- Email: [email protected]
A SINGLE SENSOR SELECTION THEOREM FOR RATIONAL STATE SYSTEMS
14th IFAC Symposium on System Identification, Newcastle, Australia, 2006
A SINGLE SENSOR SELECTION THEOREM FOR RATIONAL STATE SYSTEMS Sette Diop
Labratoire des Signaux et Systèmes CNRS – Supélec – Univ. Paris Sud Plateau de Moulon 91192 Gif sur Yvette cedex France E-mail: [email protected]
Abstract: In this Communication we show that for any dynamics given by a rational state vector equation d x/dt = f (u, x) there always is a scalar linear observation y = α1 x1 + α2 x2 + · · · + αn xn which makes the state x observable provided that the coefficients α1 , α2 , · · · , αn are allowed to be nonconstants. Moreover, any such scalar observation makes the system observable if, and only if, the coefficients are linearly independent over constants in a differential algebraic sense. Keywords: Observability; Identifiability; Parameter estimation; Nonlinear control systems; Algebraic systems theory;
1. INTRODUCTION
the coefficients of that linear combination are al lowed to be time-varying.
Given a dynamic system x˙ = f (u, x)
The precise context of this result is the differential algebraic approach of observation problems. The proof relies on the following two facts
(1)
with input u , and state x it is of interest to select measurements (when there is such an opportu nity) y = h(u, x)
• the system’s dynamics is given in the explicit form as in (1) (and not in possible implicit form). • f is a rational function of its arguments with coefficients in a differential field.
(2)
such that the state becomes observable with re spect to (u, y). This problem has been addressed in the parameter estimation context for linear sys tems in many works, see for instance pioneering ones (Mehra, 1976; Friedland, 1977). The basic aim of this sensor selection task is to improve the observability or identifiability property of the given dynamics (1).
The differential algebraic approach of observa tion problems dates back to late eighties and early nineties with works of (Pommaret, 1986; Fliess, 1987; Glad and Ljung, 1990; Diop and Fliess, 1991a; Diop and Fliess, 1991b). See (Diop, 2002) for a recent survey. The main point of this approach, as first clarified in (Diop and Fliess, 1991a) , is that a quantity, say z , of a system is observable with respect to some other one, say w (which is supposed to be available in some time interval), if each component of z is a
In this Communication we show that the measure ments selection problem has a solution as simple as a single sensor delivering a linear combination of values of the state components provided that
832
2. THE RANK CONDITION FOR DIFFERENTIAL ALGEBRAIC SYSTEMS WITH NONCONSTANT COEFFICIENTS
solution of a (non differential) algebraic equation with coefficients eventually depending on w and finitely many of w ’s time derivatives. The theory applies to models of systems in terms of differ ential algebraic equations only but which may be implicit in the variables to be observed. In this approach the identifiability of constant parame ters is simply viewed as the observability of these parameters in the system equations supplemented by differential equations expressing the fact that the time derivatives of the parameters are zero.
A general observability test is derived for polyno mial systems Pi (w, z, ζ) = 0 ,
(4)
where the Pi ’s are (finitely) many differential polynomials in w , z , ζ with coefficients in a differential field k which does not necessarily consist of constants. The variable w stands for the data, z for the variable to be observed, and ζ for the remaining variables which may eventually be present in the system’s description and which are neither part of data nor being observed. This test reduces to a general rank condition which is a necessary and sufficient condition for the observability of z with respect to w.
A quite general rank condition which applies to implicit differential algebaric systems has been obtained in (Diop and Fliess, 1991a). When spe cialized to rational systems (1) this rank condi tion is similar to (but is not formally the same as) the rank condition found in (Hermann and Krener, 1977). When the output equation (2) is scalar and linear in x as follows
y = α1 x1 + α2 x2 + · · · + αn xn
i = 1, 2, . . .
Notation 1. If z = (zj )1≤j≤n and r = (rj )1≤j≤n ∈ Nn then the family (j ) (j ) z1 1 , z2 2 , . . . , zn(jn ) 0≤j1 ≤r1 ,0≤j2 ≤r2 ,...,0≤jn ≤rn
(3)
[r]
will be denoted by z . The symbol Z [r] will stand for the corresponding family of indeterminates.
then this rank condition reduces to the rank of the Wronskian matrix of the coefficients α. The existence of a scalar output (3) which makes the state of (1) observable then results from a classical theorem on the linear dependence of vectors of functions over constants.
