hello Osama, have you investigated the "adaptive" control approach? For instance, consider x(t)=f(x(t))+g(x(t))u(t)+d(t)

where x(t) is the state vector, u(t) is the control input, f(x(t))) and g(x(t)) are known nonlinear functions, d(t) is the unknown disturbance. We consider We consider that the upper bound of the disturbance d(t) is unknown, but its time derivative d d(t)/dt ) is known. Hope this helps.

Hello Dieudonne,

Thanks for the prompt response, alongisde the adaptive approach, is there another approach that uses Lyapunov analysis?

Osama

When the upper bound of the disturbance is unknown, but its time derivative is known, this poses a more complex control problem. In such cases, several adaptive or robust control techniques can be used to compensate for the unknown disturbance. Here's how this scenario can be addressed:

1. Adaptive Sliding Mode Control (ASMC): Traditional sliding mode control relies on knowledge of the upper bound of the disturbance to design the control law. However, when this bound is unknown, adaptive strategies can be employed. In ASMC, the control gain is adjusted online based on the system's response, effectively "learning" the disturbance's characteristics without prior knowledge of its upper bound. Knowing the time derivative of the upper bound can be leveraged to refine this adaptation.

2. Backstepping with Adaptive Law: In backstepping control, when the upper bound of the disturbance is unknown, an adaptive law can be incorporated to estimate this bound in real time. The known derivative of the upper bound can be utilized to enhance the adaptation mechanism, leading to a better estimation and compensation for the disturbance.

3. Disturbance Observers: Another approach is using a disturbance observer. When the exact bound is unknown, the disturbance observer estimates the disturbance and its evolution over time. If the derivative of the disturbance's upper bound is known, this information can be used to improve the observer's performance, allowing for more accurate compensation.

4. Fractional-Order Control: In some cases, fractional-order control strategies can be applied to systems with unknown disturbances. The known derivative of the upper bound of the disturbance can be incorporated into the fractional-order controller design to improve robustness against such disturbances.

In all cases, the key is to use the known time derivative of the upper bound to guide the adaptation or estimation process, ensuring that the controller can react effectively to the unknown disturbance.

Thanks for the detailed and rich response, I appreciate it!

Could you provide me some references for the third method, papers on disturbance observers while using upper bound of the time-derivative of the unknown disturbance.

Thanks !

If you don't know the upper bound of a disturbance, you can either use a conservative estimate (e.g., 100 or 1000), but this may make your control signal unusable.

A better approach is to use adaptive control, such as adaptive sliding mode, adaptive backstepping, or other adaptive schemes, depending on the structure of your system.

This way, you don’t need to know the exact upper bound of the disturbance; you only assume that it is bounded and that its derivative is also bounded.

Additionally, there are methods like Yucelen’s Derivative-Free Adaptive Control, which eliminate the need to assume a bounded derivative.

Other methods includes observers to estimate the disturbances, such as ADRC control.

I hope this answers your question.