Asian Journal of Control, Vol. 00, No. 0, pp. 1–8, Month 2008
Published online in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/asjc.0000

Monotone Smoothing Splines using General Linear Systems
Masaaki Nagahara and Clyde F. Martin
ABSTRACTIn this paper, a method is proposed to solve the problem of monotonesmoothing splines using general linear systems. This problem, also called
monotone control theoretic splines, has been solved only when the curve
generator is modeled by the second-order integrator, but not for other cases.
The difficulty in the problem is that the monotonicity constraint should besatisfied over an interval which has the cardinality of the continuum. To
solve this problem, we first formulate the problem as a semi-infinite quadratic
programming, and then we adopt a discretization technique to obtain a finitedimensional quadratic programming problem. It is shown that the solutionof the finite-dimensional problem always satisfies the infinite-dimensionalmonotonicity constraint. It is also proved that the approximated solution
converges to the exact solution as the discretization grid-size tends to zero.
An example is presented to show the effectiveness of the proposed method.
Key Words: Smoothing splines, optimal control, semi-infinite optimization,
quadratic programming, model predictive control

I. Introduction
Control theoretic splines are either interpolatingor smoothing splines, depending on the constraints andcost function, with a constraint written as a system of
linear differential equations [25]. The spline curve isobtained as the output of a given linear system. This
concept is generalization of the smoothing spline by
Wahba [17, 29] in that a richer class of smoothing
curves can be obtained relative to polynomial curves.
They have been proved to be useful in trajectory
planning [8, 22, 28], mobile robots [26], contour
modeling of images [15], probability distribution
estimation [3, 4], and so on. For more applications anda rather complete theory of control theoretic splines, see
the book [10].
Manuscript received July 8, 2008.
M. Nagahara is with Graduate School of Informatics, Kyoto
University, Kyoto 606-8501, JAPAN. (nagahara@ieee.org).
C. F. Martin is with Department of Mathematics & Statistics,
Texas Tech University, Texas, USA.(clyde.f.martin@ttu.edu).
M. Nagahara is the corresponding author.
This research was supported in part by Grand-in-Aid forYoung Scientists (B) of Japan Society for the Promotion of
Science under Grant No. 22760317.

On the other hand, monotone control theoretic
splines are also important in deriving a model or an
estimation of a parameter such as the growing rate of
an individual [7, 9, 10]. Also, there have been a few
applications of monotone smoothing in the statistical
literature. The work of Ramsay [21] and the discussion
associated with the paper seems to be the first use. A
more recent paper by Kelly [16] is a nice application.
Monotonicity is achieved by adding a non-negative(or non-positive) derivative constraint on the outputof the target linear system over a time interval.
Since the interval is an infinite subset of [0, ∞), thisconstraint becomes infinite-dimensional, which makes
the problem difficult to solve. In [7, 9], this has beensolved when the target linear system is a secondorder integrator 1/s2 . However, for general cases,
no solution has been obtained. More recently, thesmoothing problem has been solved with inequality
constraints [1], in which inequality constraints aredefined only on sampling points, and the constraints
may be violated between sampling points.
In this paper, we propose a new method
for constructing monotone splines guaranteeing theinequality constraint on a whole time interval, say[0, T ], with more general linear systems. To solve

c 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control SocietyPrepared using asjcauth.cls [Version: 2008/07/07 v1.00]

