Control Theoretic Spline: A bridge between control and signal processing

The 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 (t₁, y₁),..., (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/s² 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.