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Periodic smoothing splines
Automatica 44 (2008) 185 – 192 www.elsevier.com/locate/automatica
Brief paper
Periodic smoothing splines夡 Hiroyuki Kano a,∗ , Magnus Egerstedt b , Hiroyuki Fujioka a , Satoru Takahashi c , Clyde Martin d a Department of Information Sciences, Tokyo Denki University, Hatoyama, Hiki-gun, Saitama 350-0394, Japan b School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA c Department of Intelligent Mechanical Systems Engineering, Kagawa University, 2217-20, Hayashi-Cho, Takamatsu-City, 761-0396, Japan d Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA
Received 21 August 2006; received in revised form 26 April 2007; accepted 1 May 2007 Available online 13 August 2007
Abstract Periodic smoothing splines appear for example as generators of closed, planar curves, and in this paper they are constructed using a controlled two point boundary value problem in order to generate the desired spline function. The procedure is based on minimum norm problems in Hilbert spaces and a suitable Hilbert space is defined together with a corresponding linear affine variety that captures the constraints. The optimization is then reduced to the computationally stable problem of finding the point in the constraint variety closest to the data points. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Spline smoothing; Periodic; Two-point boundary value problem; Constrained optimization
1. Introduction In this paper we consider the problem of constructing periodic smoothing splines. For interpolating cubic splines there are standard numerical procedures that are quite effective. However the problem of periodic smoothing splines is more general and requires additional machinery. The need for such splines arises whenever there is a need to construct closed curves in the plane (Kano, Fujioka, Egerstedt, & Martin, 2005; Takahashi, Ghosh, & Martin, 2002). We show how this problem can be addressed as an optimal control problem, whose solution is the so-called generalized splines, i.e. belonging to a rich set of splines that include polynomial, trigonometric, and exponential splines (Schumaker, 1981; Wahba, 1990). It should be noted that the problem of constructing periodic splines has been previously addressed, 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Maria Elena Valcher under the direction of Editor Roberto Tempo. ∗ Corresponding author. Tel.: +81 49 296 2911; fax: +81 49 296 6403. E-mail addresses: [email protected] (H. Kano), [email protected] (M. Egerstedt), [email protected] (H. Fujioka), [email protected] (S. Takahashi), [email protected] (C. Martin).
0005-1098/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2007.05.011
for example in Eubank (1999), Unser and Blu (2005), Wahba (1990), and some routines can be found in a spline library.1 However, apart from the fact that previous results mainly focused on polynomial splines, the solution methods tend to rely heavily on the particular spline-type under investigation in that a unified framework was missing in which a large class of splines could be systematically produced. The view taken in this paper is that the framework of linear systems theory provides the tool needed to address the problem of generalized, smoothing, periodic splines in a unified, systematic, and numerically stable manner. In this paper we give a very general construction of periodic splines based on Hilbert space methods developed by Martin and collaborators in a series of papers, (Martin, Sun, & Egerstedt, 2001; Sun, Egerstedt, & Martin, 2000; Zhou, Egerstedt, & Martin, 2005, 2001; Zhou & Martin, 2004). We use a specific technique developed in Zhou et al. (2005). That is, we use the dynamics of a controlled two point boundary value problem to generate the spline curve, given by x˙ = Ax + bu,
y = cx, x(0) − x(T ) = 0
with x ∈ Rn , and u, y ∈ R, and with the output y defining the spline curve. We assume that the system is controllable and observable. Because we are interested in so-called maximal 1 www.physics.lsa.umich.edu/akerlof/Spline/Spline/spline.doc.
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H. Kano et al. / Automatica 44 (2008) 185 – 192
smoothing (maximal degree of continuity of the derivatives at the interpolation points) we will assume that
where Q and R are positive definite matrices. The problem we are interested in is
cb = cAb = · · · = cAn−2 b = 0,
min J (u, x0 ),
(1)
or, without loss of generality, we assume that A and b are in control canonical form and c = (1, 0, . . . , 0). Moreover, the particular structure assumed on the matrices A, b, c is not necessary, strictly speaking. We assume this structure in order to achieve maximal smoothing, but it should be pointed out already at this point that the framework developed in this paper holds for any minimal (i.e. completely controllable and observable) linear control system with scalar inputs and outputs. As a final note regarding the system under consideration, it should be stressed that this system is to be thought of as a generator of spline curves. We are not investigating the issue of trajectory planning per se. Rather, the tool developed in this paper draws its motivation from the fields of data analysis and statistics, even though the results will provide some new insights into the area of endpoint-constrained, linear optimal control as an added benefit. 2. The periodic splines problems We are given a controllable linear system x˙ =Ax +bu, y =cx and a data set D = {(ti , i ) : i = 1, . . . , N}, where we let tN be equal to the final time T. Our goal is to find a square-integrable control signal u ∈ L2 [0, T ] such that |y(ti ) − i | is small for all i = 1, . . . , N, and we require that y(0) = y(T ), y(0) ˙ = y(T ˙ ), . . ., up to a suitable order, i.e. we want the spline curve to be smoothly periodic. An easy way to enforce this is to require that x(0) = x(T ). There are essentially two ways in which this problem can be formulated, and in order to see this, we first need to establish some notation. Let A(ti −s) b, ti − s > 0, ce fi (s) = 0, ti − s 0, and let yˆ = (y(t1 ), . . . , y(tN )) ,
ˆ = (1 , . . . , N ) ,
given a positive definite matrix R. In this paper, we use ( ) to denote transpose, and with this notation, as well as yi = y(ti ), we have i = 1, . . . , N,
(2)
where x(0) = x0 , and where the inner products are given by T i , x0 R = i Rx 0 and fi , uL = 0 fi (t)u(t) dt. Problem 1. Let T J (u, x0 ) = u2 (t) dt + x0 Rx 0 + (yˆ − ˆ ) Q(yˆ − ˆ ), 0
subject to the constraint x(0) = x(T ) as well as the dynamics in Eq. (2). Problem 2. Let T u2 (t) dt + (yˆ − ˆ ) Q(yˆ − ˆ ), J (u, x0 ) = 0
where Q is a positive definite matrix. The problem is min J (u, x0 ), u,x0
subject to the constraint x(0) = x(T ) and the dynamics in Eq. (2). The difference between Problems 1 and 2 is that there is an extra cost associated with the free initial x0 in Problem 1, through the term x0 Rx 0 . As we will see, this difference will make Problem 1 easier to solve than Problem 2. The basic idea of the construction of solutions is to define a linear variety V in a Hilbert space that contains all of the constraints. As we will see in the next section, it is possible to interpret the data as a point (p) in the Hilbert space, which reduces the problem to that of finding the point on the linear variety that is closest (in the sense of the norm in the Hilbert space) to the data point. We know that we can construct this point by finding the orthogonal complement of the linear variety that defines the affine variety and constructing the intersection of the affine variety with the orthogonal complement. In other words, “all” we need to do in order to solve the problem is to compute V ∩ (V⊥ + p). In this process we follow Luenberger (1969). Before constructing this intersection two things must be verified. The first is that the linear variety V is nonempty and the second is that it is closed. Both follow from the fact that V is the graph of a continuous mapping from an appropriate product space to RN . 3. Problem 1
i = R −1 eA ti c ,
yi = i , x0 R + fi , uL ,
u,x0
We begin by considering the periodic spline as a particular boundary value problem, and for this we will use the methods of Zhou et al. (2005). Let the boundary condition be given by x(0) − x(T ) = 0.