The following result is detailed in (Diop, 2002). Theorem 2. Let X be a system with variables w , z = z1 , z2 , . . . , zn , and ζ. Let rj ∈ N (1 ≤ j ≤ n) be given. Let Q1 , Q2 , · · · , Qσ ∈ kw Z [r] be a set of generators of the ideal of definition of kw z [r] over kw. Then z is observable with respect to w if, and only if, the Jacobian matrix
∂Qi 1≤i≤σ (k ) 1 ≤ j ≤ n, 1 ≤ kj ≤ rj ∂z j
In addition, the single sensor selection problem thus obtained implies that if the rational function f is with coefficients in a differential field of con stants (say, k = R) then any set of functions of the time, α1 (t), α2 (t), . . . , αn (t) , which is linearly independent over k , will make the state observ able.
j
with σ rows and
n
(rj + 1) columns is of rank
j=1
This result is thought to be fundamental in ob server design tasks where it may be desirable to reshape output equations in order to improve observability conditions.
n
The paper is organized as follows. The next sec tion is a preparation for the proof of the main result. The rank condition above mentioned is recalled and its validity is extended to the case where the ground field of coefficients of the sys tem is not necessarily of constants. Then in sec tion Section 3 the main result is state and proved. The paper ends with some concluding remarks.
The previous test is then specialized to rational state systems
x˙ = f (u, x) , y = h(u, x) .
Most of the basic notions of differential algebra used throughout are not recalled here. The reader who is not familiar with them may refer to (Ritt, 1950; Kolchin, 1973). An account of some of these notions is contained in (Diop, 2002).
The proof of the rational case already appeared in (Diop, 2002) but for constant coefficient field k. In view of the main result of this Communication we need the same rank condition for rational systems but for coefficient fields not necessarily
(rj + 1) over kw z [r] .
j=1
The generating polynomials previously mentioned in the general test or Theorem 2 are here explicitly computed.
833
coefficients of P by their respective derivatives respectively at order 1 , 2 , etc.
of constants. However the proof of the rational case is lengthy and would not fit in the available space here. This is why, in the sequel, we restrict ourselves to the polynomial case.
Note that this formula is nothing but a counter part of Lie derivatives: Authors usually consider the functions hj as free of u and the functions fi and hj as with constant coefficients so that in the left hand side of the latter equation the first sum as well as the last term are absent.
Let k be an ordinary differential field (not neces sarily of constants). Let X be a system with state description x˙ i = fi (u, x) (1 ≤ i ≤ n) , (5) yj = hj (u, x) (1 ≤ j ≤ p) ,
This construction of Qp+i from Qi is iterated in order to get Q2p+i (1 ≤ i ≤ p) as the remainder of the derivative of Qp+i (1 ≤ i ≤ p). And so on.
where the fi ’s and hj ’s are (nondifferential) poly nomials in their arguments with coefficients in k. Let
By their definition, σ ku, y (Xj )1≤j≤n (i ∈ N) . Qσi ∈ I (X )
(1)
Pi (U, X, Y ) = Xi − fi (U, X) (1 ≤ i ≤ n); Pn+j (U, X, Y ) = hj (U, X) − Yj (1 ≤ j ≤ p).
Conversely, let P
be the differential polynomials defining X . Let σ : k {U, Y } → k {u, y} be the substitution map which sends U into u and Y into y , where k {u, y} is the differential k-subalgebra of k {u, x, y} gen erated over k by u and y. Let P be in k {U, X, Y } and P σ denote the element of k {u, y} {X} ob tained by regarding P as a differential polynomial in X with coefficients in k {U, Y } and by applying σ σ to each of these coefficients, and let I (X ) stand for the differential ideal of k {u, y} {X} consisting of P σ (P ∈ I (X )). The ideal of def inition, a , of ku, ku, y is equal to y(x) over σ ku, y (Xi )1≤i≤n . Note that the set I (X ) A consisting of the Pi (1 ≤ i ≤ n) form an autoreduced set with respect to any ranking of k {U, X, Y } such that U, Y and their derivatives all are lower than X.
σ
P ∈ I (X )
where Bi (i ∈ N) are in I (X ) k {U, Y } (Xi )1≤i≤n . Since the differential ideal of k {U, X, Y } gener element in ated by Pi (1 ≤ i ≤ n) has no nonzero common with I (X ) k {U, Y } (Xj )1≤i≤n (this results from an obvious degree argument), the first sum in the previous equality must be zero. This ends the proof that Qσi (i ∈ N) form a basis of a. Let us summarize what precedes in the following statement which has its own interest, and is the clue of the Hermann-Krener observability test we are rederiving now.
(1 ≤ i ≤ p),
we then let Qp+i be the remainder of the deriva tive of Qi (1 ≤ j ≤ n) with respect to the previously mentioned autoreduced set, A. The polynomial Qp+i is merely the derivative of Qi in (1) which Xj is eliminated by substituting Pj + fj
Lemma 3. A set of generators of the ideal of definition of ku, y(x) over ku, y is given by Q1 (u, X, y), Q2 (u, X, y), . . . , Qp (u, X, y), Qp+1 (u, X, y), Qp+2 (u, X, y), . . . , Q2p (u, X, y), .............................................