2Asian Journal of Control, Vol. 00, No. 0, pp. 1–8, Month 2008

this, we first formulate the problem as a semiinfinite quadratic programming problem [13, 19].
Then we adopt a discretization technique [14, 27] toreduce the infinite-dimensional constraints to a finite
dimensional constraints. This discretization is executedby partitioning the time interval into finitely many
subintervals and estimating the constraint on a finite
grid. The finite-dimensional quadratic programming
problem can be solved with efficient algorithms [2].
We also discuss the limiting properties of theapproximated solution. Whereas the approximatedsolution is constrained on a finite grid, we show thatit satisfies the original infinite-dimensional constraint
provided the grid size is sufficiently small. We also
prove that the approximated solution converges to theexact solution as the discretization grid-size tends tozero. By these properties, the proposed method can besafely adopted for monotone control theoretic splines.
We also construct an illustrative example to show theeffectiveness of our method.
The method proposed in this paper can be directly
applied to model predictive control (MPC), in particular
nonuniformly-sampled-data MPC with continuous-timeinequality constraints. For the study, there are many
researches; MPC with uniform sampling and piecewise
constant control [20, 12], with constraints only on thesampling points [6], and with nonuniformly sampling
but unconstrained [24]. To our knowledge, there isno method for nonuniformly-sampled-data MPC withrelatively general control inputs which guarantees
continuous-time constraints as we will consider in thispaper.
The organization of this paper is as follows:
Section II defines the problem of monotone control
theoretic splines. In Section III, we formulate theproblem as a semi-infinite quadratic programming.
Section IV is the main section, where discretization
approach is introduced and the limiting properties are
discussed. Section V suggests a formula for computing
inner products used in our optimization. A numericalexample is illustrated in Section VI, and a conclusion ismade in Section VII.
NotationIn this paper, we use the following notation. R,
Rn , and Rn×m are respectively the set of real numbers,
n-dimensional real vectors and n × m matrices. We
denote L2 [0, T ] by the Lebesgue space consisting of all
square integrable real functions on [0, T ] ⊂ R, endowedwith the inner product T
x, y :=x(t)y(t)dt, x, y ∈ L2 [0, T ].
0

For a matrix (a vector) M , M  is the transpose of
M . For a vector v = [v1 , . . . , vn ] ∈ Rn , v  0 meansvi ≤ 0 for i = 1, 2, . . . , n, and for two vectors v and w,
v  w if v − w  0. For a vector v ∈ Rn ,v1 :=n

|vi |,

v∞ := max |vi |.
i=1,...,ni=1

II. Monotone Smoothing Splines
In this paper, we consider the following problem ofmonotone smoothing splines:
Problem 1 (Monotone smoothing splines) Given
linear time-invariant systemẋ = Ax + Bu,

y = Cx,ax(0) = x0 ∈ Rn ,where x(t) ∈ Rn is the state vector, u(t) ∈ R is thecontrol input, y(t) ∈ R is the plant output, A ∈ R n×n ,B ∈ Rn×1 , C ∈ R1×n , and also given a data set
D := (t1 , α1 ), (t2 , α2 ), . . . , (tm , αm ) ,with time instants 0 < t1 < t2 < · · · < tm =: T andnoisy data α1 , . . . , αm ∈ R on the time instants, findthe control u ∈ L2 [0, T ] and the initial condition x(0) =x0 ∈ Rn that minimizes the following cost function:J(u) := λ Tu(t)2 dt +

0

m



2
wi y(ti ) − αi ,

(1)

i=1

where λ and wi ’s are given positive numbers (weights),
subject to the monotonicity constraintẏ(t) ≥ 0,∀t ∈ [0, T ].

(2)

The cost function (1) considers elimination ofGaussian noise on the data {α1 , . . . , αm } and limitationon the L2 norm of the control u. This formulation is anextension of Wahba’s smoothing spline [17, 29].
This problem has been solved in [7, 9] only when
C(sI − A)−1 B =1.
s2

However, for other cases, the problem has not beensolved.
Remark 1 One might think that the problem can besolved via a technique of model predictive control(MPC). The difficulty in using MPC is that we have toc 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control SocietyPrepared using asjcauth.cls

3

M. Nagahara and C. F. Martin: Monotone Smoothing Splines

treat the constraint (2) which is infinite dimensional. If
we discretize this constraint asẏ(ti ) ≥ 0,
where θ := [η1 , . . . , ηm , x0 ] , α := [α1 , . . . , αm ] , and2(λI + GW )G 2GW F,
2F  W G
2F  W F
q := −2 G F W α,φt1 , φt1  φt2 , φt1  . . . φtm , φt1 


..
..
..
..
G := 
,.
.
.
.
φt1 , φtm  φt2 , φtm  . . . φtm , φtm  At 
Ce 1
 .. 
F :=  .  , W := diag {w1 , . . . , wm } .

P :=i = 1, 2, . . . , m

to use a standard MPC formulation, the original
constraint (2) will be guaranteed only on the sampling
points t1 , . . . , tm , not on the whole [0, T ]. As mentioned
in Section 1, there is no method for the problem givenin (1) and (2) within the framework of MPC.