(3)
We note that since x(T ) = e
AT
T
x(0) +
eA(T −s) bu(s) ds,
0
the specific dependence on x(T ) can be removed and the boundary constraint simply becomes T AT (I − e )x(0) − eA(T −s) bu(s) ds = 0. (4) 0
H. Kano et al. / Automatica 44 (2008) 185 – 192
It remains to construct the intersection V ∩ (V⊥ + p) to find the optimal point, where p = (0, 0, ˆ ) ∈ H. This construction is technically more complicated than for the smoothing spline without periodicity constraints, as reported in Zhou et al. (2005). The unique point in the intersection is defined as the solution of the following system of four equations in the unknowns u, x0 , y and , obtained by identifying x and x˜ with x0 , d with y, ˆ and d˜ with yˆ − ˆ .
We now define the following Hilbert space H = L2 [0, T ] × Rn × RN with norm
T
(u; x0 ; y) 2 = 0
u2 (t) dt + x0 Rx 0 + y Qy.
We define the linear constraint variety, V ⊂ H, to be V = (u; x; d) : di = i , xR + fi , uL ,
T
(I − eAT )x −
187
u=−
eA(T −s) bu(s) ds = 0 .
N
yˆ − ˆ , ei Q fi − b eA (T −s) ,
(6)
i=1
0
N
We first note that V is a closed subspace of H since it is the graph of a continuous function restricted to a closed linear variety, and the closed graph theorem applies. We now construct the orthogonal complement V⊥ . And, ˜ : from the definition it directly follows that V⊥ = {(u; ˜ x; ˜ d) ˜ ˜ xR + d, dQ = 0}, which allows ∀(u; x; d) ∈ Vu, ˜ uL + x, us to state the following lemma:
yi = i , x0 R + fi , uL .
Lemma 1.
We begin by eliminating x0 and u from Eq. (9) by substituting Eqs. (6) and (7). After some manipulation we have
V⊥ =
˜ : x˜ = − (u; ˜ x; ˜ d)
N
˜ ei Q i + R −1 (I − eA T ), d,
i=1
u˜ = −
N
A (T −t)
˜ ei Q fi − b e d,
for all ∈ Rn , where ei is the unit vector with zeros everywhere except for the ith position. Proof. From the expression of the orthogonal complement above, we directly get that
N ⊥ ˜ : x˜ + ˜ ei Q i , x ˜ x; ˜ d) d, V = (u; + u˜ +
i=1 N
˜ ei Q fi , u = 0 , d,
˜ ei Q i = R −1 (I − eAT ) d,
and
where G = R with = (1 2 · · · N ), and F is the Gramian T f (s)f (s) ds (10) F= 0
with f (s) = (f1 (s)f2 (s) · · · fN (s)) . Note that since the fi ’s are linearly independent, F is invertible. Moreover, since i = R −1 eA ti c we can let
and the lemma follows.
(5)
+ ,
(11)
where
T
=−
f (s)b eA (T −s) ds.
We will now use Eq. (8) to obtain a second equation in and y. ˆ Substituting u in Eq. (6) and x0 in (7) into (8), we have N AT −1 AT yˆ − ˆ , ei Q i + R (I − e ) 0 = (I − e ) − i=1
yˆ = − GQ(yˆ − ˆ ) − F Q(yˆ − ˆ ) + E R −1 (I − eAT )
˜ ei Q fi = (−b eA (T −t) ), d,
i=1
(9)
0
i=1
N
(8)
yi = − ei GQ(yˆ − ˆ ) − ei F Q(yˆ − ˆ ) + i (I − eAT ) T − fi (s)b eA (T −s) ds,
From this we conclude that
u˜ +
eA(T −s) bu(s) ds,
0
for the appropriate E, in order to obtain
following the construction in Zhou et al. (2005). Now, the relationship does not hold for all x and u but only for those x and u for which Eq. (4) holds. Multiplying by any ∈ Rn (and its transpose ), we can rewrite Eq. (4) as
N
T
0 = (I − eAT )x0 −
i=1
x˜ +
(7)
i=1
= R −1 (eA t1 c , . . . , eA tN c ) = R −1 E,
R −1 (I − eAT ) , xR + (−eA(T −t) b) , uL = 0.
yˆ − ˆ , ei Q i + R −1 (I − eAT ) ,
0
i=1
x0 = −
−
T
e 0
A(T −s)
b −
N i=1
A (T −s)
y−ˆ ˆ , ei Q fi − b e
ds.
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H. Kano et al. / Automatica 44 (2008) 185 – 192
We make the following observation: N
yˆ − ˆ , ei Q i =
i=1
N
1
i ei Q(yˆ − ˆ )
0.8
i=1
= R −1 EQ(yˆ − ˆ ).
N i=1
T
y(t)
T A(T −s)
bf i (s)ei ds = , it holds that Noting − N i=1 0 e
0.6
0.4
−e
A(T −s)
0
byˆ − ˆ , ei Q fi (s) ds = Q(yˆ − ˆ ). 0.2
Using these two constructions we then have 0
0 = (I − eAT )(−R −1 EQ(yˆ − ˆ )) + (I − eAT )R −1 (I − eAT ) − Q(yˆ − ˆ ) + ,
(12)
where is the controllability Gramian
T
=
eA(T −s) bb eA (T −s) ds.
0.2
0.3
0.4
0.5 t
0.6
0.7
0.8
0.9
1
Fig. 1. Periodic spline in Problem 1: Depicted are y(t) (solid line) and ˆ (stars).
solution to Problem 1.
By combining these two expressions (11) and (12) linking yˆ and , we obtain Eq. (14). It can be shown that the coefficient matrix in (14) is invertible under the controllability assumption of the system x˙ = Ax + bu. Using Eq. (14) we can solve for yˆ and for . These values can be used in Eqs. (6) and (7) to uniquely determine the optimal control and the optimal initial condition. We see that the optimal estimate of the data is obtained independently of the control. It is necessary to ask the question “In what sense is the spline periodic?” We state and prove the following theorem in answer to that question. Theorem 2. The function y(t) of the above construction can be extended periodically to the entire positive real line and the extension is 2n−1 times continuously differentiable everywhere with exception of the points nT : n ∈ Z where are guaranteed only n − 1 continuous derivatives. We leave the proof to the reader. An example of this procedure is seen in Fig. 1, where 0 1 0 A= , b= , c = (1 0), T = 1 0 0 1 t2 = 0.3,
t3 = 0.5,
ˆ = (0.8 0.2 0.5 1 0.3) Q = 104 I5 ,
0.1
(13)
0
t1 = 0.2,
0
t4 = 0.7,
t5 = 0.8
I +(G+F )Q
−E R −1 (I −eAT ) −
((I −eAT )R −1 E+ )Q −(I −eAT )R −1 (I −eAT ) − yˆ × (G + F )Qˆ = . (14) ((I − eAT )R −1 E + )Qˆ 4. Problem 2 We begin by considering the periodic spline as a particular boundary value problem. We will use the methods related to but not the same as in the above construction. Again let the boundary condition be given by x(0) − x(T ) = 0,
(15)
and the dynamics be the same as in Problem 1. Then as before, the boundary constraint is given by (4). Note that the initial data becomes a parameter in the constraint. We now define a Hilbert space to be H = L2 [0, T ] × RN
R = 104 I2 ,
with norm (Ip = p × p identity matrix).
