I suggest you follow https://au.mathworks.com/videos/introduction-to-state-space-equations-1547129824178.html

https://au.mathworks.com/videos/trimming-and-linearization-part-1-what-is-linearization--1543918523971.html

Hi, see

https://books.google.co.ma/books?id=NMx7E4vEj9AC&pg=PA506&lpg=PA506&dq=settling+time+for+a+non+linear+state+space+model&source=bl&ots=8Y-s1Aow52&sig=ACfU3U0aaHd7LghG7JZRY_wJrJRXdsWzSA&hl=fr&sa=X&ved=2ahUKEwi4veL2mcrqAhUPXRoKHRkbAp0Q6AEwCnoECAkQAQ#v=onepage&q=settling%20time%20for%20a%20non%20linear%20state%20space%20model&f=false

and

https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-30-feedback-control-systems-fall-2010/lecture-notes/MIT16_30F10_lec05.pdf

Hi Debamoy Datta

If it is a control-affine nonlinear system and its model is accurately known,

x' = f(x) + B·u

then it is relatively easy to design a controller

u = B–1·[– f(x) – K·x]

so that the closed-loop system becomes

x' = f(x) + B·{B–1·[– f(x) – K·x]}

x' = f(x) – f(x) – K·x

x' = – K·x.

From the linearized dynamics, x' = – K·x, the control gain K can be easily determined from the formula linked to the desired settling time.

Simple Proof:

V = ½·x² ≥ 0

V' = x·x'

V' = x·[f(x) + B·u]

V' = x·[– K·x]

V' = – K·x²

V' = – 2·K·V ≤ 0

Note: The error dynamics is skipped in this brief demo. You can force the state x to converge to an arbitrary reference point, xref, by considering the error, e = x – xref.

Dear Debamoy Datta

The time differential may be written as dt=(dt/dx).dx=(1/xdot)dx, for a scalar system. For multiple-state system you may use the norm as: dt=(1/|xdot|).|dx|, where: |xdot|= sqrt(xdot1^2+xdot2^2+...), and, |dx|=sqrt(dx1^2+dx2^2+...), whilst xdot1 and xdot2 and .. are already known in terms of x and u from the state space equations or simply the state space model, where x is the state variable, and u is the control variable. Hence, to calculate the settling time, you may scribe:

T=int(dt,0..tf)=int[(1/|xdot|),0..xf], where tf denotes final time and xf is the final state, and int abbreviates integral operator or simply the integration operation.

Since you are looking for settling time of the stabilization process, you may change the bounds of integral and integrate from x0, to zero, whilst x0 denotes the initial state or simply initial condition. I show you an example:

Assume the nonlinear scalar system as: xdot=-x^3+u, while the optimal control-law for stabilization is already known as: u=x^3-x*sqrt(x^4+1). To calculate the settling time you may write:

T=tf=int(dt,0..tf)=int[(1/|xdot|),x0..0].

It simply means:

T=tf=int(1/|-x*sqrt(x^4+1)|). dx,

while integrated from x0 to zero, as the bounds of integral. You have the integrand (the function under integral) as a function of the state variable (x), and you have the upper bound and lower bound of integration as x0 and zero, hence you can conveniently find the numerical value of the integral.

Warning: Against this scalar example, the calculation of settling time for multiple state systems is a bit tricky, but not difficult, and it is similar to what I described above for multi-state system with regard to norm concept.

Hi Saeb AmirAhmadi Chomachar

I think your method requires iteration, where the control design procedure for {u} is repeatedly executed until the state variable {x} converges to a reference equilibrium point {xref} at the desired settling time {ts}.

Dear Yew-Chung Chak

Thank you for the comment. I have presented my formulation mainly for an analytical control methodology, where control-law is analytically known as an algebraic expression in terms of only x. Furthermore, I think it is also available for a wide range of methods other than analytical methods as control-Lyapunov function (CLF) methodology. The state space model is run for example in MATLAB, then after simulation you calculate the settling time based on my formulation using state and control signal numerical values given as MATLAB data, and then employ the MATLAB sum syntax to numerically calculate the formula. I think there is no need for iteration, only after simulation you use state and control signal numerical values as an offline data to calculate the integral related to settling time.

Hi Saeb AmirAhmadi Chomachar

Take your nonlinear system as example:

x' = – x3 + {x3 – x·√(x4 + 1)}; x(0) = 1

The system converges at around 6 sec. If the desired settling time is 3 sec, wouldn't the control law need to be tuned via iteration until the desired settling time is achieved?

For instance, the tuned control law, u = x3 – 3·x·√(x4 + 1), should make the system converges at around 3 sec.

Dear Yew-Chung Chak ,

Can you provide me some details regarding error dynamics and what happens when it is not in the control affine form.

Your answer was very helpful.

The error dynamics is shown in your Wikipedia link, where you'll find the 2nd derivative of the error, e'' (because the example uses the mass–spring–damper system).

Non-affine nonlinear systems {x' = f(x, u)} is a very broad class, where the control input {u} cannot be expressed in the affine form. The feedback linearization method as shown above cannot be directly applied. Thus, controlling {x' = f(x, u)} is essentially a demanding task. Only a certain class can be transformed into a direct solvable form.

x' = cos(u) – 4·sin(u)

x' = – √(17)·sin[u – tan–1(¼)]

Dear Yew-Chung Chak

I replicate Mr. Debamoy Datta question again:

How do I find settling time for a non linear state space model?

He asked "how do I find?" NOT "how do I tune?". So my own interpretation is, Mr. Debamoy Datta , probably wants to numerically calculate the settling time after a control process, NOT to tune the settling time by applying a control law. And my answer was based on this interpretation.

Hi Saeb AmirAhmadi Chomachar

I'm just trying to say that your method can probably be used to compute the settling time as well as to tune the controller, especially for the non-affine nonlinear systems, where the direct feedback linearization may fail to do so. Perhaps we let Debamoy Datta to clarify:

"I am trying to design a controller using C.L.F for a particular problem...

... and I want to find the settling time for the controller that I would design for convergence near a desired equilibrium point that I want to define myself."

dear Saeb AmirAhmadi Chomachar ,'

The calculation of settling time for multiple state system involves |dx|=sqrt(dx1^2+dx2^2+...)

so, for finding the T, would a vector representation be helpful or some other way may make my life easier and I do want to find the settling time as you mentioned as well as tune it and my system is multiple state system.

Thank you your Saeb AmirAhmadi Chomachar your answer was indeed very helpful and would go a long way.

Dear Debamoy Datta

Dear Yew-Chung Chak

Thank you for your recent comments.

Dear Debamoy Datta

Although calculating the settling time for a multiple state nonlinear system is not difficult, whereas I think, tuning the settling time, even for a linear system in Laplace domain, is sometimes a demanding task. In nonlinear systems, particularly for multi-dimensional nonlinear systems, the control designer is usually convinced to only find the control-law (analytic or numeric), and leaves the performance characteristics (as settling time) unanswered, due to some challenges of nonlinear control, already known to control systems society.

dear Saeb AmirAhmadi Chomachar ,

dear Yew-Chung Chak

Thank you for your valuable comments.

Hi

Non-affine nonlinear systems {x' = f(x, u)} is a very broad class, where the control input {u} cannot be expressed in the affine form. The feedback linearization method as shown above cannot be directly applied. Thus, controlling {x' = f(x, u)} is essentially a demanding task. Only a certain class can be transformed into a direct solvable form.

x' = cos(u) – 4·sin(u)

or other examples

thanks