(1)
for Xj (1 ≤ j ≤ n) (The linear combination of Pj (1 ≤ j ≤ n) which appears reduces to zero when we take the remainder, so that it can be ignored.) Explicitly, Qp+i is as follows ∂Qi ∂Qi (1) (1) Qp+i = Uj + fj − Yi ∂Uj ∂Xj + Qi •
In addition, it comes from the Hilbert basis the orem that only finitely many Qi suffice to gen erate the ideal a. That is, there is some μ in N such that the first μ rows of the previous list of Qi (u, X, y) generate the ideal of definition of ku, y(x) over ku, y. According to Theorem 1, the observability of X is equivalent to the fact that the ku, y(x)-matrix
1≤j≤n
(1 ≤ i ≤ p) ,
where the notations P• ≡ P(1) ,
P(2) ,
...
i∈N
1≤i≤n, j∈N
Starting with
1≤j≤m
ku, y (Xj )1≤i≤n .
As an element of I (X ) , P may easily be written in the form (j) Ai,j Pi + Bi Qi . P =
Let us now inductively define some polynomials which will turnout to be generators of the ideal σ ku, y (Xi )1≤i≤n of ku, y (Xi )1≤i≤n . I (X ) Qi (U, X, Y ) = Pn+i (U, X, Y )
(6)
for a differential polynomial P stand for the differential polynomials obtained by replacing the
is of rank n.
834
∂Qi (u, x, y) ∂Xj
1≤i≤μ p 1≤j≤n
and sufficient, that the Jacobian matrix of y [n] with respect to x ⎛ ⎞ ∂y ⎜ ∂x ⎟ ⎜ ⎟ ⎜ ∂ y˙ ⎟ ⎜ ⎟ ⎜ ∂x ⎟ ∂y [n] ⎜ ⎟ = ⎜ ⎟ .. ⎜ ⎟ ∂x ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ ⎝ ∂y (n−1) ⎠
Now it is a basic fact that the above rank is equal to the rank of the first np rows. We thus have the following result. Corollary 4. If X possesses a state description as above, then X is observable if, and only if, the following ku, y(x)-matrix
∂Qi (u, y, x) ∂Xj
1≤i≤n p 1≤j≤n
∂x ⎛
(which is formally the counterpart of the matrix of Lie derivatives which appears in the Herman n-Krener observability test) is of rank n.
The previous formal way of writing the analogues of Lie derivatives will be relaxed in the next section where the time derivative of the output will be written in lieu of the formal polynomials Q previously considered.
⎛
⎞ α1 ⎜ ⎟ α = ⎝ ... ⎠ . αn
/∂x is merely ∂y = α . ∂x The second row of the same matrix is ∂f (u, x) ∂ y˙ = α˙ + α . ∂x ∂x Therefore, ⎛
The first row of ∂y
Theorem 5. Let the state of a system X be given by (1)
with a vector rational function f of input u , state x , and with coefficients in a differential field k. Let m and n be the respective numbers of components of u and x. Let K be a differential extension field of k. If K contains nonconstants then there always is a scalar output αi xi
⎞
y = α x
3. THE SINGLE SENSOR SELECTION THEOREM
n
∂y ∂xn ∂ y˙ ∂xn
Now y may be written as where
y=
···
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ · · · ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (n−1) ⎠ ⎝ ∂y (n−1) ∂y (n−1) ∂y ··· ∂x1 ∂x2 ∂xn be of rank n over Ku, y(x) where we have con sidered the complete system as with coefficients in K ⊇ k.
The main difference between this rank condition and the one in (Hermann and Krener, 1977) is that the rank is not over k (which is usually R) but over a much bigger field (and, here the rank condition is a necessary and sufficient condition).
x˙ = f (u, x)
∂y ∂x2 ∂ y˙ ∂x2
∂y ∂x1 ∂ y˙ ∂x1 .. .
[n]
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
α
⎞
⎟ ⎟ ⎟ ⎟ ⎟ ∂ y ¨ ⎟ ∂y [n] ⎟ = rk rkku,y(x) ⎟ ∂x ku,y(x) ⎟ ∂x ⎟ .. ⎟ . ⎟ ⎟ ⎠ ∂y (n−1) ∂x by substituting the linear combination ∂ y˙ ∂y ∂f (u, x) − = α˙ ∂x ∂x ∂x over ku, y(x) of the first two rows for the second row of ∂y [n] /∂x .
