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A Backstepping-like Approach to Adaptive Control in Discrete Time
European Journal of Control (2000)6:298-321 © 2000 EUCA
European Journal of
Control
A Backstepping-like Approach to Adaptive Control in Discrete Time S. Monaco 1, D. Normand-Cyrot 2 ,* and A. Madani 2 lDipartimento di Informatica & Sistemistica, Universita di Roma "La Sapienza", Via Eudossiana 18,00184 Roma, Italy; 2Laboratoire des Signaux et Systemes, CNRS, Supelec Plateau de Moulon, 91192 Gif sur Yvette, France
Nonlinear adaptive control in discrete time is still largely open. The existing results deal with particular cases and are mainly based upon traditional estimation techniques issued from the linear literature. In this paper, combining a backstepping strategy and "a posteriori" estimation concepts, a novel approach for achieving adaptive stabilization is proposed. It can directly be applied to a class of nonlinear discrete-time systems that, analogous to the continuous-time context, are said to be in a parametric strict-feedback form. Necessary and sufficient geometric conditions guaranteeing the equivalence of a general discrete-time dynamics to the parametric strict-feedback form or to some related classes of dynamics are also proposed. Keywords: Adaptive control; Lyapunov techniques; Nonlinear discrete-time dynamics
1. Introduction A great deal of progress was made during the last decade in the field of nonlinear adaptive control in continuous time [12,13,18,22] whereas the problem is still widely open in discrete time where the available results are essentially based upon traditional linear estimation methods [1,8,14]. Adaptive control of linear systems is largely based upon the certainty equivalence principle: a good estimator is combined with the controller that may achieve stabilization or tracking when the parameters are perfectly known. Even if such an approach is still more
Work partially supported under a grant of the Italian Space Agency. *Email: [email protected] Correspondence and offprint requests to: S. Monaco, Dipartimento di Informatica & Sistemistica, Universita di Roma "La Sapienza", Via Eudossiana 18,00184 Roma, Italy. Email: [email protected]
questionable in the nonlinear situation because parameter estimation is generally not achieved, nevertheless, in discrete time, the absence of finite escape time phenomena reinforces its use. Several specific difficulties must be faced to develop nonlinear adaptive control in discrete time. Thus, only few results [5,11,23, ... ] dealing with particular classes of systems exist. The majority of these [9,20,24,25] combines a classical estimator (gradient, least squares, ... ) with a control that achieves the linearization of the system when the parameters are known. Stability and boundedness are therefore guaranteed under restrictive growth conditions on the nonIinearities. More recently, a control scheme which separates the estimation and compensation phases has been proposed in [26,27]. The control strategy is first used to ensure parametric estimation and then to fulfil control objectives. The success of nonlinear control design based on a feedback linearization technique first suggested to develop adaptive versions of these control strategies. This has been initialized in [7,21,22] for applications to satellites. The transposition to discrete time produces serious problems as the evolution of parametric dependency under feedback and coordinate change is very difficult to handle as nonlinear compositions of functions are involved. However, when considering sampled dynamics, we show in [6] how to guarantee nonlinear adaptive feedback linearization through multirate control design. To achieve adaptive stabilization rather than output tracking is still more difficult as nonlinear discrete-time stabilization is not yet completely solved (see [2-4]) and stabilization through multirate techniques would need to be of order n (the dimension of the state vector).
Received 7 May 1999. Accepted in revisedform 15 June 2000. Recommended by J. Tsinias and A.S. Morse
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Adaptive Control in Discrete Time
We here propose a design procedure which is inspired by step-by-step Lyapunov techniques developed in continuous time as the backstepping one and its adaptive version (see for example [13]). Recalling, at first, that a simple transposition to discrete time of step-by-step Lyapunov techniques results in nonfeasible controllers since they are noncausal, we here show how this problem can be overcome by combining successive parametrizations and "a posteriori" variables as introduced in a linear context [14,15]. A step-by-step technique is performed to design successive parametric adaptation laws in terms of a posteriori estimates while the control is computed at the last step making use of all the previously generated estimates. The approach is discussed with reference to nonlinear discrete-time dynamics which exhibit suitable state representations and parametric dependencies which, analogous to the continuous-time context, will be said to be in parametric strict-feedback forms [16]. The geometric framework proposed in [19] to unify continuous time, discrete time and sampled studies, is here used to discuss the equivalence to these particular classes. Necessary and sufficient geometric conditions ensuring the equivalence of a given discretetime dynamics to one of these canonical forms are given. The present paper generalizes some preliminary studies proposed in [16,17]. The paper is organized as follows. Difficulties and obstacles for the direct application of Lyapunov techniques are illustrated in Section 2 with reference to two elementary academic examples. The main arguments of the proposed solution are also detailed in these cases studies. The design procedure is then generalized in Section 3 to systems satisfying the discrete-time parametric strict-feedback form (DTPSFF). Section 5 discusses necessary and sufficient conditions for the equivalence to the above-mentioned classes of systems. Some examples are worked out throughout the paper. In particular, in Section 4, the feasibility and performances of the method are illustrated through a two-dimensional example.
2. How to Set Backstepping in Discrete Time In this section, we show how to modify the backstepping procedure developed in continuous time [13] to overcome two main difficulties occurring when dealing with discrete-time dynamics. This is worked out in two simple examples drawing a parallel between discrete-time and continuous-time cases. The main aspects of the proposed solution are also described.
2.1. A First Elementary Example
Given the continuous-time one-dimensional dynamics depending on a constant, though unknown, parameter B x(t)
= u(t) + Bcp(x(t)),
cp(O)
= 0,
(1)
the control u(t) = -kx(t) - B(t)cp(x(t))
k 2:
°
(2)
leads, for B(t) = B, to the closed-loop stable linear dynamics x(t) = -kx(t). When B(t) i= B, one obtains x(t) = -kx(t) + 8(t)cp(x(t))
with 8(t) = B- B(t).
(3) The stability analysis of (3) can be carried out by choosing the Lyapunov function
(4) whose time derivative along the closed-loop dynamics (3) is given by: V(t)
= x(t)x(t) + 8(t)8(t) = x(t) [-kx(t) + 8(t)cp(x(t))] = -kx
2
- 8(t)B(t)
(t) + 8(t) [x(t)cp(x(t)) - B(t)].
(5)
Because of the structure of the right-hand side of (5), which we should classify as linear with respect to 8, the choice B(t)
= x(t)cp(x(t))
(6)
leads to
(7) thus guaranteeing the stability of the dynamics (3) with adaption law (6). The transposition of this scheme to a nonlinear discrete-time dynamics of the same form leads to a first difficulty which stands in the nonlinearity with respect to 8 of the Lyapunov increment even if the dynamics under study exhibits a linear dependance on the unknown parameter. To do so, consider the first-order dynamics, x(k + 1)
= u(k) + Bcp(x(k)),
(8)
It is immediately verified that the control (dead-beat control when B(t) = B),
u(k)
=
-B(k)cp(x(k))
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leads to the perturbed closed-loop dynamics x(k + 1)
=
B(k)
with B(k)
=
with p(k) positive, given by
8 - B(k).
(9)
p2 (k )
A stability analysis may be carried out by choosing the function so that ( 10)
However, by computing the increment over one step, 6. V(k) = V(k + 1) - V(k), one gets
+ !(B2(k + 1) - B2(k))
By computing 6. V(k), one obtains
!B
2(k)
-B(k+I)({}(k+I)-B(k)).
It can be immediately verified that p(k) is decreasing
V(k,p(k),B(k)) ~ !p2(k) +!p-I(0)B2(k) ~ O.
-!x2(k) -!(B(k+ 1) _(}(k))2 +
with p(O) > O.
and upper bounded by p(O). This guarantees that V(k, p, B) is positive definite as a function of p and B and lower bounded
6.V(k) = !B2(k)
=
p-l (k + 1) = p-I (k) +
(11)
6.V(k) = _!p2(k) -!(B(k+ I) -B(k))2p- 1 (k) +B2(k+ I)
x ({}(k + 1) - (}(k)) The design of an adaptation law to cancel the last row in the expression of 6. V(k) is not easy because (II) is nonlinear with respect to B(k) and contains both B(k + 1) and B(k). This first difficulty is always verified in discrete time even if the dynamics (8) is linear with respect to 8. It should be pointed out that, for this one-dimensional case, the problem can still be solved according to [11] where an adaptive control law which globally achieves stabilization without any growth restriction on the nonlinearity has been proposed in terms of a modified least-squares estimator. A different solution which has local validity is proposed below and generalized in Section 3. Consider again the one-dimensional discrete-time dynamics (8) and set p(k + I)
=
:=
(12)
thus introducing the a posteriori variable p(k): a copy of the x-dynamics (9), now expressed in terms of the one step ahead parametric error B(k + I). Let us denote by c(k) := p(k) - x(k), the corresponding a posteriori error with dynamics given by
c(k + I) = (B(k + I) - B(k))
Let us now consider the function
(k)
- p-I(k)({}(k + 1) - {}(k))J. It is now possible to choose the adaptation law in such a way to render 6. V(k) negative; i.e.
{}(k + 1) = (}(k)
+ p(k)
(14)
with p(k + 1) from (9), (12) and (14) given by 1
p(k + 1) = 1 + p(k)
(15)
The adaptation law (14) can be expressed in terms of quantities measurables at time k + 1 as p(k)
+ 1 + p(k)
Since p(k) > 0, one obtains 6.V(k)
= _!p2(k) -!p-l(k)(B(k+ 1) -B(k))2 :::; 0
(16)
which implies V(k + 1) :::; V(k) - !p2(k)
p(k+I)-x(k+l) :=
1
+B(k+ I)[
, , 8(k + 1) = 8(k)
(8 - {}(k + I))
r
= -!p2(k) -!(B(k+ 1) -B(k))2
(13)
-!p-I(O)(B(k) -B(k+ 1)(
( 17)
It follows that V(k) is decreasing, and admits a limit when k ---+ 00. Taking the limit in (17), one gets
V(oo) :::; V(oo) - b(oo) with b(k)
~
0, thus proving
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that b(k) is bounded and b( 00) = O. It follows that p(k) and (e(k + I) - e(k)) are bounded and that p(oo) = 0 and B( 00) = Cst. Because of the convergence to a constant value of the parameter estimates, the convergence to zero of x(k) follows from (13) provided
estimates at time k, measurable data at time k are needed. (iv) The proposed method can be directly applied to higher-dimensional dynamics provided that the constant unknown parameters are matched with the control, i.e. the so called parametric strict matching form,
1-
B(k+ I)
=
I +p(k)
and thus, in general, parameter estimation does not hold, i.e. e( 00) i= O. The following arguments show how to guarantee at least locally the boundedness of
xn-I(k+ 1) =xn(k) xn(k + 1) =
(18) Figure 1 illustrates the result of a simulation when
Another typical difficulty appears when the unknown parameters do not match the control variables. To illustrate this problem, let us consider the continuoustime dynamics XI (t) X2(t)
By imposing 2
a C:'::: 2p(O)M' one guarantees that x(k) E Io: and thus the boundedness of
= -8(k)
p(k)
= B(k) + I + p(k)
locally, with respect to x and B, asymptotically stabilizes the dynamics =
u(k)
+ B
( 19)
The backstepping Lyapunov approach consists in considering X2 as a virtual control to stabilize the subsystem XI setting X2(t) = -kxl (t) - B(t)
X2 - a(xl, 8),
Setting ZI := XI, one computes ZI (t)
= XI (t) = Z2(t) + a(xi (t), 8(t)) + B
Z2(t) = u(t) -
Remarks.
(i) The previous arguments put in light the local validity of the result both with respect to x and B. (ii) The same arguments hold true by considering a stabilizing control of the form u(k) = dx(k) - 8(k)
= X2(t) + B
where
p(k) p(k+ I) = I +p(k)
x(k + I)
+ B
aa(~~:~;~(t)) (X2(t) + B(t)
_ aa(xl ~t), 8(t)) (}(t). aB(t)
(21)
The control law u(t)
=
-ZI
(t) - k2z 2(t) +
aa(xi (t), 8( t)) aXI (t) (X2(t)
d:'::: 1.
(iii) Causality is respected in the sense of instantaneous computations, i.e. to compute the
+ 8( t)
(22)
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3
0
2.5
-1
2
5
x1
10
15
5
0
10
15
10
15
0 -2 0.5 -4
0
0
5
P
10
15
5
u
1.5
Fig. 1. Dynamics (8) with B = 1.5 and i.e. x(O) = I, 0(0) = 3, p(O) = I.
with k 2
0, leads to the closed-loop error dynamics
;:::
21 (t)
=
-klz l (t)
+ Z2(t) + B(t)cp(xl (t)),
22(t) = -Z[ (t) - k 2z2(t) _ oa(xl (t), B(t)) O-() ( ()) ax 1(t) t cP XI t .
(23)
This guarantees the stability of the error dynamics. Using the LaSalle's invariance principle, one can show that z(t)---+O when t---+oo and consequently x(t)---+O when t---+oo. With this in mind let us consider the second-order discrete-time dynamics
(k + I) x2(k + I)
=
XI
Choosing the function V2 -_ 2:12 z)
12
10-
+ 2: z2 + 2:
'
(24)
whose time derivative along the closed-loop dynamics (23) is given by
V2(t)
= -klzT(t) -
_
k2Z~(t) + B(t) (ZI (t)CP(XI (t))
() oa(xl (t), B(t)) ( ()) _ 0:"( )) Z2 taX) (t) cP XI t t.
(27)
(28) Setting ZI
= XI,
one computes the z-dynamics
+ I) = z2(k) + B(k)cp(xl (k)) z2(k + 1) = x2(k + I) + B(k + I )cp(XI (k + 1))
= ZI (t)CP(XI (t)) oa(xI(t),B(t)) ( ()) - Z2 () t OXI(t) cP XI t
+ Ocp(xl(k))
Considering X2 as a virtual control, XI goes to zero by setting X2 = -Ocp(XI)' However, as X2 is not a control variable, one introduces the z2-variable as the difference between X2 and its computable value, i.e.
ZI
one immediately sets,
B(t)
=
x2(k) u(k).
(k
= u(k) + B(k + I)CP(Z2(k) (25)
+ B(k)cp(Xl (k))).
+ I) is no longer linearly parametrized with respect to B(k). In addition, the design of a control to linearize the zz-dynamics is, It must be noted that z2(k
which leads to
(26)
(29)
Adaptive Control in Discrete Time
303
in fact, not achievable as it would be noncausal due to the presence of B(k + I). We show in the sequel that the introduction of suitable a posteriori variables following the lines of the first example makes it possible to overcome the problem. Consider again the dynamics (27) and now set a coordinates change as in (28) but one step backward in time with respect to the parametric estimation, ZI (k) = XI (k)
+ B(k -
z2(k) = x2(k)
(30)
l)cp(xl (k)),
so that now
Let us now introduce the two a posteriori variables PI and P2 with the dynamics of PI defined as the one of ZI but in terms of the one step ahead in time parametric error
Let the candidate Lyapunov function be
Ai 2 Ai -I -2 V I (k,PI(k),8(k)) =2PI(k) +2 PI (k)e (k),
with PI (0) > 0 and PI I (k + 1) = PI I (k) By computing ~ VI (k) one obtains
Ai 2
Ai 2
~Vl(k) =2P2(k) -2PI(k)
x pjl (k)
l
[1 + cp2(k)PI(k)r x [XI (k + I) + P2(k) - x2(k) - B(k)cp(k)]
(34)
which is now a function of measurable quantities at time k+ 1. Let us now compute from (30) the dynamics of Z2
z2(k + 1)
=
u(k)
+ B(k)cp(xi (k + 1)).
Since the unknown parameter enters nonlinearly in cp(XI (k + I)), it is now necessary to perform a new linear parametrization of cp(XI (k + 1)) which, in general, induces an overparametrization. To do so, assuming the existence of a finite parametrization, we rewrite B(k)cp(xi (k + 1)) = B(k)cp(X2(k) + 8cp(xl (k))) as
B(k)cp(X2(k))
+ 8~¢>(xl (k), x2(k), B(k))
Ai (-)2 -2 8(k+ 1) -8(k) 1)
It is now possible to choose the adaptation law in such a way to cancel the last addendum in the right hand side of ~ VI (k): it is
B(k)
+ PI (k)cp(xi (k))PI (k +
1),
(33)
thus obtaining
~ VI (k) = ~i p~(k) - ~i PT(k) - ~i (8(k + I) -
(36)
with 82, a finite linear parametrization involving powers of 8; ¢>(O,·,·) = O. Denoting below ¢>(XI (k), x2(k), B(k)) as ¢>(k) and substituting (36) into (35), gives
+ B(k)cp(X2(k)) + 8~¢>(k). (37)
u(k)
+ cp2(XI (k)).
