An introduction to the passivity — Passivity [1] is a type of dissipativity [2] with a specific definition of the supply rate that accumulates in the storage. The framework of passivity follows a bilateral “input” and the “output” compared to others that treated them as flows of signals uni-directionally. This feature makes it a close companion…

A physical interpretation of H∞ stability — In complement to the series of posts about the design of an H∞ controller, the dissipativity of the H∞ controlled system will be revealed in this article. The concept of dissipative will fill in the gaps between Lyapunov stability analysis and the H∞ performance and provides an insightful interpretation. …

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∞…

Passive dynamics and the PD-liked controller — Before diving into an H∞ controller design, it is helpful to have a big picture of how the attitude of a rigid body evolves with and without external torques. Consequently, a PD (proportional-differential) controller corresponding to the “attitude error” and the “angular speed” will be proposed with guaranteed stability. …

System modeling and the output function — In this series of posts, a typical workflow that constructs a nonlinear H∞ controller will be demonstrated for a fully-actuated rigid body (e.g. satellite). The intent of this type of controller is to regulate the attitude to a desired one with a reasonable controlling effort [1]. We will first formulate…

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∞…

The hidden “feedback” in Quantum mechanics — Preface We have demonstrated the underlying optimality in classic Newtonian mechanics in this post. In another post, we’ve also shown that Stochastic Optimal Control Problem(SOCP) can be solved by simulating a diffusing process [1] (interpreted as a classical Path Integral as the figure depicts below). …

Utilizing the time-reversed Feynman–Kac equation — Preface It is well known that an Optimal Control Problem (OCP) may be decomposed into smaller solvable optimal problems with the Principle of Optimality [1]. Under a continuous-time setup, the “solved” sub-problem will leads to a PDE called Hamilton–Jacobi–Bellman equation (HJB) that shapes the optimal cost-to-go function (readers may refer to…

A Dynamic Programming approach — Preface From our previous post, we’ve shown the Lagrange and Hamiltonian mechanics govern the optimal trajectories of the extremum action problem [1]. We also point out an observation that will be the topic of this post. Classic dynamics system is a solution of an optimal control problem Optimal control problem (OCP) A typical optimal control…