(3)
i=1
with α1 , α2 , . . . , αn in K , which makes x observ able with respect to (u, y). Moreover, for y as in output (3) to make X observable it is sufficient that the associated α ’s be linearly independent over the subfield of constants of K. PROOF. Let y [n] denote the vector ⎛ ⎞ y ⎜ y˙ ⎟ ⎜ ⎟ y [n] = ⎜ . ⎟ . ⎝ .. ⎠
α˙
More generally, by the Leibniz formula, i i (i−j) (j) α x , y (i) = α(i) x + j
y (n−1) By Corollary 4 , for the output (3) to make x observable with respect to (u, y) it is necessary,
j=1
835
and by the fact that x(j) is in ku(x) for all j ≥ 1 , it is clear that ⎛ ⎞ α ⎜ α˙ ⎟ ⎜ ⎟ ∂y [n] ⎜ α ¨ ⎟ rkku,y(x) = rk ⎜ ⎟ ku,y(x) ⎜ .. ⎟ ∂x ⎝ . ⎠ α(n−1)
one sensor to observe its state, but only a single sensor is enough if that sensor is allowed to be with time varying coefficients. Moreover, in that case, almost any time-varying linear combination of the state components will lead to the observ ability of x; the only restriction on the choice of the time-varying coefficients being that their Wronskian be non singular.
by an immediate induction on the row’s index of the Jacobian matrix. Now we note that the matrix ⎛ α ⎜ α˙ ⎜ W(α) = ⎜ .. ⎝ .
REFERENCES Diop, S. (2002). From the geometry to the algebra of nonlinear observability. In: Contemporary Trends in Nonlinear Geometric Control The ory and its Applications (A. Anzaldo-Mene ses, B. Bonnard, J. P. Gauthier and F. Mon roy-Perez, Eds.). World Scientific Publishing Co.. Singapore. pp. 305–345. Diop, S. and M. Fliess (1991b). On nonlin ear observability. In: Proceedings of the Eu ropean Control Conference (C. Commault, D. Normand-Cyrot, J. M. Dion, L. Dugard, M. Fliess, A. Titli, G. Cohen, A. Benveniste and I. D. Landau, Eds.). Hermès. Paris. pp. 152–157. Diop, S. and M. Fliess (1991a). Nonlinear observ ability, identifiability, and persistent trajecto ries. In: Proceedings of the IEEE Conference on Decision and Control. IEEE Press. New York. pp. 714–719. Fliess, M. (1987). Quelques remarques sur les observateurs non linéaires. In: Proceedings Colloque GRETSI Traitement du Signal et des Images. GRETSI. pp. 169–172. Friedland, B. (1977). On the calibration problem. IEEE Trans. Automat. Control 22 , 899–905. Glad, S. T. and L. Ljung (1990). Model structure identifiability and persistence of excitation. In: Proceedings of the IEEE Conference on Decision and Control. IEEE Press. New York. Hermann, R. and A. J. Krener (1977). Nonlinear controllability and observability. IEEE Trans. Automat. Control 22 , 728–740. Kaplansky, I. (1976). An Introduction to Differen tial Algebra. Second Edition. Hermann. Paris. Kolchin, E. R. (1973). Differential Algebra and Al gebraic Groups. Academic Press. New York. Mehra, R. K. (1976). Optimization of measure ment schedules and sensor designs for lin ear dynamic systems. IEEE Trans. Automat. Control 21 , 55–64. Pommaret, J. F. (1986). Géométrie différentielle algébrique et théorie du contrôle. C. R. Acad. Sci. Paris Sér. I 302 , 547–550. Ritt, J. F. (1950). Differential Algebra. American Mathematical Society. Providence. Seidenberg, A. (1952). Some basic theorems in dif ferential algebra (characteristic p , arbitrary). Trans. Amer. Math. Soc. 73 , 174–190.
⎞ ⎟ ⎟ ⎟ ⎠
α(n−1) is square , and of order n , and does not involve neither u nor x. Therefore it is of rank n over Ku, y(x) if, and only, its determinant is nonzero. We next note that the determinant of W(α) is simply a differential polynomial in α1 , α2 , . . . , αn with coefficients in the field of constants of K. We then refer to the following theorem Theorem 6. If G is a nonzero differential polyno mial in n indeterminates with coefficients in a differential field containing nonconstant elements then G possesses a zero (z1 , . . . , zn ) over K. For a proof see (Ritt, 1950; Seidenberg, 1952; Kolchin, 1973; Kaplansky, 1976) for instance. This terminates the proof of the first assertion in our theorem. The second assertion follows from the following. The n elements α1 , α2 , . . . , αn of K are said to be linear dependent over constants if there is a nontrivial relation c1 α1 + c2 α2 + . . . + cn αn = 0 with constant coefficients. It is a classical result that Theorem 7. α1 , α2 , . . . , αn are linearly dependent over constants if, and only if, the Wronskian matrix W(α) is singular. For a proof see the same references (Ritt, 1950; Seidenberg, 1952; Kolchin, 1973; Kaplansky, 1976) for instance. This ends the proof of our theorem. 2 4. CONCLUDING REMARKS An immediate consequence of Theorem 5 is that if system (1) is with coefficients in a field of con stants (say k = R) then we may need more than
836
A SINGLE SENSOR SELECTION THEOREM FOR RATIONAL STATE SYSTEMS Sette Diop
Labratoire des Signaux et Systèmes CNRS – Supélec – Univ. Paris Sud Plateau de Moulon 91192 Gif sur Yvette cedex France E-mail: [email protected]
Abstract: In this Communication we show that for any dynamics given by a rational state vector equation d x/dt = f (u, x) there always is a scalar linear observation y = α1 x1 + α2 x2 + · · · + αn xn which makes the state x observable provided that the coefficients α1 , α2 , · · · , αn are allowed to be nonconstants. Moreover, any such scalar observation makes the system observable if, and only if, the coefficients are linearly independent over constants in a differential algebraic sense. Keywords: Observability; Identifiability; Parameter estimation; Nonlinear control systems; Algebraic systems theory;
1. INTRODUCTION
the coefficients of that linear combination are al lowed to be time-varying.
