First-order sliding mode controllers provide robustness to measurable bounded, not necessarily differentiable, disturbances, but they are discontinuous.
In contrast, absolutely continuous high-order sliding mode controllers require (weakely) differentiability, or Lipschitz condition, of disturbances.
How to induce a sliding mode even in presence of non-differentiable Hölder disturbances with a continuous controller?
there is more type of sliding mode such as (fast terminal sliding mode) that the control law is continuous and appliciable for non-differniable disturbances. in addittion by using atan function instead of sgn for solving dis continuous problem
The hyperbolic tangent function (tanhks) can be used in this case as the smooth approximation of the discontinuous signum function (sign(s)). Stability properties of the system with the smooth implementation can be analyzed and the closed-loop system can be shown to be globally uniformly ultimately bounded with respect to a compact set around the origin. This set can be made arbitrarily small by increasing the slope k of the tanh(k s) function at s=0.
Thanks to Prof. Keshavarz and Reyhanoglu for your valuable answers.
One one hand, terminal sliding modes do not provide a sliding motion in presence of a wide variety of non-differentiable disturbances, just against disturbances that are vanishing with respect to a power law of the sliding variable (a fantastic property for unknown disturbances).
On the other hand, by any approximation of the discontinuous signum function, the establishment of an exact sliding motion in finite-time cannot be guaranteed.
Let me simplify the problem for the sake of clarity: consider the 1-D system
$$
\dot x = u + \phi,
$$
where u is the control input and \phi is a non-differentiable and unknown disturbance, not restricted to be vanishing wrt a power exponent of |x|, a whimsical assumption. Then, the objective is to enforce EXACTLY the invariant $x = 0$ in finite-time by means of a continuous $u$.
dear A.J. Muñoz-Vazquez
1-there are many references that they use fast terminal sliding mode for non vanish disturbances. in addition, in the fast terminal sliding mode, the control law use a nonlinear function of sliding surface instead of sgn(s) function in and it is finite time.
2- you can use some reduced order observer for detecting the upper bound of disturbances then use chain sliding mode for un matched disturbances.
3- in some paper designed a control law for victory of disturbance effect and state variables moved in a safe region after that for achieving finite time stability they designed terminal sliding mode controller ,
good luck
It is clear that topological properties of sliding mode controllers need to be intrinsically related with such of disturbances. If such conditions are not met, you cannot demonstrate mathematically the existence of a finite-time sliding motion.
The goal of this question is to open discussion in such a topic. Can you cite some reference of FTSM for non-vanishing disturbances?
One answer to this question can be found in
A.J. Muñoz-Vázquez, V. Parra-Vega and A. Sáncez-Orta (2015), Continuous fractional sliding mode-like control for exact rejection of non-differentiable Hölder disturbances, IMA Journal of Mathematical Control and Information. DOI: 10.1093/imamci/dnv064
It's not possible to have a continuous control with discontinuous uncertainty.
However for continuous control, pls find below my approach for Hiher order supertwinsting, Below paper under review.
https://www.researchgate.net/publication/280590093_Higher_order_supertwisting_algorithm_for_perturbed_chains_of_integrators_of_arbitrary_order
Article $L_p$-Stabilization of Integrator Chains Subject to Input Sa...
Article Higher order supertwisting algorithm for perturbed chains of...
Prof. Mohamed Harmouche you are right, good answer.
However, by using fractional order operators, the fractional integral of the signunm function provides a continuous control that is robust to non-differentiable (but Hölder continuous, nor discontinuous) disturbances, you can look at my work.
Hölder continuous are in middle of the way of discontinuous and differentiable disturbances, and no continuous integer order controllers are available for solving this problem.
Thank for your valuable answer, with regards
The questions is not stated correctly. The correct answer is: if you want to have the convergence in the infinity norm, you need to have the controllers with the same or worst smoothness properties as as the perturbation. If the perturbations are Lipshitz soper-twisting is OK. If they are Holder you need fractional degrees and so on. If the perturbations are discontinuous only first order sliding mode can help.
Thanks Prof. Fridman
As a matter of fact that is the correct answer to my question. Thanks for your valuable opinion. Accordingly, we proceeded in our previous works to solve this issue in such a way.
With regards, sincerely yours
Aldo-Jonathan Muñoz-Vázquez
Please refer to these articles:
https://www.researchgate.net/publication/313576646_From_PID_to_Nonlinear_State_Error_Feedback_Controller
https://www.researchgate.net/publication/312046377_Improved_Sliding_Mode_Nonlinear_Extended_State_Observer_based_Active_Disturbance_Rejection_Control_for_Uncertain_Systems_with_Unknown_Total_Disturbance
https://www.researchgate.net/publication/309655805_On_the_Improved_Nonlinear_Tracking_Differentiator_based_Nonlinear_PID_Controller_Design
Article From PID to Nonlinear State Error Feedback Controller
Article Improved Sliding Mode Nonlinear Extended State Observer base...
Article On the Improved Nonlinear Tracking Differentiator based Nonl...
Well, pleased to see the fractional order idea is used for additional benefit. It will be nice to do a fair comparison so that, it could be claimed that, using fractional order control strategy is the only pathway to do good tracking. Is this true?
Dear Profr YQ Chen,
Yes, that is the point, using fractional order is in some sense the unique path. For instance if we have the system
\dot x = u + d,
and we just know that d is a-Holder continuous function, with 0
If the perturbation is discontinuous like dry friction it is necessary to take into account that if the motions are not changing the direction frequently and there exsits strictly positive dwell time, when the motions are not changing directions, the super-twisting controller can reconverge. Morever, in the presence of the relay sliding mode controllers also need time to reconverge. So that is why to apply STA or not apply depends on actuators time constantsArticle When is it reasonable to implement the discontinuous sliding...
.
Yes Prof. Fridman, I agree. In addition it depends of the (topology) measure space, in the usual topology, as you mention, it depends on the rate of such direction changing.
Furthermore, in the case of fractional controllers , a further study is still needed to verify that a fractional-integral based controller provides better properties in the cases of high or slow actuator dynamics.
Inspired in your recent contributions, I am really interested in conducting the frequency analysis based on descriptive functions and harmonic balance.
- Badges
- Science topic
- Similar topics
- Engineering
- Electrical Engineering