Kiểm soát và ổn định thích ứng dự toán cho các hệ thống phi tuyến P11 pot

This is advantageous in practice because the tools available to design output-feedback controllers without a separation principle are rather limited and work for restricted classes of no

First, for systems with parametric uncertainties, we will provide a brief account of the adaptive output-feedback backstepping technique developed

by Kanellakopoulos, Kokotovi6, and Morse (for more details, the reader is referred to [115]) Next, we will show how to employ the separation principle previously developed to extend the techniques seen in Chapters ? and 8 to the output feedback framework Recall that, in the spirit of a separation principle, one seeks to construct output-feedback controllers recovering the performance of given state-feedback controllers This is advantageous in practice because the tools available to design output-feedback controllers without a separation principle are rather limited and work for restricted classes of nonlinear systems (see, e.g., Section 10.3) while, by separating the state estimation from the control design phase, one can exploit available

&ate feedback design tools for quite general cla.sses of systems On the other hand, the implementation of a state estimator requires some knowledge about the plant and, in the adaptive control case, restricts the class of uncertainties affecting the system These issues play a major role now and will somewhat limit the generality of the results illustrated in the previous chapter

As we did earlier, we will separately study stabilization and tracking and, for the latter problem, we will once again make the distinction between

363

Jeffrey T Spooner, Manfredi Maggiore, Ra´ul Ord´o˜nez, Kevin M Passino

Copyright  2002 John Wiley & Sons, Inc ISBNs: 0-471-41546-4 (Hardback); 0-471-22113-9 (Electronic)

364

a, full information setting, where the state of the system and the error performance measure e are available for feedback, and a partial information setting, where both =I; and e are not directly measurable

11.2 Control of Systems in Adaptive Tracking Form

In this section we consider systems in output-feedback form (10.5) affected

by a parametric uncertainty Specifically, we assume that the output- dependent nonlinearities gj(y) are unknown, but they can be exactly rep- resented by a linear in the parameter approximator with no representation error In other words, we consider the class of systems

La?1 = 22 + 90,l (Y> + 5 ojSj,l (Y)

xr = Xr+l + So,r(Y) + C@j~j,r(y) + dm~(y)u

j=l

x* = @dY) + 2 @g&y) + doa(y)u

j=l

Y = Xl,

where each gi,j and a(y) are locally Lipschitz functions, gi,j(O) = 0, g(y) #

0 for all y E R, and r = ~2 - m is the relative degree of the system The vector 8 E RP and the scalars di, i = 1, m are unknown and such that the polynomial p(s) = d,sm + + dls + do is Hurwitz, thus implying that the zero dynamics of the system are globally exponentially stable We a.ssume that the sign of dm is known and, without loss of generality, we assume that dm > 0

We seek to find an adaptive controller that, despite the presence of the unknown parameters 8, and di, makes y asymptotically track a smooth reference trajectory r(t) while guaranteeing boundedness of all internal vari-

ables ForOQLAet

I 0

so that (11.1) can be rewritten in vector form as

ci = Az + go(y) + k6$gj(y) + da(y)u

j=l

y = cx

1

0 -1

7

of one of the available techniques which employs the so-called K-filters to solve the tracking problem

The main idea behind the design is to replace the state of the sys- tem, which is not measurable, by a o-dependent estimate and employ the certainty equivalence principle to design an adaptive controller If 8 were known, then we could use the observer

i = A? + go(y) + 2 Ojgj(y) + da(y)u + L(y - Et)

j=l

(11.4)

so that Z = x - 2 decays as & = (A - LC)Z where L is chosen so that

(A - LC) is Hurwitz K-filters help defining a state estimate which, when 0

is known, converges exponentially to the actual state Consider the filters

6 = (A - LC>lj + Sj(Y), 1 I j 5 p

vj = (A - LC)vj + En-ja(y)u, 0 < j < - _ m,

(11.5)

where L = [Zl, , EJT is a vector chosen so that (A - LC) is Hurwitz (L

exists because (A, C) is an observable pair), and Ei denotes the 6th basis

vector for R” When 6 is known, the state of the system can be expressed

ii = (A - LC)k (11.6) The filters in (11.5) are used in our analysis to replace the unknown state of the system by known quantities and using the certainty equivalence princi- ple to replace the unknown 8 in (11.5) by an estimate 8 Similarly to what

we did in Section 10.3, nonlinear damping will be employed to account for the presence of the estimation error J;

el subsystem Let el = 21 - r and consider the subsystem defined by

'l = x2 +gO,l(Y)+ kgj,l($/) -7;

j=l

Note that x2 is not available for feedback and hence cannot be chosen as the virtual control input In order to define a suitable virtual control input, express x2 in terms of the states of the filters in (11.5) and rewrite &I as