Given a dynamic system x˙ = f (u, x)
The precise context of this result is the differential algebraic approach of observation problems. The proof relies on the following two facts
(1)
with input u , and state x it is of interest to select measurements (when there is such an opportu nity) y = h(u, x)
• the system’s dynamics is given in the explicit form as in (1) (and not in possible implicit form). • f is a rational function of its arguments with coefficients in a differential field.
(2)
such that the state becomes observable with re spect to (u, y). This problem has been addressed in the parameter estimation context for linear sys tems in many works, see for instance pioneering ones (Mehra, 1976; Friedland, 1977). The basic aim of this sensor selection task is to improve the observability or identifiability property of the given dynamics (1).
The differential algebraic approach of observa tion problems dates back to late eighties and early nineties with works of (Pommaret, 1986; Fliess, 1987; Glad and Ljung, 1990; Diop and Fliess, 1991a; Diop and Fliess, 1991b). See (Diop, 2002) for a recent survey. The main point of this approach, as first clarified in (Diop and Fliess, 1991a) , is that a quantity, say z , of a system is observable with respect to some other one, say w (which is supposed to be available in some time interval), if each component of z is a
In this Communication we show that the measure ments selection problem has a solution as simple as a single sensor delivering a linear combination of values of the state components provided that
832
2. THE RANK CONDITION FOR DIFFERENTIAL ALGEBRAIC SYSTEMS WITH NONCONSTANT COEFFICIENTS
solution of a (non differential) algebraic equation with coefficients eventually depending on w and finitely many of w ’s time derivatives. The theory applies to models of systems in terms of differ ential algebraic equations only but which may be implicit in the variables to be observed. In this approach the identifiability of constant parame ters is simply viewed as the observability of these parameters in the system equations supplemented by differential equations expressing the fact that the time derivatives of the parameters are zero.
A general observability test is derived for polyno mial systems Pi (w, z, ζ) = 0 ,
(4)
where the Pi ’s are (finitely) many differential polynomials in w , z , ζ with coefficients in a differential field k which does not necessarily consist of constants. The variable w stands for the data, z for the variable to be observed, and ζ for the remaining variables which may eventually be present in the system’s description and which are neither part of data nor being observed. This test reduces to a general rank condition which is a necessary and sufficient condition for the observability of z with respect to w.
A quite general rank condition which applies to implicit differential algebaric systems has been obtained in (Diop and Fliess, 1991a). When spe cialized to rational systems (1) this rank condi tion is similar to (but is not formally the same as) the rank condition found in (Hermann and Krener, 1977). When the output equation (2) is scalar and linear in x as follows
y = α1 x1 + α2 x2 + · · · + αn xn
i = 1, 2, . . .
Notation 1. If z = (zj )1≤j≤n and r = (rj )1≤j≤n ∈ Nn then the family (j ) (j ) z1 1 , z2 2 , . . . , zn(jn ) 0≤j1 ≤r1 ,0≤j2 ≤r2 ,...,0≤jn ≤rn
(3)
[r]
will be denoted by z . The symbol Z [r] will stand for the corresponding family of indeterminates.
then this rank condition reduces to the rank of the Wronskian matrix of the coefficients α. The existence of a scalar output (3) which makes the state of (1) observable then results from a classical theorem on the linear dependence of vectors of functions over constants.
