2 research outputs found
Structured decentralized control of positive systems with applications to combination drug therapy and leader selection in directed networks
We study a class of structured optimal control problems in which the main
diagonal of the dynamic matrix is a linear function of the design variable.
While such problems are in general challenging and nonconvex, for positive
systems we prove convexity of the and optimal control
formulations which allow for arbitrary convex constraints and regularization of
the control input. Moreover, we establish differentiability of the
norm when the graph associated with the dynamical generator is weakly connected
and develop a customized algorithm for computing the optimal solution even in
the absence of differentiability. We apply our results to the problems of
leader selection in directed consensus networks and combination drug therapy
for HIV treatment. In the context of leader selection, we address the
combinatorial challenge by deriving upper and lower bounds on optimal
performance. For combination drug therapy, we develop a customized subgradient
method for efficient treatment of diseases whose mutation patterns are not
connected.Comment: 11 pages, 7 figure
Geometric Programming for Optimal Positive Linear Systems
This paper studies the parameter tuning problem of positive linear systems
for optimizing their stability properties. We specifically show that, under
certain regularity assumptions on the parametrization, the problem of finding
the minimum-cost parameters that achieve a given requirement on a system norm
reduces to a \emph{geometric program}, which in turn can be exactly and
efficiently solved by convex optimization. The flexibility of geometric
programming allows the state, input, and output matrices of the system to
simultaneously depend on the parameters to be tuned. The class of system norms
under consideration includes the norm, norm, Hankel norm, and
Schatten -norm. Also, the parameter tuning problem for ensuring the robust
stability of the system under structural uncertainties is shown to be solved by
geometric programming. The proposed optimization framework is further extended
to delayed positive linear systems, where it is shown that the parameter
tunning problem jointly constrained by the exponential decay rate, the
-gain, and the -gain can be solved by convex
optimization. The assumption on the system parametrization is stated in terms
of posynomial functions, which form a broad class of functions and thus allow
us to deal with various interesting positive linear systems arising from, for
example, dynamical buffer networks and epidemic spreading processes. We present
numerical examples to illustrate the effectiveness of the proposed optimization
framework