, and (14) has

, gain with respect to input and output ,then

(36)

If, in addition, property H2 holds, and functions ,are locally bounded with respect to uniformly in

, then the following holds.P4: The following limiting relation holds:

(37)

Proofs of Theorem 1 and subsequent results are given in theAppendix.

Prior to discussing the results of Theorem 1, we wish tocomment on Assumption 5. Because function specifiesthe desired target set and is determined by in(1) and function , uncertainty models and thechoice of the goal function are interrelated through theconditions for existence of a function satisfying (30).When , ,and , theseconditions follow from:

(38)


As a condition for existence of , this relation takesinto account structural properties of (1) and (12). Indeed, let

, and consider partial derivatives ,with respect to vector .

Let for some the following hold:

(39)

where symbol denotes a scalar function of and . Then(39) guarantees that (38) and, subsequently, Assumption 5holds. Hence, whether Assumption 5 holds, depends, roughlyspeaking, on how partition enters the arguments of functions

, . In case , Assumption5 holds for arbitrary . If , dependon just a single component of , for instance , (39) holdsand function can be derived explicitly by integration

(40)

In all other cases, existence of the required function fol-lows from (38).

Notice that Assumption 5 holds in the relevant problem set-tings for arbitrary . Consider for instance[27], where the class of systems is restricted to (41)

(41)

The system state in (41) has dimension. Hence, according to (40), and in case functions

, there will always exist a functionsatisfying (30) with .

When , the problem of finding satis-fying (30) can be avoided [or converted into one with an alreadyknown solution such as (38) and (40)] by the embedding tech-nique proposed in [34]. The main idea of this method is to in-troduce an auxiliary (forward-complete) system

(42)

such that for all

(43)

and . Then (12) can be rewritten as9

(44)

9In general, the L [t ;1]-norm of " (t) depends on ��. Therefore, given thatbounds of �̂�� may not be available a priori, proving that the L [t ;1]-norm of" (t) is bounded is not always possible. In this case, a modified control (11),where the term f (x; �̂��; t) is replaced with f (x (t)� h (t)� x (t); �̂��; t),could be used to render (12) into (44) with " (t) = " (t)+ "(t) 2 L [t ;1].

where , and. In principle, the dimension of could

be reduced to 1 or 0. As soon as this is ensured, Assumption 5will be satisfied, and the results of Theorem 1 follow. Sufficientconditions for embedding in general cases are provided in [34].For systems in which the parametric uncertainty can be reducedto vector fields with low-triangular structure the embedding isgiven in [37].

An alternative way to construct (42) with the desired prop-erties is to use (possible, high-gain, discontinuous) robust ob-servers. In order to illustrate this approach, consider the rathergeneral case when function in (1) is given as

, and function is bounded. Further, letus assume the existence of continuous functions ,

such that

(45)

As a candidate for yet unknown tracking system (42), we selectthe following:

(46)

where function and auxiliary input arethe design parameters. Subtracting equations for in (1) from(46) yields

(47)

where . Let us finally choose the function in (47)such that the system is strictly passive with apositive definite storage function :

(48)

According to [38]10 (48) guarantees that there always existsinput in (46) such that . Takinginto account (45), we can conclude that (43) holds with

. This implies that the original error model(12) can be converted into (44), which satisfies Assumption 5

for the corresponding in (44).Let us now briefly comment on the results of Theorem 1.

The theorem ensures a set of relevant properties for both con-trol (P2, P3) and parameter estimation problems (P1, P4). Theseproperties, as illustrated with (32)–(37), provide conditions forboundedness of the solutions , zeroing thegoal function , and exact compensation of the uncertaintyterm even in the presence of unknown disturbances

. All this follows from the factthat , which inturn is guaranteed by properties (20), (21), (23) of the function

in Assumptions 3, 4. Estimate (23) in Assumption 4is particulary important for allowing potentially unbounded dis-turbances from . When no disturbances are present it

10In [38] one extra assumption on the function f (��) in (47) is imposed. Itis required that the system _�� = f (��) + �� is strongly zero-detectable with re-spect to inputs �� and output y . In our case, however, the limiting relationslim y (t) = 0, lim ��(t) = 0 are not necessary. Therefore, as fol-lows from the proof of Theorem 2 in [38 p. 1484, Theorem 2], in order to showjust ky (t)k 2 L [t ;1] the assumption of strong zero-delectability can beomitted.


is possible to show that P1–P4 hold without involving Assump-tion 4.

