From Artstein-Sontag Theorem
to the Min-Projection Strategy
A. Bacciotti and L. Mazzi
Dipartimento di Matematica, Politecnico di Torino
Torino - Italy
Given a finite or countable family of continuous vector fields , we show that there exists a feedback such that the switched system is globally asymptotically stable whenever there exists a smooth control Liapunov function such that for all , , for some .
Key Words: Families of vector fields, stability, discontinuous feedback, Filippov solutions
1. Introduction
One of the main problems in mathematical control theory is to design appropriate control strategies, in such a way that all the trajectories of a given system are driven as near as possible to an equilibrium (or, more generally, to a desired steady state). Control Liapunov functions are one of the typical tools used to elaborate these strategies. To focus the problem in a classical perspective, we first recall the case of a finite-dimensional, continuous-time, time-invariant, affine system
(1)
where represents the state variable, represents the input variable, and the vector fields are locally Lipschitz continuous. Without loss of generality, we assume that the desired equilibrium is the origin.
The so-called Artstein-Sontag Theorem (Artstein, 1983; Sontag, 1989 and 1990) states that a smooth (except possibly at the origin) stabilizer can be actually constructed provided that a smooth (at least ) control Liapunov function can be found. Roughly speaking, the point is that in the affine case, the inequality
(2)
can be fulfilled by choosing a value of which depends continuously on the value of , for each . Note that the naive idea of determining by solving the minimum problem
(3)
does not work, since it does not guarantee in general the continuity requirement.
Unfortunately, if we allow to vary only on a preassigned set , the Artstein-Sontag Theorem is not always applicable, since its proof is based on the implicit assumption that varies freely on . It often happens in applications that is a finite set, for instance when the control is exerted by means of a digital device. In this case, if we set
we may interpret the system as a switched system, by defining .
More generally, consider a family of continuous and complete vector fields of , being any finite or countable set of indices. Let be the set of switching signals, that is piecewise constant maps . According to Sunand Ge, 2005, one says that is pointwise stabilizable if it is asymptotically controllable and the input signal can be taken in for each .
Assuming now the existence of a smooth Liapunov function, it is natural to conjecture the existence of a feedback law which stabilizes at the origin. The delicate aspect of this conjecture is that since is expected to be discontinuous, the general theory of existence and continuability of solutions of ODE's cannot be applied to the closed loop system
(4)
One needs to resort to some generalized notion of solution: in the literature there are many possible choices but a common agreement has not yet clearly emerged. As a matter of fact, any result about discontinuous feedback stabilization inevitably depends on the adopted notion of solution (see Ceragioli, 2006). Note that the sampling-and-holding approach followed in Clarkeet al., 1997 is useless here, since in general the constraint set is preassigned.
In PettersonandLennarston, 2001, state depending discontinuous feedback laws have been constructed using the so-called min-projection strategy (see also GeromelandColaneri, 2006), which basically consists in exploiting (3). The authors assume that the control Liapunov function is quadratic, and that its derivative with respect to can be dominated by a quadratic negative definite form. When is finite, they prove that the origin is exponentially stable with respect to Filippov solutions of the closed loop system.
The stabilization result we present in this paper can be viewed as an improvement of Theorem 1 of PettersonandLennarston, 2001, since our Theorem holds for countable families of continuous vector fields (as opposed to finite families). Moreover, our Liapunov function is positive definite but needs not to be quadratic; its derivative with respect to the system is negative definite, but not necessarily dominated by a quadratic form as in PettersonandLennarston, 2001. As a consequence, we obtain asymptotic (not exponential) stability. Finally, we deal with solutions in Krasowski sense. Therefore, we obtain asymptotic stability with respect to a set of solutions larger than the one of Filippov solutions used in PettersonandLennarston, 2001. Although it is well known that Krasowski or Filippov solutions are not always convenient for stabilization problems, they offer the advantage of a well developed theory of existence and stability (see Filippov, 1988; Hájek,1979; Clarkeet al., 1998; PadenandSastry, 1987; see also Petterson, 2003; Pogromskyet al. 2003; Skafidaset al. 1999 for other applications of Filippov solutions in the switched systems literature).
The idea of looking for the existence of regions where at least one of the vector fields is such that has been used in Hu et al., 2002 as well; however, the authors restrict themselves to planar linear systems with centers or foci, and they use only quadratic Liapunov functions.
In the next section we expose the problem with some details, and state the result. The proof is given in Section 3 and Section 4 contains a few illustrative examples.
2. Statement of the result
Let be a family of continuous vector fields of , where is a finite or countable set of indices, endowed with the discrete topology. As mentioned in the introduction, we seek stabilizing feedback laws of the form . Let . Of course, for and . It is clear that the right hand side of the closed loop system (4) may be not continuous at the boundary points of the 's. Let us denote by the open ball of radius and center .
Recall that a Krasowski solution of (4) is any absolutely continuous curve satisfying the differential inclusion
(5)
Recall also that the origin is globally asymptotically stable with respect to Krasowski solutions of (4) if
(a) such that implies for all and all Krasowski solutions of (4);
(b) for all Krasowski solutions of (4).
