Positive systems
[T. Kaczorek. Positive 1D and 2D Systems. Springer-
Verlag, 2002] 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, 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. For instance, an
asymptotically stable
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. T ...
positive
linear time-invariant system always admits a
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δΠ...
quadratic
Lyapunov function
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s s ...
, which makes these systems more numerical tractable in the context of Lyapunov analysis.
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.
[http://advantech.gr/med07/papers/T19-027-598.pdf ]
Conditions for positivity
A continuous-time linear system
is positive if and only if A is a
Metzler matrix.
A discrete-time linear system
is positive if and only if A is a
nonnegative matrix
In mathematics, a nonnegative matrix, written
: \mathbf \geq 0,
is a matrix in which all the elements are equal to or greater than zero, that is,
: x_ \geq 0\qquad \forall .
A positive matrix is a matrix in which all the elements are strictly gre ...
.
See also
*
Metzler matrix
*
Nonnegative matrix
In mathematics, a nonnegative matrix, written
: \mathbf \geq 0,
is a matrix in which all the elements are equal to or greater than zero, that is,
: x_ \geq 0\qquad \forall .
A positive matrix is a matrix in which all the elements are strictly gre ...
*
Positive feedback
Positive feedback (exacerbating feedback, self-reinforcing feedback) is a process that occurs in a feedback loop which exacerbates the effects of a small disturbance. That is, the effects of a perturbation on a system include an increase in th ...
References
{{Reflist
Control theory
Systems theory