We assumed in Section 1 that the system x˙ = Ax + bu, y = cx is controllable and observable. In addition to that it is natural to employ minimal system for generating splines, the controllability guarantees the invertibility of the coefficient matrix in (14) implying the existence and uniqueness of an optimal
T
(u; y) 2 =
u2 (t) dt + y Qy.
0
Since we no longer have a cost associated with the initial condition, x0 cannot be part of the Hilbert space over which the minimum norm problem is solved. In fact, the constraint
H. Kano et al. / Automatica 44 (2008) 185 – 192
variety is now affine, parameterized by x0 , and it is given by Vx0 = (u; d) : i , x0 R = di − fi , uL ,
T
(I − eAT )x0 −
eA(T −s) bu(s) ds = 0 .
0
If we now let V be the linear constraint variety associated with x0 = 0 above, we note that, as before, V is a closed subspace of H. Moreover, from Luenberger (1969), we know that the unique minimizer is given by the intersection Vx0 ∩ (V⊥ + p), where V⊥ = {(v; w) : ∀(u; d) ∈ Vv, uL + w, dQ = 0}.
We now construct the intersection (V⊥ + p) ∩ Vx0 , which is determined by the solution of the following three equations which come from V⊥ + p and Vx0 . N
yˆ − ˆ , ei Q fi − b eA (T −s) ,
(16)
i=1
i , x0 R = yi − fi , uL , T eA(T −s) bu(s) ds. (I − eAT )x0 =
where QF Q S= − Q
−Q
(20)
.
Now using Eq. (19) we have yˆ − ˆ x0 =C , ˆ −
−1
−
E
−I
I − eAT
0
,
and thus we have an expression for T x0 2 u (s) ds = (x0 , ˆ )C SC . 0 ˆ
i=1
u=−
Then we can calculate the square integral of u as T yˆ − ˆ 2 u (s) ds = (yˆ − ˆ , )S , 0
where I + FQ C= Q
Using the same technique as for Problem 1 we have N ⊥ A (T −s) V = (v; w) : v = − w, ei Q fi − b e .
189
(17) (18)
0
Since yˆ − ˆ is written as x0 yˆ − ˆ = (I, 0)C , ˆ we can now write the total cost J (u, x0 ) as a function of the initial position Jˆ(x0 ). I x0 ˆ J (x0 ) = (x0 , ˆ ) C SC + C Q(I, 0)C . 0 ˆ Thus we obtain
x0
We use the first equation to eliminate u from the second and third equation. After some manipulation we have the following system of equations: I + F Q − FQ yˆ E x0 = , (19) AT Q I −e − Q ˆ
Jˆ(x0 ) = (x0 , ˆ )V
where, as before, F is the Gramian of the fi ’s in (10) and is the controllability Gramian in (13). Note that this coefficient matrix is invertible, since I + F Q − Q 0 , = M1 Q 0 −I −
and V can be computed as
where
Q V = C S + 0 V =
,
ˆ
E
−I
I − eAT
0
0
0
C,
Q−1 + F
−1
E
−I
I − eAT
0
.
and the matrix M1 defined by −1 Q +F M1 =
The cost function Jˆ(x0 ) is quadratic in x0 , and an optimum x0 minimizing the cost is obtained as a solution of the following linear equation: x0 (In , 0)V = 0. ˆ
is positive-definite. We now reconsider u from Eq. (16), and obtain yˆ − ˆ u = (−f (s)Q, −b eA (T −s) ) .
This equation in x0 has unique solution if and only if the coefficient matrix of x0 , denoted by V1 , In V1 = (In , 0)V 0
190
H. Kano et al. / Automatica 44 (2008) 185 – 192
is invertible. Moreover, since V1 is written as V1 =
E
Q−1 + F
I − eAT
−1
E
1
I − eAT
0.8
rank(eA t1 c · · · eA tN c I − eA T ) = n.
(21)
The condition for full rank in Eq. (21) is interesting. The first N columns arise from an observability problem that was studied in Martin and Smith (1987). The problem is to recover the initial data from x˙ = Ax, y = cx, x(0) = x0 when the observations are made at discrete time points, t1 , . . . , tN . There are no known necessary and sufficient conditions in terms of A, c and t1 , . . . , tN for the first N columns to have full rank. However, we will see that the last set of n columns given by I − eA T simplifies the condition (21). We introduce the following set of complex numbers √ I = {j2k/T | j = −1, k ∈ Z}.
y(t)
we see that the solution is unique whenever
0.4
0.2
0 0
The solution to Problem 2 is computed for the same example as in the previous section and the results are shown in Fig. 2. Periodic splines can be used to construct closed curves in the plane, and in Figs. 3 and 4 we show a result applied to contour modeling of jellyfish for an image frame from a real digital movie file.2 We used the same second order system (A, b, c) as in the previous examples, and the other parameters are T = 360, N =36 and Q=10−2 I36 . Using standard image processing techniques, the data set (ti , i ), i = 1, 2, . . . , N is obtained from the image in Fig. 4. Namely, ti and i are obtained as the polar coordinates of sampled boundary pixels with the origin at the centroid of jellyfish, where ti = 10 × (i − 1) denotes angles measured every 10◦ and i the radial distance in pixels. The solution to Problem 2 is obtained as in Fig. 3, from which the contour is reconstructed as superposed in Fig. 4. 2 Educational Image Collections, Information-technology Promotion Agency (IPA), Japan. http://www2.edu.ipa.go.jp/gz/.
0.2
0.3
0.4
0.5 t
0.6
0.7
0.8
0.9
1
95 90 85
Lemma 3. The condition (21) holds if and only if rank (c I − A ) = n for all ∈ I.
Theorem 4. There always exists a unique optimal solution to Problem 2.
0.1
Fig. 2. Periodic spline in Problem 2: Depicted are y(t) (solid line) and ˆ (stars).
Note that the matrix I − eAT is singular whenever A has an eigenvalue in I. Then we can show that the following lemma holds (the proof omitted).
80 y(t)
We have shown that, due to the controllability assumption, (19) has a unique solution in yˆ and for any initial condition x0 . On the other hand, the assumption of observability, which is equivalent to rank (c I − A ) = n for all ∈ C (e.g. Hautus, 1969), guarantees the existence of unique optimal x0 by Lemma 3. Thus, under the assumption of controllability and observability, we obtain the following theorem:
0.6
75 70 65 60 55 50
0
50
100
150
200 t
250
300
350
400
Fig. 3. Periodic spline in Problem 2 for the data from jellyfish image: Depicted are y(t) (solid line) and ˆ (stars).
