Adaptive Control System Part 7 pot

Unlike the state feedback case, the minimum-phase and therelative degree-one conditions are not sucient to achieve adaptive passivation full-if only the output feedback is allowed.. 6.4

Furthermore, a direct application of LaSalle's invariance principle ensures

that all the trajectories ……t†; x…t†; ^…t†† converge to the largest invariant set E contained in the manifold f…; x; ^† j…; x† ˆ …0; 0†g Therefore, x…t† tends to zero as t goes to 1 In other words, the cascade system (6.34) is globally

adaptively stabilized by (6.41) and (6.42) Finally, Proposition 3.1 ends theproof of Theorem 3.1

Remark 3.2 It is of interest to note that, if rank fg…0†…g1…0† ‡
g…0††g ˆ l ˆ

dim , then

lim

t!‡1 j^…t† j ˆ 0 Indeed, on the set E, we have g…0†…g1…0† ‡
g…0††… ^…t†† ˆ 0 which, in turn, implies that ^…t† ˆ  So, E ˆ f…0; 0; †g.

The following corollary is an immediate consequence of Theorem 3.1 wherethe -system in (6.34) is void

Corollary 3.1 Consider a linearly parametrized nonlinear system in the form

_x ˆ f …x† ‡
f …x† ‡ g…x†…u ‡
g…x††

If f f ; g; hg is strictly C 1-passive with a positive de®nite and proper storagefunction, and if (6.35) holds, then system (6.44) is strongly adaptively C1-feedback passive with a proper storage function V

6.3.2 Recursive adaptive passivation

In this section, we show that the adaptive passivation property can be

propagated via adding a feedback passive system with linearly appearingparameters This is indeed the design ingredient which was used in [19].More precisely, consider a multi-input multi-output nonlinear system of theform (6.26) with linear parametrization:

_ ˆ f10…† ‡ f1…† ‡ …G10…† ‡
G1…††y _z ˆ q…; z; y†

_y ˆ f20…x† ‡ f2…x† ‡ …G20…x† ‡
G2…x††u

…6:45†

with x ˆ …T; zT; yT†T,  2 Rn 0, z 2 Rn m and y, u 2 Rm Denote G2…x; † ˆ

G20…x† ‡
G2…x†.

Proposition 3.2 If the -subsystem of (6.45) with y considered as input is AFP,

if G2 is globally invertible for each , then the interconnected system (6.45) isalso AFP Furthermore, under the additional condition that the z-system is

BIBS (bounded-input bounded-state) stable and is GAS at z ˆ 0 whenever …; y† ˆ …0; 0†, if the -system has a UO-function V1 and a COCS-function 1associated with its AFP property, then the whole composite system (6.45) alsopossesses a UO-function V2 and a COCS-function 2 associated with its AFPproperty

Remark 3.3 Under the conditions of Proposition 3.2, it follows from

Theorem 2.2 that the …z; y†-system in (6.45) is feedback passive for every

frozen  and each 

Proof Introduce the extra integrator _^ ˆ 0where 0is a new input to be builtrecursively By assumption, there exist a smooth positive semide®nite function

V1…; ^†, smooth functions #1and 1as well as a nonnegative function 1and acontinuous function h1 such that the time derivative of the function

which implies h11…; ^; † is ane in  Then, there exist smooth functions ^h11

and
h11 such that

h11…; ^; † ˆ ^h11…; ^† ‡
h11…; ^†… ^† …6:49†

Consider the nonnegative functions

V2ˆ V1…; ^† ‡1

2jy #1…; ^†j2 …6:50†

V2ˆ V1‡1

In view of (6.47) and the de®nition of y and 1, the time derivative of V2 along

the solutions of (6.45) and _^ ˆ 0 satis®es

such that V2satis®es a di€erential dissipation inequality like (6.47)

To this end, set

On the basis of Corollary 3.1 and Proposition 3.1, a repeated application ofProposition 3.2 yields the following result on adaptive backsteppingstabilization.