T

Wl = [w + clel + to,2 + 90,l - i-,61,2 + gl,l, - - - ,tp,2 + Qp,l,

and rewrite (11.8) as

el = el - clel + 22 + d, [w,T@ + vm,J (11.9j

Since 6l is not known, we use the certainty equivalence principle to define

a stabilizing function for the virtual control urn,2 which is used to define the second component of the error system

Vl (Xl) = -wp, e2 = urn,2 - Vl (21)

(11.10)

X1 =

[ Y,~O:~I,.-.,~~,VO, ,Vrn,~‘,r,~ 1 -

Recall that L has been chosen so that the matrix A - LC is Hurwitz and let

P be the positive definite and symmetric solution of the Lyapunov equation

P(A - LC) + (A - LC)‘P = -I Consider now the function

Vl = tef + +(*l - J’)Tr-l((jl - G) + -5-P& (11.11)

Cl where cr > 0 and I’ is any positive definite symmetric matrix, and calculate its time derivative

Jkl = -neT - cle: + el& + dmel

(11.15)

and choose uum,3 as the virtual control input As we did earlier, in order

to use the certainty equivalence principle we isolate the terms containing unknown para.meters and we define a suitable regressor vector To this end, let

8” = [& , , &do , , dmlT

T- 8U

w2 - - -g [r1,2 + SdYL - * - 7 cp,2 + $41 (Y), vo,2, - - - ) %7,,2] -

and rewrite (11.15) as

62 = urn,3 - l2Vm,l - $$ [CO,2 + 90,1(y) + 521 + W,TQ2

- 2 PC0 + So(Y) + L(Y - Go)]

The choice of the stabilization function

u&2) = -kce2 - c2 au1

( >

2 e2 - dmel + kv,,1 + au1

ei subsystem ( 1 < i < T ) The procedure a,bove is iterated r times

At step i, we write the dynamics of ei = vm,i - vi-i(zi-I), where xi-1 =

[G-2) r liel), Ji-l], as

ei = Vnx,i+l - bI,,l - ay dUi-.1 [to,2 + go,1 (Y> + 3321 + w’e2

- ?qp [Ato + go(Y) + L(Y - Clo)]

* of3Ui-l

-

x - KA - LCE.j + Sj(Y)l j-1 - aeii

-

(11.23) where ztm,i+i is the virtual control input and

WT

dU&l

Using the certainty equivalence principle, we approximate O2 by fii and define the i-th stabilizing function

dui-1 ( )

2 Y&i) = -tcei - ci ei - ei-1 + ziV,7a,l +

T/i = E-1 + Jjef + i(S2 - @)TI+1(t32 - ei> + -iz’PZ 1 (11.25)

ci

By letting ei+l = vm,i+l - vi(zi) we get

i n / i-l * \ /n \ 2 i/i< -KC+ "

j=l dvi _

-22ei + eiWtT(6” - 6”) + (e2 - &)Tr-l(ji

( >

2 itJ:2ei < Ci - IL&l”

whose time derivative contains now the control input u

We conclude this section by pointing out that, despite its simplicity, this method suffers from the over-parameterization problem: at each step

of the backstepping design, dynamics of order p + m + 1 for updating @ have been introduced, and thus the total dynamic order of the controller is

~(p + m + 1)) while the number of unknown parameters is only p + m + 1

To overcome this problem, a more advanced adaptive control design exists which employs the so-called “tuning functions.” The reader is referred to [115] for a complete treatment of this topic

11.3 Separation Principle for Adaptive Stabilization

While adaptive output-feedback backstepping provides a systematic tool

to achieve global stabilization and tracking by output feedback, the class

of systems for which the technique can be applied is quite restrictive since the nonlinearities and uncertainty may only depend upon the output y

On the other hand, an interesting feature of the output-feedback control techniques introduced in Chapter 10 is the generality of the class of systems taken in consideration Here, we concentrate on the stabilization problem and we seek to exploit the generality of the tools in Chapter 10 to extend the results of Chapters 7 and 8 to the output-feedback framework

Consider the nonlinear system

(11.33)

where n(t, x) represents either a modeling uncertainty or a time-varying disturbance Suppose that, using the tools of Chapters 7 and 8, we have designed an adaptive controller for (11.33)