Fig. 4. A jellyfish image frame and the contour reconstructed from the periodic spline y(t), t ∈ [0, 360] in Problem 2 by (y(t) cos 2t/360, y(t) sin 2t/360).
H. Kano et al. / Automatica 44 (2008) 185 – 192
Finally, one can note that it may be possible to obtain a solution to this problem as a limiting solution to Problem 1 as R tends to zero. However, this line of inquiry is not pursued further and we simply state this as a possibility for the future. 5. Conclusions In this paper we present a method for generating periodic, smoothing splines using linear optimal control. In particular, we show that such curves can be obtained in a quite general fashion by viewing the smoothing cost as an inner product in a suitable space, at the same time as periodicity is enforced by limiting the solutions to lie in a particular linear variety. The resulting methodology is numerically efficient and robust, and it unifies a number of contributions in the general area of smoothing splines through the use of different linear systems as generators of curves of different characteristics, including polynomial, exponential, and trigonometric smoothing splines.
References Eubank, R. L. (1999). Nonparametric regression and spline smoothing. Second edition. Statistics: Textbooks and Monographs. New York: Marcel Dekker. Hautus, M. L. J. (1969). Controllability and observability conditions of linear autonomous systems. Nederl. Akad. Wetensch. Proc. Ser. A 72 = Indag. Math., 31, 443–448. Kano, H., Fujioka, H., Egerstedt, M., & Martin, C. F. (2005). Optimal smoothing spline curves and contour synthesis. In Proceedings of the 16th IFAC world congress, Prague, Czech Republic, July 2005. Luenberger, D. G. (1969). Optimization by vector space methods. New York, London, Sydney: Wiley. Martin, C., & Smith, J. (1987). Approximation, interpolation and sampling. Differential Geometry: The Interface Between Pure and Applied Mathematics, 68, 227–252. Martin, C. F., Sun, S., & Egerstedt, M. (2001). Optimal control, statistics and path planning. 33(1–3), 237–253. Schumaker, L. L. (1981). Spline functions: Basic theory. New York: Wiley. Sun, S., Egerstedt, M., & Martin, C. F. (2000). Control theoretic smoothing splines. IEEE Transactions on Automatic Control, 45(12), 2271–2279. Takahashi, S., Ghosh, B. K., & Martin, C. F. (2002). Boundary location using control theoretic splines. Transactions of the Society of Instrument and Control Engineers, 38(3), 293–298. Unser, M., & Blu, T. (2005). Generalized smoothing splines and the optimal discretization of the wiener filter. IEEE Transactions on Signal Processing, 53, 2146–2159. Wahba, G. (1990). Spline models for observational data. In CBMS-NSF regional conference series in applied mathematics, (Vol. 59). Philadelphia, PA: SIAM. Zhou, Y., Egerstedt, M., & Martin, C. (2005). Control theoretic splines with deterministic and random data. In Proceedings of the 44th IEEE conference on decision and control and European control conf. ECC (pp. 362–367). Seville, Spain, December 2005. Zhou, Y., Egerstedt, M., & Martin, C. F. (2001). Optimal approximation of functions. Communications in Information and Systems, 1(1), 101–112. Zhou, Y., & Martin, C. F. (2004). A regularized solution to the Birkhoff interpolation problem. Communications in Information and Systems, 4(1), 89–102.
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Hiroyuki Kano was born in Hyogo, Japan. He received the B.E. and M.E. degrees in Mechanical Engineering from Kyoto Institute of Technology, Japan, in 1970 and 1972, respectively, and the Ph.D. degree in Electrical Engineering—Systems Science from Polytechnic Institute of New York (now Polytechnic University), USA, in 1976. From 1976 to 1992, he was with the International Institute for Advanced Study of Social Information Sciences, Fujitsu Limited. Since 1992, he has been a Professor in the Department of Information Sciences, Tokyo Denki University. His research interests include control and estimation theory, motion planning and control of robots, and signal processings. He is a coauthor of the book ‘Matrix Riccati Equations for Control Engineers’, Asakura Pub. Co., 1996. Magnus B. Egerstedt was born in Stockholm, Sweden, and is an Associate Professor in the School of Electrical and Computer Engineering at the Georgia Institute of Technology, where he has been on the faculty since 2001. He also holds an adjunct appointment in the Division of Interactive and Intelligent Computing with the College of Computing at Georgia Tech. Magnus Egerstedt received the M.S. degree in Engineering Physics and the Ph.D. degree in Applied Mathematics from the Royal Institute of Technology, Stockholm, in 1996 and 2000 respectively. He also received a B.A. degree in Philosophy from Stockholm University in 1996. He spent 2000–2001 as a Postdoctoral Fellow at the Division of Engineering and Applied Science at Harvard University. Dr. Egerstedt’s research interests include optimal control as well as modeling and analysis of hybrid and discrete event systems, with emphasis on motion planning, control, and coordination of mobile robots, and he has authored over 100 papers in the areas of robotics and controls. Magnus Egerstedt is a Senior Member of the IEEE, he received the ECE/GT Outstanding Junior Faculty Member Award in 2005, and the CAREER award from the U.S. National Science Foundation in 2003. Hiroyuki Fujioka was born in Mie, Japan, in 1975. He received the B.E., M.E. and Dr.E. degrees from Ritsumeikan University, Japan, in 1997, 1999 and 2004, respectively. In 2002, he joined as an Instructor the Department of Information Sciences, Tokyo Denki University, where he is currently an Assistant Professor. He is a recipient of the 2004 Sunahara Prize and Young Investigators Award from the Institute of Systems, Control and Information Engineers (ISCIE). His research interests include theory and applications of splines, and task planning and control of robots. Satoru Takahashi was born in Tokyo in 1965. He received the B.S. and the M.S. degrees, both in Mathematical Sciences, from Tokyo Denki University in 1989 and 1991, respectively. He finished the Doctoral Programs in Mathematical and Information Sciences of Tokyo Denki University in 1997 and received the Ph.D. in 2001. From 1991 to 1993 he was a Researcher of the Nano Technology Research Center, Canon INC., Japan. From 1997 to 1999 he served as an Instructor in the Department of Information Sciences, Tokyo Denki University. From 1999 to 2000, he was a Visiting Researcher in the Systems, Science and Mathematics Department at Washington University. In 2001, he served as a Research Assistant in the Mathematics and Statistics Department at Texas Tech University. Since 2003, he has been an Associate Professor in the Intelligent Mechanical Systems Engineering Department at Kagawa University with research interests in the area of Machine Vision, Robotics and Mechatronics.
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Clyde F. Martin’s research interests include control theory, the applications of algebraic and differential geometry to problems in numerical analysis, and the development and analysis of mathematical models in agriculture, the environment and medicine. He has organized or co-organized more than 10 international conferences and has served on the program committee or board of directors of several others. In November of 2001 he received an honorary doctorate for his work in engineering from the Royal Institute of Technology in Stockholm Sweden. He has received distinguished alumni awards from both Emporia State University and the University of Wyoming. He has directed more than 60 students to advanced degrees and has published more than 300 papers in a variety of disciplines.
