ON TWO DEFINITIONS OF OBSERVATION SPACES 1

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1 ON TWO DEFINITIONS OF OBSERVATION SPACES Yuan Wang Eduardo D. Sontag Department of Mathematics Rutgers University, New Brunswick, NJ ABSTRACT This paper establishes the equality of the observation spaces defined by means of piecewise constant controls with those defined in terms of differentiable controls. This research was supported in part by US Air Force Grant AFOSR keywords: Observation space, shuffle product, input-output system, generating series.

2 Introduction Since their introduction in the mid 70 s (see [5] and [], as well as [7] for the discrete time analogue), observation spaces for nonlinear control systems ẋ = f(x)+ u i g i (x), y = h(x) () have played a central role in the understanding of realization theory. For the system (), one defines the observation space F as the linear span of the Lie derivatives L X L Xk h, where each X i is either f or one of the g i s. (Here we are taking states x(t) in a manifold, f,g,...,g m vector fields, and h a function from the manifold to IR, the output map.) It is known that many important properties of systems, such as the possibility of simulating such a system by one described by linear vector fields (the bilinear immersion problem, []), are characterized by properties of this space. It was shown in [8] that a different type of observation space is much more important when one studies questions of input output equations satisfied by (), i.e. equations of the type E(y (k) (t),...,y (t),y(t),u (k) (t),...,u (t),u(t)) = 0 (2) that hold for all those pairs of functions (u( ),y( )) that arise as solutions of (). This alternative observation space is obtained by taking the derivatives y(t),y (t),... as functions of initial states, over all u(t),u (t),... This space is obtained by considering differentiable controls and time-derivatives, while the space previously considered is based on derivatives with respect to switching times in piecewise constant controls. The central fact used in [8] in order to relate i/o equations to realizability is the equality of the two observation spaces defined in the above manners. This equality is fundamental not only for the results in that paper, which hold under the assumption that the spaces are finite-dimensional, but also for the far more general results recently announced in [9]. However, the techniques used in [8] are based on a topological argument, involving closure in the weak topology, which does not in any way extend to the more general case of infinite dimensional observation spaces. Since the latter are the norm rather than the exception (unless the system can be simulated by a bilinear system to start with), one needs to establish the equality of these two types of spaces using totally different combinatorial techniques. That is the purpose of this paper. In the next section we provide background material on generating series. We use this formalism because in applications one does not want to restrict to systems [] but one rather wants to treat the case of arbitrary input/output operators. Then we introduce rigorously the two spaces and establish their equality. An important role is played by an analoge of the main

3 result in [4]. Finally we extend our results to families of operators and then give a translation of the results into the language of systems (). 2 Generating Series Let m be a fixed integer and I = {0,,..., m}. For any integer k, we define I k to be the set of all sequences ( i 2... i k ), where i s I, s k. For k = 0, we use I 0 to denote the set whose only element is the empty sequence φ. Let I = I k. (3) k 0 Then I is a free monoid with the composition rule: ( i 2...i k )(j j 2...j l )=( i 2...i k j j 2...j l ). If ι I l, then we say that the length of ι, denoted by ι, isl. Consider now the alphabet set P = {η 0,η,..., η n } and P, the free monoid generated by P, where the neutral element of P is the empty word, denoted by, and the product is concatenation. Let P k = {η i η i2...η ik : i s m, s k} for each k 0. We define P to be the IR-algebra generated by P, i.e, the set of all polynomials in the variables η i s. A power series in the noncommutative variables η 0,η,...,η n is a formal power series c = c, φ + c, η ι η ι, (4) k= ι I k where η ι = η i η i2 η il if ι = i 2 i l, and c, η ι IR. Note that c is a polynomial if only finitely many c, η ι s are non-zero. A power series is nothing more than a mapping from I to IR; as we shall see later, however, the algebraic structures suggested by the series formalism are very important. We use S to denote the set of all power series. For c, d S and γ IR, γc + d is defined as the following: γc + d, η ι = γ c, η ι + d, η ι. Thus, S forms a vector space over IR. We shall say that the power series c is convergent if c, η ι KM k k! for each ι I k, and each k 0, (5) where K and M are some constants. Let T be a fixed value of time and let U T be the set of all essentially bounded functions u :[0,T] IR m endowed with the L norm. We write u for max{ u i :, i m} if 2