CeAtm

III. Formulation by Semi-Infinite Quadratic
Programming
We here formulate the problem given in the previous section by semi-infinite quadratic programming.
We first assume that we search the optimal control u inthe following subspace of L 2 [0, T ] as taken in [25]:where φt ∈ L2 [0, T ] is defined byφt (τ ) :=∂φt (τ )=∂t

CAeA(t−τ ) B,
0, if t ≤ τ.

if t > τ,(5)
By using this, the derivative ẏ(t) can be calculated asẏ(t) = Φ(t) θ, where
φ̇t , φt1 
φt1 (t) .. ..
.
Φ(t) :=  + CB  .  .
 φ̇t , φtm  φtm (t)

0eA t A C 

i=1

t > τ,t ≤ τ.

φ̇t (τ ) :=





m

2
Vm := u ∈ L [0, T ] : u =ηi φti , ηi ∈ R ,CeA(t−τ ) B,
0,

Let φ̇t (τ ) be the derivative of φt (τ ), that is,(3)This characterization is based on the representer
theorem [17, 29, 23] for the regularized cost functional(1) with the kernel k(t, τ ) = φt , φτ ; the optimal
estimation y(t) is represented by

The constraint (2) now becomes Φ(t) θ ≥ 0 for allt ∈ [0, T ]. In summary, Problem 1 is formulated by thefollowing:
Problem 2 Find θ ∈ Rm+n that minimizesf (θ) =1 
θ P θ + q  θ,
2

(6)subject toy(t) =m

ηi k(t, ti ) =

i=1

m


g(θ, t) = −Φ(t) θ ≤ 0,

ηi φt , φti .

i=1

One can consider other base functions {φi }Mi=1 thanM(3) and define the subspace VM := span{φi }i=1 whereM
m. This is related to basis pursuit denoising [5],with which we are not concerned in this paper.
By assuming that the control u is in the subspace
Vm , the cost function (1) can be described as [25]
J

m

i=1

ηi φti=1 θ P θ + q  θ + α W α,2

for all t ∈ [0, T ].

Note that the term α W α in (4) is omitted in (6)since this term is constant (independent of θ) anddoes not change the optimal solution. Problem 2 is asemi-infinite quadratic programming problem. By the
optimal solution θ ∗ to this problem, we obtain the
optimal control input u ∗ byu∗ (t) =mθ∗ [i]φti (t),

i=1

(4)

(7)

where θ∗ [i] is the i-th element of θ ∗ .

c 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control SocietyPrepared using asjcauth.cls

4Asian Journal of Control, Vol. 00, No. 0, pp. 1–8, Month 2008

IV. Discretization Approach to Monotone
Splines
4.1. Discretization
The difficulty in Problem 2 is that the inequality
constraint (7) is infinite dimensional. We here
introduce an approximation technique for such an
infinite-dimensional constraint. To reduce this to a
finite dimensional one, we adopt the technique of
discretization [14, 27].
We first divide the interval [0, T ] into M
subintervals:[0, T ] = [T0 , T1 ] ∪ · · · ∪ (TM−1 , TM ],where {Ti} is the discretization grid satisfying
0 = T0 < T1 < · · · < TM = T.

Then we evaluate the function g(θ, t) in (7) on the grid
points t = T0 , T1 , . . . , TM . To guarantee the constraint(7), we choose real numbers  > 0 and r > 0 and adopt
the following constraints:g(θ, T0 ) ≤ −, . . . , g(θ, TM ) ≤ −, − r1  θ  r1
(8)
where 1 = [1, 1, . . . , 1] ∈ Rm+n . The second inequality is component wise (see Notation in Section 1), andmeans θ∞ ≤ r. The finite-dimensional constraints (8)can be represented as a matrix inequality,Hθ  v,v :=Φ(T0 ) . . . Φ(TM )

I



r1

− . . . − r1

Remark 3 Splines with another constraint than monotonicity such as concavity, i.e., ÿ(t) ≤ 0, can be also
solved by the same method.
4.2. Convergence analysis
Put N := m + n. Define feasible sets F for
the original optimization and F(M, , r) for theapproximation respectively by
F := θ ∈ RN : g(θ, t) ≤ 0, t ∈ [0, T ] ,(10)F(M, , r) := θ ∈ RN : g(θ, t) ≤ −,t ∈ {T0 , . . . , TM }, θ∞ ≤ r . (11)To prove our first proposition, we assume the following:
Assumption 1 The function Φ(t) is Lipschitz continuous, that is, there exists a real number µ > 0 such that
Φ(t) − Φ(s)1 ≤ µ|t − s|,for all t, s ∈ [0, T ].