(11.34)

where Y, represents the adaptive control law and f~ is the parameter update law The controller may have been created using either a direct or indirect a,pproach It may also be defined using an approximator such as a fuzzy system or neural network We will require, however, that the controller

is Lipschitz continuous in x As in the non-adaptive output-feedback case, this will allow us to study the stability of the closed-loop system when some

of the states are replaced by their estimates Since we are dealing with adaptive stabilization for the moment (rather than tracking), the vector x containing the measurable signals is simply given by x Therefore, for the sake of clarity, throughout the rest of this section we will replace z by x Letting 8 = 6 - 8 represent, as usual, the parameter estimation error,

we assume that (11.34) has the property that there exists a continuously differentiable Lyapunov-like function Va (II:, e”) such that

where $(lxI), $(@I), i = 1,2, are class-K,, and B, is the compact set where the approximator is assumed to be valid (see Chapters 7 and 8) The reader will notice that the adaptive controllers developed in Chapter 7 and Chapter 8 enjoy the properties in (11.35) Furthermore, as pointed out there, the second inequality in (11.35) implies that if k2 is small enough the tra.jectories of the closed-loop system are uniformly ultimately bounded (UUB) and asymptotically approach the compact set

(dB, denotes the boundary of the set BY) so that {Z E R", 6 E RP : V, 5

(-YZ} is the largest level set of V, guaranteeing, for all 6, that x E B, Notice tha(t when V, > ~1, we find & 2 0 by (11.35) Suppose k2 is small enough that ~1 < CQ, and let c be any scalar satisfying 1 < c 5 a&i From the

definition of a2 we have that cai 5 V, 5 a2 implies Va < -kz(c - 1) < 0,

thus implying that the closed-loop trajectories (z(t),@t)) enter the set

1 x E R”,e E RP : Va(x,8) 5 cal

is positive definite with respect to the set 0, proper on the set {Va 5 a&

and such that its time derivative is negative on the set {cyi < V, < az> In other words, we want to have

(i) Vi = 0 * [xT,tiT]’ EN

(ii) Vi = co on {Va = a~}

(iii) V,l < 0 on {ai < V, < a~}

Choose

v; = max{O, V, - al>

which clearly satisfies (i) and (ii) above Now calculate the time derivative

of Vi on the set {ai < V, < CQ}

‘rl T;, = av,l * F v, = a2 - a1 (-ilclv, + k2) < 0,

and thus Vi satisfies (iii) above Compare now the properties of VL just de- fined to those of V in (lO.llZ), (10.113) and observe that the only difference between the two is that Vd is continuous but not continuously differentiable

It is easy to prove that the following function

where WA- (x, S) is a positive definite function with respect to N = {Va 5

(~1) a,nd proper on 2) = {Va < CQ}

In the spirit of the separation principle introduced in Chapter 10 we now try to find an adaptive output-feedback controller that relies on the

estimation of x to recover the performance of (11.34) When using output feedback, the adaptive stabilization problem becomes more complex and more restrictive assumptions than the state feedback case are needed In what follows we assume that (11.33) has special structures which will allow

us to define a separation principle for adaptive systems

11.3.1 Full State-Feedback Performance Recovery

Partition the state of the plant in two parts, z = [xiT, xzTIT, where ~1 f

[ 1 [ h1(x1,x2,4 Y2 52 1 7 (11.40) where yi E Rp’ , so that y E RP, and p = pi + n2 The components of the vector yi will be denoted by yr,i, i = 1, , pr , and a similar notation will be used, when needed, for other vectors Assume further that the observability mapping associated with the upper subsystem is a diffeomorphism and does not depend on the uncertainty A, that is,

p1, , u1 7 - - * 7 Urn7 -tUrn (fim-1) ] T E RnU, Cf& ki

= ril, n, 2 fii + + am, 0 5 fii 5 max{kr , , t;,}, and the functions ‘pi are defined as in (10.81)

Example 11.1 For single-input single-output systems, the situation de- scribed above arises when, for instance, the plant is described by a

differential equation in input-output form of the type

the input-output system is transformed into the state-space represen- tation

&,l = x1,2

&,TQ = f(XlJJ2,Udv

?2,1 = x2.2

(11.44)

i2,n2 = u, which has the form (11.39) Notice that since the control input u is known, the state 22 of the lower subsystem is available for feedback

as it can be obtained by using a chain of ~22 - 1 integrators Moreover, the observability mapping of the upper subsystem is simply given by the identity mapping

ye,1 = %(x1) = [Xl 7 1, - - - ,XlrnJT7 which is independent of the disturbance A and is clearly a diffeo- morphism Thus, the input-output system satisfies the assumptions