Note that if (a) holds, then the origin is an equilibrium solution in the sense of Krasowski for (4) (to this respect see also Xuet al., 2007); moreover, all the local Krasowski solutions corresponding to a small initial state can be continued on .
Theorem 1 Together with the family of vector fields , a positive definite, radially unbounded and smooth (at least ) function is given. Assume that
(6)
Then, there exists a discontinuous feedback such that the origin is globally asymptotically stable for (4) with respect to Krasowski solutions.
Remark 1 A function satisfying (6) is called a control Liapunov function for . By virtue of the converse theorem proven in Clarkeet al., 1998 (see also Bacciotti andRosier, 2001), the existence of a control Liapunov function is necessary for stabilization in Krasowski sense.
3 The proof
In order to prove our main result, we need to recall two results concerning the Krasowski differential inclusion associated to a discontinuous system .
Lemma 1 Given a differential system , where is measurable, we have the following identity:
For a proof of this Lemma, see ShevitzandPaden, 1994. We remark that defined in (4) is measurable if and only if is measurable.
Lemma 2 Let be the set valued map introduced in (5). If there exist a smooth (at least ), radially unbounded, positive definite function and a real valued, continuous, positive definite function such that
then the origin is globally asymptotically stable with respect to Krasowski solutions of (4).
For a proof of this result, see for instance Filippov, 1988 or Bacciotti andRosier, 2001.
Proof of Theorem 1. We consider the covering of given by the family of closed sphere hulls of radius , :
Claim. There exists such that , such that .
If this were not true, we should have that , such that ,
.
Since is compact, there exists a subsequence , such that . By continuity,
for all , a contradiction.
Therefore the claim is verified. We set
We observe that the sequence is decreasing, while is increasing, and . We define now a function
satisfies the following properties:
1. , since for all , , .
2. is positive definite and continuous for all , by construction.
3. For all there exists such that
(7)
This is true, since for all , , and, by the claim, , for some .
We define a covering of given by the closed sets
where , . We define now a feedback as:
We have The set is clearly measurable for each and for each , so that is a measurable function.
We observe that may be equivalently defined as , when . yields a discontinuous differential system
By definition, for all , there exists such that and
In order to be able to apply Lemma 2, we need to show that for all and for all , . First of all, we consider a vector obtained as a limit , for some . By construction, for all
Since and are continuous functions, we have that
By Lemma 1, if it is a convex combination of vectors obtained as above, therefore the same inequality holds for any . By Lemma 2 our main result is proven.
4 Examples
The proof of Theorem 1 shows that, whenever it is possible to construct a function satisfying (7), we are able to construct a stabilizing feedback . (The claim basically states that such a function exists whenever (6) is fulfilled.)
As illustrated by the following examples, in some cases it is not difficult to find a function independently of the construction given in Theorem 1.
Given a family of vector fields and a control Liapunov function for , we denote by the set
Example 1 Let us consider the pair of planar linear systems
and the candidate control Liapunov function . It is easily seen that
so that condition (6) is fulfilled. If we choose , we are led to the following stabilizing feedback:
Some trajectories of the closed loop system are plotted in Fig. 1. Stabilization of this system can be also addressed with the method of Wicks et al., 1998: in particular, this system is quadratically stable (see for instance Liberzon, 2003 for the definition of quadratic stability). In Wicks et al., 1998, it is also shown how to avoid the sliding mode induced by the feedback along the discontinuity surfaces, by resorting to memory-feedback and hysteresis.
Example 2 Let us consider now the pair of planar systems
and take again . We have
Condition (6) is fulfilled. If we choose , we obtain the stabilizing feedback
Some trajectories of the closed loop system are plotted in Fig. 2. The existence of a discontinuous stabilizing feedback can be also deduced from AnconaandBressan, 1999; indeed, the system is easily seen to be asymptotically controllable. However, we note that the approach in AnconaandBressan, 1999 does not lead to an easy and explicit construction.
Example 3 In this last example we consider the pair of planar linear systems
It is not difficult to show that in this case condition (6) is satisfied by no quadratic function . On the contrary, taking we have
Regions and overlap, and Condition (6) is fulfilled. By taking , we obtain the stabilizing feedback
Some trajectories of the closed loop system are drawn in Fig. 3. Note that method of Wicks et al., 1998 does not work in this case, and not even its generalization proposed in Bacciotti, 2004: in particular, we note that this system is not quadratically stable.
5 Final remarks
Besides the notion of pointwise stabilizability recalled above, in Sunand Ge, 2005 the authors introduce also the notion of consistent stabilization, meaning that the system is asymptotically controllable by a switching signal which is independent of the initial state . In this case, replacing by , (1) can be reviewed as an asymptotically stable time-varying system. If, in addition, the stability is uniform, the existence of a time-varying, smooth Liapunov function is guaranteed (see Bacciotti and Rosier, 2001). Using the same construction as in the proof of Theorem 1, it should be possible to define stabilizing rules depending both on time and on the state variable and taking values in the constraint set .
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Figura 1: Some trajectories of the system (Example 1)
Figura 2. Some trajectories of the system (Example 2)
Figura 3. Some trajectories of the system (Example 3)
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