Brief paper
Periodic smoothing splines夡 Hiroyuki Kano a,∗ , Magnus Egerstedt b , Hiroyuki Fujioka a , Satoru Takahashi c , Clyde Martin d a Department of Information Sciences, Tokyo Denki University, Hatoyama, Hiki-gun, Saitama 350-0394, Japan b School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA c Department of Intelligent Mechanical Systems Engineering, Kagawa University, 2217-20, Hayashi-Cho, Takamatsu-City, 761-0396, Japan d Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA
Received 21 August 2006; received in revised form 26 April 2007; accepted 1 May 2007 Available online 13 August 2007
Abstract Periodic smoothing splines appear for example as generators of closed, planar curves, and in this paper they are constructed using a controlled two point boundary value problem in order to generate the desired spline function. The procedure is based on minimum norm problems in Hilbert spaces and a suitable Hilbert space is defined together with a corresponding linear affine variety that captures the constraints. The optimization is then reduced to the computationally stable problem of finding the point in the constraint variety closest to the data points. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Spline smoothing; Periodic; Two-point boundary value problem; Constrained optimization
1. Introduction In this paper we consider the problem of constructing periodic smoothing splines. For interpolating cubic splines there are standard numerical procedures that are quite effective. However the problem of periodic smoothing splines is more general and requires additional machinery. The need for such splines arises whenever there is a need to construct closed curves in the plane (Kano, Fujioka, Egerstedt, & Martin, 2005; Takahashi, Ghosh, & Martin, 2002). We show how this problem can be addressed as an optimal control problem, whose solution is the so-called generalized splines, i.e. belonging to a rich set of splines that include polynomial, trigonometric, and exponential splines (Schumaker, 1981; Wahba, 1990). It should be noted that the problem of constructing periodic splines has been previously addressed, 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Maria Elena Valcher under the direction of Editor Roberto Tempo. ∗ Corresponding author. Tel.: +81 49 296 2911; fax: +81 49 296 6403. E-mail addresses: [email protected] (H. Kano), [email protected] (M. Egerstedt), [email protected] (H. Fujioka), [email protected] (S. Takahashi), [email protected] (C. Martin).
0005-1098/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2007.05.011
for example in Eubank (1999), Unser and Blu (2005), Wahba (1990), and some routines can be found in a spline library.1 However, apart from the fact that previous results mainly focused on polynomial splines, the solution methods tend to rely heavily on the particular spline-type under investigation in that a unified framework was missing in which a large class of splines could be systematically produced. The view taken in this paper is that the framework of linear systems theory provides the tool needed to address the problem of generalized, smoothing, periodic splines in a unified, systematic, and numerically stable manner. In this paper we give a very general construction of periodic splines based on Hilbert space methods developed by Martin and collaborators in a series of papers, (Martin, Sun, & Egerstedt, 2001; Sun, Egerstedt, & Martin, 2000; Zhou, Egerstedt, & Martin, 2005, 2001; Zhou & Martin, 2004). We use a specific technique developed in Zhou et al. (2005). That is, we use the dynamics of a controlled two point boundary value problem to generate the spline curve, given by x˙ = Ax + bu,
y = cx, x(0) − x(T ) = 0
with x ∈ Rn , and u, y ∈ R, and with the output y defining the spline curve. We assume that the system is controllable and observable. Because we are interested in so-called maximal 1 www.physics.lsa.umich.edu/akerlof/Spline/Spline/spline.doc.
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H. Kano et al. / Automatica 44 (2008) 185 – 192
smoothing (maximal degree of continuity of the derivatives at the interpolation points) we will assume that
where Q and R are positive definite matrices. The problem we are interested in is
cb = cAb = · · · = cAn−2 b = 0,
min J (u, x0 ),
(1)
or, without loss of generality, we assume that A and b are in control canonical form and c = (1, 0, . . . , 0). Moreover, the particular structure assumed on the matrices A, b, c is not necessary, strictly speaking. We assume this structure in order to achieve maximal smoothing, but it should be pointed out already at this point that the framework developed in this paper holds for any minimal (i.e. completely controllable and observable) linear control system with scalar inputs and outputs. As a final note regarding the system under consideration, it should be stressed that this system is to be thought of as a generator of spline curves. We are not investigating the issue of trajectory planning per se. Rather, the tool developed in this paper draws its motivation from the fields of data analysis and statistics, even though the results will provide some new insights into the area of endpoint-constrained, linear optimal control as an added benefit. 2. The periodic splines problems We are given a controllable linear system x˙ =Ax +bu, y =cx and a data set D = {(ti , i ) : i = 1, . . . , N}, where we let tN be equal to the final time T. Our goal is to find a square-integrable control signal u ∈ L2 [0, T ] such that |y(ti ) − i | is small for all i = 1, . . . , N, and we require that y(0) = y(T ), y(0) ˙ = y(T ˙ ), . . ., up to a suitable order, i.e. we want the spline curve to be smoothly periodic. An easy way to enforce this is to require that x(0) = x(T ). There are essentially two ways in which this problem can be formulated, and in order to see this, we first need to establish some notation. Let A(ti −s) b, ti − s > 0, ce fi (s) = 0, ti − s 0, and let yˆ = (y(t1 ), . . . , y(tN )) ,
ˆ = (1 , . . . , N ) ,
given a positive definite matrix R. In this paper, we use ( ) to denote transpose, and with this notation, as well as yi = y(ti ), we have i = 1, . . . , N,
(2)
where x(0) = x0 , and where the inner products are given by T i , x0 R = i Rx 0 and fi , uL = 0 fi (t)u(t) dt. Problem 1. Let T J (u, x0 ) = u2 (t) dt + x0 Rx 0 + (yˆ − ˆ ) Q(yˆ − ˆ ), 0
subject to the constraint x(0) = x(T ) as well as the dynamics in Eq. (2). Problem 2. Let T u2 (t) dt + (yˆ − ˆ ) Q(yˆ − ˆ ), J (u, x0 ) = 0
where Q is a positive definite matrix. The problem is min J (u, x0 ), u,x0
subject to the constraint x(0) = x(T ) and the dynamics in Eq. (2). The difference between Problems 1 and 2 is that there is an extra cost associated with the free initial x0 in Problem 1, through the term x0 Rx 0 . As we will see, this difference will make Problem 1 easier to solve than Problem 2. The basic idea of the construction of solutions is to define a linear variety V in a Hilbert space that contains all of the constraints. As we will see in the next section, it is possible to interpret the data as a point (p) in the Hilbert space, which reduces the problem to that of finding the point on the linear variety that is closest (in the sense of the norm in the Hilbert space) to the data point. We know that we can construct this point by finding the orthogonal complement of the linear variety that defines the affine variety and constructing the intersection of the affine variety with the orthogonal complement. In other words, “all” we need to do in order to solve the problem is to compute V ∩ (V⊥ + p). In this process we follow Luenberger (1969). Before constructing this intersection two things must be verified. The first is that the linear variety V is nonempty and the second is that it is closed. Both follow from the fact that V is the graph of a continuous mapping from an appropriate product space to RN . 3. Problem 1
i = R −1 eA ti c ,
yi = i , x0 R + fi , uL ,
u,x0
We begin by considering the periodic spline as a particular boundary value problem, and for this we will use the methods of Zhou et al. (2005). Let the boundary condition be given by x(0) − x(T ) = 0.
