To remove the chattering you can either reduce the amplitude of the switching function, incorporate Higher order SMC algorithm in your conventional SMC. The Higher order SMC really helpd in developing a much smoother controller in same SMC framework. You should really try that. There are some good works on that topic in the link below by Arie. Levant.    http://www.tau.ac.il/~levant/

Dear Muhammad;

Thank's for your helpfully comments. We would like to design second order SMC. Do you have any comments for second order SMC???

Yes i could help you with that.

Here is a good presentation that you could understand.

Here is the file

thank's for your comments.

You also can to follow my publications related to sliding mode controller.

Hello Dr.Piltan

How are you?

in the my mind,to achive that target. you have several choise.a) using adaptive gain b) using thershold adaptive c) using PSO optimization algurithm.d) using neural network.

i implementated tha sliliding mode observer based on RBF network, and see that the chattering is very small.

but becuase of that there is uncertainties, u do not use optimization. becuae in pso you must to reach to fixed point, but becuase of uncertainties, you can not to achive to fixed point. in the last perfer to use adaptive low and NN.

Dear Javad

Hi,

You said correctly, adaptive method is good method but it has some limitation such as timing and micro-based implementation. You said correctly I also writting many papers based on adaptive methodology but I'd like to find the other method. 

Hello again Dr.Piltan

in the one paper. i see the new method. that name is "dynamis sliding mode"

if i do not make mistake, in that method, use two surface sliding mode , that second surface sliding mode, used for elimination the chattering. i try to send that source to night. please give me time. it is very honor for me to speaking with you :) .

There is a good edited volume titled "Sliding Mode Control in Engineering" (Editor: Wilfrid Perruquetti) and published by Marcel Dekker (2002). The chapters on "Higher Order Sliding Mode Control" are very informative and related to this discussion.

Dear Vaidyanathan;

Thank's for your suggestion. Do you know about Terminal sliding mode controller to reduce or eliminate the chattering?

I think this method maybe can to solve this problem.

Tray  to replace the  relay ( the "sign function") by a saturation with high slope in the linear part.

Dear Prof

In this state we lost stability, is it correct?

The above solution has been proposed in several papers  but , of course, the controller is different. Chattering is caused by the implementation of singn(x) in a neighborhood about x=0. This problem  disappears with a  saturation since the control is linear in x=0 ( u=Kx=0 for x=0). For x>=(smal lpositive epsilon) the controls are  the same in both cases ( u=sign(x)). epsilon=1/k , k being the slope of the saturation at the linear part provided that the saturation saturates at M=1 to be coincident  with the sign function or epsilon=M/k if it saturates at a value M.

Concerning the stability, it is not guaranteed hat it is always preserved but it can be preserved in many cases. What is sometines called the equivalent control in the literature , which is an average control, can be very similar in both cases.

It is true that robustness can be lost  for small signal and  some kind  of oscillation about zero can appear . But real sliding-mode also has  fast switching about  zero  ( the chaterring) and very high- frequency modes as a result.

Higher order sliding mode control is a good option. You may apply reaching law based control (in reaching phase) to control the chattering effect.

The researchers have given very valuable feed back.

You can reduce chattering by using sat(.) in place of sign(.), but at the expense of robustness. In this case there will be a trade-off between smoothness of responce and the robustness of closed loop system.

For removing chattering theoretically you can use smooth sliding modes (e.g, that of stesshel). But in that case you have to use disturbance observer for maintaining robustness.

regards

Ali Arshad

Theoretically, the chattering effect is due to the presence of unmodeled dynamics in the systems. Since the model used to design the controller can never capture all the system dynamics, it is not possible to obtain a "chattering free" sliding mode control.

However, the chattering can be reduced. Due that a common problem is the controller design for systems with relative degree equal to one, the sliding mode is usually implemented with a relay represented for the sign function. A widely used alternative is the use of approximations to the sign function. Saturation and sigmoids functions are often used, offering a continuous or smooth control signal but losing the invariance property of the sliding mode control.

I guess there are two approaches to implement robust controllers with reduced chattering. Besides, they can be used in a complementary way. The first is the Integral Sliding Mode Control (ISMC) and, the second, the use of High Order Sliding Mode (HOSM) algorithms.

With ISMC, the control signal can be composed by a linear term to stabilize the system, and the relay signal passed through a filter, resulting in a continuous control signal with low excitation of the unmodeled dynamics. Of course, the ISMC concept offers plenty of additional design alternatives.

