*spline*has been widely used in signal processing, numerical computation, statistics, etc. In particular, the

*smoothing spline*gives a smooth curve that has the best fit to given noisy data with the following optimization:

where (

*t*

_{1},

*y*

_{1}),..., (

*t*

_{n},

*y*

_{n}) are given data,

*y*(

*t*) is the curve to be estimated, and

*r*is the regularization parameter that specifies the trade-off between the fidelity to the data (1st term) and the smoothness of the curve (2nd term). The optimal curve is given by a linear combination of shifted cubic splines (see G. Wahba's book for details).

The smoothing spline can be rewritten in terms of control theory. We assume

*y*(

*t*) is the

*output*of the double integrator for some input

*u*(

*t*), namely,

where "

*s*" is the differential operator (or the symbol of the Laplace transform). Then we rewrite the optimization problem as

Then, a control theoretic spline is defined by replacing 1/

*s*

^{2}with a

*general transfer function*

*P*(

*s*), that is

For example, you can obtain a

*general*smoothing spline that minimizes

by solving the control theoretic spline optimization described above with

An important fact is that the optimal solution (or optimal control),

*u*(

*t*), is given by a linear combination of

*exponential*functions (including polynomial functions) that are specified by the impulse response (i.e., the inverse Laplace transform) of

*P*(

*s*).

A control theoretic spline can be explained as drawing a curve with a robot hand modeled by

*P*(

*s*); see the top picture. This illustrates that the control theoretic spline is a

*bridge*between control and signal processing.

The original paper on control theoretic spline can be downloaded from here.

Control theoretic spline with a monotonicity constraint is discussed in this paper.

Compressive sampling approach to control theoretic spline is proposed in this paper.

See also this blog entry.