Corollary 3.2 [26] Any system in strict-feedback form

_xiˆ x i‡1 ‡ i…x1; ; xi†; 1  i  n 1

_xnˆ u ‡ n…x1; ; xn† …6:66†

is globally adaptively (quadratically) stabilizable

6.3.3 Examples and extensions

We close this section by illustrating our adaptive passivation algorithm withthe help of cascade-interconnected controlled Dung equations Possibleextensions to the output-feedback case and nonlinear parametrization arebrie¯y discussed via two elementary examples

6.3.3.1 Controlled Cung equations

Consider an interconnected system which is composed of two (modi®ed)Dung equations in controlled form, i.e

x1‡ 1_x1‡ 1x1‡ 2x3

1 ˆ u1

x2‡ 2_x2‡ 3x2‡ 4x3

where 1; 2 > 0 are known parameters,  ˆ …1; 2; 3:4† is a fourth-order

vector of unknown constant parameters and u2 is the control input Theinterconnection constraint is given by

Obviously, the …z1; y1†-system in (6.69) is AFP by means of the change of

parameter update law and adaptive controller

_^1ˆ 1y1z1‡ 11; 1> 0_^2ˆ 2y1z3

1 is a UO-function for the …z1; y1†-system which satis®es

the di€erential dissipation equality

1…^1 1†2‡21

2…^2 2†2

It is easy to check that the conditions of Proposition 3.2 hold A directapplication of our adaptive passivation method in the proof of Proposition 3.2gives our passivity-aimed adaptive stabilizer for system (6.69), or the originalsystem (6.67):

_^1 ˆ 1…2y1‡ y2 ^1z1 ^2z3

1†z1

_^2 ˆ 2…2y1‡ y2 ^1z1 ^2z3

1†z3 1

_^3 ˆ 3…y1‡ y2 ^1z1 ^2z3

1†z2; 3> 0_^4 ˆ 4…y1‡ y2 ^1z1 ^2z3

6.3.3.2 Adaptive output feedback passivation

The adaptive passivation results presented in the previous sections rely on state feedback (6.31) In many practical situations, we often face systems whosestate variables are not accessible by the designer except the information of themeasured outputs Unlike the state feedback case, the minimum-phase and therelative degree-one conditions are not sucient to achieve adaptive passivation

full-if only the output feedback is allowed This is the case even in the context of(nonadaptive) output feedback passivation, as demonstrated in [40] using thefollowing example:

_z ˆ z3‡  _ ˆ z ‡ u

y ˆ 

…6:72†

It was shown in [40, p 67] that any linear output feedback u ˆ ky ‡ v, with

k > 0, cannot render the system (6.72) passive In fact, Byrnes and Isidori [1]proved that the system (6.72) is not stabilizable under any C1 output feedbacklaw As a consequence, this system is not feedback passive via any C1 outputfeedback law though it is feedback passive via a C1 state feedback law.However, system (6.72) can be made passive via the C0 output-feedbackgiven by

Forming the derivative of V with respect to the solutions of (6.72), usingYoung's inequality [8] gives

_z ˆ z3‡  _ ˆ z ‡ u ‡ '…y; †

y ˆ 

…6:76†

where  is a vector of unknown constant parameters Assume that thenonlinear function ' checks the following concavity-like condition

(C) For any y and any pair of parameters …1; 2†, we have

y'…y; 2† y'…y; 1†  y@'@…y; 2†…2 1† …6:77†

Non-trivial examples of ' verifying such a condition include all linear

parametrization (i.e '…y; † ˆ '1…y†) and some nonlinearly parametrized functions like '…y; † ˆ '2…y†…exp…'3…y†† ‡ '1…y†† where y'2…y†  0 for all

y 2 R.

Consider the augmented storage function

V ˆ V…z; † ‡1

2…^ †T 1…^ † …6:78†

where ^ is an update parameter to be preÂcised later

By virtue of (6.73) and (6.75), we have

6.4 Small gain-based adaptive control

Up to now, we have considered nonlinear systems with parametric uncertainty.The synthesis of global adaptive controllers was approached from an input/output viewpoint using passivation±a notion introduced in the recent literature

of nonlinear feedback stabilization The purpose of this section is to addressthe global adaptive control problem for a broader class of nonlinear systemswith various uncertainties including unknown parameters, time-varying andnonlinear disturbance and unmodelled dynamics Now, instead of passivationtools, we will invoke nonlinear small gain techniques which were developed inour recent papers [21, 20, 16], see references cited therein for other applications

6.4.1 Class of uncertain systems

The class of uncertain nonlinear systems to be controlled in this section isdescribed by

where u in R is the control input, y in R is the output, x ˆ …x1; ; xn† is the

measured portion of the state while z in Rn 0 is the unmeasured portion of thestate  in Rlis a vector of unknown constant parameters It is assumed that the

i's and q are unknown Lipschitz continuous functions but the 'i's are knownsmooth functions which are zero at zero

The following assumptions are made about the class of systems (6.82).

