Ross' lemma, named after
I. Michael Ross,
is a result in computational
optimal control. Based on generating
Carathéodory- solutions for
feedback control
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...
, Ross' -lemma states that there is fundamental
time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a s ...
within which a control solution must be computed for
controllability Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.
Controllability and observabi ...
and
stability
Stability may refer to:
Mathematics
* Stability theory, the study of the stability of solutions to differential equations and dynamical systems
** Asymptotic stability
** Linear stability
** Lyapunov stability
** Orbital stability
** Structural st ...
. This time constant, known as Ross' time constant,
is proportional to the inverse of the
Lipschitz constant of the
vector field that governs the dynamics of a
nonlinear control system.
Theoretical implications
The proportionality factor in the definition of Ross' time constant is dependent upon the magnitude of the disturbance on the plant and the specifications for feedback control. When there are no disturbances, Ross' -lemma shows that the open-loop optimal solution is the same as the closed-loop one. In the presence of disturbances, the proportionality factor can be written in terms of the
Lambert W-function.
Practical applications
In practical applications, Ross' time constant can be found by numerical experimentation using
DIDO
Dido ( ; , ), also known as Elissa ( , ), was the legendary founder and first queen of the Phoenician city-state of Carthage (located in modern Tunisia), in 814 BC.
In most accounts, she was the queen of the Phoenician city-state of Tyre (t ...
. Ross ''et al'' showed that this time constant is connected to the practical implementation of a Caratheodory- solution.
That is, Ross ''et al'' showed that if feedback solutions are obtained by
zero-order hold
The zero-order hold (ZOH) is a mathematical model of the practical signal reconstruction done by a conventional digital-to-analog converter (DAC). That is, it describes the effect of converting a discrete-time signal to a continuous-time signa ...
s only, then a significantly faster
sampling rate
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples".
A sample is a value of the signal at a point in time and/or sp ...
is needed to achieve controllability and stability. On the other hand, if a feedback solution is implemented by way of a Caratheodory- technique, then a larger sampling rate can be accommodated. This implies that the computational burden on generating feedback solutions is significantly less than the standard implementations. These concepts have been used to generate collision-avoidance maneuvers in
robotics
Robotics is an interdisciplinarity, interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist human ...
in the presence of uncertain and incomplete information of the static and dynamic obstacles.
See also
*
Ross–Fahroo lemma
*
Ross–Fahroo pseudospectral method
References
{{DEFAULTSORT:Pseudospectral Optimal Control
Numerical analysis
Control theory
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