With the assumptions above and the definition of T/a’ we can use the ideas developed in Section 10.7 to estimate xi from yi and hence recover the performance of full state feedback controllers (recall that we assumed x2 to be available for feedback) Let A0 denote the known nominal value

of the vector A, possibly dependent on xi and 22 (if we have no CL priori knowledge about A, then we can set A0 = 0)) and given any scalar c > 0, let 0, denote the level set (I/d 5 c} and notice that 0, is compact In a’nalogy to what was done in Section 10.7, let Qzl denote the projection of the set 0, onto Rnl defined as

x1 -

%- - x1 E Rn’ : [x:,x,‘, gTIT E Q,, for all x2 E Rn2 and all e E RP

> (11.45)

Adaptive Output-Feedback Control

Using a separation principle and the fact that ~2 is directly available for feedback, we can now employ the estimate of ~t;r in the state-feedback adap- tive controller (11.34) to define an adaptive output-feedback controller To this end, recall the definition of Qzs in Section 10.6, pick any two positive scalars cl < c2 and a convex compact set C satisfying

of L, K, A, C, Ixl, &“), just recall that its performance is governed by the choice of two design parameters, 71 and 72 In preparation for the stability analysis, let x, = [x~,xZ,~]~, 2: = [$‘tx2,8]T, and

so that the closed-loop system using state feedback can be rewritten as

and the closed-loop system using the output-feedback controller (11.47) reads as

2, = I3(x&, u&q), A) (11.49)

Observe now that ~(x~,x~,u,(x&~) = F(x,,v,(x,), A) and that, by the Lipschitz continuity of fr , f”, fs, and v,, there exists a positive scalar 3/ such that, for all xn in 0,, and ?F E Xl’(C ),

Furthermore, note that the gradient of VL is bounded on any compact set and let A denote the upper bound on its norm over the compact set 0,,, that is, for all x, in 0,,,

8V;

I I K < Alxal -

cl

We are now ready to state the closed-loop stability theorem

Theorem 11.1: Consider the closed-loop system formed by the plant (I 1.39) and the adaptive output-feedback controller (1 I 46), (11.4 7) Then, given any triple (cl, ~2, e) such that 0 < cl < c2 and 0 < E < 73 0

$(q)/A7, there exist scalars r$ E (0, l] and T& E (0, l] such that, for

cl} is bounded within n,, = (Vi 5 cz> and asymptotically approaches the positively invariant set L!d,, where d, = y2 o yyl (A 3/ 6)

Proof: The logic of this proof is the same of that in Lemma 10.2 From the assumptions made on the observability mapping 3cr associated with the xi subsystem, we have that (11.46) is an estimator for x1 enjoying the properties of Theorem 10.3 and Lemma 10.4 As in the proof of Lemma 10.2, notice that Vj(x,(O)) < cr < c2 and there exists a positive exit time from the set a,, Since $-is constrained to lie inside ?-&’ (C ), ]?r] has

an upper bound which does not depend on qr and r72, and hence there exists a uniform upper bound 5Yr to the exit time which is independent

of qr and 55, and such that VL(x, (t)) < ~2, for all - t E [O, 571) and for all 17l,r/2 E (OJj Ch oose To such that 0 < 57’ < Tl Then, in the time interval

t E [0, Tl), we can apply Theorem 10.3, part (ii), and conclude that for any positive E: there exist q$, r/G E (0, l] such that, for all vi E (0, $1 and all

772 E (O,v$l, 1%; -xi] 5 c,Vt E [To,Tl) Hence, for all t E [To,Tl), we have that Vi(x,(t)) < c2 and #‘(t) - xi (t)l 5 E It now remains to show that this latter fact guarantees tha,t the domain R,, is positively invariant and, hence, Tl = co To this end, take the time derivative of Vj(xa) along the

which is strictly negative, we have that the set 52,, is positiv;ly invariant, and hence Tl = 00 and the closed-loop trajectories x&t) are bounded Furthermore, from the last inequality in (11.50) we conclude that the tra- jectories x,(t) asymptotically approach the set {Vi < d, >, thus concluding