+ Ai8(k + 1)[cp(XI (k))PI (k +
(35)
We can now design the certainty equivalence stabilizing control law
+pjl(k)(B(k) -B(k+ 1))].
=
=
positive and given as in the first
IS
pi (k)cp2 (XI (k)) PI(k+ 1) =PI(k) -1 +cp2(XI(k))PI(k)
B(k + 1)
PI(k+ 1)
z2(k + 1) = u(k)
Al E JR,
where PI (k) example by
8cp·(k), one computes
8(k))2p]l(k).
Denoting CP(XI (k)) as cp(k), substituting Eq. (33) into (32) and taking into account that XI (k + 1)- x2(k) =
= -B(k)cp(X2(k)) -
B~(k)¢>(k),
(38)
where B2 (k) is an estimate of 82 at time k so that the closed-loop z2-dynamics becomes
z2(k + 1)
= 8~(k)¢>(k),
(39)
where 82(k) = 82 - 02(k) is the estimation error over 82 at time k. Define now the dynamics of the aposteriori variable P2 (k + I) which is again the z2-dynamics one step ahead in time with respect to 82
P2(k + I)
:=
e~(k + I)¢>(k),
(40)
and the resulting a posteriori error Pi(k + 1)zi(k + 1) := Ei(k + 1) thus obtaining from (31), (32), (39) and (40) the error dynamics
EI(k
+ 1) = E2(k) -
[B(k
+ I) -
B(k - I)]cp(k) (41 )
(42)
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Let the candidate Lyapunov function be
V2(k)
= V, (k) + +
Arguing as in the previous example it follows that
A22
2 p~(k)
V2(k+
~~e2(k)IP2](k)e2(k)
A~ > AT,
with
- ~i P-' (0) (e(k) A2'
where the gain matrix P2 (k) is given by
P (k 2
+
+ ¢(k)¢l(k).
The computation of flV2 (k) gives flV2 (k)
Ai
2 (k)
2 PI
+
- ~i (e(k+
Ai - A~ 2 (k) 2
P2
+ P2 1(k)(e2(k + I) - e2(k))] I I - 2A~ (02(k + I) - 02(k)) P2 (k)
(e2 (k + I) - e2 (k)).
It is now possible to choose the adaptation law in such a way to render fl V2 (k) negative; it is
e2(k + I) e2(k) + P2(k)¢(k)P2(k + I) =
(43)
which leads to fl V2(k)
=-
~i phk) + Ai ; A~ p~(k)
- ~i (e(k + I) - 2A~ (02(k + x
+ I) -
I
e2(k)).
I
we deduce that parametric estimation does not hold. As far as the local boundedness of tp(x) and ¢(x, e) is concerned, we preliminarily note that for any positive constant C ;::: 0, =?
A~p~(k) :::; 2C,
A~e:(k)ei(k) ~ 2Ai,MC,
I
I) - 02(k)) P2 (k)
(e2 (k + I) - e2 ( k)).
Substituting (43) into (40), because of (37), (38) and (30) one gets
P2(k + I)
-
O2 (k))
V2 (k) is monotonically decreasing and admits a limit for k ---. 00. It follows that the (p;(k))'s as well as the (e;(k+I)-e;(k))'s are bounded for i=I,2 and limk->oop;(k) = 0, limk->ooe;(k) = Cst; for i= 1,2 (0, := 0). From Eqs (41) and (42), because their triangular structure, we can deduce the boundedness and convergence to zero of z, (k) and z2(k) provided the boundedness of ¢(x" X2, e) and tp(x,) is guaranteed. For, from (42), z2(k) is bounded and Iimk->oo z2(k) = limk->oo P2(k) = 0, while,consequently,from(41),z, (k) is bounded and limk->oo (k) = Iimk->oo PI (k) = O. Then, through the triangular coordinates change (30), we get limk->oo x, (k) = Iimk->oo z, (k) = 0 and from the boundedness of tp(x,) again, we conclude the boundedness of x2(k) and limk->oo x2(k) = limk->oo z2(k) = 0 since tp(O) = O. Since
V2(0) ~ C
e(k))2PI ' (k) -
e(k + 1))2
z,
I) - e(k))2 pl '(k)
+A~e~(k+ 1)[¢(k)p2(k+ I)
x
.
x p 2 1 (0)(e2(k
to guarantee that P21(k) is always positive definite with P2 (0) > 0 and
= _
-
- 22 (0 2 (k + I) -
I) = P (k) _ P2(k)¢(k)q/(k)P~(k) 2 1+ ¢t(k)P2(k)¢(k)
P2 ' (k + I) = P2 1(k)
~T PT(k) + Ai; A~ p~(k)
I):::; V2(k) -
= [I + ¢t(k)P2(k)¢(k)rl x (x2(k + I) + e(k)tp(x, (k + I))),
(44) i.e. P2 can be computed from measurable quantities at time k+ I.
i
= 1,2,
(45)
where A;,M denotes the maximum eigenvalue of P;(O). Let I Q ; be neighborhoods of 0 in IR such that x~ ~ a}, let M] be a bound of tp2(Xl) over IQI' let A:h be a bound of ¢t (XI, X2, e)¢(x" X2, e) over I Q1 x I Q2 and any bounded value of Note that, Nh depends on C, (denoted as A:h( C) when necessary). Moreover, for any initialization e(O) which satisfies (45), then e(k) := 0 - B(k) remains bounded as e(k) is bounded. Finally, let Me be a bound of 02 . It is now possible to show the existence of C such that, under suitable intializations of the control algorithm as in (45), if Xi(O) E I Q ; apd xl(l) E IQI' then x(k) remains in I Q1 x I Q2 and O(k) is bounded thus implying the boundedness of tp(k) and ¢(k).
e.
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Adaptive Control in Discrete Time
To do so, from (45), because of (39), one computes
z~(k + I) = (B~(k)¢(k))2
:::; 2)..2,:['h
with
C
(k PI
2
Then, since from (31),
(k
xl(k + 2) := zl(k + 2) = z2(k + I) + B(k)
2(k
+
2)
:::;
4)..2,MM2 C )..2 2
+
4)..I,MMI C
XI
ai
(46) Since fl (0) < 0, it is possible, by continuity, to find C> such that (46) holds true. This guarantees that XI (k + 2) remains in I and thus the boundedness of
a,
x2(k + I)
:= XI (k
+ 2) - B(k)
x2(k + I) < 8)..2,MM2C + 8)..I,MM I C + 2M M . -)..2 2
e
)..2 I
Since M I does not depend on choose a2 and M 1 so that
a2,
1)=X2(k+I)+B(k)
1
3. Adaptive Control of Parametric Strict-feedback Forms For the sake of simplicity, the approach is worked out on a particular class of discrete-time dynamics: the discrete-time parametric strict-feedbackform. Some slightly more general classes of systems can also be treated according to the same arguments as discussed in the concluding section.
+ I) = x2(k + I) = XI
(k
~a~ - 2MeM~) .. I)..~
<
8()..1)..2,MM2 + )..2)..I,MM I ) -
°
B)
which guarantees that x2(k + I) remains in I a2 too and thus the boundedness of ¢ over the evolution of x results to be verified. In conclusion, we can state that the control law u(k) = -B(k)
= B2(k) + P2(k)¢(k)p2(k + I)
+
i=n
xll(k + I) =
L c/x/(k) +
(49)
/=1
where
(48)
B2(k + I)
+
XIl-J (k + I) = xll(k)
°
C-
and B, asymptotically stabi-
it is possible to
Moreover, similar arguments as before, ensure the existence of C> such that :=
X
(k + I)
(47)
!2(C)
B(k)
Definition.3.1. DTPSFF The discrete-time dynamics defined on R" as
one gets to 2
= Xl (k + I) + P2(k) - x2(k) I +
= x2(k) + B
)..2 I
By imposing xi(k + 2) :::; and recalling that M2 depends on C, we get the condition ( 2)..2)..2 fl(C):=CI 12 <0. 4()..i)..2,MM2(C) + )..~)..I,MMI)-
°
+
P2
I)
locally, with respect to lizes the dynamics
we get with PI (0) := )..I,M, xI
+
(0, B) = 0, ... ,
°
(50)
will be referred to as the discrete-time parametric strictfeedback form. Such a dynamics is triangular with respect to the state variables; the control acts linearly with a constant coefficient in the last state equation only (i.e. the system has relative degree equal to n for YI :=xd. To achieve stabilization, we propose a step-by-step procedure which is worked out in the sequel assuming, without loss of generality, that the functions
PI(k)
PI(k+ I) =pJ(k) - I
+
P (k + I) = P (k) _ P2(k)¢(k)¢t(k)P~(k) 2 2 1+ ¢t(k)P2(k)¢(k)
Step 1. When B is known and manipulating the variable X2 as a control variable, one could stabilize XI
306
S. Monaco et at.
by setting
Easy computations lead to
~ VI (k) = ~T p~(k) - ~T pf(k)
X2 = -B1'"n(xI). However, as X2 is not the control, we define a new variable Z2 as the difference between X2 and its computable value delayed of n - I steps backward in time, i.e.
-
~Tpil(k)(iJ(k+ I) -iJ(k))2
+>.fO(k+ I) ['"n(XI(k))PI(k + I) +pil(k) x (iJ(k
where B(k - n + I) is an estimate of B at time k-(n-I). The objective being stabilization, we set Zl (k) = XI (k), thus obtaining
zl(k+ I)
=
z2(k)
+ Ol(k -
B(k + I)
(55)
~VI(k) = ~T p~(k) _ ~T pf(k) _ ~T pil(k) -
x (B(k
+ I) + P2(k) - x2(k) - '"YI (k)B(k) I + PI (kh?(k) XI (k + I) + P2(k) - z2(k) - , I (k) L7~/ PI (k + I) + P2(k)
2
ih (k - i)Pl (k - i + I)
+ PI (kh?(k)
-,I
- z2(k) (k)(B(k) - B(k - n + I)) 1 + PI (kh?(k)
estimate, i.e.
(56)
which is a function of measurable variables at time k+ I.
Introducing also the a posteriori errors Cl := PI - ZI, c2 := P2 - Z2, we get the a posteriori error dynamics = C2 (k) - [B( k
+ I)
- B(k - n + I)
x ,(Xl (k)).
Step 2. Let us now compute the dynamics of from (51) as
z2(k + I)
r
(54)
Let the candidate Lyapunov function be
=
PI (k) I +PI(kh?(XI(k))'
so that Pi l (k + I) = Pi' (k) + ,?(XI (k)) and Pi l (k) is always positive and lower bounded by PI (0) > O.
= x)(k) + B1'2(XI(k), x2(k)) + B1(k - n + 2hl(xl(k + I))
Z2
(57)
which is nonlinear with respect to the unknown parameters B in (XI (k + I )). It is thus necessary to perform a new linear parametrization of (XI (k + I)) which generally induces an overparametrization. To do so, assume the existence of a finite linear parametrization, B2 , involving powers of B, such that it becomes possible to rewrite in (57),
,I
where PI (k) is given by
PI(k+ I)
-
B(k)) .
= XI (k
XI (k
+ I)
+ I) -
Substituting (55) into (53) and taking into account of (49) one obtains
I
C I (k
= B(k) + PI (k)'"n (Xl (k) )PI (k + I)
gIves
n + Ihl(xI(k)),
where O(l) := B- B(l) is the estimation error at time I. Let us now introduce the a posteriori variables PI and P2 and define the dynamics of PI as the dynamics (52), in terms of a one step ahead parameter
+ I)
iJ(k))].
The choice of the adaptation law
(52)
PI (k
+ I) -
B1'2(XI (k), x2(k))
,I
+ {)l(k -
n + 2h (Xl (k + I))
as
()l(k - n + 2h (X2(k)) ()l(k - n + 2)),
+ B~r2(XI (k),
x2(k),
(58)
307
Adaptive Control in Discrete Time
where fz(O,xz,')='z(O,xz) and thus fz(O,O,·)= IZ(O,O) = O. Substituting (58) into (57) with fz(k) := fz(xi (k), Xz (k),fF (k - n + 2)), one gets
zz(k + 1)
= x3(k) + (Y(k -
n + 2)rI (xz(k))
where Pz(k) is given by
that Pi l (k+l)=Pi'(k)+fz(k)q(k) and Pi ' (k) is always positive definite if Pz(O) > O. It follows that so
+ B~fz(k). As before, let us now introduce the variable Z3 as the difference between X3 and its computable value delayed n - 2 steps backward in time with respect to Bz, i.e.
~Vz(k) = - ~~ p~(k) + >.~ ; >.~ p~(k) + ~~ p~(k)
z3(k) = x3(k) - Qz({j(k - n + 2), (jz(k - n + 2), xI(k),xz(k))
-
(59)
~~ PI] (k)(8(k + 1) -
8(k))z
+ >'~8~(k + 1)[fz(k)pz(k + I)
with
+ P"i l (k) (8z(k + 1) - 8z(k))]
Qz({j(k - n + 2), (jz(k - n + 2), XI (k), xz(k)) :=
- 2>.~ (Bz(k +
-{jl(k - n + 2)rI (xz(k)) - (jHk - n + 2)f z(k).
-
1) - Bz(k))
I
x P"i I (k) (8z(k + 1) - 8z (k)).
Because of (59), the dynamics of Zz becomes (60)
To cancel the term between the brackets, we can choose the adaptation rule for Bz
We can now define the dynamics of the a posteriori variable pz as the dynamics of Zz anticipated at time k + 1 in 8~, i.e. which leads to
For the a posteriori error dynamics, setting P3 - Z3, one gets,
~ Vz(k) = - ~~ p~(k) + >.~ ; >.~ p~(k) + ~~ p~(k)
C3 :=
-
cz(k + 1) = c3(k) - [(jz(k + 1) - (jz(k - n + 2)ff z(k).
~~ Pl I (k)(8(k+ I) -
- 2>.~ (Bz(k + 1) - Bz(k))
(62)
Let the candidate Lyapunov function be X
Vz(k)
=
VI (k) +
~~ p~(k) + ~~ 8Hk) Pi I (k)8z(k)
with >.~ > >.~,
pz
k (
1) = xz(k
+
P"i' (k) (8z(k
+ 1) -
I
8z(k) ) .
From Eqs (63) and (61) and taking into account of (49), (51), (58), (59) one obtains
+ 1) + P3(k) - x3(k) + hi (k + 1) I
8(k))z
II (xz(k))){j(k - n + 2) - q(k){jz(k)
+ f~(k)Pz(k)fz(k)
_ zz(k + 1) + P3(k) - z3(k) - q(k) '5:.,;,z Pz(k - i)fz(k - i)pz(k - i + 1) 1 + q(k)Pz(k)fz(k)
-
zz(k + 1) + P3(k) - z3(k) - q(k)({jz(k) - (jz(k - n + 2)) 1+ q(k)Pz(k)fz(k)
(64)
s.
308
this enables us to conclude that (63) is computable at time (k + I). Step n - 1. According to the results of the first n - 2 steps, one gets
Zn-I (k) = Xn-I (k) - cx n_2(B(k - 2), ... , Bn-2(k-2)I(k), ... ,Xn-2(k))
Monaco
el
at.
Let now define the dynamics of the a posteriori variable Pn-l as
Pn-I (k + I) thus getting for dynamics,
= Pn(k) + e~_l (k + I)f n- l (k), Cn
:=
Pn - Zn, the a posteriori error
Cn-I (k + 1)
(65)
with
= cn(k) - [Bn-l(k+ I) - Bn-l(k -I)ffn-l(k).
CX n-2(-) := -B1(k - 2)')'1 (x n-2(k)) - B~(k - 2)r2 (X n-3 (k), Xn-2 (k), B(k - 2)) - ...
X
As before, the choice Vn- l (k)
- B~_2(k - 2)f n- 2 (Xl (k), ... , x n-2(k),
= Vn- 2(k) + A;_I P~-l (k) + A;_I e~_l (k) x r;;~1 (k)en- l (k)
B(k - 2), ... , Bn- 3 (k - 2)),
with A~_I > A~_2' enables us to compute
so that one computes
Zn-I (k + 1)
=
xn(k) + B1')'n-1 (Xl (k), ... , Xn-l (k)) - cx n_2(B(k - 1), ... ,Bn- 2(k - I),
Bn- l (k + I)
= Bn- 2(k) + Pn- l (k)rn-l (k)Pn-1 (k + I)
with Pn-l (k + I) computable at time k + I by
xl(k+ I), ... ,xn-2(k+ I)).
