##### Department of Mathematics,

University of California San Diego

****************************

### Math 278 - Numerical Analysis Seminar

## Emre Mengi

#### UCSD

## Measuring degree of controllability of a linear dynamical system

##### Abstract:

A linear time-invariant dynamical system is
controllable if its trajectory can be adjusted
to pass through any pair of points by the proper selection of an
input.
Controllability can
be equivalently characterized as a rank
problem and therefore cannot be verified
reliably numerically in finite precision.
To measure the degree of controllability of a system
the *distance to uncontrollability* is introduced as the
spectral or
Frobenius norm of the
smallest perturbation yielding an uncontrollable system.
For a first order system we present a polynomial time
algorithm to find the nearest uncontrollable system
that improves the computational costs of the previous techniques.
The algorithm locates the global
minimum of a singular value optimization problem
equivalent to the distance to uncontrollability.
In the second part for higher-order systems we derive a singular-value
characterization and exploit
this characterization for the computation of the higher-order distance
to
+uncontrollability to low
precision.

### November 21, 2006

### 10:00 AM

### AP&M 7321

****************************