(3)
We note that since x(T ) = e
AT
T
x(0) +
eA(T −s) bu(s) ds,
0
the specific dependence on x(T ) can be removed and the boundary constraint simply becomes T AT (I − e )x(0) − eA(T −s) bu(s) ds = 0. (4) 0
H. Kano et al. / Automatica 44 (2008) 185 – 192
It remains to construct the intersection V ∩ (V⊥ + p) to find the optimal point, where p = (0, 0, ˆ ) ∈ H. This construction is technically more complicated than for the smoothing spline without periodicity constraints, as reported in Zhou et al. (2005). The unique point in the intersection is defined as the solution of the following system of four equations in the unknowns u, x0 , y and , obtained by identifying x and x˜ with x0 , d with y, ˆ and d˜ with yˆ − ˆ .
We now define the following Hilbert space H = L2 [0, T ] × Rn × RN with norm
T
(u; x0 ; y) 2 = 0
u2 (t) dt + x0 Rx 0 + y Qy.
We define the linear constraint variety, V ⊂ H, to be V = (u; x; d) : di = i , xR + fi , uL ,
T
(I − eAT )x −
187
u=−
eA(T −s) bu(s) ds = 0 .
N
yˆ − ˆ , ei Q fi − b eA (T −s) ,
(6)
i=1
0
N
We first note that V is a closed subspace of H since it is the graph of a continuous function restricted to a closed linear variety, and the closed graph theorem applies. We now construct the orthogonal complement V⊥ . And, ˜ : from the definition it directly follows that V⊥ = {(u; ˜ x; ˜ d) ˜ ˜ xR + d, dQ = 0}, which allows ∀(u; x; d) ∈ Vu, ˜ uL + x, us to state the following lemma:
yi = i , x0 R + fi , uL .
Lemma 1.
We begin by eliminating x0 and u from Eq. (9) by substituting Eqs. (6) and (7). After some manipulation we have
V⊥ =
˜ : x˜ = − (u; ˜ x; ˜ d)
N
˜ ei Q i + R −1 (I − eA T ), d,
i=1
u˜ = −
N
A (T −t)
˜ ei Q fi − b e d,
for all ∈ Rn , where ei is the unit vector with zeros everywhere except for the ith position. Proof. From the expression of the orthogonal complement above, we directly get that
N ⊥ ˜ : x˜ + ˜ ei Q i , x ˜ x; ˜ d) d, V = (u; + u˜ +
i=1 N
˜ ei Q fi , u = 0 , d,
˜ ei Q i = R −1 (I − eAT ) d,
and
where G = R with = (1 2 · · · N ), and F is the Gramian T f (s)f (s) ds (10) F= 0
with f (s) = (f1 (s)f2 (s) · · · fN (s)) . Note that since the fi ’s are linearly independent, F is invertible. Moreover, since i = R −1 eA ti c we can let
and the lemma follows.
(5)
+ ,
(11)
where
T
=−
f (s)b eA (T −s) ds.
We will now use Eq. (8) to obtain a second equation in and y. ˆ Substituting u in Eq. (6) and x0 in (7) into (8), we have N AT −1 AT yˆ − ˆ , ei Q i + R (I − e ) 0 = (I − e ) − i=1
yˆ = − GQ(yˆ − ˆ ) − F Q(yˆ − ˆ ) + E R −1 (I − eAT )
˜ ei Q fi = (−b eA (T −t) ), d,
i=1
(9)
0
i=1
N
(8)
yi = − ei GQ(yˆ − ˆ ) − ei F Q(yˆ − ˆ ) + i (I − eAT ) T − fi (s)b eA (T −s) ds,
From this we conclude that
u˜ +
eA(T −s) bu(s) ds,
0
for the appropriate E, in order to obtain
following the construction in Zhou et al. (2005). Now, the relationship does not hold for all x and u but only for those x and u for which Eq. (4) holds. Multiplying by any ∈ Rn (and its transpose ), we can rewrite Eq. (4) as
N
T
0 = (I − eAT )x0 −
i=1
x˜ +
(7)
i=1
= R −1 (eA t1 c , . . . , eA tN c ) = R −1 E,
R −1 (I − eAT ) , xR + (−eA(T −t) b) , uL = 0.
yˆ − ˆ , ei Q i + R −1 (I − eAT ) ,
0
i=1
x0 = −
−
T
e 0
A(T −s)
b −
N i=1
A (T −s)
y−ˆ ˆ , ei Q fi − b e
ds.
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H. Kano et al. / Automatica 44 (2008) 185 – 192
We make the following observation: N
yˆ − ˆ , ei Q i =
i=1
N
1
i ei Q(yˆ − ˆ )
0.8
i=1
= R −1 EQ(yˆ − ˆ ).
N i=1
T
y(t)
T A(T −s)
bf i (s)ei ds = , it holds that Noting − N i=1 0 e
0.6
0.4
−e
A(T −s)
0
byˆ − ˆ , ei Q fi (s) ds = Q(yˆ − ˆ ). 0.2
Using these two constructions we then have 0
0 = (I − eAT )(−R −1 EQ(yˆ − ˆ )) + (I − eAT )R −1 (I − eAT ) − Q(yˆ − ˆ ) + ,
(12)
where is the controllability Gramian
T
=
eA(T −s) bb eA (T −s) ds.
0.2
0.3
0.4
0.5 t
0.6
0.7
0.8
0.9
1
Fig. 1. Periodic spline in Problem 1: Depicted are y(t) (solid line) and ˆ (stars).
solution to Problem 1.
By combining these two expressions (11) and (12) linking yˆ and , we obtain Eq. (14). It can be shown that the coefficient matrix in (14) is invertible under the controllability assumption of the system x˙ = Ax + bu. Using Eq. (14) we can solve for yˆ and for . These values can be used in Eqs. (6) and (7) to uniquely determine the optimal control and the optimal initial condition. We see that the optimal estimate of the data is obtained independently of the control. It is necessary to ask the question “In what sense is the spline periodic?” We state and prove the following theorem in answer to that question. Theorem 2. The function y(t) of the above construction can be extended periodically to the entire positive real line and the extension is 2n−1 times continuously differentiable everywhere with exception of the points nT : n ∈ Z where are guaranteed only n − 1 continuous derivatives. We leave the proof to the reader. An example of this procedure is seen in Fig. 1, where 0 1 0 A= , b= , c = (1 0), T = 1 0 0 1 t2 = 0.3,
t3 = 0.5,
ˆ = (0.8 0.2 0.5 1 0.3) Q = 104 I5 ,
0.1
(13)
0
t1 = 0.2,
0
t4 = 0.7,
t5 = 0.8
I +(G+F )Q
−E R −1 (I −eAT ) −
((I −eAT )R −1 E+ )Q −(I −eAT )R −1 (I −eAT ) − yˆ × (G + F )Qˆ = . (14) ((I − eAT )R −1 E + )Qˆ 4. Problem 2 We begin by considering the periodic spline as a particular boundary value problem. We will use the methods related to but not the same as in the above construction. Again let the boundary condition be given by x(0) − x(T ) = 0,
(15)
and the dynamics be the same as in Problem 1. Then as before, the boundary constraint is given by (4). Note that the initial data becomes a parameter in the constraint. We now define a Hilbert space to be H = L2 [0, T ] × RN
R = 104 I2 ,
with norm (Ip = p × p identity matrix).