The following result is detailed in (Diop, 2002). Theorem 2. Let X be a system with variables w , z = z1 , z2 , . . . , zn , and ζ. Let rj ∈ N (1 ≤ j ≤ n) be given. Let Q1 , Q2 , · · · , Qσ ∈ kw Z [r] be a set of generators of the ideal of definition of kw z [r] over kw. Then z is observable with respect to w if, and only if, the Jacobian matrix
∂Qi 1≤i≤σ (k ) 1 ≤ j ≤ n, 1 ≤ kj ≤ rj ∂z j
In addition, the single sensor selection problem thus obtained implies that if the rational function f is with coefficients in a differential field of con stants (say, k = R) then any set of functions of the time, α1 (t), α2 (t), . . . , αn (t) , which is linearly independent over k , will make the state observ able.
j
with σ rows and
n
(rj + 1) columns is of rank
j=1
This result is thought to be fundamental in ob server design tasks where it may be desirable to reshape output equations in order to improve observability conditions.
n
The paper is organized as follows. The next sec tion is a preparation for the proof of the main result. The rank condition above mentioned is recalled and its validity is extended to the case where the ground field of coefficients of the sys tem is not necessarily of constants. Then in sec tion Section 3 the main result is state and proved. The paper ends with some concluding remarks.
The previous test is then specialized to rational state systems
x˙ = f (u, x) , y = h(u, x) .
Most of the basic notions of differential algebra used throughout are not recalled here. The reader who is not familiar with them may refer to (Ritt, 1950; Kolchin, 1973). An account of some of these notions is contained in (Diop, 2002).
The proof of the rational case already appeared in (Diop, 2002) but for constant coefficient field k. In view of the main result of this Communication we need the same rank condition for rational systems but for coefficient fields not necessarily
(rj + 1) over kw z [r] .
j=1
The generating polynomials previously mentioned in the general test or Theorem 2 are here explicitly computed.
833
coefficients of P by their respective derivatives respectively at order 1 , 2 , etc.
of constants. However the proof of the rational case is lengthy and would not fit in the available space here. This is why, in the sequel, we restrict ourselves to the polynomial case.
Note that this formula is nothing but a counter part of Lie derivatives: Authors usually consider the functions hj as free of u and the functions fi and hj as with constant coefficients so that in the left hand side of the latter equation the first sum as well as the last term are absent.
Let k be an ordinary differential field (not neces sarily of constants). Let X be a system with state description x˙ i = fi (u, x) (1 ≤ i ≤ n) , (5) yj = hj (u, x) (1 ≤ j ≤ p) ,
This construction of Qp+i from Qi is iterated in order to get Q2p+i (1 ≤ i ≤ p) as the remainder of the derivative of Qp+i (1 ≤ i ≤ p). And so on.
where the fi ’s and hj ’s are (nondifferential) poly nomials in their arguments with coefficients in k. Let
By their definition, σ ku, y (Xj )1≤j≤n (i ∈ N) . Qσi ∈ I (X )
(1)
Pi (U, X, Y ) = Xi − fi (U, X) (1 ≤ i ≤ n); Pn+j (U, X, Y ) = hj (U, X) − Yj (1 ≤ j ≤ p).
Conversely, let P
be the differential polynomials defining X . Let σ : k {U, Y } → k {u, y} be the substitution map which sends U into u and Y into y , where k {u, y} is the differential k-subalgebra of k {u, x, y} gen erated over k by u and y. Let P be in k {U, X, Y } and P σ denote the element of k {u, y} {X} ob tained by regarding P as a differential polynomial in X with coefficients in k {U, Y } and by applying σ σ to each of these coefficients, and let I (X ) stand for the differential ideal of k {u, y} {X} consisting of P σ (P ∈ I (X )). The ideal of def inition, a , of ku, ku, y is equal to y(x) over σ ku, y (Xi )1≤i≤n . Note that the set I (X ) A consisting of the Pi (1 ≤ i ≤ n) form an autoreduced set with respect to any ranking of k {U, X, Y } such that U, Y and their derivatives all are lower than X.
σ
P ∈ I (X )
where Bi (i ∈ N) are in I (X ) k {U, Y } (Xi )1≤i≤n . Since the differential ideal of k {U, X, Y } gener element in ated by Pi (1 ≤ i ≤ n) has no nonzero common with I (X ) k {U, Y } (Xj )1≤i≤n (this results from an obvious degree argument), the first sum in the previous equality must be zero. This ends the proof that Qσi (i ∈ N) form a basis of a. Let us summarize what precedes in the following statement which has its own interest, and is the clue of the Hermann-Krener observability test we are rederiving now.
(1 ≤ i ≤ p),
we then let Qp+i be the remainder of the deriva tive of Qi (1 ≤ j ≤ n) with respect to the previously mentioned autoreduced set, A. The polynomial Qp+i is merely the derivative of Qi in (1) which Xj is eliminated by substituting Pj + fj
Lemma 3. A set of generators of the ideal of definition of ku, y(x) over ku, y is given by Q1 (u, X, y), Q2 (u, X, y), . . . , Qp (u, X, y), Qp+1 (u, X, y), Qp+2 (u, X, y), . . . , Q2p (u, X, y), .............................................