4 u i is the i-th component of u, and u i is the essential supernorm of u i. For each u U T and ι I l, we define inductively the functions V ι = V ι [u] C[0, T]by V φ = and V i i l+ [u](t) = t 0 u i (s)v i2 i l+ (s) ds, (6) where u i is the i-th coordinate of u(t) for i =, 2,...,m and u 0 (t). It can be proved that each map U T C[0, T], u V ι [u] is continuous with respect to L norm in U T, C 0 norm in C[0, T]. Suppose c is convergent and let K and M be as in (5). Then for any T<(Mm+ M) (7) the series of functions F c [u](t) = c, η ι V ι [u](t) (8) is uniformly and absolutely convergent for all t [0, T] and all those u U T such that u (cf [3]). In fact, (8) is absolutely and uniformly convergent for all t [0, T] provided T u < (Mm+ M). For each nonnegative T, let V T = {u U T : u < }. (9) We say that T is admissible for c if T satisfies (7). Since each operator u V ι [u] is continuous, it follows that F c : V T C[0, T] is continuous if T is admissible for c. We call F c an input/output map defined on V T. Thus every convergent power series defines an i/o map. On the other hand, the power series c is uniquely determined by F c in the following sense: Lemma 2. Suppose that c and d are two convergent power series. If F c = F d on V T for any T>0, then c = d. Proof. It s enough to show that if c is convergent and F c =0onV T for some small T, then c = 0. Consider piecewise constant controls in V T, and use the notation u =(µ,t )(µ 2,t 2 ) (µ k,t k ) to denote the piecewise constant control whose value is µ i in the time interval i i t j, t j j=0 i=0 3

5 where and µ j =(µ j,µ 2j,...,µ mj ) IR m, µ ij <, j k, i m t 0 =0. By assumption, for any µ i,t i, such that t i <T, F c [(µ,t )(µ 2,t 2 ) (µ k,t k )](t) = 0, where t = t i. Take y = F c [u] as a function of µ,...,µ k and t,...,t k. Then k s y = 0 (0) t t k t=0+ µ i j µ isjs for all,...,i s,j,...j s, where the evaluation at t + means that we evaluate at t +,...,t+ k.we claim that, for,...,i s,j,...,j s given such that j r j q if r q, k s y = c, η t t k t=0+ µ i j µ l...η lk, () isjs µ=0 µ=0 where l p = { ir if k (p ) = j r 0 if k (p ) / {j,...,j s }. To see this, write y(t) = c, η ι V ι (t). Then, directly from the definition (6), k t t k t=0 + y = c, η l...η lk µ l k...µ l k. (2) One can see that if {(,j ),,(i s,j s )} {(l,k),,(l k, )} and l p = 0 for p/ {j,...,j s } then s µ=0 µ µ i j µ l k µ l isj k =, s otherwise, s µ=0 µ µ i j µ l k µ l isj k =0. s Combining this fact and (2), we get (0). It follows immediately that if F c [u] = 0 for all piecewise constant controls, then c = 0. 3 Observation spaces To each monomial α = η ι, we associate a shift operator c α c defined by α c, η ι = c, αη ι for η ι P. 4

6 Note that α2 α =(α α 2 ) c. It was shown in [8] that if c is convergent and T is admissible for c, then α c is also convergent and T is also admissible for α c for any α P. Using this notation, we get the following fundamental formula [2], which follows from the defition (6): d dt F c[u](t) =F η c[u](t) + m u j (t) F 0 η j c [u](t) (3) j= for any u V T which is continuous. Formula (3) implies, by induction, that if u V T is of class C k, then F c [u] is of class C k. In realization theory, the concept of observation spaces plays a very important role. One may define observation spaces in two ways. Let s now introduce the first approach. To each convergent power series c, we define the observation space F to be the space spanned by all the power series α c over IR, i.e., F (c) = span IR {α c : α P }. (4) It is well known that F c can be realized by a bilinear system if and only if dimf (c) < ; see e.g. []. To define the second type of observation spaces, we need to introduce the shuffle product on P (cf [6]). The shuffle product on P is defined in the following way. First, inductively on the length of of words in P, we let z = z =z for any z P ; wz w z =(w w z )z +(wz w )z for any w, w P,z,z P. Notice that the shuffle product is commutative: w w 2 = w 2 w, for any w,w 2 P. If c = c, η κ η κ and d = d, η ι η ι are polynomials, then c d := n κ + η =n c, η κ d, η ι η ι η κ. The following lemma can be proved by induction n: Lemma 3. Suppose w,...,w n P and w i = w i z i with w i P,z i P. Then n (w w s w n )z s = w w 2 w n. s= 5

