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 positive
linear time-invariant system
In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined ...
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 δ ...
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 se ...
, 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 In control theory, a state observer or state estimator is a system that provides an estimate of the internal state of a given real system, from measurements of the input and output of the real system. It is typically computer-implemented, and pro ...
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 In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero):
: \forall_\, x_ \geq 0.
It is named after the American economist Lloyd Metzler.
Metzler matrices appear in sta ...
.
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 gr ...
.
See also
*
Metzler matrix In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero):
: \forall_\, x_ \geq 0.
It is named after the American economist Lloyd Metzler.
Metzler matrices appear in sta ...
*
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 gr ...
*
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 the ...
References
{{Reflist
Control theory
Systems theory