Dear Alaa

For a controllable nonlinear system, a linear controller, especially when augmented with integral controller, is always powerful in stabilizing a nonlinear system especially with suitable selection for the controller parameters. However the crucial questions which it needed answers are;

i) when the system suffers from vanishing perturbations, it is important to estimate the area of attraction. When the system state initiated in the area of attraction the linear controller will be able to regulate the error asymptotically to the origin. Accordingly the controller will be more effective for larger area of attraction and vice versa.

ii) what is the ultimate boundedness for the system response when the system suffers from non-vanishing or from mismatched perturbations. In this case the error between the system output and the desired output is not asymptotically stable and the linear controller with the integral term is no longer able to regulate the error to zero except when the perturbation is constant. Accordingly the controller will be more effective for small ultimate boundedness and vice versa.

Mr.Alaa,

It is said in the literature that you need to bring in controllers with H-Infinity norms only if the PID cannot control. One aspect of control that becomes difficult with PID is fine tuning the same for a highly non linear system.

Dear Alaa Al-Ibadi,

Yes you can use PID controller for nonlinear system and also if your system is strongly nonlinear it's better to use nonlinear PID controller by the way we can now design this controller we can easy found it in literature. Also i suggest to you look at attach  publications.

Best regards

Article Design of an enhanced nonlinear PID controller

Conference Paper A design of nonlinear PID controllers with a neural-net base...

Chapter Symbolic computing aided design of nonlinear PID controllers

Article Design of Nonlinear PID Controllers

Article Design of nonlinear PID controllers based on hyper-stability criteria

Article Parameter selection for a new class of nonlinear PID controller

Article Design of Nonlinear PID Controllers for Nonlinear Plants

Dear colleagues. Thank you very much for your answers. 

You may do so via the gain scheduling approach..

Yes as  Mohamed-Mourad Lafifi says, you can apply nonlinear controller including nonlinear PD or even PID using sliding mode control, or passivity-based control..etc. these methods are nonlinear control method working based on nonlinear PID concepts. 

I suggest using nonlinear controller since there is many methods available. Also linearizing the system is not always a good idea.

Linearizing the non-linear plant is not always needed while designing PID controller. To develop a robustly stable closed loop system with PID, you can set stability constraints, for which various papers have been published. But, in presence of complex non-linearities, the PID may not generate good responses due to its oversimplification characteristics, noise due to derivative action, long settling time,etc. But, you can also add an improved version of PID which is ADRC (Active disturbance rejection technique) to the non-linear systems.

You can use nonlinear PID, otherwise you have to linearize your system.

Not being familiarized with the specific system in question, it is hard to provide a recommendation. In general terms, todays digital technology allows you to break down your system into segments which are easier to handle by a control algorithm specifically tuned for that segment. Remember that PID is not the solution to all situations but an inherited legacy from the analog world. Digital Technology provides the means for implementing very well adapted and sophisticated solutions at very low cost which may or may not utilized the PID concept depending which is more advantageous for you.

I would say the answer is a very strong "it depends". 

How nonlinear is your system?  Is it unstable?  Does it have non-minimum phase behavior?  Can you linearize about operating points faster than the system dynamics move between those points?

The issue with nonlinear systems is having a proscriptive model that predicts how the system will behave under feedback.  However, if you *can* linearize the system and if your PID *can* achieve high gain and phase margins in that linearized model, then one might believe that the PID (adjusted for each operating region) can work. 

Recall that PIDs effectively give you a lead and a lag (or a notch if you adjust things right), and so you have to understand where your nonlinear system needs lead for stability (PD) and where it needs lag (PI) for steady state response.  With nonlinear systems, much of this is determined experimentally.

Also, understanding the digital implementation of your PID in some detail will allow you to not throw away bandwidth unnecessarily.  (I have a couple of papers in that general area.)

Article Semi-automatic tuning of PID gains for Atomic Force Microscopes

Conference Paper A Unified Framework for Analog and Digital PID Controllers

 Dear Alaa

For a controllable nonlinear system, a linear controller, especially when augmented with integral controller, is always powerful in stabilizing a nonlinear system especially with suitable selection for the controller parameters. However the crucial questions which it needed answers are;

i) when the system suffers from vanishing perturbations, it is important to estimate the area of attraction. When the system state initiated in the area of attraction the linear controller will be able to regulate the error asymptotically to the origin. Accordingly the controller will be more effective for larger area of attraction and vice versa.

ii) what is the ultimate boundedness for the system response when the system suffers from non-vanishing or from mismatched perturbations. In this case the error between the system output and the desired output is not asymptotically stable and the linear controller with the integral term is no longer able to regulate the error to zero except when the perturbation is constant. Accordingly the controller will be more effective for small ultimate boundedness and vice versa.

for non linear systems u can use PID controller with fuzzy controllers to improve the performance

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...

Please refer to this article:

https://www.researchgate.net/publication/318726178_An_Improved_Active_Disturbance_Rejection_Control_for_a_Differential_Drive_Mobile_Robot_with_Mismatched_Disturbances_and_Uncertainties

Conference Paper An Improved Active Disturbance Rejection Control for a Diffe...