I) _ Zn-I (k + I) + Pn(k) - zn(k) - f~_l (k)(B n- 1(k) - Bn- I (k - I))
(k Pn-l
+
l+f~_I(k)Pn_l(k)fn_l(k)
Zn-l (k + I)
Suppose again it is possible to linearly parametrize zn-l(k+ I) so that
B1')'n-I(Xl(k), .. "xn-l(k)) - CX n-2(·)
= B1(k -Ih
+ B~(k - 1)r2(Xn-2(k),xn~1 (k), 1)) + ... + B~_2(k - l)fn - 2(x2(k), ... ,
+ Pn(k)
- zn(k) - f~_l (k)P n- 1(k - l)f n_ 1(k - I)Pn-1 (k) I + f~_l (k)Pn-1 (k)f n- l (k)
Step n and the control design. From (66), one computes n
zn(k+ I) = u(k)
+ Lc;x;(k) +81')'n(x(k)) ;=1
x (Xn-l (k)) B(k -
xn_l(k),B(k-I), ... ,Bn- 3 (k-I)) + B~_l f
n- l
(Xl (k), X2(k), ... , Xn-l (k),
Suppose that it is possible again to linearly parametrize the dynamics of zn(k + 1) so that
B1')'n(x(k)) - CXn-1 (-)
=
B/(kh (xn(k)) + B~(k)f2(Xn-1 (k), xn(k), B(k))
B(k-I), ... ,Bn-2(k-l)),
+ ... + e~_, (k)rn-I (x2(k), ... ,
= 0. It follows that Zn-l (k + 1) = zn(k) + e~_1 (k - I)f n - l (k)
where f n-l (0, ... )
xn(k), B(k), -
with f
n- I
(k)
:= f
n- l
(Xl (k), ... , Xn-I (k),
B(k-l), ... ,Bn- 2(k-I)) zn(k) =xn(k) -cxn-I(B(k-I), ... 'on-l(k-I), xl(k)" .. ,Xn-l(k)), . (66)
where fn(O, ... )
=
B~fn(XI (k),
, Bn-2(k)) , xn(k),
B(k), ... , Bn- l (k)), 0, thus obtaining n
zn(k+ I) = u(k)
+ Lc;x;(k) +B/(kh(xn(k)) ;=1
+ BHk)f2(x n-l(k),xn(k),B(k))
and
CXn-1 (-)
:=
-B1(k - I )'"n (Xn-l (k)) - B~ (k - I )f 2 X
(x n_2(k),xn_l(k),B(k - 1)) - ...
-B~_I(k-l)fn-I(Xl(k),... ,Xn-l(k),
B(k - I), ... ,Bn- 2(k - I)).
+ ...
+ B~_l (k)f n_1(x2(k), ... , xn(k), B(k), ... , Bn- 2(k))
+ B~fn(Xl (k), Bn- l (k)).
... , xn(k), B(k), ... ,
309
Adaptive Control in Discrete Time
The certainty equivalence stabilizing control can now be easily computed as a function of the data available at time k,
that
~Vn(k) = _ ~T PT(k) + AT ; A~ p~(k) + ...
n
u(k)
LCiXi(k) - (i(k)~'/l(xn(k)) i=l
== -
+
- 0&(k)f 2(x n-1 (k), xn(k), B(k)) - ...
-
- 0~-1 (k)f n-1 (x2(k), ... ,xn(k), O(k), ... ,
A~_l - A~ 2(k) 2
Pn
~T pi! (k)(O(k + 1) A~
- 2: (02(k + 1) A
A
0(k))2 t
1
O2(k)) Pi (k)
On-2(k)) - O~(k)f n(X1 (k), ... ,Xn(k), O(k), ... , (67)
x
(B 2 (k + 1) - O2 (k)) ...
-
~~ (On (k + 1) -
On (k)/ p;;! (k)
x (On(k + 1) - On(k))
Denoting fn(k):== f n(X1(k),X2(k), ... ,xn(k),O(k), 02(k), ... ,On-1 (k)), the closed-loop zn-dynamics and
+ A~B~(k + 1)[fn(k)Pn(k + 1)
the a posteriori dynan1ics become respectively
- p;;l (k)(On(k + 1) - On(k))]. (68) We can now render ~Vn(k) negative by choosing the following adaptation law for On
and (69)
thus obtaining so getting, with En: == Pn - Z n, recalling (54) and (62), the overall triangular a posteriori error dynamics
c!(k+ 1) = c2(k) - [O(k+ 1) - O(k- n+ 1)f')'(k) 102 (k
+ 1) = 103 (k)
- [02(k + 1) - O2(k - n + 2)
fr (k)
-
2
~T pi!(k)(O(k + 1) -
O(k)r
A) t - 2A~ (A02(k + 1) - 02(k) X
-?
Let the candidate Lyapunov function be
Vn(k) = Vn-! (k)
+
Pi! (k) (02(k + 1) - 02(k)) ...
?p~(k) + ?
Bn(k)t
(On(k+ 1) -On(k)r
x p;;! (k) (On (k + 1) -On (k)) ::::; O.
x p;;l (k)Bn(k) with A~ > A~_l and gain matrix Pn(k) given by
P (k 1) n +
==
P (k) _ Pn(k)fn(k)f~(k)Pn(k) n 1 + r~(k)Pn(k)rn(k) ,
so that p;;l (k + 1) == p;;l (k) + fn(k)f~(k) and p;;l (k) is always positive definite if Pn(O) > O. It follows
Substituting (71) into (69) and taking into account of (68), one obtains
Pn(k + 1)
==
[1
+ f~(k)Pn(k)f n(k) ] -1 zn(k + 1) (72)
which is a function of measurable quantities at time
k + 1 with zn(k + 1) given by (66) at time k + 1.
s.
310
any positive constant C 2: 0,
Arguing as before it follows that
AT 2 A~ - AT 2 2 P2(k) - ... Vn(k+ 1) ~ Vn(k) -2PI(k) -
Vn(O) ~ C =* A~p~(k) ~ 2C, 2 -t
-
AiOi(k)Oi(k)
>.~ -2>'~-l p~(k) - ~T pi
-
l
(0)
~
t
1
-2(02(k+ 1) -02(k)) P2 (0) x
(0 2 (k + 1) - O2 (k)) ...
-
~ (On(k + 1) -
X
(On(k + 1) - On(k)).
On(k))1 p;;l (0)
i.e. Vn(k) is monotonically decreasing and admits a limit for k ---+ 00. It follows that the (Pi(k))'s and the ([)i(k + 1) - [)i(k))'s are bounded and that limk----*oo Pi(k) == and limk----*oo [)i(k) == Cst for
°
i==l, ... ,n.
~
2Ai,MC,
. 1
==
1, ... , n,
(74)
where Ai,M denotes the maximum eigenvalue of Pi(O). For i == 1, ... , n, let L ai be a neighborhood of in lR such that xT ~ aT, let M i be a bound of ri(XI, ... ,Xi)2 over L al x ... x La. and let M i be a bound of f:(XI, , Xi, 0, ... , Oi-l)fi(XI, . .. , Xi, 0, .. ., Oi-l) over L al x X L ai for any bounded value of Oi; (M I :== MI as f l :== rl and M i depends on C, denoted M i ( C) if needed). We note that, for any initialization [)(O) which satisfies (74), then Oi(k):== 0 - [)i(k) remains bounded as [)i(k) remains bounded since ~ V n is negative. Finally, let Me be a bound for 02 . It is now possible to show the existence of C such that, under suitable intializations of the control algorithm as in (74), if xi(h) E L ai for i == (1, ... ,n) and }i == (0, ... ,n - i), then x(k) :== (Xl (k), ... ,xn(k)) remains in La :== L al X ... X Lan and Oi(k) is bounded thus implying the boundedness of fi(k). Let us preliminarily note that the closed loop z-dynan1ics satisfies the relation
°
x (O(k + 1) - O(k))2
A~ ~
Monaco et at.
From the triangular expressions of the a posteriori error dynamics (70), we can conclude for i == n to i == 1 that the boundedness and convergence to zero of the ([)i(k + 1) - [)i(k))'s imply the boundedness and convergence to zero of the errors (Ei(k))'s provided the boundedness of the (fi)'s. Then, one concludes the boundedness of the (zi(k))'s and lin1k----*oo Pi (k) == limk----*oo Zi(k) == 0. Then, from the coordinates change defined by (51), (59) and (66)
n
zl(k+n)
==
L:[)i(k)fi(k+n -1). i=l
Arguing as for the two preliminary examples and since Xl (k + n) == Zl (k + n), one gets the immediately
Zl (k) == Xl (k) ==
x2(k)
+ {)t(k -
n + l)rl (k)
z3 (k) ==
X3 (k)
+ ot (k -
n + 2)rl (x2(k))
Z2(k)
By imposing xi(k condition over C
+ n)
::;
ai,
one gets the following
+ {)~(k - n + 2)r 2 (k) (75)
°
zn(k)
==
xn(k)
+ ()t(k -
l)rl(X n-l(k))
+ O~(k - 1)f2(x n-2(k), Xn-l (k), O(k - 1))
(75) can be verified for c> by continuity arguments since!l (0) < 0. This ensures that Xl (k + n) remains in L al and thus the boundedness of rl. Moreover, denoting rj(k) :== rj(XI (k), ... ,xj(k)), and since
(73) we can iteratively conclude for i == 1 to i == n that lin1k----*oo xi(k) == due to the triangularity of the coordinates change (73) and because rl (0) == r 2 (0,0, 0) == ... == f n-l (0, 0, ... ,On-2) == 0. ~ As far as the boundedness off n(x, 0, ... ,On-I), ... , f 2 (XI, X2, 0), rl (Xl), is concerned, one proceeds as in the first two preliminary examples noting that for
°
Xj(k + n -} + 1)
==
xj-l(k + n -} + 2) -ot rj _l (k+n-}+l)
xn(k + 1)
==
Xn-l (k) - ot rn - l (k + 1),
311
Adaptive Control in Discrete Time
one gets also,
e(k+ 1)
= e(k) +Pl(k)'''fI(k)PI(k + 1)
e2(k + I)
=
e2(k) + P2(k)r2(k)p2(k + I)
en(k + I)
= en(k) + Pn(k)rn(k)Pn(k + I)
PI(k+ I)
=
XJ (k + n - j + 1) :::; Yn ~2AMM'C L.. "2' A;
;=1
i-I
+ l:y-p+IMeMp p=1
P (k + I)
=
2
n-I
+
PI(k)
I
l: 2
n
-
+ PI (kh?(xI (k))
P (k) _ P2(k)r2(k)Q(k)P2(k) 2 1+ Q(k)P2(k)r 2(k)
p+ 1MeMp.
p=l
xJ
n-
aJ,
By imposing (k + j + 1) :::; provided the (O))'s and the (A1;)'s are such that for (j = 2, ... , n), i-I
aJ - l:y-p+IMeMp > 0
(76)
with Pl(k+l) 21
p=1
(k + I) - 2:7=dO,(k) - O,(k - n + 1)]tC(k - i + I) 1 + PI (khr(k)
P2(k+ I)
which can be verified because of the triangular structures of the bi)'s, one gets, for j = (2, ... ,n), the following conditions over C
::; O.
22(k + I) - 2:7-2[O,(k) - O,(k - n + 2)]tr,(k - i + 2) 1 + r~(k)P2(k)r2(k)
(77)
Since jj(O) < 0, by continuity arguments, it exists C> 0 such that (77) holds true. Then, it remains proved that there exists C such that the (x;)'s remain in the (a;)'s thus implying the boundedness of the (f;)'s. This ends the proof. In conclusion, we can state that the control law
where r l := change Zl
(k)
=
/'1
:=
and 01 := 0, and the coordinates
XI (k)
z2(k)
= x2(k) + et(k -
n + Ihl(k)
zn(k)
= xn(k) + et(k - Ih (Xn-I (k))
n
u(k) = -
l: G;x;(k) -
et(khl (xn(k))
;=1
- e~(k)r2(Xn-l(k),xn(k),e(k)) - ... - e~(k)rn(xI (k), ... , xn(k),
+ e~(k -1)r2(Xn-2(k),xn-l(k),e(k - I))
x e(k), ... , en- I (k))
+ ... + e~_1 (k -
l)rn _ 1(k)
S. Monaco
312
locally, with respect to x and 8, asymptotically stabilizes the dynamics (k + 1) = x2(k) + 81'YI (k) x2(k+ 1) = x3(k) +8l'Y2(k)
el
al.
with PI (0) > 0, it is possible to choose the adaptation law as
XI
to get b.VI(k)
~~ p~(k) - ~~ pi(k) - ~~ (B(k+ 1)
=
n
xn(k + 1)
= u(k) + L
c;
x;(k)
+ 8l'Yn(k).
-B(k))2pil (k).
;=1
A stabilizing control law is given by
4. An Example
u(k) =
Let us consider the following example on n~?,
-XI
thus getting
+ 1) = x2(k) + 8xi(k) x2(k + 1) = u(k) + XI (k) + 8(X2(k) + 8xi(k))
XI (k
z2(k + 1) = B2(k)lr(x(k), B(k)) P2(k+ 1) = B2(k+ l)lr(x(k),B(k)),
(78) which can be steered to zero in two steps when 8 is known. The choice of x2(k) = -8xi(k) would get XI (k + 1) to zero, but as X2 is not the control and 8 is unknown we set
and the a posteriori error dynamics
cI(k + 1) = c2(k) - [B(k + 1) - B(k - 1)]xi(k) c2(k + 1)
= XI (k) z2(k) = x2(k) + B(k - l)xi(k), ZI
(k) - B(k)x~(k) - B~(k)r(x(k), B(k)),
= -
[B2(k + 1) - B2(k)fr(x(k),B(k)).
(k)
V2(k)
and compute the z-dynamics zl(k + 1)
=
z2(k + 1)
=
=
Setting, for A~ > A~ and P2(0) > 0,
+ B(k - l)xi(k) u(k) + XI (k) + 8[X2(k) + (h~(k)] + B(k)x~(k + 1) u(k) + Xl (k) + B(k)x~(k)
P 2 (k
z2(k)
=
= VI (k) + ~~ p~(k) + ~~ B2 (k)lp:; I (k)B2(k)
+ 1) P (k) _ P2(k)r(x(k), ~(k))rl(x(k), B(k)):i(k) 2 1 + fL(x(k), 8(k))P2(k)r(x(k), 8(k)) ,
the following adaptation law
+ 8~r(x(k), B(k)) with
can now be chosen to render b. V2(k) negative. The control law so far computed u(k)
and 82 = (8, 82 Setting
r an overparametrization of 8.
PI(k + 1) VI (k, PI (k), B(k)) PI (k + 1)
=
-XI
B(k + 1)
(k) - B(k)x~(k) - B~(k)r(x(k), B(k))
= B(k) + PI (k)xi(k)PI (k + 1)
B2(k + 1)
= B2(k) + P2(k)r(x(k), B(k))P2(k + 1)
= P2(k) + B(k + l)xi(k)
PI (k + 1)
=
= ~~ pi(k) + ~~ pi l (k)B2(k)
P2(k+ 1)
p~(k)xt(k)
= PI (k) - 1 + xt(k)PI (k)
=
PI (k) 1 + xt(k)PI (k)
P (k) _ P2(k)r(x(k), ~(k))rl(x(k),B(k)):~(k) 2 1 + fL(x(k), 8(k))P2(k)r(x(k), 8(k))
313
Adaptive Control in Discrete Time
with
f(x(k), e(k)) = (x2(k) PI (k +1 )
P2
k (
+ 2e(k)X2(k)xT(k), xT(k) + e(k)xi(k))l
xI(k + I) - (eHk) - eHk - I))f(k - I) - (e(k) - e(k - I))xi(k) I + PI (k)xi(k)
=----'--'-----"'-------'--~-'-----------''-'-------'--'---:-:'-:----:;--:-:-:---'---------'-----'----'--'-----'--'----'--
) _ x2(k+ I) + e(k)xi(k + I) 1 + P(k)P2(k)f(k)
+1
locally, with respect to x and B, asymptotically stabilizes the dynamics (78). Some simulation results are reported in Figs 2-4 which also illustrate the deadbeat stabilization and the possible estimation of the parameter (Fig. 4).
5. Necessary and Sufficient Conditions for the Equivalence to Strict-feedback Forms
5.1. The Geometric Context Consider a general discrete-time dynamic on nonlinearly parametrized by the vector BEn c ffi.P,
F(B,((k),u(k)),
(79)
I 0=0
' (80)
and compute its expansion with respect to u around 0, th us getting
(81 ) with G?(B,·) := GO(B, ,,0) and
.) '= 8i-lGO(B, " u) I GO(B I ,. 8u i - 1 u=o
i
> 1.