The use of HOSM is considered the best alternative to obtain continuous and "smooth enough" control signals. The so-called super-twisting algorithm is a common alternative to the use of the sign function. For higher relative degrees, there are higher-order algorithms as the nested sliding mode algorithms of the quasi-continuous sliding mode algorithms.

If you need a very smooth control signal, you can increase the relative degree of the designed sliding mode manifold with the use of a chain of k integrators an apply a higher-order method, obtaining a k-1 smooth control signal.

For the ISMC, I recommend the reading of the Vadim Utkin book on electromechanical systems and his 1996 paper. For HOSM, it is recommended to visit the Arie Levant and Leonid Fridman homepages. Both can be easily found in google with "Levant sliding mode" and "Fridman UNAM".

Please, do not hesitate to keep asking.

Regards!

Juan Diego

Dear Prof

Please read the following paper. We have tried to do the same by keeping both robustness and chattering alleviation of sliding mode controllers by replacing the discontinuous sign function by smooth continuous scalar sign function.

Singla, Mithun, Leang-San Shieh, Gangbing Song, Linbo Xie, and Yongpeng Zhang. "A new optimal sliding mode controller design using scalar sign function." ISA transactions, Vol. 53, Issue 2, March 2014, p 267-279.

Thanks

Mithun

Dear Juan Diego Sánchez Torres

Thank's for your complete answer. Your idea are good about two methods but I 'd like to find the other method. I think the main objective to control of nonlinear systems is design linear behavior controller, is it correct?

So if we design higher order SMC the controller behavior can be more nonlinear,  is it correct?

Regarding to above discussion we cannot use HSMC.

Please, don't hesitate to keep asking.

Regards

Farzin Piltan

The idea to impose a linear dynamics on a nonlinear system is usually proposed. However, for performance objectives, sometimes it is desired to impose some nonlinear dynamics for the closed loop system. Indeed, sometimes nonlinear behaviour is induced in linear systems.

Again, it is worth to note that the Integral Sliding Mode Control (ISMC) and the use of High Order Sliding Mode (HOSM) algorithms can be used in a complementary way. For example, you can proposed a integral sliding mode manifold and reach it with a HOSM algorithm.

I suggest, that if you want to impose a linear dyanamis on a perturbed system, you should try with ISMC. For this case, when the manifold is reached, the sliding mode dynamics will be given by the same system but without the matched disturbances, therefore you could use a feedback linearization controller (or any other that you want), for the non-disturbed system. Again, this integral sliding mode manifold can be implemented, for example, with a super-twisting. If well is a high order algorithm it does not requires on-line derivative calculations or any complicated implementation.

Juan Diego.

In 2011 I wrote a paper related to hybrid integral SMC,you said correctly it can reduce the chattering. As you know sliding surface slope  played an important role to reduce the chattering. I'd like to reduce it regarding to sliding surface slope role. 

If you are interested to reduce chattering  regarding to sliding surface slope role, you can read this paper:

"Interval type-2 fuzzy-neural network indirect adaptive

sliding mode control for an active suspension system",

Nonlinear Dynamics,

Majid Moradi Zirkohi · Tsung-Chih Lin

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...

with the linear methods we can answer to the

linearly treated signals and not the nonlinear conditions. In

the classical digital approach, that considers the affected

signal and the treating function linear in a reduced time or

sampling period the outcome is not relevant.

Even if we take in consideration the nonlinearity, we

should define a variable nonlinearity or a dynamic

nonlinearity.

In the digital and adaptive method, the static

nonlinearity can be treated if the nonlinearity function is

known.

Recovering the weak and lost sources in the Sea Acoustic. Available from: https://www.researchgate.net/publication/278771170_Recovering_the_weak_and_lost_sources_in_the_Sea_Acoustic [accessed Jul 06 2018]. Article Recovering the weak and lost sources in the Sea Acoustic

Recovering the weak and lost sources in the Sea Acoustic

Samad F. PAKZAD,

2

Our approach

The Intermodulation, is a Gaussian function in it’s

reduced simplest schematically study. The Correlation is a

multi-order function

The theoretical way of treating this kind of signal is:

•

To decompose it or de-correlate it with a time

variant complementary function,

•

To apply it to a complementary Gaussian function

that is variable in time.

With this method we have succeeded to recover the

signals lost under the noise, our improved SIGNAL TO

NOISE ratio is better than 30 dB for the very weak signals.

Refer to this article to see how we can remove non-linearity noise from the right signal.

