Adaptive Control System Part 11 potx

AssumptionA1 is perhaps the most restrictive of all assumptions, and is made here toaccomplish the ®rst step in the design of stable adaptive NLP-systems.Our objective is to construct th

for measurement, p; i2 R, and 2 Rl are unknown parameters.

i: R‡ ! Rm, and ' : R‡ ! Rl are known and bounded functions of thestate variable Ap is nonlinear in p, and fi is nonlinear in both i and i Ourgoal is to ®nd an input u such that the closed loop system has globally boundedsolutions and so that Xptracks as closely as possible the state Xmof a referencemodel speci®ed in equation (9.2), where r is a bounded scalar input We makethe following assumptions regarding the plant and the model:

(A1) Xp…t† is accessible for measurement.

(A2) i2 i, where iX ‰min;i; max;i Š, and min;i and max;i are known; p is

unknown and lies in a known interval P ˆ pmin; pmax ‰ Š  R.

(A3) …t† and '…t† are known, bounded functions of the state variable Xp.

(A4) f is a known bounded function of its arguments

(A5) All elements of A…p† are known, continuous functions of the parameter p.

(A6) bmˆ bp where is an unknown scalar with a known sign and upper

bound on its modulus, jjmax

(A7) …A…p†; bp † is controllable for all values of p 2 P, with

A…p† ‡ bmgT…p† ˆ Am

where g is a known function of p

(A8) Am is an asymptotically stable matrix in Rnwith

det…sI Am† X Rm…s† ˆ …s ‡ k†R…s†; k > 0

Except for assumption (A1), the others are satis®ed in most dynamicsystems, and are made for the sake of analytical tractability Assumption(A2) is needed due to the nonlinearity in the parametrization Assumptions(A3)±(A5) are needed for analytical tractability Assumptions (A6) and (A7)are matching conditions that need to be satis®ed in LP-adaptive control aswell (A8) can be satis®ed without loss of generality in the choice of thereference model, and is needed to obtain a scalar error model Assumption(A1) is perhaps the most restrictive of all assumptions, and is made here toaccomplish the ®rst step in the design of stable adaptive NLP-systems.Our objective is to construct the control input, u, so that the error,

E ˆ Xp Xm, converges to zero asymptotically with the signals in the closedloop system remaining bounded The structure of the dynamic system inequation (9.1) and assumptions (A6) and (A7) imply that when , p, , and

controller in (9.42), with the actual parameters replaced by their estimatestogether with a gradient-rule for the adaptive law, will not suce We thereforepropose the following adaptive controller:

wi are the corresponding wi's that realize the min-max solutions in (9.51) and

(9.52), and jjmaxdenotes the maximum modulus of The stability property ofthis adaptive system is given in Theorem 4.1 below

Theorem 4.1 The closed loop adaptive system de®ned by the plant in (9.1), thereference model in (9.2) and (9.43)±(9.52) has globally bounded solutions if

^p…t0 † 2 P and ^i…t0† 2 i8 i In addition, lim t!1e"…t† ˆ 0.

Proof For the plant model in (9.1), the reference model in (9.2) and the

control law in (9.43), we obtain the error di€erential equation

which is very similar to the error model in (9.37) De®ning the tuning error, e",

as in (9.44), we obtain that the control law in (9.43), together with the adaptivelaws in (9.45)±(9.52) lead to the following Lyapunov function:

This follows from Corollary 4.2 by showing that _V  0 This leads to the

global boundedness of e", ~, ~p, ~ and ~i for i ˆ 1; ; m Hence ecis boundedand by Lemma 4.2, E is also bounded As a result, i; ' and the derivative _ecare bounded which, by Barbalat's lemma, implies that e" tends to zero.Theorem 4.1 assures stable adaptive control of NLP-systems of the form in(9.3) with convergence of the errors to within a desired precision " The proof

of boundedness follows using a key property of the proposed algorithm Thiscorresponds to that of the error model discussed in Section 9.4.1, which is given

by Lemma 3.2 As mentioned earlier, Lemma 3.2 is trivially satis®ed inadaptive control of LP-systems, where the inequality reduces to an equalityfor !0 determined with a gradient-rule and a0ˆ 0 For NLP-systems, an

inequality of the form of (9.18) needs to be satis®ed This in turn necessitatesthe reduction of the vector error di€erential equation in (9.53) to the scalarerror di€erential equation in (9.54)

