The talk Numerical Linear Algebra in Computational Control, by Zlatko Drmač (University of Zagreb), is held on Monday February 24, 2014 at 8:30 in room K4.
Control theory provides interesting and challenging problems to numerical linear algebra. Modern theoretical developments and exciting engineering applications demand efficient and numerically sound algorithms implemented as robust and accurate numerical software. Our aim is to illustrate how some recent developments in accurate linear algebra (accurate algorithms for eigenvalues and singular values) improve numerical computations in control theory.
Advanced applications are based on high level packages, such as Matlab, and computing engines such as numerical software libraries LAPACK, SLICOT. We stress the importance of reliable numerical software and call for more mathematical rigor in the implementation phase (coding) and testing. To illustrate a problem, we show how adding just one "WRITE(*,*) variable" statement to a mission critical code based on the above mentioned libraries, or changing compiler options, completely changes the computed key parameters of a given linear time invariant (LTI) system. Such situations may occur only at certain distance to singularity, and some computational tasks (such as e.g. revealing a numerical rank) are usually performed and are crucial (and interesting) on data close to singularity. And, since many phenomena are possible when close to singularity, any ill-behavior of the software is usually attributed to ill-conditioning, bailed out by backward stability, and the true problem may remain inconspicuous. (We give an example of rank revealing QR factorization software (LINPACK, LAPACK, SLICOT, Matlab, SciLab,...) instability that had been circulating undetected in all relevant matrix computation libraries for more than thirty years.)
This is certainly undesired behavior, even if such computation remains backward stable, and even if the computation is doomed to fail, due to ill--conditioning. We show the advantages of using state of the art matrix perturbation theory in rigorous numerical linear algebra algorithms development.
We use other examples (computing certain canonical forms in control, model order reduction algorithms) to show how some instabilities undetected spoil the accuracy, and that they are removable by modifications inspired by error analysis and perturbation theory.