Research · 04 / 06

Positive systems

Systems whose state is confined to the nonnegative orthant and which map nonnegative inputs to nonnegative outputs. They appear naturally wherever physical variables are themselves positive.

Linear positive systems

Many physical systems naturally involve positive variables, so representing them as positive dynamical systems is natural. Interesting examples include biological systems, epidemiological systems, and communication networks. Linear positive systems take the form

$$ \dot{x}(t) = A\, x(t), \quad A\ \mathrm{Metzler}, \quad x_0 \geq 0 $$

and have remarkable properties. Their stability is exactly characterized by sum-separable Lyapunov functions of the form

$$ V_\ell(x) = \sum_{i=1}^n v_i x_i, \quad v_i > 0 $$

or

$$ V_q(x) = \sum_{i=1}^n d_i x_i^2 = x^T D x, \quad D = \mathrm{diag}(d_i) > 0. $$

The linear form leads to stability conditions cast as a linear program. The quadratic form leads to a semidefinite program, simpler than the usual one for general linear systems thanks to the diagonal structure of $D$. Such Lyapunov functions also enable structured state-feedback design in a nonconservative way, a problem whose general instances are known to be NP-hard. Recently, $L_1$ and $L_\infty$ gains of linear positive systems have been shown to be computable exactly via linear programming, in sharp contrast with the poor tractability of $L_\infty$ for general linear systems.

Positive systems with delays

Linear positive systems with delays take the form

$$ \dot{x}(t) = A\, x(t) + A_h\, x(t - h), \qquad x(s) = \phi(s) \geq 0,\ s \in [-h, 0] $$

where $A$ is Metzler and $A_h$ is nonnegative. As in the undelayed case, the structure leads to tractable analysis: the system is stable for any positive delay $h$ if and only if the matrix $A + A_h$ is Hurwitz.

Related publications

For the full list, see the publications page (search for "positive" or "Metzler").