We note that the tuning functions a

i and !

i in the adaptive controller have

to be chosen di€erently depending on whether f is concave/convex or not, sincethey are dictated by the solutions to the min±max problems in (9.51) and (9.52).The concavity/convexity considerably simpli®es the structure of these tuningfunctions and is given by Lemma 3.1 For general nonlinear parametrizations,

the solutions depend on the concave cover, and can be determined usingLemma 4.3.

Extensions of the result presented here are possible to the case when only ascalar output y is possible, and the transfer function from u to y has relativedegree one [10]

9.4.3 Adaptive observer

As mentioned earlier, the most restrictive assumption made to derive thestability result in Section 9.4.2 is (A1), where the states were assumed to beaccessible In order to relax assumption (A1), the structure of a suitableadaptive observer needs to be investigated In this section, we provide anadaptive observer for estimating unknown parameters that occur nonlinearlywhen the states are not accessible

The dynamic system under consideration is of the form

ypˆ W…s; p†u; W…s; p† ˆ

Pn

iˆ1bi…p†sn 1

sn‡Pniˆ1ai…p†sn i …9:56†

and the coecients ai…p† and bi…p† are general nonlinear functions of an

unknown scalar p We assume that

(A2-1) p lies in a known interval P ˆ pmin; pmax ‰ Š.

(A2-2) The plant in (9.56) is stable for all p 2 P.

(A2-3) ai and biare known continuous functions of p

It is well known [1] that the output of the plant, yp, in equation (9.56)satis®es a ®rst order equation given by

…9:60†

f …p† ˆ ch 0…p†; c…p†T; d0; d…p†TiT …9:61†

for some functions c0…
†, c…
†, d0…
†,and d…
†, which are linearly related to bi…
†

and ai…
†  in (9.58) and (9.59) is an …n 1†  …n 1† asymptotically stable matrix and …; k† is controllable.

Given the output description in equation (9.57), an obvious choice for anadaptive observer which will allow the on-line estimation of the nonlinear

parameter p, and hence ai's and bi's in (9.56), is given by

The stability properties are summarized in Theorem 4.2 below:

Theorem 4.2 For the linear system with nonlinear parametrization given in(9.56), under the assumptions (A2-1)±(A2-3), together with the identi®cationmodel in (9.62), the update law in (9.64) ensures that our parameter estimation

problem has bounded errors in ~p if ^p…t0 † 2 P In addition, lim t!1e"…t† ˆ 0.

Proof The proof is omitted since it follows along the same lines as that ofTheorem 4.1

We note that as in Section 9.4.2, the choices of a0 and !0 are di€erentdepending on the nature of f When f is concave or convex, these functions aresimpler and easier to compute, and are given by Lemma 3.1 For generalnonlinear parametrizations, these solutions depend on the concave cover and

as can be seen from Lemma 3.4, are more complex to determine

9.5 Applications

9.5.1 Application to a low-velocity friction model

Friction models have been the focus of a number of studies from the time ofLeonardo Da Vinci Several parametric models have been suggested in theliterature to quantify the nonlinear relationship between the di€erent types offrictional force and velocities One such model, proposed in [13] is of the form

F ˆ FC sgn… _x† ‡ …FS FC† sgn… _x†e …v_xs†2‡ F

where x is the angular position of the motor shaft, F is the frictional force, FCrepresents the Coulomb friction, FS stands for static friction, Fvis the viscousfriction coecient, and vs is the Stribeck parameter Another steady statefriction model, proposed in [14] is of the form :