7 Now consider for each q, the following set of 2 q matrices: j j 2 j q S q = { : i s,j s Z, i s m, (0, ) (j, ) (i q,j q )}, (5) i 2 i q where is the lexicographgic order on the set {(i, j) : i, j Z}. For each element j j 2 j q in S q and n q + j r, we define i 2 i q Γ j j q i q (n) =η k 0 η i X j η i2 X j 2 η iq X jq X=, (6) where k = n q j s. The evaluation is interpreted as follows: first introduce a new variable X, then perform all shuffles, and finally delete X from the result. Note that (6) is different from η i η i2 η iq, for example, η 0 η X X= = η 0 η +2η η 0 while η 0 η = η 0 η + η η 0. For w P and c S, we define ψ c (w) =w c. For any polynomial d = d, η κ η κ, we define ψ c (d) = d, η κ η κ c. Now let X j =(X j,...,x mj )bem indeterminates over IR, for j 0. For any n>0, let c n (X 0,...,X n )=ψ c (η n 0 )+ n q=0 ( s! s p! ψ c Γ j ) j q i q (n) X i j X iqjq, (7) j j 2 j q where the second sum is taken over all those S q such that j s + q n, i 2 i q and where s,...,s p are integers so that ( j j 2 j q i 2 i q ) = β β β 2 β 2 β p β p α α }{{} α 2 α 2 α }{{} p α p }{{} s s 2 and (α,β ) < (α 2,β 2 ) < < (α p,β p ). For n = 0, we define c 0 := c. s p We are now ready to introduce the second type of observation space associated to c, F 2 (c). 6

8 This is defined as follows: F 2 (c) = span IR {c n (µ 0,...,µ n ): µ i IR m, 0 i n, n 0}. (8) We will see below that the elements of F 2 (c) are closely related to the derivatives of F [u](t) with respect to time. A central fact that will be needed in the proof of our main result is that the coefficients of the generating series can be partitioned into infinitely many sets of finitely many elements such that the coefficient of each monomial u (j ) u (j 2) i 2 u (jp) i p appearing when computing the derivatives y (s) only depends on elements of one of these sets. This can be proved directly, but the following lemma gives a useful expression. This formula is an analogue, proved by using different techniques, of a similar formula proved for state space systems, given in the paper [4]. Lemma 3.2 If u V T is of class C n and T is admissible for c, then we have d n dt n F c[u](t) =F cn(u(t),...,u n (t))[u](t). (9) Before proving this formula, we look at an example to illustrate its meaning. Example 3.3 For n =2,wehave c 2 (X,X 2 ) = m ψ c (η0)+ 2 ψ c (Γ 0 i (2)) X i0 i= + m ψ c (Γ 00 m ij (2))X i0 X j0 + 2 ψ c(γ 00 ii (2))Xi0 2 + ψ c (Γ i (2))X i i<j i= i= = (η 0 η 0 ) c + ) ((η 0 η i ) c +(η η 0 ) c X i0 + ( ) (η i η j ) c +(η j η i ) c X i0 X j0 + (η i η i ) cxi0 2 i<j + η i cx i. Thus, for n = 2, formula (9) becomes: y (t) = F c2 (u(t),u (t))[u](t) =F (η0 η 0 ) c[u](t)+ ) (F (η0 η i ) c[u](t)+f (ηi η 0 ) c[u](t) u i (t) + ( ) F (ηi η j ) c[u](t)+f (ηj η i ) c[u](t) u i (t)u j (t)+ F (ηi η i ) c[u](t) u 2 i i<j + F η i c [u](t) u i(t). (20) Proof of lemma 3.2: For each η ι P, define θ c (η ι )=F η d, ηκ η κ, define ι c and for any polynomial d = θ c (d) = d, η κ θ c (η ι )= d, η κ F η κ c. 7

9 Then (9) is equivalent to y (n) (t) = dn dt n F c[u](t) = n s q=0 S! s p! θ c(γ j j q i q (n))(t) u (j ) q (t) u jq (i q) (t), (2) in the other words, y (n) (t) is a polynomial in u(t),...,u (n) (t) whose coefficients are the θ c (η ι )(t) s, and the coefficient of u (j ) (t) u (jq) i q (t) iny (n) (t) is ( s! s p! θ c Γ j ) j q i q (n) (t). (22) Note that the right side of (22) can also be written as ( s! s p! θ c (η0 k if u (j ) u (jq) ( i q = u (β ) ) s ( α u (βp) α p s η α X β s 2 η α2 X β 2 ) sp, where s p η αp X βp ) X= ) (t) w s w 2 s 2 w 3 s p w p = w w 2 w 2 w }{{ 2 w } 3 w 3 w }{{} p w p. }{{} s s 2 s p 8