Recovering the weak and lost sources in the Sea Acoustic. Available from: https://www.researchgate.net/publication/278771170_Recovering_the_weak_and_lost_sources_in_the_Sea_Acoustic [accessed Jul 06 2018].

Recovering the weak and lost sources in the Sea Acoustic. Available from: https://www.researchgate.net/publication/278771170_Recovering_the_weak_and_lost_sources_in_the_Sea_Acoustic [accessed Jul 06 2018].

Recovering the weak and lost sources in the Sea Acoustic

Samad F. PAKZAD,

It is IMPOSSIBLE to eliminate chattering if you would like to keep theoretically exact compenstation of matched uncertainties. You can just adjust the chattering. No more.

To adjust the chattering you should consider the main reason of chattering. The methods of chattering adjustment depend of types of non- idealities.

See:

Theory of Oscillators

Andronov, A. A.; Vitt, A. A.; Khaikin, S

Publicado por Addison-Wesley Publishing (1966)

ISBN 10: 0080099815ISBN 13: 9780080099811

One way to avios it, is using a transition zone between the modes. See: “Continuous sliding control applied to subsea oil and gas separation” as an exemple.

El calculo fracional.......

I do agree to prof Lenoid Fridman's reply. Chattering can be suppressed only. Alot of significant strategies have been proposed in the open literature.

I think explicit SMC is one option. Please consider the following: Article Chattering free full-order sliding-mode control

Dear Professor Qudrat Khan ,

One way to reduce the chattering effect is using higher-order Sliding mode control such as Using a Super twisting algorithm (STA) with Sliding Mode Control to eliminate the drawback of SMC while keeping its other parameters.

Once more.

1. Super-twisting cannot completely substitute discontinuous sliding mode controller because discontinuous SMC can compensate bounded uncertainties. They can be discontinuous.

Super-twisting is able compensate LIpschitz continuous uncertainties only.

2. The values of parameters characterizing non-idealities (actuator´s time constants) for which ST controllers adjusting the chattering better than discontinuous ones, are explicitly calculated in

https://www.researchgate.net/publication/327515503_When_is_it_reasonable_to_implement_the_discontinuous_sliding-mode_controllers_instead_of_the_continuous_ones_F

This means it is not correct to say that the super-twisting adjusts the chattering every time, see also:

https://www.researchgate.net/publication/282556270_Discussion_Aspects_of_High_Order_Sliding_Mode_Control

Prof Fridman, direct estimation and rejection of uncertainty can reomve the chattering completely. For instance, SMC based on disturbance observer (DO) or uncertainty and disturbance estimator (UDE).

>For instance, SMC based on disturbance observer (DO) or uncertainty and > >disturbance estimator (UDE).

It is not completely true.

1. When the input matrix is uncertain( it happens practically every time), it is imposible to compensate disturabces theoretically exactly.

2. In the case, when the input matrix is precesely known, to estimate the uncertainties it is necessary to estimate one derivative more w.r.t. to design an SMC. The difference in precisions could crucial in the presence of noises and system uncertainties.

See for all details:

Article Robust Control With Exact Uncertainties Compensation: With o...

See this paper: B. Xian, D.M. Dawson, M.S. de Queiroz, and J. Chen, “A Continuous Asymptotic Tracking Control Strategy for Uncertain Nonlinear Systems,” IEEE Transactions on Automatic Control, Vol. 49, No. 7, pp. 1206- 1211, July 2004. Has been named RISE by Dixon's group.

You cant eliminate shattering, but you can minimize it.

You cant eliminate shattering, but you can minimize it.

You cant eliminate shattering, but you can minimize it.

You cant eliminate shattering, but you can minimize it.

You cant eliminate shattering, but you can minimize it.

You cant eliminate shattering, but you can minimize it.

The paper [1] is explain the very basic fundamental for how to suppress chattering phenomenon. And various methods are given in the following reference as stated below

Reference:

[1] Utkin, Vadim. "Chattering problem." IFAC Proceedings Volumes 44, no. 1 (2011): 13374-13379.

Farzin Piltan

As the matter of fact chattering MUST not be eliminated because elimination of chattering means a trauma to the sense of SMC.

However, in control control engineering we always do tradeoffs between robustness and performance. Thus, the same is true for SMC also. An attempt to reduce chattering will compromise robustness.

To me HOSM is the good way of reducing chattering.

I agree with Prof. Fridman that the chattering of SMC cannot be eliminated completely. Many research works on the chattering free topic only did the minimizing, even it may approach nearly zero in practice.