(82)
Denoting by F6(B,.) the j-times compOSItIon of Fo(B, '), let Gf(B,·) be the transport of G7(B,·) along the free evolution
Fb(e, '), i.e.
(83) or according to the Ad operator notation introduced in [10],
Gf(B, () Rn ,
=
°
O(B. ).= 8F(B, F(B, . ,u)-I, U + c:) G ' ,u . 8 c:
Following the classification introduced in the continuous-time context [13], we introduce in this section two slightly different strict-feedback forms in discrete time which exhibit suitable structures with respect to the free dynamics and the parameter dependencies. The study is performed in the geometric context introduced in [19]. Necessary and sufficient conditions of equivalence under parameter independent coordinates changes are given. To distinguish between two different forms in discrete-time is necessary because a linear parametrization is no longer maintained under coordinates change even being no parameter dependent. Moreover, even limiting the parametric uncertainties to the drift term, parametric uncertainties affect the control dependent part of the dynamics after coordinates change. In particular, this difficulty motivates the introduction of the discrete-time parametric strict feedback form (DTPSFF) of Definition 3.1. With this in mind, we will recall some results from [19] about linear equivalence and feedback linear equivalence which will be generalized to parametrized dynamics.
((k+ I)
where ((, u) ERn x R represent respectively the state and control variables, F(B,',') : R n x R ---+ R n is analytic in its arguments, Fo(B,·) := F(B,·, 0) is assumed to be invertible for any value of BEn and (e = is assumed to be the equilibrium state. For u sufficiently small to guarantee the invertibility of F(B,', u) too, let GO(B,·, u) : Rn ---+ Rn be the parametrized control dependent vector field, defined as
:=
AdF~(II,() dI(B, ().
When B = 0, we recover the canonical family of vector fields G{(() := Gf (0, (), for i? 1, j? introduced in [19].
°
314
S. Monaco et al. 0.5 r----~-------,
o
0.5
o -0.5
L - -_ _~_ _~_ ____'
o
5
10
x,
15
1.6,..-----------, 0.5 1.4 0
15
-0.5
5
0
. \Ll
3
.. . . - . - . . -
2.5 1.5
5
10
15
1.5 0
5
Xl
-
.- . .
.
15
8(22)
(0) = 0.8, xz(O) = -0.9 with 0(0) = 1.6, p(O) = 20, Pii(O) = 10, Pij(O) = 0,
o
-~.~•.
-1 -2
.
15
10
8 (21)
Fig. 2. Dynamics (78), B = 1 and PI (0) = pz(O) = O.
..
- - - -
..........•........... ;
2
o
10
u
'-------~---~-------'
o
5
x,
10
-2 '-----_ _
o
15
~
5
~
x
2
10
_ _-.J 15
1lL=..
o: .
1.4
-
1.2 0
5
8
10
15
0
5
0
5
u
10
15
10
15
2.5
2
1
1.5
1 0
2 ..
. .
........- . . ~
1.5 1 10
5 8(21)
15
8(22)
Fig. 3. Dynamics (78), B=I and XI(O) = I, xz(O) = -I with 0(0) = 1.5, p(O) = 20, Pii(O) = 10, Pij(O) = 0, Pl(O) = pz = 1(0).
315
Adaptive Control in Discrete Time 5,----------,-------,----------.,
10 ,-----~---~-----,
o -5 '-----_ _
o
~
~
5
_ _----J
15
10
x,
-20
0
10 0
0.6
...
-10 0.4 L_----========--_~ 10 15 5 o
o
-20
. v----
0
5
10
· · · ·
.. . .
:
"
5
15
u
.
10
15
10
15
0.4
0.2
o
15
10
5
5
0(21)
Fig. 4. Dynamics (78), B = 0.4 and P2(0) = o.
XI
(0) = 3, X2(0) = -3 with B(O) = 0.6, p(O) = 20, Pii(O) = 10, Pij(O) = 0,
Remark As shown in [19], the discrete-time driftinvertible dynamics (79) admits the exponential representation (( k + 1)
= eU(k)d'(O,.,lI(k)) [ I d ] ,
(84)
1Fo(o.(k))
where ucjl(e,·, u) : IR" -+ IR" is a smooth vector field parametrized by u and and Id denotes the identity function. More precisely, one has
e i
ucjl(e, " u) =
u L 1Bi(G, (e, '), ... , G (e, .)), 0
0 i
i2':1 I.
where Bi(G?, ... , G?) is a Lie bracket polynomial of degree i in the canonical vector fields One obtains for the first terms
GJ.
GOl' B , ..-
0(22)
'-GO2' B 2·-
PI
(0) =
Let us now recall from [19] some results about linear and linear feedback equivalences. The discrete-time dynamics (79) is locally linear equivalent (i.e. linearizable through coordinates change) around (e = 0, for = 0, if and only if,
e
(i) G~(() = 0 for k 2: 2, (ii) [G?, G;](() = 0 for i 2: 0, (iii) Rank[G?,Gj, ... ,G7- I J(O) =n. The discrete-time dynamics (79) is locally linear feedback equivalent (i.e. linearizable through static feedback and coordinates change) around (e = 0, for = 0, if and only if,
e
(i) G~(() E span{G?(()} for k 2: 2, (ii) The distribution span{G~, Gj, ... , involutive (iii) Rank[G?, Gl, ... , G';-'](O) = n
G7- 2 }
IS
or equivalently, This fact puts in light how a discrete-time representation can be computed starting from a family of functions through a formal exponential form according to (84).
(i) G~(() E span{G~(()} for k 2: 2, (ii) The distributions (f:= span{ G~, Gj, ... , G\} o:: : i ::::: n - 1 are involutive and of constant rank equal to i + 1 around O.
316
S. Monaco et al.
5.2. The Parametric Strict-feedback form, PSFF
5.2.1. An Example
Definition 5.1. The discrete-time dynamics on R"
Consider the nonlinear discrete-time dynamics
= x2(k) + 'P1(x,(k),e) x2(k+ I) = x3(k) + 'P2(X,(k),X2(k),e) x,(k+ I)
(,(k+ 1)
=
(2(k) +2«(I(k)
+ (e-
1)(~(k))2
+u(k)(1 +«(,(k)+(e-I)(~(k))2) (2(k+I)=(,(k)+(e-l)(~(k).
XIl-l (k + I)
+ 'P1l-I (XI (k),···, XII-I (k), e) xll(k + I) = 'Po(x(k)) + 'P1l(x(k), e) + (3(x(k), =
xll(k)
u(k), e)
By computations
Fo(e, () = «(2
= 0, 'PI (0, e) = 0,
(3(x, 0, e)
=
... , 'P1l(0, e)
+ 2«(1 + (e -
o 0 G (e,(,u) = G,(e,() = (1
= 0,
1)(~)2,
(I+(e-l)(~)t
(85)
with 'P;'s and (3 analytic functions such that
'Po(O)
(91)
2 a + (2) 0(1'
while, for the nominal dynamics (e
= 0), we have
o{3(~uu, e) lu=o i- 0
0,
will be denoted as the parametric strict-feedback form (PSFF). Theorem 5.1. The discrete-time dynamics (79) is locally equivalent to the parametric strict-feedback form (85) under parameter independent coordinates change if and only if
It can easily show that this dynamics has a strong relative degree equal to 2 with respect to the function >.«() = (2 and the system is linear feedback equivalent. Following (83) and (89), we compute
G:«():= Ad Fo () G~«)
(i) The system is linear feedback equivalent for e = 0 (ii) The following parametric strict-feedback conditions (PSFF) hold
x
for k 2: I
G\ (e,·) E (;/
for 0 ~ i ~ n - 2, (86)
hence gives
G?«()
- 2(~)2)
(4(2~+~) 0(, 0(2
LGo(>. 0 Fo)«() I
GZ(e, () E span{G~«()}
= (I + «(I
=
(I
+ (~),
= 0/0(1. The computation of
G?(e, ()
where
G\ (e, () := Ad Fo (II,() G\ «() -
Ad Fo ()
G\ «() (87)
with
G\ «() := Ad Fo () G\-I «(),
(88)
is a coordinates change underwhich the dynamics (91) is transformed into
and ~
I
0
G1«():=L (>.o~-') xG 1 «(), G'i 0 Ie >. : JR"
-+
(89) XI
JR, an analytic function solution of
a>. x [Go ... G"- 2 ](0) = 0 o(
The parametric strict-feedback conditions are thus satisfied and
I'
~~ x G'r' (0) i- O.
(k
+ I) = x2(k) + ex~(k)
x2(k + I)
=
xl(k)
+ (x2(k) + ex~(k))2
+ (1 + (x2(k) + ex~(k))2)u(k)
'
(90)
The proof of the theorem is given in the appendix.
which is in the parametric strict-feedback form. Remark. In terms of the exponential representations, it can be easily verified that the dynamics (91) admits
317
Adaptive Control in Discrete Time
5.3.1. An Example
the exponential representation
((k + 1)
= eu (k)G?(8,l [IdJl
FO(B,((k))
Consider the discrete-time nonlinear dynamics
.
(J(k+ I) Under the coordinates change x := (O, it takes the form
=
+ ()((I (k)
(2(k)
+ ((,(k) - (~(k) (~(k))2) + u(k))2 (94)
- (i(k))2
+ ()((2(k) + ()((, (k)
-
(2(k+l)
= (, (k) -
(~(k)
- (~(k))2)
with
+ ()((2(k) + ()((, (k)
+ u(k).
By computations,
FoUl, () = ((2
+ 8((,
(n 2+ ((I -
-
(J -
5.3. The Discrete-time Parametric Strict-feedback Form-DTPSFF
Theorem 5.2. The discrete-time dynamics (79) is locally equivalent to the discrete-time parametric strict-feedback form (49) under parameter independent coordinates change, if and only if (i) The system is linear equivalent for () = 0, (ii) The following DTPSFF conditions hold:
GZ((),O
= G~(() = 0 for
6; ((), .) E (/ where
The proof is given in the appendix.
(n
-
(n 2»)2)
)
while, for the nominal dynamics, one has
and
The computation ofGl (() gives Gl (() = 8/8(,. In addition we note that Fo(O = FQJ(O and consequently one has for k 2': 0,
Gfk(() = G~(() Gfk+' (0
= Gl (0
One easily verifies that [G?, G: 1 = 0 and that Rank [G?,Gi](O) = 2. Then, the nominal system is linear equivalent. The DTPSFF conditions (92) are satisfied too; for one has
o
k 2': 2
0::; i ::; n - 2,
(? + 8((2 + 8((1 -
2
and
To verify the conditions of Theorem 5.1, the computation of A solving (90) is required, which can be a difficult task. Easily checkable conditions can be obtained with reference to a slightly more restrictive discretetime dynamics, the one given in Definition 3.\ called the discrete-time parametric strict-feedback form, DTPSFF. The drift term exhibits the same triangular structure as the PSFF, but the control enters linearly and independently from the parameters in the last state equation only. Moreover, in this last equation, the dynamics which does not depend on the parameters, cPo(x), is linear in x. Necessary and sufficient conditions for the existence of a parameter independent coordinates change under which any dynamics of the form (79) is transformed into the DTPSFF (49) are given hereafter.
G~((),O
(? + 8((2 + 8((,
AdFo(O,(lG,
8
8
= (I + 2()(2) 8(1 + () 8(2'
(92) and therefore 6~(() = ()G~((). .One can now easily build the coordinates change
318
S. Monaco et al.
which gives the DTPSFF studied in Section 4 Xl
(k + I)
x2(k + I)
= x2(k) + Ox~(k) = Xl (k)
+ O(x2(k) + Ox~(k)) + u(k).
7. Appendix Proof of Theorem 5.1.: From (i) there exists an analytic function>. : lRn solution of (90) such that
Remark. It can be easily verified that (94) admits the exponential representation ((k
+ I) = eu(kld!(O,l[Idj
x(k + I)
0
Fo), ... , d( >.
lR
0 p;;-l )
are linearly independent and consequently
l'O(O,«(k))
which, under the coordinates change, the form
d>', d( >.
->
X
:= ((), takes
= eUG'!(-) [Idjl''"0 (O,x(k))
with
= (() = [>.
X
>.
0
Fo ... >. 0 F3- l
r
is a coordinates change with (O) = O. Under the action of (95), the dynamics (79), when o= 0, is transformed into xi(k
+ I)
= Xi+l(k),
I:S i:S n - 1
xlI(k + 1) = ipo(x(k)) + (3o(x(k), u(k))
with
6. Conclusions In this paper, the problem of local adaptive stabilization of discrete-time nonlinear systems has been addressed with reference to suitable classes of dynamics. Through simple examples, classical difficulties of nonlinear adaptive control in discrete time have been pointed out. A design procedure, which parallels the continuous-time backstepping techniques, has been proposed. The main difficulties which lie in the loss of linearity with respect to the unknown parameters and the loss of causality are overcome using the concept of a posteriori errors well known in the linear adaptive control literature. Still, overparametrization techniques are needed to set linearly parametrized error dynamics. The extension to multi-input dynamics, technically more complicated, follows the same lines. The design procedure, which is proposed for dynamics which exhibit the discretetime parametric strict-feedback form, can be extended to more general classes of dynamics as an example when (3(x(k), u(k), 0) = (3o(x(k), u(k)) in (85). In that case, under the assumption that (3o(O, u) i=- 0 and thanks to the implicit function theorem, one recovers the DTPSFF through a simple transformation on the input variable.
(95)
(3o(x(k), O) = 0
ipo(O) = 0,
and
(96) 8{3o(x, u)j
8ul u=o i=- O. In the X coordinates, for 0 = 0, the following exponential representation is obtained
(97) with Fo(x)
=
0
F o 0 -l (x)
= [X2 ... XII ipo(xW,
(98)
(99) As a consequence, given diately obtains
G? defined in (89), one imme(100)
and thus
-.
{ 8
8 }
Q'=span ~""'~ , UX II
uX n -/
(101 )
O:Si:Sn-l.
Acknowledgements In the new coordinates, one has The authors are grateful to Stefano Battilotti and an anonymous reviewer for their useful comments which improved the stability analysis.
(102)
Adaptive Control in Discrete Time
319
A(X) = XI, for the dynamics (85) and hence for the dynamics (79).
while 1
AdFo(II,()G\ «() = [Jx
1"'(0 1"'(0
so that from (87)
Proof of Theorem 5.2. Condition (i) ensures the existence of a coordinates change x =
n
xn(k+ I) Condition (86), rewritten becomes
the x coordinates,
III
E span{ ",f) , ... , '" f) .} UX n
UX n -
0:::;
x(k + I) = eU (k)<7i() [Id] _
i:::; n - 2. with
Since, in x one easily computes
o
(107)
1Folx)
1
(103)
G\ (x) =
(106)
i=1
Consequently, in the x coordinates the exponential representation can be computed
6\ «(,O)t-l(X)
[Jc<
= LCiX;(k) +u(k).
Fo(x) =
0
I
F o 0
t
Ci
X/(k)l
o
(108) ..-. line n - i
I
*
(109)
*
It follows that
and
-. = span { -f), " ' , -f)- } , (]
o JxCFo(x)) x
G\ (x) =
f)X n
f)Xn-i
(110)
o I
..-. line n - i - I
where
*
* where "*" corresponds 'to nonzero elements, (103) implies
In the original coordinates, one has AdF(II,()G\«()
= [Jx
(104) 1
AdFo(()G\(() = [Jx
In addition, since
= ... = !Pn(O, 0) = O.
(lOS)
The necessity of conditions (i) and (ii) of Theorem 5.1 can be easily checked. If there exists a coordinates change that transforms (79) into (85), then one easily shows that conditions (i) and (86) are satisfied, with
1"'(0
thus
[Jc<
6\ (0, ()] I,~-l(x) = AdPo(lI,x)G\ (x) - AdPo(x)G\ (x)]
1",«)
,
320
S. Monaco et al.
and (92), in the x coordinates, becomes
E span {
~f) ""'~}
VX n
0::; i::; n - 2.
vX n -/
( III)
Since, in the x coordinates, one easily computes
o
G\ (x) =
o 1
+-
line n - i
*
* and
o Jx(Fo(x)) x G~ (x) =
o 1
+-
line n - i - I
* * (92) clearly implies
Fo(B,x) - Fo(x) = [cpl(x\,B)·· .CPn(x, B)r. (112) In addition, since (O)
=
0 and Fo(B, 0)
=
0, one has ( 113)
The necessity of conditions (i) and (ii) of Theorem 5.2 can be easily checked. If there exists a change of coordinates that transforms (79) into (49), then one easily checks that condition (i) and (92) are satisfied for the dynamic (49) and hence for the dynamic (79).
