Nonlinear H∞ Attitude Controller for Satellites: part3

A Hamilton Jacobi Inequality based approach

In this last part, an H∞ controller will be proposed for attitude regulation and validated through simulation. The derivation relies on properly modeled dynamics (part1) and the insight gained from the design of a PD-liked controller (part2). As with most nonlinear controller design procedures, the construction of a nonlinear H∞ Attitude controller requires intensive algebraic manipulation (and some luck). If you are not interested in ad-hoc stuff, you may skip the corresponding sections and head directly to the final results.

Basic introduction of a Nonlinear H∞ controller

Modeling of a rigid body motion under disturbances, space debris perhaps

The central idea of a nonlinear H∞ controller is a performance-motivated design, in which the induced gain from disturbance and/or initial condition to the output function is expected to be limited. Roughly speaking, an H∞ controller is not a specific type nor an architecture, but a specification that could be met.

Problem setup for a non-linear H∞ controller design, yes I agree with you that it should be named L₂ controller but we prefer the fancy greek symbol :)

Hamilton Jacobi Inequality with a state feedback controller

For a state feedback-controlled system, satisfying HJI with the corresponding Lyapunov function is a sufficient condition to achieve the H∞ performance [1], in other words, a limited “L₂-gain” is guaranteed. The controlled version of HJI is stated as the following without derivation, a non-formal proof could be found in this previous post.

Modified Hamilton Jacobi Inequality with state feedback controlling, “V” is the Lyapunov candidate, and “u*” is the control effort applied everywhere

The H∞ Attitude Controller

The dynamics and the output function constructed for attitude regulation were already discussed in this post and the results are given as shown.

Dynamics for rigid body and the output function for the regulation problem

By trail & error, some luck, insight from the PD-liked stabilizing controller proposed in this post, and reference [2], a valid Lyapunov candidate was patched and written as:

The Lyapunov function and the state feedback control law, “R” is treated as identity without loss of generality

Notice that the corresponding controller shared an identical form with the PD-liked design. Consequently, the controlled HJI can be expressed explicitly:

HJI with the Lyapunov function given explicitly

The trick here is to restrict the free coefficients “k₁,₂,₃” such that the cross-coupling part“ωσ” was canceled out and left with two chunks of isolated quadratic terms.

Upper bound of the HJI with a restriction on coefficients

The problem left here is to determine the range of the coefficient which provides a negative upper bound for the HJI:

Sufficient condition of the coefficients (controller gain) for a negative HJI

On the other hand, to maintain a positive Lyapunov function, utilizing the well-known result of the Schur complement (referring to the appendix of [3]):

Sufficient condition of the coefficients for a positive Lyapunov function

By substituting the coefficient “k₃” with “k₁,₂”, two restrictions for a negative HJI and positive Lyapunov can be merged as the final result:

Sufficient condition for both a negative HJI and positive Lyapunov function, the former one is stricter

There is only a one-sided limitation of the Lyapunov function coefficient (controller gain), thus the H∞ performance is achievable for any output function coefficient “α,β”, sweet.

Comparison to a PD-liked controller

A PD-liked attitude controller was proven to be stable for any positive gain from the previous part of this series. In contrast, there are more restrictions for an H∞ controller. This observation is not a surprise since every Lyapunov function for HJI is sufficiently a Lyapunov function for basic stability analysis without disturbances. This relation provides a chance to construct a valid Lyapunov function for HJI from the existing stabilizing controller(s) which is demonstrated in this example.

A valid Lyapunov function satisfying the HJI also fulfills the stability check

Simulation results

To emphasize the characteristic of an H∞ controller, externally applied disturbance with a finite duration will be introduced in the simulation. The parameters used in the simulation are “M=diag([2, 1, 3])”, “x_init=[0,0,0,1,0,0,0]”, “disturbance=[40,40,40], for 0.5 second”, “α=100”, “β=1000”, and the suppression ratio “γ=1.2”. A proper control gain achieving an H∞ performance is chosen as “k₁=50”, “k₂=32”.

Simulation setup of a rigid body motion, disturbances will be injected from the port at LHS

Controlled 3-DOF spacecraft with H∞ gain “k₁=50”, “k₂=32”

The accumulated disturbance and output function, clearly the accumulated 1.44*||output||²<||dis||², which is guaranteed by the H∞ performance

As a comparison, let’s try a pair of gains which does not meet the sufficient condition derived in this post, “k₁=50”, “k₂=5”. Beware that this design might not meet the sufficient condition, but it may still satisfy the HJI and achieve an H∞ performance, it’s just that our analysis couldn’t tell.

Controlled 3-DOF spacecraft with arbitrary gain “k₁=50”, and “k₂=5”, might still satisfy an H∞ performance, who knows?

The accumulated disturbance and output function, note that the H∞ performance is almost violated

Conclusion and takeaway

In this part, a nonlinear H∞ controller design is proposed for the attitude regulation problem. The construction of the controlled Lyapunov function (usually the most struggling part) is inspired by the stabilizing PD-liked controller, and the corresponding sufficient condition was also resolved as inequalities. Despite the overly complicated derivation, the actual implementation of the H∞ controller for a rigid body is simple, this only holds for the case where the Lyapunov function was almost artificially designed.

This will be the end of this series, til next time.

References

[1] A. van der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control. Springer International Publishing, 2018.

[2] R. J. Adams, J. M. Buffington, A. G. Sparks, and S. S. Banda, Robust Multivariable Flight Control. Springer London, 2012.

[3] S. Boyd, S. P. Boyd, L. Vandenberghe, and C. U. Press, Convex Optimization. Cambridge University Press, 2004.