(A1) For each 1  i  n, there exist an unknown positive constant p 

i and twoknown nonnegative smooth functions i1, i2such that, for all …z; x; u; t†

j
i…x; z; u; t†j  p 

i i1…j…x1; ; xi†j† ‡ p 

i i2…jzj† …6:83†

Without loss of generality, assume that i2…0† ˆ 0.

(A2) The z-system with input x1 has an ISpS-Lyapunov function V0, that is,there exists a smooth positive de®nite and proper function V0…z† such

that

@V0

@z …z†q…t; z; x1†  0 0…jx1j† ‡ d0 8 …z; x1† …6:84†where 0 0are class K1-functions and d0is a nonnegative constant.The nominal model of (6.82) without unmeasured z-dynamics and externaldisturbances
i was referred to as a parametric-strict-feedback system in [26]and has been extensively studied by various authors±see the texts [26, 32] andreferences cited therein The robustness analysis has also been developed to aperturbed form of the parametric-strict-feedback system in recent years [48, 22,

31, 35, 51] Our class of uncertain systems allows the presence of moreuncertainties and recovers the uncertain nonlinear systems considered pre-viously within the context of global adaptive control

The theory developed in this section presupposes the knowledge of partial state information and the virtual control coecients Extensions to the cases ofoutput feedback and unknown virtual control coecients are possible at theexpense of more involved synthesis and analysis ± see, for instance, [17, 18] Anillustration is given in subsection 6.4.4 via a simple pendulum example

x-6.4.2 Adaptive controller design

to be chosen later, p  max fp?

i; p? 2

i j1  i  ng is an unknown constant and the

time-varying variables ^, ^p are introduced to diminish the e€ects of parametricuncertainties.

With the help of Assumption (A1), the time derivative of V1 along thesolutions of (6.82) satis®es:

1† is the value of the derivative of  at x2

1 In the sequel,  is chosensuch that 0 is nonzero over R‡

Since 11 is a smooth function and 11…jx1j† ˆ 11…0† ‡ jx1jR01 0

It is shown in the next subsection that a similar inequality to (6.94) holds for

each …x1; ; xi†-subsystem of (6.82), with i ˆ 2; ; n.

6.4.2.2 Recursive steps

Assume that, for a given 1  k < n, we have established the following property (6.95) for the …x1; ; xk†-subsystem of system (6.82) That is, for each

1  i  k, there exists a proper function Vi whose time derivative along thesolutions of (6.82) satis®es

In (6.95), "j> 0 …1  j  i† are arbitrary, cj > n j …2  j  i† are design

parameters, #j…1  j  i†, i and $i are smooth functions and the variables

The above property was established in the preceding subsection with k ˆ 1.

In the sequel, we prove that (6.95) holds for i ˆ k ‡ 1.

Consider the Lyapunov function candidate

‡Xk‡1

jˆ1 j2

…jzj†2…6:99† From (6.93) and (6.96), it is seen that …w1; ; wk; wk‡1 † ˆ …0; ; 0; 0† if and only if …x1; ; xk; xk‡1 † ˆ …0; ; 0; 0† Recall that  0 …x2† 6ˆ 0 by selection.

With this observation in hand, given any "k‡1> 0, lengthy but simplecalculations imply the existence of a smooth nonnegative function ^k‡1 suchthat

parametric-strict-feedback systems without unmodelled dynamics, introducethe notation

k‡1 ˆ k‡

'k‡1

By induction, at the last step where k ˆ n in (6.95), if we choose the

following parameter update laws and adaptive controller

i"n i‡1 …n i‡1†1 …0†2 …6:109†

As a major di€erence with most common adaptive backstepping design

procedures [26, 32], because of the presence of dynamic uncertainties z, we areunable to conclude any signi®cant stability property from the inequality(6.109) Another step is needed to robustify the obtained adaptive back-stepping controllers (6.106) and (6.107).