(9)

where
H :=

These constraints are finite dimensional and can be
included easily in our quadratic programming. In
general, equality constraints on the output y and thederivative ẏ on time instants in [0, T ] can be represented
by linear constraints as above.

−I

 



,

.

In summary, the problem of monotone control
theoretic splines for arbitrarily linear time-invariant
system {A, B, C} can be approximately described by
the following quadratic programming: find θ ∈ R m+n
which minimizes (6) subject to (9). This is a standard
quadratic programming and can be efficiently solved bynumerical softwares such as MATLAB.
Remark 2 We can also include equality constraints
into our optimization. For example, if we want tohave y(T ) = 1 and ẏ(T ) = 0, then our constraints are
represented by
φT , φt1 , . . . , φT , φtm , CeAT θ = 1,φ̇T , φt1 , . . . , φ̇T , φtm , CAeAT θ = 0.

Remark 4 Systems whose relative degree is higher
than or equal to 2 satisfy the above assumption. Firstly,by the definition of φ̇t (τ ) in (5), we can say that for j =d1, 2, . . . , m, dt
φ̇t , φtj  is bounded on [0, T ], and hence
φ̇t , φtj  is Lipschitz on [0, T ]. Secondly, the function
eA t A C  is Lipschitz. Thirdly, we have CB = 0 since
the relative degree ≥ 2. These facts result in that Φ(t) is
Lipschitz.
Define the discretization grid-size byImax (M ) := max |Ti − Ti−1 | : i = 1, 2, . . . , M .
We then have the following proposition:
Proposition 1 Suppose that Assumption 1 is satisfied.
Suppose also that M ,  and r are chosen such that
Imax (M ) ≤


.
rµ

(12)

Then we have F(M, , r) ⊆ F .

c 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control SocietyPrepared using asjcauth.cls

5

M. Nagahara and C. F. Martin: Monotone Smoothing Splines

Proof.
Let θ ∈ F(M, , r). By the definition ofF(M, , r) in (11), θ satisfiesg(θ, Ti ) ≤ −,

Proof. Since int F is nonempty, there exist θ̄ ∈ int F .
It follows that there exists δ > 0 such thatg(θ̄, t) ≤ −δ,i = 0, 1, . . . , M.

Then, for any t ∈ [0, T ], there exists i ∈ {1, . . . , M }such that t ∈ [Ti−1 , Ti ]. By Assumption 1 and thecondition θ∞ ≤ r in (11), we haveg(θ, t) − g(θ, Ti ) = − (Φ(t) − Φ(Ti )) θ ≤ rµ(Ti − t).

g(θ, t) ≤ rµ(Ti − t) + g(θ, Ti ) ≤ rµImax (M ) −  ≤ 0.

The last inequality is due to (12). Thus, θ ∈ F and2hence F(M, , r) ⊆ F .
By this proposition, we can guarantee theconstraint (7) by searching the parameter θ in the finitedimensional feasible set F(M, , r) provided that the
number M is sufficiently large to satisfy (12).
Next, we analyze the limiting property of theapproximated solution when the discretization grid size
tends to zero. To prove the property, we assume thefollowing:
Assumption 2 The interior of F
int F := θ ∈ RN : g(θ, t) < 0, for all t ∈ [0, T ]θξ := ξ θ̄ + (1 − ξ)θ∗ ∈ int F

for all ξ ∈ (0, 1]. Fix arbitrarily ξ ∈ (0, 1]. Since f is aconvex function, we haveg(θξ , t) = −Φ(t) θξ= ξg(θ̄, t) + (1 − ξ)g(θ∗ , t)
≤ −ξδ,

where the last inequality is due to inequality (13) andg(θ∗ , t) ≤ 0 for θ∗ ∈ F defined in (10). Letr̄ := max θξ ∞ξ∈(0,1]and fix arbitrarily r ≥ r̄. Then we haveθξ ∈ F(M (ξδ, r), ξδ, r).

