Observability by using viability kernels

OBSERVABILITY BY USING VIABILITY KERNELS

Khalid Kassara

Systems and Control Group

University Hassan II, Casablanca 20100, Morocco kassarak@ams.org

ABSTRACT

The intent of this talk is to propose an alternative way to deal with observability of systems governed by ODEs, in a more general setting than the standard out- put equation. We show that the starting instants from which the system is observabe correspond to points of single-valuedness of a multifunction I( · ), which involves the viability kernel of output domain under the aug- mented system. The approach may be used either for global or local observability, to which available results on single-valuedness of multifunctions shall be applied to I( · ) in order to get necessary and/or sufficient char- acterizing conditions. Several examples are provided in order to illustrate the method.

1. MAIN FACTS

Observability is one of the old concepts of control the- ory. It addresses the issue of whether the initial state of a system can be uniquely determined, out of the information provided by the current output.

The aim of this note to provide an alternative set- valued approach within a more general framework in regard to the output expression, and which can be in- troduced in the following.

Let n be an integer and f : R

× R

→ R

be a continuous function. For t

> 0, consider the system governed by the ordinary differential equation,

˙

z(t) = f (t, z(t)), t ∈ [t

, t

], (1a) with output expressed as follows,

(t, z(t)) ∈ Θ, t ∈ [t

, t

], (1b) where t

∈ [0, t

) and Θ is a closed subset of R

× R

, which we call the output domain.

One immediately can see that this new setting en- compasses the standard situation in which both obser- vation θ and state z satisfy the output equation,

θ(t) = h(t, z(t)), t ∈ [t

, t

],

for some smooth function h. For this, it is obvious that the output domain can be given as follows,

Θ .

= { (t, z) ∈ R

× R

| h(t, z) − θ(t) = 0 } .

Moreover, Eq. (1b) may serve to describe incom- plete, partial or uncertain informations on the state of the system. Following are the classical statements of observability, which we adapt in light of the above motivational ideas.

Let t

∈ [0, t

), z

, z

and z

belong to R

and Σ be a subset of R

. We say that state z

generates output (1b) on the horizon [t

, t

], whenever the constrained ODE (1) has a solution ¯ z which satisfies ¯ z(t

) = z

. For notation,

z

f

Θ on [t

, t

].

The states z

and z

are said to be indistinguishable on [t

, t

] if both generate the output (1b) on the interval [t

, t

].

System (1) is said to be :

(a) Σ − observable on the horizon [t

, t

] if there are no pairs of distinct indistinguishable states on [t

, t

], which are included in subset Σ.

(b) observable on the horizon [t

, t

] if it is R

− observable on [t

, t

].

(c) locally observable on the horizon [t

, t

] around state z

, if there exists a neighborhood W of z

such that system (1) is W − observable on [t

, t

].

(d) observable from near ¯ t ∈ (0, t

), if there exists a neighborhood J of ¯ t such that for all t

∈ J , System (1) is observable on [t

, t

]. It is said to be continuously observable from near ¯ t if, further, the function t

∈ J → z

, (where z

f

Θ on [t

, t

]) is well defined and continuous.

Let us define the following multifunction by, I(t) .

= { z | z

Θ on [t, t

] } , (2)

for each t ∈ [0, t

]. Then, according to (a) we get, (1) is Σ-observable on [t

, t

] ⇐⇒ card(I(t

) ∩ Σ) ≤ 1.

Consider the augmented system,

t ˙ = 1, z ˙ = f (t, z), (3) and let K be the viability kernel of system (3) under the output domain Θ, which we call the observability kernel of System (1). We refer to [1, 2, 8, 3] for viability theory and viability kernels. It merely follows that,

I(t) = { z | (t, z) ∈ K} ,

By the above analysis, we see that observability con- nects to viability kernels and single-valuedness of mul- tifunctions. The latter notion, which is mostly used in variational analysis, consists of studying points where a multifunction is single-valued, as done in the papers [4, 6, 7, 9]. Their main theorems will be applied to multifunction I( · ) in order to state our observability results.

Figure 1: A possible representation (in dark) of the observ- ability kernel on the horizon [0 t

] : here note the system is observable on both [0 t

] and on all the horizons [t t

] for t > t

. It is observable from near t

and locally observable on [t

t

] around x

. But it is unobservable on [t

t

].

The most relevant result, for our considerations, is ob- tained by using the characterization of single-valuedness by means of premonotonicity and lower semicontinuity of multifunctions, as done in [6]. Providing the starting instants from near which the system is continuously ob- servable. To that end we need consider the implication below,

z

f

Θ on [π

(y

) t

] and z

f

Θ on [π

(y

) t

],

⟨ y

− y

, z

− z

⟩ ≥ −⟨ y

⇓ − y

, τ (y

) − τ(y

) ⟩ , (4)

for couples (z

, z

), (y

, y

) ∈ R

× R

, such that π

(y

) ∈ [0, t

] for i = 1 or 2 and a function τ : R

→ R

, and π

standing for the first projection on R

.

Then we are in a position to state the following.

Theorem 1 Let t ¯ belong to [0 t

), y ¯ .

= (¯ t, 0, . . . , 0)

and

¯

z generate output (1b) on the horizon [¯ t t

]. Then System (1) is continuously observable from near ¯ t iff the statements below are satisfied,

(a) There exist two neighborhoods U (of y), and ¯ V (of z) and a continuous function ¯ τ : U → V , such that implication (4) holds true for all couples (y

, z

) ∈ U × V and i = 1, 2.

(b) Whenever a sequence (t

)

converges to a point near t, ¯ there exists a sequence (z

)

which con- verges to some z near z ¯ such that z

Θ on [t

t

], for all q.

2. REFERENCES

[1] J. P. Aubin. Viability theory, second ed., Boston:

Modern Birkh¨ auser Classics, 2009.

[2] J. P. Aubin, P. Saint-Pierre, Viability kernels and capture basins for analyzing the dynamic behav- ior: Lorenz attractors, Julia sets, and Hutchin- sons maps, Differential Equations, Chaos and Vari- ational Problems, Birkh¨ auser, 2008.

[3] N. Bonneuil. Computing the viability kernel in large state dimension, J. Math. Anal. Appl. 2006, 323:

1444-1454.

[4] F. Deutch, I. Singer. On single-valuedness of convex set-valued maps, Set-Valued Anal. 1993, 1: 97-103.

[5] K. Kassara, Observability by using viability kernels, J. Control Theory Appl., to appear.

[6] A. B. Levy, R. A. Poliquin. Characterizing the single-valuedness of multifunctions, Set-Valued Anal. 1997, 5: 351-364.

[7] K. Nikodem, D. Popa. On single-valuedness of set- valued maps satisfying linear inclusions, Banach J.

Math. Anal.2009, 3: 44-51.

[8] P. Saint-Pierre, Approximation of the viability ker- nel, Appl. Math. Optim. 1994, 29: 187-209.

[9] E. H. Zarantonello. Dense single-valuedness of

monotone operators, Israel J. Math. 1973, 15: 158-

166.