F ˆ FC sgn… _x† ‡ sgn… _x†…FS FC†

1 ‡ … _x=vs †2 ‡ Fv_x …9:66†

Equations (9.65) and (9.66) show that while the parameters FC; FS and Fvappear linearly, vs appears nonlinearly As pointed out in [14], these param-eters, including vs, depend on a number of operating conditions such aslubricant viscosity, contact geometry, surface ®nish and material properties.Frictional loading, usage, and environmental conditions introduce uncertain-ties in these parameters, and as a result these parameters have to be estimated.This naturally motivates adaptive control in the presence of linear andnonlinear parametrization The algorithm suggested in Section 9.4.2 in thischapter is therefore apt for the adaptive control of machines with suchnonlinear friction dynamics

In this section, we consider position control of a single mass system in thepresence of frictional force F modelled as in equation (9.65) The underlyingequations of motion can be written as

where f … _x; † is convex for all _x > 0 and concave for all _x < 0 We choose a

where  and !nare positive values suitably chosen for the application problem

It therefore follows that a control input given by

with a0 and !0 corresponding to the min±max solutions when sign…e"† f is

concave/convex, suce to establish asymptotic tracking

We now illustrate through numerical simulations the performance that can

be obtained using such an adaptive controller We also compare its ance with other linear adaptive and nonlinear ®xed controllers In all thesimulations, the actual values of the parameters were chosen to be

k ˆ 1, and adaptive laws as in equations (9.72)±(9.74) with " = 0.0001 ^…0†

was set to 1370 corresponding to an initial estimate of ^vs= 0.027 m/s, which is50% larger than the actual value Figure 9.5 illustrates the tracking error,

e ˆ x xm,the control input, u, and the error in the frictional force,

eF ˆ F F, where F is given by (9.65) and ^^ F is computed from (9.65) byreplacing the true parameters with the estimated values e, u and eF aredisplayed both over [0, 6 min] and [214 min, 220 min] to illustrate the nature

of the convergence We note that the position error converges to about

5  10 5rad, which is of the order of ", and eF to about 5  10 3N Thediscontinuity in u is due to the signum function in f in (9.68)

Simulation 2 To better evaluate our controller's performance, we simulatedanother adaptive controller where the Stribeck e€ect is entirely neglected in thefriction compensation That is

as well as for T ˆ [214 min, 220 min] As can be observed, the maximum

position error does not decrease beyond 0.01 rad It is worth noting that thecontrol input in Figure 9.6 is similar to that in Figure 9.5 and of comparable

magnitude showing that our min±max algorithm does not have any tinuities nor is it of a `high gain' nature Note also that the error, eF does notdecrease beyond 0.5 N.

discon-Simulation 3 In an attempt to avoid estimating the nonlinear parameters vs,

in [15], a friction model which is linear-in-the-parameters was proposed Thefrictional force is estimated in [15] as

^

F ˆ ^FCsgn… _x† ‡ ^FSj _xj1=2sgn… _x† ‡ ^Fv_x …9:79†

with the argument that the square-root-velocity term can be used to closelymatch the friction-velocity curves and linear adaptive estimation methodssimilar to Simulation 2 were used to derive closed loop control The resulting

performance using such a friction estimate and the control input in equation(9.78) is shown in Figure 9.7 which illustrates the system variables e, u and eF

for T ˆ [0, 6 min] and for T ˆ [360 min, 366 min] Though the tracking error

remains bounded, its magnitude is much larger than those in Figure 9.5obtained using our controller

9.5.2 Stirred tank reactors (STRs)

Stirred tank reactors (STRs) are liquid medium chemical reactors of constantvolume which are continuously stirred Stirring drives the reactor medium to auniform concentration of reactants, products and temperature The stabiliza-tion of STRs to a ®xed operating temperature proves to be dicult because a

few physical parameters of the chemical reaction can dramatically alter thereaction dynamics De®ning X1 and X2 as the concentration of reactant andproduct in the in¯ow, respectively, r as the reaction rate, T as the temperature,

h as the reaction heat released during an exothermic reaction, d as thevolumetric ¯ow into the tank, T ˆ T Tamb, it can be shown that threedi€erent energy exchanges a€ect the dynamics of X1 [16]: (i) conductive heatloss with the environment at ambient temperature, Tamb, with a thermal heattransfer coecient, e,(ii) temperature di€erences between the in¯ow andout¯ow which are Tamb and T respectively, (iii) a heat input, u, which acts

as a control input and allows the addition of more heat into the system Thisleads to a dynamic model