Corollary 1: Let (1), (12), (29), and (31) be given, ,and Assumptions 3, 5 hold. Then, the following holds.

P5: is nonincreasing and properties P1–P411

of Theorem 1 hold with .In addition to the fact that is not

required to be bounded from below as in (23), Corollary 1 en-sures that is not growing with time when .Its practical relevance is: to guarantee desired convergence (18)with a much weaker, local, version of Assumption 4.

Let us formulate conditions ensuring convergence of the esti-mates to in the closed loop system (1), (12), (29), and (31).When the mathematical model of the uncertainties is linear in itsparameters, i.e., , the usual requirementfor convergence is that signal is persistently exciting[15].

Definition 1 (Persistent Excitation): Let a functionbe given. Function is said to be per-

sistently exciting iff there exist constants and suchthat for all the following holds:

(49)

Checking (49) often necessitates knowledge of signal as afunction of time. In the closed loop system, however, relevantsignals in the model of uncertainty can depend onstate , initial conditions, uncertainties, parameters of the feed-back, and initial time . In order to take such dependence intoaccount the notion of uniform persistent excitation has been sug-gested [26].

Definition 2 (Uniform Persistent Excitation): Let functionbe given, and be a solution

of (1), where the vector stands for parameters of (1)and feedback (11), (29), (31). Function issaid to be uniformly persistently exciting iff there exist constants

and such that for all , ,the following holds:

In the linear case, persistent excitation of signal , i.e.,inequality (49), implies that the following property holds:

(50)

In other words, the difference is propor-tional to the distance in parameter space for some

. When dealing with nonlinear parameterization,it is useful to have a similar characterization which takes modelnonlinearity into account. So it is natural to replace the linearterm in (50) with its nonlinear substi-tute , as has been done, for

11In this case, the bound for k (x(t); t)k will be different fromthe one given by (35) in Theorem 1. Its new estimate is given by (81) in theAppendix .

example, in [27] for systems with convex/concave parametriza-tion. It is also natural to replace proportion in (50)with a nonlinear function. We, therefore, use the following mod-ified notion of nonlinear persistent excitation.

Definition 3 (Nonlinear Persistent Excitation): Functionis said to be persistently

excited with respect to parameters iff there existconstant and functionsuch that for all , the following holds:

(51)

Properties (49) and (51) in Definitions 1 and 3 provide al-ternative characterizations of excitation in dynamical systems.While (49) accounts for properties of the signals in the un-certainty, (51) reflects the possibility to detect parametric mis-matches from the difference .The following theorem presents corresponding alternatives forparameter convergence in (1), (13), (29), and (31).

Theorem 2 (Convergence): Let (1), (13), (29), and (31) sat-isfy Assumptions 1–3. Let, in addition, Assumption 5 hold with

. Then , .Moreover the limiting relation:

is ensured if is locally bounded in uniformly in , andone of the following alternatives hold:

1) function is persistently exciting, and hypothesisH3 holds;

2) function is nonlinearly persistently exciting,i. e. it satisfies condition (51); it satisfies hypotheses H1,H2; function satisfies H4; functionis locally bounded in uniformly in ;

In case alternative 1) is satisfied, the estimates con-verge to exponentially fast. If, in addition, is uni-formly persistently exciting and Assumption 4 holds, then con-vergence is uniform. The rate of convergence can be estimatedas follows:

(52)

Theorem 2 considers error models (13) without a disturbanceterm but can straightforwardly be extended to ones withdisturbance (12). As follows from alternative 1), the parameterestimation subsystem is exponentially stable when ispersistently exciting. This allows (sufficiently small) additivedisturbances in the right-hand side of (13). In case the excitationis uniform, convergence of the estimates to a neighborhoodof is guaranteed for every by the inverseLyapunov stability theorems [39]. In case of alternative 2), (51)guarantees convergence (18) without invoking Assumption 4 or


H3. However, convergence may not be robust, which seems tobe a natural tradeoff between generality of nonlinear parame-terizations and robustness with respect to unknowndisturbances .

V. EXAMPLES

We apply our method to two systems with nonlinearly pa-rameterized uncertainties and unstable target dynamics. In ex-ample A, we consider the problem of optimal slip identifica-tion in brake control systems. Example B considers the problemof signal transmission between a pair of nonlinear oscillatorswithout requiring stable synchrony. The emphasis on practicalrelevance in these examples means that we will not dwell on theissue of illustrating the validation of every technical assumptionwe made. Validation of Assumption 1 is trivial in both exam-ples; other assumptions will be illustrated. The control functionin the first example is identification-based, and there is no needfor using control (11) in the second. Nevertheless, we will showthat both systems can be transformed into the error-model form(12), to which the results of Theorem 1 and Theorem 2 apply.

