Adaptive Control System Part 8 pdf

Themain novelties of our design are: i the temporal and algorithmic separation of the parameter estimation task from the control task, and ii the ment of an active identi®cation procedur

considerations, a new class of uncertain nonlinear systems with unmodelleddynamics has been considered in the second part of this chapter A novelrecursive robust adaptive control method by means of backstepping and smallgain techniques was proposed to generate a new class of adaptive nonlinearcontrollers with robustness to nonlinear unmodelled dynamics.

It should be mentioned that passivity and small gain ideas are naturallycomplementary in stability theory [5] However, this idea has not been used innonlinear control design We hope that the passivation and small gainframeworks presented in this chapter show a possible avenue to approachthis goal

Acknowledgements This work was supported by the Australian ResearchCouncil Large Grant Ref No A49530078 We are very grateful to LaurentPraly for helpful discussions that led to the development of the result insubsection 6.4.4.2

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of Cascaded Nonlinear Systems', IEEE Trans Autom Contr., 37, 1386±1388.[30] Mareels, I and Hill, D J (1992) `Monotone Stability of Nonlinear FeedbackSystems', J Math Systems Estimation Control, 2, 275±291

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Active identi®cation for

In this chaper we propose a novel approach which removes this obstacle andyields global stability and tracking for systems that can be transformed into anoutput-feedback, strict-feedback, or partial-feedback canonical form Themain novelties of our design are: (i) the temporal and algorithmic separation

of the parameter estimation task from the control task, and (ii) the ment of an active identi®cation procedure, which uses the control input toactively drive the system state to points in the state space that allow theorthogonalized projection estimator to acquire all the necessary informationabout the unknown parameters We prove that our algorithm guaranteescomplete (for control purposes) identi®cation in a ®nite time interval, whosemaximum length we compute

develop-Thus, the traditional structure of concurrent on-line estimation and control

is replaced by a two-phase control strategy: ®rst use active identi®cation, andthen utilize the acquired parameter information to implement any controlstrategy as if the parameters were known

7.1 Introduction

In recent years, a great deal of progress has been made in the area of adaptivecontrol of continuous-time nonlinear systems [1], [2] In contrast, adaptivecontrol of discrete-time nonlinear systems remains a largely unsolved problem.The few existing results [3, 4, 5, 6] can only guarantee global stability underrestrictive growth conditions on the nonlinearities, because they use techniquesfrom the literature on adaptive control of linear systems [7, 8] Indeed, it hasrecently been shown that any discrete-time adaptive nonlinear controller using

a least-squares estimator cannot provide global stability in either the nistic [9] or the stochastic [10] setting The only available result whichguarantees global stability without imposing any such growth restrictions isfound in [11], but it only deals with a scalar nonlinear system which contains asingle unknown parameter

determi-The backstepping methodology [1], which provided a crucial ingredient forthe development of solutions to many continuous-time adaptive nonlinearproblems, has a very simple discrete-time counterpart: one simply `looks ahead'and chooses the control law to force the states to acquire their desired valuesafter a ®nite number of time steps One can debate the merits of such adeadbeat control strategy [12], especially for nonlinear systems [13], but it seemsthat in order to guarantee global stability in the presence of arbitrary non-linearities, any controller will have to have some form of prediction capability

In the presence of unknown parameters, however, it is impossible to calculatethese `look-ahead' values of the states Furthermore, since these calculationsinvolve the unknown parameters as arguments of arbitrary nonlinear func-tions, no known parameter estimation method is applicable, since all of themrequire a linear parametrization to guarantee global results This is the biggestobstacle to providing global solutions for any of the more general discrete-timenonlinear problems

In this chapter we introduce a completely di€erent approach to this problem,which allows us to obtain globally stabilizing controllers for several classes ofdiscrete-time nonlinear systems with unknown parameters, without imposingany growth conditions on the nonlinearities The major assumptions are thatthe unknown parameters appear linearly in the system equations, and that thesystem at hand can be transformed, via a global parameter-independentdi€eomorphism, into one of the canonical forms that have been previouslyconsidered in the continuous-time adaptive nonlinear control literature [1].Another major assumption is that our system is free of noise; this allows us

to replace the usual least-squares parameter estimator with an orthogonalizedprojection scheme, which is known to converge in ®nite time, provided theactual values of the regressor vector form a basis for the regressor subspace.The main diculty with this type of estimator is that in general there is no way

to guarantee that this basis will be formed in ®nite time The ®rst steps towards

removing this obstacle were taken in preliminary versions of this work [14, 15].

