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## 2 Stabilizability

Definition 2.1   The control system (1.1) is globally asymptotically controllable (or GAC) provided that there exists a nonincreasing function such that and a function such that the following holds. Let be given. Then, given any , there exists a control function so that the associated trajectory which satisfies also satisfies

(i)
;

(ii)
;

(iii)
.

We say that a function is positive definite if and whenever . Also, is said to be proper provided that as .

Definition 2.2   A control Lyapunov pair is a pair of continuous, positive definite functions , with also proper, such that for each one has the infinitesimal decrease condition
 (2.1)

We refer to the in a control Lyapunov pair as a control Lyapunov function (or CLF).

The infinitesimal decrease condition is equivalent to one in Dini derivate terms, which has been used by several authors in characterizing nonsmooth CLF's; see Sontag [47] as well as Sontag and Sussman [49]. This Dini formulation is

 (2.2)

The equivalence of (2.1) and (2.2) is due to a deep theorem of Subbotin [51], which links the proximal and directional generalized solution concepts for partial differential equations. An alternate proof of Subbotin's theorem was derived by Clarke and Ledyaev via a generalized mean value inequality [11]; see also [20].

Recently, Rifford [44] proved the following result.

Theorem 2.3   The control system is GAC iff there exists a CLF which is locally Lipschitz on .

The above theorem extends an important result of [47] (see also [49]) where the CLF produced is only continuous. In this regard, Theorem 2.3 provides an answer to a conjecture attributed to Sontag and Sussman. Furthermore, Theorem 2.3 extends a result of Clarke, Ledyaev, Rifford, and Stern [12], where the CLF produced is locally Lipschitz, but only on the complement of a ball around the origin.

Theorem 2.3 and Theorem 3 of [12] yield a general result on robust stabilization. The asserted stabilization is framed in terms of the sublevel sets of a CLF. We introduce the following sublevel set notation for given :

Theorem 2.4   Assume that the control system is GAC, and let be a CLF as in Theorem 2.3. Let be specified. Then for any sufficiently small , there exist a feedback along with positive numbers , and such that, for every there exists as follows: for any partition of with
 (2.3)

the error bounds
 (2.4)

and
 (2.5)

imply that any -trajectory satisfying also satisfies
 (2.6)

 (2.7)

 (2.8)

Remark 2.5

1. In view of the nature of the CLF , the preceding result implies that for any , there exists a feedback defined on a neighborhood of which stabilizes every initial point in the ball to the ball in a robust manner.

2. The feedback is constructed via a technique dubbed proximal aiming in [14] and [20]. This is a projective method akin to the extremal aiming concept of Krasovskii and Subbotin in differential game theory [34], [35]. The feedback will now be described in rough terms.

In view of the continuity of , given , there exists such that

 (2.9)

Since is Lipschitz on (we denote this Lipschitz rank by ) and is continuous, it follows that (2.9) is equivalent to
 (2.10)

For appropriately defined positive integer and positive constants and , one considers a family of shells''

for (along with specialized definitions for ). It can be arranged that the family is disjoint.

Let ( ) and choose . Then the construction implies , and by definition, . A nonsmooth analysis argument then yields the existence of such that

 (2.11)

We emphasize that in establishing (2.11), the Lipschitz nature of is crucial. In [13], where a nonrobust version of Theorem 2.4 was proven, only a continuous CLF was available, and a different proof technique was employed (involving quadratic inf-convolutions, also required in the next section).

Now, , and in view of (2.10) there exists such that

We then set . It transpires that under the constructions we have sketched, the -trajectory emanating from moves towards at a positive rate, with the process resetting and continuing as increases, with the -trajectory meeting the requirements of the theorem.

3. The double sided bound on the meshsize in (2.3) says that the partition is reasonably uniform.'' The upper bound requires that the step size be small enough to ensure stabilization, which is rather natural. On the other hand, the lower bound may appear somewhat surprising. Its purpose is to preclude a possible chattering phenomenon; each step must be big enough to counteract the measurement error by means of the attractive effect arising from the infinitesimal decrease property used in the construction.

4. The maximum allowable external disturbance error is proportional to the quantity . That this is natural can be seen heuristically by considering the (very) special case where is smooth and is a continuous feedback such that

Then the perturbed system

is stabilized by this feedback if

which is a condition similar to a bound involving .

5. Other types of error, such as a disturbance entering into the dynamics as in

can easily be reduced to the case of an external disturbance as in the preceding theorem, due to the continuity of in the variable .

6. Ledyaev and Sontag have approached the robustness question in the presence of small measurement errors and an arbitrarily fast sampling rate by alternate means in [37], [38].

• In [37], the control with guide'' method of Krasovskii and Subbotin has been adapted to achieve robustness of stabilizing feedback with respect to -small measurements and arbitrary high sampling rate. This method entails utilizing an internal model of the system in conjunction with an arbitrary stabilizing discontinuous feedback (not necessarily the ones which are constructed by using nonsmooth CLFs).

• In [38] it was established that there is a relationship between regularity and the above-mentioned type of robustness. In particular, it was shown that the existence of a smooth CLF is equivalent to the existence of stabilizing feedback which is robust with respect to small measurement errors and arbitrary high sampling rate. This result was proven for a general nonlinear control system with persistent disturbances. It gives a characterization of the existence of CLFs in terms of regular (that is, robust feedback) in a manner analogous to the well-known result of Artstein [1] which implies that for systems which are affine in control, the existence of a smooth CLF is equivalent to the existence of a smooth feedback. Thus, this result demonstrates that if we do not restrict the stepsize in the sampling from below and require that a stabilizing feedback should be robust in the sense mentioned above, then there is a smooth CLF.

The interesting applied aspect of results from [38] is a straightforward construction of discontinuous stabilizing feedback which is robust in the above-mentioned strong sense.

Next: 3 Fixed duration state Up: clark Previous: 1 INTRODUCTION
2003-08-05