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European Journal of
Control
A Backstepping-like Approach to Adaptive Control in Discrete Time S. Monaco 1, D. Normand-Cyrot 2 ,* and A. Madani 2 lDipartimento di Informatica & Sistemistica, Universita di Roma "La Sapienza", Via Eudossiana 18,00184 Roma, Italy; 2Laboratoire des Signaux et Systemes, CNRS, Supelec Plateau de Moulon, 91192 Gif sur Yvette, France
Nonlinear adaptive control in discrete time is still largely open. The existing results deal with particular cases and are mainly based upon traditional estimation techniques issued from the linear literature. In this paper, combining a backstepping strategy and "a posteriori" estimation concepts, a novel approach for achieving adaptive stabilization is proposed. It can directly be applied to a class of nonlinear discrete-time systems that, analogous to the continuous-time context, are said to be in a parametric strict-feedback form. Necessary and sufficient geometric conditions guaranteeing the equivalence of a general discrete-time dynamics to the parametric strict-feedback form or to some related classes of dynamics are also proposed. Keywords: Adaptive control; Lyapunov techniques; Nonlinear discrete-time dynamics
1. Introduction A great deal of progress was made during the last decade in the field of nonlinear adaptive control in continuous time [12,13,18,22] whereas the problem is still widely open in discrete time where the available results are essentially based upon traditional linear estimation methods [1,8,14]. Adaptive control of linear systems is largely based upon the certainty equivalence principle: a good estimator is combined with the controller that may achieve stabilization or tracking when the parameters are perfectly known. Even if such an approach is still more
Work partially supported under a grant of the Italian Space Agency. *Email: [email protected] Correspondence and offprint requests to: S. Monaco, Dipartimento di Informatica & Sistemistica, Universita di Roma "La Sapienza", Via Eudossiana 18,00184 Roma, Italy. Email: [email protected]
questionable in the nonlinear situation because parameter estimation is generally not achieved, nevertheless, in discrete time, the absence of finite escape time phenomena reinforces its use. Several specific difficulties must be faced to develop nonlinear adaptive control in discrete time. Thus, only few results [5,11,23, ... ] dealing with particular classes of systems exist. The majority of these [9,20,24,25] combines a classical estimator (gradient, least squares, ... ) with a control that achieves the linearization of the system when the parameters are known. Stability and boundedness are therefore guaranteed under restrictive growth conditions on the nonIinearities. More recently, a control scheme which separates the estimation and compensation phases has been proposed in [26,27]. The control strategy is first used to ensure parametric estimation and then to fulfil control objectives. The success of nonlinear control design based on a feedback linearization technique first suggested to develop adaptive versions of these control strategies. This has been initialized in [7,21,22] for applications to satellites. The transposition to discrete time produces serious problems as the evolution of parametric dependency under feedback and coordinate change is very difficult to handle as nonlinear compositions of functions are involved. However, when considering sampled dynamics, we show in [6] how to guarantee nonlinear adaptive feedback linearization through multirate control design. To achieve adaptive stabilization rather than output tracking is still more difficult as nonlinear discrete-time stabilization is not yet completely solved (see [2-4]) and stabilization through multirate techniques would need to be of order n (the dimension of the state vector).
Received 7 May 1999. Accepted in revisedform 15 June 2000. Recommended by J. Tsinias and A.S. Morse
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Adaptive Control in Discrete Time
We here propose a design procedure which is inspired by step-by-step Lyapunov techniques developed in continuous time as the backstepping one and its adaptive version (see for example [13]). Recalling, at first, that a simple transposition to discrete time of step-by-step Lyapunov techniques results in nonfeasible controllers since they are noncausal, we here show how this problem can be overcome by combining successive parametrizations and "a posteriori" variables as introduced in a linear context [14,15]. A step-by-step technique is performed to design successive parametric adaptation laws in terms of a posteriori estimates while the control is computed at the last step making use of all the previously generated estimates. The approach is discussed with reference to nonlinear discrete-time dynamics which exhibit suitable state representations and parametric dependencies which, analogous to the continuous-time context, will be said to be in parametric strict-feedback forms [16]. The geometric framework proposed in [19] to unify continuous time, discrete time and sampled studies, is here used to discuss the equivalence to these particular classes. Necessary and sufficient geometric conditions ensuring the equivalence of a given discretetime dynamics to one of these canonical forms are given. The present paper generalizes some preliminary studies proposed in [16,17]. The paper is organized as follows. Difficulties and obstacles for the direct application of Lyapunov techniques are illustrated in Section 2 with reference to two elementary academic examples. The main arguments of the proposed solution are also detailed in these cases studies. The design procedure is then generalized in Section 3 to systems satisfying the discrete-time parametric strict-feedback form (DTPSFF). Section 5 discusses necessary and sufficient conditions for the equivalence to the above-mentioned classes of systems. Some examples are worked out throughout the paper. In particular, in Section 4, the feasibility and performances of the method are illustrated through a two-dimensional example.
2. How to Set Backstepping in Discrete Time In this section, we show how to modify the backstepping procedure developed in continuous time [13] to overcome two main difficulties occurring when dealing with discrete-time dynamics. This is worked out in two simple examples drawing a parallel between discrete-time and continuous-time cases. The main aspects of the proposed solution are also described.
2.1. A First Elementary Example
Given the continuous-time one-dimensional dynamics depending on a constant, though unknown, parameter B x(t)
= u(t) + Bcp(x(t)),
cp(O)
= 0,
(1)
the control u(t) = -kx(t) - B(t)cp(x(t))
k 2:
°
(2)
leads, for B(t) = B, to the closed-loop stable linear dynamics x(t) = -kx(t). When B(t) i= B, one obtains x(t) = -kx(t) + 8(t)cp(x(t))
with 8(t) = B- B(t).
(3) The stability analysis of (3) can be carried out by choosing the Lyapunov function
(4) whose time derivative along the closed-loop dynamics (3) is given by: V(t)
= x(t)x(t) + 8(t)8(t) = x(t) [-kx(t) + 8(t)cp(x(t))] = -kx
2
- 8(t)B(t)
(t) + 8(t) [x(t)cp(x(t)) - B(t)].
(5)
Because of the structure of the right-hand side of (5), which we should classify as linear with respect to 8, the choice B(t)
= x(t)cp(x(t))
(6)
leads to
(7) thus guaranteeing the stability of the dynamics (3) with adaption law (6). The transposition of this scheme to a nonlinear discrete-time dynamics of the same form leads to a first difficulty which stands in the nonlinearity with respect to 8 of the Lyapunov increment even if the dynamics under study exhibits a linear dependance on the unknown parameter. To do so, consider the first-order dynamics, x(k + 1)
= u(k) + Bcp(x(k)),
(8)
It is immediately verified that the control (dead-beat control when B(t) = B),
u(k)
=
-B(k)cp(x(k))
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leads to the perturbed closed-loop dynamics x(k + 1)
=
B(k)
with B(k)
=
with p(k) positive, given by
8 - B(k).
(9)
p2 (k )
A stability analysis may be carried out by choosing the function so that ( 10)
However, by computing the increment over one step, 6. V(k) = V(k + 1) - V(k), one gets
+ !(B2(k + 1) - B2(k))
By computing 6. V(k), one obtains
!B
2(k)
-B(k+I)({}(k+I)-B(k)).
It can be immediately verified that p(k) is decreasing
V(k,p(k),B(k)) ~ !p2(k) +!p-I(0)B2(k) ~ O.
-!x2(k) -!(B(k+ 1) _(}(k))2 +
with p(O) > O.
and upper bounded by p(O). This guarantees that V(k, p, B) is positive definite as a function of p and B and lower bounded
6.V(k) = !B2(k)
=
p-l (k + 1) = p-I (k) +
(11)
6.V(k) = _!p2(k) -!(B(k+ I) -B(k))2p- 1 (k) +B2(k+ I)
x ({}(k + 1) - (}(k)) The design of an adaptation law to cancel the last row in the expression of 6. V(k) is not easy because (II) is nonlinear with respect to B(k) and contains both B(k + 1) and B(k). This first difficulty is always verified in discrete time even if the dynamics (8) is linear with respect to 8. It should be pointed out that, for this one-dimensional case, the problem can still be solved according to [11] where an adaptive control law which globally achieves stabilization without any growth restriction on the nonlinearity has been proposed in terms of a modified least-squares estimator. A different solution which has local validity is proposed below and generalized in Section 3. Consider again the one-dimensional discrete-time dynamics (8) and set p(k + I)
=
:=
(12)
thus introducing the a posteriori variable p(k): a copy of the x-dynamics (9), now expressed in terms of the one step ahead parametric error B(k + I). Let us denote by c(k) := p(k) - x(k), the corresponding a posteriori error with dynamics given by
c(k + I) = (B(k + I) - B(k))
Let us now consider the function
(k)
- p-I(k)({}(k + 1) - {}(k))J. It is now possible to choose the adaptation law in such a way to render 6. V(k) negative; i.e.
{}(k + 1) = (}(k)
+ p(k)
(14)
with p(k + 1) from (9), (12) and (14) given by 1
p(k + 1) = 1 + p(k)
(15)
The adaptation law (14) can be expressed in terms of quantities measurables at time k + 1 as p(k)
+ 1 + p(k)
Since p(k) > 0, one obtains 6.V(k)
= _!p2(k) -!p-l(k)(B(k+ 1) -B(k))2 :::; 0
(16)
which implies V(k + 1) :::; V(k) - !p2(k)
p(k+I)-x(k+l) :=
1
+B(k+ I)[
, , 8(k + 1) = 8(k)
(8 - {}(k + I))
r
= -!p2(k) -!(B(k+ 1) -B(k))2
(13)
-!p-I(O)(B(k) -B(k+ 1)(
( 17)
It follows that V(k) is decreasing, and admits a limit when k ---+ 00. Taking the limit in (17), one gets
V(oo) :::; V(oo) - b(oo) with b(k)
~
0, thus proving
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that b(k) is bounded and b( 00) = O. It follows that p(k) and (e(k + I) - e(k)) are bounded and that p(oo) = 0 and B( 00) = Cst. Because of the convergence to a constant value of the parameter estimates, the convergence to zero of x(k) follows from (13) provided
estimates at time k, measurable data at time k are needed. (iv) The proposed method can be directly applied to higher-dimensional dynamics provided that the constant unknown parameters are matched with the control, i.e. the so called parametric strict matching form,
1-
B(k+ I)
=
I +p(k)
and thus, in general, parameter estimation does not hold, i.e. e( 00) i= O. The following arguments show how to guarantee at least locally the boundedness of
xn-I(k+ 1) =xn(k) xn(k + 1) =
(18) Figure 1 illustrates the result of a simulation when
Another typical difficulty appears when the unknown parameters do not match the control variables. To illustrate this problem, let us consider the continuoustime dynamics XI (t) X2(t)
By imposing 2
a C:'::: 2p(O)M' one guarantees that x(k) E Io: and thus the boundedness of
= -8(k)
p(k)
= B(k) + I + p(k)
locally, with respect to x and B, asymptotically stabilizes the dynamics =
u(k)
+ B
( 19)
The backstepping Lyapunov approach consists in considering X2 as a virtual control to stabilize the subsystem XI setting X2(t) = -kxl (t) - B(t)
X2 - a(xl, 8),
Setting ZI := XI, one computes ZI (t)
= XI (t) = Z2(t) + a(xi (t), 8(t)) + B
Z2(t) = u(t) -
Remarks.
(i) The previous arguments put in light the local validity of the result both with respect to x and B. (ii) The same arguments hold true by considering a stabilizing control of the form u(k) = dx(k) - 8(k)
= X2(t) + B
where
p(k) p(k+ I) = I +p(k)
x(k + I)
+ B
aa(~~:~;~(t)) (X2(t) + B(t)
_ aa(xl ~t), 8(t)) (}(t). aB(t)
(21)
The control law u(t)
=
-ZI
(t) - k2z 2(t) +
aa(xi (t), 8( t)) aXI (t) (X2(t)
d:'::: 1.
(iii) Causality is respected in the sense of instantaneous computations, i.e. to compute the
+ 8( t)
(22)
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3
0
2.5
-1
2
5
x1
10
15
5
0
10
15
10
15
0 -2 0.5 -4
0
0
5
P
10
15
5
u
1.5
Fig. 1. Dynamics (8) with B = 1.5 and i.e. x(O) = I, 0(0) = 3, p(O) = I.
with k 2
0, leads to the closed-loop error dynamics
;:::
21 (t)
=
-klz l (t)
+ Z2(t) + B(t)cp(xl (t)),
22(t) = -Z[ (t) - k 2z2(t) _ oa(xl (t), B(t)) O-() ( ()) ax 1(t) t cP XI t .
(23)
This guarantees the stability of the error dynamics. Using the LaSalle's invariance principle, one can show that z(t)---+O when t---+oo and consequently x(t)---+O when t---+oo. With this in mind let us consider the second-order discrete-time dynamics
(k + I) x2(k + I)
=
XI
Choosing the function V2 -_ 2:12 z)
12
10-
+ 2: z2 + 2:
'
(24)
whose time derivative along the closed-loop dynamics (23) is given by
V2(t)
= -klzT(t) -
_
k2Z~(t) + B(t) (ZI (t)CP(XI (t))
() oa(xl (t), B(t)) ( ()) _ 0:"( )) Z2 taX) (t) cP XI t t.
(27)
(28) Setting ZI
= XI,
one computes the z-dynamics
+ I) = z2(k) + B(k)cp(xl (k)) z2(k + 1) = x2(k + I) + B(k + I )cp(XI (k + 1))
= ZI (t)CP(XI (t)) oa(xI(t),B(t)) ( ()) - Z2 () t OXI(t) cP XI t
+ Ocp(xl(k))
Considering X2 as a virtual control, XI goes to zero by setting X2 = -Ocp(XI)' However, as X2 is not a control variable, one introduces the z2-variable as the difference between X2 and its computable value, i.e.
ZI
one immediately sets,
B(t)
=
x2(k) u(k).
(k
= u(k) + B(k + I)CP(Z2(k) (25)
+ B(k)cp(Xl (k))).
+ I) is no longer linearly parametrized with respect to B(k). In addition, the design of a control to linearize the zz-dynamics is, It must be noted that z2(k
which leads to
(26)
(29)
Adaptive Control in Discrete Time
303
in fact, not achievable as it would be noncausal due to the presence of B(k + I). We show in the sequel that the introduction of suitable a posteriori variables following the lines of the first example makes it possible to overcome the problem. Consider again the dynamics (27) and now set a coordinates change as in (28) but one step backward in time with respect to the parametric estimation, ZI (k) = XI (k)
+ B(k -
z2(k) = x2(k)
(30)
l)cp(xl (k)),
so that now
Let us now introduce the two a posteriori variables PI and P2 with the dynamics of PI defined as the one of ZI but in terms of the one step ahead in time parametric error
Let the candidate Lyapunov function be
Ai 2 Ai -I -2 V I (k,PI(k),8(k)) =2PI(k) +2 PI (k)e (k),
with PI (0) > 0 and PI I (k + 1) = PI I (k) By computing ~ VI (k) one obtains
Ai 2
Ai 2
~Vl(k) =2P2(k) -2PI(k)
x pjl (k)
l
[1 + cp2(k)PI(k)r x [XI (k + I) + P2(k) - x2(k) - B(k)cp(k)]
(34)
which is now a function of measurable quantities at time k+ 1. Let us now compute from (30) the dynamics of Z2
z2(k + 1)
=
u(k)
+ B(k)cp(xi (k + 1)).
Since the unknown parameter enters nonlinearly in cp(XI (k + I)), it is now necessary to perform a new linear parametrization of cp(XI (k + 1)) which, in general, induces an overparametrization. To do so, assuming the existence of a finite parametrization, we rewrite B(k)cp(xi (k + 1)) = B(k)cp(X2(k) + 8cp(xl (k))) as
B(k)cp(X2(k))
+ 8~¢>(xl (k), x2(k), B(k))
Ai (-)2 -2 8(k+ 1) -8(k) 1)
It is now possible to choose the adaptation law in such a way to cancel the last addendum in the right hand side of ~ VI (k): it is
B(k)
+ PI (k)cp(xi (k))PI (k +
1),
(33)
thus obtaining
~ VI (k) = ~i p~(k) - ~i PT(k) - ~i (8(k + I) -
(36)
with 82, a finite linear parametrization involving powers of 8; ¢>(O,·,·) = O. Denoting below ¢>(XI (k), x2(k), B(k)) as ¢>(k) and substituting (36) into (35), gives
+ B(k)cp(X2(k)) + 8~¢>(k). (37)
u(k)
+ cp2(XI (k)).
