Seeking Nonlinear H∞ controller for Differential-Drive robot

There is none

Don’t get it wrong, a diff-drive robot can still reach any point on a flat plane, it is just that a smooth state-feedback controller doesn’t exist. We will start our journey by first reviewing the dynamic characteristic of a diff-drive robot, giving an introductive idea of what is an H∞ controller, and finally show the non-existence of such for a diff-drive robot. Some fundamental knowledge of dynamic systems is sufficient for this article.

Differential-Drive Robot [1]

A differential-drive robot with two driving wheel

A diff-drive robot is a very common configuration for cleaning robots and is also an example that suffers from a non-holonomic constraint. Assume the wheel is driving without any sideslip, a nonlinear state-space representation that embedded the non-holonomic constraint is shown as follows.

Dynamic equation for the differential-drive robot with external disturbances

The position of the robot “x,y” is defined at the center of the wheels while the heading direction is measured from the x-axis. The control inputs are pre-mapped from the rotational speed of the wheels to the longitudinal speed “v” and the rotation speed “ω” correspondingly. At last, the dubious term “Gε” may be viewed as a disturbance and will later play its role.

At this point, readers familiar with state-space design may notice the system is not controllable in the sense that “A=0, rank([B, AB, A²B]) < 3”.

Hamilton Jacobi Inequality [2]

An H∞ controller is motivated by designing a system that suppresses the H∞ gain defined on the transfer function, which is identical to the induced 2-norm (L₂ gain) for a linear system. We will stick with the latter one which can be extended to a non-linear system as the following problem states.

A typical dynamic system with disturbance “ε” and output “z”, and the L₂ gain requirement given as “γ²”

The requirement (3rd equation above) will be full-filled if the following Hamilton Jacobi Inequality is satisfied.

The Hamilton Jacobi Inequality for a nonlinear H∞ problem, with a positive function V(x)

We will prove this relation informally as below and readers may refer to [2] for more information. Starting from a positive function dependent on states “V(x)” which “V(0) = 0”, the time derivative of “V” can be expanded and substituted with HJI.

The time derivative of the “V”, the 3rd equation comes from a substitution of HJI to the second equation

Integrating each side along the time axis, and assuming “x0=0”, we will reach a result of a limited worst-case L₂ gain “γ”.

Integrate both side with time, notice V(x0)=V(0)=0

Nonlinear state feedback H∞ controller

With some mild augmentation [3], the aforementioned HJI problem could be transformed from a system design problem to a state-dependent (feedback) controller design problem. If this is the case, the best action of a controller should drive the HJI in a minimizing direction, which effectively leads to a smaller L₂ gain “γ” (more robust to the disturbances).

Augmented “autonomous” system with a state-dependent controller “Bu” and the new output concatenate with the weighted control signal “Ru”

The HJI for the augmented system is straightforward after substituting system dynamic “f” and output “h”. A best-played controller “u*” should seek a minimized result, which is trivial for an affine input case as the second equation depicts. As a result, we are left with an HJI under a best-performed controller and an H∞ controller exists if some “V” satisfied the equation.

HJI for the state-dependent controlled system, and the “optimal” control signal

HJI for Differential-Drive Robot

For a diff-drive robot, the corresponding HJI can be written as the first equation as shown below. It follows that the quadratic weighting of the second term is necessary to be a negative semi-definite matrix.

HJI for diff-drive robot and the necessary condition, assuming a diagonal weighting matrix “R=diag(rv,rw)”

The condition that guarantees a negative semi-definite matrix may be judged by the eigenvalues which are simple for a low dimension matrix [4]. Focusing on the upper left block, the sum of the two eigenvalues should be negative and the product should be positive.

The condition for a semi-negative definite upper left matrix, in which no such parameters exist

Finally, we have concluded that the HJI will never be satisfied and the H∞ controller doesn’t exist for a diff-drive robot. This is similar to the well-known fact stated as Theorem 13.1 in [1].

A system “q_dot = G(q)*u” with “rank (G(0)) < dim(q)” cannot be stabilize to the origin by a continous time-invariant feedback control law

References

[1] K. M. Lynch and F. C. Park, Modern Robotics. Cambridge University Press, 2017.

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

[3] J. A. Ball, J. W. Helton and M. L. Walker, “H/sup infinity / control for nonlinear systems with output feedback,” in IEEE Transactions on Automatic Control, vol. 38, no. 4, pp. 546–559, April 1993, doi: 10.1109/9.250523.

[4] S. Boyd, L. E. Ghaoui, E. Feron, και V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104), 1994.