In those papers we developed procedures for selecting the value of the controlinput during the initial identi®cation period in a way that drives the systemstate towards points in the state space that generate a basis for this subspace in

a speci®ed number of time steps In this chapter we integrate those procedureswith the orthogonalized projection estimator to construct a true activeidenti®cation scheme, which produces a parameter estimate in a familiarrecursive (and thus computationally ecient) manner, and at each time instantuses the current estimate to compute the appropriate control input As a result,

we guarantee that all the parameter information necessary for control purposeswill be available after at most 2nr steps for output-feedback systems and

…n ‡ 1†r steps for strict-feedback systems, where n is the dimension of the

system and r is the dimension of the regressor subspace If the number ofunknown parameters p is equal to r, as it would be in any well-posedidenti®cation problem, this implies that at the end of the active identi®cationphase the parameters are completely known If, on the other hand, p > r, then

we only identify the projection of the parameter vector that is relevant to thesystem at hand, and that is all that is necessary to implement any controlalgorithm In essence, our active identi®cation scheme guarantees that all theconditions for persistent excitation will be satis®ed in a ®nite time interval: inthe noise-free case and for the systems we are considering, all the parameterinformation that could be acquired by any identi®cation procedure in anyamount of time, will in fact be acquired by our scheme in an interval which ismade as short as possible, and whose upper bound is computed a priori Thefact that our scheme attempts to minimize the length of this interval isimportant for transient performance considerations, since this will preventthe state from becoming too large during the identi®cation phase

Once this active identi®cation phase is over, the acquired parameterinformation can be used to implement any control algorithm as if theparameters were completely known As an illustration, in this chapter we use

a straightforward deadbeat strategy The fact that discrete-time systems (evennonlinear ones) cannot exhibit the ®nite escape time phenomenon, makes itpossible to delay the control action until after the identi®cation phase and still

be able to guarantee global stability

7.2 Problem formulation

The systems we consider in this section comprise all systems that can betransformed via a global di€eomorphism to the so-called parametric-output-

where  2 Rpis the vector of unknown constant parameters and i, i ˆ 1; ; n

are known nonlinear functions The name `parametric-output-feedback form'denotes the fact that the nonlinearities i that are multiplied by unknown

parameters depend only on the output y ˆ x1, which is the only measuredvariable; the states x2; ; xnare not measured It is important to note that thefunctions i are not restricted by any type of growth conditions; in fact, theyare not even assumed to be smooth or continuous The only requirement is thatthey take on ®nite values whenever their argument x1 is ®nite; this excludesnonlinearities like x 1

1 1, for example, but it is necessary since we want toobtain global results This requirement also guarantees that the solutions of(7.1) (with any control law that remains ®nite for ®nite values of the statevariables) exist on the in®nite time interval, i.e there is no ®nite escape time.Furthermore, no restrictions are placed on the values of the unknown constantparameter vector  or on the initial conditions However, the form (7.1) alreadycontains several structural restrictions: the unknown parameters appearlinearly, the nonlinearities are not allowed to depend on the unmeasuredstates, and the system is completely noise free: there is no process noise, nosensor noise, and no actuator noise

Our control objective consists of the global stabilization of (7.1) and theglobal tracking of a known reference signal yd…t† by the output x1…t†.

For notational simplicity, we will denote i;t ˆ i…x1…t†† for i ˆ 1; n.

x1at time t ‡ 2 In other words, given any initial conditions x1…0† and x2…0†, we

have no way of in¯uencing x1…1† through u…0† The best we can do is to drive

x1…2† to zero and keep it there The control would simply be a deadbeat

controller, which utilizes our ability to express future values of x1as functions

of current and past values of x1 and u:

(1) The well-known problems of poor inter-sample behaviour resulting fromapplying deadbeat control to sampled-data systems do not arise here, since

we are dealing with a purely discrete-time problem

(2) Deadbeat control can result to instability when applied to generalpolynomial nonlinear systems As an example, consider the system

If we implement a deadbeat control strategy to track the reference signal

yd…t† ˆ 2 t, one of the two possible closed-form solutions yields

chapter, such issues do not even arise, owing to the special structure of oursystems which guarantees that boundedness of x1; ; xi automaticallyensures boundedness of xi‡1, since xi‡1 …t† ˆ xi…t ‡ 1† T i…x1…t††.

Of course, when  is unknown, the controller (7.4) cannot be implemented.Furthermore, it is clear that any attempt to replace the unknown  with anestimate ^ would be sti¯ed by the fact that  appears inside the nonlinearfunction 1 Available estimation methods cannot provide global results forsuch a nonlinearly parametrized problem, except for the case where 1 isrestricted by linear growth conditions

7.2.2 Avoiding the nonlinear parametrization

Our approach to this problem does not solve the nonlinear parametrizationproblem; instead, it bypasses it altogether Returning to the control expression(7.4), we see that its implementation relies on the ability to compute the term

Since this computation must happen at time t, the argument x1…t ‡ 1† is not yet

available, so it must be `pre-computed' from the expression

2…x† ‡ 1…~x† …7:9†

since then we would be able at time t to compute the terms

and from them the control (7.4)

Hence, our main task is to compute the projection of  along vectors of theform (7.9) To achieve this, we proceed as follows:

Regressor subspace: First, we de®ne the subspace spanned by all vectors ofthe form (7.9):

