Homogeneous state feedback stabilization of homogeneous systems

We show that for any asymptotically controllable homogeneous system in Euclidean space (not necessarily Lipschitz at the origin) there exists an homogeneous control Lyapunov function and a homogeneous, possibly discontinuous state feedback law stabilizing the corresponding sampled closed loop system. We also show the relation between the degree of homogeneity and the bounds on the sampling rates which ensure asymptotic stability.


Introduction
In this paper we consider the problem of asymptotic state feedback stabilization of homogeneous control systems in R". This problem has been considered by a number of authors during the last years, see e.g. Homogeneous systems appear naturally as local approximations to nonlinear systems, cf. e.g. [13]. In order t o make use of this approximation property in the design of locally stabilizing feedbacks for nonlinear systems the main idea lies in the construction of homogeneous feedbacks, i.e. feedback laws that preserve homogenit,y for the resulting closed loop system. Utilizing a corresponding homogeneous Lyapunov function, those laws can then be shown to be locally stabilizing also for the approsimated nonlinear system, cf. [13,17,191. Regarding the existence of homogeneous stabilizing feedback laws, it was shown in 1141 that if the system admits a continuous, but not necessarily homogeneous, stabilizing state feedback law, tohen there exists a homogeneous dynamic feedback stabilizing the system. Unfortunately, if we are looking for static state feedback laws, it is in general not true that any continuously stabilizable homogeneous system is stabilizable by a continuous and homogeneous state feedback law, as the esamples in [22] show. Even worse, there exist homogeneous systems, e.g. Brockett's classical example [2], whichalthough asymptotically controllable-do not admit a stabilizing continuous state feedback law at all. Especially Brockett's results inspired the search for alternative feedback concepts. In the present paper we are going to use discontinuous state feedback laws for which the corresponding closed loop systems are defined as sampled systems. Although a classical concept, it has recently received new attention, see e.g. the survey [23]. In particular, it was shown in [4] that (global) asymptotic controllability is equivalent to the existence of a (globally) stabilizing discontinuous state feedback law for the sampled closed loop system. Stability in this context means asymptotic stability for the sampled trajectories (i.e. the feedback is evaluated only at discrete sampling times with the values being used until the next sampling time) where-in general-the intervals between two sampling times have to tend to zero close to the equilibrium and far away from it. A related but slightly different concept of a discontinuous feedback is the notion of discrete feedback introduced in 161; here also sampled trajectories are considered, but with fixed intersampling times. With this approach it was possible to show in [9] that for semilinear systems asymptotic controllability is equivalent to (exponential) discrete feedback stabilizability.
In the present paper we will combine these two concepts in the framework of homogeneous systems. As in [9] we use a spectral characterization of asymptotic controllability by means of Lyapunov exponents, and obtain stability results for fixed sampling rates; as in [4] we construct the feedback based on a suitable (and here also homogeneous) control Lyapunov function, and obtain stability not only for fixed intersampling times but for all sufficiently small ones. Furthermore, and this is a key feature of our construction, the resulting stabilizing state feedback law is homogeneous, thus rendering the corresponding closed loop system homogeneous. All this will be done just under the assumption that the corresponding homogeneous system is asymptotically controllable.
We start this paper by defining two classes of homogeneous systems in Section 2. Section 3 provides the concepts of asymptotic controllability and stabilization by means of sampled feedback laws. After stating our main theorem at the end of this section, we sketch the main arguments of its proof in Section 4. We refer to the full version of this paper [ll] for a detailed proof.

Homogeneous systems
We consider a class of systems .(t) = g(z(t), 4 t ) ) (2.1) on Rn where w(.) E W , and W denotes the space of measurable and locally essentially bounded functions from R to W c R". We assume that the vector field g is continuous, g(., w) is locally Lipschitz on R n \ (0) for each w E W , and satisfies the following property. Yote that the trajectories of (2.1) may tend to infinity in finite time if r > 0 and that uniqueness of the trajectory may not hold if r < 0, however it holds away from the origin. As long as uniqueness holds (i.e. if r 2 0 or the trajectory does not cross the origin) we denote the (open loop) trajectories of (2.1) by x(t. for 20 E R". If uniqueness fails to hold z(., 20, tu(.)) shall denote one possible trajectory; in this case we implicitely assume the definitions of Section 3, below, to be valid for all possible trajectories. Now we introduce and discuss a class of auxiliary which will turn out to be useful for our analysis: Consider on R" where ti(.) E U , and U denotes the space of measurable functions from Iw to some compact set U c R". We assume that the vector field f is continuous, f (., U) is locally Lipschitz on Rn \ (0) for each U E U , and satisfies the following property. Note that this definition implies f ( 0 , u ) = 0 for all U E U . We denote the trajectories of (2.5) with initial value zo a t the time t = 0 and control function ti(.) E U again by x(t,zo,u(.)). Observe that also the trajectories of (2.5) may escape in finite time if r > 0 and that uniqueness of the trajectory may not hold in the origin if r < 0 (here again we use the convention as for the trajectories of (2.1)). As long as the trajectories exist and uniqueness holds we obtain from Definition 2.2 that z ( t , haze, t i ( @ ' . ) ) = A,z(art, 2 0 , U(.)) (2.7) for all zo E R".
Besides being useful auxiliary systems for our stabilization problem for homogeneous systems, homogeneousin-the-state systems themselves form an interesting class of systems.  [9]; all results in this paper can easily be adapted to that case.
The connection between homogeneous and homogeneous-in-the-state systems is easily seen: Given some homogeneous system (2.1) satisfying we define f(+, U) := dz, Aiv(z)u).
(2.8) Then it is immediate from the property of the di- Homogeneous and homogeneous-in-the-state systems can be considerably simplified applying suitable coordinate and time transformations. We will make use of this procedure for homogeneous-in-the-state systems: Using the dilated norm N a straightforward construction (see [ l l ] ) shows the existence of a coordinate transformation y = Q(z), which is continuous on Iw" and C1 on E%"\(O} and satisfies

