Estimation of the system transfer function for certain linear systems

A non-transitory computer-readable medium containing instructions that, when loaded into memory and executed by a processor, cause execution of a method for producing an estimate  of a system transfer matrix A for a linear system, wherein the linear system is characterized by y=Ax+e, where x is an input vector, e is a noise vector and y is the output vector resulting from input vector x, the method comprising:

• selecting a set of calibration inputs Xcal=[x1, . . . , xm] where each of the xm is an input vector that is a calibration pattern;
• applying the calibration inputs Xcal to the linear system;
• observing the calibration outputs Ycal resulting from the calibration inputs Xcal according to Y=AX+E, where Ycal−=[y1, . . . , ym] where each of the ym is an output vector resulting from the corresponding input calibration pattern xm, and E−=[e1, . . . , em] where each of the em is a noise vector;
• identifying, by the processor executing the instructions, a subspace of A, assuming that a column span of A equals a row span of A;
• calculating, by the processor executing the instructions, an estimate  within the subspace of A, the estimate  calculated from the calibration outputs Ycal and calibration inputs Xcal and subject to a constraint that a row span of the estimate  equals a column span of the estimate Â;
• observing outputs Y1 produced by the linear system;
• and
• based on solving an inverse problem for the calculated estimate Â, estimating, by the processor executing the instructions, corresponding inputs X1 that produced the outputs Y1.

A system for producing an estimate  of a system transfer matrix A for a linear system, wherein the linear system is characterized by y=Ax+e, where x is an input vector, e is a noise vector and y is the output vector resulting from input vector x, the system comprising:

• a computer system comprising a memory and a processor;
• a selection module that selects a set of calibration inputs Xcal[x1, . . . , xm] where each of the xm is an input vector that is a calibration pattern;
• an observation module that captures the calibration outputs Ycal resulting from the application of the calibration inputs Xcal to the linear system according to Y=AX+E, where Ycal−=[y1, . . . , ym] where each of the is an output vector resulting from the corresponding input calibration pattern xm, and E−=[e1, . . . , em] where each of the em is a noise vector
• wherein the processor executes instruction stored in the memory, that identifies a subspace of A, assuming that a column span of A equals a row span of A;
• and that further calculates an estimate A within the subspace of A, the estimate  calculated from the calibration outputs Ycal and calibration inputs Xcal and subject to a constraint that a row span of the estimate  equals a column span of the estimate Â;
• the observation module further capturing outputs Y1 produced by the linear system;
• and
• the processor executing further instruction stored in the memory, for solving an inverse problem for the calculated estimate  to estimate corresponding inputs X1 that produced the outputs Y1.