+ Ai8(k + 1)[cp(XI (k))PI (k +
(35)
We can now design the certainty equivalence stabilizing control law
+pjl(k)(B(k) -B(k+ 1))].
=
=
positive and given as in the first
IS
pi (k)cp2 (XI (k)) PI(k+ 1) =PI(k) -1 +cp2(XI(k))PI(k)
B(k + 1)
PI(k+ 1)
z2(k + 1) = u(k)
Al E JR,
where PI (k) example by
8cp·(k), one computes
8(k))2p]l(k).
Denoting CP(XI (k)) as cp(k), substituting Eq. (33) into (32) and taking into account that XI (k + 1)- x2(k) =
= -B(k)cp(X2(k)) -
B~(k)¢>(k),
(38)
where B2 (k) is an estimate of 82 at time k so that the closed-loop z2-dynamics becomes
z2(k + 1)
= 8~(k)¢>(k),
(39)
where 82(k) = 82 - 02(k) is the estimation error over 82 at time k. Define now the dynamics of the aposteriori variable P2 (k + I) which is again the z2-dynamics one step ahead in time with respect to 82
P2(k + I)
:=
e~(k + I)¢>(k),
(40)
and the resulting a posteriori error Pi(k + 1)zi(k + 1) := Ei(k + 1) thus obtaining from (31), (32), (39) and (40) the error dynamics
EI(k
+ 1) = E2(k) -
[B(k
+ I) -
B(k - I)]cp(k) (41 )
(42)
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Let the candidate Lyapunov function be
V2(k)
= V, (k) + +
Arguing as in the previous example it follows that
A22
2 p~(k)
V2(k+
~~e2(k)IP2](k)e2(k)
A~ > AT,
with
- ~i P-' (0) (e(k) A2'
where the gain matrix P2 (k) is given by
P (k 2
+
+ ¢(k)¢l(k).
The computation of flV2 (k) gives flV2 (k)
Ai
2 (k)
2 PI
+
- ~i (e(k+
Ai - A~ 2 (k) 2
P2
+ P2 1(k)(e2(k + I) - e2(k))] I I - 2A~ (02(k + I) - 02(k)) P2 (k)
(e2 (k + I) - e2 (k)).
It is now possible to choose the adaptation law in such a way to render fl V2 (k) negative; it is
e2(k + I) e2(k) + P2(k)¢(k)P2(k + I) =
(43)
which leads to fl V2(k)
=-
~i phk) + Ai ; A~ p~(k)
- ~i (e(k + I) - 2A~ (02(k + x
+ I) -
I
e2(k)).
I
we deduce that parametric estimation does not hold. As far as the local boundedness of tp(x) and ¢(x, e) is concerned, we preliminarily note that for any positive constant C ;::: 0, =?
A~p~(k) :::; 2C,
A~e:(k)ei(k) ~ 2Ai,MC,
I
I) - 02(k)) P2 (k)
(e2 (k + I) - e2 ( k)).
Substituting (43) into (40), because of (37), (38) and (30) one gets
P2(k + I)
-
O2 (k))
V2 (k) is monotonically decreasing and admits a limit for k ---. 00. It follows that the (p;(k))'s as well as the (e;(k+I)-e;(k))'s are bounded for i=I,2 and limk->oop;(k) = 0, limk->ooe;(k) = Cst; for i= 1,2 (0, := 0). From Eqs (41) and (42), because their triangular structure, we can deduce the boundedness and convergence to zero of z, (k) and z2(k) provided the boundedness of ¢(x" X2, e) and tp(x,) is guaranteed. For, from (42), z2(k) is bounded and Iimk->oo z2(k) = limk->oo P2(k) = 0, while,consequently,from(41),z, (k) is bounded and limk->oo (k) = Iimk->oo PI (k) = O. Then, through the triangular coordinates change (30), we get limk->oo x, (k) = Iimk->oo z, (k) = 0 and from the boundedness of tp(x,) again, we conclude the boundedness of x2(k) and limk->oo x2(k) = limk->oo z2(k) = 0 since tp(O) = O. Since
V2(0) ~ C
e(k))2PI ' (k) -
e(k + 1))2
z,
I) - e(k))2 pl '(k)
+A~e~(k+ 1)[¢(k)p2(k+ I)
x
.
x p 2 1 (0)(e2(k
to guarantee that P21(k) is always positive definite with P2 (0) > 0 and
= _
-
- 22 (0 2 (k + I) -
I) = P (k) _ P2(k)¢(k)q/(k)P~(k) 2 1+ ¢t(k)P2(k)¢(k)
P2 ' (k + I) = P2 1(k)
~T PT(k) + Ai; A~ p~(k)
I):::; V2(k) -
= [I + ¢t(k)P2(k)¢(k)rl x (x2(k + I) + e(k)tp(x, (k + I))),
(44) i.e. P2 can be computed from measurable quantities at time k+ I.
i
= 1,2,
(45)
where A;,M denotes the maximum eigenvalue of P;(O). Let I Q ; be neighborhoods of 0 in IR such that x~ ~ a}, let M] be a bound of tp2(Xl) over IQI' let A:h be a bound of ¢t (XI, X2, e)¢(x" X2, e) over I Q1 x I Q2 and any bounded value of Note that, Nh depends on C, (denoted as A:h( C) when necessary). Moreover, for any initialization e(O) which satisfies (45), then e(k) := 0 - B(k) remains bounded as e(k) is bounded. Finally, let Me be a bound of 02 . It is now possible to show the existence of C such that, under suitable intializations of the control algorithm as in (45), if Xi(O) E I Q ; apd xl(l) E IQI' then x(k) remains in I Q1 x I Q2 and O(k) is bounded thus implying the boundedness of tp(k) and ¢(k).
e.
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Adaptive Control in Discrete Time
To do so, from (45), because of (39), one computes
z~(k + I) = (B~(k)¢(k))2
:::; 2)..2,:['h
with
C
(k PI
2
Then, since from (31),
(k
xl(k + 2) := zl(k + 2) = z2(k + I) + B(k)
2(k
+
2)
:::;
4)..2,MM2 C )..2 2
+
4)..I,MMI C
XI
ai
(46) Since fl (0) < 0, it is possible, by continuity, to find C> such that (46) holds true. This guarantees that XI (k + 2) remains in I and thus the boundedness of
a,
x2(k + I)
:= XI (k
+ 2) - B(k)
x2(k + I) < 8)..2,MM2C + 8)..I,MM I C + 2M M . -)..2 2
e
)..2 I
Since M I does not depend on choose a2 and M 1 so that
a2,
1)=X2(k+I)+B(k)
1
3. Adaptive Control of Parametric Strict-feedback Forms For the sake of simplicity, the approach is worked out on a particular class of discrete-time dynamics: the discrete-time parametric strict-feedbackform. Some slightly more general classes of systems can also be treated according to the same arguments as discussed in the concluding section.
+ I) = x2(k + I) = XI
(k
~a~ - 2MeM~) .. I)..~
<
8()..1)..2,MM2 + )..2)..I,MM I ) -
°
B)
which guarantees that x2(k + I) remains in I a2 too and thus the boundedness of ¢ over the evolution of x results to be verified. In conclusion, we can state that the control law u(k) = -B(k)
= B2(k) + P2(k)¢(k)p2(k + I)
+
i=n
xll(k + I) =
L c/x/(k) +
(49)
/=1
where
(48)
B2(k + I)
+
XIl-J (k + I) = xll(k)
°
C-
and B, asymptotically stabi-
it is possible to
Moreover, similar arguments as before, ensure the existence of C> such that :=
X
(k + I)
(47)
!2(C)
B(k)
Definition.3.1. DTPSFF The discrete-time dynamics defined on R" as
one gets to 2
= Xl (k + I) + P2(k) - x2(k) I +
= x2(k) + B
)..2 I
By imposing xi(k + 2) :::; and recalling that M2 depends on C, we get the condition ( 2)..2)..2 fl(C):=CI 12 <0. 4()..i)..2,MM2(C) + )..~)..I,MMI)-
°
+
P2
I)
locally, with respect to lizes the dynamics
we get with PI (0) := )..I,M, xI
+
(0, B) = 0, ... ,
°
(50)
will be referred to as the discrete-time parametric strictfeedback form. Such a dynamics is triangular with respect to the state variables; the control acts linearly with a constant coefficient in the last state equation only (i.e. the system has relative degree equal to n for YI :=xd. To achieve stabilization, we propose a step-by-step procedure which is worked out in the sequel assuming, without loss of generality, that the functions
PI(k)
PI(k+ I) =pJ(k) - I
+
P (k + I) = P (k) _ P2(k)¢(k)¢t(k)P~(k) 2 2 1+ ¢t(k)P2(k)¢(k)
Step 1. When B is known and manipulating the variable X2 as a control variable, one could stabilize XI
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by setting
Easy computations lead to
~ VI (k) = ~T p~(k) - ~T pf(k)
X2 = -B1'"n(xI). However, as X2 is not the control, we define a new variable Z2 as the difference between X2 and its computable value delayed of n - I steps backward in time, i.e.
-
~Tpil(k)(iJ(k+ I) -iJ(k))2
+>.fO(k+ I) ['"n(XI(k))PI(k + I) +pil(k) x (iJ(k
where B(k - n + I) is an estimate of B at time k-(n-I). The objective being stabilization, we set Zl (k) = XI (k), thus obtaining
zl(k+ I)
=
z2(k)
+ Ol(k -
B(k + I)
(55)
~VI(k) = ~T p~(k) _ ~T pf(k) _ ~T pil(k) -
x (B(k
+ I) + P2(k) - x2(k) - '"YI (k)B(k) I + PI (kh?(k) XI (k + I) + P2(k) - z2(k) - , I (k) L7~/ PI (k + I) + P2(k)
2
ih (k - i)Pl (k - i + I)
+ PI (kh?(k)
-,I
- z2(k) (k)(B(k) - B(k - n + I)) 1 + PI (kh?(k)
estimate, i.e.
(56)
which is a function of measurable variables at time k+ I.
Introducing also the a posteriori errors Cl := PI - ZI, c2 := P2 - Z2, we get the a posteriori error dynamics = C2 (k) - [B( k
+ I)
- B(k - n + I)
x ,(Xl (k)).
Step 2. Let us now compute the dynamics of from (51) as
z2(k + I)
r
(54)
Let the candidate Lyapunov function be
=
PI (k) I +PI(kh?(XI(k))'
so that Pi l (k + I) = Pi' (k) + ,?(XI (k)) and Pi l (k) is always positive and lower bounded by PI (0) > O.
= x)(k) + B1'2(XI(k), x2(k)) + B1(k - n + 2hl(xl(k + I))
Z2
(57)
which is nonlinear with respect to the unknown parameters B in (XI (k + I )). It is thus necessary to perform a new linear parametrization of (XI (k + I)) which generally induces an overparametrization. To do so, assume the existence of a finite linear parametrization, B2 , involving powers of B, such that it becomes possible to rewrite in (57),
,I
where PI (k) is given by
PI(k+ I)
-
B(k)) .
= XI (k
XI (k
+ I)
+ I) -
Substituting (55) into (53) and taking into account of (49) one obtains
I
C I (k
= B(k) + PI (k)'"n (Xl (k) )PI (k + I)
gIves
n + Ihl(xI(k)),
where O(l) := B- B(l) is the estimation error at time I. Let us now introduce the a posteriori variables PI and P2 and define the dynamics of PI as the dynamics (52), in terms of a one step ahead parameter
+ I)
iJ(k))].
The choice of the adaptation law
(52)
PI (k
+ I) -
B1'2(XI (k), x2(k))
,I
+ {)l(k -
n + 2h (Xl (k + I))
as
()l(k - n + 2h (X2(k)) ()l(k - n + 2)),
+ B~r2(XI (k),
x2(k),
(58)
307
Adaptive Control in Discrete Time
where fz(O,xz,')='z(O,xz) and thus fz(O,O,·)= IZ(O,O) = O. Substituting (58) into (57) with fz(k) := fz(xi (k), Xz (k),fF (k - n + 2)), one gets
zz(k + 1)
= x3(k) + (Y(k -
n + 2)rI (xz(k))
where Pz(k) is given by
that Pi l (k+l)=Pi'(k)+fz(k)q(k) and Pi ' (k) is always positive definite if Pz(O) > O. It follows that so
+ B~fz(k). As before, let us now introduce the variable Z3 as the difference between X3 and its computable value delayed n - 2 steps backward in time with respect to Bz, i.e.
~Vz(k) = - ~~ p~(k) + >.~ ; >.~ p~(k) + ~~ p~(k)
z3(k) = x3(k) - Qz({j(k - n + 2), (jz(k - n + 2), xI(k),xz(k))
-
(59)
~~ PI] (k)(8(k + 1) -
8(k))z
+ >'~8~(k + 1)[fz(k)pz(k + I)
with
+ P"i l (k) (8z(k + 1) - 8z(k))]
Qz({j(k - n + 2), (jz(k - n + 2), XI (k), xz(k)) :=
- 2>.~ (Bz(k +
-{jl(k - n + 2)rI (xz(k)) - (jHk - n + 2)f z(k).
-
1) - Bz(k))
I
x P"i I (k) (8z(k + 1) - 8z (k)).
Because of (59), the dynamics of Zz becomes (60)
To cancel the term between the brackets, we can choose the adaptation rule for Bz
We can now define the dynamics of the a posteriori variable pz as the dynamics of Zz anticipated at time k + 1 in 8~, i.e. which leads to
For the a posteriori error dynamics, setting P3 - Z3, one gets,
~ Vz(k) = - ~~ p~(k) + >.~ ; >.~ p~(k) + ~~ p~(k)
C3 :=
-
cz(k + 1) = c3(k) - [(jz(k + 1) - (jz(k - n + 2)ff z(k).
~~ Pl I (k)(8(k+ I) -
- 2>.~ (Bz(k + 1) - Bz(k))
(62)
Let the candidate Lyapunov function be X
Vz(k)
=
VI (k) +
~~ p~(k) + ~~ 8Hk) Pi I (k)8z(k)
with >.~ > >.~,
pz
k (
1) = xz(k
+
P"i' (k) (8z(k
+ 1) -
I
8z(k) ) .
From Eqs (63) and (61) and taking into account of (49), (51), (58), (59) one obtains
+ 1) + P3(k) - x3(k) + hi (k + 1) I
8(k))z
II (xz(k))){j(k - n + 2) - q(k){jz(k)
+ f~(k)Pz(k)fz(k)
_ zz(k + 1) + P3(k) - z3(k) - q(k) '5:.,;,z Pz(k - i)fz(k - i)pz(k - i + 1) 1 + q(k)Pz(k)fz(k)
-
zz(k + 1) + P3(k) - z3(k) - q(k)({jz(k) - (jz(k - n + 2)) 1+ q(k)Pz(k)fz(k)
(64)
s.
308
this enables us to conclude that (63) is computable at time (k + I). Step n - 1. According to the results of the first n - 2 steps, one gets
Zn-I (k) = Xn-I (k) - cx n_2(B(k - 2), ... , Bn-2(k-2)I(k), ... ,Xn-2(k))
Monaco
el
at.
Let now define the dynamics of the a posteriori variable Pn-l as
Pn-I (k + I) thus getting for dynamics,
= Pn(k) + e~_l (k + I)f n- l (k), Cn
:=
Pn - Zn, the a posteriori error
Cn-I (k + 1)
(65)
with
= cn(k) - [Bn-l(k+ I) - Bn-l(k -I)ffn-l(k).
CX n-2(-) := -B1(k - 2)')'1 (x n-2(k)) - B~(k - 2)r2 (X n-3 (k), Xn-2 (k), B(k - 2)) - ...
X
As before, the choice Vn- l (k)
- B~_2(k - 2)f n- 2 (Xl (k), ... , x n-2(k),
= Vn- 2(k) + A;_I P~-l (k) + A;_I e~_l (k) x r;;~1 (k)en- l (k)
B(k - 2), ... , Bn- 3 (k - 2)),
with A~_I > A~_2' enables us to compute
so that one computes
Zn-I (k + 1)
=
xn(k) + B1')'n-1 (Xl (k), ... , Xn-l (k)) - cx n_2(B(k - 1), ... ,Bn- 2(k - I),
Bn- l (k + I)
= Bn- 2(k) + Pn- l (k)rn-l (k)Pn-1 (k + I)
with Pn-l (k + I) computable at time k + I by
xl(k+ I), ... ,xn-2(k+ I)).
