There are 4 issues I have seen in this discussion:

1. Backstepping can compensate theoretically exactly only uncertainties decreasing together with the state variables. No comparison between backstepping and SMC in this sense.

2. Matched and unmatched uncertainties

For perturbed chain of integrators there are no unmatched perturbations. If you will differentiate the output till the system order all the perturbations will be matched! That is why the only reasonable to control the systems with unmatched uncertainties is a combination between backstepping and SMC, where backstepping compensates state dependent part uncertainties and SMC(see Estrada et al IJRNC,27(4),2017,DOI : 10.1002/rnc.3590,Automatica 2010(11), TAC 2010(11), J. Davila TAC 2013, Ferreira et al Journal of Franklin Institute, 2014, 351(4),doi:/10.1016/j.jfranklin.2013.12.011, Automatica, September 2015, Vol. 59, 10.1016/j.automatica.2015.06.020)

will compensate the unvanished disturbances.

3. SMC can not be smooth. Sliding mode (by definition!) is a motion on the sliding surface!

4. Chattering

4.a. There is no way to keep the sliding mode and eliminate the chattering.

4.b. It is wrong opinion that continuous HOSM controllers (like super-twisting,like other super-twisting like continuous algorithms HOSM Moreno DOI:10.1007/978-3-319-62464-8_8, Kamal et al DOI:10.1016/j.automatica.2016.02.001, Torres et al doi :10.1016/j.automatica.2017.02.035, Laghrouche et al TAC,2017) everytime produce less chattering that discontinuous ones (see https://www.researchgate.net/publication/317229996_Is_It_Reasonable_to_Substitute_Discontinuous_SMC_by_Continuous_HOSMC).

It is true for the systems with fast actuators only.

This is basically a very broad question, in the sense that it very much depends on your system, i.e., the actuator type, the dynamic model, nonlinearities and uncertainties.

For instance, if your system does not have any uncertainties, implementation of the SMC and Backstepping will have similar results. In case that you have negligible nonlinearity level, neglecting the nonlinear dynamics and using SMC is recommended. 

SMC gives you more freedom to tune the parameters to obtain the result you want, using the system gain and the sliding surface that you can design. But using the SMC you have to be careful about the chattering that may occur, if you are going to test your system experimentally. If your multi-degrees of freedom system is independent in its degrees, you will have N independent systems. If not and you are facing a MIMO problem, there are multiple works out there to design a MIMO SMC or Backstepping controller.

To effectively evaluate the performance of the sliding mode based control and backstepping control, we must identify the fundamental differences between them:

1) Using sliding mode or sliding surface is an effective way to transform a relatively high order system to a lower order system (with respect to the introduced sliding mode). This greatly facilitates the controller design since we have much experience in designing control and analyzing stability for low-order systems. From an analysis perspective, the relation between the sliding mode and the original variables of the system must be clarified before we can derive the system stability, e.g., input-output analysis is often adopted for this purpose which involves stability concepts such as BIBO and ISS. 

2) Backstepping control is known as a construction approach in the sense that it has a systematic way of constructing the Lyapunov function along with the control input design. To ensure stability or the negativeness of the derivative of the every-step Lyapunov function, it usually requires the cancellation of the indefinite cross-coupling terms, which may be the most prominent difference between backstepping control and sliding mode based control. While this cancellation results in the perfect-looking of the derivative of the Lyapunov function, it does not necessarily mean that good performance is ensured.

Some researchers have ever said that the (unnecessary)  cross-coupling cancellation of the backstepping approach will give rise to relatively high control activity and possibly bad robustness while this usually does not happen in sliding mode based control (based on input-output analysis). Therefore, I would like the sliding mode method.

Dear Hanlei Wang;

Thank's for your perfect response. I also like SMC, so you think can we writing a comparison study paper between SMC and backstepping??

Regards

To Farzin Piltan:

     I think it would be interesting if a comparison between SMC and backstepping approaches is given by writing a journal paper.  It seems that there are not such studies in the literature.

Best

I think that a discontinous  method link SMC compared with a continuous method link Backstepping, requires to precise the criterion for comparison. Also, it is needed to specify the class of non-linear systems, the control objective and the control configuration. One possibility was suggested by Hanlei Wag in which the objective is to reduce the order of a class of non-linear systems in that case both techniques are applicable, arriving to a reduced system which is stabilized by another method.

SMC, as remarked by Rene Galindo and many other researchers, is commonly related to the so-called discontinuous control. However, in my opinion, the nature of SMC itself does not tell us that it is bound to be discontinuous, e.g., sliding-mode based adaptive control is not only continuous but also smooth. The reason for this common recognition may be that SMC appears so frequently in variable-structure robust control literature.