(1)
for Xj (1 ≤ j ≤ n) (The linear combination of Pj (1 ≤ j ≤ n) which appears reduces to zero when we take the remainder, so that it can be ignored.) Explicitly, Qp+i is as follows ∂Qi ∂Qi (1) (1) Qp+i = Uj + fj − Yi ∂Uj ∂Xj + Qi •
In addition, it comes from the Hilbert basis the orem that only finitely many Qi suffice to gen erate the ideal a. That is, there is some μ in N such that the first μ rows of the previous list of Qi (u, X, y) generate the ideal of definition of ku, y(x) over ku, y. According to Theorem 1, the observability of X is equivalent to the fact that the ku, y(x)-matrix
1≤j≤n
(1 ≤ i ≤ p) ,
where the notations P• ≡ P(1) ,
P(2) ,
...
i∈N
1≤i≤n, j∈N
Starting with
1≤j≤m
ku, y (Xj )1≤i≤n .
As an element of I (X ) , P may easily be written in the form (j) Ai,j Pi + Bi Qi . P =
Let us now inductively define some polynomials which will turnout to be generators of the ideal σ ku, y (Xi )1≤i≤n of ku, y (Xi )1≤i≤n . I (X ) Qi (U, X, Y ) = Pn+i (U, X, Y )
(6)
for a differential polynomial P stand for the differential polynomials obtained by replacing the
is of rank n.
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∂Qi (u, x, y) ∂Xj
1≤i≤μ p 1≤j≤n
and sufficient, that the Jacobian matrix of y [n] with respect to x ⎛ ⎞ ∂y ⎜ ∂x ⎟ ⎜ ⎟ ⎜ ∂ y˙ ⎟ ⎜ ⎟ ⎜ ∂x ⎟ ∂y [n] ⎜ ⎟ = ⎜ ⎟ .. ⎜ ⎟ ∂x ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ ⎝ ∂y (n−1) ⎠
Now it is a basic fact that the above rank is equal to the rank of the first np rows. We thus have the following result. Corollary 4. If X possesses a state description as above, then X is observable if, and only if, the following ku, y(x)-matrix
∂Qi (u, y, x) ∂Xj
1≤i≤n p 1≤j≤n
∂x ⎛
(which is formally the counterpart of the matrix of Lie derivatives which appears in the Herman n-Krener observability test) is of rank n.
The previous formal way of writing the analogues of Lie derivatives will be relaxed in the next section where the time derivative of the output will be written in lieu of the formal polynomials Q previously considered.
⎛
⎞ α1 ⎜ ⎟ α = ⎝ ... ⎠ . αn
/∂x is merely ∂y = α . ∂x The second row of the same matrix is ∂f (u, x) ∂ y˙ = α˙ + α . ∂x ∂x Therefore, ⎛
The first row of ∂y
Theorem 5. Let the state of a system X be given by (1)
with a vector rational function f of input u , state x , and with coefficients in a differential field k. Let m and n be the respective numbers of components of u and x. Let K be a differential extension field of k. If K contains nonconstants then there always is a scalar output αi xi
⎞
y = α x
3. THE SINGLE SENSOR SELECTION THEOREM
n
∂y ∂xn ∂ y˙ ∂xn
Now y may be written as where
y=
···
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ · · · ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (n−1) ⎠ ⎝ ∂y (n−1) ∂y (n−1) ∂y ··· ∂x1 ∂x2 ∂xn be of rank n over Ku, y(x) where we have con sidered the complete system as with coefficients in K ⊇ k.
The main difference between this rank condition and the one in (Hermann and Krener, 1977) is that the rank is not over k (which is usually R) but over a much bigger field (and, here the rank condition is a necessary and sufficient condition).
x˙ = f (u, x)
∂y ∂x2 ∂ y˙ ∂x2
∂y ∂x1 ∂ y˙ ∂x1 .. .
[n]
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
α
⎞
⎟ ⎟ ⎟ ⎟ ⎟ ∂ y ¨ ⎟ ∂y [n] ⎟ = rk rkku,y(x) ⎟ ∂x ku,y(x) ⎟ ∂x ⎟ .. ⎟ . ⎟ ⎟ ⎠ ∂y (n−1) ∂x by substituting the linear combination ∂ y˙ ∂y ∂f (u, x) − = α˙ ∂x ∂x ∂x over ku, y(x) of the first two rows for the second row of ∂y [n] /∂x .