Example A: Adaptive Brake Control

Consider the problem of minimizing the braking distance fora wheel rolling along a surface. The surface properties can varydepending on the current position of the wheel. The wheel dy-namics is given by the following equations [40]:

(53)

is the longitudinal velocity, is the angular velocity,12 is the wheel slip, , is the

mass of the wheel, is the moment of inertia, is the radius ofthe wheel, is the control input (brake torque), isa function specifying the tyre-road friction force depending onthe surface-dependent parameter and the load force . Thisfunction can be derived from the steady-state behavior of theLuGre tyre-road friction model [3]

(54)

(55)

where , are Coulomb and static friction coefficients, isthe Stribeck velocity, is the normalized rubber longitudinalstiffness, is the length of the road contact patch. In order toavoid singularities in the solutions at we assume, assuggested in [40], that control input no longer applies whenvelocity reaches a small neighborhood of zero (we stopped assoon as became less than 1 m/s). Moreover, given that func-tions (54), (55) are bounded for the relevant set of the system pa-

12Given this functional relation, variablex in (53) can be viewed as an outputof the reduced system with state (x ; x ). Even though it is not necessary toconsider variable x as the state variable, we will show later that it is the veryvariable to be controlled. For this reason, we included the differential equationfor x explicitly into the system model (53).

rameters, it is always possible to design control functionin (53) such that

(56)

for all , .In order to minimize the braking distance, we have to ensure

that the tyre-road friction is always maximal. Thecorresponding control problem, hence, is to steer the wheel slip

to optimal value

(57)

If the value of would be known the feedback

(58)

could steer the variable into a small neighborhood of ex-ponentially fast. The problem, however, is that the value ofis not available. While the majority of the model parameters,including state variables , and load , can be estimateda priori or measured the tyre-road parameter depends on theroad surface. Therefore, on-line identification of is necessary.

In order to estimate parameter by measuring the values ofvariables and , we invoke Theorem 2. This requiresan error dynamics of type (13). To satisfy this requirement weintroduce the following auxiliary variable :

(59)

where is the estimate of . Denoting andtaking into account (53), (59) we obtain

(60)

The desired dynamics of (60), therefore, is

(61)

where is in . Let us check applicability of The-orem 2 to the extended system (53), (59). Notice first that state

of (53) is always bounded according to the physical laws gov-erning the dynamics of (53). In addition, boundedness ofimplies that is bounded. Hence Assumption 1 holds forthe extended system. System (61), obviously, has

gain. Therefore, Assumption 2 holds.Let us check Assumptions 3, 4. Taking into account prop-

erty (56) and (22), (54), (55) we can conclude that functionin (60) satisfies these assumptions with

. Because , Assumption 5 is satis-fied with . Finally, notice that ispersistently exciting. Then according to (29), (60), a parameteradjustment algorithm is defined as follows:

(62)

According to Theorem 2 the estimates (62) converge to expo-nentially fast in the domain specified by (56). Notice that algo-


Fig. 3. (a) Trajectories of (53). Solid lines show trajectories of (53) with algo-rithm (62) and one-line estimation of x ; the dashed line shows the values ofx as a function of time: x (�(s(t))). (b) Solid line shows braking distance d

a function of x in (53) and (58) with preset constant values of x 2 [0:1; 0:3]and known �. The dashed line marks the braking distance in the system withon-line estimation of x , �.

rithm (62) is a parametric linear proportional-integral scheme.Hence, it can be implemented with standard PI controllers.

We simulated (53)–(62) with the following setup of param-eters and initial conditions: , , ,

, , , , ,, , , , ,

. The road parameter, , was defined by the fol-lowing piece-wise constant function of the wheel positionon the road at time :

(63)

Fig. 3 illustrates the effectiveness of estimation algorithm(62). Estimates approach the actual values of parametersufficiently fast for the controller to calculate the optimal slipvalue and steer the system toward this point in real brakingtime. The braking distance in a system with on-line estimationof according to (57) is substantially shorter, compared to one

in which the values of were kept constant (in the interval[0.1,0.3]) and the tyre-road parameter is known.