S0ˆDRf 1…x† ‡ 2…~x†; 8 x 2 R; 8 ~x 2 Rg …7:12†

Note that the known nonlinear functions 1 and 2 need to be evaluatedindependently over all possible values of their arguments This is necessarybecause we are not imposing any smoothness or continuity assumptions on

these functions However, for any reasonable nonlinearities, determining thissubspace will be a fairly straightforward task which of course can be performedo€-line The dimension of S0, denoted by r0, will always be less than or equal

to the number of unknown parameters p: r0 p In fact, in any reasonably

posed problem we will have r0ˆ p, since r0< p means that we are consideringmore parameters than are actually entering the system equations; in that case,complete parameter identi®cation cannot be achieved with any method orinput, since the regressor vector cannot acquire the values necessary to identifysome of the parameters Hence, if r0< p, then the number of unknownparameters can be reduced to r0without any loss of information or generality.Projection measurements Clearly, in order to be able to implement the control(7.4), all we need to know about  is its projection on the subspace S0 But how

do we acquire this projection? From (7.3) we see that at time t, using themeasurements x1…t†; x1…t 1†; x1…t 2† and the known value of the control

u…t 2†, we can compute the following projection:

T

2…x1…t 2†† ‡ 1…x1…t 1††

Hence, if the values of x1 are such that the corresponding values of the vector

2…x1…t 2†† ‡ 1…x1…t 1†† eventually form a basis for the subspace S0, wewill obtain all the necessary information about  But how do we guaranteethat this identi®cation phase will be of ®nite duration?

Active identi®cation Instead of allowing the system state to drift on its own,

we use the control input u to drive the output x1 to values which result inlinearly independent vectors 2;t 2‡ 1;t 1and form a basis for S0 in at most2nr0steps (where n is the dimension of the system state and r0the dimension ofthe nonlinearity subspace) But how can we determine the values of u that willresult in such basis vectors in the presence of unknown parameters? Thisseemingly hopeless dilemma can be resolved by the following observation,which will be clari®ed further later on:

The expression (7.4) is not computable if and only if at least one of thevectors 2;t 1‡ 1;t and 2;t‡ 1;t‡1 is independent of the past values

meas-ured projections is equivalent to the knowledge that new independentdirections are being generated by the system

In other words, whenever our identi®cation process gets `stuck', that is, thesystem does not generate new directions over the next few steps, then theprojection information we have already acquired is enough for us to compute avalue of control which will get the system `unstuck' and will generate a newdirection after at most 2n (in this case 4) steps: this is the time it takes to changethe arguments of both 1 and 2 and measure the resulting projection

Orthogonalized projection estimation All the projection information of  isautomatically incorporated into the parameter estimate ^ produced by anorthogonalized projection algorithm This means that after the active identi-

®cation phase is complete, all the terms appearing in (7.10) and (7.11) can becomputed simply by replacing  by its estimate ^ This allows us to proceedwith the implementation of the controller (7.4) or any other control strategy as

if the parameters were known

Clearly, this two-stage process depends critically on the fact that, contrary totheir continuous-time counterparts, discrete-time nonlinear systems cannotexhibit ®nite escape times, as long as their nonlinearities take on ®nite valueswhenever their arguments are ®nite This property allows us to postponeclosing the loop with a controller until after the ®nite-duration identi®cationphase has been completed

7.3 Active identi®cation

Let us now elaborate further on the above outlined approach by presenting indetail its two most challenging ingredients, namely the pre-computationscheme and the input selection for active identi®cation To do this, we return

to the general output-feedback form (7.1) and rewrite it in the following scalarform:

will globally stabilize the system (7.1) and yield x1…t† ˆ yd…t† ; t  n.

Clearly, the implementation of the control law (7.15) requires us to calculate(at time t) the projection of the unknown  along the vectorPnkˆ1 k;t‡n k Thismeans that we need to compute the value ofPnkˆ1 k;t‡n kat time t Rewriting

of the states x1…t ‡ 1†; ; x1…t ‡ n 1† at time t To see how to calculate these

states, let us return to equation (7.14) and express x1…t ‡ 1†; ; x1…t ‡ n 1†

the vectorPnkˆ1 k;t‡1 kare known at time t, the key to successfully calculating(at time t) the value of x1…t ‡ 1† depends on whether we are able to compute

the projection of the unknown  along the vectorPnkˆ1 k;t‡1 kat time t.Next, let us examine what we need to calculate the value of x1…t ‡ 2† at time

t From (7.17), the value of x1…t ‡ 2† is equal to the sum of u…t n ‡ 2† and

TPn

u…t n ‡ 2† and TPn

acquired (at time t) The value of u…t n ‡ 2† is known at time t, while from

we see that the value ofPnkˆ1 k;t‡2 k depends on x1…t ‡ 1† This means that

pre-computing the value of x2…t ‡ 2† requires the values of both x1…t ‡ 1† and

TPn

calculation of x1…t ‡ 1† at time t requires us to compute (at time t) the value of

TPn

pre-compute (at time t) the values of

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