It is said when a simple PID controller can do the job, there is no need for bringing the H-Infinity machinery for controller. The only problem when dealing with high end dynamic systems is it might take more time for fine tuning the PID.

Just remind that if you are following the strategy to linearize the process in different working points and the switch between various PID parameters (called gain schedulink) then you need to solve the problem of mutual tracking of the state of all coupled controllers in the group in order the get bumpless, i.e. smooth control action when switching between different PIDs.

of course you can use PID for nonlinear systems see the following links for example

Conference Paper Robust self-tuning PID controller for nonlinear systems

https://www.sciencedirect.com/science/article/pii/S0959152401000415

Definitely, PI & PID controllers can use for nonlinear systems.

yes it can be but the question is whether it would be efficient and effective .... there is non linear PID.

Yes, this can be done by getting appropriate Input/Output data for the real system via various input signals (black -box method). This data can then be used to form Transfer functions and then tuning the gains theoretically to observe the response characteristics. A choice can be made from the response and choose the respective gains. These gains can then be tuned further, before applying to the real system.

I think you should apply gain scheduling technique for tuning of PID controller gain for nonlinear system.

Dear Pushkar

You can use PID controller with some robust control technique.

It is well known that most of the processes in the industry are 90 % controlled by PID controllers. While the majority of these processes are nonlinear ones. Nonlinear systems are highly effected by the magnitude of its input signals. The Fractional order PID (FOPID) counterpart controller can adapt in a more realistic manner with nonlinear systems.

Yes, you can use PID controller for non-linear system.

I am curious about Osama's statement that a fractional order PID can adapt in a more realistic manner than that of a conventional PID. First of all, this implies that the controller in question adapts on the fly to the size of the input signal. There is nothing inherent in a PID -- fractional or otherwise -- that accounts for adaptation. There is also nothing that prevents this from being added in.

Furthermore, adaptation typically happens on a time scale far slower than the signal time scale, so as to operate on parameter rather than signal variations. Without this "time scale decomposition" one has difficultly knowing what the parameters should track unless one has been able to isolate physical system parameters that directly contribute to fast time scale errors in the system operation. This is not trivial to do and so it's by no means a safe assumption to say that adaptation can happen on the same time scale as the signals.

Finally, for all of their ability to save one PID coefficient by making the second PID term have a fractional order, there is a matter of implementation. One can implement a PID in an analog form, but it's pretty damn hard to do adaptation in the analog world (I mean for real, not in simple mathematics only proofs). One can implement conventional PID controllers using 3 coefficients analogously to the 3 coefficient analog PID controllers (I've written some stuff about this). However, to approximate a fractional order PID in a digital system requires some sort of rational polynomial approximation. One fractional order guru told me , "Sure, we can approximate it with about 20 taps." So, we have used fractional order PIDs to go from the cumbersome 3 coefficients to 2 coefficients (plus an

exponent) and now to implement it we need 20 coefficients. Please explain to me where the savings are?

This is why I believe that the statement that somehow the fractional order PID handles nonlinear systems more naturally requires a lot more justification.

see the paper in this link

https://booksc.xyz/book/8920503/c5a961

When a simple PID can do the job you don't need the h-infinity norms.

To Daniel Abramovitch

Thanks for the comments on my answer. Firstly, I really did not meant by adapt to be adaptive in its way of controlling rather than in a more specific meaning to compensate in a more effective way, that is by its nature of fractional derivative and integral actions. It is examined on time variant systems and on nonlinear systems, and if well tuned, Quality of control and system performance will greatly enhanced, in comparison to the use of the PID controller.

Secondly, i mentioned the magnitude of the input signal, which will be the control signal to the plant, to be taken in account, during the design of nonlinear systems weather using linear or nonlinear controllers.

Fuzzy-PID controller can be made by generating the rule-base to control nonlinear processes.

To Osama Awad:

Thank you for your response, but I still do not understand how it relates to my original comment on your statements. I also do not see any support for your statement that "quality of control and system performance will [be] greatly enhanced, in comparison to the use of the PID controller." It is a statement made simply as an axiom, wihtout explaining the logic, the "why" of the statement.

In my understanding of a fractional order PID, it's one claim to fame is that by making the exponent of the integrator or differentiator non-integer, a wider range of magintude/phase combinations are possible than with an integer order PID. I do not see how this provides superior control than a conventional PID, or a PID with a biquad loop shaping filter attached. The latter would require a 4th order rational function of polynomials (9 coefficients). Implementing a fractional order PID would require a 20-30 tap polynomial approximation.

What am I missing here?

Fuzzy-PID controller can be made by generating the rule-base to control nonlinear processes and if combine it with a model reference adaptive control it would be better.

You should linearize the systems around the fixed point if you want to use linear methods (e.g. PID) for nonlinear systems.

The designed controller would apply to the nonlinear plant, but it is efficient in just a domain for your initial condition. (Domain of attraction)

Yes, you can always do that. Please do the linearization around the equilibrium point then use the PID controller. Else go with the hit and trial run, you may control with PID.

yes we can used, but utilized an approaches to adjuster the parameters of PID(gradient method).In addition, describe the nonlinear system by a mathematical model to calculate the sensitivity and increase the speed of convergence.

Yes we can used that