We assumed in Section 1 that the system x˙ = Ax + bu, y = cx is controllable and observable. In addition to that it is natural to employ minimal system for generating splines, the controllability guarantees the invertibility of the coefficient matrix in (14) implying the existence and uniqueness of an optimal
T
(u; y) 2 =
u2 (t) dt + y Qy.
0
Since we no longer have a cost associated with the initial condition, x0 cannot be part of the Hilbert space over which the minimum norm problem is solved. In fact, the constraint
H. Kano et al. / Automatica 44 (2008) 185 – 192
variety is now affine, parameterized by x0 , and it is given by Vx0 = (u; d) : i , x0 R = di − fi , uL ,
T
(I − eAT )x0 −
eA(T −s) bu(s) ds = 0 .
0
If we now let V be the linear constraint variety associated with x0 = 0 above, we note that, as before, V is a closed subspace of H. Moreover, from Luenberger (1969), we know that the unique minimizer is given by the intersection Vx0 ∩ (V⊥ + p), where V⊥ = {(v; w) : ∀(u; d) ∈ Vv, uL + w, dQ = 0}.
We now construct the intersection (V⊥ + p) ∩ Vx0 , which is determined by the solution of the following three equations which come from V⊥ + p and Vx0 . N
yˆ − ˆ , ei Q fi − b eA (T −s) ,
(16)
i=1
i , x0 R = yi − fi , uL , T eA(T −s) bu(s) ds. (I − eAT )x0 =
where QF Q S= − Q
−Q
(20)
.
Now using Eq. (19) we have yˆ − ˆ x0 =C , ˆ −
−1
−
E
−I
I − eAT
0
,
and thus we have an expression for T x0 2 u (s) ds = (x0 , ˆ )C SC . 0 ˆ
i=1
u=−
Then we can calculate the square integral of u as T yˆ − ˆ 2 u (s) ds = (yˆ − ˆ , )S , 0
where I + FQ C= Q
Using the same technique as for Problem 1 we have N ⊥ A (T −s) V = (v; w) : v = − w, ei Q fi − b e .
189
(17) (18)
0
Since yˆ − ˆ is written as x0 yˆ − ˆ = (I, 0)C , ˆ we can now write the total cost J (u, x0 ) as a function of the initial position Jˆ(x0 ). I x0 ˆ J (x0 ) = (x0 , ˆ ) C SC + C Q(I, 0)C . 0 ˆ Thus we obtain
x0
We use the first equation to eliminate u from the second and third equation. After some manipulation we have the following system of equations: I + F Q − FQ yˆ E x0 = , (19) AT Q I −e − Q ˆ
Jˆ(x0 ) = (x0 , ˆ )V
where, as before, F is the Gramian of the fi ’s in (10) and is the controllability Gramian in (13). Note that this coefficient matrix is invertible, since I + F Q − Q 0 , = M1 Q 0 −I −
and V can be computed as
where
Q V = C S + 0 V =
,
ˆ
E
−I
I − eAT
0
0
0
C,
Q−1 + F
−1
E
−I
I − eAT
0
.
and the matrix M1 defined by −1 Q +F M1 =
The cost function Jˆ(x0 ) is quadratic in x0 , and an optimum x0 minimizing the cost is obtained as a solution of the following linear equation: x0 (In , 0)V = 0. ˆ
is positive-definite. We now reconsider u from Eq. (16), and obtain yˆ − ˆ u = (−f (s)Q, −b eA (T −s) ) .
This equation in x0 has unique solution if and only if the coefficient matrix of x0 , denoted by V1 , In V1 = (In , 0)V 0
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H. Kano et al. / Automatica 44 (2008) 185 – 192
is invertible. Moreover, since V1 is written as V1 =
E
Q−1 + F
I − eAT
−1
E
1
I − eAT
0.8
rank(eA t1 c · · · eA tN c I − eA T ) = n.
(21)
The condition for full rank in Eq. (21) is interesting. The first N columns arise from an observability problem that was studied in Martin and Smith (1987). The problem is to recover the initial data from x˙ = Ax, y = cx, x(0) = x0 when the observations are made at discrete time points, t1 , . . . , tN . There are no known necessary and sufficient conditions in terms of A, c and t1 , . . . , tN for the first N columns to have full rank. However, we will see that the last set of n columns given by I − eA T simplifies the condition (21). We introduce the following set of complex numbers √ I = {j2k/T | j = −1, k ∈ Z}.
y(t)
we see that the solution is unique whenever
0.4
0.2
0 0
The solution to Problem 2 is computed for the same example as in the previous section and the results are shown in Fig. 2. Periodic splines can be used to construct closed curves in the plane, and in Figs. 3 and 4 we show a result applied to contour modeling of jellyfish for an image frame from a real digital movie file.2 We used the same second order system (A, b, c) as in the previous examples, and the other parameters are T = 360, N =36 and Q=10−2 I36 . Using standard image processing techniques, the data set (ti , i ), i = 1, 2, . . . , N is obtained from the image in Fig. 4. Namely, ti and i are obtained as the polar coordinates of sampled boundary pixels with the origin at the centroid of jellyfish, where ti = 10 × (i − 1) denotes angles measured every 10◦ and i the radial distance in pixels. The solution to Problem 2 is obtained as in Fig. 3, from which the contour is reconstructed as superposed in Fig. 4. 2 Educational Image Collections, Information-technology Promotion Agency (IPA), Japan. http://www2.edu.ipa.go.jp/gz/.
0.2
0.3
0.4
0.5 t
0.6
0.7
0.8
0.9
1
95 90 85
Lemma 3. The condition (21) holds if and only if rank (c I − A ) = n for all ∈ I.
Theorem 4. There always exists a unique optimal solution to Problem 2.
0.1
Fig. 2. Periodic spline in Problem 2: Depicted are y(t) (solid line) and ˆ (stars).
Note that the matrix I − eAT is singular whenever A has an eigenvalue in I. Then we can show that the following lemma holds (the proof omitted).
80 y(t)
We have shown that, due to the controllability assumption, (19) has a unique solution in yˆ and for any initial condition x0 . On the other hand, the assumption of observability, which is equivalent to rank (c I − A ) = n for all ∈ C (e.g. Hautus, 1969), guarantees the existence of unique optimal x0 by Lemma 3. Thus, under the assumption of controllability and observability, we obtain the following theorem:
0.6
75 70 65 60 55 50
0
50
100
150
200 t
250
300
350
400
Fig. 3. Periodic spline in Problem 2 for the data from jellyfish image: Depicted are y(t) (solid line) and ˆ (stars).