Q -l ( a k y ) = A,Q-'(y) and DQ(A,z) = akAh,lDQ(z).
Thus defining J(y, U ) := D$(Q-'(y))f(Wl(y), U ) we obtain J ( a k y 2 u ) = a 7 a k f ( y , u ) , which implies f is homogeneous-in-the-state with respect to the standard dilation A, = aId, with mimimal power k = 1, and with degree r = y.
Furthermore, setting f ( y , U ) = f [ y , u)IIyII-Y (which defines a time transformation for f) we obtain a system with degree r = 0.
We will first prove our results for systems of the form and then indicate how to retranslate the results to the general case.

Asymptotic controllability and feedback stabilization
In this section we give the precise definitions of asymptotic controllability and feedback stabilization. For this purpose we briefly describe the idea of sampling and the concept of control Lyapunov functions. We formulate t,he concepts for system ( 2 . l ) , with the obvious modifications, however, all definitions also apply to system (2.5).  [2] is discussed which-in suitable coordinates-is in fact a homogeneous system. Furthermore, even if stabilizing continuous feedback laws exist, it is possible that no such law is homogeneous, as the examples in [22] show. However, using discontinuous feedbacks for the solutions of the classical closed loop system x = g ( i , F ( z ) ) the usual existence and uniqueness results might not hold. In order to obtain a meaningful solution for the closed loop system we use the following concept of a sampled closed loop system. The next definition introduces control Lyapunov functions which will be vital for the construction of the feedback. Here we make use of the lower directional derivatives, see e.g. The following definition now describes the stability concepts we will use in this paper. Recall that a function  Note that each of the concepts (ii)-(iv) implies (i) which is equivalent to the s-stability property as defined in [1], cf. also 123, Sections 3.1 and 5.11. Hence any of these concepts implies global stability for the (possibly nonunique) limiting trajectories as h + 0. The difference "only" lies in the performance with fixed sampling rate. From the applications point of view, however, this is an important issue, since e.g. for an implementation of a feedback using some digital controller arbitrary small sampling rates in general will not be realizable. Furthermore if the sampling rate tends to zero the resulting stability may be sensitive to measurement errors, if the feedback is based on a non-smooth clf, see [18, 231. In contrast to this it is quite straightforward to see that for a fixed sampling rate the stability is in fact robust to small errors in the state measurement (small, of course, relative to the norm of the current state of the system) if the corresponding clf is Lipschitz, cf. [23, Theorem E]. For a detailed discussion of these concepts see also [lo].
The main result of this paper is the following theorem on the existence of a homogeneous clf V and a homogeneous stabilizing feedback F .

Sketch of Proof
We first sketch how to prove Part (b) of the theorem for systems of type (2.9). Afterwards, we sketch the proof of Part (b) for general homogeneous-in-the-state systems and finally, we indicate how to obtain Part (a) from Part (b). For the details of this proof we refer to We start by characterizing asymptotic controllability of (2.9). For this purpose we introduce the finite time exponential growth rate (cf. [9,121) It follows immediately from (2.9) that z ( t , azo, U ( . ) ) = az (t,xo,u(.)) and thus the growth rates satisfy (see [5] for more information about these objects) we obtain U < 0 if and only if the system is asymptotically We will now use this inequality for the construction of a homogeneous Lyapunov function for system (2.9). First observe that the projectmion of (2.9) onto S"-l is well defined due to the homogenity is asymptotically controllable then for each p E (0,IuI) there exists 6, > 0 such that for all 6 E (0,6,] and all SO E S"-' the inequality Sva(s0) < -p holds.
The function VO is homogeneous with degree 1 (with respect to the standard dilation) and by a dynamic programming argument on proves that VO is a clf which satisfies Based on Vo we can now construct the stabilizing feedback law for system (2.9). To this end for /.!I > 0 we con- Furthermore there exists a feedback law F : Rn + U satisfying F ( a z ) = F ( z ) for all z E EX", a > 0 and constants h > 0 and C > 0 such that any 7r-trajectory corresponding to some partition 7r with d ( n ) 5 h satisfies Ill7r(t,xo1 F)ll 5 Ce-ptllzoll. Thus, we have obtained the desired result for systems of type (2.9). In order to prove Theorem 3.5(b) it remains to retranslate this result to general homogeneous-int he-st ate systems.
Obviously, if the system defined by f is asymptotically controllable, then the transformed system defined by f is asymptotically controllable. Thus from the above considerations we obtain v = L$ and F = F satisfying the assertion for f which is homogeneous-in-the-state with A, = aId, k = 1 and T = 0.
We start by showing the result for the system defined by Finally, we show Theorem 3.5(a). To this end, recall that for each homogeneous system (2.1) we find an associated homogeneous-in-the-state system by (2.8). In fact, one can show that this system inherits the asymptotic controllability property. Hence from Theorem 3.5(b) we obtain a clf and a feedback Fl for the homogeneous-in-the-state system. Setting V = VI and F ( x ) = A j~(~) F l ( x ) we immediately obtain the assertion.