From the name of "SMC", it is preferable to be referred to as a methodology for transforming complex systems to simple systems (like backstepping).

You are right Hanlei Wang, not all the SMC are discontinuous. You mention adaptive sliding-mode control, and another one is integral sliding mode. So, the comparison must be with such controls.   

SMC is the best robust controller. But how to solve a chattering problem in SMC? Is there any method to solve this problem? The problem occurred because of the high - frequency switching of a SMC

Dear Prof

Yes, you said correctly about SMC and to reduce the chattering with regards to improve the stability and robustness several methods discussed in the literature and you also can to follow my papers as the following link:

http://iranssp.com/english/?page_id=36

high order sliding mode controllers reduce chattering, however, a compromise with the sampling period or with the integration step is introduced.  

Dear Rene

You said correctly, higher order SMC can reduce the chattering but one of the main objectives to design any controller is design a linear behavior for nonlinear system. Therefore your solution can create a new challenge. Is it correct???

Regards

This may provide further insight:

An approach proposed by professor K.J. Hedrick (Universiy of Berkeley Calfornia, Department of Mechanical Engineering) and his co-workers about mid-1990, called “multi-surface” sliding mode control. It mainly consists to build sliding surfaces step by step for systems under strict feedback (sort of triangular) form. This approach has then been modified by the introduction of filters under the name of “Dynamic surface control” which is very similar to back stepping approach. This latter approach has the advantage of avoiding the “term explosion” a problem well known in backstepping.

For controlling multi DOF of actuators you can try DSP based processors. We have used similar type of situation and I recommend you to try TMS320F28335. How many actuators are you thinking to control?

There is no conflict between the two methods. Sliding mode methods consist of the enforcing of the motion on a proposed manifold. Thus, two problems are derived: (i) which manifold (ii) how to reach the manifold. Some times, this manifold has relative degree higher than one, for those cases backstepping or block control methods are highly useful. See the Loukianov,s papers in this subject.

The nature of these methods are different.

Sliding mode is proper for systems which satisfy matching condition.

But Backstepping relaxes the matching condition. Or it is applicable for systems with non-companion form or systems with unmatched perturbations. 

May I add some details about chattering ? We have shown in the past recent years, that the so-called digital chattering on both the input and the output, that is created solely by the time-discretization (see the works by Z. Galias and co-workers), can be suppressed if a correct method is empoyed, instead of the classical explicit Euler discretized controller. Instead, an implicit discretized input has to be used. This is developed in several articles (Systems and Control Letters 2010, IEEE TAC 2012, 2016, 2017, CEP 2016, IEEE TCST 2014, SIAM Cont. Opt. 2018 etc) with experimental validations. As fas as first-order SMC with co-dimension one sliding surface is concerned, the implicit controller is easy to implement, since it consists of a projection of a known, scalar quantity, on the interval [-1,+1]. The only thing you should be very careful about, is the calculation of the sliding variable \sigma, which is a function of the plant's state. Usually in most applications, \sigma depends on the measured output, and on its derivatives. The derivatives may be obtained by differentiation. Filters come into the play and have to be correctly tuned, for otherwise the whole SMC design may collapse (with both explicit or implicit methods...). The implicit discretization method has been shown to preserve the global Lyapunov stability, finite-time convergence, robustness properties, and most importantly it allows to give a rigorous and precise meaning to the notion of discrete-time sliding surface (no need for so-called quasi-sliding surface). When the system is inside the sliding surface, then it becomes insensitive to the choice of the gain (you can increase the gain without changing the controller magnitude and chattering, which is something impossible to do with an explicit discretized input). Moreover we have checked experimentally and numerically, that large sampling periods can be chosen for the implicit method, without deteriorating too much the closed-loop behaviour. Again this is not at all the case with the explicit discretization. As a last comment, it is noteworthy that the implicit method keeps the original continuous-time structure, without adding any parameter or dynamics. Finally I would like to end by noting that contrarily to what is announced in most papers, adding a saturation is not the miracle cure to chattering. It is more like a dressing on an unstoppable bleeding...

If I understand well Bernard Brogliato answer, chattering depends mainly on the discretization method: Euler, implicit discretized input, Tustin, exact discretization for linear systems, etc. Also, how much information does we need to know about \sigma on the SMC with implicit discretized input?, it is like a measured or know disturbance?.

