2 Stabilizability

Definition 2.1   The control system (1.1) is globally asymptotically controllable (or GAC) provided that there exists a nonincreasing function $% M:(0,\infty )\rightarrow (0,\infty )$ such that $\lim_{R\downarrow 0}M(R)=0,$ and a function $T:(0,\infty )\times (0,\infty )\rightarrow \lbrack 0,\infty )$ such that the following holds. Let $0<r<R$ be given. Then, given any $% \Vert \alpha \Vert \leq R$, there exists a control function so that the associated trajectory which satisfies $x(0)=\alpha$ also satisfies

(i)

$\lim_{t\rightarrow \infty }x(t)=0$;

(ii)

$\forall \,t\geq 0,~~\Vert x(t)\Vert \leq M(R)$;

(iii)

$\forall \,t\geq T(r,R),~~\Vert x(t)\Vert \leq r$.

We say that a function $V:\mathbb{R}^{n}\rightarrow \lbrack 0,\infty )$ is positive definite if $V(0)=0$ and $V(x)>0$ whenever $x\neq 0$. Also, $V$ is said to be proper provided that $V(x)\rightarrow \infty$ as $\Vert x\Vert \rightarrow \infty$.

Definition 2.2   A control Lyapunov pair is a pair of continuous, positive definite functions $(V,W)$, with $V$ also proper, such that for each $x\neq 0$ one has the infinitesimal decrease condition

$$\min_{u\in U}\langle \zeta ,f(x,u)\rangle \leq -W(x)\quad \forall \,\zeta \in \partial _{P}V(x).$$ (2.1)

We refer to the $V$ 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

$$\min_{u\in U}DV(x;f(x,u))\leq -W(x)\quad \forall \, x\neq 0.$$ (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 $(\ref{de1})$ is GAC iff there exists a CLF which is locally Lipschitz on $\mathbb{R}^{n}$.

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 $c\geq 0$:

$$% latex2html id marker 1610S(c):=\{x\in \mathbb{R}^{n}:V(x)\leq c\}.$$

Theorem 2.4   Assume that the control system $(\ref{de1})$ is GAC, and let $V$ be a CLF as in Theorem 2.3. Let $0<a<b$ be specified. Then for any sufficiently small $\gamma >0$, there exist a feedback $k:S(b)+\gamma B_{n}\rightarrow U$ along with positive numbers $\delta _{0}$, $T$ and $E_{q}$ such that, for every $\delta \in (0,\delta _{0})$ there exists $E_{p}(\delta )>0$ as follows: for any partition $\pi$ of $[0,\infty)$ with

$$\frac{\delta }{2}\leq t_{i+1}-t_{i}\leq \delta ,\quad i\geq 0,$$ (2.3)

the error bounds

$$\Vert p(t_{i})\Vert \leq E_{p}(\delta ),\quad i\geq 0$$ (2.4)

and

$$\Vert q\Vert _{\infty }\leq E_{q}$$ (2.5)

imply that any $\pi$-trajectory $x_{\pi }$ satisfying $x_{\pi }(0)\in S(b)+\gamma B_{n}$ also satisfies

$$x_{\pi }(t_{i})+p_{i}\in S(b)+\gamma B_{n}\quad \forall \,i\geq 0;$$ (2.6)

$$x_{\pi }(t)\in S(b)+2\gamma B_{n}\quad \forall \,t\geq 0;$$ (2.7)

$$x_{\pi }(t)\in S(a)+\gamma B_{n}\quad \forall \,t\geq T.$$ (2.8)

Remark 2.5    

1. In view of the nature of the CLF $V$, the preceding result implies that for any $0<r<R$, there exists a feedback defined on a neighborhood of $RB_{n}$ which stabilizes every initial point in the ball $% RB_{n}$ to the ball $rB_{n}$ in a robust manner.

2. The feedback $k$ 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 $k$ will now be described in rough terms.

   In view of the continuity of $W$, given $\eta >0$, there exists $\omega >0$ such that
   

   $$\min_{u\in U}\langle \zeta ,f(x,u)\rangle \leq -\omega \quad... ...a \in \partial _{P}V(x),\quad \forall \,x\in S(b)+\eta B_{n}.$$ (2.9)

   
   
   Since $V$ is Lipschitz on $S(b)+\eta B_{n}$ (we denote this Lipschitz rank by $L_{V}$) and $f$ is continuous, it follows that (2.9) is equivalent to
   

   $$\min_{u\in U}\langle \zeta ,f(x,u)\rangle \leq -\omega \quad... ...a \in \partial _{L}V(x),\quad \forall \,x\in S(b)+\eta B_{n}.$$ (2.10)

   
   

   For appropriately defined positive integer $N$ and positive constants $% \gamma$ and $\beta$, one considers a family of ``shells''
   

   $$\Omega _{i}:=\{S(b-i\beta )+\gamma B_{n}\}\backslash \{S(b-(i+1)\beta )+\gamma B_{n}\}$$
   
   
   for $0\leq i\leq N-1$ (along with specialized definitions for $i=N,N+1$ ). It can be arranged that the family is disjoint.

   Let $x\in \Omega _{i}$ ( $i=0,1,\ldots ,N$) and choose $s=s_{x}\in proj_{S(b-i\beta )}$. Then the construction implies $x\neq s$, and by definition, $x-s\in N_{S(b-i\beta )}^{P}(x)$. A nonsmooth analysis argument then yields the existence of $\lambda >0$ such that
   

   $$\lambda (x-s)\in \partial _{L}V(s).$$ (2.11)

   
   

   We emphasize that in establishing (2.11), the Lipschitz nature of $V$ 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, $\lambda \Vert x-s\Vert \leq L_{V}$, and in view of (2.10) there exists $u\in U$ such that
   

   $$\langle x-s,f(s,u)\rangle \leq \frac{-\omega }{2L_{V}\Vert x-s\Vert }.$$
   
   

   We then set $k(x)=u$. It transpires that under the constructions we have sketched, the $\pi$-trajectory emanating from $x$ moves towards $S(b-i\beta )$ at a positive rate, with the process resetting and continuing as $i$ increases, with the $\pi$-trajectory meeting the requirements of the theorem.

3. The double sided bound on the meshsize in (2.3) says that the partition $\pi$ 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 $E_{q}$ is proportional to the quantity $\omega /L$. That this is natural can be seen heuristically by considering the (very) special case where $V$ is smooth and $k$ is a continuous feedback such that
   
   $$\langle \nabla V(x),f(x,k(x))\rangle \leq -W(x).$$
   
   
   Then the perturbed system
   
   $$\dot{x}=f(x,k(x))+q$$
   
   
   is stabilized by this feedback if
   
   $$\Vert q\Vert _{\infty }<\frac{W(x)}{\vert\nabla V(x)\Vert },$$
   
   
   which is a condition similar to a bound involving $\omega /L$.

5. Other types of error, such as a disturbance $d(\cdot )$ entering into the dynamics as in
   
   $$\dot{x}=f(x,k(x)+d),$$
   
   
   can easily be reduced to the case of an external disturbance $q(\cdot )$ as in the preceding theorem, due to the continuity of $f(x,u)$ in the variable $u$.

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 $L_{\infty }$-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 $L_{\infty }$ 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.