6.4.2.3 Small gain design step

The above design steps were devoted to the x-subsystem of (6.82) with zconsidered as the disturbance input The e€ect of unmeasured z-dynamics hasnot been taken into account in the synthesis of adaptive controllers (6.106) and(6.107) The goal of this section is to specify a subclass of adaptive controllers

in the form of (6.106), (6.107) so that the overall closed loop system isLagrange stable Furthermore, the output y can be driven to a small vicinity

of the origin if the control design parameters are chosen appropriately.First of all, the design function 1 as introduced in subsection 6.4.2.1 isselected to satisfy

i"n i‡1 …n i‡1†1 …0†2 …6:116†

Let vand vbe two class-K1 functions such that

v…jzj†  V0…z†  v…jzj† …6:117†

Given any 0 < "2< c, (6.114) ensures that

_

Vn max

2

Given any 0 < "4< 1, we obtain

1 "4



…6:127†

Under the above choice of the design functions  and 1, the stability properties

of the closed loop system (6.82), (6.106) and (6.107) will be analysed in the nextsubsection

6.4.3 Stability analysis

If we apply the above combined backstepping and small-gain approach to theplant (6.82), the stability properties of the resulting closed loop plant (6.82),(6.106) and (6.107) are summarized in the following theorem

Theorem 4.1 Under Assumptions (A1) and (A2), the solutions of the closedloop system are uniformly bounded In addition, if a bound on the unknownparameters p

i is available for controller design, the output y can be driven to

an arbitrarily small interval around the origin by appropriate choice of thedesign parameters

Proof Letting ~ ˆ ^  and ~p ˆ ^p p, it follows that Vnis a positive de®nite

and proper function in …x1; ; xn; ~; ~p† Also, _~ ˆ _^ and _~p ˆ _^p Decompose the

closed-loop system (6.82), (6.106) and (6.107) into two interconnected

sub-systems, one is the …x1; ; xn; ~; ~p†-subsystem and the other is the z-subsystem.

We will employ the Small Gain Theorem 2.5 to conclude the proof

Consider ®rst the …x1; ; xn; ~; ~p†-subsystem From (6.118) and (6.119), it

follows that a gain for this ISpS system with input ~V0and output Vnis given by

1…s† ˆc 2

21

 1 v

1

condition (6.23) as stated in Theorem 2.5 is satis®ed between 1and 2 Hence,

a direct application of Theorem 2.5 concludes that the solutions of theinterconnected system are uniformly bounded The second statement ofTheorem 4.1 can be proved by noticing that the drift constants in (6.119)and (6.127) can be made arbitrarily small

Remark 4.1 It is of interest to note that the adaptive regulation methodpresented in this section can be easily extended to the tracking case Roughlyspeaking, given a desired reference signal yr…t† whose derivatives y …i†r …t† of order

up to n are bounded, we can design an adaptive state feedback controller so

that the system output y…t† remains near the reference trajectory yr…t† after a

considerable period of time

Remark 4.2 Our control design procedure can be applied mutatis mutandis

to a broader class of block-strict-feedback systems [26] with nonlinear

-…x1; ; xi; 1; ; i 1; z† is considered as the input Similar conditions to

(6.83) are required on the disturbances
xi and
i

6.4.4 Examples and discussions

We demonstrate the e€ectiveness of our robust adaptive control algorithm bymeans of a simple pendulum with external disturbances Along the way, weshow that our combined backstepping and small gain control design procedurecan be extended to cover systems with unknown virtual control coecients.Moreover, we shall see that the consideration of dynamic uncertaintiesoccurring in our class of systems (6.82) becomes very natural when theoutput feedback control problem is addressed Then, in the subsection6.4.4.2, we compare the above adaptive design method with the dynamicnormalization-based adaptive scheme proposed in our recent contribution[18] via a second order nonlinear system

6.4.4.1 Pendulum example

The following simple pendulum model has been used to illustrate severalnonlinear feedback designs (see, e.g., [3,24]):

ml  ˆ mg sin  kl _ ‡1lu ‡
0…t† …6:131† where u 2 R is the torque applied to the pendulum,  2 R is the anticlockwise

angle between the vertical axis through the pivot point and the rod, g is theacceleration due to gravity, and the constants k, l and m denote a coecient offriction, the length of the rod and the mass of the bob, respectively
0…t† is a time-varying disturbance such that j
0…t†j  a0for all t  0 It is assumed that

these constants k, l, m and a0 are unknown and that the angular velocity _ isnot measured

Using the adaptive regulation algorithm proposed in subsection 6.4.3, wewant to design an adaptive controller using angle-only so that the pendulum is

kept around any angle  <  ˆ 0  .