I think it comes down to the tuning of control gains for both controllers. With a proper selection of the nonlinear PD gains, the disturbance can be satisfactorily rejected. Moreover, for 2nd-order systems, it is possible to build a Nonlinear PD Controller that performs as good as the Sliding Mode Controller does, at least from the theoretical perspective.

If you want to make a fair comparison, then ensure that the control gains for each controller are optimized to give best pointing accuracy under the same test environment.

Here is a simple simulation for a double integrator system:

x'' = 0.1·sin(π·t) + ui

where

u1 = − (2·x' + x)

u2 = − 0.1·tanh((2·x' + x)/0.1)

Initial conditions: x = 0.1, x' = 0.0

Divya,

Rather difficult to answer this right away since you have not provided any detail as of yet !

Could you tell us a bit about the following:

-Sanjay

Fully agree with Sanjoy Roy.

It can be tunning problem, but without knowing the model exactly, it is hard to tell.

Hi Yew-Chung, thanks for replying. Makes sense to me. But, I am not optimizing the gains as of now so I am just using trial and error for it such that

Thanks for your update, Divya. Here is your proposed SMC:

u = J·(– λ·qe' + ωd' – K·usat) + (ω×)·J·ω – umax

At the first glance, your SMC looks pretty standard for a rigid spacecraft, except that it should not contain the external disturbance, umax. I'd probably shift {K·usat} to the outside of the brackets for the convenience of tuning Ksw against the disturbance umax, and also add the term {Kp·s} to speed up the convergence (hopefully to improve the steady-state error):

u = J·(– λ·qe' + ωd') + (ω×)·J·ω – Kp·s – Ksw·usat

where the switching gain, Ksw = κ·diag(1, 1, 1) and κ > 0.0012.

Other than tuning λ and K, you also need to tune ɛ in the saturation function, by gradually decreasing it without inducing the chattering. Please check the computation of error quaternion, qe = qd ⊗ q as well.

How does your Nonlinear PD controller look like?

I think you mean u_SMC = J·(– λ·qe' + ωd') + (ω×)·J·ω – J.Kp·s – J.Ksw·usat. Also , what should be the value of Kp? Additionally, the multiplicative quaternion error I am using is qerr=q ⊗ qd^(-1) since qerr is in the body reference frame. where q ⊗ =[q4*eye(3)-crossm(q(1:3)) q(1:3); -q(1:3)' q(4)]. The NL PD controller I am using is

u_PD=-kp*sign(qerr(4))*qerr(1:3)-kd*(1±qerr(1:3)'*qerr(1:3))*w

usmc1 = J·(– λ·qe' + ωd') + (ω×)·J·ω – Kp·s – Ksw·usat

usmc2 = J·(– λ·qe' + ωd') + (ω×)·J·ω – J·kp·s – J·ksw·usat

Thanks for your reply, Divya. Both usmc1 and usmc2 are actually the same, where Kp = J·kp and Ksw = J·ksw . Note that J, kp, ksw, Kp, Ksw ∈ ℝ3×3. Based on my experience, the reason you get usmc2 is because of the selection V = sT·s. If you choose V = sT·J·s, then you will get usmc1.

For the selection of value of Kp, you may start with Kp = 1. If you find the convergence rate is unlikely too fast, then gradually decrease it.

Thanks Yew. What values can lamda take?

I'm trying find out the real intent behind your question about λ. From the mathematical theory of first-order linear ordinary differential equation, λ > 0. In sliding mode control, the value of λ affects the transient response because λ is the parameter that influences the exponential decay. I usually set as λ = 1 as the benchmark for SMC design.

yes, this is what i understood, that lamda=1. But with this value it is hard to get the accuracies I require. So, i just tried increasing this value and with 2.7 I got to some required accuracies but there is chattering in the control input.

I don't know how you ran the system. Maybe the value of ɛ is too low. When it is too low, usat acts like a signum function and causes chattering. You may set {λ = 1, Kp = 1, Ksw > umax, ɛ = 0.1} as benchmark and then adjusting each one by gradually increasing λ, Kp, Ksw and decreasing ɛ.

Also, ensure that the step size, h is small enough to capture the dynamics.

Thanks Yew, I figured that out and I am getting the required accuracy w/o chattering.