10 We now use induction to prove the lemma. From (3) we see that the conclusion is true for n =. Suppose the conclusion is true for n. Consider the coefficient of u (j ) u (jq) i q in y (n).by induction from formula (3) it can be seen that j s +q n. First we assume that j s +q<n. Let k = n j s q. Suppose u (j ) u (jq) i q =(u (β ) α ) s (u (βp) α p ) sp, where (α,β ) < < (α p,β p ). Further, we assume that β r = 0 for r l. Let ŷ (t) = p r= θ(w r )(t) u (β s ) α τ v α (βr) sr r u (βp) sp α p, r where v α (βr) sr r = u sr α r if β r =0 ( u (βr) α r ) sr u β r α r if β r, and τ r = s! s p!if u (β s ) α v α (βr) sr r u (βp) sp ( α p = u (β ) α ) s (u (β p ) α p ) s p, and, w r = η k 0 η k 0 s η α (s r ) η αr s p η αp X βp if β r =0 s η α s r η αr X βr η αr X βr s p η αp X βp if β r 0. Note that the coefficient of u (j ) u (jq) i q in { d θ(w r )(t) u (β s ) α dt τ v α (βr) sr } r u (βp) sp α p r is if r l, and, if r>l. Let y (t) =ŷ (t) + s (s r )! s p! θ(w rη r )(t), s! (s r )! s p! θ(w r)(t), ( s! s p! θ c Γ j ) j q i q (n ) u (j ) (t) u (jq) (t). i q By induction assumption, the coefficient of u (j ) u (jq) i q in y (n) (t) is the same as in y (t). Thus, 9

11 this coefficient is θ c (w)(t), where w = { l s! (s r )! s p! (ηk 0 r= p + r=l+ s! (s r )! s p! ηk 0 + s!s 2! s p! (ηk 0 s η α2 s η α s η α s p η αp X βp )η 0 } s r η αr s p η αp X βp )η αr s r η αr X βr η αr X βr s p η αp X βp X=. (23) Notice that and w r r w 2 = w t w 2 r t w 2 r t=0 { w r w 2 X β w 2 X β } { } X= = (w r w 2 X β w 2 X β )X X= = {( r ) } w t w 2 X β w 2 X β r t w 2 X β X r X=. t=0 Applying lemma (3.) to (23), we get w = = { η k s 0 η α s! s p! s! s p! Γj j q i q (n). s p η αp X βp } X= In the case q + j s = n, the proof is virtually the same except that k = 0, which leads to the fact that the coefficient of u (j ) u (jq) i q in y (n ) is 0, so the last term in (23) dissappears. 4 Main Result In last section we defined Γ j j q i q (n) and c n (X 0,...,X n ). One can see that c n (X 0,...,X n ) is a polynomial on the X i s with coefficients belonging to F (c). Thus, c n (µ 0,...,µ n )isa linear combination of elements of F (c) for each fixed value of (µ 0,...,µ n ). Therefore, F 2 (c) F (c). But in fact, these two spaces are the same as we can see in the following theorem. Theorem If c is a power series, then F (c) =F 2 (c). 0

12 Proof. We ve shown that F 2 (c) F (c). The other direction is however much less trivial. Now for fixed positive integers k and,i 2,...,i q such that i 2 i q m, let S k (,i 2,...,i q )= σ(0,...,0,i }{{},i 2,...,i q ): σ S n, where n = k + q and S n is the permutation group on a set of n elements. Let Ω k (,i 2,...,i q )= k { } (η l η l2 η ln ) c : (l,...,l n ) S k (,i 2,...,i q ). Then F (c) = span IR {d Ω k (,i 2,...,i q ): k 0,q 0}. Thus the theorem can be proved by showing that Ω k (,i 2,...,i q ) F 2 (c) (24) for any k, q, and (,i 2,...,i q ). Now fix k and (,i 2,...,i q ) and put the lexicographic order on Ω k (,i 2,...,i q ) according to the order of (l,l 2,...,l n ). Write the elements of Ω k (,i 2,...,i q ) ordered as Y,Y 2,...,Y r. Let ˆΩ k (,i 2,...,i q )= ( {d = ψ c Γ j ) } j q i q (k) : j s 0. Then we have ˆΩ k (,i 2,...,i q ) F 2 (c). Put the lexicographic order on ˆΩ k (,i 2,...,i q ) according to the order of ( j s,j,...,j q ). Notice that for each element d i ˆΩ k (,i 2,...,i q ), there exist some positive integers a ij such that r d i = a ij Y j. j= Let A be the matrix of r columns and infinitely many rows whose (i, j)-th entry is a ij, i.e, A =(a ij ). We claim that A is of full column rank in the sense that there is no nonzero vector v IR n such that Av = 0. Suppose there is some v 0 such that Av = 0. Construct a polynomial e in the following way: e, η l η lt =0 if (l,...,l t ) / S k (,i 2,...,i q ) and e, η l η lt = v i