This gives the following inequality:∗f (θξδ,r) ≤ f (θξ ) ≤ f (θ∗ ) + ξ∆.

Remark 5 Assumption 2 is slightly restrictive. A
sufficient condition for this is that A is a Metzler
matrix∗ and CA  0 (CA ≡ 0). By the positive systemtheory [11], A is Metzler if and only if e At  0 for allt ∈ R. Let u ≡ 0. Then, we have ẏ(t) = CAeAt x0 ≥0 for all t ∈ R. Since CAeAt x0 is continuous on R,
we have ẏ(t) = CAeAt x0 > 0 for all t ∈ [0, T ], if wechoose sufficient large x0  0. Therefore, int F isnonempty in this case.

and hence

Then we have the following theorem:

We thus have

θ∈F

∗
θ,r
:=

arg minθ∈F(M(,r),,r)f (θ).

∗
Then we have f (θ,r) → f (θ∗ ) as  → +0 and r → ∞.
∗Amatrix A is called a Metzler matrix if all the off-diagonalcomponents are nonnegative.

(14)

where ∆ := f (θ̄) − f (θ∗ ). Note that ∆ ≥ 0 by thedefinition of θ ∗ . Also, for all t ∈ [0, T ], we haveis nonempty.

Theorem 1 Suppose that Assumptions 1 and 2 aresatisfied. Let M (, r) be the minimum value among all
M ’s satisfying (12). For f (θ) given in (6), define(13)

Then, since F is a convex set, we havef (θξ ) ≤ ξ(θ̄) + (1 − ξ)f (θ∗ ) = f (θ∗ ) + ξ∆,

By this inequality, we haveθ∗ := arg min f (θ),∀t ∈ [0, T ].

(15)

On the other hand, by Proposition 1, we haveF ⊇ F(M (ξδ, r), ξδ, r),∗f (θ∗ ) ≤ f (θξδ,r
).

(16)

Combining (15) and (16) gives∗f (θ∗ ) ≤ f (θξδ,r) ≤ f (θ∗ ) + ξ∆.

∗∗lim f (θ,r) = lim f (θξδ,r) = f (θ∗ ).

→+0
r→∞

ξ→+0
r→∞

2

Remark 6 By Theorem 1, the semi-infinite programming we consider here is weakly discretizable [19].
∗Note that this does not necessarily imply that θ ,r→ θ∗ .
By Theorem 1, the proposed approximation method
given in the previous section can be safely adopted formonotone control theoretic splines.

c 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control SocietyPrepared using asjcauth.cls

6Asian Journal of Control, Vol. 00, No. 0, pp. 1–8, Month 2008

V. Computing Inner ProductOutput y(t)To compute the Grammian G and the matrix Φ(t),we have to compute the following inner products for
fixed s, t ∈ [0, T ]:

φ̇s , φt  =

0

 T
0

1.4

Amplitudeφs , φt  =

 T

1.6

φs (τ )φt (τ )dτ,φ̇s (τ )φt (τ )dτ.

1.2

1

0.8

0.6

These values can be easily computed by matrix
exponentials [18]:
 h

 As
−A τ −Aτ
φs , φt  = B ee
C Cedτ eAt B0 =: vs F22
F12 vt ,

0

1

2

3
4
Time (sec)

5

6

7

Fig. 1. Data (circles) and estimation y(t) by the proposed method(solid), and a conventional discretization method (dash)Derivative of y

The inner product φ̇s , φt  is also obtained by φ̇s , φt  = vs H22
H12 vt , 
H11 H12A
A C  C:= exp
h .
0
−A
0
H22

0.7

Proposed
Conventional

0.6
0.5
0.4
Amplitude

where h := min(s, t), vτ := eAτ B (τ = t, s), and thematrices F22 and F12 are defined by 
F11 F12A
CC:= exp
h .
0 F22
0
−A

0.3
0.2
0.1
0
−0.1
0

1

2

3
4
Time (sec)

