MATLAB Codes for L1 Control Theoretic Smoothing Splines

Introduction

M. Nagahara and C. F. Martin,
L1 Control Theoretic Smoothing Splines,
submitted to IEEE Signal Processing Letters.

MATLAB Codes

Demo file that produces the figures in the paper:
demo.m

MATLAB function for L1 control theoretic splines
L1cts_coef.m

Gramian data file
GHdata.mat
This file gives gramian matrix G and intial condition matrix H for the optimization.
Since the computation of these matrix needs quite a lot of time, so one can use this file to reduce computational time.

Note that to execute the codes, you need Control System Toolbox and CVX.
We have made sure that the code run with no errors on MATLAB version R2012b.
On older versions of MATLAB, the code might have unexpected errors.

Usage

`[theta,x,G,H] = L1cts_coef(P,Ds,lambda,W,p)`
or
`[theta,x,G,H] = L1cts_coef(P,Ds,lambda,W,p,initial_state,G,H,q)`
computes the optimal coefficients of the control theoretic smoothing splines that minimizes L1-Lp cost function
Jp = lambda*||theta||_1 + ||W(G*theta-y)||_p^p

Inputs
`P`:
dynamical system (SS or TF object).
`Ds`:
data set; `Ds = [T,Y]` where `T` is a column vector of sampling instants, and `Y` is a column vector of observed data.
`lambda`:
regularization parameter (>0)
`W`:
wight matrix for empirical risk.
`p`:
norm index for empirical risk (p = 1 or 2)
`initial_state`:
if this value is not 0, then the initial state of P becomes also a design variable. default = 0;
`G` (optional):
Gramian matrix for computation. default = [];
`H` (optional):
Initial state matrix for computation. default = [];
`q` (optional):
if q = 2 then this function computes
min lambda*||theta||_2^2 + ||W(G*theta-y)||
that is, the L2 optimal control theoretic spline. default = 1;

Outputs
`theta`:
the optimal coefficient vector
`x`:
the optimal initial state
`G`:
Gramian matrix
`H`:
Initial state matrix