Inverse optimal design of input-to-state stabilizing homogeneous controllers for nonlinear homogeneous systems
Kaixin Lu, Ziliang Lyu, Haoyong Yu
TL;DR
The paper tackles inverse optimal control for nonlinear homogeneous systems with ISS and IOS guarantees by introducing a cost functional that penalizes the output via $y^T R_2(x) y$ in addition to state and input penalties. Using homogeneity and Legendre-Fenchel transforms, it constructs the weighting functions $l(x)$, $R_1(x)$, $R_2(x)$ and the disturbance penalty $\gamma_0$, and derives a sufficiency result (Theorem 1) that yields an inverse gain law $u^*(x)$ minimizing the modified cost. The control law takes the form $u^*(x)= -\frac{\beta\kappa}{2\\vartheta^2} R(x)^{-1}(L_{G_1}V)^T$, with $V$ a homogeneous Lyapunov function, and ensures ISS and finite-gain $L_2$ stability when $\gamma(s)=\frac{1}{\\mu}s^2$. Theorem 2 shows that if the disturbance-free, homogeneous system is homogeneously stabilizable, the inverse optimal gain assignment problem is solvable (locally) for homogeneous systems, providing a constructive design path. The framework highlights the HJI equation with an extra output penalty term $h(x)^T R_2(x) h(x)$ and provides a constructive route for inverse optimal design under homogeneity, with potential applicability to robust nonlinear control where output performance matters.
Abstract
This work studies the inverse optimality of input-to-state stabilizing controllers with input-output stability guarantees for nonlinear homogeneous systems. We formulate a new inverse optimal control problem, where the cost functional incorporates penalties on the output, in addition to the state, control and disturbance as in current related works. One benefit of penalizing the output is that the resulting inverse optimal controllers can ensure both input-to-state stability and input-output stability. We propose a technique for constructing the corresponding meaningful cost functional by using homogeneity properties, and provide sufficient conditions on solving the inverse optimal gain assignment problem. We show that homogeneous stabilizability of homogeneous systems in the case without disturbance is sufficient for the solvability of inverse optimal gain assignment problem for homogeneous systems.