5

6

7

VI. Example
We here show an example of monotone control
theoretic splines. Assume that the system is given by
P (s) =1.
s2 (s2 + 1)

(17)

Note that for the second order integrator 1/s2 the exact
optimal solution can be obtained [7, 9], but for higher
order systems as in (17), there has been no method toconstruct monotone smoothing splines. Let the original
curve beyorig (t) = 1.5 − e−t ,and the noisy data is obtained byti = i,αi = yorig (ti ) + εi ,i = 1, . . . , 7,

where εi is an independently and identically distributed random variable of the Gaussian distribution
N (0, 0.05). The estimation result is shown in Fig. 1.
We also calculate the estimation by a standard method

Fig. 2. Derivative of estimation y(t) by the proposed method (solid),
and a conventional discretization method (dash)

as used in [1] with the monotone constraint only on
the points t = 0, 1, . . . , 7. This figure shows that ourestimation works well while the conventional one shows
oscillations in the intervals (3, 4) and (5, 6), whichindicates that the derivative is negative. To see this more
precisely, we show the derivative ẏ(t) in Fig. 2. We
can see that the derivative by the conventional method
is non-negative on the sampling points t = 0, 1, . . . , 7,
however the constraint is violated in the intervals (3, 4)
and (5, 6) (See Fig. 3 for an enlarged plot). On theother hand, the derivative by our method is always
non-negative on the whole interval t ∈ [0, 7]. Fig. 4
shows the input u(t) for the both methods. By thisfigure, we can see that the input signal u(t) of theconventional method shows large gain on the intervals(3, 4) and (5, 6) where the conventional method violates
the monotonicity constraint.

c 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control SocietyPrepared using asjcauth.cls

M. Nagahara and C. F. Martin: Monotone Smoothing Splines

Derivative of y
0.05

Proposed
Conventional

0.04

Amplitude

0.03

0.02

0.01

0

−0.01
2.5

3

Time (sec)

3.5

4

Fig. 3. Enlarged plot of Fig. 2
Input u(t)0.5
0.4

Proposed
Conventional

0.3

Amplitude

0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
0

1

2

3
4
Time (sec)

5

6

7

Fig. 4. Input u(t): proposed (solid) and conventional (dash)VII. Conclusion
In this paper, we have proposed a new method forsolving the problem of monotone splines using generallinear systems. By using discretization technique,the problem is approximately described as a finitedimensional quadratic programming. The optimal
parameters can be efficiently obtained by numerical
optimization softwares such as MATLAB. The spline
curve generated by the obtained parameters isguaranteed to satisfy the monotonicity constraint over
the time interval of interest. We also proved limiting
properties of approximated solutions. Future work
may include quantification of the error introduced by
the proposed approximation, and also construction ofmonotone control theoretic splines for MIMO (multiinput multi-output) systems.
REFERENCES

7

1. Bell B. M., J. V. Burke, and G. Pillonetto,“An inequality constrained nonlinear KalmanBucy smoother by interior point likelihood
maximization,” Automatica, Vol. 45, No. 1,
pp. 25–33 (2009).
2. Boyd S. and L. Vandenberghe, Convex Optimization, Cambridge (2004).
3. Charles J. K., Probability Distribution Estimation
Using Control Theoretic Splines, Doctoral Dissertation, Texas Tech University (2009).
4. Charles J. K., S. Sun, and C. F. Martin,“Cumulative distribution estimation via control
theoretic smoothing splines,” Three Decades ofProgress in Control Sciences, Springer, pp. 83–92(2010).
5. Chen S. S., D. L. Donoho, and M. A. Saunders,“Atomic decomposition by basis pursuit,” SIAM
Journal on Scientific Computing, Vol. 20, No. 1,
pp. 33–61 (1998).
6. Diehl M., H. G. Bock, J. P. Schlöder, R. Findeisen,
Z. Nagy, and F. Allgöwer, “Real-time optimization and nonlinear model predictive control
of processes governed by differential-algebraic
equations,” Journal of Process Control, Vol. 12,
No. 4, pp. 577–585 (2002).
7. Egerstedt M. and C. F. Martin, “Monotonesmoothing splines,” Proceedings of the Mathematical Theory of Networks and Systems, pp. 1020–
1028 (2000).
8. Egerstedt M. and C. F. Martin, “Optimal trajectory
planning and smoothing splines,” Automatica,
Vol. 37, No. 7, pp. 1057–1064 (2001).
9. Egerstedt M. and C. F. Martin, “Optimal control
and monotone smoothing splines,” New Trends
in Nonlinear Dynamics and Control, Springer,
pp. 279–294 (2003).
10. Egerstedt M. and C. F. Martin, Control Theoretic
Splines, Princeton University Press (2010).
11. Farina L. and S. Rinaldi, Positive Linear Systems:
Theory and Applications, Wiley-Interscience(2000).
12. Grüne L., D. Nešić, and J. Pannek, “Model
predictive control for nonlinear sampled-datasystems,” Assessment and Future Directions of
Nonlinear Model Predictive Control, Springer,
pp. 105–113 (2007).
13. Hettich R. and K. O. Kortanek, “Semi-infinite
programming: theory, methods, and applications,”
SIAM Review, Vol. 35, No. 3, pp. 380–429 (1993).
14. Hettich R. and G. Gramlich, “A note on an
implementation of a method for quadratic semiinfinite programming,” Mathematical Programming, Vol. 46, No. 2, pp. 249–254 (1990).