13 if (l,...,l t ) S k (,i 2,...,i q ) and (η l...η lt ) c corresponds to the i-th element of Ω k (,i 2,...,i q ). By the definitions of A and d, we know that e n (µ 0,...,µ n ) = 0 for any n. Therefore, d n dt n F e[u](0) = F dn(µ 0,...,µ n )[u](0) = 0 for any n and any analytic control u, which implies that F e [u] = 0 for any analytic controls. Since analytic controls are dense in V T (under the L topology), it follows that F e 0. By lemma (2.), e = 0. Thus, v = 0, a contradiction to the assumption. Hence, A is of full column rank. Now let A s be the subspace of IR r spanned by the first s row vectors of A. Then A A 2 A s. Since A r IR r for any r, there exists some s 0 > 0 such that A s = A s0 for every s s 0. Let A be the s 0 r submatrix of A consisting the first s 0 rows of A. Then A = TA for some matrix T. Therefore rank A = r. By the construction of A, we know that Y d Y A 2. = d 2. Y r d r From the facts that d i F 2 (c) and A is of full column rank, we get the conclusion that Y i F 2 (c) for each i, therefore, (24) holds. Since k, q and (,i 2,...,i q ) were arbitrary, we get the desired conclusion F (c) =F 2 (c). 5 Families of Series and Systems In this section we consider families of power series. Let Λ be a index set. We say that c is a family of power series (parameterized by λ Λ) if c := {c λ : λ Λ}, where c λ is a power series for each fixed λ. A family c can also be viewed as a power series with coefficient belonging to 2

14 the ring of functions from Λ to IR, i.e, c = c,η ι η ι, where c,η ι :Λ IR, c,η ι (λ) c λ,η ι. Let S be the set of all families of power series. For c, d S and γ IR,γc + d is defined to be the family of power series {γc λ + d λ : λ Λ}. ThusS forms a vector space over IR. We say that c is a convergent family if each member of the family is convergent. For any monomial α P, α c is defined to be the family {α c λ : λ Λ}. For any n 0, c n (X 0,...,X n ) is defined to be the family { } c λ n(x 0,...,X n ): λ Λ, where X i =(X i,...,x im ) are m indeterminates over IR, i 0. Appling lemma (3.2), we have that d n dt n F c λ [u](t) =F c λ n (u(t),...,u n (t))[u](t), (25) for each λ. As in the case of single power series, we associate to c two types of observation spaces in the following way: F (c) := span IR {α c : α P }. F 2 (c) := span IR {c n (µ 0,...,µ n ): µ i IR m, 0 i n, n 0}. Note that F (c) (respectively, F 2 (c)) is formally analogous to F (c) (respectively, F 2 (c)) studied before. Using c and d instead of c and d in the proof of theorem, we get the following result: Theorem 2 If c is a family of power series, then F (c) =F 2 (c). Now consider a state space system ẋ = g 0 (x)+ u i g i (x) (26) y = h(x) where x(t) X, ac ω manifold, g 0,g,...,g m are C ω vector fields, and h a C ω function from X to IR. One type of observation space associated with (26) is F := span IR {L gi L gik h : k 0}. For µ 0,...,µ k given, we let, for each x X, y µ 0 µ k (x) := dk dt k t=0 y x(t), 3

15 where y x (t) is the output corresponding to initial state x and any C input u such that u (j) (0) = µ j for 0 j k. We associate to (26) a second type of observation space, as follows: F 2 := span IR {y µ µ k : µ i IR m,k 0}. By a fundamental formula due to Fliess (see [3]), the input/output map of (26) can be written as y(t) =F c [u](t), where the family c is defined by c,η i η ik = L gik L gi h, or, equivalently, for the output corresponding to the initial state x, y x (t) =F c x[u](t), where Thus, and, therefore, c x,η i η ik = L gik L gi h(x). L gik L gi h = c,η i η ik = (η i η ik ) c,φ, F = { d,φ : d F (c)}. By (25), we know that y µ 0 µ k x = F c x k (µ 0,...,µ k )[u](0) = c x k(µ 0,...,µ k ),φ. Hence, F 2 = { d,φ : d F 2 (c)}. So the following conclusion follows immediately from theorem (2): Corollary 5. For the state space system (26), F = F 2. 6 References. M. Fliess and I. Kupka, A finiteness criterion for nonlinear input-output differential systems, SIAM J.Control and Opt. 2 (983):

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