Example B: Asynchronous Message Transmission

The present example illustrates the applicability of our ap-proach to problems where the target dynamics is not necessarilystable. We consider a pair of coupled nonlinear forced oscilla-tors (master) and (slave). Their inputs depend on param-eters in and in . Parameter can be under-stood as a “message” transmitted by the master. Parameter inthe slave subsystem should track the message transmitted by

. Traditionally, this problem is solved within observer-basedapproaches under the conditions that there is a stable synchronybetween and at and the oscillators are linearly pa-rameterized in .

As our present does not require stable synchrony ator linearity in , it enables secure asynchronous communicationwhen the oscillators are chaotic. Nonlinear parametrization ofthe uncertainty extends the problem of message transmissioninto the realm of modelling biologically relevant circuits.

Let us consider a pair of coupled oscillators, for instance,well-known Hindmarsh-Rose model neurons [41]

(64)

where ,, . Vari-

able is the coupling time-varying coefficient,is the nonlinear transfor-

mation of the input signal , and are the thresholds or,“messages”. The value of defines the upper bound for “stim-ulation” . Function is a plausible model ofthe synaptic gates which open when the value of exceedsthreshold . For the current choice of we set whichallows chaotic bursting in (64).

A sufficient condition for synchronization between theand subsystems at is (see [42] fordetails). The problem, however, is how to design as a functionof the state of (64), input and time such that tracks thevalues of for below .

First, we notice that , are bounded if ,are bounded: both and are semipassive with

a quadratic storage function, which is positive definite and ra-dially unbounded outside a bounded domain in the state spaceof (64) [42]. For the error function , we chose the difference

. Because boundedness of the state for all , ,is already ensured by strict semipassivity of (64), checking As-sumption 1 is not necessary for application of our method. Letus consider the dynamics of

(65)


Fig. 4. Trajectories of (64) and (67) as functions of time t. Input signal r(t) was set as follows: r(t) = sin(0:001t). White areas mark the time instances wherethe coupling variable c(t) exceeds the critical value c = 21:5: c(t) = 21:55 ensuring synchronization between theM and S subsystems at �̂ = �. The shadeddomains correspond to the time intervals in which conditions for synchronization are violated: c(t) = 0:05. Even though subsystemsM and S fail to synchronizeover these intervals, message � sent byM is successfully tracked by S .

Given that , , , are bounded, As-sumption 2 is satisfied for (65) with , where isarbitrary small. Function in this case can be definedas

Notice also that is differentiable. Hence, according toHadamard’s lemma, the difference canbe expressed as

(66)

Given that is strictly monotone in , the value of

is positive. Furthermore, is ultimately bounded.Therefore, Assumption 3 is satisfied with .Because , Assumption 5 is satisfied with

.Using (29) and (31) we write the adaptation algorithm for

as

(67)

According to Theorem 1 and Corollary 1 the variable isbounded and nonincreasing. Hence, taking into account (66)

and the fact that is separated from zero forall bounded , , Assumption 4 holds for the nonlinearity

in (64) and (65) with bounded and algorithm(67). Moreover, is uniformly persistentlyexciting. Therefore, Theorem 2 applies and converges to

in (64) exponentially fast. In other words, message astransmitted can be recovered exponentially fast using algorithm(67). This property is illustrated in Fig. 4.

VI. CONCLUSION

We proposed an adaptive control method that does not rely onassumptions of Lyapunov stability of the target dynamics. In-stead, the target dynamics is described by an input-output map-ping which is not necessarily continuous. Hence, because con-tinuity in this context is equivalent to stability, stability of thetarget dynamics is not necessary in our approach. An additionaladvantage is that knowledge of the Lyapunov function is notneeded for the design of an adaptation algorithm in the case ofstable dynamics.

The proposed method allows models of uncertainties to benonlinearly parameterized. However, we required them to besmooth and monotonic functions of linear functionals of the pa-rameters. Provided examples illustrate that this class has a broadspectrum of applications.

APPENDIX

PROOFS OF THE THEOREMS AND AUXILIARY RESULTS

Proof of Theorem 1: Let us first show that property P1holds. Consider solutions of (1), (12), (29), and (31) passingthrough the point , for .13 Let us calcu-late formally the time-derivative of function

13In accordance with the formulation of the theorem, interval [t ; T ] is theinterval of existence of the solutions


Notice that

(68)

According to Assumption 5. Then, taking (68) into

account, we can obtain

(69)

Notice that according to the proposed notation we can rewritethe term in (69) in thefollowing way:

Hence it follows from (29) and (69) that

Derivative can therefore be written in the followingmanner:

(70)

Consider the following positive-definite function:

(71)

Its time-derivative can be derived according to (70) as follows:

(72)

Let , then consider the difference .Applying Hadamard’s lemma we represent this dif-ference in the following way:

, .