Chattering may have other sources, but the discretization method is one of them, and it seems to be an important one (according to my own experience and the experimental results we reported in several publications). Concerning the information needed about \sigma: this is not an easy question. I have no precise answer to give. The main weakness of SMC, I believe, is that the plant's state is needed to construct \sigma. Only the super-twisting algorithm relaxes this. In our Control Engineering Practice paper (Huber et al, CEP, vol.46,pp.129-141, 2016), a method for tuning the "dirty" filters that are used to differentiate the output signal, is described. All I can say is that we had to go through such a filter tuning process to obtain (very) good results. What I can say also, is that if you apply an explicit method (either with Euler or with ZOH discretizations), then you will obtain always chattering at both the ouput \sigma, and at the control input (bang-bang controller). Adding a saturation, as I said above, may cure the desease, but this is not automatic at all, and may require a long tuning process to get satisfactory results (with, usually, a very small sampling time). You may have a look at section 3.2.4 in Huber et al, "Experimental results on implicit and explicit time-discretizaton...", in Recent Trends in Sliding Mode Control, IET Control, Robotics and Sensors Series, vol.102, pp.207-233, 2016, for more details. Of course all this will strongly depend on your particular application, and all its features (dimension of the plant, co-dimension of the sliding surface, quality of the output measure, type of actuators, etc). It seems difficult to say more.

There are 4 issues I have seen in this discussion:

1. Backstepping can compensate theoretically exactly only uncertainties decreasing together with the state variables. No comparison between backstepping and SMC in this sense.

2. Matched and unmatched uncertainties

For perturbed chain of integrators there are no unmatched perturbations. If you will differentiate the output till the system order all the perturbations will be matched! That is why the only reasonable to control the systems with unmatched uncertainties is a combination between backstepping and SMC, where backstepping compensates state dependent part uncertainties and SMC(see Estrada et al IJRNC,27(4),2017,DOI : 10.1002/rnc.3590,Automatica 2010(11), TAC 2010(11), J. Davila TAC 2013, Ferreira et al Journal of Franklin Institute, 2014, 351(4),doi:/10.1016/j.jfranklin.2013.12.011, Automatica, September 2015, Vol. 59, 10.1016/j.automatica.2015.06.020)

will compensate the unvanished disturbances.

3. SMC can not be smooth. Sliding mode (by definition!) is a motion on the sliding surface!

4. Chattering

4.a. There is no way to keep the sliding mode and eliminate the chattering.

4.b. It is wrong opinion that continuous HOSM controllers (like super-twisting,like other super-twisting like continuous algorithms HOSM Moreno DOI:10.1007/978-3-319-62464-8_8, Kamal et al DOI:10.1016/j.automatica.2016.02.001, Torres et al doi :10.1016/j.automatica.2017.02.035, Laghrouche et al TAC,2017) everytime produce less chattering that discontinuous ones (see https://www.researchgate.net/publication/317229996_Is_It_Reasonable_to_Substitute_Discontinuous_SMC_by_Continuous_HOSMC).

It is true for the systems with fast actuators only.

However, it is interesting to note that SMC can be completely insensible to nonparametric matched perturbations (at least ideally), while backstepping can compensate the state dependent part (parametric uncertainties) only for adaptive backstepping where the parametric uncertainties enter linearly in the state (or am I wrong?). Also, it seems for me that for a general "nonparametric unmatched" perturbation neither SMC or BS could effectively work for compensation.

Yes you are wrong. If unmatched perturbations are differentiable they becames to be matched after differentiation, i.e. they can be compensated with SMC

Very interesting Prof. Fridman! I will read your papers. When you say "backstepping compensates state dependent part uncertainties and SMC will compensate the unvanished disturbances." I am curious about how HOSMC compensate unvanished unmatched disturbances, because regular (1st order SMC can't do that). Suppose a chain of 4 or 5 integrators with a senoidal perturbation in the first one. Due to the derivatives, this senoidal signal appears extremely amplified in the last integrator (with the control input). And a regular 1st order SMC can't exactly compensate the perturbation based on its derivative, even for large switching gains. I am curious about how HOSMC do that. I will read the papers.

2 more recently published journal articles showing that continuous HOSM controllers (like super-twisting) NOT EVERYTIME producing less chateering and conditions when it happens are here

Article Design of Super-Twisting control gains: a Describing Functio...

Article When is it reasonable to implement the discontinuous sliding...

Please share me the best answer might you get...

Some new perspectives have been added concerning this topic, and in particular, the definition of SMC is emphasized (for instance, its discontinuous nature). In my opinion, any concept might not be kept to be absolutely invariant without taking into account the development of the relevant fields. Though sliding mode is perhaps introduced in the context of discontinuous control, its power has been expanded over the past decades and in many new fields, for instance, consensus and synchronization of nonlinear uncertain systems.

I personally think that sliding mode itself might at best be distinguished or separated from the so-called discontinuity (which is tightly associated with the introduction of SMC, though). This is based on an important fact that the most fundamental role of a concept is to help us understand and control (physical) systems in a better way. The concept itself, as the time goes, may also need to be improved or expanded gradually.

They are not mutually exclusive ...