I) _ Zn-I (k + I) + Pn(k) - zn(k) - f~_l (k)(B n- 1(k) - Bn- I (k - I))
(k Pn-l
+
l+f~_I(k)Pn_l(k)fn_l(k)
Zn-l (k + I)
Suppose again it is possible to linearly parametrize zn-l(k+ I) so that
B1')'n-I(Xl(k), .. "xn-l(k)) - CX n-2(·)
= B1(k -Ih
+ B~(k - 1)r2(Xn-2(k),xn~1 (k), 1)) + ... + B~_2(k - l)fn - 2(x2(k), ... ,
+ Pn(k)
- zn(k) - f~_l (k)P n- 1(k - l)f n_ 1(k - I)Pn-1 (k) I + f~_l (k)Pn-1 (k)f n- l (k)
Step n and the control design. From (66), one computes n
zn(k+ I) = u(k)
+ Lc;x;(k) +81')'n(x(k)) ;=1
x (Xn-l (k)) B(k -
xn_l(k),B(k-I), ... ,Bn- 3 (k-I)) + B~_l f
n- l
(Xl (k), X2(k), ... , Xn-l (k),
Suppose that it is possible again to linearly parametrize the dynamics of zn(k + 1) so that
B1')'n(x(k)) - CXn-1 (-)
=
B/(kh (xn(k)) + B~(k)f2(Xn-1 (k), xn(k), B(k))
B(k-I), ... ,Bn-2(k-l)),
+ ... + e~_, (k)rn-I (x2(k), ... ,
= 0. It follows that Zn-l (k + 1) = zn(k) + e~_1 (k - I)f n - l (k)
where f n-l (0, ... )
xn(k), B(k), -
with f
n- I
(k)
:= f
n- l
(Xl (k), ... , Xn-I (k),
B(k-l), ... ,Bn- 2(k-I)) zn(k) =xn(k) -cxn-I(B(k-I), ... 'on-l(k-I), xl(k)" .. ,Xn-l(k)), . (66)
where fn(O, ... )
=
B~fn(XI (k),
, Bn-2(k)) , xn(k),
B(k), ... , Bn- l (k)), 0, thus obtaining n
zn(k+ I) = u(k)
+ Lc;x;(k) +B/(kh(xn(k)) ;=1
+ BHk)f2(x n-l(k),xn(k),B(k))
and
CXn-1 (-)
:=
-B1(k - I )'"n (Xn-l (k)) - B~ (k - I )f 2 X
(x n_2(k),xn_l(k),B(k - 1)) - ...
-B~_I(k-l)fn-I(Xl(k),... ,Xn-l(k),
B(k - I), ... ,Bn- 2(k - I)).
+ ...
+ B~_l (k)f n_1(x2(k), ... , xn(k), B(k), ... , Bn- 2(k))
+ B~fn(Xl (k), Bn- l (k)).
... , xn(k), B(k), ... ,
309
Adaptive Control in Discrete Time
The certainty equivalence stabilizing control can now be easily computed as a function of the data available at time k,
that
~Vn(k) = _ ~T PT(k) + AT ; A~ p~(k) + ...
n
u(k)
LCiXi(k) - (i(k)~'/l(xn(k)) i=l
== -
+
- 0&(k)f 2(x n-1 (k), xn(k), B(k)) - ...
-
- 0~-1 (k)f n-1 (x2(k), ... ,xn(k), O(k), ... ,
A~_l - A~ 2(k) 2
Pn
~T pi! (k)(O(k + 1) A~
- 2: (02(k + 1) A
A
0(k))2 t
1
O2(k)) Pi (k)
On-2(k)) - O~(k)f n(X1 (k), ... ,Xn(k), O(k), ... , (67)
x
(B 2 (k + 1) - O2 (k)) ...
-
~~ (On (k + 1) -
On (k)/ p;;! (k)
x (On(k + 1) - On(k))
Denoting fn(k):== f n(X1(k),X2(k), ... ,xn(k),O(k), 02(k), ... ,On-1 (k)), the closed-loop zn-dynamics and
+ A~B~(k + 1)[fn(k)Pn(k + 1)
the a posteriori dynan1ics become respectively
- p;;l (k)(On(k + 1) - On(k))]. (68) We can now render ~Vn(k) negative by choosing the following adaptation law for On
and (69)
thus obtaining so getting, with En: == Pn - Z n, recalling (54) and (62), the overall triangular a posteriori error dynamics
c!(k+ 1) = c2(k) - [O(k+ 1) - O(k- n+ 1)f')'(k) 102 (k
+ 1) = 103 (k)
- [02(k + 1) - O2(k - n + 2)
fr (k)
-
2
~T pi!(k)(O(k + 1) -
O(k)r
A) t - 2A~ (A02(k + 1) - 02(k) X
-?
Let the candidate Lyapunov function be
Vn(k) = Vn-! (k)
+
Pi! (k) (02(k + 1) - 02(k)) ...
?p~(k) + ?
Bn(k)t
(On(k+ 1) -On(k)r
x p;;! (k) (On (k + 1) -On (k)) ::::; O.
x p;;l (k)Bn(k) with A~ > A~_l and gain matrix Pn(k) given by
P (k 1) n +
==
P (k) _ Pn(k)fn(k)f~(k)Pn(k) n 1 + r~(k)Pn(k)rn(k) ,
so that p;;l (k + 1) == p;;l (k) + fn(k)f~(k) and p;;l (k) is always positive definite if Pn(O) > O. It follows
Substituting (71) into (69) and taking into account of (68), one obtains
Pn(k + 1)
==
[1
+ f~(k)Pn(k)f n(k) ] -1 zn(k + 1) (72)
which is a function of measurable quantities at time
k + 1 with zn(k + 1) given by (66) at time k + 1.
s.
310
any positive constant C 2: 0,
Arguing as before it follows that
AT 2 A~ - AT 2 2 P2(k) - ... Vn(k+ 1) ~ Vn(k) -2PI(k) -
Vn(O) ~ C =* A~p~(k) ~ 2C, 2 -t
-
AiOi(k)Oi(k)
>.~ -2>'~-l p~(k) - ~T pi
-
l
(0)
~
t
1
-2(02(k+ 1) -02(k)) P2 (0) x
(0 2 (k + 1) - O2 (k)) ...
-
~ (On(k + 1) -
X
(On(k + 1) - On(k)).
On(k))1 p;;l (0)
i.e. Vn(k) is monotonically decreasing and admits a limit for k ---+ 00. It follows that the (Pi(k))'s and the ([)i(k + 1) - [)i(k))'s are bounded and that limk----*oo Pi(k) == and limk----*oo [)i(k) == Cst for
°
i==l, ... ,n.
~
2Ai,MC,
. 1
==
1, ... , n,
(74)
where Ai,M denotes the maximum eigenvalue of Pi(O). For i == 1, ... , n, let L ai be a neighborhood of in lR such that xT ~ aT, let M i be a bound of ri(XI, ... ,Xi)2 over L al x ... x La. and let M i be a bound of f:(XI, , Xi, 0, ... , Oi-l)fi(XI, . .. , Xi, 0, .. ., Oi-l) over L al x X L ai for any bounded value of Oi; (M I :== MI as f l :== rl and M i depends on C, denoted M i ( C) if needed). We note that, for any initialization [)(O) which satisfies (74), then Oi(k):== 0 - [)i(k) remains bounded as [)i(k) remains bounded since ~ V n is negative. Finally, let Me be a bound for 02 . It is now possible to show the existence of C such that, under suitable intializations of the control algorithm as in (74), if xi(h) E L ai for i == (1, ... ,n) and }i == (0, ... ,n - i), then x(k) :== (Xl (k), ... ,xn(k)) remains in La :== L al X ... X Lan and Oi(k) is bounded thus implying the boundedness of fi(k). Let us preliminarily note that the closed loop z-dynan1ics satisfies the relation
°
x (O(k + 1) - O(k))2
A~ ~
Monaco et at.
From the triangular expressions of the a posteriori error dynamics (70), we can conclude for i == n to i == 1 that the boundedness and convergence to zero of the ([)i(k + 1) - [)i(k))'s imply the boundedness and convergence to zero of the errors (Ei(k))'s provided the boundedness of the (fi)'s. Then, one concludes the boundedness of the (zi(k))'s and lin1k----*oo Pi (k) == limk----*oo Zi(k) == 0. Then, from the coordinates change defined by (51), (59) and (66)
n
zl(k+n)
==
L:[)i(k)fi(k+n -1). i=l
Arguing as for the two preliminary examples and since Xl (k + n) == Zl (k + n), one gets the immediately
Zl (k) == Xl (k) ==
x2(k)
+ {)t(k -
n + l)rl (k)
z3 (k) ==
X3 (k)
+ ot (k -
n + 2)rl (x2(k))
Z2(k)
By imposing xi(k condition over C
+ n)
::;
ai,
one gets the following
+ {)~(k - n + 2)r 2 (k) (75)
°
zn(k)
==
xn(k)
+ ()t(k -
l)rl(X n-l(k))
+ O~(k - 1)f2(x n-2(k), Xn-l (k), O(k - 1))
(75) can be verified for c> by continuity arguments since!l (0) < 0. This ensures that Xl (k + n) remains in L al and thus the boundedness of rl. Moreover, denoting rj(k) :== rj(XI (k), ... ,xj(k)), and since
(73) we can iteratively conclude for i == 1 to i == n that lin1k----*oo xi(k) == due to the triangularity of the coordinates change (73) and because rl (0) == r 2 (0,0, 0) == ... == f n-l (0, 0, ... ,On-2) == 0. ~ As far as the boundedness off n(x, 0, ... ,On-I), ... , f 2 (XI, X2, 0), rl (Xl), is concerned, one proceeds as in the first two preliminary examples noting that for
°
Xj(k + n -} + 1)
==
xj-l(k + n -} + 2) -ot rj _l (k+n-}+l)
xn(k + 1)
==
Xn-l (k) - ot rn - l (k + 1),
311
Adaptive Control in Discrete Time
one gets also,
e(k+ 1)
= e(k) +Pl(k)'''fI(k)PI(k + 1)
e2(k + I)
=
e2(k) + P2(k)r2(k)p2(k + I)
en(k + I)
= en(k) + Pn(k)rn(k)Pn(k + I)
PI(k+ I)
=
XJ (k + n - j + 1) :::; Yn ~2AMM'C L.. "2' A;
;=1
i-I
+ l:y-p+IMeMp p=1
P (k + I)
=
2
n-I
+
PI(k)
I
l: 2
n
-
+ PI (kh?(xI (k))
P (k) _ P2(k)r2(k)Q(k)P2(k) 2 1+ Q(k)P2(k)r 2(k)
p+ 1MeMp.
p=l
xJ
n-
aJ,
By imposing (k + j + 1) :::; provided the (O))'s and the (A1;)'s are such that for (j = 2, ... , n), i-I
aJ - l:y-p+IMeMp > 0
(76)
with Pl(k+l) 21
p=1
(k + I) - 2:7=dO,(k) - O,(k - n + 1)]tC(k - i + I) 1 + PI (khr(k)
P2(k+ I)
which can be verified because of the triangular structures of the bi)'s, one gets, for j = (2, ... ,n), the following conditions over C
::; O.
22(k + I) - 2:7-2[O,(k) - O,(k - n + 2)]tr,(k - i + 2) 1 + r~(k)P2(k)r2(k)
(77)
Since jj(O) < 0, by continuity arguments, it exists C> 0 such that (77) holds true. Then, it remains proved that there exists C such that the (x;)'s remain in the (a;)'s thus implying the boundedness of the (f;)'s. This ends the proof. In conclusion, we can state that the control law
where r l := change Zl
(k)
=
/'1
:=
and 01 := 0, and the coordinates
XI (k)
z2(k)
= x2(k) + et(k -
n + Ihl(k)
zn(k)
= xn(k) + et(k - Ih (Xn-I (k))
n
u(k) = -
l: G;x;(k) -
et(khl (xn(k))
;=1
- e~(k)r2(Xn-l(k),xn(k),e(k)) - ... - e~(k)rn(xI (k), ... , xn(k),
+ e~(k -1)r2(Xn-2(k),xn-l(k),e(k - I))
x e(k), ... , en- I (k))
+ ... + e~_1 (k -
l)rn _ 1(k)
S. Monaco
312
locally, with respect to x and 8, asymptotically stabilizes the dynamics (k + 1) = x2(k) + 81'YI (k) x2(k+ 1) = x3(k) +8l'Y2(k)
el
al.
with PI (0) > 0, it is possible to choose the adaptation law as
XI
to get b.VI(k)
~~ p~(k) - ~~ pi(k) - ~~ (B(k+ 1)
=
n
xn(k + 1)
= u(k) + L
c;
x;(k)
+ 8l'Yn(k).
-B(k))2pil (k).
;=1
A stabilizing control law is given by
4. An Example
u(k) =
Let us consider the following example on n~?,
-XI
thus getting
+ 1) = x2(k) + 8xi(k) x2(k + 1) = u(k) + XI (k) + 8(X2(k) + 8xi(k))
XI (k
z2(k + 1) = B2(k)lr(x(k), B(k)) P2(k+ 1) = B2(k+ l)lr(x(k),B(k)),
(78) which can be steered to zero in two steps when 8 is known. The choice of x2(k) = -8xi(k) would get XI (k + 1) to zero, but as X2 is not the control and 8 is unknown we set
and the a posteriori error dynamics
cI(k + 1) = c2(k) - [B(k + 1) - B(k - 1)]xi(k) c2(k + 1)
= XI (k) z2(k) = x2(k) + B(k - l)xi(k), ZI
(k) - B(k)x~(k) - B~(k)r(x(k), B(k)),
= -
[B2(k + 1) - B2(k)fr(x(k),B(k)).
(k)
V2(k)
and compute the z-dynamics zl(k + 1)
=
z2(k + 1)
=
=
Setting, for A~ > A~ and P2(0) > 0,
+ B(k - l)xi(k) u(k) + XI (k) + 8[X2(k) + (h~(k)] + B(k)x~(k + 1) u(k) + Xl (k) + B(k)x~(k)
P 2 (k
z2(k)
=
= VI (k) + ~~ p~(k) + ~~ B2 (k)lp:; I (k)B2(k)
+ 1) P (k) _ P2(k)r(x(k), ~(k))rl(x(k), B(k)):i(k) 2 1 + fL(x(k), 8(k))P2(k)r(x(k), 8(k)) ,
the following adaptation law
+ 8~r(x(k), B(k)) with
can now be chosen to render b. V2(k) negative. The control law so far computed u(k)
and 82 = (8, 82 Setting
r an overparametrization of 8.
PI(k + 1) VI (k, PI (k), B(k)) PI (k + 1)
=
-XI
B(k + 1)
(k) - B(k)x~(k) - B~(k)r(x(k), B(k))
= B(k) + PI (k)xi(k)PI (k + 1)
B2(k + 1)
= B2(k) + P2(k)r(x(k), B(k))P2(k + 1)
= P2(k) + B(k + l)xi(k)
PI (k + 1)
=
= ~~ pi(k) + ~~ pi l (k)B2(k)
P2(k+ 1)
p~(k)xt(k)
= PI (k) - 1 + xt(k)PI (k)
=
PI (k) 1 + xt(k)PI (k)
P (k) _ P2(k)r(x(k), ~(k))rl(x(k),B(k)):~(k) 2 1 + fL(x(k), 8(k))P2(k)r(x(k), 8(k))
313
Adaptive Control in Discrete Time
with
f(x(k), e(k)) = (x2(k) PI (k +1 )
P2
k (
+ 2e(k)X2(k)xT(k), xT(k) + e(k)xi(k))l
xI(k + I) - (eHk) - eHk - I))f(k - I) - (e(k) - e(k - I))xi(k) I + PI (k)xi(k)
=----'--'-----"'-------'--~-'-----------''-'-------'--'---:-:'-:----:;--:-:-:---'---------'-----'----'--'-----'--'----'--
) _ x2(k+ I) + e(k)xi(k + I) 1 + P(k)P2(k)f(k)
+1
locally, with respect to x and B, asymptotically stabilizes the dynamics (78). Some simulation results are reported in Figs 2-4 which also illustrate the deadbeat stabilization and the possible estimation of the parameter (Fig. 4).
5. Necessary and Sufficient Conditions for the Equivalence to Strict-feedback Forms
5.1. The Geometric Context Consider a general discrete-time dynamic on nonlinearly parametrized by the vector BEn c ffi.P,
F(B,((k),u(k)),
(79)
I 0=0
' (80)
and compute its expansion with respect to u around 0, th us getting
(81 ) with G?(B,·) := GO(B, ,,0) and
.) '= 8i-lGO(B, " u) I GO(B I ,. 8u i - 1 u=o
i
> 1.