From the fact that the constant d, can be made arbitrarily small, we have that Theorem 11.1 proves practical stability of the set Ji\/ (defined

by (11.36)) un er output-feedback control Recall that, when cl + 00, we d find f&, 7 the estimate of the domain of attraction for ode, coincides with the set {Va _< az}, the estimate of the domain of attraction for {V, 5 al}

using state feedback Hence, by choosing the estimator design parameters

71 and r]z sufficiently small, the adaptive output-feedback controller (11.47) recovers the performance of the adaptive state-feedback controller

Example 11.2 An important class of nonlinear systems satisfying the assumptions of Theorem 11.1 is given by systems in adaptive tracking form (11.1) In order to see that, let u’ = a(y)u and note that (11.1) can be rewritten in input-output form as

(11.51) where &, 0 < i < p, are smooth functions - - The proof of this fact

is rather straightforward and is left as an exercise Note now that (11.51) is a particular type of system in input-output form (11.42) and therefore it can be represented in the form (11.39), where n is simply a vector of unknown constant parameters,

and

il,, = ‘@0(X1> + 27/)i(Xl)Qi + mc diXa,i+l + d,,,I/’

x2,1 = x2.2

where U denotes the m-th time derivative of u’ By adding a chain

of m integrators at the input side we have that U is our new control input and 22 is available for feedback In order to solve the adaptive state-feedback stabilization problem, it is sufficient to drive the state

of the upper subsystem to the origin The exponential sta,bility of the zero dynamics of (11.52) guarantees then that the state of the lower subsystem is bounded and converges to the origin as well

Next, we will show that we can use the tools of Chapter 7 to develop

an adaptive controller for the upper subsystem in (11.52) Since we are considering a stabilization problem, we take e = xi and we rewrite the upper subsystem as

e = a(t,x1)+ pu, (11.53) where the time dependence in a comes from the presence of x2(t), viewed as an external measurable signal Assumption 7.1 is satisfied with k2 = 0 since, if n was known, a static stabilizing controller would

be given by

’ u,(x,A) 52 -dr $0(x1) + k$i(xl)@i + mc dixz,i+l + kTxl )

m

where k E R” is any vector such that the polynomial sn + knSn-’ + + Icr is Hurwitz Notice that, when U = v,, the upper subsystem becomes linear

il = AOX

and A0 is a Hurwitz matrix with eigenvalues given by the roots of

sn + knsnwl + + kl Hence, Vs = x[Pxi, where P is the solution

of

A,TP + PA0 = -I,

is a Lyapunov function for the closed-loop system Further, Assump- tion 7.2 is satisfied with c = l/d, and p = [0, ,O, llT Next, we

where q > 0,

ip(Xl,XZ) = - [~~(x~)+JCT51,~1(xl),.~~,lllp(xl~rx~~1~~~~ I X2,m T 7

I and 8 is the estimate of the vector

Now defining an adaptive output-feedback controller using the re- sults of this section is a rather straightforward task and is left as an exercise Here, we just highlight the fact that the output-feedback controller has the property the the origin of the closed-loop system is semiglobally practically stable

The example above shows that the tools developed in this section and Chapter 7 can be applied instead of the adaptive output-feedback back- stepping technique seen earlier for the stabilization of systems in adap- tive tracking form The control design here is simpler than the recursive backstepping design, but the stability result is slightly weaker (semiglobal practical stability rather than global asymptotic stability) An advantage

of Theorem 11.1, besides the design simplicity, is that it deals with more general classes of nonlinear systems

11.3.2 Partial State-Feedback Performance Recovery

In this section we consider the situation, which frequently arises in nonlin- ear control, when the adaptive state-feedback controller uses partial state feedback to achieve a given control objective Suppose that the plant can

be partitioned as the interconnection of two subsystems

7 ,Yp7- ,Y~p l) 1 T = %(x1,$ (11.57) where ki and s are defined in the usual way

Example 11.3 For single-input single-output systems, the situation de- scribed above arises when, for instance, the system is described by differential equations with a part in lower-triangular form

(51 = fl Kl > + 91 (Cl )&?

(5 7x1-1 = f rt-1 (6 17 7 Jnl-1) +972, -l (Il7~~~rtn~ l)trl (5 121 = fnl (C) + Qnl-1 (t) (Al(t) + nl(E)u%)

f2(5,x242(0)

Y = (5,

(11.58) which, letting xr = [[I, ,&JT, has the form (11.56) The func- tions gi are bounded away from zero, the unknown functions & (xl), A2(Xl), Al are assumed to be smooth, and the x2-subsystem is assumed to be input-to-state practically stable with respect to x1 The class of systems represented by (11.58) is similar to that con- sidered in Section 8.3.2 (there, however, the x2 subsystem was not