Feedback control via ODE variable coefficients?

[forkandwait]

If one has a simple variable coefficient process like y'(t) = r(t)*y(t), is there a way to control it to a set point by feedback hitting the variable coefficients in r(t)?

I am interested in feedback control of population processes. y'(t) = r*y(t) is simple proportional growth with a constant growth term. I assume one would control this process by "mortality" -- reducing y(t) - u(t) when necessary to keep it under the setpoint. (A delay ODE is the next step...)

I have not seen any reference to adjusting variable coefficients in the controls books I have looked at. I wonder if it is possible to somehow make r(t) a "real" ODE variable so that the conventional models work?

I don't really know what I am talking about here (a lower division ODE class, plus lots of scattered reading is my only background), so feel free to answer appropriately.

Thanks!

 

[MisterX]

You might want to look at logistic growth, which can be used to model population growth with a so-called carrying capacity.

Logistic function on WikipediaThe ODE you posted is separable and has a solution family
$y = Ce^{\int_0^t r(t')dt'}$
This may not serve the purpose you have in mind very well I think.

 

[bikengr]

how to

If one has a simple variable coefficient process like y'(t) = r(t)*y(t), is there a way to control it to a set point by feedback hitting the variable coefficients in r(t)?

Seems straightforward to me. You make the function r depend on y, for example r= -K*(y- y0) where y0 is the desired population. This makes the derivative of y negative if y is above y0, and positive if it is below, so it shoots stably to the value y0. You can embellish this all kinds of ways to get faster convergence, interesting oscillations, etc.

The main point here is that r is not strictly a function of t only, but it depends on y and the desired population goal y0.

 

[MisterX]

I don't think that's quite right. Remember r(t) gets multiplied by y in the equation specified.


Anyway, perhaps the simplest system would be the differential equation which is associated with Newton's law of cooling.

$y' = -K(y - y_0)$

The set point would be $y_0$. When $y_0$ is constant, y will exponentially decay to the set point, y_0.

$y = y_0 + Ce^{-Kt}$

A block diagram is shown below. Note that this simple system may not work well if you want to hit a moving target (y_0 not constant).

 

[alphy]

I appreciate your interest in feedback control of population processes and your curiosity about using variable coefficients in the control process. While it is not a commonly discussed topic in control theory, there are some approaches that do consider variable coefficients in feedback control.

One approach is to use adaptive control, where the control parameters are adjusted based on the changing coefficients of the system. This requires a model of the system and knowledge of the coefficients, which may not always be available in real-world scenarios.

Another approach is to use model predictive control, where the control inputs are optimized over a finite time horizon based on a model of the system and its variable coefficients. This can be a more robust approach as it does not require knowledge of the exact coefficients, but it may be computationally intensive.

In terms of your specific example of controlling y'(t) = r(t)*y(t), one could potentially use a combination of both approaches. For example, adaptive control could be used to adjust the control inputs based on the changing growth rate, while model predictive control could be used to optimize the control inputs over a finite time horizon.

However, as you mentioned, incorporating a delay ODE into the system would add another level of complexity and may require more advanced control techniques. It would also be important to consider the limitations and uncertainties in the model and the potential effects of the control inputs on the system.

Overall, while incorporating variable coefficients in feedback control can be a challenging task, there are some approaches that can be used. It would be important to carefully consider the specific system and its dynamics before determining the most appropriate control strategy.