According to Assumption 5, termis negative semidefinite; hence, using Assumptions 3,

4 and (12), we can estimate derivative as

(73)

It follows immediately from (73) and (71) that

(74)

Therefore, . Furthermore

In particular

(75)

Hence, , as a sumof two functions from . In order to estimate the upperbound of norm from(75), we use the Minkowski inequality

and then apply the triangle inequality to the functions from

(76)

Therefore, property P1 is proven.Let us prove property P2. In order to do this we have to check

first if the solutions of the closed loop system are defined forall , i.e., they do not reach infinity in finite time. We


prove this by a contradiction argument. Indeed, let there ex-ists time instant such that . It follows fromP1, however, that .Furthermore, according to (76) the norm

can be bounded from above by a contin-uous function of , , and . Let us denotethis bound by symbol . Notice that does not depend on

. Consider (12) for

Given that both ,and taking Assumption 2 into account, we au-

tomatically obtain that . In partic-ular, using the triangle inequality and the fact that function

in Assumption 2 is nondecreasing in, we can estimate the norm as follows:

(77)

According to Assumption 1 the following inequality holds:. Hence

(78)

Because a superposition of locally bounded functions is locallybounded, we conclude that is bounded. This,however, contradicts to the previous claim that .Taking (74) into account as well as the fact thatin (74) is bounded from above by , we can derivethat both and are bounded for every .Moreover, according to (77), (78), and (74) these bounds are(locally bounded) functions of initial conditions and parame-ters. Therefore, , .Inequality (35) follows immediately from (76), (15), and thetriangle inequality. Property P2 is proven.

Let us show that P3 holds. It is assumed that (14) has, gain. In addition, we have

just shown that. Hence, taking into account (12) we conclude that

, . On the other hand, given that, are locally bounded with respect to their

first two arguments uniformly in and that ,, , , signal

isbounded. Then implies that is boundedand P3 is guaranteed by Barbalat’s lemma.

To complete the proof of the theorem (property P4) considerthe time-derivative of function

Taking into account that , are locally bounded;function is continuously differentiable in , ;derivative is locally bounded with respectto , uniformly in ; functions , arelocally bounded with respect to uniformly in , then

is bounded. Given thatwe conclude by

applying Barbalat’s lemma thatas . The theorem is proven.

Proof of Corollary 1: Let . Choosing functionas in (71), using (72), and invoking Assumption 3 we

obtain that

(79)

Equality (79) and that in (71) imply that the normis nonincreasing. Furthermore, (79) implies that

(80)

This proves property P1. Taking into account (80) and giventhat Assumptions 1, 2 are satisfied we can conclude that

, , and that the following es-timate holds:

(81)

Hence, P2 is also proven. Properties P3 and P4 follow by thesame arguments as in the proof of Theorem 1. Therefore, P5 isproven. The corollary is proven.

Proof of Theorem 2: According to the theorem formula-tion, Assumptions 1, 2, 3, 5 hold. Hence, applying Corollary 1we can conclude that and .Let us show that limiting relation (18) holds in case alternative

1) is satisfied. To this purpose, consider derivative

(82)


Given that and , and thatHypothesis H3 holds, the function satisfies the fol-lowing inequality for some , :

Therefore, there exists function ,such that

(83)

Notice that matrix is positive definite and symmetric. It, there-fore, can be factorized as: , where is a nonsingular

real matrix. Let us define . In these newcoordinates, (83) will have the following form:

(84)

Denoting we can rewrite (84) as fol-lows:

(85)

where function satisfies equality

(86)

for all . Taking into account that function ispersistently exciting, , and that we canobtain the following bound for quadratic form (86):

(87)

Hence, function is also persistently exciting. Noticealso that is bounded from above

In order to show that as exponentially fast weinvoke Lemma 5 from [26].

