Hi Saeb,

Your approach is similar to the feedback linearization.

If you study the dynamics of x such that u = 0, you will notice that −x³ is a stabilizing nonlinear damping term. So, you can make good use of it.

Good thinking! Keep it up.

@Yew-Chung. Hi, Good answer friend. Stay tuned. I've gotten some novel ideas to be appended here soon. Thank you for your feedback.

Hi Saeb,

I actually did something similar but my approach was a Fourier series-based control. Anyhow, I look forward to hearing your novel ideas.

@Yew-Chung. Hi, again. Look at the plot illustration here. For the dynamic system; xdot=-x^3+u, I've simply set xdot=-|x| (for absolute value dissipation) or; xdot=-x*e^t (for hyper-exponential dissipation), and respectively got u=x^3-|x| and u=x^3-x*exp(t). These are feedback stabilization laws and so worthy, although the second one is time-varying. The problem is that these feedback laws are not input-to-state stable (similar as what happens to the case of dynamic inversion), but they are simple control strategies which seem as worthwhile feedback stabilization methods.

It turns out, for the system xdot=-x^3+u, how to select xdot to artificially decrease, is what significantly affects the state closed-loop dynamics. Moreover, Lipschitz condition of the function f(x), where xdot= -f(x), is seen to have a major role on convergence time.

In the illustration, closed-loop dynamics for: f(x)=x^2, and also f(x)=sqrt(x), are plotted separately.

These control laws are not input-to-state stable (as shown in the second graph), similarly to the case usually happens to Nonlinear Dynamic Inversion (NDI) methodology.

u= u(x,t)=-x^3+x*e^t

Matlab ezsurf.

Hi Saeb,

Thanks for sharing your knowledge with us. Please take note that some of your proposed control schemes {i.e., −|x|, −x2, −√(x)} do not stabilize the dynamic system for x(0) < 0.

Consider the first-order nonlinear system

x' = f(x) + G(x)u.

If G(x) ≠ 0, the feedback control law is given by

u = G-1(x)[− f(x) + ν],

where ν is an auxiliary stabilizing control function.

In order to achieve global stabilization, the auxiliary control function must be a monotonically decreasing function. In simple terms, ν has to stay in the 2nd and 4th quadrants. The figure shows some of the typical monotonically decreasing functions.

Please also note that your proposed ν = − x*exp(t) is a straight line with time-varying negative slope and it satisfies this condition.

If you design a non-strictly decreasing function such as, ν = (x+1)2 − (x+1)3, then the closed-loop system will have two equilibrium points at −1 and 0.

@Yew-Chung Chak. Thank you friend. Good analysis. Marvelous. Now, I've known more about my ponder.

But how about your analysis with regard to xdot=-sign(x)?.

I got the insight. v=-sgn (x), stays in 2nd and 4th quadrants, hence it satisfies the criteria you mentioned. Thank you.

Hi Saeb,

Thanks for your reply. If ν stays in the 2nd and 4th quadrants, by using Lyapunov Stability theorem, it can be easily shown that V' = x*x' = x*ν ≤ 0.

Hi,

Use ADRC, see my published papers

https://www.researchgate.net/profile/Ibraheem_Ibraheem3/contributions

Hi Saeb,

Using ADRC on the self-stabilizing system, x' = − x3, might be a bit of overkill and is computationally expensive.

If you are interested, try stabilizing the following 1st-order SISO nonlinear system with ADRC, FOC, PID, SMC, or whatever kind of controller you have in your arsenal without directly canceling the nonlinearity "1/x":

x' = 1/x + u

Glossary:

ADRC – Active disturbance rejection control

FOC – Fractional-order control

PID – Proportional-integral-derivative

SMC – Sliding mode control

Dear Saeb,

Referring to your original question, your proposition about the control law is effectively the state feedback linearization.

Dear Yew-Chung Chak

Would you please give me more info about the siso system:

x' = 1/x + u

in terms of its physics. I mean what physical control system is associated to this siso system?. Also please cite a paper or textbook on this benchmark. Thank You.

Hi Saeb AmirAhmadi Chomachar ,

I'm amazed that it took you nearly 700 days to grow the interest in this problem.

I can't remember the sources, but the ODE x' = 1/x has an analytical solution. Whether this simple 1st-order system can be practically stabilized or not, is another intellectual debate.

Dear Yew-Chung Chak

you are right. That's a bit strange. I am usually working on different topics from a wide range of research projects. Sometimes, I research on fluid mechanics, sometimes, on solids mechanics, and sometimes on control and the same ilk. So, I usually switch from one topic to another and sometimes it takes a long time to switch. I occasionally browse in my RG questions and other sources of my research, hence, there are always new questions that pop up in my mind and I would be curious to find the answers. But, one of my research topics is optimal control, and, I am looking for simple and intuitive, but somehow challenging benchmarks as the siso system: xdot=1/x+u, which you exemplified. I was curious to know its physical background, nothing else. Regards, Saeb.