Dear Dr. Piltan,

if you follow my page, try and see the following it will help you:

Article: variable structure system with sliding mode controller

Ali. A. Ahmed, R.B Ahmad, AbidYahya, Hazim H.Tahir, James Quinlan

and try and see the rest ( regarding your question)

Best Regards,

Hazim H. Tahir

Dear Dr

Thank's for your comments

There are many ways to increase the stability of smc systems. However, the method to improve stability should be selected based on the stability challenges and problems encountered. For instance, if you only aim to enhance stability you can use variation of conventional SMC such integral, boundary layer, adoptive etc. The complexity brought to the system by such variation could degrade the performance so it is important to understand how much stability is satisfactory for the system. If the conventional SMC does not provide a stable response for the system during the intended performance time and singularities occur, you can change the sliding surface to TSMC (terminal) or NTSMC(Non-singular) to obtain your desired performance with acceptable stability.

Dear Samira

You said correctly.

As you know, SMC is a robust and stable controller but conventional SMC has chattering problems. To solve this challenge some researchers introduced some methods but some of these method can reduce or eliminate the stability. whats your idea???

the methods that are usually used to address the chattering phenomenon can address the stability problems if the switching gain and initial conditions are chosen properly. However, they dont guarantee a stable performance for any finite time. As I said earlier the stability challenges and corresponding solutions should be addressed based on applications. For instance, a system with short performance time may be very well controlled using a conventional sliding mode controller. The variation of conventional SMC can be used for meditating chattering and enhancing the stability (given the gain and initial conditions are set properly). However, for a system that should operate for long periods of time (like a satellite that should complete at least an orbit (exp: 6000s) ) a modified conventional SMC is not enough.

Dear Samira

Thank's for your comments. for example if you want to design conventional chattering free SMC, what is your idea? 

Regards

I suggest a boundary layer SMC with sgn function replaced by a saturation or even tan function. If that is not enough, integral augmented SMC is a simple yet effective solution. If the chattering is persistent, you can try adoptive that is more complex and adds burden to the computational procedure. I suggest taking a step at a time and start by basic chattering solutions as mentioned above. Take note that gain tuning is an important stage. After the chattering meditation method has been applied, a gain tuning procedure is essential. Sometime you will achieve the desired performance by changing the gain by trial and error. Fuzzy solutions can come in handy if you prefer an automatic gain tuning. In addition, take note that choosing proper initial condition can significantly influence your result.

Excellent replies by Samira Eshghi & special thanks to Farzin Piltan for initiating good research discussions regularly.

Best wishes

Sundar

Sliding mode control is inherently robust to matched disturbances which means that the controller inherently compensates for matching disturbance.

In general given a system d^nx/dt^n = f(x,t)+D-u where n is the relative degree and D a disturbance a sliding mode controller is a convergence function C(x, xDot, ...d^n-1 x/dt^n-1) function so that:

d^n/dt^n +  C(x, xDot, ...d^n-1 x/dt^n-1 = f(x,t) +D =f(x,t)+D converges to the origin no matter what the RHS term is and provided that there is enough control authority.

Going to extremes you can consider f(x,t)+D as a disturbance you may in which case the controller implicitly compensates for f(x,t)+D.

You may also compensate in full or at least partially for estimated value of f(x,t).

If you do so you may have a less strained controller i.e. gains smaller because in this case the sliding mode controller only has to compensate for f(x,t)-Estimated(f,x,t) +D.

In this case you need to use either an algebraic representation of f(x,t) or eventually an estimators.

The good news is that you do not need a full and exact compensation of f(x,). Since the SMC will implicitly compensate for what you are not compensating explicitly, so you do not need to go though excruciating pain to exactly compensate for f(x,t).

Dear All

There is no measure to say "increase stability". 

Once a system is being stabilized (whether linear or non linear), the goal is to incorporate robustness and to enhance robustness.

Unfortunate for control theoriticians and practitioners that the phenominon known as THE WATER BED EFFECT is always accompanied with any sort of controller which says that at the same time you can't increase both the performance and robustness.

The robustness of a SMC in the presence of uncertainties and/or matched disturbances is guaranteed with the selection of sliding manifold and hence the sliding mode dynamics. Assume the case of a linear system and take the sliding manifold as the linear combination of states which is MONIC and HURWITZ. then a measure of performance is determined by the poles of this polynomial when sliding modes are established i.e. s =0 and the robustness will be guaranteed by the magnitude of the controller gain as compared to the upper bound of the uncertainty and/or disturbance.

                                            REGARDS

IMRAN KHAN YOUSUFZAI