Fig. 4. A jellyfish image frame and the contour reconstructed from the periodic spline y(t), t ∈ [0, 360] in Problem 2 by (y(t) cos 2t/360, y(t) sin 2t/360).
H. Kano et al. / Automatica 44 (2008) 185 – 192
Finally, one can note that it may be possible to obtain a solution to this problem as a limiting solution to Problem 1 as R tends to zero. However, this line of inquiry is not pursued further and we simply state this as a possibility for the future. 5. Conclusions In this paper we present a method for generating periodic, smoothing splines using linear optimal control. In particular, we show that such curves can be obtained in a quite general fashion by viewing the smoothing cost as an inner product in a suitable space, at the same time as periodicity is enforced by limiting the solutions to lie in a particular linear variety. The resulting methodology is numerically efficient and robust, and it unifies a number of contributions in the general area of smoothing splines through the use of different linear systems as generators of curves of different characteristics, including polynomial, exponential, and trigonometric smoothing splines.
References Eubank, R. L. (1999). Nonparametric regression and spline smoothing. Second edition. Statistics: Textbooks and Monographs. New York: Marcel Dekker. Hautus, M. L. J. (1969). Controllability and observability conditions of linear autonomous systems. Nederl. Akad. Wetensch. Proc. Ser. A 72 = Indag. Math., 31, 443–448. Kano, H., Fujioka, H., Egerstedt, M., & Martin, C. F. (2005). Optimal smoothing spline curves and contour synthesis. In Proceedings of the 16th IFAC world congress, Prague, Czech Republic, July 2005. Luenberger, D. G. (1969). Optimization by vector space methods. New York, London, Sydney: Wiley. Martin, C., & Smith, J. (1987). Approximation, interpolation and sampling. Differential Geometry: The Interface Between Pure and Applied Mathematics, 68, 227–252. Martin, C. F., Sun, S., & Egerstedt, M. (2001). Optimal control, statistics and path planning. 33(1–3), 237–253. Schumaker, L. L. (1981). Spline functions: Basic theory. New York: Wiley. Sun, S., Egerstedt, M., & Martin, C. F. (2000). Control theoretic smoothing splines. IEEE Transactions on Automatic Control, 45(12), 2271–2279. Takahashi, S., Ghosh, B. K., & Martin, C. F. (2002). Boundary location using control theoretic splines. Transactions of the Society of Instrument and Control Engineers, 38(3), 293–298. Unser, M., & Blu, T. (2005). Generalized smoothing splines and the optimal discretization of the wiener filter. IEEE Transactions on Signal Processing, 53, 2146–2159. Wahba, G. (1990). Spline models for observational data. In CBMS-NSF regional conference series in applied mathematics, (Vol. 59). Philadelphia, PA: SIAM. Zhou, Y., Egerstedt, M., & Martin, C. (2005). Control theoretic splines with deterministic and random data. In Proceedings of the 44th IEEE conference on decision and control and European control conf. ECC (pp. 362–367). Seville, Spain, December 2005. Zhou, Y., Egerstedt, M., & Martin, C. F. (2001). Optimal approximation of functions. Communications in Information and Systems, 1(1), 101–112. Zhou, Y., & Martin, C. F. (2004). A regularized solution to the Birkhoff interpolation problem. Communications in Information and Systems, 4(1), 89–102.
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Hiroyuki Kano was born in Hyogo, Japan. He received the B.E. and M.E. degrees in Mechanical Engineering from Kyoto Institute of Technology, Japan, in 1970 and 1972, respectively, and the Ph.D. degree in Electrical Engineering—Systems Science from Polytechnic Institute of New York (now Polytechnic University), USA, in 1976. From 1976 to 1992, he was with the International Institute for Advanced Study of Social Information Sciences, Fujitsu Limited. Since 1992, he has been a Professor in the Department of Information Sciences, Tokyo Denki University. His research interests include control and estimation theory, motion planning and control of robots, and signal processings. He is a coauthor of the book ‘Matrix Riccati Equations for Control Engineers’, Asakura Pub. Co., 1996. Magnus B. Egerstedt was born in Stockholm, Sweden, and is an Associate Professor in the School of Electrical and Computer Engineering at the Georgia Institute of Technology, where he has been on the faculty since 2001. He also holds an adjunct appointment in the Division of Interactive and Intelligent Computing with the College of Computing at Georgia Tech. Magnus Egerstedt received the M.S. degree in Engineering Physics and the Ph.D. degree in Applied Mathematics from the Royal Institute of Technology, Stockholm, in 1996 and 2000 respectively. He also received a B.A. degree in Philosophy from Stockholm University in 1996. He spent 2000–2001 as a Postdoctoral Fellow at the Division of Engineering and Applied Science at Harvard University. Dr. Egerstedt’s research interests include optimal control as well as modeling and analysis of hybrid and discrete event systems, with emphasis on motion planning, control, and coordination of mobile robots, and he has authored over 100 papers in the areas of robotics and controls. Magnus Egerstedt is a Senior Member of the IEEE, he received the ECE/GT Outstanding Junior Faculty Member Award in 2005, and the CAREER award from the U.S. National Science Foundation in 2003. Hiroyuki Fujioka was born in Mie, Japan, in 1975. He received the B.E., M.E. and Dr.E. degrees from Ritsumeikan University, Japan, in 1997, 1999 and 2004, respectively. In 2002, he joined as an Instructor the Department of Information Sciences, Tokyo Denki University, where he is currently an Assistant Professor. He is a recipient of the 2004 Sunahara Prize and Young Investigators Award from the Institute of Systems, Control and Information Engineers (ISCIE). His research interests include theory and applications of splines, and task planning and control of robots. Satoru Takahashi was born in Tokyo in 1965. He received the B.S. and the M.S. degrees, both in Mathematical Sciences, from Tokyo Denki University in 1989 and 1991, respectively. He finished the Doctoral Programs in Mathematical and Information Sciences of Tokyo Denki University in 1997 and received the Ph.D. in 2001. From 1991 to 1993 he was a Researcher of the Nano Technology Research Center, Canon INC., Japan. From 1997 to 1999 he served as an Instructor in the Department of Information Sciences, Tokyo Denki University. From 1999 to 2000, he was a Visiting Researcher in the Systems, Science and Mathematics Department at Washington University. In 2001, he served as a Research Assistant in the Mathematics and Statistics Department at Texas Tech University. Since 2003, he has been an Associate Professor in the Intelligent Mechanical Systems Engineering Department at Kagawa University with research interests in the area of Machine Vision, Robotics and Mechatronics.
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Clyde F. Martin’s research interests include control theory, the applications of algebraic and differential geometry to problems in numerical analysis, and the development and analysis of mathematical models in agriculture, the environment and medicine. He has organized or co-organized more than 10 international conferences and has served on the program committee or board of directors of several others. In November of 2001 he received an honorary doctorate for his work in engineering from the Royal Institute of Technology in Stockholm Sweden. He has received distinguished alumni awards from both Emporia State University and the University of Wyoming. He has directed more than 60 students to advanced degrees and has published more than 300 papers in a variety of disciplines.