Positive systems

Positive systems^[1][2] constitute a class of systems that has the important property that its state variables are never negative, given a positive initial state. These systems appear frequently in practical applications,^[3][4] as these variables represent physical quantities, with positive sign (levels, heights, concentrations, etc.).

The fact that a system is positive has important implications in the control system design.^[5] For instance, an asymptotically stable positive linear time-invariant system always admits a diagonal quadratic Lyapunov function, which makes these systems more numerical tractable in the context of Lyapunov analysis.^[6]

It is also important to take this positivity into account for state observer design, as standard observers (for example Luenberger observers) might give illogical negative values.^[7]

Conditions for positivity

A continuous-time linear system ${\displaystyle {\dot {x}}=Ax}$ is positive if and only if A is a Metzler matrix.^[1]

A discrete-time linear system ${\displaystyle x(k+1)=Ax(k)}$ is positive if and only if A is a nonnegative matrix.^[1]

See also

• Metzler matrix
• Nonnegative matrix
• Positive feedback