Lemma 1: Let (85) be given, (87) hold (uniformly), andin (85) be bounded . Then (85) is

(uniformly) exponentially stable and, furthermore

(88)

According to Lemma 1 solutions of (85) converge to the originexponentially fast with a rate of convergence defined by (88),where

(89)

Taking (88), (89) into account and observing that, we can estimate as follows:

(90)

Given that and using (90), we derivethe following bounds for :

This proves alternative 1) of the theorem.Let us prove alternative 2). It follows immediately from

Corollary 1 of Theorem 1 that

(91)

Furthermore, given that, and is

locally bounded in uniformly in , we can conclude that

as . Let us divide the a following unionof subintervals: , , ,

, . The value of is chosen to satisfy, where is the constant from Definition 3. The fact thatas ensures that

(92)

In order to show this let us integrate (82)

(93)


Applying the Cauchy-Schwartz inequality to (93) and subse-quently using the mean value theorem we can obtain the fol-lowing estimate:

(94)

Given that limiting relation (91) holds, , andis locally bounded uniformly in we can conclude from

(94) that limiting relation (92) holds.Let us choose a sequence of points from : such

that , . As follows from the nonlinear persistentexcitation condition (51), for every , there exists apoint such that the following inequality holds:

(95)

Let us consider the following differences:. It follows immediately from

H1, H2, and (92) that

(96)

Taking into account (96) and (91) we can derive that

(97)

According to (97) and (95), sequence isbounded from above and below by two sequences convergingto zero. Hence, . Notice that

which implies that

(98)

In order to show that notice that, .

Hence, applying the triangle inequalityand using (92), (98) we can conclude

that is bounded from above and below by two func-tions converging to zero. Hence, as and(18) holds. The theorem is proven.

ACKNOWLEDGMENT

The authors would like to thank anonymous reviewers fortheir constructive critique and many useful comments. The au-thors are very grateful to Profs. V. Terekhov, H. Nijmeijer, R.Ortega, and Dr. A. Pavlov for helpful discussions.

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Ivan Yu. Tyukin (M’00) received the M.S. degreewith Honors in system engineering, and the Ph.D. andD.Sc. degrees in system analysis, control, and infor-mation processing from Saint-Petersburg State Uni-versity of Electrical Engineering, Saint-Petersburg,Russian Federation, in 1998, 2001, and 2006, respec-tively.

He was then a Visiting Researcher with theLinkoping University, a Research Fellow withthe Ford Research Laboratory, Dearborn, and theUniversity of Michigan-Dearborn, and an Assistant

Professor with Saint-Petersburg State University of Electrical Engineering,Department of Automation and Control during 2000–2001. From 2001 until2007, he was a Research Scientist with the Laboratory for Perceptual Dy-namics, Brain Science Institute, RIKEN, Japan. In 2007, he received a post as aRCUK Academic Fellow with the Department of Mathematics, the Universityof Leicester, U.K. His research interests include adaptive and nonlinear control,nonlinear dynamics, cooperative processes, mathematical neuroscience, andanalysis of emergent phenomena in nonlinear systems.

Danil V. Prokhorov (SM’02) received the M.S.degree with Honors from the State Academy ofAerospace Engineering (former LIAP), St. Peters-burg, Russia, in 1992, and the Ph.D. in electricalengineering from Texas Tech University, Lubbock,in 1997.

He was with Ford Research Laboratory, Dearborn,MI, from 1997 until 2005. He had been engaged inapplication-driven studies of neural networks andtheir training algorithms. He is currently a researchmanager with Toyota Technical Center. His research

interest is in machine learning algorithms and their applications to decisionmaking under uncertainty.

Dr. Prokhorov was awarded the INNS Young Investigator in 1999. He was theIJCNN 2005 General Chair and IJCNN 2001 Program Chair. He is the AssociateEditor of the IEEE TRANSACTIONS ON NEURAL NETWORKS and ComputationalIntelligence Magazine. He has been a reviewer for numerous journals and con-ferences, a program committee member of many conferences, a Guest Editor oftwo special issues of journals, and a panel expert for the NSF every year since1995.

Cees van Leeuwen received the M.S. and Ph.D. de-grees from the University of Nijmegen, the Nether-lands, in 1986 and 1989, respectively.

From 1989 to 1999, he was an Assistant Pro-fessor with teaching assignments in the InformationSciences Department, University of Amsterdam,the Netherlands. He was awarded a Research Pro-fessorship by the University of Sunderland, U.K.,in 2000. This position was continued as a VisitingProfessorship in 2005. He also holds a VisitingProfessorship with ECONA, Italy. Since 2001, he

is leading the Laboratory for Perceptual Dynamics, Brain Science Institute,RIKEN, Japan. His research interests include visual perception and cognition,neural networks, dynamical systems, and neuroscience.

Dr. van Leeuwen is the Editor of two scientific journals.

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