c 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control SocietyPrepared using asjcauth.cls

8Asian Journal of Control, Vol. 00, No. 0, pp. 1–8, Month 2008

15. Kano H., M. Egerstedt, H. Fujioka, S. Takahashi,
and C. F. Martin, “Periodic smoothing splines,”
Automatica, Vol. 44, No. 1, pp. 185–192 (2008).
16. Kelly C. and J. Rice, “Monotone smoothingwith application to dose-response curves and theassessment of synergism,” Biometrics, Vol. 46,
pp. 1071–1085 (1990).
17. Kimeldorf G. and G. Wahba, “Some resultson Tchebycheffian spline functions,” Journal of
Mathematical Analysis and Applications Vol. 33,
No. 1, pp. 82–95 (1971).
18. Loan C. F. V., “Computing integrals involving thematrix exponential,” IEEE Trans. on AutomaticControl, Vol. 23, No. 3, pp. 395–404 (1978).
19. López M. and G. Still, “Semi-infinite programming,” European Journal of Operational
Research, Vol. 180, No. 2, pp. 491–518 (2007).
20. Magni L. and R. Scattolini, “Model predictive
control of continuous-time nonlinear systems withpiecewise constant control,” IEEE Trans. Automatic Control, pp. 49, No. 6, pp. 900–906 (2004).
21. Ramsay J. O., “Monotone regression splines inaction,” Statistical Science, Vol. 3, pp. 425–461(1988).
22. F. J. Rubio, F. J. Valero, J. L. Suñer, and V. Mata,“Simultaneous algorithm for trajectory planning,”
Asian Journal of Control, Vol. 12, No. 4, pp. 468–
479 (2010).
23. Schölkopf B. and A. J. Smola, Learning withKernels, The MIT Press (2002).
24. Sheng J., T. Chen, and S. L. Shah, “Generalized
predictive control for non-uniformly sampled
systems,” Journal of Process Control, Vol. 12,
No. 8, pp. 875–885 (2002).
25. Sun S., M. B. Egerstedt, and C. F. Martin, “Control
theoretic smoothing splines,” IEEE Trans. on
Automatic Control Vol. 45, No 12, pp. 2271–2279(2000).
26. Takahashi S. and C. F. Martin, “Optimal control
theoretic splines and its application to mobile
robot,” Proc. of the 2004 IEEE CCA, pp. 1729–
1732 (2004).
27. Teo K. L. and X. Q. Yang, “Computational
discretization algorithms for functional inequality
constrained optimization,” Annals of Operations
Research, Vol. 98, pp. 215–234 (2000).
28. Tsai M.-S., C. L. Yen, and H. T. Yau,
“Development of a hybrid iterative learning
control for contouring NURBS curves,” Asian
Journal of Control, Vol. 13, No. 1, pp. 107–125(2011).
29. Wahba G., Spline Models for Observational Data,
SIAM (1990).
c 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control SocietyPrepared using asjcauth.cls