(82)
Denoting by F6(B,.) the j-times compOSItIon of Fo(B, '), let Gf(B,·) be the transport of G7(B,·) along the free evolution
Fb(e, '), i.e.
(83) or according to the Ad operator notation introduced in [10],
Gf(B, () Rn ,
=
°
O(B. ).= 8F(B, F(B, . ,u)-I, U + c:) G ' ,u . 8 c:
Following the classification introduced in the continuous-time context [13], we introduce in this section two slightly different strict-feedback forms in discrete time which exhibit suitable structures with respect to the free dynamics and the parameter dependencies. The study is performed in the geometric context introduced in [19]. Necessary and sufficient conditions of equivalence under parameter independent coordinates changes are given. To distinguish between two different forms in discrete-time is necessary because a linear parametrization is no longer maintained under coordinates change even being no parameter dependent. Moreover, even limiting the parametric uncertainties to the drift term, parametric uncertainties affect the control dependent part of the dynamics after coordinates change. In particular, this difficulty motivates the introduction of the discrete-time parametric strict feedback form (DTPSFF) of Definition 3.1. With this in mind, we will recall some results from [19] about linear equivalence and feedback linear equivalence which will be generalized to parametrized dynamics.
((k+ I)
where ((, u) ERn x R represent respectively the state and control variables, F(B,',') : R n x R ---+ R n is analytic in its arguments, Fo(B,·) := F(B,·, 0) is assumed to be invertible for any value of BEn and (e = is assumed to be the equilibrium state. For u sufficiently small to guarantee the invertibility of F(B,', u) too, let GO(B,·, u) : Rn ---+ Rn be the parametrized control dependent vector field, defined as
:=
AdF~(II,() dI(B, ().
When B = 0, we recover the canonical family of vector fields G{(() := Gf (0, (), for i? 1, j? introduced in [19].
°
314
S. Monaco et al. 0.5 r----~-------,
o
0.5
o -0.5
L - -_ _~_ _~_ ____'
o
5
10
x,
15
1.6,..-----------, 0.5 1.4 0
15
-0.5
5
0
. \Ll
3
.. . . - . - . . -
2.5 1.5
5
10
15
1.5 0
5
Xl
-
.- . .
.
15
8(22)
(0) = 0.8, xz(O) = -0.9 with 0(0) = 1.6, p(O) = 20, Pii(O) = 10, Pij(O) = 0,
o
-~.~•.
-1 -2
.
15
10
8 (21)
Fig. 2. Dynamics (78), B = 1 and PI (0) = pz(O) = O.
..
- - - -
..........•........... ;
2
o
10
u
'-------~---~-------'
o
5
x,
10
-2 '-----_ _
o
15
~
5
~
x
2
10
_ _-.J 15
1lL=..
o: .
1.4
-
1.2 0
5
8
10
15
0
5
0
5
u
10
15
10
15
2.5
2
1
1.5
1 0
2 ..
. .
........- . . ~
1.5 1 10
5 8(21)
15
8(22)
Fig. 3. Dynamics (78), B=I and XI(O) = I, xz(O) = -I with 0(0) = 1.5, p(O) = 20, Pii(O) = 10, Pij(O) = 0, Pl(O) = pz = 1(0).
315
Adaptive Control in Discrete Time 5,----------,-------,----------.,
10 ,-----~---~-----,
o -5 '-----_ _
o
~
~
5
_ _----J
15
10
x,
-20
0
10 0
0.6
...
-10 0.4 L_----========--_~ 10 15 5 o
o
-20
. v----
0
5
10
· · · ·
.. . .
:
"
5
15
u
.
10
15
10
15
0.4
0.2
o
15
10
5
5
0(21)
Fig. 4. Dynamics (78), B = 0.4 and P2(0) = o.
XI
(0) = 3, X2(0) = -3 with B(O) = 0.6, p(O) = 20, Pii(O) = 10, Pij(O) = 0,
Remark As shown in [19], the discrete-time driftinvertible dynamics (79) admits the exponential representation (( k + 1)
= eU(k)d'(O,.,lI(k)) [ I d ] ,
(84)
1Fo(o.(k))
where ucjl(e,·, u) : IR" -+ IR" is a smooth vector field parametrized by u and and Id denotes the identity function. More precisely, one has
e i
ucjl(e, " u) =
u L 1Bi(G, (e, '), ... , G (e, .)), 0
0 i
i2':1 I.
where Bi(G?, ... , G?) is a Lie bracket polynomial of degree i in the canonical vector fields One obtains for the first terms
GJ.
GOl' B , ..-
0(22)
'-GO2' B 2·-
PI
(0) =
Let us now recall from [19] some results about linear and linear feedback equivalences. The discrete-time dynamics (79) is locally linear equivalent (i.e. linearizable through coordinates change) around (e = 0, for = 0, if and only if,
e
(i) G~(() = 0 for k 2: 2, (ii) [G?, G;](() = 0 for i 2: 0, (iii) Rank[G?,Gj, ... ,G7- I J(O) =n. The discrete-time dynamics (79) is locally linear feedback equivalent (i.e. linearizable through static feedback and coordinates change) around (e = 0, for = 0, if and only if,
e
(i) G~(() E span{G?(()} for k 2: 2, (ii) The distribution span{G~, Gj, ... , involutive (iii) Rank[G?, Gl, ... , G';-'](O) = n
G7- 2 }
IS
or equivalently, This fact puts in light how a discrete-time representation can be computed starting from a family of functions through a formal exponential form according to (84).
(i) G~(() E span{G~(()} for k 2: 2, (ii) The distributions (f:= span{ G~, Gj, ... , G\} o:: : i ::::: n - 1 are involutive and of constant rank equal to i + 1 around O.
316
S. Monaco et al.
5.2. The Parametric Strict-feedback form, PSFF
5.2.1. An Example
Definition 5.1. The discrete-time dynamics on R"
Consider the nonlinear discrete-time dynamics
= x2(k) + 'P1(x,(k),e) x2(k+ I) = x3(k) + 'P2(X,(k),X2(k),e) x,(k+ I)
(,(k+ 1)
=
(2(k) +2«(I(k)
+ (e-
1)(~(k))2
+u(k)(1 +«(,(k)+(e-I)(~(k))2) (2(k+I)=(,(k)+(e-l)(~(k).
XIl-l (k + I)
+ 'P1l-I (XI (k),···, XII-I (k), e) xll(k + I) = 'Po(x(k)) + 'P1l(x(k), e) + (3(x(k), =
xll(k)
u(k), e)
By computations
Fo(e, () = «(2
= 0, 'PI (0, e) = 0,
(3(x, 0, e)
=
... , 'P1l(0, e)
+ 2«(1 + (e -
o 0 G (e,(,u) = G,(e,() = (1
= 0,
1)(~)2,
(I+(e-l)(~)t
(85)
with 'P;'s and (3 analytic functions such that
'Po(O)
(91)
2 a + (2) 0(1'
while, for the nominal dynamics (e
= 0), we have
o{3(~uu, e) lu=o i- 0
0,
will be denoted as the parametric strict-feedback form (PSFF). Theorem 5.1. The discrete-time dynamics (79) is locally equivalent to the parametric strict-feedback form (85) under parameter independent coordinates change if and only if
It can easily show that this dynamics has a strong relative degree equal to 2 with respect to the function >.«() = (2 and the system is linear feedback equivalent. Following (83) and (89), we compute
G:«():= Ad Fo () G~«)
(i) The system is linear feedback equivalent for e = 0 (ii) The following parametric strict-feedback conditions (PSFF) hold
x
for k 2: I
G\ (e,·) E (;/
for 0 ~ i ~ n - 2, (86)
hence gives
G?«()
- 2(~)2)
(4(2~+~) 0(, 0(2
LGo(>. 0 Fo)«() I
GZ(e, () E span{G~«()}
= (I + «(I
=
(I
+ (~),
= 0/0(1. The computation of
G?(e, ()
where
G\ (e, () := Ad Fo (II,() G\ «() -
Ad Fo ()
G\ «() (87)
with
G\ «() := Ad Fo () G\-I «(),
(88)
is a coordinates change underwhich the dynamics (91) is transformed into
and ~
I
0
G1«():=L (>.o~-') xG 1 «(), G'i 0 Ie >. : JR"
-+
(89) XI
JR, an analytic function solution of
a>. x [Go ... G"- 2 ](0) = 0 o(
The parametric strict-feedback conditions are thus satisfied and
I'
~~ x G'r' (0) i- O.
(k
+ I) = x2(k) + ex~(k)
x2(k + I)
=
xl(k)
+ (x2(k) + ex~(k))2
+ (1 + (x2(k) + ex~(k))2)u(k)
'
(90)
The proof of the theorem is given in the appendix.
which is in the parametric strict-feedback form. Remark. In terms of the exponential representations, it can be easily verified that the dynamics (91) admits
317
Adaptive Control in Discrete Time
5.3.1. An Example
the exponential representation
((k + 1)
= eu (k)G?(8,l [IdJl
FO(B,((k))
Consider the discrete-time nonlinear dynamics
.
(J(k+ I) Under the coordinates change x := (O, it takes the form
=
+ ()((I (k)
(2(k)
+ ((,(k) - (~(k) (~(k))2) + u(k))2 (94)
- (i(k))2
+ ()((2(k) + ()((, (k)
-
(2(k+l)
= (, (k) -
(~(k)
- (~(k))2)
with
+ ()((2(k) + ()((, (k)
+ u(k).
By computations,
FoUl, () = ((2
+ 8((,
(n 2+ ((I -
-
(J -
5.3. The Discrete-time Parametric Strict-feedback Form-DTPSFF
Theorem 5.2. The discrete-time dynamics (79) is locally equivalent to the discrete-time parametric strict-feedback form (49) under parameter independent coordinates change, if and only if (i) The system is linear equivalent for () = 0, (ii) The following DTPSFF conditions hold:
GZ((),O
= G~(() = 0 for
6; ((), .) E (/ where
The proof is given in the appendix.
(n
-
(n 2»)2)
)
while, for the nominal dynamics, one has
and
The computation ofGl (() gives Gl (() = 8/8(,. In addition we note that Fo(O = FQJ(O and consequently one has for k 2': 0,
Gfk(() = G~(() Gfk+' (0
= Gl (0
One easily verifies that [G?, G: 1 = 0 and that Rank [G?,Gi](O) = 2. Then, the nominal system is linear equivalent. The DTPSFF conditions (92) are satisfied too; for one has
o
k 2': 2
0::; i ::; n - 2,
(? + 8((2 + 8((1 -
2
and
To verify the conditions of Theorem 5.1, the computation of A solving (90) is required, which can be a difficult task. Easily checkable conditions can be obtained with reference to a slightly more restrictive discretetime dynamics, the one given in Definition 3.\ called the discrete-time parametric strict-feedback form, DTPSFF. The drift term exhibits the same triangular structure as the PSFF, but the control enters linearly and independently from the parameters in the last state equation only. Moreover, in this last equation, the dynamics which does not depend on the parameters, cPo(x), is linear in x. Necessary and sufficient conditions for the existence of a parameter independent coordinates change under which any dynamics of the form (79) is transformed into the DTPSFF (49) are given hereafter.
G~((),O
(? + 8((2 + 8((,
AdFo(O,(lG,
8
8
= (I + 2()(2) 8(1 + () 8(2'
(92) and therefore 6~(() = ()G~((). .One can now easily build the coordinates change
318
S. Monaco et al.
which gives the DTPSFF studied in Section 4 Xl
(k + I)
x2(k + I)
= x2(k) + Ox~(k) = Xl (k)
+ O(x2(k) + Ox~(k)) + u(k).
7. Appendix Proof of Theorem 5.1.: From (i) there exists an analytic function>. : lRn solution of (90) such that
Remark. It can be easily verified that (94) admits the exponential representation ((k
+ I) = eu(kld!(O,l[Idj
x(k + I)
0
Fo), ... , d( >.
lR
0 p;;-l )
are linearly independent and consequently
l'O(O,«(k))
which, under the coordinates change, the form
d>', d( >.
->
X
:= ((), takes
= eUG'!(-) [Idjl''"0 (O,x(k))
with
= (() = [>.
X
>.
0
Fo ... >. 0 F3- l
r
is a coordinates change with (O) = O. Under the action of (95), the dynamics (79), when o= 0, is transformed into xi(k
+ I)
= Xi+l(k),
I:S i:S n - 1
xlI(k + 1) = ipo(x(k)) + (3o(x(k), u(k))
with
6. Conclusions In this paper, the problem of local adaptive stabilization of discrete-time nonlinear systems has been addressed with reference to suitable classes of dynamics. Through simple examples, classical difficulties of nonlinear adaptive control in discrete time have been pointed out. A design procedure, which parallels the continuous-time backstepping techniques, has been proposed. The main difficulties which lie in the loss of linearity with respect to the unknown parameters and the loss of causality are overcome using the concept of a posteriori errors well known in the linear adaptive control literature. Still, overparametrization techniques are needed to set linearly parametrized error dynamics. The extension to multi-input dynamics, technically more complicated, follows the same lines. The design procedure, which is proposed for dynamics which exhibit the discretetime parametric strict-feedback form, can be extended to more general classes of dynamics as an example when (3(x(k), u(k), 0) = (3o(x(k), u(k)) in (85). In that case, under the assumption that (3o(O, u) i=- 0 and thanks to the implicit function theorem, one recovers the DTPSFF through a simple transformation on the input variable.
(95)
(3o(x(k), O) = 0
ipo(O) = 0,
and
(96) 8{3o(x, u)j
8ul u=o i=- O. In the X coordinates, for 0 = 0, the following exponential representation is obtained
(97) with Fo(x)
=
0
F o 0 -l (x)
= [X2 ... XII ipo(xW,
(98)
(99) As a consequence, given diately obtains
G? defined in (89), one imme(100)
and thus
-.
{ 8
8 }
Q'=span ~""'~ , UX II
uX n -/
(101 )
O:Si:Sn-l.
Acknowledgements In the new coordinates, one has The authors are grateful to Stefano Battilotti and an anonymous reviewer for their useful comments which improved the stability analysis.
(102)
Adaptive Control in Discrete Time
319
A(X) = XI, for the dynamics (85) and hence for the dynamics (79).
while 1
AdFo(II,()G\ «() = [Jx
1"'(0 1"'(0
so that from (87)
Proof of Theorem 5.2. Condition (i) ensures the existence of a coordinates change x =
n
xn(k+ I) Condition (86), rewritten becomes
the x coordinates,
III
E span{ ",f) , ... , '" f) .} UX n
UX n -
0:::;
x(k + I) = eU (k)<7i() [Id] _
i:::; n - 2. with
Since, in x one easily computes
o
(107)
1Folx)
1
(103)
G\ (x) =
(106)
i=1
Consequently, in the x coordinates the exponential representation can be computed
6\ «(,O)t-l(X)
[Jc<
= LCiX;(k) +u(k).
Fo(x) =
0
I
F o 0
t
Ci
X/(k)l
o
(108) ..-. line n - i
I
*
(109)
*
It follows that
and
-. = span { -f), " ' , -f)- } , (]
o JxCFo(x)) x
G\ (x) =
f)X n
f)Xn-i
(110)
o I
..-. line n - i - I
where
*
* where "*" corresponds 'to nonzero elements, (103) implies
In the original coordinates, one has AdF(II,()G\«()
= [Jx
(104) 1
AdFo(()G\(() = [Jx
In addition, since
= ... = !Pn(O, 0) = O.
(lOS)
The necessity of conditions (i) and (ii) of Theorem 5.1 can be easily checked. If there exists a coordinates change that transforms (79) into (85), then one easily shows that conditions (i) and (86) are satisfied, with
1"'(0
thus
[Jc<
6\ (0, ()] I,~-l(x) = AdPo(lI,x)G\ (x) - AdPo(x)G\ (x)]
1",«)
,
320
S. Monaco et al.
and (92), in the x coordinates, becomes
E span {
~f) ""'~}
VX n
0::; i::; n - 2.
vX n -/
( III)
Since, in the x coordinates, one easily computes
o
G\ (x) =
o 1
+-
line n - i
*
* and
o Jx(Fo(x)) x G~ (x) =
o 1
+-
line n - i - I
* * (92) clearly implies
Fo(B,x) - Fo(x) = [cpl(x\,B)·· .CPn(x, B)r. (112) In addition, since (O)
=
0 and Fo(B, 0)
=
0, one has ( 113)
The necessity of conditions (i) and (ii) of Theorem 5.2 can be easily checked. If there exists a change of coordinates that transforms (79) into (49), then one easily checks that condition (i) and (92) are satisfied for the dynamic (49) and hence for the dynamic (79).
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