_X1ˆ 0exp T ‡ T1

amb

X1‡ d…X1in X1†

_X2ˆ 0exp T ‡ T1

amb

X1 dX2_

oper, we can state the problem as thetracking of the output T

m of a ®rst order model, speci®ed as_

Driving and regulating an STR to an operating temperature is confounded

by uncertainties in the reaction kinetics Speci®cally, a poor knowledge of theconstants, 0 and 1, in Arrhenius' law, the reaction heat, h, and thermal heattransfer coecient, e, makes accurately predicting reaction rates nearlyimpossible To overcome this problem, an adaptive controller where 0, 1, hand e are unknown may be necessary

9.5.2.1 Adaptive control based on nonlinear parametrization

The applicability of the adaptive controller discussed in Section 9.4.2 becomesapparent with the following de®nitions

function of  (It is linear in 1ˆ 0and exponential in 2ˆ 1 If reaction heat,

h, is unknown, it may be incorporated in 1, i.e 1ˆ h0.) Since 0, 1and h areunknown constants within known bounds, assumption (A2) is satis®ed.Assumption (A1) is satis®ed since the temperatures are measurable Thesystem state, T ˆ , complies with (A3) Furthermore f is smooth and

di€erentiable with known bounds and hence (A4) is satis®ed Finally, (A6)±

(A8) are met due to the choice of the model as in (9.81) for ˆ k q

 Since e isunknown, is unknown and is therefore estimated Since the plant is ®rstorder, the composite error ec is given by ecˆ Te Referring to the adaptivecontroller outlined in Section 9.4.2, the control input and adaptation laws are

9.5.3 Magnetic bearing system

Magnetic bearings are currently used in various applications such as machinetool spindle, turbo machinery, robotic devices, and many other contact-freeactuators Such bearings have been observed to be considerably superior tomechanical bearings in many of these applications The fact that the underlyingelectromagnetic ®elds are highly nonlinear with open loop unstable poses achallenging problem in dynamic modelling, analysis and control As a result,controllers based on linearized dynamic models may not be suitable forapplications where high rotational speed during the operation is desired Yetanother feature in magnetic bearings is the fact that the air gap, which is anunderlying physical parameter, appears nonlinearly in the dynamic model Due

to thermal expansion e€ects, there are uncertainties associated with thisparameter The fact that dynamic models of magnetic bearings include non-linear dynamics as well as nonlinear parametrizations suggests that an adaptivecontroller is needed which employs prior knowledge about these nonlinearitiesand uses an appropriate estimation scheme for the unknown nonlinearparameters.

To illustrate the presence of nonlinear parametrization, we focus on aspeci®c system which employs magnetic bearings which is a magneticallylevitated turbo pump [5] The rotor is spun through an electric motor, and

to actively position the rotor, a bias current i0 is applied to both upper andlower magnets and an input u is to be determined by the control strategy For amagnetic bearing system where rotor mass is M operating in a gravity ®eld g,the rotor dynamics is represented by a second order di€erential equation of theform

z g ˆ f1 …h0; ; z† ‡ f2…h0; ; z†u ‡ f3…h0; ; z†u 2; juj < 2i0 …9:83†

9.5.3.1 Adaptive control based on nonlinear parametrization

By examining equation (9.83), it is apparent that the parameter h0 occursnonlinearly while occurs linearly An examination of the functions f1, f2u,and f3u2 further reveals their concavity/convexity property and are

9.5.2.1 Adaptive control based on nonlinear parametrization

The applicability of the adaptive controller... the maximum modulus of The stability property ofthis adaptive system is given in Theorem 4.1 below

Theorem 4.1 The closed loop adaptive system de®ned by the plant in (9.1), thereference model... Lemma 3.2 is trivially satis®ed inadaptive control of LP-systems, where the inequality reduces to an equalityfor !0 determined with a gradient-rule and a0ˆ For NLP-systems, an

inequality