(3)
i=1
with α1 , α2 , . . . , αn in K , which makes x observ able with respect to (u, y). Moreover, for y as in output (3) to make X observable it is sufficient that the associated α ’s be linearly independent over the subfield of constants of K. PROOF. Let y [n] denote the vector ⎛ ⎞ y ⎜ y˙ ⎟ ⎜ ⎟ y [n] = ⎜ . ⎟ . ⎝ .. ⎠
α˙
More generally, by the Leibniz formula, i i (i−j) (j) α x , y (i) = α(i) x + j
y (n−1) By Corollary 4 , for the output (3) to make x observable with respect to (u, y) it is necessary,
j=1
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and by the fact that x(j) is in ku(x) for all j ≥ 1 , it is clear that ⎛ ⎞ α ⎜ α˙ ⎟ ⎜ ⎟ ∂y [n] ⎜ α ¨ ⎟ rkku,y(x) = rk ⎜ ⎟ ku,y(x) ⎜ .. ⎟ ∂x ⎝ . ⎠ α(n−1)
one sensor to observe its state, but only a single sensor is enough if that sensor is allowed to be with time varying coefficients. Moreover, in that case, almost any time-varying linear combination of the state components will lead to the observ ability of x; the only restriction on the choice of the time-varying coefficients being that their Wronskian be non singular.
by an immediate induction on the row’s index of the Jacobian matrix. Now we note that the matrix ⎛ α ⎜ α˙ ⎜ W(α) = ⎜ .. ⎝ .
REFERENCES Diop, S. (2002). From the geometry to the algebra of nonlinear observability. In: Contemporary Trends in Nonlinear Geometric Control The ory and its Applications (A. Anzaldo-Mene ses, B. Bonnard, J. P. Gauthier and F. Mon roy-Perez, Eds.). World Scientific Publishing Co.. Singapore. pp. 305–345. Diop, S. and M. Fliess (1991b). On nonlin ear observability. In: Proceedings of the Eu ropean Control Conference (C. Commault, D. Normand-Cyrot, J. M. Dion, L. Dugard, M. Fliess, A. Titli, G. Cohen, A. Benveniste and I. D. Landau, Eds.). Hermès. Paris. pp. 152–157. Diop, S. and M. Fliess (1991a). Nonlinear observ ability, identifiability, and persistent trajecto ries. In: Proceedings of the IEEE Conference on Decision and Control. IEEE Press. New York. pp. 714–719. Fliess, M. (1987). Quelques remarques sur les observateurs non linéaires. In: Proceedings Colloque GRETSI Traitement du Signal et des Images. GRETSI. pp. 169–172. Friedland, B. (1977). On the calibration problem. IEEE Trans. Automat. Control 22 , 899–905. Glad, S. T. and L. Ljung (1990). Model structure identifiability and persistence of excitation. In: Proceedings of the IEEE Conference on Decision and Control. IEEE Press. New York. Hermann, R. and A. J. Krener (1977). Nonlinear controllability and observability. IEEE Trans. Automat. Control 22 , 728–740. Kaplansky, I. (1976). An Introduction to Differen tial Algebra. Second Edition. Hermann. Paris. Kolchin, E. R. (1973). Differential Algebra and Al gebraic Groups. Academic Press. New York. Mehra, R. K. (1976). Optimization of measure ment schedules and sensor designs for lin ear dynamic systems. IEEE Trans. Automat. Control 21 , 55–64. Pommaret, J. F. (1986). Géométrie différentielle algébrique et théorie du contrôle. C. R. Acad. Sci. Paris Sér. I 302 , 547–550. Ritt, J. F. (1950). Differential Algebra. American Mathematical Society. Providence. Seidenberg, A. (1952). Some basic theorems in dif ferential algebra (characteristic p , arbitrary). Trans. Amer. Math. Soc. 73 , 174–190.
⎞ ⎟ ⎟ ⎟ ⎠
α(n−1) is square , and of order n , and does not involve neither u nor x. Therefore it is of rank n over Ku, y(x) if, and only, its determinant is nonzero. We next note that the determinant of W(α) is simply a differential polynomial in α1 , α2 , . . . , αn with coefficients in the field of constants of K. We then refer to the following theorem Theorem 6. If G is a nonzero differential polyno mial in n indeterminates with coefficients in a differential field containing nonconstant elements then G possesses a zero (z1 , . . . , zn ) over K. For a proof see (Ritt, 1950; Seidenberg, 1952; Kolchin, 1973; Kaplansky, 1976) for instance. This terminates the proof of the first assertion in our theorem. The second assertion follows from the following. The n elements α1 , α2 , . . . , αn of K are said to be linear dependent over constants if there is a nontrivial relation c1 α1 + c2 α2 + . . . + cn αn = 0 with constant coefficients. It is a classical result that Theorem 7. α1 , α2 , . . . , αn are linearly dependent over constants if, and only if, the Wronskian matrix W(α) is singular. For a proof see the same references (Ritt, 1950; Seidenberg, 1952; Kolchin, 1973; Kaplansky, 1976) for instance. This ends the proof of our theorem. 2 4. CONCLUDING REMARKS An immediate consequence of Theorem 5 is that if system (1) is with coefficients in a field of con stants (say k = R) then we may need more than
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